Datasets:

Hendrik-a

/

dancer-data

like 0

quant-ph9603021

i

consider the following experimental arrangement . a particle moves in one dimension , along the @xmath0 axis . a detector is placed in the position @xmath1 . let @xmath2 be the time at which the particle is detected , which we denote as the `` time - of - arrival '' of the particle at @xmath3 . can we predict @xmath2 from the knowledge of the initial state of the particle ? in classical mechanics , the answer is simple . let @xmath4 be the general solution of the equations of motion corresponding to initial position and momentum @xmath5 and @xmath6 at @xmath7 . we obtain the time - of - arrival @xmath2 as follows . we invert the function @xmath8 with respect to @xmath9 , obtaining the function @xmath10 . the time of arrival @xmath2 at @xmath3 of a particle with initial data @xmath5 and @xmath6 is then @xmath11 two remarks are in order . first , if @xmath12 is multivalued , we are only interested in its lowest value , since the particle is detected the first time it gets to @xmath3 . second , for certain values of @xmath5 and @xmath6 , it may happen that @xmath3 is outside the range of the function @xmath4 . this indicates that the detector in @xmath3 will never detect a particle with that initial state . the time - of- arrival is a physical variable that in a sense can take two kinds of values : either a real number @xmath13 , or the value : @xmath2 = _ never _ " . notice that in the latter case , the quantity @xmath2 formally computed from ( [ cl : tgen ] ) turns out to be complex . thus , a complex @xmath2 from ( [ cl : tgen ] ) ( for given @xmath14 ) means that the particle with initial data @xmath15 is never detected at @xmath3 . in quantum mechanics , the problem is surprisingly harder . in this case the time of arrival can be determined only probabilistically . let @xmath16 be the probability density that the particle is detected at time @xmath2 . namely , let @xmath17 be the probability that the particle is detected between the time @xmath18 and the time @xmath19 . how can we compute @xmath16 from the quantum state , e.g. from the particle s wave function @xmath20 at @xmath7 ? to the best of our knowledge , this question has not received a complete treatment in the standard literature on quantum mechanics . the problem of computing the time of detection of a particle is usually treated in very indirect manners . for instance , the probability ( in time ) of detecting a decay product say a particle escaping the potential of a nucleus can be obtained from the time evolution of the probability that the particle is still within the confining potential . alternatively , one can treat the detector that measures the time - of - arrival quantum - mechanically , and compute the probabilities for the positions of the detector pointer at a later time ; in this way one can trade a meaurement of the time - of - arrival for a measurement of position at a late fixed time . in the fifties , wigner considered the problem of relating the energy derivative of the wave function s phase shift to the scattering delay of a particle @xcite . this approach , later developed by smith @xcite and others ( see for instance gurjoy and coon @xcite ) gives the average delay , but fails to provide the full probability distribution of the time of arrival . ( smith s paper begins with : `` it is surprising that the current apparatus of quantum mechanics does not include a simple representation for so eminently observable a quantity as the lifetime of metastable entities . '' ) in the seventies , piron discussed the problem in a conference proceeding @xcite , sketching ideas related to the ones developed here . ideas related to the ones presented here were explored in ref . @xcite , but in this case too only the average time - of - arrival was obtained , and not its full probability distribution . kumar @xcite studied the quantum first passage problem in a path integral approach , but did not obtain a positive probability density . the problem has been studied in the framework of hartle s generalized quantum mechanics @xcite by using sum over histories methods . various attempts in this direction and discussions of difficulties can be found in ref . see also the recent paper ref . @xcite for a discussion of the problem and for references ; in particular , ref . @xcite discusses the difficulties one has to face in trying to compute _ sequences _ of times - of - arrival an important problem which , however , we do not address here . as stressed by hartle in @xcite , generalized quantum mechanics _ generalizes _ `` usual quantum mechanics '' ; here , on the other hand , we are interested in the question whether @xmath16 can be computed within the mathematical framework of conventional _ hamiltonian _ quantum mechanics . we see two reasons of interest for discussing the problem of computing time - of - arrival in quantum mechanics . first , it is a well posed problem in simple quantum theory , and there must be a solution . echoing smith @xcite , we do not expect that quantum mechanics could fail to predict a probability distribution that can be experimentally measured by simply placing a detector at a fixed position and noting the time at which it `` clicks '' . the problem is not just academic : it is related to the problem of computing the full probability distribution ( as opposed to the expectation value ) for the tunnelling time through a potential barrier . this problem has relevance , for instance , in computing rates of chemical reactions ( see , for example , kumar @xcite ) . second , the problem bears directly on the interpretation of quantum theories without newtonian time @xcite and thus on quantum gravity ; we shall briefly comment on this issue in closing . this paper is the first of a sequence of two . here we develop a general theory for the time - of - arrival operator , and study the free nonrelativistic particle case in detail . in a companion paper @xcite , we investigate a technique for the explicit construction of the time - of - arrival operator in more general cases , we extent our formalism to parametrized systems , and we study some less trivial models : a particle in an exponential potential and a cosmological model . in the next section we give a general argument , based on the superposition principle , for the existence of an operator @xmath21 ( the time - of - arrival operator ) such that @xmath16 can be obtained from the spectral decomposition of @xmath20 in eigenstates of @xmath22 , in the usual manner in which probability distribution are obtained in quantum theory . @xmath21 has peculiar properties that distinguish it from conventional quantum observables . we give a general argument based on the correspondence principle indicating that @xmath21 can be expressed in terms of position and momentum operators by the inverse of the classical equations of motion , eq . ( [ cl : tgen ] ) . this does not suffice in fixing the operator , since factor - ordering ambiguities can be serious . the problem of the actual construction of the operator @xmath21 in more general systems will be addressed in @xcite . in section 3 we study an explicit form of the operator in the case of a free nonrelativistic particle . we diagonalize the operator , providing a general expression for the time - of - arrival probability density @xmath16 . in particular , we calculate @xmath16 explicitly for a gaussian wave packet . in section 4 we discuss some consequences of our construction . we notice that the existence of the operator implies that the quantum mechanics of a free particle can be expressed in a `` time - representation '' basis . we derive time energy uncertainty relations . we conclude in section 5 with a general comment on the equivalence between time and position variables suggested by our results . in the appendix , we study whether the probability distribution we computed is reasonable , by comparing it with the one estimated indirectly using the schrdinger current . we find that the two agree within second order in the debroglie wavelength of the particle . the probability computed from the schrdinger current can not be physically correct to all orders because it is not positive definite ; whether or not the probability distribution computed with @xmath21 is physically correct to all orders is a question that can , perhaps , be decided experimentally .

we consider a solution to this problem based on the spectral decomposition of the particle s ( heisenberg ) state into the eigenstates of a suitable operator , which we denote as the `` time - of - arrival '' operator . we discuss the general properties of this operator . we construct the operator explicitly in the simple case of a free nonrelativistic particle , and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the schrdinger current . we derive a well defined uncertainty relation between time - of - arrival and energy ; this result shows that the well known arguments against the existence of such a relation can be circumvented . finally , we define a `` time - representation '' of the quantum mechanics of a free particle , in which the time - of - arrival is diagonal .

we study the problem of computing the probability for the time - of - arrival of a quantum particle at a given spatial position . we consider a solution to this problem based on the spectral decomposition of the particle s ( heisenberg ) state into the eigenstates of a suitable operator , which we denote as the `` time - of - arrival '' operator . we discuss the general properties of this operator . we construct the operator explicitly in the simple case of a free nonrelativistic particle , and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the schrdinger current . we derive a well defined uncertainty relation between time - of - arrival and energy ; this result shows that the well known arguments against the existence of such a relation can be circumvented . finally , we define a `` time - representation '' of the quantum mechanics of a free particle , in which the time - of - arrival is diagonal . our results suggest that , contrary to what is commonly assumed , quantum mechanics exhibits a hidden equivalence between independent ( time ) and dependent ( position ) variables , analogous to the one revealed by the parametrized formalism in classical mechanics . # 1#2 # 1 | # 2 # 1| # 1

1109.3909

i

an important class of problems in quantum systems is that of quantum quench , where a parameter in the hamiltonian varies with time , typically attaining constant values at early and late times . starting with some initial state , the problem is to determine the nature of the final state . this problem has recently attracted a lot of attention in several areas of many - body physics , particularly because of progress in cold atom experiments @xcite,@xcite,@xcite . among other things , this problem is interesting for two different reasons . the first relates to the question of thermalization . does the system evolve into some kind of steady state ? if so , is the state `` thermal '' in any sense ? the second question deals with the situation where the quench takes place across a value of the parameter where there is an equilibriium critical point . in this case , kibble - zurek type scaling arguments indicate that there are several physical quantities which are _ universal _ and determined by the critical exponents of the critical point . furthermore for two dimensional theories which are _ suddenly _ quenched to a critical point , powerful techniques of boundary conformal field theory have been used in @xcite to show that ratios of relaxation times of one point functions , as well as the length / time scales associated with the behavior of two point functions of different operators are given in terms of ratios of their conformal dimensions at the critical point , and hence universal . another related application of this phenomenon is in gravitational physics and cosmology . in this context , this is the phenomenon of particle production in time dependent backgrounds , which is relevant to many physical problems ranging from black hole evaporation to behavior of quantum fluctuations in an expanding universe . there are very few theoretical tools available to study such systems when they are strongly coupled . in this note we will explore the use of ads / cft correspondence @xcite - @xcite to this problem . ads / cft techniques have been in fact used in the past to study quantum quench , though not at critical points . in ads / cft the boundary values of bulk fields become coupling constants , so that this problem becomes that of determining a time dependent background with specified initial and boundary conditions . as is usual , a _ problem in the boundary theory becomes a classical problem in general relativity . one class of problems involve a vacuum initial state ( in the bulk this is pure @xmath11 in the distant past ) and a change of the boundary value of some field ( e.g. the dilaton or the boundary metric ) over some finite time interval , in a regime where supergravity is always valid . in the boundary field theory this corresponds to turning on a time dependent source for the dual operator . under suitable conditions , this leads to black hole formation in the bulk @xcite . the correlators at future time would then be thermal with a temperature characterized by the hawking temperature . the time scale after which this happens depends on the nature of the correlators , but turns out to be always smaller than what one would expect from a conformally invariant system evolving to a thermal state . thus , in this case thermalization of the field theory is signalled by black hole formation . another class of problems involve a similar setup , but a suitable variation of the coupling which prevents black hole formation in the supergravity regime . the coupling , however , becomes _ weak _ at some time and the bulk string frame curvature grows large , leading to a breakdown of the supergravity approximation - thus mimicking a space - like singularity @xcite . for the case of a slow variation of the coupling it turns out that the gauge theory remains well defined and may be used to show that a smooth passage through this region of small coupling is possible without formation of a large black hole . related scenarios appear in @xcite and @xcite . many interesting phenomena in gauge - gravity duality , in particular phase transitions , can be explored consistently in a _ probe approximation_. in this approximation , a certain set of bulk fields can be treated separately and independently of bulk gravity . one set of examples are probe branes in a background @xmath12 . this introduces hypermultiplets which in general live on a defect in the original field theory . when the number of such branes , @xmath13 is much smaller than the number of flux units which produce the background geometry , @xmath14 , the dynamics of these branes do not backreact on the bulk metric which remains @xmath12 . the dynamics is then described by a dbi action in a fixed background - the fields in this action are then bulk probe fields . if the boundary value of such a probe field is time dependent in a fashion similar to above , black hole formation in the bulk is invisible in the probe approximation . however , rather remarkably , thermalization of the hypermultiplet sector is still visible . this manifests itself by the formation of an apparent horizon ( which evolve into an event horizon in some situations ) in the induced metric on the brane worldvolume @xcite . fluctuations of the brane feel this induced metric and their correlators have thermal properties . none of the above works study quench dynamics near a phase transition . in this paper we take the first step towards doing that , in a simple model which displays an equilibrium critical point in a probe limit @xcite . the model involves a neutral scalar field whose mass lies in the range @xmath15 in the background of a charged @xmath2 black brane . the overall coupling is chosen to be large so that the scalar can be treated as a probe . for a given brane background ( i.e. for a given temperature and charge density in the dual field theory ) , and for a vanishing non - normalizable mode of the field ( corresponding to a vanishing source of the dual operator ) there is always a critical mass @xmath3 below which the trivial solution is unstable . in this regime the stable solution is nontrivial , signifying a nonzero expectation for the dual operator in the dual theory . @xmath16 with a vanishing source is then a critical point , which happens to have standard mean field exponents . this setup is in fact quite similar to models of holographic superconductors and fermi surfaces @xcite-@xcite . however , here the scalar field is neutral so that the dynamics of the gauge field is frozen . this model has been argued to model phase transitions in antiferromagnets . as is well known , in this mass range there are two inequivalent quantizations . our analysis is restricted to the conventional quantization of the bulk scalar , where the mode which vanishes slower at the boundary is treated as the source in the boundary theory . a similar analysis can be easily performed for the alternative quantization . we will study the time evolution of the order parameter starting from adiabatic initial conditions in the ordered phase . the passage through the critical point is done in two ways - ( i ) by keeping the bulk mass at its critical value and changing the leading term of the near - boundary expansion of the bulk field and ( ii ) by changing a bulk mass parameter with a vanishing leading term . the field theory meaning of ( i ) is clear - this corresponds to turning on a source dual to the bulk field : the critical point appears when the source vanishes and the mass is at its critical value . the meaning of ( ii ) is less direct - changing a bulk mass means changing the anomalous dimension of the corresponding operator in the field theory . as discussed in @xcite , one way to do this is in fact coupling this scalar field to another field and changing the source for this other field . away from the critical point , adiabaticity holds for a sufficiently low rate of change . we will show that the adiabatic corrections are proportional to _ linear _ time derivatives , even though the equations of motion involve only second order time derivatives . this is a well - known phenomenon and happens because of dissipation implied by the presence of a horizon . in the bulk , we need to impose regularity conditions at the horizon , which is easily done using ingoing eddington - finkelstein coordinate @xcite @xmath17 as time . @xmath17 coincides with the usual time coordinate @xmath18 on the boundary and @xmath19 , though @xmath20 . the bulk equations now involve first order derivatives @xmath21 . as expected , adiabaticity fails as one approaches the critical point at a time which we choose to be @xmath22 . the appearance of a first order time derivative is related to the fact that the dynamical critical exponent of similar models turn out to be @xmath23 @xcite . when adiabaticity fails , the system enters a scaling regime . a useful way to understand the dynamics is to decompose the bulk field in terms of eigenstates of the radial operator which appears in the linearized problem around the equilibrium solution . for any nonzero temperature of the background black brane , this operator has a zero mode which is regular at the horizon exactly at the critical point . we will show that in the critical region , the dynamics for small @xmath24 is dominated by that of this zero mode . the other modes remain adiabatic . the scaling properties can be then understood in terms of a landau - ginsburg type dynamics of the zero mode with a _ linear _ time derivative . the order parameter remains nonzero even when the instantaneous value of the coupling reaches the equilibrium critical value and first drops to zero at a time @xmath25 where @xmath24 denotes the rate of change of the coupling in the critical region . the departure of the value of the order parameter from the instantaneous equilibrium value also scales as @xmath26 . the exponents @xmath27 and @xmath28 are essentially determined by the static critical exponents . beyond the critical region , however , the other modes become important . in fact , at very late times the bulk equation may be approximated by its linearized form , so that the solutions are nothing but the usual quasinormal modes around the background black hole . as usual , the imaginary part of the lowest quasinormal mode frequency then determine the decay of late time behavior . we also present preliminary results of a numerical solution of the time dependent equations for the bulk field and extract the time dependence of the order parameter . the results are consistent with the above picture . our numerical results are not yet accurate enough to verify the scaling behavior . however this should be possible with further work . in section 2 we discuss spatially homogeneous classical quench in a landau - ginsburg type model as a prelude . in section 3 we review the main faetures of the equilibrium phase transition . in section 4 we discuss the failure of adiabaticity in quantum quench across this transition . we then show that in the critical region the dynamics for an initially slow quench is dominated by that of the zero mode , and discuss its relationship of the dynamics to the landau - ginsburg model of section 2 . in section 5 we present the results of a numerical solution of the bulk equation of motion and verify the expectations in the previous section . section 6 contains some concluding remarks . the three appendices provide details of derivation of the some of the results contained in the main text .

the model involves a probe neutral scalar field with mass - squared in the range in a charged black brane background . for a given brane background there is a critical mass - squared , such that for the scalar field condenses . the theory is critical when and the source for the dual operator vanishes . at the critical point , the radial operator for the bulk linearized problem has a zero mode . we show that in the critical region the dynamics for an initially slow variation is dominated by the zero mode : this leads to an effective description in terms of a landau - ginsburg type dynamics with a _ linear _ time derivative . starting with an adiabatic initial condition in the ordered phase , we find that the order parameter drops to zero at a time which is later than the time when or hits zero . in the critical region , , and the departure of the order parameter from its adiabatic value , scale with the rate of change , with exponents determined by static critical behavior . numerical results for the order parameter are consistent with these expectations .

we study the problem of quantum quench across a critical point in a strongly coupled field theory using ads / cft techniques . the model involves a probe neutral scalar field with mass - squared in the range in a charged black brane background . for a given brane background there is a critical mass - squared , such that for the scalar field condenses . the theory is critical when and the source for the dual operator vanishes . at the critical point , the radial operator for the bulk linearized problem has a zero mode . we study the dynamics of the order parameter with a time dependent source , or a null - time dependent bulk mass across the critical point . we show that in the critical region the dynamics for an initially slow variation is dominated by the zero mode : this leads to an effective description in terms of a landau - ginsburg type dynamics with a _ linear _ time derivative . starting with an adiabatic initial condition in the ordered phase , we find that the order parameter drops to zero at a time which is later than the time when or hits zero . in the critical region , , and the departure of the order parameter from its adiabatic value , scale with the rate of change , with exponents determined by static critical behavior . numerical results for the order parameter are consistent with these expectations . uk/11 - 06 + * quantum quench across a holographic critical point * + pallab basu and sumit r. das

hep-ph0212115

i

future linear colliders are expected to reveal the answers to many questions of modern particle physics . one of these concerns the the physics of the higgs particle and origin of electroweak symmetry breaking . the neutral scalar higgs boson is an important ingredient of the standard model ( sm ) and is the only sm elementary particle which has not been detected so far ( see for a review @xcite , @xcite ) . the lower limit on @xmath7 of approximately @xmath8 gev at @xmath9 c.l . has been obtained from direct searches at lep @xcite . current experiments are concentrating on the possibility of finding a higgs particle in the intermediate mass region @xmath10 in this region it decays mainly to a @xmath11 pair . the _ photon mode _ of the future linear collider ( lc ) , namely the collisions of the energetic polarized compton photons , will be used for the production and for the study of the higgs particle . in the intermediate mass range , the main production process is @xmath12 qcd as well as electroweak radiative corrections to this process have been studied very well and have been found to be small in this region @xcite . the main challenge , however , is to get under control the background process @xmath13 which gets extremely large qcd corrections . in this paper we discuss the process of the quark anti - quark production in the photon mode of the lc , @xmath0 . the amplitude for this process in the scalar channel contains large double logarithms at @xmath14 . at very high energies , the large logarithms spoil the perturbative predictions . therefore it is mandatory to develop a clear resummation procedure for this double and single logarithmic terms . let us stress , that the main interest in the process @xmath0 comes from the fact ( but is not limited to it ) that it represents the dominant background for the production of the higgs particle , @xmath15 @xcite . in fact , our motivations for this study are twofold : + ( 1 ) a detailed study of the process @xmath0 is very important due to phenomenological reasons mentioned above ; and + ( 2 ) in addition to that , the quark anti - quark production in photon collisions , being one of the simplest process in qcd , is important in its own right , e.g. for studying and understanding qcd effects . the born cross section for the polarized @xmath16 production in the scalar channel ( at @xmath17 ) , where the higgs will be studied as well , is suppressed by @xmath18 @xcite , ( here @xmath19 is the center of mass energy of the initial photons ) . however , the perturbative qcd corrections contains the large double logarithms of the form @xmath20 , which give a contribution to the cross section which is of the same order as the born contribution at high energies . the presence of the large correction was noticed by jikia in @xcite . the double logarithmic nature and the origin of these corrections were studied in @xcite . the authors studied the process to one and two loop accuracy . later , the form of the resummed results for the double logarithms has been argued in @xcite . these authors also claimed that the double logarithms have a``non - sudakov '' origin . in this paper , we first present an alternative way of understanding the resummation procedure for the double logarithms . the general idea of our approach , is that the only source of double logarithms is the off - shell sudakov form factor included in the triangle topologies of the one - loop box diagram . we have proved that the other types of the higher loop diagrams will either cancel in subgroups or develop a universal on - shell sudakov exponent due to the final quark anti - quark lines . in addition to that , ( 1 ) we extend this analysis to the next - to - leading - logarithmic ( nll ) accuracy , and ( 2 ) we study the mechanism of cancellations alluded to above , between the different diagrams which leads to very simple resummed results . we demonstrate the cancellation explicitly at three loops and point out the mechanism responsible which then allows for the generalization to higher loops , all up to the next to leading logarithmic accuracy . as an aside , we argue that all the large logarithms up to the next to leading level are related to the sudakov ones . this includes not only the leading ones of the form @xmath1 but also the next to leading ones of the form @xmath2 ( @xmath3 is the quark mass ) . it is this fact together with an understanding of the cancellation mechanism which allows us to develop an easy resummation procedure . the paper is organized as follows . in section 2.1 we discuss the one loop diagrams and explain which topologies are important . in section 2.2 we present a procedure for the resummation of the double logarithms . in section 2.3 we extend our analysis to the next - to - leading logarithmic level . in this section we consider all possible topologies and give the final result of the resummation . section 3 is devoted to the study of the different cancellations which are responsible for the simple resummed results of section 2.2 for the double logarithmic case . in section 4 we tackle the case of the form factors to the next to leading logarithmic accuracy and justify the results given in section 2.3 . the paper ends with some conclusions and discussions .

the only source of all the logarithms to this accuracy is the off - shell sudakov form factor included into the triangle topologies of the one - loop box diagram . we prove that any other configurations of diagrams to this accuracy , either cancel in subgroups or develop a universal on - shell sudakov exponent due to the final quark anti - quark lines . we study the mechanism of cancellations between the different diagrams , which leads to the simple resummed results . we show the cancellation explicitly at three loops for the leading and at two loops for the next - to - leading logarithms . we also point out the general mechanism responsible for it , and discuss how it can be extended to higher orders .

we study the resummation of large qcd radiative corrections up to the next to leading logarithmic accuracy to the process ; i.e. , we resum logarithms of the type and ( is the quark mass ) . the only source of all the logarithms to this accuracy is the off - shell sudakov form factor included into the triangle topologies of the one - loop box diagram . we prove that any other configurations of diagrams to this accuracy , either cancel in subgroups or develop a universal on - shell sudakov exponent due to the final quark anti - quark lines . we study the mechanism of cancellations between the different diagrams , which leads to the simple resummed results . we show the cancellation explicitly at three loops for the leading and at two loops for the next - to - leading logarithms . we also point out the general mechanism responsible for it , and discuss how it can be extended to higher orders . mctp-02 - 66 + * on the resummation of large qcd corrections to * + _ michigan center for theoretical physics + randall laboratory of physics , university of michigan , + ann arbor , michigan 48109 - 1120 , usa _ e - mail : [email protected] +e - mail : [email protected] +e - mail : [email protected]

hep-ph0212115

i

in this paper we have studied the resummation up to the next to leading logarithmic level of the qcd radiative corrections to @xmath331 production by photon photon collisions . apart from the phenomenological applications , this problem has inherent interest in providing a theoretical laboratory to study qcd effects . we showed that to the accuracy considered all logarithms are of the sudakov type . on a diagram by diagram basis other types of diagrams do give rise to next to leading logarithms but they cancel amongst each other by the dipole mechanism . we explicitly showed how the dipole mechanism works to 3 loops and outlined an all orders generalization . at the nll level we found that in the feynman gauge the only new logarithms are of collinear origin . the final expression for the resummed amplitude is given in section 2 , eq.([ampnll ] ) together with eq.([rep1],[finalb],[finalc ] ) .

we study the resummation of large qcd radiative corrections up to the next to leading logarithmic accuracy to the process ; i.e. , we resum logarithms of the type and ( is the quark mass ) . mctp-02 - 66 + * on the resummation of large qcd corrections to * + _ michigan center for theoretical physics + randall laboratory of physics , university of michigan , + ann arbor , michigan 48109 - 1120 , usa _

we study the resummation of large qcd radiative corrections up to the next to leading logarithmic accuracy to the process ; i.e. , we resum logarithms of the type and ( is the quark mass ) . the only source of all the logarithms to this accuracy is the off - shell sudakov form factor included into the triangle topologies of the one - loop box diagram . we prove that any other configurations of diagrams to this accuracy , either cancel in subgroups or develop a universal on - shell sudakov exponent due to the final quark anti - quark lines . we study the mechanism of cancellations between the different diagrams , which leads to the simple resummed results . we show the cancellation explicitly at three loops for the leading and at two loops for the next - to - leading logarithms . we also point out the general mechanism responsible for it , and discuss how it can be extended to higher orders . mctp-02 - 66 + * on the resummation of large qcd corrections to * + _ michigan center for theoretical physics + randall laboratory of physics , university of michigan , + ann arbor , michigan 48109 - 1120 , usa _ e - mail : [email protected] +e - mail : [email protected] +e - mail : [email protected]

cond-mat0608649

i

the investigation of systems involving ferromagnet ( f ) and superconductor ( s ) junctions is an active component of superconductor - based spintronics@xcite research . a broad array of interesting effects arises in s / f nanostructures , which opens doors for nanotechnologies and associated devices and applications that may offer benefits beyond current superconducting devices such as standard josephson junctions . advances in fabrication techniques permit growth of ferromagnet and superconductor layers in the form of junctions and heterostructures smooth up to the atomic scale . the arrangement of consecutive f and s layers , as in sfs junctions , results in competition between magnetic and superconducting orderings . superconducting correlations can leak into the ferromagnet while spin polarization can extend into the superconductor : these are the now well established s / f proximity effects.@xcite the phase coherence embodied in the superconducting correlations becomes modified in the f regions . the exchange energy in the ferromagnet shifts the kinetic energies of the quasiparticles constituting the cooper pairs and subsequently a new superconducting state arises whereby the center of mass momentum of the pair is nonzero.@xcite this results in a spatially decaying pair amplitude that oscillates over a characteristic length scale much smaller than the superconducting coherence length . the modulating pair amplitude within the magnet indirectly links adjacent s layers , and thus proximity effects in f cause local oscillations in physically relevant single - particle quantities , including the magnetization@xcite and density of states@xcite ( dos ) . similarly , in the s material the magnet locally polarizes the superconductor , causing a monotonic decline in the pairing correlations near the interface over an extended region . the associated spin - split andreev quasiparticle states also lead to interesting local behavior in the dos and magnetic moment in the superconductor . the nontrivial behavior of the proximity effects in these structures plays a central role in the competition between the magnetic and superconducting order . the modification of the superconducting phase coherence due to proximity effects in clean multilayers consisting of one or more successive sfs junctions is particularly striking . on the atomic level , the pair amplitude is a smoothly varying function of the spatial coordinates . depending on the values of certain parameters ( such as f layer width , @xmath3 ) the damped oscillatory pair amplitude in the f layer may arrange itself in such a manner that is energetically favorable for its sign to change from one of the s layers to the next , yielding a so - called @xmath1-junction , as first proposed long ago.@xcite if the pair amplitude does not change sign between s layers , it is an ordinary or @xmath0-junction . there is a rich and broad parameter space that then enables a certain level of control over the competing magnetic and superconducting orderings , allowing one to increase or diminish the proximity effects that dominate the relative sfs coupling . the actual equilibrium state ( @xmath0 or @xmath1 ) is dependent upon several variables , including predominantly the f region s material characteristics and the temperature , @xmath2 , all of which ultimately determine the pair amplitude modulation in the magnet . a system comprised of a larger number of sfs sequences results in a greater number of possible @xmath0 or @xmath1 junction combinations . the transitions between @xmath0 and @xmath1 states can be explored through the signatures of a variety of physical parameters . experimental study of this question has focused primarily on measurements of the critical current @xmath7@xcite and , thermodynamically , on the critical temperature@xcite of the transition to the normal state , @xmath6 . evidence of @xmath8 transitions can be seen in the sfs josephson coupling , which manifests itself in the vanishing of @xmath7 , although higher order harmonics in the current - phase relationship can modify this.@xcite measurements@xcite as a function of @xmath3 have shown that @xmath6 , which is of course smaller than @xmath9 , the critical temperature for bulk s material , oscillates as a function of @xmath3 , confirming theoretical predictions@xcite based upon the semi - classical usadel equations . intrinsically linked to this phenomenon are damped oscillations in @xmath7 as a function of @xmath3 and exchange energy in the clean@xcite and dirty limits.@xcite these changes in the critical current have been experimentally confirmed@xcite . of particular interest is ref . , which demonstrates the robustness of @xmath8 transitions by providing evidence of switching in samples with interfaces that were not atomically smooth . indeed , despite deviations as large as @xmath10 over @xmath11 of a sample , clear evidence of switching was found . near @xmath6 , and in the diffusive limit , the theory was later extended to include arbitrary interface transparency.@xcite measurements of the superconducting phase@xcite have corroborated the @xmath1 state in sfs junctions , and the predicted oscillations in several thermodynamic quantities have in many cases been found experimentally . direct evidence of dos oscillations was reported in a tunneling spectroscopy experiment,@xcite but not observed@xcite in other cases . such studies give us the valuable insight that the oscillations are correlated with @xmath12 transitions . in this work , we show that there is indeed an intimate relation between the oscillations in @xmath6 as a function of relevant parameters and the transitions from the @xmath1 to the @xmath0 state and we find good quantitative agreement with experimental data . since the possibility of having a particular junction configuration depends fundamentally on the intricate properties of the pair amplitude , the complicated and demanding task of calculating the pair potential , @xmath13 , rigorously and self - consistently becomes absolutely necessary , particularly as the inhomogeneities occur on a microscopic scale . the first step in the self - consistency process often involves an assumed simple piecewise constant form for @xmath13 , which is then iterated through the relevant equations until convergence is achieved . it is not justified to bypass the technical difficulties associated with self - consistency and to use only an assumed form for the pair potential . the final calculated @xmath13 often deviates significantly , even in overall symmetry , from the assumed form : the self - consistent @xmath13 has a complicated spatial behavior that can lead to stable states mixing @xmath0 and @xmath1 junction configurations.@xcite a self - consistently calculated pair potential minimizes , at least locally , the free energy of the system . to determine whether the calculated state is merely a local minimum of the free energy or the global one , the free energies from all possible self - consistent @xmath0- and @xmath1-junction configurations must be compared with high precision . recently developed numerical algorithms@xcite overcome the difficulties that arise in computing the small difference between much larger quantities and enable accurate computation of the differences in the values of the condensation free energy of different minima . for clean sfs junctions , a relevant set of basic parameters to consider includes @xmath3 , the exchange energy @xmath14 and @xmath2 . as these parameters vary , the @xmath0 or @xmath1-state free energies may cross at certain points in parameter space , yielding phase transitions . it has been shown@xcite that at @xmath15 , transitions occur when varying @xmath14 , @xmath3 and also the mismatch parameter @xmath4 , defined as the ratio of fermi energies in the f and s regions . this mismatch can induce a transition because at @xmath16 , when the fermi wavevectors match , the layers couple more strongly , while at small @xmath4 the coupling is effectively weaker . if the temperature varies it is also possible to have a first order transition between @xmath0 and @xmath1 junction states , as recently shown in both the clean,@xcite and dirty@xcite limits , and also predicted for short - period f / s superlattices.@xcite the temperature has been shown to have a pronounced effect on the pair amplitude in the f region of f / s structures,@xcite strongly diminishing its magnitude while maintaining its characteristic period of oscillation as @xmath2 increases . this translates into weaker coupling between adjacent s layers . if the magnet width is such that the junction is near a @xmath8 transition point at @xmath15 , increasing the temperature can result in the critical current of the junction having a non - monotonic temperature dependence.@xcite it has been argued@xcite that the transition is discontinuous in uniform samples but rounded off in samples of variable thickness.@xcite however , a transition can be observed@xcite in just a portion of samples with nonuniform thickness . these results indicate that the temperature can be used to switch between a @xmath0 and a @xmath1 state configuration . it is possible to locate regions of parameter space that give the desired transitions using the @xmath15 results as guides , however the task is still significantly demanding . such temperature transitions were found to occur in one - junction and 3-junction systems for moderate values of @xmath4.@xcite thus , a 1-junction system was found to have a @xmath17 first order transition as @xmath2 was lowered , and a @xmath18 transition was found for a 3-junction system.@xcite in each case , the free energies of a stable and a metastable state crossed at the transition temperature with differing derivatives , and therefore entropies . the existence of metastable states and an entropy discontinuity are hallmarks of first - order phase transitions . moreover , the reported latent heats were reported to be within available experimental resolution . it is therefore desirable to systematically study the coexistence of metastable states and the nature of the transition in sfs and higher - order multilayer structures . the main objective of this paper therefore , is to map out the regions of parameter space in which the different junction states are stable , and to trace the locations of the phase transitions in systems with sfs junctions . an extensive sweep of the geometric and materials parameters including @xmath3 , @xmath4 , as well as @xmath2 , is performed . to start with , it is important to know which @xmath3 and @xmath4 ranges allow more than one self - consistent state at @xmath15 . one can then check if a metastable state at low temperature becomes the equilibrium state at higher @xmath2 . by using this procedure we obtain a complete phase diagram of an sfs junction within the relevant region of @xmath19 space . to accomplish this , we use a method that can accommodate arbitrary values of the above parameters , without recourse to approximations . as discussed above , all calculations involving the pair potential must be performed using fully self - consistent algorithms , starting from the microscopic equations ( bogoliubov - degennes ( bdg ) ) . the need for a fully microscopic theory arises because the characteristic period of the pair potential oscillations approaches the atomic scale . for the nanoscale interlayer widths considered here , geometrical oscillations decisively influence the final results . we present in sec . [ sec : methods ] the microscopic equations and the associated notation relevant for systems containing sfs junctions . we review the numerical procedures involved in calculating the self - consistent pair potential and quasiparticle spectra , and the method used to calculate the primary thermodynamic quantity , the condensation free energy , @xmath20 , from the self - consistent spectrum and pair potential . we also outline a semi - analytic method to calculate @xmath6 through the linearized bdg equations . in sec . [ sec : results ] , we show that first order transitions with measurable latent heat can occur between states containing different numbers of @xmath0 and @xmath1 junctions as the temperature changes . for sfs junctions the transitions we find are from the @xmath1 to the @xmath0 state as @xmath2 increases , as found in experiment,@xcite and occur predominantly in regions where @xmath6 is low . using the @xmath6 calculated from the linearized theory and the @xmath8 phase transitions , we obtain the full phase diagram in an extended region of parameter space spanned by @xmath2 , @xmath3 , and @xmath4 . we compare our calculated oscillations in @xmath6 as a function of @xmath3 with reported nb / co experimental data@xcite and find good agreement .

the trilayer sfs junction is studied in particular detail : first order transitions between and states as a function of the temperature are located by finding where the free energies of the two phases cross . the occurrence of these transitions is mapped as a function of the thickness of the f layer and of the fermi wavevector mismatch parameter . similar first order transitions are found for systems with a larger number of layers : examples are given in the 7 layer ( 3 junction ) case . the latent heats associated with these phase transitions are evaluated and found to be experimentally accessible . the transition temperature to the normal state is calculated from the linearized bogoliubov - degennes equations and found to be in good agreement with experiment . the first order transitions are associated with dips in the transition temperature to the non - superconducting state , which should facilitate locating them .

we study the thermodynamics of clean , layered superconductor / ferromagnet nanostructures using fully self consistent methods to solve the microscopic bogoliubov - degennes equations . from these self - consistent solutions the condensation free energies are obtained . the trilayer sfs junction is studied in particular detail : first order transitions between and states as a function of the temperature are located by finding where the free energies of the two phases cross . the occurrence of these transitions is mapped as a function of the thickness of the f layer and of the fermi wavevector mismatch parameter . similar first order transitions are found for systems with a larger number of layers : examples are given in the 7 layer ( 3 junction ) case . the latent heats associated with these phase transitions are evaluated and found to be experimentally accessible . the transition temperature to the normal state is calculated from the linearized bogoliubov - degennes equations and found to be in good agreement with experiment . thus , the whole three dimensional phase diagram in space can be found . the first order transitions are associated with dips in the transition temperature to the non - superconducting state , which should facilitate locating them . results are given also for the magnetic moment and the local density of states ( dos ) at the first order transition .

1406.4819

c

we theoretically studied the transport properties of a corner - junction realized by using a two - dimensional topological insulator shaped in the form of a four - terminal qpc . the coulomb interaction effects , at mean field level , and the reactivation of the spin - orbit coupling due to the tight confinement of the electron densities within the qpc region are taken into account by implementing a self - consistent scattering field theory able to capture the coulomb blockade physics . in this framework , the charging effects are described by a low - energy scattering matrix parametrized by two self - consistent parameters , namely @xmath147 and @xmath148 , and the resulting theoretical approach takes simultaneously into account non - equilibrium effects and coulomb blockade physics beyond the linear response regime . assuming a crossed - bias configuration ( @xmath191 and @xmath319 ) , we studied the current - voltage characteristics of the device and the current - current local and non - local correlations ( noise ) . the dc transport of the device shows a coulomb blockade regime which can be removed by using the side - gate and the bias voltage , while the differential conductance presents a structure reminiscent of a single coulomb diamond whose extension along the side - gate axis is proportional to the interaction value @xmath216 . concerning the fluctuation properties of the current flowing through the device , we studied the thermal noise and the transport noise . it has been proven that the ( local ) thermal noise takes the universal value @xmath242 below a gap - dependent characteristic temperature @xmath244 , the latter property being relevant in discriminating devices with helical properties . the non - local thermal fluctuations contain instead important information on the tunneling amplitudes and thus can help in characterizing the junction properties . the coulomb blockade region has been further characterized by evaluating the fano factor which provides a measure of the transport fluctuations in a given electrode . it has been found a sub - poissonian character of the fluctuation outside the coulomb blockade region and a super - poissonian value of the fano factor as the effect of the blockade imposed by the charging energy within the qpc . the non - local transport fluctuations exhibit a double - minimum structure as a function of the applied bias . the double - minimum is peculiar of a non - vanishing coulomb interaction within the qpc and thus is not present when the charging energy of the constriction is neglected . the theory presented in this work contains all the relevant ingredients needed to describe the transport properties of a corner - junction realized using helical matter ( i.e. a topological insulator ) and allows a direct comparison with forthcoming experimental works . the experimental implementation of our proposal presents similar difficulties of the ones described in ref . [ ] even though the topological corner - junction setup does not require the use of high magnetic fields to induce edge states , the latter being an important simplification of the experimental situation . finally , it s worth mentioning that the universal nature of the thermal fluctuation of a corner - junction realized using helical matter can be an important signature to probe the emergence of a topological phase at cryogenic temperatures ( few kelvins ) . the latter point appears to be even more important in testing topological properties of the next - generation room temperature topological insulators@xcite .

we study the transport properties of a four - terminal corner junction made by etching a two - dimensional topological insulator to form a quantum point contact ( qpc ) . peculiar signatures of these effects are theoretically investigated using a scattering field theory modified to take into account the electron - electron interaction within a self - consistent mean - field approach . universal aspects of the thermal noise of the corner junction made of helical matter are also discussed .

we study the transport properties of a four - terminal corner junction made by etching a two - dimensional topological insulator to form a quantum point contact ( qpc ) . the qpc geometry enables inter - boundary tunneling processes allowing for the coupling among states with different helicity , while the tight confinement in the qpc region activates charging effects leading to the coulomb blockade physics . peculiar signatures of these effects are theoretically investigated using a scattering field theory modified to take into account the electron - electron interaction within a self - consistent mean - field approach . the current - voltage characteristics and the current fluctuations ( noise ) are derived beyond the linear response regime . universal aspects of the thermal noise of the corner junction made of helical matter are also discussed .

1104.1560

i

this article is part of a series of works where we study quantitative merging properties of time inhomogeneous finite markov chains . time inhomogeneity leads to a great variety of behaviors . moreover , even in rather simple situations , we are at a loss to study how a time inhomogeneous markov chain might behave . here , we focus on a natural but restricted type of problem . consider a sequence of aperiodic irreducible markov kernels @xmath1 on a finite set @xmath2 . let @xmath3 be the invariant measure of @xmath4 . assume that , in a sense to be made precise , all @xmath4 and all @xmath3 are similar and the behavior of the time homogeneous chains driven by each @xmath4 separately is understood . can we then describe the behavior of the time inhomogeneous chain driven by the sequence @xmath1 ? to give a concrete example , on @xmath5 , consider a sequence of aperiodic irreducible birth and death chain kernels @xmath4 , @xmath6 with @xmath7 and with reversible measure @xmath3 satisfying @xmath8 , for all @xmath9 . what can we say about the behavior of the corresponding time inhomogeneous markov chain ? remarkably enough , there is very little known about this question . what can we expect to be true ? what can we try to prove ? let @xmath10 denote the distribution , after @xmath11 steps , of the time inhomogeneous chain described above started at @xmath12 . it is not hard to see that such a chain satisfies a doeblin type condition that implies @xmath13 in the absence of a true target distribution and following @xcite , we call this property _ merging_. of course , this does not qualify as a quantitative result . extrapolating from the behavior of each kernel @xmath4 taken individually , we may hope to show that , if @xmath14 then @xmath15 the aim of this paper and the companion paper @xcite is to present techniques that apply to this type of problem . the simple minded problem outlined above is actually quite challenging and we will not be able to resolve it here without some additional hypotheses . however , we show how to adapt techniques such as singular values , nash and log - sobolev inequalities to time inhomogeneous chains and provide a variety of examples where these tools apply . in @xcite , we discussed singular value techniques . here , we focus on nash and log - sobolev inequalities . the examples treated here ( as well as those treated in @xcite ) are quite particular despite the fact that one may believe that the techniques we use are widely applicable . whether or not such a belief is warranted is a very interesting and , so far , unanswered question . this is deeply related to the notion of @xmath0-stability that is introduced here and in @xcite . the examples we present here and in @xcite are about the only existing evidence of successful quantitative analysis of time inhomogeneous markov chains . a more detailed introduction to these questions is in @xcite . the references @xcite discuss singular value techniques in the case of time inhomogeneous chains that admit an invariant distribution [ all kernels @xmath4 in the sequence @xmath16 share a common invariant distribution ] . time inhomogeneous random walks on finite groups provide a large collection of such examples ( see also @xcite for a particularly interesting example : semirandom transpositions ) . the papers @xcite are also concerned with quantitative results for time inhomogeneous markov chains . in particular , the techniques developed in @xcite are closely related to ours and we will use some of their results concerning the modified logarithmic sobolev inequality . references on the basic theory of time inhomogeneous markov chains are @xcite . for a different perspective , see also @xcite . a short review of the relevant aspects of the time inhomogeneous markov chain literature , including the use of `` ergodic coefficients '' can be found in @xcite . the vast literature on the famous simulated annealing algorithm is not very relevant for our purpose but we refer to @xcite for a recent discussion . the paper @xcite concerned with filtering and genetic algorithms describes problems that are related in spirit to the present work . let @xmath2 be a finite set equipped with a sequence of kernels @xmath17 such that , for each @xmath11 , @xmath18 and @xmath19 . an associated markov chain is a @xmath2-valued random process @xmath20 such that , for all @xmath11 , @xmath21 the distribution @xmath22 of @xmath23 is determined by the initial distribution @xmath24 and given by @xmath25 where @xmath26 is defined inductively for each @xmath11 and each @xmath27 by @xmath28 with @xmath29 ( the identity ) . if we interpret the @xmath30 s as matrices , then this definition means that @xmath31 . this paper is mostly concerned with the behavior of the measures @xmath10 as @xmath11 tends to infinity . in the case of time homogeneous chains where all @xmath32 are equal , we write @xmath33 . our main interest is in ergodic like properties of time inhomogeneous markov chains . in general , one does not expect @xmath34 to converge toward a limiting distribution . instead , the natural notion is that of merging of measures as discussed in @xcite . fix a sequence of markov kernels as above . we say the sequence is merging if for any @xmath35 , @xmath36 if the sequence @xmath16 is merging then , for any two starting distributions @xmath37 , the measures @xmath34 and @xmath38 are merging , that is , @xmath39 . since we assume the set @xmath2 is finite , merging is equivalent to @xmath40 . hence , we also refer to this property as `` total variation merging . '' total variation merging is also referred to as weak ergodicity in the literature and there exists a body of work concerned with understanding when weak ergodicity holds . see , for example , @xcite . a main tool used to show weak ergodicity is that of contraction coefficients . furthermore , in @xcite , birkhoff s contraction coefficient is used to study _ ratio ergodicity _ which is equivalent to what we will later call relative - sup merging . however , it should be noted that even for time homogeneous chains birkhoff coefficients and related methods fail to provide useful quantitative bounds in most cases . our goal is to develop quantitative results in the context of time inhomogeneous chains in the spirit of the work of aldous , diaconis and others . in these works , precise estimates of the mixing time of ergodic chains are obtained . typically , a family of markov chains indexed by a parameter , say @xmath41 , is studied . loosely speaking , as the parameter @xmath41 increases , the complexity and size of the chain increases and one seeks bounds that depend on @xmath41 in an explicit quantitative way . see , for example , @xcite . efforts in this direction for time inhomogeneous chains are in @xcite . still , there are only a very small number of results and examples concerning the quantitative study of merging as defined above for time inhomogeneous markov chains so that it is not very clear what kind of results should be expected and what kind of hypotheses are reasonable . we refer the reader to @xcite for a more detailed discussion . the following definition is useful to capture the spirit of our study . it indicates that the simplest case we would like to think about is the case when the sequence @xmath4 is obtained by deterministic but arbitrary choices between a finite number of kernels @xmath42 . we say that a set @xmath43 of markov kernels on @xmath2 is merging in total variation if for any sequence @xmath44 with @xmath45 for all @xmath46 , we have @xmath47 in the study of ergodicity of finite markov chains , the convergence toward the target distribution is measured using various notions of distance between probability measures . these include the total variation distance @xmath48 the chi - square distance ( w.r.t . @xmath49 . note the asymmetry between @xmath50 and @xmath49 . ) @xmath51 and the relative sup - distance ( again , note the asymmetry ) @xmath52 these will be used here to measure merging . in the quantitative theory of ergodic time homogeneous markov chains , the notion of mixing time plays a crucial role . for time inhomogeneous chain , we propose to consider the following definitions . fix @xmath53 . given a sequence @xmath16 of markov kernels on a finite set @xmath2 , we call max total variation merging time the quantity @xmath54 fix @xmath53 . we say that a set @xmath43 of markov kernels on @xmath2 has max total variation @xmath55-merging time at most @xmath56 if for any sequence @xmath1 with @xmath45 for all @xmath46 , we have @xmath57 , that is , @xmath58 of course , merging can be measured in ways other than total variation . also merging is a bit less flexible than mixing in this respect since there is no reference measure . one very natural and much stronger notion than total variation is relative sup - distance . for time inhomogeneous chains , total variation merging does not necessarily imply relative - sup merging as defined below . see @xcite . we say a sequence @xmath16 of markov kernels on a finite set @xmath2 is merging in relative - sup if for all @xmath35 @xmath59 with the convention that @xmath60 and @xmath61 for @xmath62 . fix @xmath53 , we call relative - sup merging time the quantity @xmath63 we say a set @xmath43 of markov kernels on @xmath2 is merging in relative - sup if any sequence @xmath1 with @xmath64 for all @xmath46 is merging in relative - sup . fix @xmath53 . we say that @xmath43 has relative - sup @xmath55-merging time at most @xmath56 if for any sequence @xmath1 with @xmath45 for all @xmath46 , we have @xmath65 , that is , @xmath66 the following problem is open . it is a quantitative version of the problem stated at the beginning of the introduction . [ pb - bdu ] let @xmath5 and @xmath67 . let @xmath68 be the set of all birth and death chains @xmath69 on @xmath70 with @xmath71 $ ] if @xmath72 , and reversible measure @xmath73 satisfying @xmath74 , @xmath9 . 1 . prove or disprove that there exists a constant @xmath75 independent of @xmath41 such that @xmath68 has total variation @xmath55-merging time at most @xmath76 . 2 . prove or disprove that there exists a constant @xmath75 independent of @xmath41 such that @xmath68 has relative - sup @xmath55-merging time at most @xmath77 . this problem is open ( in most cases ) even if one considers a sequence @xmath1 drawn from a set @xmath78 of two kernels . observe that the hypothesis that the invariant measures @xmath3 are all comparable to the uniform plays some role . how to harvest the global hypothesis of comparable stationary distributions @xmath3 is not entirely clear . see theorem [ th - bdu ] below for a partial solution . if @xmath79 and @xmath80 are not comparable , it is possible for @xmath81 and @xmath82 to have the same mixing time yet for @xmath78 to have a merging time of a higher order . assume that @xmath83 and @xmath84 are two biased random walks with equal drift , one drift to left , the other to the right . despite the fact that each of these random walks has a relative - sup mixing time of order @xmath41 , the inhomogeneous chain driven by the sequence @xmath85 has a relative - sup merging time of order @xmath86 , see @xcite . in this section , we consider a property , @xmath0-stability , that plays a crucial role in the techniques we develop to provide quantitative bounds for time inhomogeneous markov chains . this property was introduced and discussed in @xcite . it is a straightforward generalization of the property of sharing the same invariant measure . unfortunately , it is hard to check . fix @xmath87 . a sequence of markov kernels @xmath88 on a finite set @xmath2 is @xmath0-stable if there exists a measure @xmath24 such that @xmath89 where @xmath34 . if this holds , we say that @xmath88 is @xmath0-stable with respect to the measure @xmath24 . a set @xmath43 of markov kernels is @xmath0-stable with respect to a measure @xmath24 if any sequence @xmath1 such that @xmath45 for all @xmath46 is @xmath0-stable with respect to @xmath24 . if all @xmath4 share the same invariant distribution @xmath73 then @xmath16 is @xmath90-stable with respect to @xmath73 . suppose a set @xmath43 of aperiodic irreducible markov kernels is @xmath0-stable with respect to a measure @xmath24 . let @xmath73 be an invariant measure for some @xmath91 . then we must have @xmath92 hence , @xmath43 is also @xmath93-stable with respect to @xmath73 and any two invariant measures @xmath94 for kernels @xmath95 must satisfy @xmath96 the following theorem which relates to a special case of problem [ pb - bdu ] illustrates the role of @xmath0-stability . [ th - bdu ] let @xmath5 . let @xmath68 be the set of all birth and death chains @xmath69 on @xmath70 with @xmath97 \qquad\mbox{if } |x - y|\le1\ ] ] and reversible measure @xmath73 satisfying @xmath74 , @xmath9 . let @xmath16 be a sequence of birth and death markov kernels on @xmath70 with @xmath98 . assume that @xmath16 is @xmath0-stable with respect to the uniform measure on @xmath70 , for some constant @xmath87 independent of @xmath41 . then there exists a constant @xmath99 ( in particular , independent of @xmath41 ) such that the relative - sup merging time for @xmath16 on @xmath70 is bounded by @xmath100 this will be proved later in a stronger form in section [ section - nash ] . in @xcite the weaker conclusion @xmath101 was obtained using singular value techniques . here , we will use nash inequalities to obtain @xmath102 . it is possible that the set @xmath68 is @xmath0-stable with respect to the uniform measure for some @xmath0 . indeed , it is tempting to conjecture that this is the case although the evidence is rather limited ( see also the discussion in @xcite ) . if this is true , then theorem [ th - bdu ] solves problem [ pb - bdu ] . however , we do not know how to approach the problem of proving @xmath0-stability for @xmath68 . while the assumption of @xmath0-stability in theorem [ th - bdu ] is quite strong , sections 4.2 and 5 of @xcite give specific examples of families @xmath68 for which it holds . further , we note that the question of whether or not @xmath0-stability holds is extremely natural and interesting in itself .

we study time - inhomogeneous markov chains with finite state spaces using nash and logarithmic - sobolev inequalities , and the notion of-stability . we develop the basic theory of such functional inequalities in the time - inhomogeneous context and provide illustrating examples . and . .

we study time - inhomogeneous markov chains with finite state spaces using nash and logarithmic - sobolev inequalities , and the notion of-stability . we develop the basic theory of such functional inequalities in the time - inhomogeneous context and provide illustrating examples . and . .

1406.6530

i

studies of the radiative capture reactions on numerous light atomic nuclei have been continued at thermal and astrophysical energies with the model - independent pionless effective field theory ( eft(@xmath0 ) ) approach in the recent years @xcite-@xcite . the calculation of radiative capture amplitude and cross section of @xmath1 @xmath2 and @xmath5 @xmath6 are an essential input in the calculation of the parity - violating radiative capture of the above processes at the thermal energy @xcite . in the present paper , we study the @xmath1 @xmath2 process fully with the projection operator method based on the cluster - configuration space which is introduced by @xcite . we also consider the calculation of observables with m1 transition up to next - to - next - to - leading order ( @xmath3 ) with the following significant changes in comparison with the previous eft(@xmath0 ) calculation @xcite : a ) including the diagrams with radiation from external nucleon leg , external deuteron leg , and on - shell two - body bubble ( see the diagrams `` @xmath7 '' , `` @xmath8 '' , and `` @xmath9 '' in fig.[fig : nd capture ] of page ) , b ) considering both contributions corresponding to two nucleon poles before and after photon creation in the first diagram of the second row in fig.[fig : nd capture ] , c ) inserting the diagram with the radiation directly from the exchanged nucleon , d ) adding the contribution of the @xmath10 m1 transition and e ) introducing .and using the properly normalized triton wave function . the triton or helium - three wave functions consist of two parts , one is the nucleon and dibaryon cluster wave function and the other is the two nucleon structure of the dibaryon cluster . we follow the bethe - salpeter ( bs ) equation in @xcite and we use the normalization condition of the relativistic two - body vertex function and work out the nonrelativistic one which is suitable for neutron - deuteron ( @xmath11 ) scattering leading to the formation of a triton . the theoretical calculations of the observables in the @xmath1 @xmath2 process were previously performed based on model - dependent approaches @xcite . the cross section and polarization observables were studied theoretically for radiative capture reactions @xmath12@xmath13@xmath14 and @xmath12@xmath15@xmath16 at low energies @xcite . the cross section for thermal neutron radiative capture on the deuteron was measured to be @xmath17 mb @xcite , in agreement with the results of earlier experiments @xcite . in the present work , the calculation of all m1 diagrams are calculated for the incoming doublet and quartet @xmath11 channels fully in the cluster - configuration space up to @xmath3 in sec.[sec : amplitude ] . the calculation of the cross section for @xmath1 @xmath2 is presented in sec.[sec : cross section ] . in sec.[sec : numerical ] numerical aspects of the calculation of m1 amplitudes are discussed . the results and comparison with other theoretical and experimental works are explained in sec.[sec : results ] . finally , we summarize the paper and discuss future investigations in sec.[sec : conclusion ] .

we take a new look at the neutron radiative capture by a deuteron at thermal energy with the pionless effective field theory ( eft( ) ) approach . , we consider all possible one - body and two - body photon interaction diagrams . we compare the cross section of the dominant magnetic m1-transition of up to next - to - next - to - leading order with the results of the previous model - dependent theoretical calculations and experimental data .

we take a new look at the neutron radiative capture by a deuteron at thermal energy with the pionless effective field theory ( eft( ) ) approach . we present in detail the calculation of amplitudes for incoming doublet and quartet channels leading to the formation of a triton fully in the projection method based on the cluster - configuration space approach . in the present work , we consider all possible one - body and two - body photon interaction diagrams . in fact , additional diagrams that make significant changes in the results of the calculation of the total cross section in the process are included in this study . the properly normalized triton wave function is calculated and taken into consideration . we compare the cross section of the dominant magnetic m1-transition of up to next - to - next - to - leading order with the results of the previous model - dependent theoretical calculations and experimental data . the more acceptable results for cross section\:\textrm{mb}$ ] show order by order convergence and cutoff independence . no three - body currents are needed to renormalize observables up to in this process . o # 1#2_#1 -2.5pt # 1#2#3_#1

1406.6530

r

in this work , we have concentrated on the evaluation of the cross section of the @xmath1 @xmath2 process up to @xmath3 . our eft(@xmath0 ) results for the amplitudes and cross sections of the @xmath1 @xmath2 process at thermal energy , @xmath178 mev are shown in table [ tab : results ] . we compare schematically our eft(@xmath0 ) results at thermal energy for the cross section with the previous model - dependent theoretical calculations and the experimental data in fig.[fig : plot ] . cccccc @xmath179 & @xmath180 & @xmath181 & @xmath182 & @xmath183 & @xmath184 + @xmath25 & @xmath185 & @xmath186 & 0.232 & 0.065 & 0.297 @xmath187 0.196 + @xmath188 & @xmath189 & @xmath190 & 0.264 & 0.157 & 0.421 @xmath187 0.093 + @xmath191 & @xmath192 & @xmath193 & 0.273 & 0.196 & 0.469 @xmath187 0.033 + we use the power counting introduced by bedaque @xmath194 in @xcite . the eft(@xmath0 ) expansion parameter is @xmath195 , where @xmath196 and @xmath197 are the small and large parameters , so the nlo and @xmath3 diagrams enter 33@xmath198 and 11@xmath198 corrections to the leading- and next - to - leading - order amplitudes , respectively . also , with respect to our power counting , the error of @xmath3 amplitude must be less than 3.7@xmath198 of the exact value . the cross section is proportional to the square of the amplitude . it is obvious that if the systematic eft(@xmath0 ) error in the amplitude is @xmath199 , as an example , the cross section has a maximum error @xmath200 in the eft(@xmath0 ) approach . so , we expect to have a maximum error of 7@xmath198 at @xmath3 for the cross section . comparison between different theoretical results for the total cross section of the @xmath1 @xmath2 process . the points from left to right denote the results computed by experiment @xcite , our lo eft(@xmath0 ) , our nlo eft(@xmath0 ) , our @xmath3 eft(@xmath0 ) , earlier @xmath3 eft(@xmath0 ) @xcite , av14/viii(ia+mi+md ) @xcite , av18/ix(ia+mi+md ) @xcite , av14/viii(ia+mi+md+@xmath201 ) @xcite , av18/ix(ia+mi+md+@xmath201 ) @xcite , av14/viii(ia+mi+md+@xmath202 ) @xcite , av18/ix(ia+mi+md+@xmath202 ) @xcite , av18/ix(gauge inv . ) @xcite , and av18/ix(gauge inv.+@xmath203-current ) @xcite methods , respectively . the thin band indicates the error band of the experimental result of the cross section . two horizontal dashed lines determine the upper and lower limits due to our systematic eft(@xmath0 ) error at @xmath3.,width=604 ] our results in table [ tab : results ] show the convergence in our power counting from lo to @xmath3 . at nlo , 0.124 mb adds to the leading - order value and at @xmath3 0.048 mb to the next - to - leading order . our eft result for the total cross section of neutron radiative capture by a deuteron at @xmath3 , @xmath204 mb , has an error of 7@xmath198 compare with the experimental value , @xmath17 mb . we stress that the contribution of the e2 transition has not been included in our calculation for the amplitude of the @xmath1 @xmath2 reaction . the e2 transition is suppressed by two powers of the initial nucleon momentum or photon energy compared to the dominant m1 transition . therefore , this effect numerically has a contribution of @xmath205 correction in the quartet - initial - channel amplitude of @xmath1 @xmath2 and so @xmath206 in total cross section at threshold regime . also , with respect to the power counting as discussed above , we expect a maximum error @xmath207 in @xmath3 eft(@xmath0 ) results of the cross section . thus the 7@xmath198 error in our @xmath3 results is acceptable . we believe that the higher - order corrections make this discrepancy narrow . cc @xmath179 & abs[@xmath208 + @xmath25 & 0.098756 + @xmath188 & 0.045714 + @xmath191 & 0.004006 + according to table [ tab : cutoff var ] , we have computed the cutoff variation of our eft(@xmath0 ) results for the total cross section within a natural range of @xmath209 to @xmath210 mev at lo , nlo , and @xmath3 . the range of cutoff variation should be a few times the pion mass because , here , the existence of a definite @xmath40 limit in an eft calculation does not guarantee that the results found in that limit are rigorous consequences of the eft @xcite . our results in table [ tab : cutoff var ] indicate that the m1 amplitudes and the cross section of @xmath1 @xmath2 are cutoff independent and properly renormalized . the differences of our results and the previous eft(@xmath0 ) calculation of total cross section at thermal energy @xcite are due to the ignored diagrams and the @xmath82@xmath83 m1 transition effects . @xmath211 & @xmath212 & @xmath213 difference@xmath213 + @xmath25 & @xmath214 & @xmath214 & 0 + @xmath188 & @xmath215 & @xmath216 & @xmath217 + @xmath191 & @xmath218 & @xmath219 & @xmath220 + the @xmath84 coefficient corresponding to the contribution of the @xmath82@xmath83 m1 transition is small compared with @xmath85 which comes from the @xmath221@xmath83 m1 transition @xcite . so , we expect that the @xmath82@xmath83 m1 transition has a small ( and negligible ) effect at nlo results but at @xmath3 the @xmath82@xmath83 m1 transition could have a significant effect . our results for the total cross section with and without the @xmath84 coefficient effect which are summarized in table [ tab : l2 effect ] are as we expected . .the investigation of the ignored contributions in the previous eft(@xmath0 ) calculation @xcite at each order . @xmath222 indicates the total @xmath66 amplitude of m1 @xmath1 @xmath2 without the contribution of the time ordering that corresponds to the photon emitted during the nucleon exchange in the first diagram of the second line in fig.[fig : nd capture ] . @xmath223 denotes the sum of the amplitudes of the diagrams `` @xmath7 '' , `` @xmath8 '' , and `` @xmath9 '' for the incoming @xmath126 channel . also , @xmath224 is only the contribution of the `` @xmath225 '' diagram in the first line of fig.[fig : nd capture ] for the initial @xmath126 channel , respectively . @xmath226 and @xmath227 in the first column indicate the initial quartet ( @xmath228@xmath229 ) and doublet ( @xmath230@xmath231 ) channels , respectively . the results of the amplitudes are presented in units of @xmath232 . [ cols="^,^,^,^,^,^",options="header " , ] the effects of the diagrams in fig.[fig : nd capture ] which have been neglected in the previous eft(@xmath0 ) calculation @xcite have been investigated in table [ tab : ignored diagrams effects ] . the results in the third column of table [ tab : ignored diagrams effects ] are the total doublet and quartet amplitudes of the m1 @xmath1 @xmath2 transition at lo , nlo and @xmath66 . in the fourth column , we present the computed values of the contribution which is only corresponding to the nucleon pole before photon creation in the first diagram in the second line of fig.[fig : nd capture ] at each order . the fifth and sixth columns of table [ tab : ignored diagrams effects ] represent only the evaluated values for the amplitudes of the `` @xmath233 '' and `` @xmath225 '' diagrams in the first line of fig.[fig : nd capture ] , respectively , for both doublet and quartet channels . the lack of a correct calculation of the first diagram of the second line in fig.[fig : nd capture ] creates the significant errors as indicated in the fourth column of table iv at each order . the results shown in the fifth column of table [ tab : ignored diagrams effects ] indicate that the diagrams with radiation from external nucleon leg , external deuteron leg , and on - shell two - body bubble in the first line of fig.[fig : nd capture ] are lo effects and so , one expects that these diagrams have a very important effect in the final results of the amplitude of the m1 @xmath1 @xmath2 transition and could not be ignored . but the last column depicts that the diagram `` @xmath225 '' in the first line of fig.[fig : nd capture ] has a small effect at lo especially in the quartet channel as estimated in the previous eft(@xmath0 ) calculation @xcite . finally , we emphasize that our results have been evaluated using the properly normalized triton wave function .

we present in detail the calculation of amplitudes for incoming doublet and quartet channels leading to the formation of a triton fully in the projection method based on the cluster - configuration space approach . in the present work in fact , additional diagrams that make significant changes in the results of the calculation of the total cross section in the process are included in this study . the more acceptable results for cross section\:\textrm{mb}$ ] show order by order convergence and cutoff independence .

we take a new look at the neutron radiative capture by a deuteron at thermal energy with the pionless effective field theory ( eft( ) ) approach . we present in detail the calculation of amplitudes for incoming doublet and quartet channels leading to the formation of a triton fully in the projection method based on the cluster - configuration space approach . in the present work , we consider all possible one - body and two - body photon interaction diagrams . in fact , additional diagrams that make significant changes in the results of the calculation of the total cross section in the process are included in this study . the properly normalized triton wave function is calculated and taken into consideration . we compare the cross section of the dominant magnetic m1-transition of up to next - to - next - to - leading order with the results of the previous model - dependent theoretical calculations and experimental data . the more acceptable results for cross section\:\textrm{mb}$ ] show order by order convergence and cutoff independence . no three - body currents are needed to renormalize observables up to in this process . o # 1#2_#1 -2.5pt # 1#2#3_#1

physics0605117

i

this work is inspired by the problem of predicting the effects of earthquakes in cities . it is known that the most dangerous effects are produced in cities built on soft underground underlain by a hard substratum . a simple model of the city is considered herein in which the buildings are absent ( i.e. , the ground is flat ) , the soft underground is constituted by a homogeneous , soft layer overlying , and in welded contact with , a homogeneous , hard substratum . this configuration is solicited by a sh plane body wave and the object is to determine the time history of response on the ground , preferably in a numerically - efficient , physically - understandable manner .

we treat the case of a flat stress - free surface ( i.e. , the ground in seismological applications ) separating air from a homogeneous , isotropic , solid substratum overlain by a homogeneous , isotropic , solid layer ( in contact with the ground ) solicited by a sh plane body wave incident in the substratum .

we treat the case of a flat stress - free surface ( i.e. , the ground in seismological applications ) separating air from a homogeneous , isotropic , solid substratum overlain by a homogeneous , isotropic , solid layer ( in contact with the ground ) solicited by a sh plane body wave incident in the substratum . the analysis is first carried out in the frequency domain and subsequently in the time domain . the frequency domain response is _ normal _ in that no resonances are excited ( a resonance is here understood to be a situation in which the response is infinite in the absence of dissipation ) . the translation of this in the time domain is that the scattered pulse is of relatively - short duration . the duration of the pulse is shown to be largely governed by radiation damping which shows up in the imaginary parts of the complex eigenfrequencies of the configuration . three methods are elaborated for the computation of the time history and give rise to the same numerical solutions for a large variety of configurations of interest in the geophysical setting under the hypothesis of non - dissipative , dispersionless media . the method appealing to the complex eigenfrequency representation is shown to be the simplest and most physically - explicit way of obtaining the time history ( under the same hypothesis ) . moreover , it is particularly suited for the case in which modes can be excited as occurs when the incident wave is not plane or the boundary condition is not of the stress - free variety for all transverse coordinates on the ground plane .

physics0605117

c

the main result of this contribution is that the three methods give rise to the same solutions for a large variety of scattering configurations . the complex frequency pole - residue convolution method turns out to be the most interesting method since : i ) it is numerically - efficient , ii ) it is explicit as concerns the understanding and quantification of the duration of the time domain response , iii ) it can be employed even in the case in which genuine resonances ( due to mode excitation ) are produced . the part of this study concerning the complex frequency pole residue convolution method constitutes a correction of its counterpart in our previous publications ( groby and wirgin , 2005a ) and ( groby and wirgin 2005b ) . a somewhat similar approach , although applied to a fluid layer in a fluid host , is that of ( conoir , 1987 ) . work remains to be done to take into account dispersion and damping of the material in the layer . the natural follow - up of this study is to elucidate theoretically the nature of the time histories of response not only for the case ( the one treated herein ) in which the configuration is unable to excite ( e.g. , love ) modes , but also in the case in which such modes can be excited ( groby and wirgin 2005a , b ) .

three methods are elaborated for the computation of the time history and give rise to the same numerical solutions for a large variety of configurations of interest in the geophysical setting under the hypothesis of non - dissipative , dispersionless media . the method appealing to the complex eigenfrequency representation is shown to be the simplest and most physically - explicit way of obtaining the time history ( under the same hypothesis ) .

we treat the case of a flat stress - free surface ( i.e. , the ground in seismological applications ) separating air from a homogeneous , isotropic , solid substratum overlain by a homogeneous , isotropic , solid layer ( in contact with the ground ) solicited by a sh plane body wave incident in the substratum . the analysis is first carried out in the frequency domain and subsequently in the time domain . the frequency domain response is _ normal _ in that no resonances are excited ( a resonance is here understood to be a situation in which the response is infinite in the absence of dissipation ) . the translation of this in the time domain is that the scattered pulse is of relatively - short duration . the duration of the pulse is shown to be largely governed by radiation damping which shows up in the imaginary parts of the complex eigenfrequencies of the configuration . three methods are elaborated for the computation of the time history and give rise to the same numerical solutions for a large variety of configurations of interest in the geophysical setting under the hypothesis of non - dissipative , dispersionless media . the method appealing to the complex eigenfrequency representation is shown to be the simplest and most physically - explicit way of obtaining the time history ( under the same hypothesis ) . moreover , it is particularly suited for the case in which modes can be excited as occurs when the incident wave is not plane or the boundary condition is not of the stress - free variety for all transverse coordinates on the ground plane .

0707.3998

i

star formation is one of the most important astrophysical processes , but because it takes place on small physical scales and behind heavy optical obscuration , observations of the early phases of star formation have generally been limited to the milky way . only in recent years have these studies begun to be extended to extragalactic objects . now , the high sensitivity , high angular resolution , and large field of view provided by the _ spitzer space telescope _ at mid - infrared wavelengths open a new window on recently formed stars in nearby galaxies like the magellanic clouds . studying these young stellar objects ( ysos ) provides an excellent complement to observations of star formation in the milky way because the physical conditions they are subject to are quite different from those in the galaxy . in particular , the metallicity of the magellanic clouds is well below solar , and their dust content is correspondingly lower as well @xcite . most star formation today is taking place in galaxies smaller than the milky way ( for which the magellanic clouds are good prototypes ) , and star formation at high redshift occurred in low - metallicity environments , so investigating the effects of these differences on how star formation works will be an important step towards understanding how many of the stars in the universe were formed . the small magellanic cloud ( smc ) , and its more massive companion the large magellanic cloud ( lmc ) , are the two nearest star - forming dwarf galaxies . located at a distance of 61.1 kpc @xcite , and with a luminosity of @xmath5 l@xmath1 @xcite and a metallicity of @xmath6 solar @xcite , the smc is perhaps the best local analog of primitive galaxies . the smc is actively forming stars at a rate of @xmath7 m@xmath1 yr@xmath3 @xcite , and is populated by well studied regions and young star clusters . the most active star - forming region is ngc 346 ( alternately known as n66 ; * ? ? ? * ) , located toward the northern end of the smc bar . the ob association powering n66 contains 33 spectroscopically confirmed o stars , and a similar number of bright blue stars without spectra that are likely to be o stars as well @xcite in this paper , we employ _ spitzer _ observations from the _ spitzer _ survey of the small magellanic cloud ( s@xmath8mc ; @xcite , hereafter b07 ) to locate objects in n66 with mid - infrared excesses that we identify as candidate ysos . in the following section , we very briefly describe the observations and our photometry . in [ results ] , we classify the detected sources by fitting their spectral energy distributions and study their locations in color - color space . in [ discussion ] we discuss some of the implications of our results and compare to recent optical studies of n66 . we summarize our findings in [ conclusions ] .

we use _ spitzer space telescope _ observations from the _ spitzer _ survey of the small magellanic cloud ( smc ) to study the young stellar content of n66 , the largest and brightest region in the smc . the observable properties of the ysos appear consistent with those in the milky way .

we use _ spitzer space telescope _ observations from the _ spitzer _ survey of the small magellanic cloud ( smc ) to study the young stellar content of n66 , the largest and brightest region in the smc . in addition to large numbers of normal stars , we detect a significant population of bright , red infrared sources that we identify as likely to be young stellar objects ( ysos ) . we use spectral energy distribution ( sed ) fits to classify objects as ordinary ( main sequence or red giant ) stars , asymptotic giant branch stars , background galaxies , and ysos . this represents the first large - scale attempt at blind source classification based on _ spitzer _ seds in another galaxy . we firmly identify at least 61 ysos , with another 50 probable ysos ; only one embedded protostar in the smc was reported in the literature prior to the smc . we present color selection criteria that can be used to identify a relatively clean sample of ysos with irac photometry . our fitted seds indicate that the infrared - bright ysos in n66 have stellar masses ranging from 2 m to 17 m , and that approximately half of the objects are stage ii protostars , with the remaining ysos roughly evenly divided between stage i and stage iii sources . we find evidence for primordial mass segregation in the region , with the most massive ysos being preferentially closer to the center than lower - mass objects . despite the low metallicity and dust content of the smc , the observable properties of the ysos appear consistent with those in the milky way . although the ysos are heavily concentrated within the optically bright central region of n66 , there is ongoing star formation throughout the complex and we place a lower limit on the star formation rate of m yr over the last myr .

0707.3998

r

for the purposes of this study , we limited our point - source fitting to a @xmath20 region covering the full extent of the hot dust emission from n66 seen in our 24 @xmath13 m image ( from 005751 to 010037 and from @xmath21721648 to @xmath21720419 [ all coordinates j2000.0 ] ) . in this box , we detect 6544 sources at 3.6 @xmath13 m , 5836 at 4.5 @xmath13 m , 1784 at 5.8 @xmath13 m , 1718 at 8.0 @xmath13 m , 101 at 24 @xmath13 m , and 15 at 70 @xmath13 m , for a total of 8011 unique objects . a significant number of these objects have _ spitzer _ colors that are redder than normal stars should be at these wavelengths , suggesting that they may be ysos or background galaxies ( b07 ) . there are two possible approaches to determining the nature of individual _ spitzer _ sources . one could simply use the observed colors and magnitudes to classify the sources into various categories . stars of nearly all kinds ( except those with dusty atmospheres ) have colors near zero for any combination of _ spitzer _ bands because the irac and mips bands are on the rayleigh - jeans tail of their spectral energy distributions ( seds ) . ysos , because of the emission from warm ( @xmath22 k ) dust around the central protostar , have red colors throughout the mid - ir . irac colors for theoretical yso models are given by , e.g. , @xcite and @xcite . alternatively , one can compare the full seds to a variety of source models and find the best match for each object . the advantage of the color selection strategy is its simplicity , but it also has some drawbacks . it fails to use all of the available information about each object ( since we also have mips , optical , and near - ir fluxes for many of the sources ) , and it is very difficult to learn about the detailed properties of individual sources from only their colors . color selection also does not offer a way to test the assumption that sed models designed for milky way sources offer a good description of ysos that form in the metal- and dust - poor environment of the smc ; this is a disadvantage for studies seeking to determine how star formation proceeds at low metallicity , but could also be an advantage in that it enables the selection of ysos without regard to the properties of their environment . because the sed classification should be more accurate , we begin with that technique and then compare the results to color selection in [ colors ] . the sed fitting tool used for this study employs a linear regression method to find all the seds from a large grid of models that fit the data within a specified @xmath23 @xcite . the grid of models consists of 7853 stellar atmospheres @xcite encompassing all available metallicities and effective temperatures , a limited number of _ infrared space observatory _ ( iso ) spectra of galaxies @xcite and agb stars @xcite , and the 20,000 yso models from @xcite computed using the radiation transfer codes from @xcite . each yso model outputs seds for 10 viewing angles , so the yso grid effectively contains @xmath24 seds . the foreground extinction , @xmath25 , is fit simultaneously using an extinction law derived from glimpse observations ( indebetouw et al . 2006 ) . at ultraviolet and visible wavelengths , this extinction law is not appropriate for the smc , but in the near - ir and mid - ir the differences in extinction between the smc and galaxy are small @xcite . the fitter is run first using only the stellar atmosphere grid . it is then run three more times using the yso grid , iso galaxy spectra , and iso agb spectra on the sources that are not well - fit by stellar atmospheres . based on all the successful fit results , defined by their @xmath23 values ( see below ) , we can classify sources and calculate best estimates and uncertainties for each model parameter . we fit an sed for every object for which we had at least four flux measurements . to better constrain the fits at long wavelengths where most of the sources were not detected , we added 24 @xmath13 m upper limits of 1 mjy ( 5 times the limiting sensitivity of the 24 @xmath13 m data ) for each source that was not detected at 24 @xmath13 m . because there are many more model parameters than data points that go into each observed sed , the traditional reduced @xmath23 statistic can not be used to evaluate the success of the fits . as a proxy , we calculate the @xmath23 per data point ( where each sed has between 4 and 11 data points ) . given this definition , it is difficult to establish _ a priori _ the division between good and poor fits . we inspected the fits by eye and determined that fits with a @xmath23 per data point of less than 2.2 could reasonably be interpreted as good fits , while higher @xmath23 values indicated fits that were not a good match to the templates . we will therefore use this cutoff point to separate good and poor fits throughout the paper . note that this definition depends on the flux uncertainties imposed in [ observations ] . out of the 1645 sources with four or more fluxes , 1322 ( 80.4% ) are succesfully fit with stellar seds ( see above for what we mean by a `` successful fit '' ) . of the remaining 323 sources , 61 can only be fit successfully by yso models , 27 are background galaxies , 6 are agb stars , 81 can be fit by multiple classes of models , and 148 can not be fit by any of the available templates . we display one example yso sed in figure [ fits ] . for the sources that can be well - described by multiple types of templates , we used the @xmath23 value of the best fit for each template type to attempt a more accurate classification . we set a cutoff of 1.5 times the best @xmath23 value , and if , for example , the best yso fit had @xmath26 and the best galaxy fit had @xmath27 then we classified the object as a probable yso . this choice of a @xmath23 cutoff indeed results in the majority of the uncertain objects being classified as ysos and leaves very few objects with unknown types , which appears to be a reasonable outcome . while the exact value chosen for the cutoff is arbitrary , it is clear from the spatial distribution of the various object types that many of the sources classified as possible or definite galaxies must actually be ysos . the results of this paper do not depend strongly on the exact value of the cutoff . using this method , we were able to obtain probable classifications for 68 of the 81 objects of uncertain type , including 50 ysos , 13 galaxies , and 5 agb stars . thus , the final object counts are 111 ysos , 40 galaxies , and 11 agb stars . the remaining 13 objects had @xmath23 values for multiple object types that were too close to distinguish reliably . we list selected parameters from the fits for all 111 ysos in table [ ysotable ] , and we summarize the source classification results in table [ sourcetypes ] . we inspected each of the 148 poorly fit sources and the various attempted fits to their seds to determine why the fitting failed for them . we found that slightly more than 1/3 of the objects had seds at short wavelengths that appeared stellar in origin , but the long wavelength data ( often at 24 @xmath13 m ) were significantly in excess of the extrapolated photospheric emission . the fits to these sources appear to have failed for a number of reasons including slight mismatches between the optical and ir photometry ( either from variability or measurement error ) , confusion , misidentification of the long wavelength counterparts , and contamination of the photometry by underlying dust emission ( revealed by pah features ) . however , some of these sources may also have true infrared excesses indicating the presence of circumstellar material . another @xmath28 of the bad fits were generally faint sources that are spatially coincident with dust filaments in the _ spitzer _ images , again causing significant contamination of their seds by pah features in the irac bands . some of these objects are probably point sources ( either stars or ysos ) , while others may simply be unresolved dust knots . the final 1/3 of the bad fits was a mixed population whose seds could not be straightforwardly interpreted . photometric errors in one or more bands and blending were likely responsible for the failure to fit these objects , but they represented only 2.6% of the total sample . using the results of the sed fitting , we can now investigate the colors of the objects classified as ysos and as stars . we display four color - color plots in figure [ ysocolors ] to illustrate the possibilities for color selection . stars , as expected , lie in the cloud of points centered near ( 0 , 0 ) , while sources that are red in one or both colors may be ysos . note that the frequently used [ 3.6]@xmath29[4.5 ] vs. [ 5.8]@xmath30[8.0 ] color - color plot ( fig . [ ysocolors]a ) does not cleanly separate ysos from other types of sources , particularly stars with modest ir excesses and sources with pah contamination . better separation can be achieved using different combinations of colors , as shown in figure [ ysocolors]b . these plots take advantage of longer color baselines and the abrupt change in yso spectra between the 4.5 @xmath13 m and 5.8 @xmath13 m bands to distinguish ysos from stars , galaxies , and pahs . these results demonstrate that while color selection can be a useful technique for identifying ysos , it does not appear possible to obtain a yso sample that is both complete and clean with a simple set of color diagnostics . sed fitting is a more comprehensive way to determine the nature of sources if measurements in enough bands are available . nevertheless , because in many cases it is desirable to classify objects with easily applicable techniques , we use our data to provide guidance for color selection of ysos . assuming that the purple points in figure [ ysocolors ] are indeed ysos , one can select ysos with the following set of criteria : @xmath31 - [ 4.5 ] & > & 0.6\times([4.5 ] - [ 8.0 ] ) - 1.0 \\ \nonumber [ 4.5 ] - [ 8.0 ] & < & 2.8 \\ \nonumber [ 3.6 ] - [ 4.5 ] & < & 0.6\times([4.5 ] - [ 8.0 ] ) + 0.3 \\ { [ 3.6]}- [ 4.5 ] & > & -([4.5 ] - [ 8.0 ] ) + 0.85 . \label{colorselection1}\end{aligned}\ ] ] these criteria were defined to maximize completeness ; one could alternatively choose to minimize contamination at the cost of increased incompleteness , but given the distribution of sources in figure [ ysocolors]_b _ , the differences would be small . this selection is only 7% incomplete for ysos in our sample that have measured fluxes at 3.6 @xmath13 m , 4.5 @xmath13 m , and 8.0 @xmath13 m , and has a contamination of less than 27% ( the majority of the objects in this color box with uncertain classifications may still be ysos ) . approximately equivalent results can be obtained by substituting the [ 3.6]@xmath21[8.0 ] color for the [ 4.5]@xmath21[8.0 ] color , but it is slightly more difficult to avoid picking up stars on the blue end of the [ 3.6]@xmath21[8.0 ] color axis . b07 identified a set of very bright ysos across the entire smc based on their 8.0 @xmath13 m magnitudes and [ 5.8]@xmath21[8.0 ] colors . if we select the same region of the color - magnitude diagram in n66 , we find 15 sources , only 3 of which are conclusively classified as ysos . almost all of the remaining objects have seds that are not well - fit by any of the models we apply , primarily because of pah contamination . the pah contamination across the rest of the galaxy should be much lower than in n66 , so these numbers clearly represent a lower limit to the fraction of actual ysos in the b07 sample . ysos are traditionally divided into classes based on their observed spectral indices as originally defined by @xcite . because spectral indices can vary with inclination angle as well as with evolutionary state , @xcite described their yso models in terms of `` stages '' , which are analogous to the usual classes , but are based on the physical quantities that define the evolutionary stage of the models . when comparing these models to data ( rather than considering only observations ) , it therefore makes sense to use the stage system . the definitions of each stage rely on the ratio of the disk mass and envelope accretion rate to the central stellar mass . stage i sources have @xmath32 , stage ii sources have @xmath33 and @xmath34 , and stage iii sources have @xmath33 and @xmath35 , where @xmath36 is the envelope accretion rate . note that we assumed a dust - to - gas ratio of @xmath37 ( 1/5 of the milky way value ) for n66 @xcite . using this classification scheme , we have grouped the 111 definite and probable ysos in n66 into stages . the seds of most objects can be fit by more than 1 yso model , so we first selected the models that produce a @xmath23 that is within 1 of the best @xmath23 for each object . we then computed a stage for each of the acceptable models and calculated a weighted average stage , using the @xmath23 values as weights . the averages were rounded to the nearest integer to produce a classification . we find that of the 111 ysos , 33 are stage i , 50 are stage ii , and 28 are stage iii . we also calculated yso masses with the same weighted averaging scheme , and defined the uncertainty on the mass to be the weighted standard deviation of the masses of the acceptable models . we found ysos with a range of masses from @xmath38 m@xmath1 , including 19 objects that appear to be proto - ob stars ( @xmath39 m@xmath1 ) . we list the names , positions , luminosities , luminosity uncertainties ( @xmath40l , the weighted standard deviation of the luminosities of the acceptable models ) , masses , mass uncertainties ( @xmath40 m , as defined in the first sentence of this paragraph ) , stages , and stage uncertainties ( @xmath40stage , the weighted standard deviation of the stages of the acceptable models ) of all of the ysos in table [ ysotable ] . in figure [ spatialdist ] we plot the spatial distribution of the ysos in n66 . the protostars are obviously highly concentrated towards the peaks of the 8.0 @xmath13 m emission . however , there is also star formation taking place outside of the dense dust cloud that marks the optical region , particularly to the south and southeast . @xcite hypothesized that star formation in n66 has proceeded from the southwest to the center of the present - day region , but we do find a few probable ysos southwest of n66 , indicating that at least modest star formation has taken place there within the past few million years . the distribution of ysos throughout the region as a function of mass and stage is not uniform . we find that the most - embedded objects ( stage i ) are slightly more concentrated towards the center of the region than the more advanced ( and presumably older ) ysos . we also see evidence for mass segregation , with the most massive objects exhibiting a strong preference for locations close to the center ( see figure [ massdist ] ) . all but 2 of the ysos with @xmath41 m@xmath1 lie on top of bright dust filaments in the main region , and many of them are coincident with molecular peaks and optical star clusters @xcite . if these massive ysos are indeed single objects rather than multiple unresolved sources , then the mass segregation must be primordial in origin , as the ysos have not had time to move very far from their birthplaces .

we present color selection criteria that can be used to identify a relatively clean sample of ysos with irac photometry . although the ysos are heavily concentrated within the optically bright central region of n66 , there is ongoing star formation throughout the complex and we place a lower limit on the star formation rate of m yr over the last myr .

we use _ spitzer space telescope _ observations from the _ spitzer _ survey of the small magellanic cloud ( smc ) to study the young stellar content of n66 , the largest and brightest region in the smc . in addition to large numbers of normal stars , we detect a significant population of bright , red infrared sources that we identify as likely to be young stellar objects ( ysos ) . we use spectral energy distribution ( sed ) fits to classify objects as ordinary ( main sequence or red giant ) stars , asymptotic giant branch stars , background galaxies , and ysos . this represents the first large - scale attempt at blind source classification based on _ spitzer _ seds in another galaxy . we firmly identify at least 61 ysos , with another 50 probable ysos ; only one embedded protostar in the smc was reported in the literature prior to the smc . we present color selection criteria that can be used to identify a relatively clean sample of ysos with irac photometry . our fitted seds indicate that the infrared - bright ysos in n66 have stellar masses ranging from 2 m to 17 m , and that approximately half of the objects are stage ii protostars , with the remaining ysos roughly evenly divided between stage i and stage iii sources . we find evidence for primordial mass segregation in the region , with the most massive ysos being preferentially closer to the center than lower - mass objects . despite the low metallicity and dust content of the smc , the observable properties of the ysos appear consistent with those in the milky way . although the ysos are heavily concentrated within the optically bright central region of n66 , there is ongoing star formation throughout the complex and we place a lower limit on the star formation rate of m yr over the last myr .

0707.3998

c

the census of previously known embedded protostars in the smc consists of a single object discovered by @xcite . @xcite and @xcite used variability data and imaging to uncover a small sample of herbig ae / be stars in the smc , and @xcite discovered several hundred low - mass pre - main - sequence stars in n66 via isochrone fits to _ hst _ photometry . these objects , however , are generally much more evolved than the ysos identified by this study , many of which are still embedded in their natal dust clouds and are therefore faint or invisible at optical wavelengths . the distribution of ysos is concentrated towards clumps of molecular gas traced by the co(2 - 1 ) emission line @xcite and the peaks of the dust emission at 7 @xmath13 m @xcite . @xcite showed that dense h@xmath42 knots are associated with these molecular clumps and suggested that massive star formation could be taking place there . the large number of embedded ysos found in this study confirms that prediction . moreover , ir spectroscopy of the 3 brightest embedded sources detected with ground - based near - ir imaging confirm that these sources are ysos ( rubio and barb ' a , in preparation ) . the sample of stage i , ii , and iii ysos that we have identified in n66 presents the first opportunity for studying in detail a large sample of embedded ysos in another galaxy ( see also * ? ? ? * ; * ? ? ? * ) , and more importantly , one whose ism properties differ substantially from those of the milky way . the fact that we are able to identify over 100 ysos in n66 , as well as obtain successful sed fits for almost all of the other sources , suggests ( perhaps surprisingly ) that protostars in the smc resemble the yso models constructed by @xcite for milky way objects , even though the metallicity and dust - to - gas ratio are a factor of @xmath43 lower in the smc . the only potential difference between milky way and smc ysos that is evident in our results is that some of the sources are best fit by relatively cool photospheres , but the yso models with such photospheres do not have high enough luminosities to match their observed brightnesses at the distance of the smc . if these objects are actually ysos , this problem could be an indication that accretion is continuing even after these stars have reached the main sequence , which leads to expanded photospheres and lower temperatures compared to normal pre - main sequence tracks and zero - age main - sequence photospheres @xcite . an alternative possibility is that these sources actually consist of multiple cool protostars , which would explain their unusually high luminosities . the luminosities of these objects are @xmath44 orders of magnitude higher than the expected luminosities of individual ysos of the same temperature , however , which makes the multiplicity explanation appear unlikely . finally , evolved stars in the post - agb phase also have expanded cool photospheres , so it is possible that some of the `` cool luminous '' sources could be post - agb stars . additional modeling and observations of some representative objects may be necessary to resolve this issue . the 111 ysos in our sample have a combined stellar mass of 692 m@xmath1 . the observed mass function ( see figure [ imf ] ) turns over at @xmath45 m@xmath1 , indicating that incompleteness becomes serious at this point . artificial star tests in our photometry show that we are 90% complete even in the confused central region of n66 down to flux levels of 300 @xmath13jy , 200@xmath13jy , 200@xmath13jy , and 300 @xmath13jy from 3.6 - 8.0 @xmath13 m . applying these limits to the full library of yso models confirms that our incompleteness is severe below 4 m@xmath1 . if we assume that star formation in n66 follows a @xcite initial mass function ( imf ) down to 0.1 m@xmath1 , we calculate that the total mass in protostars for the entire region is @xmath46 m@xmath1 . in reality , this is a lower limit to the mass because even at 4 m@xmath1 the data are somewhat incomplete . if these ysos have all formed within the last @xmath47 myr , then the average star formation rate over that time is @xmath2 m@xmath1 yr@xmath3 . thus , we find that n66 comprises at least @xmath48% of the total current star formation in the smc . @xcite , @xcite , and @xcite used _ hst _ imaging to study the young stellar population of n66 in the optical . @xcite identified 16 subclusters of pre - main sequence stars ( their figure 8 ; several of these clusters were also pointed out by @xcite ) in the region , and we find that all but two of these also have ysos associated with them ( see below ) . @xcite estimated ages of @xmath9 myr for the first 15 of the subclusters , but our detection of ysos demonstrates that star formation has continued until the present day in these areas . subcluster 1 ( sc 1 ) is nearly coincident with an extremely bright _ spitzer _ source ( the 4th - most luminous object in the field at 8.0 @xmath13 m ) , ssts3mc 14.7725 - 72.1766 , which we are able to fit with both yso and agb seds with similar @xmath23 values . given the location of this object at the center of the ngc 346 cluster , surrounded by numerous very young massive stars , it is most likely a yso ( nevertheless , since it does not formally meet our selection criteria it is not included in our analysis ) . if so , the fitted seds suggest that this is a stage i object with a luminosity of @xmath49 l@xmath1 and a mass of 14.7 m@xmath1 . however , it is important to remember that because of the very high source density here we may actually be seeing multiple unresolved ysos . in that case , we would be likely to overestimate the mass of the most massive yso and underestimate the total mass of ysos contained in this source . this type of source confusion should not have a strong effect on the sed fitting results , because the observed sed will be dominated by that of the most massive embedded yso . a mid - ir spectrum of this object is displayed in ( * ? ? ? * peak c ) . another bright _ spitzer _ source , ssts3mc 14.7748 - 72.1749 , is located within 3 of sc 1 and has a steeply - rising mid- and far - ir sed that we are unable to fit successfully . if we remove the 24 @xmath13 m upper limit and either the 70 @xmath13 m or @xmath50- and @xmath51-band detections ( perhaps justified because of confusion in this very densely populated region ) , then this source has the sed of a very massive ( @xmath52 m@xmath1 ) , early - stage yso . we do not detect any non - stellar sources in sc 3 ( immediately south of sc 1 ) , but confusion as a result of the very bright source just north of the cluster may play a role in this non - detection . sc 2 is located very close to 2 _ spitzer _ sources . ssts3mc 14.7574 has a stellar sed with a strong 24 @xmath13 m detection ( indicating either confusion or circumstellar dust ) and ssts3mc 14.7580 - 72.1763 is formally a poor fit as a result of a low @xmath51-band flux and slight pah contamination , but appears to have an sed consistent with being an early stage yso . scs 4 - 6 each have bright infrared counterparts . sc 4 is coincident with ssts3mc 14.7605 - 72.1687 , a massive embedded yso . sc 5 is ssts3mc 14.7514 - 72.1681 , another object that can be fitted by both yso and agb seds . again , given its position in a very young cluster , it is most likely a massive stage i yso rather than an evolved star . and sc 6 contains 2 blended _ spitzer _ sources , ssts3mc 14.7371 - 72.1651 and ssts3mc 14.7380 - 72.1651 , both very high - mass stage i ysos . the caveat mentioned above about multiple unresolved sources applies here as well . located on the southern edge of sc 7 , very close to the prominent dust lane that arcs nearly halfway around n66 , is ssts3mc 14.7733 - 72.1835 , yet another very massive ( 18.6 m@xmath1 ) stage i protostar that can also be fit by agb models . slightly further to the southwest are 2 additional , lower - mass ysos , ssts3mc 14.7673 - 72.1834 and ssts3mc 14.7698 - 72.1846 . sc 8 has a faint _ spitzer _ counterpart , ssts3mc 14.7816 - 72.1802 , which shows a stellar sed in the optical and near - ir and excesses in the irac bands , but is not fit well by any of our yso models . it is possible that the optical and ir emission is coming from different sources . scs 9 - 11 are located around a quintet of ysos . the 2 brightest of these , ssts3mc 14.8112 - 72.1843 and ssts3mc 14.8130 - 72.1840 , lie within sc 10 , but it is not clear whether these are truly blended ysos as opposed to a bright , extended clump of dust . the other 3 sources classified as ysos in this region are ssts3mc 14.8041 - 72.1867 , ssts3mc 14.8068 - 72.1856 , and ssts3mc 14.8174 - 72.1866 . sc 12 contains two bright _ spitzer _ sources ( ssts3mc 14.8318 - 72.1890 and ssts3mc 14.8353 - 72.1892 ) that are not fit well by any of the available models . the rising seds of these objects towards long wavelengths strongly suggest that there are ysos present here , but blending may be a problem . this cluster is coincident with the source labeled peak i by @xcite . scs 13 - 15 lie in the northward extension of the region ( n66a ) . clusters 13 and 14 each overlap with a yso ( ssts3mc 14.8009 - 72.1663 = @xcite peak f and ssts3mc 14.8080 - 72.1578 = @xcite peak g ) , but as with sc 10 the source in sc 14 does not appear pointlike in the irac bands , and so may not be a single object . sc 15 is associated with ssts3mc 14.8205 - 72.1544 , which shows very strong pah emission but is not well - fit by yso models . finally , sc 16 contains 2 _ spitzer _ sources , but both of these are well - fit by normal stellar models , consistent with the older age of this cluster derived by @xcite . out of the 44 ysos in our sample that lie within the acs images of @xcite , @xmath53 are spatially coincident with gas or dust pillars strongly resembling those made famous by _ hst _ imaging of the `` pillars of creation '' in the eagle nebula @xcite . higher spatial resolution near - ir and mid - ir imaging of these objects may reveal exactly how the ysos are related to these features .

this represents the first large - scale attempt at blind source classification based on _ spitzer _ seds in another galaxy . we firmly identify at least 61 ysos , with another 50 probable ysos ; only one embedded protostar in the smc was reported in the literature prior to the smc . our fitted seds indicate that the infrared - bright ysos in n66 have stellar masses ranging from 2 m to 17 m , and that approximately half of the objects are stage ii protostars , with the remaining ysos roughly evenly divided between stage i and stage iii sources . we find evidence for primordial mass segregation in the region , with the most massive ysos being preferentially closer to the center than lower - mass objects . despite the low metallicity and dust content of the smc ,

we use _ spitzer space telescope _ observations from the _ spitzer _ survey of the small magellanic cloud ( smc ) to study the young stellar content of n66 , the largest and brightest region in the smc . in addition to large numbers of normal stars , we detect a significant population of bright , red infrared sources that we identify as likely to be young stellar objects ( ysos ) . we use spectral energy distribution ( sed ) fits to classify objects as ordinary ( main sequence or red giant ) stars , asymptotic giant branch stars , background galaxies , and ysos . this represents the first large - scale attempt at blind source classification based on _ spitzer _ seds in another galaxy . we firmly identify at least 61 ysos , with another 50 probable ysos ; only one embedded protostar in the smc was reported in the literature prior to the smc . we present color selection criteria that can be used to identify a relatively clean sample of ysos with irac photometry . our fitted seds indicate that the infrared - bright ysos in n66 have stellar masses ranging from 2 m to 17 m , and that approximately half of the objects are stage ii protostars , with the remaining ysos roughly evenly divided between stage i and stage iii sources . we find evidence for primordial mass segregation in the region , with the most massive ysos being preferentially closer to the center than lower - mass objects . despite the low metallicity and dust content of the smc , the observable properties of the ysos appear consistent with those in the milky way . although the ysos are heavily concentrated within the optically bright central region of n66 , there is ongoing star formation throughout the complex and we place a lower limit on the star formation rate of m yr over the last myr .

0707.3998

i

we have obtained mid - infrared imaging of the region n66 ( ngc 346 ) in the smc with the irac and mips instruments on the _ spitzer space telescope_. we detected 8011 unique sources , with photometric coverage extending from @xmath16-band to 24 @xmath13 m ( 70 @xmath13 m in a few cases ) . most of these sources have colors and seds consistent with being normal stars , but we also detect a significant population of objects that are very red in the mid - ir . sed fitting of the 1645 sources with photometric measurements in at least four bands yielded 111 objects with seds that are best fit by yso models rather than stars or background galaxies . these data represent the first significant sample of embedded ysos identified in an external galaxy . we show that these ysos can be mostly , but not completely , separated from stars on the basis of their irac colors . however , sed fitting is necessary to significantly constrain the properties of individual objects . we find that the yso models of @xcite , which were designed to represent milky way ysos with solar metallicities , fit most of the ysos in n66 well , despite its much lower metallicity and dust - to - gas ratio . these results suggest that if low metallicity causes significant changes in the star formation process , the threshold for those effects must lie below the metallicity of the smc ( 12 + log[o / h ] @xmath54 8.0 ; @xcite ) . the one possible difference between smc and milky way ysos is that the smc appears to contain a population of very luminous but cool objects that have not been seen in the milky way . these objects may have expanded photospheres and hence lower temperatures than normal because they are still accreting material from their protostellar disks when they reach the main sequence . alternatively , these sources could be multiple cool ysos that are unresolved by _ spitzer_. we calculate masses and stages ( analogous to the usual yso classes ) for each of the n66 ysos , finding a range of masses from 2.4 to 16.6 m@xmath1 ( including 19 objects with masses above 8 m@xmath1 ) . almost half ( 45% ) of the ysos are stage ii objects , 30% are stage i , and the remaining 25% are evolved stage iii sources . we examine the spatial distribution of the ysos and find that they are strongly concentrated in the center of the region where bright and dust emission is seen , but there are also small numbers of ysos in the surrounding region with much less diffuse ism emission . the most massive ysos are preferentially located closer to the center of n66 , indicating that mass segregation is taking place . we compare our yso map to the clusters of pre - main sequence stars identified in the optical by @xcite and find that all but 2 of the clusters have associated ysos . using a salpeter imf , we calculate that a total of at least 3160 m@xmath1 of ysos have been formed in the last @xmath55 yr , representing @xmath56% of the current star formation in the smc . we would like to thank the anonymous referee for a careful reading of the paper that produced valuable feedback . j.d.s . gratefully acknowledges the support of a millikan fellowship provided by the california institute of technology . this research was partially funded by nasa through an award issued by jpl / caltech ( nasa - jpl _ spitzer _ grant 1264151 awarded to cycle 1 project 3316 ) . b.w . was supported by nasa astrophysics theory program grant nng05gh35 g and the _ spitzer _ theoretical research program under subcontract 1290701 , m.r . was supported by the chilean _ center for astrophysics _ fondap no . 15010003 , and financial support from fondecyt no . 1050052 is acknowledged by r.h.b . we thank you - hua chu , robert gruendl , lynne hillenbrand , jacco van loon , adam leroy , and bob benjamin for helpful conversations , and we also thank dimitrios gouliermis for providing us with the _ hst _ photometry in advance of publication . this work is based on observations made with the _ spitzer space telescope _ , which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . this publication makes use of data products from the two micron all sky survey , which is a joint project of the university of massachusetts and the infrared processing and analysis center / california institute of technology , funded by nasa and the national science foundation . this research has also made use of nasa s astrophysics data system bibliographic services and the simbad database , operated at cds , strasbourg , france . cccccccccccc 1 & [email protected] & 00 58 00.94 & @xmath2172 05 22.3 & 1290 & 556 & 6.5 & 0.8 & iii & 0.6 & 0.26 & + 2 & [email protected] & 00 58 13.89 & @xmath2172 09 19.5 & 131 & & 5.1 & & i & & 1.76 & + 3 & [email protected] & 00 58 30.26 & @xmath2172 08 41.5 & 328 & 107 & 4.6 & 0.4 & ii & 0.4 & 0.12 & + 4 & [email protected] & 00 58 40.31 & @xmath2172 09 01.0 & 1590 & 1600 & 6.0 & 2.5 & ii & 0.5 & 0.40 & + 5 & [email protected] & 00 58 48.43 & @xmath2172 11 00.2 & 35 & 2 & 2.8 & 0.5 & i & & 4.06 & + 6 & [email protected] & 00 58 50.25 & @xmath2172 10 20.9 & 3300 & & 8.3 & & iii & & 5.53 & + 7 & [email protected] & 00 58 51.55 & @xmath2172 14 33.9 & 671 & 353 & 5.3 & 0.9 & iii & 0.5 & 0.95 & + 8 & [email protected] & 00 58 53.35 & @xmath2172 10 04.3 & 3520 & 789 & 8.9 & 1.5 & i & & 1.39 & + 9 & [email protected] & 00 58 55.56 & @xmath2172 10 24.3 & 4430 & & 9.0 & & iii & & 1.37 & + 10 & [email protected] & 00 58 55.90 & @xmath2172 14 38.1 & 261 & & 6.4 & & i & & 6.15 & + 11 & [email protected] & 00 58 58.30 & @xmath2172 10 26.9 & 4340 & & 7.9 & & i & & 3.55 & + 12 & [email protected] & 00 58 59.63 & @xmath2172 11 03.3 & 348 & & 4.5 & & ii & & 1.68 & + 13 & [email protected] & 00 59 00.22 & @xmath2172 10 47.0 & 1940 & & 7.3 & & iii & & 2.37 & + 14 & [email protected] & 00 59 00.81 & @xmath2172 10 58.6 & 388 & 1 & 4.7 & & ii & & 5.48 & + 15 & [email protected] & 00 59 01.58 & @xmath2172 10 19.8 & 2230 & & 7.5 & & ii & & 0.68 & + 16 & [email protected] & 00 59 01.96 & @xmath2172 16 17.5 & 306 & 813 & 4.6 & 1.1 & ii & 0.7 & 0.41 & + 17 & [email protected] & 00 59 02.53 & @xmath2172 10 07.4 & 5120 & 2210 & 10.3 & 0.8 & i & 0.4 & 3.69 & + 18 & [email protected] & 00 59 03.34 & @xmath2172 12 24.3 & 197 & 46 & 3.9 & 0.2 & ii & & 0.56 & + 19 & [email protected] & 00 59 04.75 & @xmath2172 11 04.7 & 684 & 7 & 5.5 & & ii & & 0.50 & + 20 & [email protected] & 00 59 06.37 & @xmath2172 13 48.0 & 369 & 214 & 4.5 & 0.7 & ii & 0.5 & 0.00 & + 21 & [email protected] & 00 59 07.19 & @xmath2172 08 43.4 & 261 & & 6.4 & & i & & 9.19 & + 22 & [email protected] & 00 59 07.90 & @xmath2172 12 32.9 & 459 & 151 & 4.9 & 0.4 & ii & & 6.59 & + 23 & [email protected] & 00 59 09.03 & @xmath2172 10 37.0 & 4430 & & 9.0 & & iii & & 1.44 & + 24 & [email protected] & 00 59 09.13 & @xmath2172 11 11.0 & 9880 & & 11.6 & & iii & & 13.16 & + 25 & [email protected] & 00 59 09.25 & @xmath2172 10 57.8 & 30300 & & 16.6 & & i & & 6.79 & + 26 & [email protected] & 00 59 09.55 & @xmath2172 10 39.6 & 1620 & & 7.0 & & iii & & 3.99 & + 27 & [email protected] & 00 59 10.40 & @xmath2172 11 34.9 & 6540 & 12500 & 7.9 & 8.3 & i & & 12.36 & + 28 & [email protected] & 00 59 12.22 & @xmath2172 09 58.8 & 5060 & & 10.1 & & i & & 9.19 & + 29 & [email protected] & 00 59 12.98 & @xmath2172 11 12.3 & 176 & 15 & 3.8 & & ii & & 0.81 & + 30 & [email protected] & 00 59 13.33 & @xmath2172 16 19.1 & 456 & 245 & 4.8 & 0.7 & iii & 0.5 & 0.01 & + 31 & [email protected] & 00 59 13.46 & @xmath2172 11 30.5 & 5990 & 2100 & 13.1 & 3.1 & i & & 6.52 & + 32 & [email protected] & 00 59 13.62 & @xmath2172 11 08.0 & 540 & 131 & 5.2 & 0.4 & ii & & 0.18 & + 33 & [email protected] & 00 59 13.85 & @xmath2172 08 47.7 & 728 & 159 & 5.6 & 0.3 & iii & & 5.52 & + 34 & [email protected] & 00 59 13.92 & @xmath2172 09 27.9 & 8380 & 1470 & 11.0 & 0.6 & ii & 0.7 & 8.32 & + 35 & [email protected] & 00 59 14.49 & @xmath2172 09 43.7 & 387 & & 4.7 & & ii & & 7.82 & + 36 & [email protected] & 00 59 14.68 & @xmath2172 11 03.4 & 13000 & 7940 & 15.1 & 2.9 & i & & 1.17 & + 37 & [email protected] & 00 59 15.11 & @xmath2172 11 02.2 & 9560 & 2180 & 11.4 & 0.8 & ii & 0.5 & 5.23 & + 38 & [email protected] & 00 59 15.36 & @xmath2172 11 30.0 & 3300 & & 8.3 & & iii & & 0.42 & + 39 & [email protected] & 00 59 16.18 & @xmath2172 11 11.6 & 210 & 61 & 4.0 & 0.2 & ii & & 1.00 & + 40 & [email protected] & 00 59 17.07 & @xmath2172 10 55.8 & 87 & 10 & 4.5 & 0.3 & i & & 6.43 & + 41 & [email protected] & 00 59 18.82 & @xmath2172 09 24.3 & 1180 & & 6.4 & & ii & & 4.08 & + 42 & [email protected] & 00 59 19.00 & @xmath2172 09 30.5 & 656 & & 5.5 & & ii & & 4.88 & + 43 & [email protected] & 00 59 19.25 & @xmath2172 14 38.8 & 453 & 174 & 4.9 & 0.8 & iii & 0.5 & 0.00 & + 44 & [email protected] & 00 59 20.95 & @xmath2172 08 41.5 & 922 & 345 & 6.2 & 0.4 & ii & 0.3 & 0.15 & + 45 & [email protected] & 00 59 23.16 & @xmath2172 08 48.4 & 4430 & 1780 & 8.8 & 1.7 & iii & 0.3 & 2.67 & + 46 & [email protected] & 00 59 23.69 & @xmath2172 08 47.1 & 204 & 60 & 3.9 & 0.3 & ii & & 2.78 & + 47 & [email protected] & 00 59 25.22 & @xmath2172 08 04.1 & 1340 & & 6.6 & & iii & & 0.40 & + 48 & [email protected] & 00 59 28.83 & @xmath2172 12 31.9 & 131 & & 5.1 & & i & & 6.78 & + 49 & [email protected] & 00 59 31.91 & @xmath2172 13 35.3 & 206 & & 5.9 & & i & & 5.28 & + 50 & [email protected] & 00 59 33.61 & @xmath2172 16 43.9 & 678 & 215 & 5.5 & 0.5 & iii & & 2.11 & + 51 & [email protected] & 00 59 36.94 & @xmath2172 09 03.2 & 77 & 11 & 4.6 & 0.3 & i & 0.4 & 3.33 & + 52 & [email protected] & 00 59 36.98 & @xmath2172 14 23.2 & 1000 & 495 & 5.8 & 1.5 & ii & 0.2 & 0.22 & + 53 & [email protected] & 00 59 44.64 & @xmath2172 12 44.0 & 2150 & & 7.5 & & iii & & 7.94 & + 54 & [email protected] & 00 59 50.48 & @xmath2172 16 35.2 & 644 & 249 & 5.4 & 0.6 & iii & 0.5 & 0.51 & + 55 & [email protected] & 00 59 52.12 & @xmath2172 15 40.8 & 529 & 376 & 5.0 & 0.9 & ii & 0.5 & 0.21 & + 56 & [email protected] & 00 59 56.59 & @xmath2172 14 43.5 & 683 & 177 & 5.9 & 0.8 & ii & 0.4 & 11.79 & + 57 & [email protected] & 01 00 15.27 & @xmath2172 16 28.7 & 1000 & & 6.2 & & iii & & 5.99 & + 58 & [email protected] & 01 00 15.72 & @xmath2172 04 43.6 & 360 & 87 & 4.6 & 0.4 & ii & & 0.26 & + 59 & [email protected] & 01 00 20.62 & @xmath2172 13 23.6 & 488 & 119 & 5.0 & 0.3 & ii & 0.5 & 7.91 & + 60 & [email protected] & 01 00 22.32 & @xmath2172 09 58.2 & 2910 & 944 & 8.0 & 0.7 & i & 0.5 & 0.40 & + 61 & [email protected] & 01 00 33.97 & @xmath2172 15 26.5 & 1750 & 99 & 8.8 & 0.4 & i & & 5.55 & + 62 & [email protected] & 00 57 57.49 & @xmath2172 14 59.7 & 12700 & 10900 & 11.4 & 5.3 & iii & & 1.95 & 3.49 + 63 & [email protected] & 00 58 02.00 & @xmath2172 05 00.7 & 2000 & 2130 & 7.0 & 2.3 & ii & 0.4 & 0.04 & 8.60 + 64 & [email protected] & 00 58 02.50 & @xmath2172 12 07.0 & 573 & 536 & 5.1 & 1.5 & ii & 0.8 & 0.02 & 4.23 + 65 & [email protected] & 00 58 09.85 & @xmath2172 08 41.2 & 160 & 107 & 4.0 & 0.8 & ii & 0.5 & 0.28 & 1.45 + 66 & [email protected] & 00 58 21.87 & @xmath2172 11 53.8 & 114 & & 5.0 & & i & & 5.85 & 10.50 + 67 & [email protected] & 00 58 23.01 & @xmath2172 14 50.0 & 95 & 93 & 4.3 & 0.9 & i & 0.4 & 0.60 & 7.06 + 68 & [email protected] & 00 58 38.23 & @xmath2172 14 37.0 & 210 & 300 & 3.7 & 1.1 & ii & 0.5 & 0.01 & 2.29 + 69 & [email protected] & 00 58 47.45 & @xmath2172 09 03.2 & 2290 & 188 & 7.6 & 0.2 & iii & & 5.99 & 14.70 + 70 & [email protected] & 00 58 49.60 & @xmath2172 06 58.0 & 1470 & 2230 & 6.5 & 1.2 & ii & 0.3 & 0.05 & 3.93 + 71 & [email protected] & 00 58 52.04 & @xmath2172 10 43.8 & 517 & 900 & 4.8 & 0.9 & ii & 0.2 & 2.20 & 7.18 + 72 & [email protected] & 00 58 52.16 & @xmath2172 07 45.4 & 260 & 317 & 4.3 & 0.6 & ii & 0.2 & 0.07 & 1.78 + 73 & [email protected] & 00 58 52.19 & @xmath2172 12 29.1 & 80 & 16 & 4.1 & 1.0 & i & & 4.49 & 12.55 + 74 & [email protected] & 00 58 52.24 & @xmath2172 12 22.6 & 37 & 69 & 2.9 & 0.8 & ii & 0.6 & 0.00 & 0.32 + 75 & [email protected] & 00 58 53.00 & @xmath2172 09 12.9 & 1640 & 307 & 7.0 & 0.4 & iii & & 0.06 & 1.01 + 76 & [email protected] & 00 58 54.36 & @xmath2172 11 44.6 & 59 & 105 & 2.4 & 4.6 & i & & 0.01 & 2.38 + 77 & [email protected] & 00 58 54.45 & @xmath2172 16 25.1 & 1170 & 861 & 6.4 & 2.3 & iii & 0.5 & 0.00 & 2.29 + 78 & [email protected] & 00 58 55.85 & @xmath2172 08 31.3 & 236 & & 4.0 & & i & & 3.05 & 11.44 + 79 & [email protected] & 00 58 56.89 & @xmath2172 09 54.5 & 9390 & 3200 & 13.5 & 1.8 & i & & 3.54 & 10.49 + 80 & [email protected] & 00 58 57.11 & @xmath2172 09 54.3 & 14800 & 7190 & 15.5 & 2.4 & i & & 0.18 & 1.96 + 81 & [email protected] & 00 58 57.19 & @xmath2172 08 41.5 & 231 & 370 & 4.1 & 1.0 & ii & 0.5 & 0.02 & 1.76 + 82 & [email protected] & 00 58 57.76 & @xmath2172 16 34.0 & 1210 & 874 & 6.4 & 0.6 & ii & 0.6 & 0.48 & 1.57 + 83 & [email protected] & 00 58 58.23 & @xmath2172 10 12.4 & 3290 & & 8.3 & & iii & & 0.32 & 1.64 + 84 & [email protected] & 00 59 01.01 & @xmath2172 13 30.1 & 257 & 469 & 4.2 & 0.8 & ii & 0.4 & 0.00 & 3.06 + 85 & [email protected] & 00 59 01.65 & @xmath2172 16 21.1 & 251 & & 6.5 & & i & & 2.77 & 8.52 + 86 & [email protected] & 00 59 01.86 & @xmath2172 16 17.4 & 726 & 1020 & 5.2 & 1.4 & ii & 0.6 & 0.00 & 6.20 + 87 & [email protected] & 00 59 04.14 & @xmath2172 11 00.1 & 2290 & 1870 & 7.0 & 1.9 & ii & 0.7 & 2.42 & 8.66 + 88 & [email protected] & 00 59 04.84 & @xmath2172 12 32.5 & 310 & 47 & 6.7 & 0.5 & i & & 0.96 & 3.43 + 89 & [email protected] & 00 59 07.54 & @xmath2172 08 12.8 & 176 & 151 & 3.6 & 0.7 & iii & 0.5 & 0.50 & 2.90 + 90 & [email protected] & 00 59 08.93 & @xmath2172 10 41.2 & 2360 & & 7.6 & & ii & & 0.20 & 6.33 + 91 & [email protected] & 00 59 09.17 & @xmath2172 08 38.7 & 201 & 914 & 3.9 & 1.2 & ii & 0.5 & 0.12 & 1.09 + 92 & [email protected] & 00 59 09.43 & @xmath2172 08 24.2 & 80 & 40 & 3.6 & 0.5 & ii & 0.5 & 0.10 & 7.14 + 93 & [email protected] & 00 59 09.60 & @xmath2172 05 27.9 & 821 & 226 & 5.8 & 0.4 & iii & 0.7 & 0.48 & 3.25 + 94 & [email protected] & 00 59 09.80 & @xmath2172 12 31.9 & 57 & 23 & 2.7 & 0.2 & ii & & 0.38 & 8.03 + 95 & [email protected] & 00 59 09.81 & @xmath2172 11 49.1 & 2270 & 1610 & 7.2 & 1.7 & iii & 0.5 & 0.49 & 9.59 + 96 & [email protected] & 00 59 09.87 & @xmath2172 12 18.8 & 162 & 74 & 3.7 & 0.3 & ii & 0.3 & 0.05 & 6.49 + 97 & [email protected] & 00 59 10.93 & @xmath2172 11 11.5 & 114 & 373 & 3.2 & 1.5 & ii & & 0.01 & 3.12 + 98 & [email protected] & 00 59 11.72 & @xmath2172 08 13.7 & 114 & & 5.0 & & i & & 6.32 & 12.05 + 99 & [email protected] & 00 59 12.13 & @xmath2172 11 38.1 & 171 & 54 & 4.1 & 0.7 & i & 0.5 & 1.69 & 8.27 + 100 & [email protected] & 00 59 12.25 & @xmath2172 11 36.7 & 305 & 228 & 4.2 & 0.8 & ii & & 0.07 & 7.34 + 101 & [email protected] & 00 59 22.23 & @xmath2172 14 45.8 & 275 & 170 & 4.7 & 0.6 & ii & 0.2 & 0.00 & 0.12 + 102 & [email protected] & 00 59 23.19 & @xmath2172 09 04.5 & 356 & 109 & 4.5 & 0.6 & ii & & 2.22 & 6.76 + 103 & [email protected] & 00 59 24.77 & @xmath2172 10 00.8 & 1160 & 90 & 7.0 & 0.7 & ii & 1.0 & 0.70 & 7.04 + 104 & [email protected] & 00 59 29.15 & @xmath2172 10 38.6 & 490 & 58 & 7.3 & 0.3 & i & & 2.46 & 5.53 + 105 & [email protected] & 00 59 29.49 & @xmath2172 08 21.6 & 123 & 45 & 4.4 & 0.8 & i & & 2.10 & 4.92 + 106 & [email protected] & 00 59 30.74 & @xmath2172 10 09.9 & 441 & 385 & 4.4 & 1.5 & iii & 0.5 & 0.97 & 9.78 + 107 & [email protected] & 00 59 31.60 & @xmath2172 14 18.6 & 540 & 963 & 4.6 & 1.6 & ii & 0.7 & 0.16 & 6.91 + 108 & [email protected] & 00 59 32.35 & @xmath2172 04 54.2 & 98 & 47 & 4.2 & 1.1 & i & & 1.85 & 6.99 + 109 & [email protected] & 00 59 53.70 & @xmath2172 06 36.3 & 201 & 215 & 4.3 & 0.7 & ii & 0.6 & 0.05 & 4.07 + 110 & [email protected] & 01 00 21.49 & @xmath2172 10 27.5 & 80 & 30 & 3.9 & 0.9 & i & 0.5 & 0.14 & 3.10 + 111 & [email protected] & 01 00 28.78 & @xmath2172 08 15.6 & 847 & 166 & 5.9 & 0.3 & iii & & 0.06 & 6.27 + all sources & 1645 + normal stars & 1322 + ysos & 61 + galaxies & 27 + agb stars & 6 + probable ysos & 50 + probable galaxies & 13 + probable agb stars & 5 + multiple good fits ; unable to classify & 13 + poor fits & 148 [ sourcetypes ]

in addition to large numbers of normal stars , we detect a significant population of bright , red infrared sources that we identify as likely to be young stellar objects ( ysos ) . we use spectral energy distribution ( sed ) fits to classify objects as ordinary ( main sequence or red giant ) stars , asymptotic giant branch stars , background galaxies , and ysos .

we use _ spitzer space telescope _ observations from the _ spitzer _ survey of the small magellanic cloud ( smc ) to study the young stellar content of n66 , the largest and brightest region in the smc . in addition to large numbers of normal stars , we detect a significant population of bright , red infrared sources that we identify as likely to be young stellar objects ( ysos ) . we use spectral energy distribution ( sed ) fits to classify objects as ordinary ( main sequence or red giant ) stars , asymptotic giant branch stars , background galaxies , and ysos . this represents the first large - scale attempt at blind source classification based on _ spitzer _ seds in another galaxy . we firmly identify at least 61 ysos , with another 50 probable ysos ; only one embedded protostar in the smc was reported in the literature prior to the smc . we present color selection criteria that can be used to identify a relatively clean sample of ysos with irac photometry . our fitted seds indicate that the infrared - bright ysos in n66 have stellar masses ranging from 2 m to 17 m , and that approximately half of the objects are stage ii protostars , with the remaining ysos roughly evenly divided between stage i and stage iii sources . we find evidence for primordial mass segregation in the region , with the most massive ysos being preferentially closer to the center than lower - mass objects . despite the low metallicity and dust content of the smc , the observable properties of the ysos appear consistent with those in the milky way . although the ysos are heavily concentrated within the optically bright central region of n66 , there is ongoing star formation throughout the complex and we place a lower limit on the star formation rate of m yr over the last myr .

cond-mat0209608

i

interest in the properties of magnetic nanoparticles has grown recently , partly because of advances in synthesis and measurement techniques and partly because of potential applications for high storage - density magnetic media and spin electronics . ferromagnetic nanoparticles with diameters of a few nanometers containing of order of 1000 or fewer atoms can now be reliably fabricated and studied with a variety of different methods@xcite . small monodomain magnetic particles have traditionally@xcite been described using classical _ micromagnetic _ theory , in which the total energy is expressed as a function of the magnetization orientation . shape and magnetocrystalline magnetic anisotropy leads to a dependence of energy on orientation , to barriers that separate minima that occur at _ easy _ magnetization orientations@xcite , and to hysteretic discontinuous changes in orientation as a function of the strength of an external magnetic field . when the size of a magnetic particle is only a few nanometers , the discrete nature of its quantum energy spectrum can be directly observable at low temperatures and starts to affect the magnetic properties of the particle . a milestone in the experimental study of _ individual _ ferromagnetic metal nanoparticles was achieved recently by deshmukh , guron , ralph _ et.al._@xcite , using single - electron tunneling spectroscopy . by exploiting the coulomb blockade effect , these experimentalists were able to resolve _ individual quantum states _ in the discrete many - body excitation spectrum of single ferromagnetic metal nanograins with sizes in the range from @xmath0 to @xmath1 nm . as in bulk ferromagnetic metals , the low - energy excitations of the nanoparticles that were probed in these experiments involve both particle - hole excitations of the electronic quasiparticles and quantized collective excitations of the magnetization - orientation field that appears in the classical micromagnetic theory . in an initial attempt to achieve an understanding of the novel physics evident in the external field dependence of the experimental excitation spectra , two of us recently@xcite analyzed a simple quantum model with long range exchange interactions . we were able to demonstrate explicitly that the low - energy excitations of a ferromagnetic metal nanoparticle are specified by the occupation numbers of its quasiparticle orbitals , as in a fermi liquid , _ and _ by the global orientation of the aligned spins of all single - occupied ( and therefore spin - polarized ) orbitals . this model is not , however , able to account realistically for the influence of spin - orbit interactions , which play the essential role in controlling the complex hysteretic behavior seen in these experiments@xcite . kleff , von delft , deshmukh and ralph@xcite have proposed that the single - electron tunneling spectroscopy experiments can be explained by accounting for non - equilibrium spin accumulation , and by assuming that the magnetic anisotropy energy of a small magnetic particle has surprisingly large fluctuations as a function of the number of electrons on the particle . this assumption leads to a non - trivial magnetic field dependence of tunneling resonances that resembles experimental behavior . more critically , these authors point out that if non - equilibrium spin and quasi - particle excitations both occur@xcite , the low energy spectra are characterized by many closely spaced resonances , consistent with experiment@xcite . in ref . we presented a possible approach toward achieving a unified and consistent quantum description of both collective and quasiparticle physics in magnetic nanoparticles . the attempts put forward in refs . to develop a quantum description of ferromagnetic metal nanoparticles do not address the microscopic origin of the magnetic anisotropy and the fluctuations of this quantity that are a necessary consequence of spin - orbit interactions in mesoscopic systems . this article addresses these issues and , more in general , investigates the changes in magnetic properties of small metallic nanoparticles that occur because of the finite spacing between quasiparticle levels near the fermi energy . our conclusions follow from qualitative arguments based on perturbation theory expressions for the influence of spin - orbit interactions on quasiparticle energy levels , and on numerical studies of a simplified model that we believe is sufficiently realistic to describe generic aspects of the interrelated mesoscopic physics of quasiparticle energy - levels and magnetic anisotropy energy in ferromagnetic metal nanoparticles . we are particularly interested in the variation of quasiparticle energies with fields on the scale of the coercive field , which is trivial in the absence of spin - orbit interactions , but entirely non - trivial in their presence . the model we study is based on a slater - koster tight binding hamiltonian and a mean - field treatment of exchange interactions . our aim is to understand the dependence of magnetic anisotropy and quasiparticle energy - level - spacing statistics on particle size and shape , external magnetic field , and , with single - electron - transistor systems in mind , also on electron number . we find that , because of the absence of strong correlations between angular momentum operator matrix elements and orbital energy differences , the size of spin - orbit induced energy shifts in nanoparticles and bulk perfect crystals are similar . on the basis of estimates for energy shifts and for avoided crossing gaps , we predict that a crossover from weak to strong quasiparticle spin - orbit scattering will occur over the range from approximately 200-atom to approximately 1000-atom nanoparticles . we find that for small particles the contribution of individual quasiparticle orbitals to the magnetocrystalline anisotropy energy has a wide distribution , characterized by a variance comparable to the spin - orbit - scattering lifetime broadening energy , @xmath2 , and a mean that is smaller by more than two orders of magnitude . surprisingly , we find no measurable correlation between the contributions to magnetic anisotropy from quasiparticles that are close in energy in this limit . as a result of the statistical properties of the quasiparticle magnetic anisotropies , the total magnetic anisotropy per atom fluctuates by several percents when the number of electrons in the nanoparticle changes by one , even for particles containing 260 atoms . finally , in agreement with experiment@xcite , we find that the quasiparticle excitation spectrum exhibits a complex non - monotonic behavior as a function of an external magnetic field , with abrupt jumps when the magnetization orientation of a nanoparticle changes discontinuously in response to the field . our analysis provides insight into mesoscopic fluctuations of the micromagnetic energy functional which appears in the classical theory of a magnetic nanoparticle . our paper is organized in the following way . in section [ model ] we introduce the model and describe the formalism . in section [ so_anis ] we analyze the qualitative change in quasiparticle energy - level statistics induced by spin - orbit interactions , and discuss the connection between the quasiparticle properties and the magnetic anisotropy of a ferromagnetic nanoparticle . fluctuations of the magnetic anisotropy as a function of spin - splitting field , atom number , and electron number are investigated in section [ fluct ] . magnetic hysteresis and the external field dependence of quasi - particle energy levels are discussed in section [ hysteresis ] . finally , in section [ conclusion ] we summarize our findings and present our conclusions .

we use a microscopic slater - koster tight - binding model with short - range exchange and atomic spin - orbit interactions that realistically captures generic features of ferromagnetic metal nanoparticles to address the mesoscopic physics of magnetocrystalline anisotropy and hysteresis in nanoparticle quasiparticle excitation spectra . calculations of the total energy as a function of magnetization direction demonstrate that the magnetic anisotropy per atom fluctuates by several percents when the number of electrons in the particle changes by one , even for the largest particles we consider . contributions of individual orbitals to the magnetic anisotropy are characterized by a broad distribution with a mean more than two orders of magnitude smaller than its variance and with no detectable correlations between anisotropy contribution and quasiparticle energy . we find that the discrete quasiparticle excitation spectrum of a nanoparticle displays a complex non - monotonic dependence on an external magnetic field , with abrupt jumps when the magnetization direction is reversed by the field , explaining recent spectroscopic studies of magnetic nanoparticles . our results suggests the existence of a broad cross - over from a weak spin - orbit coupling to a strong spin - orbit coupling regime , occurring over the range from approximately 200- to 1000-atom nanoparticles .

we use a microscopic slater - koster tight - binding model with short - range exchange and atomic spin - orbit interactions that realistically captures generic features of ferromagnetic metal nanoparticles to address the mesoscopic physics of magnetocrystalline anisotropy and hysteresis in nanoparticle quasiparticle excitation spectra . our analysis is based on qualitative arguments supported by self - consistent hartree - fock calculations for nanoparticles containing up to 260 atoms . calculations of the total energy as a function of magnetization direction demonstrate that the magnetic anisotropy per atom fluctuates by several percents when the number of electrons in the particle changes by one , even for the largest particles we consider . contributions of individual orbitals to the magnetic anisotropy are characterized by a broad distribution with a mean more than two orders of magnitude smaller than its variance and with no detectable correlations between anisotropy contribution and quasiparticle energy . we find that the discrete quasiparticle excitation spectrum of a nanoparticle displays a complex non - monotonic dependence on an external magnetic field , with abrupt jumps when the magnetization direction is reversed by the field , explaining recent spectroscopic studies of magnetic nanoparticles . our results suggests the existence of a broad cross - over from a weak spin - orbit coupling to a strong spin - orbit coupling regime , occurring over the range from approximately 200- to 1000-atom nanoparticles .

astro-ph0606615

c

we have developed a generalized post - processor analysis tool that couples ray - tracing in the kerr metric with global grmhd simulations to produce light curves and power spectra of accreting black holes . using a variety of emission models , we probe different regions of the accretion disk and are better able to understand the underlying causes of time variability in the observed flux . the optically thick blackbody emission / absorption model is particularly useful for understanding the behavior of stellar - mass black holes in the thermal dominant state , which is characterized by a broad peak in the photon energy spectrum and very low levels of variability . by fixing the black hole mass and accretion rate , we can convert from dimensionless code units to physical units for local fluid density . then , assuming a radiation - pressure dominated gas in the inner disk , we can derive a physical temperature , and thus emissivity and absorption coefficients . the temperature and scale height of the disk compare well with the novikov - thorne predictions for a schwarzschild black hole accreting at 50% @xmath263 with a small torque at the inner boundary . despite the very different assumptions used in the analytic model and the computer simulations , the agreement is close enough to provide reasonable confidence in our conversion factors for density and temperature , and thus the conclusion that the simulations can be used to understand the thermal dominant state . we have also developed new methods for analyzing the azimuthal structure of the accretion flow , determining the characteristic sizes and lifetimes of density perturbations . over a large range of radii , we found the perturbations have a nearly exponential distribution of lifetimes , with @xmath264 . also , the characteristic shapes of the hot spots appears to be self - similar throughout the keplerian regions of the disk , with @xmath265 and @xmath266 . from these short coherence times , it seems clear that the hot spots formed by mhd turbulence can not survive long enough to produce the qpos with quality factors of @xmath267 described in @xcite . thus , if the transient high frequency qpos with 2:3 frequency ratios identified in section [ hfqpos ] are in fact robust signatures of grmhd disk dynamics , they are most likely _ not _ produced by geodesic hot spots . this conclusion is supported by the fact that the qpos are seen only from a single azimuthal viewing angle at any one time , perhaps suggesting some form of localized stationary wave in the disk . finally , as part of our search for qpos in the simulation data , we have also developed a general formalism for quantifying the significance of such features in simulations . these analytic results have been combined with monte carlo calculations to estimate our confidence limits for the qpos at @xmath268 . as argued throughout this paper , we believe the mhd simulations most closely resemble the thermal state of black hole x - ray binaries . this high - luminosity state is dominated by a broad thermal peak in the energy spectrum around @xmath269 kev and a relatively small contribution from a steep power - law tail at higher energies @xcite . the timing properties are characterized by a featureless power spectrum with @xmath270 , with typical power - law index @xmath271 . the total power in this state is also small , with ( rms / mean)@xmath272 hz@xmath273 above 1 hz . our simulated power spectra predict @xmath274 , which is rather greater than the typical slope . on the other hand , just this sort of steep power spectrum has been seen in the hard state observations of @xcite . one robust prediction of the mhd / ray - tracing simulations is that , independent of the emission mechanism , the integrated rms power increases with disk inclination . in principle , this should be an observable prediction that could be tested with _ data from black hole binaries in the thermal state . of course , there are complications when comparing two different binary systems , including different relative contributions to the thermal peak or power - law tail , different total luminosities , and different black hole masses . however , with a sufficiently large number of observations , it should be possible to account for these variables and extract a correlation between integrated spectral power and inclination . other observational trends that might be explored with future mhd simulations include the observed linear relation between the x - ray flux and rms , which appears to extend to agn as well @xcite . the timing properties of black hole binaries also vary greatly between their different spectral states , a general feature that is still not understood . to explore either of these relationships , we will require more sophisticated emission models and , quite likely , more simulation data with a broader range of disk parameters . however , even with improved simulations and light curve models , there still exist some inherent difficulties in comparing theory with observation . _ rxte _ typically requires thousands of seconds of observation to detect high - frequency qpos at rms amplitudes of a few percent ( limited largely by photon counting statistics and `` dead time '' corrections ) . to achieve a similar sensitivity , our mhd simulations require hundreds of hours corresponds to generate only 0.2 seconds of real time for a @xmath96 black hole ( we have perfect photon statistics , but are limited by the inherent variance in the power spectra of short time series ) . this is more closely analogous to the time scales of agn observations . if we were to scale the simulation to an agn of mass @xmath275 , it would correspond to a sampling rate of every 40 minutes for roughly a month , assuming _ perfect _ photon counting statistics ! thus it should be no surprise that qpos are still so difficult to detect with high significance from supermassive black holes @xcite . 99 abramowicz , m. a. , & kluzniak , w. 2001 , a&a , 374 , l19 agol , e. , & krolik , j. h. 2000 , apj , 528 , 161 armitage , p. j. , & reynolds , c. s. 2003 , mnras , 341,1041 balbus , s. a. , & hawley , j. f. 1998 , rev . phys . , 70 , 1 belanger , g. , et al . 2006 , apj , submitted ; pre - print [ astro - ph/0604337 ] belloni , t. , et al . 2006 , mnras , in press ; pre - print [ astro - ph/0603210 ] benlloch , s. , et al . 2001 , apj , 562 , l121 de villiers , j .- , & hawley , j. f. 2003 , apj , 592 , 1060 de villiers , j .- p . , hawley , j. f. , & krolik , j. h. 2003 , apj , 599 , 1238 done , c. , madejski , g. m. , & zycki , p. t. 2000 , apj , 536 , 213 evans , c. r. , & hawley , j. f. 1988 , apj , 332 , 659 fabian , a. c. , et al . 1995 , mnras , 277 , l11 fuerst , s. v. , & wu , k. 2004 , a&a , 424 , 733 gammie , c. f. , mckinney , j. c. , & toth , g. 2003 , apj , 589 , 444 hawley , j. f. , & krolik , j. h. 2001 , apj , 548 , 348 hawley , j. f. , & krolik , j. h. 2002 , apj , 566 , 164 homan , j. , et al . 2005 , apj , 623 , 383 king , r. e. , et al . 2001 , phys . rev . a , 64 , 053810 laor , a. 1991 , apj , 376 , 90 machida , m. , & matsumoto , r. 2003 , apj , 585 , 429 markowitz , a. , & edelson , r. 2004 , apj , 617 , 939 matthews , a. p. , & scheuer , p. a. g. 1990 , mnras , 242 , 623 mcclintock , j. e. , & remillard , r. a. 2005 , in _ compact x - ray sources _ , ed . w. h. g. lewin & m. van del klis , preprint [ astro - ph/0306213 ] mckinney , j. c. , & gammie , c. f. 2004 , apj , 611 , 977 miller , j. m. , et al . 2001 , apj , 563 , 928 miller , j. m. , et al . 2002 , apj , 578 , 348 petri , j. 2005 , a&a , 439 , 443 pollak , g. d. , delameter , n. d. , nash , j. k. , & hammel , b. a. 1994 , jqsrt , 51 , 303 rebusco , p. 2004 , pasj , 56 , 553 remillard , r. a. , et al . 2002 , apj , 580 , 1030 reynolds , c. s. , & begelman , m. c. 1997 , apj , 488 , 109 rezzolla , l. , yoshida , si . , maccarone , t. j. , & zanotti , o. 2003 , msras , 344 , l37 rybicki , g. b. , & lightman , a. p. 1979 , _ radiative processes in astrophysics _ ( new york : wiley - interscience ) schnittman , j. d. 2005 , apj , 621 , 940 schnittman , j. d. , & bertschinger , e. 2004 , apj , 606 , 1098 schnittman , j. d. , & craxton , r. s. 2000 , phys . plasmas 7 , 2964 schnittman , j. d. , & rezzolla , l. 2006 , apj , 637 , l113 smith , m. d. , norman , m. l. , winkler , k .- h . a. , & smarr , l. 1985 , mnras , 214 , 67 strohmayer , t. e. 2001a , apj , 552 , l49 strohmayer , t. e. 2001b , apj , 552 , l169 tregillis , i. l. , jones , t. w. , & ryu , d. 2001 , apj , 577 , 475 uttley , p. 2004 , mnras , 347 , l61 uttley , p. , mchardy , i. m. , & vaughan , s. 2005 , mnras , 359 , 345 vaughan , s. , & uttley , p. 2005 , mnras , 362 , 235 vanden berk , d. e. , et al . 2001 , aj , 122 , 549 van der klis , m. 2005 , in _ compact x - ray sources _ , ed . w. h. g. lewin & m. van del klis , preprint [ astro - ph/0410551 ]

we apply three basic emission models to sample different properties of the time - dependent accretion disk . with one of these models , which assumes thermal blackbody emission and free - free absorption , we can predict qualitative features of the high - frequency power spectrum from stellar - mass black holes in the `` thermal dominant '' state .

we use a relativistic ray - tracing code to calculate the light curves observed from a global general relativistic magneto - hydrodynamic simulation of an accretion flow onto a schwarzschild black hole . we apply three basic emission models to sample different properties of the time - dependent accretion disk . with one of these models , which assumes thermal blackbody emission and free - free absorption , we can predict qualitative features of the high - frequency power spectrum from stellar - mass black holes in the `` thermal dominant '' state . the simulated power spectrum is characterized by a power law of index and total rms fractional variance of above 10 hz . for each emission model , we find that the variability amplitude should increase with increasing inclination angle . on the basis of a newly - developed formalism for quantifying the significance of quasi - periodic oscillations ( qpos ) in simulation data , we find that these simulations are able to identify any such features with ( rms / mean ) amplitudes near the orbital frequency at the inner - most stable orbit . initial results indicate the existence of transient qpo peaks with frequency ratios of nearly 2:3 at a confidence limit , but they are not generic features because at any given time they are seen only from certain observer directions . additionally , we present detailed analysis of the azimuthal structure of the accretion disk and the evolution of density perturbations in the inner disk . these `` hot spot '' structures appear to be roughly self - similar over a range of disk radii , with a single characteristic size and , and typical lifetimes .

1104.0933

i

under the influence of gravity , disk galaxies are expected to assemble in an `` inside - out '' fashion : stars form first from high - density gas in the central region of the galaxy where the potential is deepest , and subsequently at increasing galacto - centric radii ( e.g. * ? ? ? an immediate consequence of this formation scenario is that stars born at the same time and in the same region of a galaxy should have similar chemical compositions . however , observations in our galaxy suggest that these initial conditions are not maintained . @xcite argued that the sun was substantially more metal rich than nearby solar age stars and the local interstellar medium ( ism ) . a recent recalibration of the geneva copenhagen survey ( gcs ) using the infrared flux method finds no discrepancy with solar age stars @xcite , but even modern studies confirm that the age - metallicity relationships ( amrs ) of field and solar neighborhood stars are characterized by higher dispersions than expected @xcite . in addition , simple chemical evolution models that divide the galaxy into independently evolving concentric annuli predict many more low metallicity g - dwarfs in our region of the disk compared to those observed , a discrepancy known as `` the local g - dwarf problem '' @xcite . evidence seemingly in contradiction to standard galaxy chemical evolution theory is not limited to our own galaxy . metallicity gradients in disk galaxies are shallower than predicted by classical models ( e.g. , * ? ? ? @xcite and @xcite find unexpectedly old stellar populations on nearly circular orbits in the outskirts of m31 and m33 , respectively . the outermost regions of ngc300 and ngc7739 show flattened or positive abundance gradients with radius @xcite . these perplexing observations can not be readily explained within the confines of classic galaxy formation models . a natural explanation for the observational challenges above arises if the present day radii of many stars could be significantly different from their birth radii . one difficulty in establishing radial migration as a common phenomenon , from a dynamical standpoint , lies in finding a mechanism that can cause a substantial fraction of stars to migrate several kiloparsecs while retaining the observed approximately circular orbits . in a seminal paper , ( * * hereafter sb02 ) investigated the relationship between changes in stellar angular momentum and disk heating . they found that radial migration is a ubiquitous process in spiral galaxies ; stars naturally migrate ( change angular momentum ) as they resonantly interact with transient spiral waves . stars in corotational resonance ( cr ) with said waves are scattered without heating the disk and maintain their nearly circular orbits ( unlike lindblad resonance ( lr ) scattering ) . in the present paper , we examine radial migration in simulations of disk galaxies . our experiments include galactic disks evolved both in isolation and under the action of infalling satellites of the type expected in the currently favored cold dark matter ( cdm ) paradigm of hierarchical structure formation ( e.g. , * ? ? ? * ; * ? ? ? the latter set of experiments were presented in the studies of ( * ? ? ? * hereafter k08 ) and @xcite and were utilized to investigate the _ generic _ dynamical and morphological signatures of galactic disks subject to bombardment by cdm substructure . inspired by sb02 , several groups have recently investigated the potential role of radial mixing in the chemical and dynamical evolution of disk galaxies . @xcite presented the first chemical evolution model to incorporate radial migration . the rate at which stars migrate via the sb02 mechanism is left as a free parameter constrained by the metallicity distribution function ( mdf ) of solar neighborhood stars in the gcs @xcite . their model successfully reproduced , within systematic uncertainties , the observed age - metallicity distribution of stars in the gcs @xcite and the observed correlation between tangential velocity and abundance pattern described by @xcite . however , there is partial degeneracy between the magnitude of radial migration and other parameters in the model such as star - formation rates and gas inflow characteristics . furthermore , it is unclear whether the level of migration required to fit the data is consistent with theoretical expectations . numerical simulations have confirmed the occurrence of radial migration under a variety of conditions . @xcite studied the migration of stars in a simulation of an isolated milky way ( mw)-sized stellar disk formed from the cooling of a pressure - supported gas cloud in a @xmath3 dark matter halo . in their simulations , some older stars radially migrated to the outskirts of the disk while maintaining nearly circular orbits , forming a population akin to that observed in m31 and m33 @xcite . @xcite found that @xmath4 of all stars in the solar neighborhood were not born _ in situ _ ; this is a natural explanation for the observed dispersion in the amr and solar neighborhood metallicity distribution function ( mdf ) . more recently , @xcite investigated radial migration in a stellar disk perturbed by a low - mass ( @xmath5 ) orbiting satellite . their numerical simulations integrated test particle orbits in a static galactic potential and highlighted the fact that mergers and perturbations from satellite galaxies and subhalos can induce stellar radial mixing . although informative , test particle simulations in a static isothermal potential will not capture all the relevant physics of the process of stellar radial migration in disk galaxies ; the interactions between the gravitational perturbations and the self - gravity of the disk are essential to a detailed analysis of the phenomenon . in our paper , we expand on the analysis of @xcite by investigating radial migration using fully self - consistent numerical simulations both with and without satellite bombardment . there are now several established phenomena that can cause a star to populate a region of the disk different from its birth radii . stars on elliptical orbits maintain their guiding center and angular momentum ( modulo asymmetries in the potential ) but can be found over the range in galacto - centric radius defined by their pericenter and apocenter . changing a star s angular momentum , and hence its guiding center , requires direct scattering or a resonant interaction with transient patterns in the disk . the local encounters of stars with molecular clouds ( e.g. * ? ? ? * ) or lindblad resonance ( lr ) scattering between stars and spiral waves ( e.g. sb02 ) both change stellar guiding centers ( albeit to a relatively small degree ) and increase the random motions of stars over time , `` blurring '' the disk . stars scattered at cr with spiral waves can change their guiding centers by several kiloparsecs without increasing the amplitude of their radial motion . for any single spiral wave , sb02 predict that stars are scattered on each side of the cr , `` churning '' the contents of the disk . stars may undergo several encounters with transient spiral waves throughout their lifetimes . while sb02 investigate all resonant interactions between spiral waves and stars , we will refer to this special case of cr as the `` sb02 mechanism '' . as sb02 note , migration due to spiral waves can be described by blurring and churning regardless of how the waves arise ( satellites , e.g. , could induce spiral structure that would lead to migration described by sb02 ) . simulations have shown that other transient wave patterns internal to a galaxy , such as those resulting from bar propagation , can produce resonance overlap with existing spiral patterns and induce radial migration @xcite . orbiting satellites , external to the galaxy and discussed above , will have a complex interaction with the disk as they provide a means of direct scattering over a large area and also induce spiral modes in the disk . in this work , we aim to characterize radial migration induced by satellite bombardment and compare its effects on the stellar disk with those observed in secularly evolved galaxies . our investigation complements earlier and ongoing radial migration studies . we perform a simulation campaign , including numerical experiments of isolated disk galaxies with different scale heights and gas fractions , which in turn lead to different levels of spiral structure . for the first time , we examine the effect of satellite bombardment on radial migration utilizing simulations where galactic disks are subjected to a cosmologically motivated satellite accretion history . via a comparative approach , we determine how the magnitude and efficiency of radial migration depend on input physics , establish correlations between orbital parameters and migration , and present evidence that each of the three migration mechanisms is distinct in the examined parameter space . these characteristics lead to possible observational signatures that may constrain the relative importance of each migration mechanism in the milky way .

we use a suite of numerical simulations to investigate the mechanisms and effects of radial migration of stars in disk galaxies like the milky way ( mw ) . an isolated , collisionless stellar disk with a mw - like this level of migration would add considerable dispersion to the age - metallicity relation of solar neighborhood stars . in the isolated disk models , we investigate correlations between changes in radius and changes in orbital circularity or vertical energy , identifying signatures that might be used to test models and distinguish radial migration mechanisms in chemo - dynamical surveys of the mw disk . [ firstpage ] methods : numerical ; galaxy : kinematics and dynamics ; galaxy : disc ; galaxies : formation ; galaxies : evolution

we use a suite of numerical simulations to investigate the mechanisms and effects of radial migration of stars in disk galaxies like the milky way ( mw ) . an isolated , collisionless stellar disk with a mw - like scale - height shows only the radial `` blurring '' expected from epicyclic orbits . reducing the disk thickness or adding gas to the disk substantially increases the level of radial migration , induced by interaction with transient spiral arms and/or a central bar . we also examine collisionless disks subjected to gravitational perturbations from a cosmologically motivated satellite accretion history . in the perturbed disk that best reproduces the observed properties of the mw , 20% of stars that end up in the solar annulus started at , and 7% started at . this level of migration would add considerable dispersion to the age - metallicity relation of solar neighborhood stars . in the isolated disk models , the probability of migration traces the disk s radial mass profile , but in perturbed disks migration occurs preferentially at large radii , where the disk is more weakly bound . the orbital dynamics of migrating particles are also different in isolated and perturbed disks : satellite perturbations drive particles to lower angular momentum for a given change in radius . thus , satellite perturbations appear to be a distinct mechanism for inducing radial migration , which can operate in concert with migration induced by bars and spiral structure . we investigate correlations between changes in radius and changes in orbital circularity or vertical energy , identifying signatures that might be used to test models and distinguish radial migration mechanisms in chemo - dynamical surveys of the mw disk . [ firstpage ] methods : numerical ; galaxy : kinematics and dynamics ; galaxy : disc ; galaxies : formation ; galaxies : evolution

1104.0933

r

figure [ fig : sims ] presents surface density maps of the stellar distributions of our four collisionless galactic disks with different initial scale - heights @xmath61 . each simulation exhibits a distinctive combination of morphological features that could affect radial migration . the initial smooth disks are unstable to spiral instabilities arising from the swing amplification of particle shot noise , an effect that leads to emerging spiral structure as seen in the final isolated disks ( e.g. * ? ? ? owing to its smaller `` effective '' toomre @xmath22 stability parameter ( e.g. , * ? ? ? * ) , the @xmath62 disk ( hereafter , disks with this initial scale height will be referred to as `` thin '' ) develops prominent spiral structure and a strong bar . conversely , the @xmath63 disk ( hereafter , disks with @xmath63 will be referred to as `` thick '' ) has relatively little spiral structure and does not form a bar in isolation . we emphasize that the `` thick '' , @xmath63 disk is the one in best agreement with the old , thin disk of the mw , while @xmath62 is too thin ( see section [ sec : isolated_disks ] ) . the radial and vertical morphology of the perturbed disks is distinct from their isolated counterparts . both perturbed disks develop bars and are characterized by prominent flaring and much larger scale heights than those of the isolated disks ( k08 ; @xcite ) . there is evidence that some spiral structure evident in the isolated disks has been washed out in the simulations with substructure bombardment : within @xmath64 of the galactic center , local enhancements of the stellar surface density evident in the isolated disks are substantially more diffuse after the action of the infalling satellites . throughout this section , we will investigate how the growth and dissipation of spiral structure affect radial migration . while we have investigated similar maps in the four hydrodynamical simulations , we do not show them here as they are qualitatively similar to their collisionless counterparts . as expected , however , the magnitude of spiral structure increases as the gas fraction rises . we note that the differential effect of gas on the strength of spiral structure we find here is smaller than that reported in previous numerical investigations of isolated disk galaxies ( e.g. , * ? ? ? this is mainly due to the effect of stellar feedback , which causes the ism in our simulations to become turbulent and multi - phase ( see also * ? ? ? we first investigate where particles move throughout each simulation . figure [ fig : isodisks ] shows the distribution of @xmath60@xmath65 in the six isolated disk simulations , where @xmath66 and @xmath67 refer respectively to each particle s final and initial projected distance from the galactic center . each panel contains the @xmath60 distribution , the median @xmath68 ( denoted @xmath69 ) , and @xmath70 percentile @xmath68 ( denoted @xmath71 ) for two of the six isolated galaxies in our simulation suite . the isolated , collisionless simulations ( left panel ) clearly demonstrate that disk scale height and gas content affect the radial migration process . the thin disk s @xmath60 distribution ( black line ) is considerably broader than the thick disk s ( gray hatch ) . both @xmath69@xmath72 and @xmath71@xmath73 of particles in the thin disk are approximately double those found in the thick disk . the remaining panels of figure [ fig : isodisks ] compare the @xmath60distributions of the hydrodynamical simulations of our sample . we examine four isolated disks with two initial gas fractions : @xmath74 ( middle panel ) and @xmath75 ( right panel ) . the two histograms in each panel represent star formation efficiencies ( @xmath39 ) of @xmath76 ( solid line ) or @xmath77 ( gray , hatched histogram ) . gas has a strong impact on particle radial migration : @xmath69 increases from @xmath78 in the collisionless case ( all hydrodynamical simulations are of `` thick '' disks ) to @xmath79 when @xmath74 and @xmath80 when @xmath75 . the extent of radial migration is less dependent on star formation efficiency ; there are only minor differences amongst each pair of histograms in the middle and right panels . however , in both panels , @xmath69 and @xmath71 are highest when @xmath39@xmath81 . in these four hydrodynamical simulations , the time - averaged gas content of the disk is directly correlated with the percentage of particles that migrate and their typical displacement relative to their formation radius . decreasing the scale height of the stellar disk , increasing the fraction of the initial disk mass in gas , and lowering the star - formation efficiency all increase the midplane density of the disk . this enhanced density increases the coherence and self - gravity of spiral structure , supporting it against the dissipative nature of individual particles random motions ( as described by , e.g. , * ? ? ? the correlation of spiral structure strength with the fraction of particles that migrate away from their birth radii and the median migratory distance traversed suggests that the sb02 mechanism , intimately linked with spiral waves , plays a major role in the migration of particles in the isolated systems . the symmetric shapes ( to within @xmath82 about @xmath83 ) of the @xmath60 distributions are also consistent with migration via the sb02 mechanism . resonances between the bar and spiral structure could also influence the timescale for migration in the thin disk models @xcite , but the @xmath63 models do not develop bars , regardless of whether they include gas . figure [ fig : perdisks ] compares the @xmath60 distributions of the isolated collisionless disks to those of their perturbed counterparts . the @xmath69 and @xmath71 of each distribution are labeled and color - coded to match the corresponding histogram . in each panel , the most sharply peaked histogram ( red ) shows the expected @xmath60distribution from epicyclic motion alone , which we compute given the particle s initial phase space coordinates . we model the epicycle as a simple harmonic oscillator about the particle s guiding center radius ( @xmath84 ) with an amplitude set by its initial radial energy ( @xmath85 ) ( chapter 3 , * ? ? ? the energy associated with the circular component of the particle s orbit ( @xmath86 ) is set by @xmath84 , which is the radius of a circular orbit with angular momentum equal to the midplane component of the particle s angular momentum . the radial energy of the orbit is @xmath87 where @xmath88 is the vertical energy component and @xmath89 is the total energy . as @xmath85 sets the amplitude of the epicycle , we can solve for the position of the particle as function of time with respect to its guiding center radius . we choose two random phases of the epicycle oscillation , designating the first as @xmath67and the second as @xmath66 , and compute @xmath60 . the resulting `` baseline '' distribution is shown in red and is a result of observing the two random phases ( akin to our `` initial '' and `` final '' snapshots ) of the initially elliptical orbits . it does not take into account the potential heating of these initial orbits and the subsequent increase in amplitude of their radial motion ( blurring ) . the shape of the distribution does not change if we change our random seed or average a larger number of phases to obtain @xmath66 and @xmath67 . for the isolated thick disk ( right panel ) , the @xmath60 distribution from full dynamical evolution is only marginally broader than the expected baseline distribution , suggesting that these stars are not heated with time and continue to follow their initial orbits . however , satellite bombardment broadens the @xmath60 distribution dramatically , nearly doubling both @xmath69 and @xmath71 , necessitating guiding center modification of a substantial fraction of the orbits . for the thin disk , isolated evolution produces a much broader @xmath60distribution than the baseline distribution , demonstrating the impact of the bar and spiral structure in this more unstable disk . in this case , satellite bombardment only slightly increases the width of the @xmath60 distribution , despite the strong impact on disk structure that is visually evident in figure [ fig : sims ] . we suspect that the small net difference between these two histograms reflects a cancellation between two competing effects of satellite bombardment . accretion events heat the stellar disk and thereby suppress the development of spiral structure , thus reducing the level of sb02 migration . however , the accretion events also induce radial mixing directly via their dynamical perturbations . the @xmath60 histograms of both perturbed disks , while still approximately symmetric , are noticeably more asymmetric than those of the isolated disks , with @xmath90 ( @xmath91 ) more particles moving outwards than inwards in the thin ( thick ) cases , compared to @xmath92 for the isolated models . the @xmath60 distribution for all disk particles in the four collisionless simulations . each histogram show the fraction of particles in non - overlapping @xmath93 bins of @xmath60 . the isolated ( hatched , gray histogram ) and perturbed ( black line ) @xmath62 ( left panel ) and @xmath63 ( right panel ) disks are shown . the perturbed disks show greater dispersion in @xmath60 compared to isolated galaxies with the same geometry . the red histograms denote the expected @xmath60 distribution from epicyclic motion alone , given the initial orbital configurations of the particles in each disk . @xmath69 and @xmath71 are labeled and color - coded for the three distributions in each panel.,width=326 ] figure [ fig : mass_dr ] presents clear evidence that satellite - induced migration is distinct from that in the isolated experiments , acting in different environments from either epicyclic blurring or spiral - induced churning . in each panel , solid , dashed , and dotted lines show the fraction of particles at each birth radius @xmath94 that migrate by more than @xmath95 , @xmath96 , or @xmath97 , respectively . shaded regions show the scaled radial surface density profile of the initial disk . for both isolated disks , the migration probability decreases outwards and approximately traces the surface density profile . this behavior is similar to that assumed in the chemical evolution models of @xcite , who parametrized the probability that a star migrates as proportional to the mass surrounding it . however , for both perturbed disks the probability of migration is flat or increasing outwards beyond @xmath94@xmath98 , where the disk potential weakens , and it definitely does not trace the mass distribution . to better understand this difference between satellite- and spiral - induced radial mixing , we now investigate how changes in angular momentum and energy are correlated with change in radius . total energy and angular momentum are the classic two - dimensional integrals of motion @xcite . figure [ fig : lindblad ] shows lindblad diagrams for the two initial and four final states of the collisionless simulations , plotting particle specific angular momentum projected along the axis of symmetry ( @xmath99 ) versus specific binding energy ( @xmath100 ) . particles are grouped into 50 distinct linear bins of @xmath100 . we calculate the median @xmath99 in each bin . red , yellow , and orange regions connect the @xmath101 , @xmath102 , and @xmath103 percentile @xmath99 intervals ( centered on each median @xmath99 ) , respectively , across all energy bins . the @xmath76 most discrepant particles in @xmath99 for a given @xmath100 are plotted as individual points . using each disk s rotation curve , we plot @xmath99 and @xmath100 for circular orbits in the midplane of the disk ( green line ) . here , @xmath104 , where @xmath105 is the circular velocity at radius @xmath106 and @xmath107 , where @xmath108 is the potential in the disk midplane at radius @xmath106 . particles on a circular orbit have the maximum @xmath99 allowed given their energy . the fraction of particles that migrate more than @xmath109 ( solid line ) , @xmath110 ( long dashed line ) , and @xmath97 ( dotted line ) as a function of formation radius . results are binned such that the migration probability is calculated for particles in non - overlapping @xmath109 wide annuli . lines connect the migration fraction in each bin ( x - coordinates are bin centers , from @xmath111 to @xmath112 ) . the gray regions are the radial mass profiles of the initial disks normalized such that the total mass contained in the first annulus equals @xmath113 on this scale . the migration probability follows the mass distribution in the isolated disks but is anti - correlated with mass in the perturbed disks.,width=326 ] by construction , particles are initially ( left column of figure [ fig : lindblad ] ) on nearly circular orbits with a small radial velocity component . the red region , falling close to the circular orbit curve in both initial disks , confirms that @xmath114 is small . particles significantly displaced from the green curve in the initial states either have extreme @xmath115 or are on highly inclined orbits ( @xmath99 is projected along the @xmath116 axis ) . due to this inclination effect , the initial thick disk has a lower median @xmath99 as a function of @xmath100 than the thin disk despite having the same @xmath115 distribution . in the isolated thin disk ( upper middle panel of figure [ fig : lindblad ] ) , the range of @xmath99 grows relative to the initial values at every energy @xmath100 . the change is largest at lower energies . recall from figure [ fig : sims ] that the isolated thin disk develops a bar , which is composed of particles on radial orbits with relatively low @xmath99 , in the most bound region of the disk . thus , this relatively large change towards less circular orbits at low energies can be associated with bar formation . in contrast , the distribution of @xmath99 in the isolated thick disk is basically equivalent to its initial state . in the isolated thick disk there is no bar , relatively little spiral structure develops , and we find the least radial migration among the four simulations . ignoring the region of the lindblad diagram influenced by members of the bar , the isolated disk diagrams suggest that the extensive radial migration seen in the thin disk and associated with guiding center modification decreases the angular momentum of particles . the two perturbed disks ( right column of figure [ fig : lindblad ] ) show larger changes in their lindblad diagrams . bar formation in both simulations can explain the significantly lower median @xmath99 at lower energies . at higher energies , corresponding to less bound particles further out in the disk , both disks show a much larger dispersion in @xmath99 than their isolated counterparts . additionally , there is substructure in the lindblad diagrams ( groups of relatively low @xmath99 in a narrow range of energy ) not seen in the isolated disks . satellite bombardment in the perturbed disk simulations has a qualitatively discernible impact on the angular momentum distribution of the disk . to quantify changes in @xmath99 , we introduce the circularity ( @xmath117 ) quantity ( e.g. , * ? ? ? * ) . for a particle of energy @xmath118 and angular momentum @xmath119 in the @xmath116-direction , we define circularity as @xmath120 , the ratio of @xmath119 to the specific angular momentum the particle would have if it were on a circular orbit with energy @xmath118 ( obtained using the circular orbit curve ) . circular orbits have @xmath121 and radial orbits have @xmath122 . negative circularities correspond to retrograde orbits . we note that this method is formally different from that of @xcite . the circular orbit curve may not represent the highest @xmath99 for a given @xmath100 due to shot noise in the potential or gravitational potential asymmetries . strictly defining the maximum @xmath99 in each energy bin ( as in @xcite ) results in the same qualitative trends discussed later . our method benefits from the use of a mathematically constructed and reproducible rotation curve and systematically decreases @xmath123@xmath124 at the @xmath125 level . the inset of each panel in figure [ fig : lindblad ] shows the circularity distribution of each simulation . figure [ fig : ecc_circ ] plots the relation between circularity and eccentricity for orbits near the solar radius ( @xmath126 ) in the @xmath63 disk . circularity is related to eccentricity via the particle s radial and vertical energies as well as its orbital inclination ( since @xmath99 is a projected quantity ) . orbits in the midplane of the disk have the highest circularity for a given eccentricity . shaded regions in figure [ fig : ecc_circ ] show the @xmath102 and @xmath103 percentile circularity as a function of eccentricity at the solar radius . for reference in interpreting figure [ fig : lindblad ] and subsequent figures , it is worth noting that changes of @xmath127 in @xmath117 typically correspond to quite noticeable changes in eccentricity . the boundaries of these regions are not smooth due to binning and small number statistics for initially high eccentricity particles . possible values of circularity as a function of eccentricity in the solar annulus ( @xmath128 ) of the @xmath63 disk . the top edge of the dark region represents orbits in the midplane of the disk . the dark region encompasses the @xmath102 percentile of particle circularity in the solar annulus of each simulation ; the bottom edge of the light region corresponds to the @xmath103 percentile circularity . , width=326 ] returning to figure [ fig : lindblad ] , we see that the two isolated disks show markedly different evolution of their circularity distributions . in the isolated thin disk , the fraction of stars with @xmath117 @xmath129 ( rightmost bin ) drops from @xmath130 to @xmath131 , and the median @xmath117 drops from @xmath132 to @xmath133 . particles in the bar predominantly populate the newly formed low circularity tail . in the thick isolated disk , on the other hand , the @xmath117 distribution is nearly identical to the initial disk s , with median @xmath117 dropping only @xmath134 . this lack of evolution in the circularity distribution is fully consistent with cr scattering . as described by sb02 , individual stars may exchange angular momentum across crs ( churning ) while the overall distribution would remain unchanged . however , our results from section [ sec : radmix ] show that the delta r distribution of this model , which reflects individual particles and their orbits , is nearly identical to that expected from observing the particles initially elliptical orbits . the circularity distribution of the isolated , thick disk combined with its delta r distribution imply that , on average , individual particle guiding centers are not significantly modified . the circularity distributions of the perturbed disks are demonstrably altered from their isolated counterparts but are similar to one another . the most common circularities are now in the range @xmath135 in both disks , with a decrease for @xmath117@xmath136 . the thin disk starts with more @xmath117 @xmath137 particles because of its smaller orbital inclinations , and its circularity distribution evolves more strongly , with median @xmath117 dropping from @xmath132 to @xmath138 vs. @xmath139 to @xmath140 for the perturbed thick disk . notably , the perturbed thin and thick disks evolve to similar @xmath60(figure [ fig : perdisks ] ) and @xmath117 distributions despite starting with different scale heights . figure [ fig : lindzoom ] tracks the changes of selected individual particles in the ( @xmath100 , @xmath99 ) space of the lindblad diagram . while @xmath100 and @xmath99 are not individually conserved in the presence of a non - axisymmetric perturbation , the jacobi invariant @xmath141 is , where @xmath142 is the pattern speed of the perturbation , assumed to be static and small ( sb02 ; @xcite ) . if @xmath143 , then @xmath144 . the sb02 mechanism operates at the corotation resonance of the star / particle and the spiral wave , requiring that @xmath145 . thus , in the galaxy , particles should move parallel to the line that is tangent to the circular orbit curve at their binding energy prior to scattering . in other words , the sb02 mechanism requires that changes in energy be accompanied by changes in angular momentum that preserve the orbital shape , modulo differences in the slope of the circular orbit curve over the range @xmath146 $ ] of a given particle . the initial and final @xmath100 , @xmath99 pairs for ten randomly selected particles in each simulation with initial energy @xmath147 and angular momentum @xmath148 ( large , filled square ) . for clarity we require that plotted particles lose at least @xmath149 in energy during the simulation . the final @xmath100,@xmath99 pairs of the ten particles are plotted as open squares . lines connect the initial and final @xmath150 points of each particle . the @xmath150 curve populated by circular orbits in the initial state of each simulation is indicated by the thick black curve . the dashed line is tangent to the circular orbit curve at the initial energy of all ten particles.,width=326 ] figure [ fig : lindzoom ] zooms in on the area of the lindblad diagram designated by the two small boxes drawn on the initial state diagrams in figure [ fig : lindblad ] . we randomly select ten particles within the same initial @xmath100 and @xmath99 range ( filled , dark squares in figure [ fig : lindzoom ] ) and plot their final @xmath100 and @xmath99 . we require @xmath151 to ensure that the individual tracks are visible . in the isolated thin disk ( top left ) , most particles move parallel to the circular orbit curve tangent ( dashed line ) ; this relationship between @xmath152 and @xmath153 is consistent with corotational resonant scattering ( equation 2 , sb02 ) . when particles lose energy in the isolated thick disk ( bottom left ) , their @xmath99 typically remains closer to the circular orbit curve than in the isolated thin disk . the changes in energy and angular momentum are relatively small , indicating that the guiding centers are modified to a modest extent @xcite . note that these particles are in the top @xmath76 of @xmath154 ; most particles in this particular experiment have @xmath155 . the randomly selected particles in both perturbed disks have slopes @xmath156 that are steeper than the slope of the tangent to the circular orbit curve at corotation . this distinct coupling of @xmath153 and @xmath152 , combined with our results concerning the migration probability as a function of @xmath94 ( section [ sec : radmix ] ) , offer compelling evidence that migration in the perturbed disks is driven , at least in part , by a mechanism that does not operate in the isolated systems . figure [ fig : lindzoom ] shows that the orbital characteristics of many particles in the perturbed disks are modified in a fashion inconsistent with either epicyclic motion or a single spiral wave scattering event . median change in circularity @xmath123 as a function of @xmath60 for the entire disk ( open squares ) and for particles with @xmath157 ( filled squares ) . particles are sorted into non - overlapping @xmath158 bins of @xmath60 . error bars mark the @xmath159 and @xmath160 percentile @xmath123 in each bin . histograms at the bottom of each panel represent the relative fraction of particles in each bin for the case @xmath157 . , width=326 ] we now examine the correlation between migration and changes in orbital properties . since the metallicity of star - forming gas increases with time and decreases with radius , any such correlations also imply observable correlations between the present orbital parameters and metallicities of stars as a function of age and galacto - centric radius . figure [ fig : dr_dcirc ] shows the median change in circularity , @xmath161 , as a function of radial migration distance @xmath60 in the four collisionless simulations ( open squares ) . in the isolated thick disk ; changes in circularity are tiny ( median @xmath162 at any @xmath60 ; figure [ fig : lindzoom ] shows that particles in this simulation stay close to the circular velocity curve even if they change energy . in the other three simulations , particles with negative @xmath60 show substantial drops in circularity ( typical median @xmath163 ) , which are almost certainly associated with bar formation . the bars that form in these three simulations increase the orbital eccentricities of their members . to focus on behavior in the disk proper , the filled squares in each panel show the median @xmath123vs . @xmath60 for those particles that start and end the simulation at @xmath164 , beyond the extent of the bar . error bars mark the inter - quartile range ( @xmath165 to @xmath166 percentile ) at each @xmath60 . in the isolated thin disk , particles that migrate inwards ( @xmath60@xmath167 ) experience a modest decrease in circularity , stronger for more negative @xmath60 ( and @xmath153 ) , consistent with the tracks shown in figure [ fig : lindzoom ] . particles that migrate outwards ( @xmath60@xmath168 ) increase their energy and typically traverse areas of the lindblad diagram with relatively little change in the slope of the circular orbit curve . following the tangent to the circular orbit curve , particles will not significantly change their circularity in such a scenario ( note the relative lack of low circularity particles at @xmath169 in figure [ fig : lindblad ] ) . the perturbed disks show a decrease in median circularity at every @xmath60 , and a much wider inter - quartile range indicating a greater range of orbital inclinations and eccentricities . circularity drops more strongly for particles that have experienced strong radial migration , either inward or outward . the minimum in these curves is slightly offset to positive @xmath60 because dynamical heating slightly puffs up the disk radially , decreasing the potential at a given radius , thereby increasing its total energy , moving particles to the right on the lindblad diagram , and lowering the circularity for particles with @xmath60@xmath170 . figure [ fig : dr_dcirc ] indicates that stars with anomalous chemistry for their age and current position should have preferentially more eccentric orbits . the trend is smaller than the inter - quartile range , but similar in magnitude . like figure [ fig : dr_dcirc ] , but the change in maximum vertical displacement , @xmath171 , is plotted in place of the change in circularity . open squares show all disk particles , filled squares show those with @xmath66 , @xmath67 @xmath172 , and error bars mark @xmath159 and @xmath160 percentile at a given @xmath60.,width=326 ] @xcite and @xcite discuss the possible role of radial migration in producing thick disks like the ones observed in the milky way @xcite and other edge - on galaxies @xcite . figure [ fig : dr_dzmax ] is similar to figure [ fig : dr_dcirc ] , but instead of @xmath123 it plots the change in vertical energy , quantified by the maximum distance @xmath173that a particle can reach from the disk plane . we compute @xmath173approximately from each particle s vertical velocity component assuming that the final potential of each simulation is static , namely @xmath173@xmath174 where @xmath116 is the vertical position of the particle at the final output , @xmath175 is the velocity along the @xmath116 axis , @xmath176 is the gravitational constant , and @xmath177 is the surface density of the disk at radius @xmath106 assuming all the mass of the disk is in the midplane . the change in @xmath173 is @xmath171@xmath178 where the subscripts @xmath179 and @xmath180 refer to the initial and final simulation snapshots , respectively . the two isolated simulations show a shallow linear trend between @xmath171and @xmath60 ( open squares for all particles ; filled squares for those with @xmath67,@xmath66@xmath181 ) . when particles move outwards ( inwards ) through the disk , they experience a weaker ( stronger ) gravitational potential , thus increasing ( decreasing ) @xmath173 . however , changes in @xmath173are small , less than @xmath182 pc even when we consider the quartile range at the extremes of @xmath60 . the perturbed disks , by contrast , show larger changes in the median @xmath173 and a dramatic increase in the inter - quartile range at all @xmath60 . k08 show that the perturbed @xmath183 pc disk develops a two - component vertical structure in quantitative agreement with the observed thin / thick disk structure of the mw . the perturbed @xmath11 pc disk increases its scale height ( to @xmath184 at the solar annulus ) , but it can still be described by a single component model . this vertical heating by satellite perturbations is evident in figure [ fig : dr_dzmax ] . particles that have large positive @xmath60 have the largest increase in @xmath173 , which is plausibly a consequence of moving outwards to regions of lower disk surface density and thus weaker vertical restoring force . for @xmath60 @xmath167 , the trend of median @xmath171 with @xmath60 is approximately flat , suggestion a cancellation between the effects of increased @xmath177 at smaller @xmath106 and direct satellite - induced heating of those particles with the largest excursions . figure [ fig : dr_dzmax ] implies that stars with high metallicity for their age and present location should have preferentially larger @xmath173 , though the scatter is larger than the trend . the distribution of formation radius ( @xmath94 , in ) for all particles that are in the solar annulus ( @xmath185 ) at the end of each simulation . histograms report the fraction of solar annulus particles emigrating from non - overlapping @xmath93 annuli in @xmath94 . particles within the dashed lines remained in the solar annulus throughout the simulation . both thin disks and the perturbed thick disk show a broad range of @xmath94 at the solar annulus . the gray histogram in each panel is the @xmath94 distribution for those particles that are in the solar annulus and at large heights above the plane ( @xmath186 ) in the final simulation output.,width=326 ] the solar neighborhood is easier to study than other regions of the galaxy , since high - precision spectroscopy is easier for brighter stars and parallax and proper motion measurements are more accurate at smaller distances . some of the most detailed chemo - dynamic surveys , such as the gcs @xcite and the radial velocity experiment ( rave , * ? ? ? * ) concentrate on the solar neighborhood . in this section , we repeat some of our earlier analysis specifically for stars that reside in the solar annulus ( @xmath187 ) at the end of each simulation . this focus on the solar annulus also removes much of the impact of the bars that dominate evolution of the inner disk ( @xmath188 ) in three of our simulations , though some particles from the bar region can migrate as far as the solar radius , and resonances between the bar and spiral structure may increase migration frequency @xcite . figure [ fig : rform ] shows the radius of formation ( @xmath94 ) distribution of particles residing in the solar annulus , marked by the vertical dashed lines . only the isolated thick disk simulation predicts a final solar annulus dominated by stars born in the solar annulus , with tails extending @xmath189@xmath190 on either side . the broad @xmath60 distributions of the other three simulations show that their stars migrate to the solar annulus from a wide range of formation radii . the global @xmath60 ( figure [ fig : perdisks ] ) and solar annulus @xmath94 distributions of the isolated and perturbed thin disks are remarkably similar , despite the differences in migration mechanisms discussed in section [ sec : orbits ] . in the isolated thin disk , @xmath191 of solar annulus particles originated at @xmath94@xmath192 and @xmath193 at @xmath94@xmath194 . corresponding numbers for the perturbed thin disk are @xmath195 and @xmath91 . the @xmath94 distribution of the perturbed thick disk is slightly narrower , but it still broad with respect to the isolated thick disk . the change in circularity @xmath123 as a function of @xmath60 for particles ending the four collisionless simulations in the solar annulus ( @xmath196 ) . particles are sorted into non - overlapping @xmath158 bins of @xmath60 . we plot the median ( squares ) and the @xmath159 and @xmath160 percentile ( error bars ) @xmath123 in each bin . histograms at the bottom of each panel represent the relative fraction of particles in each bin.,width=326 ] figure [ fig : dcirc_dr_soln ] shows the correlations between @xmath123and @xmath60 for particles that end in the solar annulus . consistent with results for the full disk ( figure [ fig : dr_dcirc ] ) , the solar annulus particles in the isolated thick disk show no significant change in median @xmath117 regardless of @xmath60 . in the isolated thin disk , the range of @xmath123 is much larger , with a modest decrease in median @xmath117 . the median @xmath123 drops at large positive @xmath60 because particles move to less inclined orbits as they migrate outwards and disk heating has slightly modified the galaxy s circular velocity curve , allowing outward moving particles to potentially increase circularity . in the perturbed thick disk , there is a strong and nearly linear trend between @xmath123 and @xmath60 . particles that migrated to the solar annulus from the outer disk have experienced substantial drops in circularity , while particles migrating from the inner disk show only modest decreases . the range of @xmath123 is large at all @xmath60 . results for the perturbed thin disk are intermediate between those of the isolated thin and perturbed thick disks : quasi - linear trends at large @xmath68 but a flat plateau at intermediate @xmath68 . this behavior is plausibly a consequence of two different mechanisms contributing to migration , with spiral - induced mixing dominating at intermediate @xmath60 and satellite - induced mixing dominating at the extremes . figure [ fig : dzmax_dr_soln ] shows the solar annulus correlations of @xmath171 with @xmath60 , analogous to figure [ fig : dr_dcirc ] for the full disk . for the two isolated disks there is a clear linear trend of median @xmath171 with @xmath60 as expected from the arguments in section [ sec : orbits ] : particles that migrate outward move to a region of lower disk surface density , so if their vertical velocities are not systematically changed by the radial migration they will attain higher @xmath173 . both perturbed disks show signs of the strong vertical heating induced by satellite accretion . the median @xmath171 is @xmath197 almost independent of @xmath60 , and the range of @xmath171 is much larger than in the isolated disks . since inwardly migrating particles experience a higher vertical potential at @xmath66 than @xmath67 , they must experience more vertical heating than outwardly migrating particles to keep the @xmath171 trend flat . like figure [ fig : dcirc_dr_soln ] , but the change in vertical displacement @xmath171 is plotted in place of the change in circularity.,width=326 ] returning to figure [ fig : rform ] , gray histograms represent the @xmath94 distributions of particles that end the simulations in the solar annulus at high @xmath116 , @xmath186 . in the isolated thick disk and both perturbed disks , the @xmath94 distribution of high @xmath116 particles resembles that of all solar annulus particles . in these three models , therefore , selecting high- @xmath116 particles does not isolate a population with atypical radial migration . in the isolated thin disk , on the other hand , the @xmath94 distribution of high-@xmath116 particles is strongly skewed towards low formation radii , with a peak at @xmath94@xmath198 . here , only initially hot particles that migrate significantly outwards via churning or interactions with the bar and experience a weaker potential have enough vertical energy to overcome the local restoring force and reach high @xmath116 . only @xmath199 of the solar annulus is at high @xmath116 in the isolated thin disk ( truly the tail of the initial vertical velocity dispersion ) ; this number rises to @xmath200 and @xmath201 in the thin and thick perturbed disks , respectively . the radial migration mechanisms in the perturbed disks ensure that even the high @xmath116 population of the solar annulus comes from a broad range of @xmath94 . measurements of the age - metallicity relation for high-@xmath116 stars could be a valuable diagnostic for distinguishing models of radial migration and vertical heating .

we also examine collisionless disks subjected to gravitational perturbations from a cosmologically motivated satellite accretion history . in the perturbed disk that best reproduces the observed properties of the mw , 20% of stars that end up in the solar annulus started at , and 7% started at . the probability of migration traces the disk s radial mass profile , but in perturbed disks migration occurs preferentially at large radii , where the disk is more weakly bound . thus , satellite perturbations appear to be a distinct mechanism for inducing radial migration , which can operate in concert with migration induced by bars and spiral structure .

we use a suite of numerical simulations to investigate the mechanisms and effects of radial migration of stars in disk galaxies like the milky way ( mw ) . an isolated , collisionless stellar disk with a mw - like scale - height shows only the radial `` blurring '' expected from epicyclic orbits . reducing the disk thickness or adding gas to the disk substantially increases the level of radial migration , induced by interaction with transient spiral arms and/or a central bar . we also examine collisionless disks subjected to gravitational perturbations from a cosmologically motivated satellite accretion history . in the perturbed disk that best reproduces the observed properties of the mw , 20% of stars that end up in the solar annulus started at , and 7% started at . this level of migration would add considerable dispersion to the age - metallicity relation of solar neighborhood stars . in the isolated disk models , the probability of migration traces the disk s radial mass profile , but in perturbed disks migration occurs preferentially at large radii , where the disk is more weakly bound . the orbital dynamics of migrating particles are also different in isolated and perturbed disks : satellite perturbations drive particles to lower angular momentum for a given change in radius . thus , satellite perturbations appear to be a distinct mechanism for inducing radial migration , which can operate in concert with migration induced by bars and spiral structure . we investigate correlations between changes in radius and changes in orbital circularity or vertical energy , identifying signatures that might be used to test models and distinguish radial migration mechanisms in chemo - dynamical surveys of the mw disk . [ firstpage ] methods : numerical ; galaxy : kinematics and dynamics ; galaxy : disc ; galaxies : formation ; galaxies : evolution

1405.6965

i

conformal mapping techniques are powerful tools for analyzing two dimensional models , and have been applied to a variety of problems @xcite including aerodynamics @xcite , fluid flows @xcite , free boundary problems @xcite potential theory / electrostatics @xcite , elasticity @xcite and even computer vision @xcite . abstractly , a problem of interest in the physical plane " is transformed into a simpler problem in the pre - image plane " by an appropriately chosen complex analytic function , i.e. a conformal map @xcite relating the pre - image plane to the physical plane . in addition to generating analytic solutions , conformal mapping methods have also been applied in conjunction with numerical techniques to solve various problems of physical interest @xcite . an important issue in applying numerical ( discretized ) conformal maps to two dimensional problems where the domain has sharp regions / near singularities is the phenomenon of _ crowding_. crowding refers to the tendency for image nodes to accumulate in the physical plane near regions of high boundary curvature and leave low curvature regions extremely coarsely represented . crowding plagues all standard numerical conformal mapping methods when used to generate image domains with highly curved boundaries @xcite , and there are specialized algorithms to mitigate the effects of crowding for conformal mappings to _ known _ polygonal domains @xcite . in this paper , we consider a model electrostatic free boundary problem where the interface can develop regions of extremely high curvature . we develop a matched asymptotics method for conformal maps , that completely avoids the computational difficulties associated with crowding for _ a priori unknown domains_. an interesting interplay between interface geometry and induced stresses exists in electromechanical systems . when a conducting interface is deflected by an external field , the induced charge distribution /the local electrostatic pressure is higher in the parts of the sheet with high curvature . this in turn causes even higher deflections and sharper curvatures . this can produce extremely sharp and nearly singular shapes , as well as _ the pull - in instability _ , a runaway effect due to the positive feedback between curvature and electrostatic forces , whereby no equilibrium solutions exist beyond a critical forcing strength . for these reasons we choose an electromechanical model as a test case for applying our techniques for multi - scale conformal maps . this model also has connections to recent studies that have been primarily driven by the development of microelectromechanical systems ( mems ) , including micro - scale capacitors and actuators @xcite . the paper is organized as follows : in section [ sec : model ] we present the basic electromechanical model , a two fluid system where the interface is governed by a balance between electrostatic forces , gravity and surface tension . in section [ sec : conformal ] we present a reformulation of our model problem in terms of conformal mappings , and discuss the advantages of such a reformulation . in section [ sec : collocation ] we present the basic numerical method for constructing numerical conformal maps , and apply it to situations where the free boundary is not sharp " . the method breaks down as the interface develops sharp features , and in section [ sec : multi_scale ] we present our matching technique for conformal maps that applies in a regime where the free boundary is sharp . we present a concluding discussion in section [ sec : discussion ] . we consider a two fluid system an infinite conducting fluid initially in the half - space @xmath0 , and an infinite dielectric fluid in the half space @xmath1 . gravity acts in the @xmath2 direction , and the density of the upper fluid @xmath3 is less than the density of the lower fluid @xmath4 , so the interface is gravitationally stable . the interface is deflected by a line charge with strength @xmath5 per unit length , along the @xmath6 direction , placed in the upper fluid at a distance @xmath7 above the undeflected interface . since all the induced charge on the fluid will reside on its surface , we can equally well consider a model with an infinite conducting sheet on an elastic foundation . the conducting sheet is initially at @xmath8 and is subsequently pulled up by a line charge placed at @xmath9 . the restoring force from the foundation can be modelled as a linear function of the deflection from @xmath10 , and this agrees with the expression for the hydrostatic pressure on the interface in the two - fluid model , which is also a linear function of the deflection . the setup is depicted in fig . [ fig : generic_model ] , and by translation invariance in the @xmath11-direction , it suffices to consider a two dimensional problem in the @xmath12 plane with a one dimensional interface @xmath13 . the pressure balance along the deflected interface @xmath13 is @xmath14 where @xmath15 is the surface tension of the interface , @xmath16 is the curvature of the interface at @xmath17 , @xmath18 is the dielectric permittivity of the upper fluid , @xmath19 is the unit normal to the interface , and @xmath20 is the magnitude of the electric field on the conducting interface . when forced by a line charge , the system is closed by the equations @xmath21 the electric field @xmath22 vanishes as @xmath23 , so we get @xmath24 as @xmath23 , _ i.e _ gravity stabilizes the interface for large @xmath25 . in order to simplify the equations , we pick units to make @xmath26 . this is equivalent to the non - dimensionalization : @xmath27 for the lengths where @xmath28 is the capillary length , and @xmath29 we will henceforth work exclusively with the non - dimensionalized equations so there is no confusion in dropping the primes . this yields the nondimensional equations [ e : h_gov_pt ] @xmath30 the system has a reflection symmetry @xmath31 , and for simplicity we will restrict our attention to symmetric solutions satisfying @xmath32 . the system has two dimensionless parameters , a non - dimensional charge per unit length ( i.e forcing ) @xmath5 and a non - dimensional aspect ratio @xmath7 , the ratio of the vertical length scale ( the charge location ) and the horizontal length scale , the capillary length . the local electrostatic pressure @xmath33 at each point on the sheet depends on the entire deflection profile @xmath34 through the boundary conditions on @xmath35 . this non - local coupling prevents the direct calculation of closed form solutions of , and a numerical treatment is required instead .

we use conformal maps to study a free boundary problem for a two - fluid electromechanical system , where the interface between the fluids is determined by the combined effects of electrostatic forces , gravity and surface tension . we successfully apply our method to the electromechanical model problem and discuss how it generalizes to other situations .

we use conformal maps to study a free boundary problem for a two - fluid electromechanical system , where the interface between the fluids is determined by the combined effects of electrostatic forces , gravity and surface tension . the free boundary in our system develops sharp corners / singularities in certain parameter regimes , and this is an impediment to using existing single - scale " numerical conformal mapping methods . the difficulty is due to the phenomenon of crowding , i.e. the tendency of nodes in the preimage plane to concentrate near the sharp regions of the boundary , leaving the smooth regions of the boundary poorly resolved . a natural idea is to exploit the scale separation between the sharp regions and smooth regions to solve for each region separately , and then stitch the solutions together . however , this is not straightforward as conformal maps are rigid global " objects , and it is not obvious how one would patch two conformal maps together to obtain a new conformal map . we develop a multi - scale " ( i.e. adaptive ) conformal mapping method that allows us to carry out this program of stitching conformal maps on different scales together . we successfully apply our method to the electromechanical model problem and discuss how it generalizes to other situations .

1405.6965

c

in this paper , we develop an adaptive numerical conformal mapping method for efficiently computing multiple scale structures in 2 dimensional problems . the central result is that , with our method , we can use conformal mapping techniques to analyze structures that have widely disparate length scales . this is a significant improvement over existing numerical conformal mapping methods for free boundary problems , because they are essentially single - scale " and are plagued by the problem of crowding when applied to multiple - scale problems . as we discuss in sec . [ sec : matching ] , there are good reasons why one would not even consider matched asymptotics for conformal maps , and to our knowledge , this is the first work that successfully combines numerical conformal mappings with the method of matched asymptotics . we consider a model system is governed by three basic forces gravity , electrostatics and surface tension . the interface is governed by multiple dominant balances between different forcing / restoring mechanisms , so the free boundary does develop a multiple scale structure . the model problem is thus a good test bed for developing our method , although it is not ( to our knowledge ) directly relevant to applications . we reformulate the problem in terms of conformal mappings it converts a non - local problem into a problem that is governed by local equations , except some of the equations are in real " space an some are in reciprocal " space ( i.e in the fourier transform domain ) . since the fast fourier transform ( fft ) allows us to efficiently convert from real to reciprocal space and vice - versa , we can get efficient numerical methods for approximating solutions to our problem in terms of discretized conformal maps . this approach leads to the collocation method as described in section [ sec : collocation ] . the fft requires evenly spaced nodes ( in both real and reciprocal space ) , so we are no longer free to choose the node locations in our discretization . this is a particular concern in multiple - scale problems , since the nodes tend to concentrate in regions of high curvature , leaving the rest of the domain under - resolved . this phenomenon , called crowding , is a significant impediment to the using conformal mapping methods for multiple - scale problems . the underlying issue , namely the stiffness " of the governing equations , also shows up in other methods ( boundary integral / boundary element etc . ) , and is responsible for the challenges of numerically studying free boundary problems with widely disparate scales . in our model , we find that the numerical collocation method no longer converges to a solution once the tip curvature gets sufficiently large . the obvious " physical parameters of the model are the non - dimensional charge strength @xmath5 and location @xmath7 . however , the correct " asymptotic parameters in the sharp - tip regime are the uncorrected " tip height @xmath175 and the tip - charge separation @xmath147 . in section [ sec : concentrated ] we use physical arguments to determine the charge distribution on the interface and deduce the appropriate scaling for the the physical parameters in terms of the asymptotic parameters . we emphasize that the analysis in this section _ did not use conformal maps _ , and as such the ideas also extend to three dimensional versions the two - fluid or mems models . using scalings in terms of the asymptotic parameter @xmath147 , we reduce the governing equation to an _ outer _ equation that describes the balance between gravity and surface tension , and an _ inner _ equation that describes the balance between electrostatic forces and surface tension . asymptotic equations _ have non - trivial symmetries , and we determine _ discretized conformal map solutions _ of these equations with appropriate normalization using the collocation method as in section [ sec : collocation ] . it is important that we pick the discretization appropriately to reflect the known structure of the solutions ( a corner in the interface for the outer solution , asymptote to a pair of straight lines with a known angle for the inner solution ) . because each region has a single balance , we are not solving for a multiple - scale structure , and the collocation method works well ( sections [ sec : outer ] and [ sec : inner ] ) . in applying this method to other problems , it is crucial that we identify the correct " asymptotic parameters and scaling relations for the physical quantities , since this knowledge is required to derive the asymptotic limiting equations for the system . we then use the relevant symmetries of the asymptotic equations ( the method is powerful precisely because conformal maps have a rich group of symmetries ) to determine the appropriate matching conditions that allows us to combine distinct conformal maps on different scales to form a single composite multi - scale " conformal map . in section [ sec : matching ] , we discuss this issue in detail for our model system . the key insight is that the natural domain for discretized multi - scale conformal maps is not a single complex plane . rather , it is two ( or perhaps more for other problems ) sets of evenly spaced points on unit - circles , that are related by non - trivial mapping functions ( conformal automorphisms of the unit disk ) . each discretized unit circle resolves one scale ( i.e region with one dominant balance ) in the solution , and the requirement that the functions defined on the various circles agree ( asymptotically ) on their mutual overlaps allow us to patch them together to construct a composite multi - scale conformal map . since each unit circle is discretized evenly , we can still exploit the efficiency of the fft in solving the asymptotic equations . and the the morphing of the unit circle by conformal automorphisms allows us to ( in effect ) pick the nodes in our discretization adaptively and thus resolve structures with disparate length scales in the solution ( see figure [ fig : two - circles ] ) . this idea seems related to the cross - ratios of delaunay triangulation ( crdt ) algorithm for computing schwarz - christoffel transformations for nonconvex / multi - armed polygonal domains @xcite . the crdt algorithm is designed to combat crowding , and a key idea in is to use multiple maps into unit circles that are related by conformal automorphisms . these automorphisms are chosen to blow - up " portions of the unit circle to ensure that nodes are not crowded ( locally ) . however , unlike our method of asymptotic matching of conformal maps , the crdt algorithm exploits the invariance of the cross ratio @xcite under conformal automorphisms to ( implicitly ) construct a composite conformal map . further work is needed to explicate the precise connection between our multi - scale method for free boundary problems and the crdt algorithm for given multi - scale " polygonal domains .

the free boundary in our system develops sharp corners / singularities in certain parameter regimes , and this is an impediment to using existing single - scale " numerical conformal mapping methods . we develop a multi - scale " ( i.e. adaptive ) conformal mapping method that allows us to carry out this program of stitching conformal maps on different scales together .

we use conformal maps to study a free boundary problem for a two - fluid electromechanical system , where the interface between the fluids is determined by the combined effects of electrostatic forces , gravity and surface tension . the free boundary in our system develops sharp corners / singularities in certain parameter regimes , and this is an impediment to using existing single - scale " numerical conformal mapping methods . the difficulty is due to the phenomenon of crowding , i.e. the tendency of nodes in the preimage plane to concentrate near the sharp regions of the boundary , leaving the smooth regions of the boundary poorly resolved . a natural idea is to exploit the scale separation between the sharp regions and smooth regions to solve for each region separately , and then stitch the solutions together . however , this is not straightforward as conformal maps are rigid global " objects , and it is not obvious how one would patch two conformal maps together to obtain a new conformal map . we develop a multi - scale " ( i.e. adaptive ) conformal mapping method that allows us to carry out this program of stitching conformal maps on different scales together . we successfully apply our method to the electromechanical model problem and discuss how it generalizes to other situations .

cond-mat0309509

i

the prospect of manipulating the electron spin to store and transport information in semiconductor devices has led to renewed interest in the physics of transition metal ( tm ) impurities in semiconductors - an area which was intensively studied in the eighties @xcite . current interest @xcite in achievement of ferromagnetism ( fm ) at ambient temperatures has led to the investigation of the mechanism that stabilizes fm in transition - metal doped semiconductors . one useful approach to obtain an understanding of the electronic properties of these systems is the _ first principles electronic structure approach _ where one focuses on the explicit electronic and spin wavefunctions of the system . variational minimization of the total energy , determines within the underlying approximations of the spin density functional theory some of the basic features of the states involved , such as the extent of localization , the magnitude of the spin interactions as well as the identity of disorder and compensating defects ( antisites ; interstitials ) . however , the approach does have the drawback of underestimating the extent of electron correlations in addition to being a zero - temperature approach . while comparison with experiment ( _ e.g. _ ferromagnetic temperature vs. alloy composition @xcite ) can be used to assess the extent to which electron correlations are underestimated , the first - principle results have to be generally mapped onto a model hamiltonian to calculate finite temperature properties . while model hamiltonians @xcite have been widely used to describe the properties of these systems , the underlying assumptions in choosing a particular form for the hamiltonian are rarely justified in their own right . generally @xcite , one renormalizes away the electronic degree of freedom and retains only the spin degree of freedom for the transition metal impurity . one then assumes a local interaction between the transition metal impurity and the free carrier ( usually rkky - like ) , and then solves for various physical properties of these systems . in the present work we use first - principles calculations to examine whether the assumptions made in model hamiltonian treatments are consistent with an _ ab - initio _ description of the electronic structure of these systems . our detailed results are then cast in the language of a simple electronic structure model , which could be used in an informed construction of model hamiltonians . we start by identifying the main physical quantities that come into play in determining the electronic structure of these systems . when a trivalent cation site such as ga of a iii - v semiconductor is replaced by divalent mn , an acceptor level ( denoted as e(0/- ) ) is generally created in the band gap . if the fermi level @xmath7 lies below this e(0/- ) level , then mn is charge neutral , _ i.e. _ , its formal oxidation + 3 equals that of the ga atom being replaced . in this case there is a hole in the mn - related orbital . if , on the other hand , the fermi level is above e(0/- ) , then the mn - related orbital captures an electron from the fermi sea ( _ i.e. _ creating a hole there ) , becoming negatively charged ( _ i.e. _ oxidation state mn@xmath8 ) . in this case the hole resides in the fermi sea . the mn - induced hole for @xmath9 e(0/- ) features prominantly in contemporary theories of ferromagnetism the model hamiltonians involve three entities - the host crystal states , the transition metal atom , and the impurity - induced hole state . there are approximations made at various levels which involve decoupling various degrees of freedom . at the first level , one decouples the orbital degrees of freedom associated with the transition metal atom , describing it with a localized spin - only part . the spin is interacting with a hole system through a local exchange interaction . at the next level of approximation , one reduces the problem to that of the transition metal spin interacting with the hole spin , assuming that the host crystal is unperturbed . the main assumptions made in such approaches , which we wish to examine , are : \(i ) _ the hole resides in a bulk - like , hydrogenic , delocalized state . _ this picture is based on the assumption that the perturbing potential @xmath10-@xmath11 generated by the impurity is dominated by a long - ranged coulomb part , as a result of which only a small percentage of the charge resides in the wigner - seitz cell and the rest is distributed over a large portion of the host crystal . in this `` host - like hole '' picture , one reduces the problem to a quasi - hydrogenic form in which the acceptor state is designated via quantum numbers ( @xmath12 , @xmath13 ... ) of the host lattice . in such cases the wavefunction of the acceptor level is delocalized , and can essentially be constructed from the host crystal @xmath14 states . this picture is motivated by the fact that divalent , post transition metal atom elements such as zn@xmath8 form in iii - v semiconductors quasi - hydrogenic acceptor states @xcite with small binding energies . similarly , extrinsic @xmath13-type doping of ii - vi dilute magnetic semiconductor cdmnte @xcite also form hydrogenic hole states . however , unlike zn in gaas or extrinsic @xmath13-type cdmnte , the mn atom introduced into iii - v s has chemically active @xmath5 orbitals @xcite , so it is not obvious that the acceptor state it forms in gan , gap or gaas would qualify as a delocalized host - like hydrogenic state . indeed , the microscopic features determining the localization of the hole wavefunction , such as the @xmath5-character of the acceptor level must be considered . such interactions could change the symmetry ( @xmath0 vs. @xmath1 ) of the hole state , hence its coupling to the host . the pertinent quantum designation of the hole state is impurity - like ( @xmath0 , @xmath1 ) not host - like effective - mass @xcite . \(ii ) _ the host valence band maximum ( vbm ) levels are unperturbed by the transition metal impurity . _ in this view , the host band structure represented in the model hamiltonian could be described by a @xmath15.@xmath13 model , valid for the pure host crystal and is decoupled from the part of the hamiltonian involving the host + hole system . however , since one of the symmetry representations of the mn @xmath5 orbitals in tetrahedral sites ( @xmath0 , @xmath1 ) is the same as that of the zincblende vbm ( @xmath0 ) , such states could couple , hence become mutually perturbed . \(iii ) _ the spin of the hole couples to the spin of the transition metal impurity via an interatomic local exchange interaction @xmath16 . _ as only the spin degree of freedom of the transition metal atom is considered , while the orbital degree of freedom is ignored , the free carriers feel the effective magnetic field produced by the transition metal impurity spin . this is modelled as a _ local _ exchange interaction , @xmath16 , between the transition metal impurity and the spin of the free carrier . hence , the magnitude of @xmath16 determines the energy scale of ferromagnetic ordering . areas visited by the free carrier are rendered ferromagnetic . however certain materials @xcite are found to show activated behavior in their transport implying no free charge carriers , yet they exhibit ferromagnetism . the current model which requires delocalized carriers can not explain ferromagnetism in such systems . in what follows , we use first principles calculations to examine the validity of assumptions ( i)-(iii ) for 3@xmath5 impurities v , cr , mn , fe , co and ni in gaas , as well as for mn and cr in gan , gap and gasb . we then construct a qualitative model that explains our numerical results . we find that \(i ) the mn - induced hole could have significant 3@xmath5 character . the assumption of a `` delocalized hydrogenic hole '' is not supported by first - principle calculations . the depth of the acceptor level ( reflecting its localization ) and the coupling of the 3@xmath5 impurity orbitals to the hole change markedly with the host crystal in the series gan @xmath2 gap @xmath2 gaas @xmath2 gasb . the hole generated by introducing mn in gan is found to have significant 3@xmath5 character , while in gasb the hole is found to have primarily host character . further , the symmetry of the hole depends on the combination of the host crystal with the impurity atom . for example while in gaas : co the hole has dominantly @xmath0 character , the corresponding isoelectronic impurity znse : fe has a hole with @xmath1 symmetry . \(ii ) the presence of the transition metal impurity perturbs the valence band of the host semiconductor . the extent of the perturbation depends on the relative position of the impurity generated levels ( referenced to the valence band maximum of the host ) which have the same symmetry as the valence band maximum . in gaas : v which has levels with @xmath1 symmetry in the bandgap , the perturbation is small , while for gaas : mn the perturbation is large . \(iii ) the interaction between the spin of the tm atom and the spin of the host - like hole has a predominantly non - local part . this is evidenced by the strong stabilization of the ferromagnetic state for mn and cr pairs in gaas at @xmath17 8 @xmath18 separation . this interaction induces a spin - polarization of the host - like states . the direction of the spin - polarization depends on the relative energy position of the cation vacancy generated ( host - like ) states with respect to the impurity states . furthermore , the band - theoretic description of cr in gap shows a partially occupied mid - gap band , and the wavefunctions associated with this mid - gap state are localized . yet , even in the absence of free carriers , our total energy calculations predict a ferromagnetic ground state to be strongly stabilized , while no long range magnetic order is expected .

we use first - principle calculations for transition metal impurities v , cr , mn , fe , co and ni in gaas as well as cr and mn in gan , gap and gasb to identify the basic features of the electronic structure of these systems . the microscopic details of the hole state such as the symmetry and the orbital character , as well as the nature of the coupling between the hole and the transition metal impurity are determined . this could help in the construction of model hamiltonians to obtain a description of various properties beyond what current first - principle methods are capable of . thus , a dbh - cfr level anticrossing exists along this host - crystal series . consequently , the identity of the hole carrying orbital changes .

we use first - principle calculations for transition metal impurities v , cr , mn , fe , co and ni in gaas as well as cr and mn in gan , gap and gasb to identify the basic features of the electronic structure of these systems . the microscopic details of the hole state such as the symmetry and the orbital character , as well as the nature of the coupling between the hole and the transition metal impurity are determined . this could help in the construction of model hamiltonians to obtain a description of various properties beyond what current first - principle methods are capable of . we find that the introduction of a transition metal impurity in iii - v semiconductor introduces a pair of levels with symmetry - one localized primarily on the transition metal atom referred to as crystal - field resonance ( cfr ) and the other localized primarily on the neighboring anions referred to as the dangling bond hybrid ( dbh ) . in addition , a set of nonbonding states with symmetry , localized on the transition metal atom are also introduced . each of the levels is also spin - split . considering mn in the host crystal series gan gap gaas gasb , we find that while in gan the hole resides in the level deep in the band gap , in gaas and gasb it resides in the level located just above the valence - band maximum . thus , a dbh - cfr level anticrossing exists along this host - crystal series . a similar anticrossing occurs for a fixed host crystal ( _ e.g. _ gaas ) and changing the 3 impurity along the 3 series : v in gaas represents a dbh - below - cfr limit , whereas mn corresponds to the dbh - above - cfr case . consequently , the identity of the hole carrying orbital changes . the symmetry ( vs. ) , the character ( dbh vs. cfr ) as well as the occupancy of the gap level , determine the magnetic ground state favored by the transition metal impurity . lda+u calculations are used to model the effect of lowering the energy of the mn state by varying u. we find that this makes the dbh state more host - like , and at the same time , diminishes ferromagnetism . while the spin - splitting of the host valence band in the presence of the impurity has been used to estimate the exchange coupling between the hole and the transition metal impurity , we show how using this would result in a gross underestimation of the coupling .

cond-mat0309509

r

we now divide the main features of the first principle calculations into three main entities introduced in section i : a. the nature of the impurity - induced level in the gap b. the impurity - induced valence - band resonances c. the perturbed host vbm then , in section v , we will provide a simple model that explains all of our numerical results qualitatively . figure 1 shows the transition - metal local dos for v , cr , mn , fe , co and ni in gaas , projected into irreducible representations @xmath0 and @xmath1 and spin directions + and - . the vbm is at the zero of the energy . the gga band gap of pure gaas is found to be 0.3 ev ; all the impurities v - ni introduce levels into this band gap . we first discuss the nature of these gap levels , and then the circumstances how and when a hole is present in them . for spin - up @xmath40 ( shaded region ) , spin down @xmath41 ( solid black line ) and @xmath1 ( dashed line ) symmetries . the zero of energy represents the valence band maximum of the host . the number of k - points used is 64.,width=528 ] from fig . 1 we see that the sequence of levels occupied for cr in gaas are @xmath40 , @xmath42 , @xmath41 and @xmath40 in the order of increasing energy . for a free atom one would expect levels of one spin channel to be filled up before levels of the other spin channel ; the deviation that one observes here reflects solid state effects . the two sets of @xmath40 and @xmath41 levels that we find for each impurity are suggestive of bonding / antibonding combinations arising from hybridization . we therefore determined the atoms on which each of the @xmath0 states are localized by computing atom - projected dos . bonding states with a large wavefunction amplitude on the tm site are referred to as `` crystal field resonances '' ( cfr ) @xcite , whereas antibonding @xmath0 states with low contribution on the tm which are localized instead on the four nearest as atoms are referred to as the dangling bond hybrid ( dbh ) . the full explanation of the genesis of these states will be provided in sec . v. we see that , \1 . _ symmetry of gap levels and lowest unoccupied levels _ : substituting cr , mn and co in gaas introduce levels with up - spin character and @xmath0 symmetry in the band gap . these levels are partially occupied by 1 , 2 and 2 electrons for neutral cr , mn and co , respectively : cr@xmath43 ( @xmath44 ) , mn@xmath43 ( @xmath45 ) and co@xmath43 ( @xmath45 ) . the levels introduced by v@xmath43 ( @xmath46 ) and fe@xmath43 ( @xmath47 ) are fully occupied . the first unoccupied levels have @xmath40 and @xmath48 symmetry for v and fe respectively . _ d character of gap levels _ : the transition metal projected partial density of states for different transition metal impurities in gaas given in fig . 1 indicates that the gap level / first unoccupied level is strongly @xmath5-like for the early transition metal impurities v and cr , while for the heavier 3@xmath5 elements _ e.g. _ mn these levels have less @xmath5 character . an increased @xmath5 character of the gap level would imply increased spatial localization of the wavefunction in the vicinity of the impurity . and @xmath49+@xmath50 about the tm impurity for mn in gan ( filled squares ) , cr in gaas ( open inverted triangles ) and mn in gaas ( filled circles ) compared with the result for an electron gas ( solid line ) . @xmath51 is the lattice constant of the host supercell.,width=528 ] \3 . _ degree of localization of gap levels _ : we quantify the degree of localization by plotting in fig . 2 the charge q(r@xmath52)=@xmath53 enclosed between concentric spheres with radius @xmath49 and @xmath54 centered about the impurity atom . the integrated charge between the spheres is plotted as a function of r@xmath52 . for comparison , we show also the result expected for a homogeneous charge distribution ( electron gas ) , where the charge density at any point in the cell is given by reciprocal volume @xmath55 . we see that q(r@xmath52 ) for the tm impurities has little similarity to the results for an electron gas . changing the impurity from mn to cr in gaas , we see an increase in the charge density localized in the vicinity of the impurity . we find that till a radius which includes second neighbors of the tm atom , the integrated charge for cr is higher than for mn . further , we find that the enclosed charge in the vicinity of the impurity atom is higher in gan : mn than in gaas : mn and the decay of the wavefunction is faster . point for @xmath3 , @xmath56 ( upper panel ) and @xmath4 ( lower panel ) states for v - co impurities in gaas.,width=528 ] \4 . _ the negative exchange splitting of the gap levels _ : having established the identity of the gap levels , we now investigate their spin splittings . in fig . 3 we have plotted the spin splittings of the cfr and dbh levels at @xmath14 point for the impurities v - co in gaas obtained from an analysis of their eigenvalues / eigenfunctions . for v and cr the spin splitting of the dbh levels is positive _ i.e. _ @xmath57 states are at lower energies compared to @xmath58 . however , for mn , fe and co the splitting is negative with the @xmath58 states at lower energies compared to @xmath57 . a similar negative exchange splitting was observed earlier for the te states in mnte @xcite , the ce states in cefe@xmath21 @xcite , and the mo states in sr@xmath21femoo@xmath59 @xcite . the explanation was that the @xmath13 states of te in cdte , the @xmath5 states of ce in cefe@xmath21 and the @xmath5 states of mo in sr@xmath21femoo@xmath59 are sandwitched in between the 3@xmath5 states of the transition metal atom . the @xmath13-@xmath5 hybridization results in an exchange splitting of these states opposite in direction to that on the transition metal atom . indeed we see from fig . 1 that for the cases where the dbh states are bracketed by the spin split cfr states , the spin splitting is negative . _ enhanced exchange splitting for @xmath3 states _ : from fig . 3 , we see that the exchange splitting of the @xmath3 states is larger than that of the @xmath56 states for mn , fe and co impurities . having discussed the existence of impurity - induced levels in the band gap of the host semiconductor , we next discuss the location of these levels . _ acceptor transitions for gap levels _ : single particle lda or gga levels do not have any rigorous meaning . we thus calculate _ transition _ energies , @xmath60(@xmath22,@xmath61 ) which correspond to the value of the fermi energy @xmath7 at which the defect changes from a charge state @xmath22 to @xmath61 . table ii provides the calculated and measured @xcite acceptor / donor transition energies for various transition metal impurities in gaas . the calculated acceptor levels for mn and cr in gasb , gaas , gap and gan are plotted in fig . 4 where the host band edges are aligned according to their calculated unstrained valence band offsets @xcite . we see that as the electronegativity of the host crystal increases in the sequence gasb @xmath2 gaas @xmath2 gap @xmath2 gan , its bulk ionization energy (= position of vbm with respect to vacuum ) increases . the acceptor level is thus farther away from the vbm of gan than it is from the vbm of gaas . thus , gan : mn and gap : cr have more localized hole states whereas gasb : mn has more delocalized holes . this behavior , whereby the acceptor energy level does not follow the host valence band energy ( as in the case of hydrogenic impurities ) characterizes localized states @xcite . \7 . _ multiplet states and violation of isovalency rule _ : we use the level occupancies ( fig . 1 ) as well as the net magnetic moments that we obtain for different transition metal impurities in gaas ( table iii ) to obtain the multiplet configuration describing the ground state . these are given in table iii . we also provide the multiplet configuration observed from experiment @xcite and find that there is agreement in all cases . it is interesting to compare the ground state multiplets of two isoelectronic cases - znse : fe@xmath8 and gaas : co@xmath62 in their neutral charge states . in both cases we expect an electron configuration of @xmath20 . normally , one would expect to find equal multiplets for isoelectronic cases @xcite ( isovalency rule ) . this expectation is based on the fact that we are looking at a low transition metal impurity concentration regime where basic crystal field theory ideas are expected to be sufficient to explain the observed ordering of energy levels . however , we find in gaas : co@xmath62 the configuration @xmath63t@xmath21 , i.e the hole is in the @xmath4 level ( @xmath64 ) , whereas in znse : fe@xmath8 we find @xmath65e , i.e the hole is in the @xmath66 level ( @xmath67 ) . the reason for the difference is that the stronger @xmath13-@xmath5 hybridization for gaas : co pushes the @xmath57 levels to higher energies , so that the @xmath68 levels are occupied first and the hole resides in the @xmath57 level . we thus conclude that the isovalency rule is not applicable , and one can not assume that the hole is in a `` generic '' @xmath5 state . _ fm vs. afm ground state and their relation to the symmetry of the gap levels _ : having summarized the nature of the level induced in the gap by the introduction of the transition metal impurity , we now analyze when a ferromagnetic state is favored . in table iii we provide the energy difference between the ferromagnetic and antiferromagnetic energies for two tm atoms at nearest @xmath69e@xmath70 and fourth neighbor @xmath69e@xmath71 fcc positions in a 64 atom supercell of gaas . we find that ( a ) when the level in the gap is fully occupied as in v@xmath43 and fe@xmath43 , the favored ground state is antiferromagnetic . ( b ) when the level in the gap is partially occupied and has @xmath0 symmetry as in cr@xmath43 , mn@xmath43 , the ferromagnetic state is lower in energy . this is also the case for electron doped v@xmath72 in gaas which is strongly ferromagnetic . although co@xmath43 ( @xmath63t@xmath21 ) also has a hole in the @xmath0 level , the system is at the brink of a ferromagnetic - to - nonmagnetic transition . ( c ) when the level in the gap has @xmath1 symmetry , as in the case of electron doped fe@xmath72 , the stability of the fm state is weaker . evidently the _ symmetry _ of the hole carrying state strongly determines the magnetic order . _ we conclude that fm is stabilized strongly only when the hole resides in the level with @xmath0 symmetry . _ ( note , viz . v , that in @xmath73 symmetry @xmath0 states are strongly bonded to their neighboring atoms , whereas the lobes of the @xmath1 orbitals point in between the nearest - neighbor atoms . ) figure 1 shows that in addition to the gap levels , the introduction of a transition metal atom gives rise to resonance levels that lie deep within the valence band of the host semiconductor . in most model hamiltonian theories @xcite , one usually ignores the orbital degree of freedom of the transition metal impurity , and the presence of the impurity is included only as a localized spin of value 5/2 . in our calculations we find that the degree of localization of such deep resonances ( thus , the possibility of depicting them as local point - like spin ) varies sharply with the position of the impurity in the periodic table . for heavier tm such as fe and mn , the deeper resonance level has significant tm @xmath5 character ( being crystal - field resonances ) , while for the early tm impurities in gaas , one finds that the deeper @xmath0 levels have significantly less tm character ( being dangling bond hybrids ) . this is discussed next . _ anticrossing of the two @xmath0 levels in different host materials : _ level anticrossing is evident when keeping the impurity atom fixed , and , changing the host semiconductor . considering the example of mn , we find that by changing the host from gasb to gan , the dbh and the cfr exhibit anticrossing . this is not the only difference : we find that the exchange splitting of the dbh levels is in the same direction as the cfr levels ( positive ) in gan : mn , in contrast to gaas : mn . further , in gan : mn the @xmath74 levels lie above the @xmath75 levels , unlike the case in gaas : mn . the reason is evident from fig . 4 which shows that the vbm of gan is much deeper than the vbm of gaas . since the free mn@xmath8 ion has its @xmath5 orbitals _ above _ the gan vbm , but _ below _ the vbm of gasb or gaas , an anticrossing occurs along the gan @xmath2 gap @xmath2 gaas @xmath2 gasb series . this is illustrated in fig . 5 which shows that in gan : mn for the up spin channel , the upper @xmath0 is more localized than the lower @xmath0 , whereas in gaas : mn the localization sequence is reversed . this clarifies a confusion that existed in the literature @xcite regarding the question of whether the gap level is localized or not . our result shows that the answer depends on the host . these results also clarify the nature of the acceptor transition for mn in different materials . gan : mn can be viewed as a @xmath76-like case since its configuration is @xmath77 ( hole in @xmath78 ) , and the ( 0/- ) acceptor transition is from a mn configuration @xmath76 to @xmath6 . on the other hand for mn in gap , gaas and gasb we have the configuration ( @xmath6 + hole ) , _ i.e. _ @xmath79 and the acceptor transition is from a mn configuration of ( @xmath6+hole ) to @xmath6 . projected partial density of states evaluated within a sphere of radius 1.2 @xmath18 for a mn impurity in gan ( upper panel ) and gaas ( lower panel ) . the number of k - points used is 64 . , width=528 ] \2 . _ occupancy of the valence band resonances and comparison with photoemission _ : table iii gives the calculated occupancy of the crystal field resonances of the 3@xmath5 impurities in gaas ( in square brackets ) . these levels are found to have a configuration of `` @xmath6 '' for cr , mn and fe . experimentally the position of these levels can be detected by valence band photoemission @xcite . by suitably tuning the photon energy so that the photo - ionization cross - section is maximum for the tm - related states , an electron can be ionized from these deep cfr levels . @xcite used resonant valence band photoemission and showed that the the cfr levels for mn in gaas are located at @xmath80 - 4 ev . a direct comparison of the position of these levels with the single - particle density of states calculated for gaas : mn places these energies at @xmath80 - 2 - 3 ev . the lda error in the position of these states is because of the self - interaction correction ( sic ) that places these energies too high @xcite . as pointed out earlier @xcite for the 3@xmath5 states in ii - vi s , the experimental result should be compared with the total energy difference between the configurations @xmath76 and @xmath6 and not with the bare single particle eigenvalues . alternatively , the lda error can be empirically corrected by using the simplified lda+u version of sic . in fig . 6 we plot the mn @xmath81 partial density of states as a function of u for gaas : mn . as @xmath82 increases , the position of the mn @xmath3 level is pushed deeper into the gaas valence band . agreement with xps for the @xmath81 being at @xmath80 - 4 ev occurs for u @xmath17 2 ev . at the same time , the mn character of the dbh state at @xmath7 ( not shown ) decreases with increasing @xmath82 . as the dbh hole becomes more delocalized , the e@xmath83-e@xmath84 stabilization energy ( insert to fig . 6 ) is reduced ; a `` host - like hole '' obtained for unphysically large u leads to nearly vanishing fm stabilization energy . clearly , the picture of `` host like hole '' is invalid for gaas : mn , since for the @xmath82 that leads to agreement with xps the dbh hole is still localized to some extent , whereas for very large @xmath82 , when the hole is delocalized , there is no ferromagnetism . projected partial density of states evaluated within a sphere of radius 1.2 @xmath18 for on - site coulomb interaction strengths u=0 , 6 and 10 and 15 for mn in gaas . the number of k - points used is 4x4x4 . the inset shows the variation in e(fm)-e(afm ) for two mn atoms at nearest neighbor positions.,width=528 ] having studied the impurity - induced levels in the gap and deep in the host valence band , we next examine the perturbation of the host states , especially the host valence band maximum by the presence of the impurity atom . figure 7 shows the up and down spin band dispersions for 3 @xmath85 cr doped gap supercell ( panels ( a ) and ( b ) ) . the band dispersion of the gap host without the impurity has been provided in panel ( c ) for comparison . the thickness of the lines depicting these bands has been made proportional to the cr @xmath5 character of the states . we see that cr introduces a new band within the band gap of gap . in a band - theoretic picture , this system is metallic , with the fermi energy within the impurity band . interestingly , ( 1 ) the host band dispersions are significantly altered by the presence of the impurity . in particular the vbm is found to have significant tm @xmath5 character for the 3@xmath85 cr concentration represented by the supercell . ( 2 ) a cr - induced spin - splitting of the valence band maximum is observed . effects ( 1 ) and ( 2 ) suggest that the host vbm is sufficiently perturbed by the transition metal . using 64 k - points . the positions of the as atoms ( filled circles ) with respect to the mn atoms is shown in the inset.,width=528 ] another way of detecting perturbations in the host bands is to examine the host projected dos of the system containing the impurity . in fig . 8 we have plotted the as @xmath13 partial density of states projected onto different as atoms labelled 1 - 4 for a gaas supercell containing 2 mn atoms . the as atom labelled 1 has one mn nearest neighbor , while the as atom labelled 2 has two mn nearest neighbors . the as atoms show a strong polarization which increases with the number of mn neighbors . the as atoms labelled 3 and 4 which are far away from the mn atoms show a reduced polarization . to pictorially see the perturbation in the vbm states , we compare in fig . 9 the wavefunction squared along two chains in the ( 110 ) plane for the valence band maximum of the pure gaas host ( panel ( a ) ) as well as the vbm ( _ i.e. _ state below dbh ) of the system with the mn impurity in the up ( panel ( b ) ) and down ( panel ( c ) ) spin channels . the upper chain in panels ( b ) and ( c ) contains the perturbing mn impurity . the perturbation of the vbm in the presence of the impurity can be assessed by comparing the perturbed charge density for each spin channel with the unperturbed charge density of the host lattice . we find that the perturbations are significant in the chain containing the mn atom , while in adjoining chains , the perturbation is limited in extent . further the perturbations are stronger in the up spin channel than in the down spin channel . and each contour is 1.6 times larger . , width=336 ] to evaluate the 3@xmath5-induced spin - splitting in the vbm , we reference the up and down spin vbm eigenvalues of the impure system to the corresponding vbm of the pure host semiconductor . this is done by aligning the average potentials on ga atoms far away from the impurity for the two systems . the presence of the impurity band with @xmath0 symmetry above the vbm for mn and cr impurities complicates the identification of the valence band maximum . we associate the highest occupied triply degenerate eigenvalues at @xmath14 point with the impurity band , and the next deeper set as @xmath86 . the shift with respect to the pure host @xmath87=@xmath88(gaas : mn)-@xmath32 ( gaas ) and @xmath89=@xmath90(gaas : mn)-@xmath32 ( gaas ) is given in table iv for the impurities v , cr and mn in different host semiconductors . we find that the perturbation of the host vbm is smaller in the down spin channel compared to the up spin channel . this is consistent with what we find from the charge density plotted in fig . the spin splitting of the valence band maximum depends strongly on the transition metal impurity and host semiconductor combination . considering the case of impurities in gaas , we find that while the spin splitting associated with the introduction of v is only 0.06 ev , it increases to 0.39 ev for mn . keeping the impurity fixed ( mn ) , and varying the semiconductor host ( gaas to gan ) , we find the splitting decreases from 0.39 ev to 0.1 ev . the small valence band splittings in the case of v in gaas as well as mn in gan compared with that for mn in gaas is because of the larger energy separation between the interacting @xmath0 states in the former cases compared to the latter . we are now in a position to examine whether the physical picture of the electronic structure of 3@xmath5 impurities as assumed in model hamiltonian theories ( reviewed in sec . i ) is consistent with first - principles calculations ( outlined in sec . \(i ) _ the nature of the tm - induced hole state _ : a 3@xmath5 impurity in a iii - v semiconductor generates two sets of states with @xmath0 symmetry , and one set of states with @xmath1 symmetry in each spin channel . while one set of @xmath0 states are localized on the tm atom ( cfr ) , the other are localized on the host anion atoms next to the impurity ( dbh ) . these states cfr and dbh exhibit anticrossing for a fixed tm as a function of the host anion gan @xmath2 gap @xmath2 gaas @xmath2 gasb , or for a fixed host as a function of the impurity v @xmath2 mn . the localization of the hole state decreases as we move from mn in gan to mn in gap , and then to mn in gasb . not all impurities introduce holes . in gaas , v@xmath43 and fe@xmath43 have no hole ; cr@xmath43 , mn@xmath43 and v@xmath72 have @xmath0 holes ; and fe@xmath72 has an @xmath1 hole . in all cases , however , the hole is non - hydrogenic , manifesting significant admixture of 3@xmath5 character and showing deep acceptor levels whose energies do not follow the host vbm . this implies that the neglect of the short - range part of the impurity potential and the consequent expansion of the acceptor wavefunction in terms of a single host wavefunction are questionable . the effective mass of the hole state is therefore different from that of the host , as observed in recent experiments @xcite . the exchange splitting of the cfr states is different for the @xmath0 states from that for the @xmath1 states . while the splitting for the @xmath1 states is larger than that for the @xmath0 states for v and cr in gaas , the order is reversed for mn , fe and co. this reversal in the order of the spin splitting of the cfr states is accompanied by a reversal in the sign of the spin splitting of the dbh states . the identity of the hole state - both the symmetry as well as the character depends on the impurity - host combination . while the hole carrying orbital for fe in znse has @xmath1 symmetry , the hole is found to be located in an orbital with @xmath0 symmetry for the isovalent doping of co in gaas . \(ii ) _ the nature of the host vbm _ : the introduction of the transition metal perturbs the valence band of the host crystal . we find this perturbation to be large when the state in the gap has @xmath4 character . this is because the effective coupling is larger since the dbh states have strong host character . we find that the vbm is spin - split in the presence of 3@xmath5 impurity , and that the vbm in the up - spin channel is perturbed more strongly than in the spin down channel in the presence of the impurity . \(iii ) _ ferromagnetism and symmetry _ : impurities with fully occupied dbh - like @xmath0 gap states such as v@xmath43 , fe@xmath43 show antiferromagnetism . partial occupation of the @xmath4 as in cr@xmath43 , mn@xmath43 or v@xmath72 show ferromagnetism . partially occupied @xmath1-like level like in fe@xmath72 show weak or no ferromagnetism . \(iv ) _ ferromagnetism and hole localization _ : using lda+u as an artificial device to explore the consequences of delocalized host - like - hole states we find ( insert to fig . 8) that in this limit there is reduced ferromagnetism . we find that despite the well known gga - lda band gap error , as well as the underestimation of the location of deep cfr states due to sic , these first principle calculations provide us with the correct spin multiplets . lda+u changes some details ( cfr locations ) , but does not alter the basic picture emerging from gga / lda when the hole is dbh - like as in gaas : mn .

we find that the introduction of a transition metal impurity in iii - v semiconductor introduces a pair of levels with symmetry - one localized primarily on the transition metal atom referred to as crystal - field resonance ( cfr ) and the other localized primarily on the neighboring anions referred to as the dangling bond hybrid ( dbh ) . in addition , a set of nonbonding states with symmetry , localized on the transition metal atom are also introduced . each of the levels is also spin - split . considering mn in the host crystal series gan gap gaas gasb , we find that while in gan the hole resides in the level deep in the band gap , in gaas and gasb it resides in the level located just above the valence - band maximum . a similar anticrossing occurs for a fixed host crystal ( _ e.g. _ gaas ) and changing the 3 impurity along the 3 series : v in gaas represents a dbh - below - cfr limit , whereas mn corresponds to the dbh - above - cfr case .

we use first - principle calculations for transition metal impurities v , cr , mn , fe , co and ni in gaas as well as cr and mn in gan , gap and gasb to identify the basic features of the electronic structure of these systems . the microscopic details of the hole state such as the symmetry and the orbital character , as well as the nature of the coupling between the hole and the transition metal impurity are determined . this could help in the construction of model hamiltonians to obtain a description of various properties beyond what current first - principle methods are capable of . we find that the introduction of a transition metal impurity in iii - v semiconductor introduces a pair of levels with symmetry - one localized primarily on the transition metal atom referred to as crystal - field resonance ( cfr ) and the other localized primarily on the neighboring anions referred to as the dangling bond hybrid ( dbh ) . in addition , a set of nonbonding states with symmetry , localized on the transition metal atom are also introduced . each of the levels is also spin - split . considering mn in the host crystal series gan gap gaas gasb , we find that while in gan the hole resides in the level deep in the band gap , in gaas and gasb it resides in the level located just above the valence - band maximum . thus , a dbh - cfr level anticrossing exists along this host - crystal series . a similar anticrossing occurs for a fixed host crystal ( _ e.g. _ gaas ) and changing the 3 impurity along the 3 series : v in gaas represents a dbh - below - cfr limit , whereas mn corresponds to the dbh - above - cfr case . consequently , the identity of the hole carrying orbital changes . the symmetry ( vs. ) , the character ( dbh vs. cfr ) as well as the occupancy of the gap level , determine the magnetic ground state favored by the transition metal impurity . lda+u calculations are used to model the effect of lowering the energy of the mn state by varying u. we find that this makes the dbh state more host - like , and at the same time , diminishes ferromagnetism . while the spin - splitting of the host valence band in the presence of the impurity has been used to estimate the exchange coupling between the hole and the transition metal impurity , we show how using this would result in a gross underestimation of the coupling .

astro-ph9611077

i

mapping the large - scale structure ( lss ) of the universe is one of the major goals of observational cosmology . traditionally this is performed using large surveys of galaxies , either the projected 2-dimensional distributions or the 3-dimensional surveys in redshift space . the main shortcoming of the galaxy surveys is that they trace light , while most of the matter appears to be dark . one therefore needs to translate the galaxy power spectrum into the matter power spectrum and in order to do so one has to make some assumptions on the nature of galaxy biasing . even in the simplest model this can be achieved only up to an unknown biasing parameter @xmath9 , which at present can not be theoretically estimated . while sophisticated n - body and hydro - dynamical simulations will eventually provide some answers to this question , at present the biasing relation between the light and matter remains rather poorly understood and prevents one from drawing definitive conclusions on the amplitude and distribution of mass fluctuations in the universe from galaxy survey data . it is clearly important to seek ways to estimate the large - scale structure that are insensitive to biasing . several observable tracers have been proposed that probe directly the underlying mass distribution : cosmic microwave background ( cmb ) anisotropies , gravitational lensing , peculiar velocity flows and abundances of massive nonlinear objects . this paper focuses on tracing the dark matter with gravitational lensing , in particular the effects of weak lensing by large scale structure on background galaxies . weak lensing magnifies and shears the images of distant galaxies . the shear induces an ellipticity in the image of an intrinsically circular galaxy . background galaxies are of course not circular , but by averaging over the observed ellipticities of a large number of galaxies , the induced ellipticity can be measured , and related to the mass fluctuations along the line of sight and to the spatial geometry of the universe . ellipticities of distant background galaxies averaged over several arcminute windows are sensitive to the mass power spectrum on scales of @xmath10 mpc . for a given spectrum of mass fluctuations , it is sensitive to the cosmological parameters @xmath8 and @xmath11 . thus while strong lensing which leads to multiple images probes non - typical regions of the universe which contain massive halos , weak lensing provides a different and a more direct measure of the mass fluctuations on large scales . the first calculations of the shear signal due to weak lensing that used modern models for the large scale structure power spectrum were those of @xcite and @xcite , based on the pioneering work by gunn ( 1967 ) . our work generalizes the results of these authors to include the effects of nonlinear evolution of the matter fluctuations for flat as well as open and @xmath12dominated cosmologies . villumsen ( 1996 ) has considered some aspects of the linear calculation for open models . very recently , bernardeau et al . ( 1996 ) , kaiser ( 1996 ) and stebbins ( 1996 ) have also made the linear calculation for different cosmologies . our work extends the results of the above authors by using the nonlinear power spectrum , which makes a significant difference on small angular scales of @xmath13 . for the nonlinear calculation we have used the prescription of @xcite as implemented by @xcite and @xcite which provide accurate power spectra for different models , valid from the linear to the strongly nonlinear regime . one can take two different approaches in interpreting a possible measurement of the shear signal . the first is to work within the framework of a physical model for the dark matter and background cosmology . such a model is best normalized to cobe and then its small scale predictions can be compared to the observational constraints of e.g. cluster or damped ly-@xmath14 system abundances , strong lensing statistics or peculiar velocity flows , or in our case the shear amplitude on a given angular scale . another possibility , which is less model dependent , is to compare the constraints from different tracers on the same physical scale . this way one can test the gravitational instability assumption that both tracers probe the same underlying power spectrum and place constraints on the cosmological parameters which scale differently with the tracers . at the present the cmb data constrain the power spectrum only on very large scales , where there are no available data from other tracers . while several current cmb experiments are approaching the scales probed by other tracers , cosmological parameters such as baryon density , reionization epoch and hubble constant become important and complicate the power spectrum reconstruction . the test mentioned above is possible between the cluster abundances ( or strong lensing statistics of multiple images at large separation , both of which trace essentially the same property ) and the peculiar velocities . unfortunately this method can not give model independent constraints on cosmological parameters , because the two tracers scale roughly equally with the matter density @xmath8 ( peebles 1980 , white , efstathiou & frenk 1993 ) . a similar comparison can also be made between weak lensing observations and the tracers discussed above . for example , the shear amplitude on @xmath1 angular scale and rich cluster abundance both probe power spectrum scales around @xmath15 mpc , so the ratio of the two is roughly independent of the shape of the power spectrum . one could hope that using such comparisons would lead to an estimate of the mean density of the universe , because the two tracers scale differently with it . unfortunately as shown in this paper this is not the case so that this is not a promising method of obtaining the mean density . nevertheless , weak lensing can also give important constraints on the power spectrum both on smaller and larger scales , where other tracers give less stringent constraints . on the smaller angular scales , our calculations provide the enhancement in the amplitude and the change in the scalings with @xmath16 and @xmath8 that arise due to nonlinear evolution . once reliable detections of the shear at different angular scales are available , these results can be used to constrain cosmological parameters and the matter power spectrum . the principal advantage of working with weak lensing is that no assumptions about the formation of observable structures need to be made to connect the mass power spectrum to the observed shear . the only unknown parameter is the redshift distribution of source galaxies , and in principle this can also be determined observationally . in 2 and the appendix , the formalism for the weak lensing calculation is presented following seljak ( 1995 , 1996 ) , who generalized the work by @xcite and @xcite to a non - flat universe and non - linear regime . the derivation presented here is complementary to recent derivations by bernardeau et al . ( 1996 ) and kaiser ( 1996 ) . the linear and nonlinear evolution of the power spectrum is introduced in 2.1 , and the resulting dependence on cosmological parameters for power law spectra is obtained in 2.2 . in 3 realistic cdm - like spectra are used to predict the rms shear . we provide accurate power law fits for the dependence of the shear on the parameters @xmath17 , @xmath3 , @xmath16 and @xmath8 for different cosmological models . 3.1 provides predictions of the shear for cobe and cluster - abundance normalized spectra for the range of angular scales which are being probed by current and forthcoming observations . the effects of non - gaussianity in the distribution of shear are considered in 4 where we compute the skewness of the distribution . this is followed by a discussion and conclusions in 5 .

weak lensing by large scale structure induces correlated ellipticities in the images of distant galaxies . nonlinear effects in the evolution of the power spectrum significantly enhance the ellipticity for for the rms ellipticity is , which is nearly twice as large as the linear prediction . the scaling with cosmological parameters also changes due to nonlinear effects . by measuring the correlations on small ( nonlinear ) and large ( linear ) angular scales ,

weak lensing by large scale structure induces correlated ellipticities in the images of distant galaxies . the two - point correlation is determined by the matter power spectrum along the line of sight . we use the fully nonlinear evolution of the power spectrum to compute the predicted ellipticity correlation . we present results for different measures of the second moment for angular scales , and for alternative normalizations of the power spectrum , in order to explore the best strategy for constraining the cosmological parameters . normalizing to observed cluster abundance the rms amplitude of ellipticity within a radius is , almost independent of the cosmological model , with being the median redshift of background galaxies . nonlinear effects in the evolution of the power spectrum significantly enhance the ellipticity for for the rms ellipticity is , which is nearly twice as large as the linear prediction . this enhancement means that the signal to noise for the ellipticity is only weakly increasing with angle for , unlike the expectation from linear theory that signal to noise is strongly peaked on degree scales . the scaling with cosmological parameters also changes due to nonlinear effects . by measuring the correlations on small ( nonlinear ) and large ( linear ) angular scales , different cosmological parameters can be independently constrained to obtain a model independent estimate of both power spectrum amplitude and matter density . nonlinear effects also modify the probability distribution of the ellipticity . using second order perturbation theory we find that over most of the range of interest there are significant deviations from a normal distribution . # 1 # 1

1105.1949

c

in this paper the microphase separation in binary ( a , b ) bottlebrush polymers with rigid backbones driven by decrease of temperature ( i.e. , variation of the solvent quality from theta solvents to poor solvents ) was studied , considering also the variation of grafting density along the backbone of these cylindrical brushes . two complementary theoretical methods were used , namely , the first approach has the merit that arbitrarily long chain lengths of the side chains can be considered , as well as infinitely long backbones ( so that end effects or effects of boundary conditions along the backbone do not matter ) . besides , the approach results in some explicit expressions for the correlation functions under study . however , the disadvantage of the approach is that it is basically a linear stability analysis around the homogeneous state , strongly nonlinear effects are out of consideration . the md approach , on the other hand , in principle can take both nontrivial correlations and nonlinear effects into account , but in practice ( to avoid excessive requests for computer time resources ) is limited to rather short length of the side chains , and also the length of the ( rigid ) backbone is finite ( end effects then are avoided by a periodic boundary conditions , but the price to be paid is that reciprocal space in the z - direction along the backbone is discretized ) . also , non - negligible statistical errors ( and systematic errors due to insufficient length of the md trajectories , which affect the establishment of full thermal equilibrium ) restrict the accuracy of the results that can be gotten . whereas the rpa treatment assumes that the monomer density in the radial direction perpendicular to the cylinder axis is uniform and constant ( up to the cylinder radius , while it is zero outside of the cylinder surface ) , in the model studied by md this assumption clearly does not apply , as has been shown in our previous work @xcite . rather , it was found that there occur density oscillations very close to the cylinder axis ( reminiscent of the layering of fluid particles adjacent to the hard wall ) , and for the chain lengths one can study only for a rather small regime of radial distances the density profile then is approximately constant . one approaches rather fast the regime where the density continuously drops to zero , similar to the density profiles across interfaces in phase - separated polymer solvent systems , where the density decreases from its value in the melt towards zero over a distance of several monomer diameters . this low temperature behavior in the simulation comes closest to the situation assumed in the rpa calculations . at higher temperatures , close to the @xmath55 temperature , the interfacial profile is so broadened that a flat region where the density has reached the melt density no longer can be identified . given the different temperature and chain length regimes for which theory and simulation access the microphase separation conditions , it is very gratifying that there is not only a good qualitative agreement between both , but furthermore the wavelength of instability towards the microphase separation agrees quantitatively between both . qualitatively , the analytical theory describes similar trends as seen in the simulation : there is a competition between microphase separation developing periodic order in axial direction , i.e. , the system tends to develop long range order with a particular periodicity ( the wavelength of this periodic modulation being controlled by the gyration radius of the side chains , while the strength of the modulation depends on temperature and grafting density , as the analytical theory compellingly shows , and as is confirmed by our simulations ) and microphase separation of the janus - type , characterized by @xmath440 ( the cross section of the dumbbell , depending on the ratio @xmath5 , in the strongly segregated regime ) . the theory shows that the instability leading to the latter type of order prevails if the grafting density is sufficiently high ( and/or the side chain length sufficiently long ) . both the snapshot pictures of the simulated bottle - brushes ( figs . 4,5 ) and the quantitative analysis of the simulations in terms of the correlation functions of density fluctuations @xmath327 [ with @xmath441 and concentration fluctuations ( or their fourier transforms ) are compatible with such an interpretation . we emphasize that both the simulation and the analytical theory have considered a particularly symmetric case , where both types of side chains have equal length ( @xmath442 ) and also their number was taken exactly the same ( and moreover the same solvent quality was chosen for both types of chains , @xmath443 ) . clearly , when one would like to discuss particular experimental systems , these rather special conditions need to be relaxed . also , we have restricted attention to the case where the side chains are perfectly flexible while the backbone was assumed to be rigid . for many real bottle - brush systems it is probably more realistic to assume backbones that are semi - flexible rather than completely rigid , and also the local intrinsic stiffness of the side chains may play a role . thus , the treatment presented in our paper can be taken as a first step towards a more complete description of less complex but interesting systems only . * acknowledgments*. i.e. thanks the alexander von humboldt foundation and the ministry of education and science of the russian federation for support of this work . p.e.t . thanks the max planck institute for polymer research for supporting him with a max planck fellowship . ( color online ) snapshot picture of bottle - brush polymers at @xmath444 for @xmath283 ( upper part ) and @xmath285 ( lower part ) . a and b monomers are distinguished by different color ( or light grey vs. dark grey , respectively),title="fig:",height=234 ] ( color online ) snapshot picture of bottle - brush polymers at @xmath444 for @xmath283 ( upper part ) and @xmath285 ( lower part ) . a and b monomers are distinguished by different color ( or light grey vs. dark grey , respectively),title="fig:",height=226 ] i.m . lifsitz , a.yu . grosberg and a.r . khokhlov , rev . phys . * 50 * , 683 ( 1978 ) ; a. yu . grosberg and a. r. khokhlov , _ statistical physics of macromolecules _ , american institute of physics , new york , 1994 .

starting from this analogy , leibler s theory of microphase separation in block copolymer melts is generalized to derive a description of the system in the weak segregation limit . various correlation functions are analyzed to study the conditions when ( local ) janus cylinder - type ordering and when ( local ) microphase separation in the direction along the cylinder axis occurs . both the analytical theory and the simulations give evidence for short range order due to a tendency towards microphase separation in the axial direction , with a wavelength proportional to the side chain gyration radius , irrespective of temperature and grafting density , for a wide range of these parameters .

when two types of side chains ( a , b ) are densely grafted to a ( stiff ) backbone and the resulting bottle - brush polymer is in a solution under poor solvent conditions , an incompatibility between a and b leads to microphase separation in the resulting cylindrical brush . the possible types of ordering are reminiscent of the ordering of block copolymers in cylindrical confinement . starting from this analogy , leibler s theory of microphase separation in block copolymer melts is generalized to derive a description of the system in the weak segregation limit . also molecular dynamics simulation results of a corresponding coarse - grained bead - spring model are presented . using side chain lengths up to n = 50 effective monomers , the ratio of the lennard - jones energy parameter between unlike monomers and monomers of the same kind is varied . various correlation functions are analyzed to study the conditions when ( local ) janus cylinder - type ordering and when ( local ) microphase separation in the direction along the cylinder axis occurs . both the analytical theory and the simulations give evidence for short range order due to a tendency towards microphase separation in the axial direction , with a wavelength proportional to the side chain gyration radius , irrespective of temperature and grafting density , for a wide range of these parameters .

0805.3751

i

let @xmath2 denote the unique simply connected complete @xmath1-dimensional riemannian manifold with constant sectional curvature @xmath3 , which we call the hyperbolic @xmath1-space . associated to a complete finite - total - curvature -@xmath0 ( constant mean curvature one ) conformal immersion @xmath4 of a riemann surface @xmath5 are two meromorphic maps called the hyperbolic gauss map and the secondary gauss map , which we denote by @xmath6 and @xmath7 respectively ( to be defined in the next section ) . using these two gauss maps , we can define two characteristics of the surface @xmath8 : 1 . it is known that @xmath5 is biholomorphic to a compact riemann surface with a finite number of points removed , and hence each end is conformally a punctured disk . therefore we may consider the order of the hyperbolic gauss map @xmath6 at each end , and an end is called _ regular _ if @xmath6 has at most a pole singularity at this end . if @xmath6 has an essential singularity , the end is called _ irregular_. 2 . although @xmath6 is single - valued on @xmath5 , the secondary gauss map @xmath7 might be multi - valued on @xmath5 , so we can have a nontrivial monodromy representation defined on the first fundamental group of @xmath5 . this monodromy group is a subgroup of @xmath9 , and if all members of this group can be diagonalized by the same conjugation , we say that the surface @xmath8 is _ reducible_. otherwise , we say that @xmath8 is _ irreducible_. ( irreducibility depends on a global behavior of the surface but not on individual ends . ) if a -@xmath0 immersion is reducible , the surface can be deformed preserving its hyperbolic gauss map @xmath6 and hopf differential ( @xmath10-part of the second fundamental form , see section [ sec : prelim ] ) . on the other hand , an irreducible surface is the only surface with given hyperbolic gauss map and hopf differential . recent progress in the theory of -@xmath0 surfaces in @xmath2 has led to the discovery of many new examples of these surfaces . many examples with regular ends are now known , and various properties of these surfaces are understood . bryant @xcite found a local description for these surfaces in terms of holomorphic data that initiated this recent progress . the last two authors @xcite@xcite developed the theory using bryant s description to find many examples and properties , and work in this direction has been continued by small @xcite , the authors @xcite@xcite , costa - sousa neto @xcite , earp - toubiana @xcite@xcite , yu @xcite@xcite , levi - rossman @xcite , barbosa - berard @xcite , do carmo and gomes and lawson and thorbergsson and silveira @xcite , and others . regarding properties of the ends of embedded examples , collin , hauswirth and rosenberg @xcite have recently shown that any embedded -@xmath0 surface of finite total curvature is either a horosphere or all of its ends are asymptotic to catenoid cousin ends . in @xcite it is further shown that any irregular end can not be embedded , and the limit points of such an end are dense at infinity . recently , pacard and pimentel @xcite established a method for attaching small handles between tangent horospheres and deforming to produce -@xmath0 surfaces , and this construction produces many embedded -@xmath0 surfaces of any genus . also , karcher @xcite has recently constructed periodic -@xmath0 surfaces with fundamental domains in several different types of compact quotients of @xmath2 . a typical example of an irregular end is the end of the enneper cousin , a surface first constructed by bryant @xcite . after that the last two authors @xcite constructed examples of genus zero and two irregular ends , and also many reducible -@xmath0 surfaces of genus zero whose ends are all irregular , using deformations from minimal surfaces . ( the conclusion of remark 4.4 in @xcite contains an error . the number of ends should be @xmath11 , and hence the genus of @xmath12 is zero . ) recently , daniel @xcite has investigated irregular ends from the viewpoint of nevanlinna theory . after @xcite , no further surfaces with irregular ends and finite total curvature had been constructed . ( however , such an example with infinite total curvature can be found in @xcite . ) in particular , until now no irreducible -@xmath0 surfaces with irregular ends and finite total curvature had been known . the purpose of the paper is to construct countably many @xmath0-parameter families of genus zero -@xmath0 surfaces with irregular ends and finite total curvature , which have either dihedral or platonic symmetries . we further show that examples with dihedral symmetries , and the simplest example with tetrahedral symmetry , must be irreducible . all of our examples have irregular ends of finite type in the sense of daniel @xcite . to do the construction , we start with the meromorphic data for the genus zero irreducible -@xmath0 surfaces with regular ends found in @xcite and @xcite and modify this data to make surfaces with irregular ends . the spirit of the construction is similar to the construction of trinoids in @xcite , where -@xmath0 surfaces with prescribed gauss maps are constructed by reflecting spherical triangles , and we use monodromy killing arguments like in @xcite and @xcite , but the techniques are brought to bear more intricately here . in section [ sec : prelim ] we give necessary preliminaries . as our construction is done by reflecting abstract spherical triangles , we discuss this in section [ sec : reflection ] , and introduce a method to construct -@xmath0 surfaces with irregular ends ( theorem [ thm : gen ] ) , which is proved in section [ sec : proof ] . as an application of the theorem , we construct examples of genus zero with either dihedral or platonic symmetries in section [ sec : platonic ] . finally , in section [ sec : torus ] , we construct a -@xmath0 surface of genus @xmath0 with four irregular ends , which is the first known example with positive genus whose ends are all irregular .

the purpose of this paper is to construct countably many-parameter families of genus zero - surfaces with irregular ends and finite total curvature , which have either dihedral or platonic symmetries . for all the examples we produce , we show that they have finite total curvature and irregular ends . for the examples with dihedral symmetry and the simplest example with tetrahedral symmetry , we show irreducibility . moreover , we construct a genus one - surface with four irregular ends , which is the first known example with positive genus whose ends are all irregular .

with the developments of the last decade on complete constant mean curvature ( - ) surfaces in the hyperbolic-space , many examples of such surfaces are now known . however , most of the known examples have regular ends . ( an end is irregular , resp . regular , if the hyperbolic gauss map of the surface has an essential singularity , resp . at most a pole , there . ) there are some known surfaces with irregular ends , but they are all either reducible or of infinite total curvature . ( the surface is reducible if and only if the monodromy of the secondary gauss map can be simultaneously diagonalized . ) up to now there have been no known complete irreducible - surfaces in with finite total curvature and irregular ends . the purpose of this paper is to construct countably many-parameter families of genus zero - surfaces with irregular ends and finite total curvature , which have either dihedral or platonic symmetries . for all the examples we produce , we show that they have finite total curvature and irregular ends . for the examples with dihedral symmetry and the simplest example with tetrahedral symmetry , we show irreducibility . moreover , we construct a genus one - surface with four irregular ends , which is the first known example with positive genus whose ends are all irregular .

quant-ph0303152

c

within the frame of macroscopic qed , we have derived an expression for the vdw potential of an atomic system near an arbitrary configuration of dispersing and absorbing bodies . it generalizes the results obtained by means of normal - mode expansion and may be regarded as a foundation of the results inferred from linear response theory . we have performed the calculations for both the minimal - coupling scheme and the multipolar - coupling scheme and shown that the results are essentially the same . we have applied the theory to the vdw interaction between an atom and a sphere . from the integral expression , we have derived the correct long - distance law corresponding to the retardation limit and recovered the short - distance law corresponding to the non - retardation limit . in particular , replacing in the long - distance law the polarizability of the sphere with that of an atom just yields the vdw potential between two atoms . on the other hand , for sufficiently small distances of the atom from the sphere the vdw potential approaches the potential observed for an atom near a planar body . it is worth noting that in the long - distance limit it is the static permittivity that enters the vdw potential . hence material absorption has no effect on it . however , with decreasing distance of the atom from the sphere the relevant frequencies extend for a finite ( increasing ) interval and material absorption becomes substantial , thereby diminishing the strength of the force . in this article , we have restricted our attention to ground - state systems and calculated the vdw potential in lowest - order of perturbation theory with respect to the interaction of the atomic system with the medium - assisted electromagnetic field . the theory allows of course extensions in several respects . as a consequence of the lowest - order perturbation theory , the energy denominators that enter the polarizability of the atomic system are the unperturbed ones , without consideration of the level shift and broadening caused by the presence of the bodies . in fact , the polarizability of an atomic system is expected to drastically change when it becomes close to a macroscopic body and the spontaneous decay thus becomes purely radiationless , with the decay rate being proportional to @xmath210 @xcite . since the level broadening is essentially determined by the spontaneous - decay rate , the polarizability becomes distance - dependent an effect that needs careful consideration . since the electromagnetic field in ( linear ) magnetic media can be quantized analogously @xcite , another interesting extension of the theory be the inclusion in it of composite materials characterized by both a complex permittivity and a complex permeability . interestingly , such materials , which have been fabricated recently , are left - handed . last not least the underlying quantization scheme renders it also possible to extend the theory to atoms and molecules in excited states and treat the motion of driven atomic systems . we would like to thank ludwig knll for valuable discussions . this work was supported by the deutsche forschungsgemeinschaft .

it is shown that the minimal - coupling scheme and the multipolar - coupling scheme lead to essentially the same formula for the van der waals potential . as an application , closed expressions for the long - distance ( retardation ) and short - distance ( non - retardation ) limits are given , and the effect of material absorption is discussed .

within the frame of macroscopic quantum electrodynamics in causal media , the van der waals interaction between an atomic system and an arbitrary arrangement of dispersing and absorbing dielectric bodies including metals is studied . it is shown that the minimal - coupling scheme and the multipolar - coupling scheme lead to essentially the same formula for the van der waals potential . as an application , the vdw potential of an atom in the presence of a sphere is derived . closed expressions for the long - distance ( retardation ) and short - distance ( non - retardation ) limits are given , and the effect of material absorption is discussed .

1003.1615

r

the elastic " process for the @xmath73 reaction is shown in fig . the @xmath1 gets absorbed on a pair of protons in the target nucleus and produces a @xmath2-@xmath3 pair . this pair is detected in coincidence in the final state . no measurements are made on the recoiling nucleus , hence the measurements are inclusive in that sense . the recoiling nucleus is left in a _ two hole _ state centred around the excitation corresponding to the summed binding energy ( @xmath74 ) of nucleons 1 and 2 in the nucleus a. let us denote different states of b around this excitation by @xmath13 . the inclusive absorption probability for protons in shells ( @xmath75 ; @xmath76 ) in the nucleus is then given by @xmath77 where @xmath78 and @xmath79 is the excitation energy of the state @xmath13 in @xmath80 . @xmath81 denotes the kinetic energy of the particle @xmath82 . bar on the sum in the above expression denotes the average and sum over the spins in the initial and final states respectively . the transition matrix element @xmath83 is given by @xmath84^{1/2}\,\int d\xi\,\,dx_1\,\,dx_2\,\ , \psi_{b\,,\,n}^*(\xi)\,\,\chi^{-*}(x_1,x_2)\,\,h_{12}^{k^-}\,\ , \psi_a(\xi , x_1,x_2)\,\,\phi_k,\ ] ] where @xmath85 are the active number of absorbing proton pairs in shell ( @xmath75 ; @xmath76 ) in the target nucleus , @xmath86 . @xmath87 is the nuclear wave function , and @xmath88 is the elastically scattered @xmath2 and @xmath3 wave function . @xmath89 is the absorption vertex , and it depends only on the proton coordinates , @xmath90 and @xmath91 in the nucleus a. @xmath92 represents collectively the coordinates of the @xmath93 nucleons . to proceed further we note that since the excited states ` @xmath13 ' in the nucleus @xmath80 are the hole states corresponding to two nucleons , they are not likely to have much energy spread . hence , in the energy delta function in eq . ( 8) we replace @xmath79 by @xmath94 . with this we obtain @xmath95\,\sum_n \bar{\sum _ \sigma}\,|m_{fi}|^2,\ ] ] with @xmath96 $ ] , the phase space factor , given by @xmath97\,=\,\frac{1}{(2\pi ) ^5}\,\,\delta(m_i\,-\,t_\lambda\,-\,t_p\ , -\,t_b\,-\,b_{12})\,\,d\vec{p_\lambda}\,\,d\vec { p_p},\ ] ] and @xmath98 . sum over ` @xmath13 ' is now performed using the closure relation " , yielding @xmath99\,\,\bar{\sum _ \sigma}\,\int d\xi\,\,\left|\,\int dx_1\,\,dx_2\,\,\chi^{-*}(x_1,x_2)\,\,h_{12}^{k^-}\,\ , \psi_a(\xi , x_1,x_2)\,\,\phi_k\,\right|^2.\end{aligned}\ ] ] since we are interested only in the inclusive absorption strength we take a simple description of the target nucleus , where the absorbing protons move in shell model orbitals @xmath100 and @xmath101 and the core of @xmath93 nucleons is a spectator . with this description , above expression reduces to @xmath102\,\,\bar{\sum _ \sigma}\,\,\left|\ , \int dx_1\,\,dx_2\,\,\chi^{-*}(x_1,x_2)\,\,h_{12}^{k^-}\,\ , \psi_{n_1l_1m_1;n_2l_2m_2}(x_1,x_2)\,\,\phi _ k\,\right|^2,\ ] ] where @xmath103 is the properly anti - symmetrized two proton wave function in the nucleus . in the @xmath104 representation it is written as @xmath105 for two protons in the same shell antisymmetry requires that @xmath106 . further on , performing sum over @xmath107 and @xmath108 contained in @xmath109 in eq . ( 13 ) , we get @xmath110 the correlation functions @xmath54 and @xmath55 mentioned in eq . ( 3 ) and after it are absorbed here in the absorption vertex @xmath111 . to proceed further , let us now utilize the fact that the momentum @xmath113 appearing in the absorption vertex @xmath111 has large magnitude and a short range . because of this , in the expression for @xmath114 we can factorize the expectation value of @xmath111 from the rest , and write , @xmath115\,\,\times \\ \nonumber & & \left[\,\frac{n_{l_1\,l_2}}{(2l_1 + 1)(2l_2 + 1)}\,\,\sum_{lm}\ , \left|\,\int d\vec{r_1}\,\,d\vec{r_2}\,\,\chi ^{- * } ( \vec{r_1},\vec{r_2})\,\,\phi_k(\vec{r_1})\,\,\phi_{lm } ( \vec{r_1},\vec{r_2})\,\,\delta(\vec{r_1}-\vec { r_2})\,\right|^2 \,\right].\end{aligned}\ ] ] two expressions in the square brackets in above can be identified with two terms of eq . ( 4 ) , i.e. @xmath116,\ ] ] and @xmath117\,\,n_{l_1\,l_2}\frac { 1}{(2l_1 + 1)(2l_2 + 1)}\,\times \\ & & \nonumber \sum_{lm}\,\,\left|\,\int d\vec{r_1}\,\,d\vec{r_2}\,\ , \chi^{-*}(\vec{r_1},\vec{r_2})\,\,\phi_k(\vec{r_1})\,\ , \phi_{lm}(\vec{r_1},\vec{r_2})\,\,\delta(\vec{r_1}-\vec{r_2})\ , \right|^2.\end{aligned}\ ] ] as the energies of the proton and the lambda following @xmath1 absorption is around 160 mev or so , we describe the scattering of these particles by the recoiling nucleus using eikonal approximation . the basic assumption in this description is that the propagating particle is mainly scattered in the forward direction . taking z - axis parallel to the proton momentum , @xmath119 , the proton distorted wave , @xmath120 is written in eikonal approximation as @xmath121 where the distortion function @xmath122 is given in terms of an optical potential , @xmath123 by @xmath124,\ ] ] where @xmath125 . for writing the @xmath3 distorted wave we recall that the @xmath3 moves opposite to the proton . therefore , the momentum vector @xmath126 is anti - parallel to the chosen z - axis . the distortion factor , @xmath127 for @xmath3 therefore becomes @xcite @xmath128.\ ] ] combining @xmath129 for the proton and the lambda we then get @xmath130.\ ] ] if we make the @xmath131 approximation for @xmath132 and assume forward scattering for @xmath133 , we get @xmath134,\ ] ] where @xmath135 and @xmath136 are respectively the total cross section and the ratio of the real to imaginary part of the scattering amplitude for the @xmath137 system . now , if we ignore the difference between the proton and the lambda elementary scattering parameters and take them as that for the better studied @xmath138 system at some mean value of the @xmath2 and @xmath3 energies , above expression simplifies to @xmath139 \\ & = & exp\,\left[\,\frac{i}{2}\,\,\sigma_t^{pn}(i+\beta _ { pn})\,\ , t(\vec{b})\,\right],\end{aligned}\ ] ] where @xmath140 is the nuclear density and @xmath141 is the total nuclear material seen by the proton and the lambda _ together _ at an impact parameter , @xmath142 . it is given by @xmath143 with @xmath144 . now if we observe eq . ( 24 ) for @xmath145 a little closely , we realize that this , in fact , is the total distortion factor for the passage of a proton from one end of the nucleus to another . this , thus , is the mathematical description for the statement made in an earlier section that , because of the peripheral localization of the @xmath1 absorption and the back - to - back emission of @xmath2 and @xmath3 , total scattering of the @xmath2 and the @xmath3 can be included in the final state by considering the passage of only one particle ( @xmath2 or @xmath3 ) through the whole nucleus . furthermore , since the effect of distortion at the energies of @xmath2 and @xmath3 in the studies here ( @xmath33 160 mev ) is mainly absorptive , and most of the measurements on @xmath146 following @xmath1absorption are of inclusive type , it will be reasonably correct to include the overall effect of the distortion in @xmath65 by multiplying it by an attenuation factor , @xmath147 , given by @xmath148 and removing the distortion factor from the integral in eq . factor @xmath149 in above has been introduced before @xmath150 to include the effect of pauli - blocking of the nucleons in the nucleus after scattering . a nuclear matter estimate for its value above twice the fermi energy is given by @xcite @xmath151 where @xmath152 , with @xmath153 denoting the fermi energy . with above treatment of distortion , eq . ( 18 ) for @xmath65 simplifies to @xmath154\,\,|\eta_a(t_p)|^2\,\,\sum_{lm}|\,f_{lm}^{l_1l_2}(q)\ , \nonumber \sum_{lm}|\,f_{lm}^{l_1\,l_2}(q)\,|^2&=&\sum_{lm}\left|\ , \sqrt{\frac{n_{l_1\,l_2}}{(2l_1 + 1)(2l_2 + 1)}}\,\,\int d\vec{r_1}\ , \,e^{-i\,\vec{q}\,\cdot\,\vec{r_1}}\,\,\phi_k(\vec{r_1})\,\ , \phi_{lm}(\vec{r_1},\vec{r_1})\,\right|^2 \\ & = & n_{l_1\,l_2}\,\,\sum_l ( l_1\,l_200\,/\,l\,0)^2\,\,|\ , g_{l_1\,l_2}^l(q)\,|^2,\end{aligned}\ ] ] with @xmath155 where @xmath156 are the nucleon radial wave functions in the nucleus . ( 26 ) ) for @xmath157li , @xmath4c and @xmath158ca nuclei.,width=415,height=340 ] the value of @xmath147 depends upon the proton energy , @xmath159 through @xmath160 and upon the nucleus through the thickness function , @xmath161 . to get an idea about the value and its variation , in fig . 3 we plot @xmath147 in the proton kinetic energy range 50 - 250 mev for three nuclei , @xmath157li , @xmath4c and @xmath158ca . for calculating the pauli blocking factor , @xmath149 , fermi momentum is taken equal to 200 mev / c . the nuclear densities are taken from ref . we find that in the energy range ( 100 - 200 mev ) of interest in the present calculations the attenuation factor approximately remains constant . the values of this factor for the three nuclei are around 0.9 , 0.7 and 0.4 respectively . prescription to describe the absorption vertex , @xmath162 is not clear and also not simple . one thing , which is definite about it is that , it involves large momentum transfer , hence , spatially it can not be localized over any extended volume . dynamically , the one - nucleon absorption mechanism , @xmath163 is understood to involve the @xmath164 , which decays in to a pion and a hyperon . the range of this vertex is determined by the @xmath164 propagator . a study in ref . @xcite suggests that the absorption probability depends upon this range , and in their estimate this range could be around 0.75 fm or so . the two nucleon mechanism , due to strong attractive @xmath1-@xmath2 interaction in the t = 0 state , involves dynamically a strongly correlated system of @xmath165 , where the @xmath1 is continuously exchanged between two protons . detailed dynamical composition of this system is determined in the @xmath166pts by the non - perturbative coupling amongst various s = @xmath201 , t = 0 channels , @xmath21 , @xmath22 , @xmath167 , @xmath24 , @xmath9 , @xmath25 . this system eventually decays in to @xmath146 . exact mechanism of this decay is not immediately obvious . however , in line with the one - nucleon @xmath1 absorption , one mechanism could be that at some stage in the @xmath165 system a @xmath164 is produced , which , as suggested in ref . @xcite , interacts with another proton through an exchange of pions or a pair of @xmath16-@xmath168 and goes over to the @xmath2-@xmath3 final state . in ref . @xcite , using the range parameter 0.2- 1.2 fm for the absorption vertex the authors estimate the decay width of 2 - 8 mev for the @xmath165 system , with the maximum width occurring for the range around 0.7 fm . thus , it appears that , even if the details of the absorption vertex is not known very clearly , two things are clear : ( i ) the magnitude of the two proton @xmath1 capture depends upon the spatial extension of the absorption vertex , and ( ii ) the probable range of the vertex is such that the variation of the capture probability with momentum , @xmath64 could not be very rapid . the ( @xmath1 , @xmath0 ) reaction measurements have been done on the distributions of the @xmath169 ( and proton ) momentum , invariant @xmath0 mass , and their angular correlations . in all these distributions , as we will discuss in the next section , each point involves a folding of @xmath66 and @xmath65 . however , the values of @xmath64 for all the measurements are large ( around 500 mev / c ) and do not have much spread ( only up to 10@xmath43 ) . because of this , the factor @xmath162 ( eq . ( 17 ) ) in the formalism enters in determining the absolute magnitude of the ( @xmath1 , @xmath0 ) process only . the shapes of different distributions are determined by the factor @xmath65 . furthermore , as the available data from the experiments exist only in arbitrary units , we have taken @xmath162 as a constant factor , denoted by @xmath170 , in our calculations . before we present the results let us make some points about the finuda experiment . the @xmath3 s are detected in this experiment by reconstructing the invariant mass of the @xmath3 decay products , @xmath2 and @xmath171 . however , the restriction on the low momentum threshold for @xmath171 in the finuda spectrometer is such that it cuts out the @xmath169 hyperons with a momentum lower than 300 mev / c . therefore , the @xmath169 from the quasifree process ( @xmath172 ) in this experiment is hardly observed . above around 400 mev / c , since the main contribution comes from two - nucleon absorptions ( @xmath173 ) , the finuda measurements have major contribution from this capture process . furthermore , the measurements are done with the @xmath2 and @xmath169 in coincidence , with the cosine of the angle between them restricted as @xmath174 cos @xmath175 . the latter constraint is put because the cross section beyond these limits is insignificant ( see fig . 4 ) . for calculations , putting all factors together from the above section , the differential absorption strength , @xmath176 for @xmath1-absorption on two protons in shell model orbitals , ( @xmath75 ; @xmath76 ) is finally written as @xmath95\,c\,|\eta_a(t_p)\,|^2\,n_{l_1\,l_2 } \,\,\sum_l(l_1\,l_2\,0\,0\,/\,l\,0)^2\,\,|\,g_{l_1\,l_2}^l(q)\,|^2,\ ] ] where all the terms are as defined in above sections . we present here calculated results using this expression for the @xmath4c target nucleus with an appropriate phase space factor , @xmath96 $ ] . the proton wave functions in @xmath4c are generated in an oscillator potential , whose length parameter , @xmath177 is taken equal to 1.67 fm . this parameter fits the elastic electron scattering data on the @xmath4c nucleus @xcite . the binding energies of protons in @xmath69 and @xmath44 shells are taken as given by the ( @xmath2 , @xmath40 ) and ( @xmath7 , @xmath178 ) reactions @xcite . for @xmath4c they are 15.96 and 34.0 mev respectively . the @xmath1 absorption is considered to occur on proton pairs in @xmath69 , @xmath44 and @xmath179 shells . the value of @xmath85 for these absorptions is taken equal to the number of possible proton pairs in these shells , as an upper limit . they are in the ratio of @xmath180 . the phase space factor , @xmath181 for calculating the angular correlation between the @xmath2 and @xmath169 and their momentum distribution is written as latexmath:[\[ps\,=\,\frac{4\,\pi\,m_p\,m_{\lambda}}{(2\pi)^5}\,\ , \frac{|\vec{p_p}|^2|\,\vec{p_\lambda}|^2}{(m_{\lambda}\,+\,m_b)\ , ( \theta_{\lambda\,p})}\,d\,cos\,(\theta_{\lambda\,p})\,d\ , in fig . 4 we present the calculated angular correlation between the @xmath3 and the proton along with the finuda measurements . we find them to agree very well with each other . the steep rise in the distribution towards @xmath183 = 180@xmath184 is the strong confirmation of the two - proton absorption mechanism . both the results are given in relative units . and @xmath2 along with the finuda measurements @xcite.,width=415,height=340 ] measurements have also been done in the finuda experiment on the momentum distributions of the proton and @xmath3 . the measured @xmath3 distribution from ref . @xcite is shown in fig . 5 . if one believes that the observed @xmath2 and @xmath3 come from the two - proton absorption vertex and do not undergo any further fsi except the elastic scattering , this distribution should be similar to the one calculated using eq . we show the calculated @xmath3 and proton momentum distributions in fig . 5 , and find that the peak position of the @xmath3 distribution nearly agrees with the corresponding measured distribution . the shape of the calculated distributions is however found to be less broad compared to the observed one . larger magnitude of measured events below 400 mev / c , which makes it broader can not be understood easily . they can not be attributed to the quasifree process either because the finuda spectrometer cuts off @xmath169 s below 300 mev / c . another source of this deficiency can be that , in our calculations we have taken @xmath162 ( denoted by @xmath170 in eq . ( 31 ) ) as constant . we examined it . as we see in the phase space expression ( eq . ( 32 ) ) , calculation of the absorption probability for each value of the @xmath185 involves an integral over @xmath186 , which in the finuda measurements lies between @xmath201.0 and @xmath200.8 . this implies in the calculation for each @xmath187 an integral over a certain range of @xmath64 and @xmath63 corresponding to this range of @xmath146 angle . for 400 , 500 and 600 mev / c values of @xmath187 ( which more or less covers fig . 5 ) this range of @xmath64 is about 510 - 495 , 520 - 505 and 515 - 500 mev / c respectively . these values , as we see are large and have about same range for all the @xmath3 momenta . therefore our assumption about the constancy of @xmath162 in the calculations should not be the cause of concern . the shape of the @xmath187 distribution is , in fact , mainly determined by @xmath65 through the @xmath63 dependence of the nuclear wave functions . the range of @xmath63 for the above three @xmath188 s in the @xmath189 are 200 - 260 , 35 - 220 , and 160 - 270 mev / c respectively . this range shows why the @xmath187 momentum distribution peaks around 500 mev / c . it also suggests that it can probably be made broader by enriching the nucleon shell model wave functions in high momentum components . however , to reproduce the width of the observed momentum distribution , we believe that it will require a considerable modification of the nuclear wave function . and @xmath2 inclusive momentum distributions with @xmath174 cos @xmath190 along with the measured @xmath169 momentum distribution ( represented by histogram).,width=415,height=340 ] next we calculate the @xmath2-@xmath169 invariant mass distribution . the phase space for this is written as @xmath191 where @xmath192 denotes the reduced mass of the @xmath193 system . the magnitudes of @xmath194 and @xmath113 are determined for a given invariant mass @xmath195 through @xmath196 and @xmath197 the angle @xmath198 between @xmath194 and @xmath113 is constrained such that @xmath199 . calculated invariant mass distribution along with the corresponding measured finuda distribution @xcite are shown in fig . we observe that , compared to the mass of @xmath200 in the free state ( 2370 mev ) the calculated distribution is down - shifted in mass by about 50 mev due to proton binding in @xmath4c . the measured mass distribution , however , is still below this by an additional 70 mev or so . this , incidentally , is around the reported calculated binding energy in the literature of the @xmath5 " module taking @xmath164 mass 27 mev below @xmath1-@xmath17 threshold . the range of the values of @xmath64 and @xmath63 for @xmath201 mev , ( which covers the calculated distribution ) are 520 - 490 and 27 - 320 mev / c respectively . this says that the values of @xmath64 are large and do not vary much , hence @xmath66 is not likely to influence the shape of the mass distribution . its shape would be mainly determined by the nuclear wave function through @xmath65 and the factor @xmath63 in the phase space . and @xmath2 along with finuda measurements . @xmath174 cos @xmath175.,width=453,height=377 ] if we include in the kinematics of our calculations an additional binding energy of 70 mev in the initial system , then , we find that the peak position in the calculated mass distribution ( not plotted here ) , of course , comes near to the peak position of the measured distribution , but the shape of the calculated distribution turns out much sharper than the experimental one . the calculated @xmath3 momentum distribution with the additional binding also gets shifted towards lower momenta . this then spoils the agreement of the calculated results with the experiments shown above in fig . 5 without any @xmath200 " binding . the angular correlation between @xmath3 and proton , however , remains unchanged . in the two - nucleon @xmath1 absorption model , with only elastic scattering of @xmath2 and @xmath3 included in the final state , the measured @xmath3-@xmath2 angular correlation distribution is reproduced well , and the momentum spectrum of @xmath3 s to a reasonable extent without introducing any additional binding of the @xmath5 " module . the calculated invariant @xmath3-@xmath2 mass distribution , however , peaks around 70 mev higher than the measured one . attributing this shift to the additional binding of the @xmath5 " module , the calculations including this binding in kinematics , obviously , reproduce the peak position of the measured mass distribution , but the shape of the distribution remains very narrow . it also spoils the agreement with the @xmath3 momentum distribution achieved without including any @xmath200 " binding . thus , in an overall conclusion it seems that the observed invariant mass distribution in the ( @xmath1 , @xmath0 ) reaction does not correspond to the peak predicted in this section with only elastic scattering included in the fsi . the knock - out scattering means that the proton or @xmath169 , emanating from vertex i , encounters a hard collision with a nucleon in the residual nucleus and knocks it out ( fig . this collision alters significantly the momentum distribution of the striking particle . to calculate the knock - out contribution we recollect that , as discussed in the earlier section , @xmath2 and @xmath169 move back to back at vertex i and the vertex itself is localized on the surface of the target nucleus . because of this only one particle goes into the nucleus , the other one moves out . for such trajectories , as discussed earlier in section a(2 ) for distorted waves in elastic scattering , here too the contribution of @xmath2 and @xmath169 to ko fsi can be included by considering the passage of only one of the two particles , but through the whole nucleus . this can be reasonably correct if the elastic scattering parameters for @xmath138 and @xmath3-@xmath17 systems at intermediate energies are not much different . considering that it is so , in the following we calculate the ko contribution induced by the proton . let @xmath202 describe the momentum distribution of the proton when it incidents on the vertex ii , where @xmath203 is the momentum of the proton at the time of leaving the vertex i. the altered momentum of this proton after the ko scattering is denoted by @xmath119 . the proton with this momentum along with the @xmath3 are detected in the experiment . the knocked - out nucleon from the nucleus is not seen . the probability for this inclusive " process is determined by the product of @xmath202 and the proton induced inclusive single nucleon knock - out cross section as @xmath204 where @xmath205 is the total proton - nucleus cross section at the momentum @xmath206 . momentum @xmath207 is given in terms of the variables at the first vertex . in terms of @xmath126 it is given by @xmath208^{1/2}\right],\ ] ] + where all the notations are self explanatory . @xmath81 represents the kinetic energy of the particle @xmath82 . function @xmath210 is obtained from eq . ( 31 ) , by first writing the phase space @xmath96 $ ] for @xmath126 as , @xmath97\,=\,\frac{m_\lambda\,m_b}{(2\pi)^5}\,\,\frac{|\vec{p_p^\prime}| \vec { p_p^\prime}|m_\lambda cos\,(\theta_{\lambda p^\prime})}\,\ , d\vec { p_\lambda},\ ] ] and then substituting this phase space expression in eq . ( 31 ) , and identifying @xmath202 with ( @xmath211 ) . this gives @xmath212 \\ & & \times\,\,\left[c\,|\eta_a(t_p)|^2\,n_{l_1\,l_2}\ , \sum_l(l_1l_200/l0)^2 |g_{l_1\,l_2}^l(p_b)|^2\right]\end{aligned}\ ] ] however , before we proceed further let us mention that eq . ( 36 ) for the knock - out contribution , which has a factorization of the vertex i and ii , holds under a certain approximation . more correctly , as shown in diagram 2(b ) these two vertices should have been correlated in space through the proton ( or lambda ) propagator between them . this propagator would be a proton scattered wave between the two vertices with an outgoing boundary condition . that is , we would have between two vertices a factor like @xmath213 where @xmath214 collectively represents all other factors . @xmath215 represents the proton interaction with the medium . this integral has two parts , one originating from the principal value and another from the energy conserving @xmath216-function part of the propagator . physically these two parts represent the off - shell and the on - shell scattering in the intermediate state . the on - shell part can be shown to be roughly proportional to the proton momentum , hence dominates at higher energies . the off - shell part dominates at lower energies ( see for example @xcite ) . in our case , since the energies of the proton ( or lambda ) are in the intermediate energy range , we have restricted ourselves to the energy conserving on - shell contribution only . this , essentially is the assumption implicit in writing eq . ( 36 ) with the proton being described at both vertices by distorted waves . proton induced single nucleon knock - out reaction at intermediate energies is a well studied subject experimentally as well as theoretically @xcite . using the notations given in diagram ( 2b ) , the expression for the ( @xmath2 , @xmath2@xmath17 ) knock - out reaction is given by @xmath217 where @xmath218 denotes the average and sum over the spins in the initial and final states respectively . @xmath219 is the momentum of the recoiling nucleus . @xmath220 and @xmath79 denote its intrinsic excitation and the excitation energy respectively . to evaluate this expression , first we integrate over @xmath221 and utilize the momentum conserving delta function , giving @xmath222 with @xmath223 , where @xmath224 is the momentum transfer from the incident proton . the transition matrix @xmath225 in above describes the t - matrix for the knock - out of a nucleon from the nucleus b and leaving the residual nucleus in a one - hole excited state denoted by ` @xmath13 ' . it is given by @xmath226 where @xmath80 and @xmath220 denote the nuclear wave functions in the initial and final states respectively . @xmath227 is the @xmath17-@xmath17 scattering amplitude . this amplitude is half off - shell if the distortion of the continuum nucleons is ignored and becomes fully off - shell if the distortions are included . however , at the energies of concern to us , the off - shell effects are known not to be significant . hence , normally the @xmath17-@xmath17 t - matrix here is taken on - shell and the energy , @xmath228 at which it is evaluated is taken corresponding to the incident momentum @xmath203 . the @xmath229 itself is related to the @xmath17-@xmath17 cross section in the centre of mass through @xmath230 where @xmath231 is the @xmath17-@xmath17 momentum in its centre of mass , and @xmath232 is the nucleon mass . the sum over ` @xmath13 ' in eq . ( 42 ) means the sum over the excited states in the recoiling nucleus consistent with the momentum conservation . since the reaction mechanism involves only a nucleon in the nucleus b , ` @xmath13 ' would have only a small spread . therefore , in the energy delta function in eq . ( 42 ) we can replace , to a reasonable approximation , @xmath79 by the binding energy , @xmath233 of the knocked - out nucleon in b. this simplifies the summation over @xmath13 in eq . ( 42 ) . using closure , we can then write @xmath234 where @xmath92 collectively denotes coordinates of all the spectator core nucleons in the nucleus @xmath80 . to proceed further we now use a simple description of the nucleus . we write the target nucleus b wave function as a product of a single nucleon ( ` n ' ) wave function in a shell with quantum numbers @xmath235 " and the core nucleus wave function @xmath236 . with this eq . ( 45 ) reduces to @xmath237 with @xmath238 where @xmath239 is the nucleon spectroscopic factor . in the present simplified description of the nuclear wave function , it is equal to the number of nucleons in the shell @xmath235 " . for the bound nucleon we use the momentum space representation , @xmath240 , where @xmath241 from the momentum conservation at the @xmath242 vertex and that following eq . ( 42 ) equals to , @xmath243 . with this identification eq . ( 46 ) then factorizes as @xmath244 \\ & & \times\,\,\left[\,\frac{1}{4}\,\,\frac{s(l)}{2l+1}\,\,\sum_{m_l } \,|\,\phi_{nlm_l}(-\,\vec{q_r})\,|^2\,\right].\end{aligned}\ ] ] the single nucleon knock - out cross section expression ( eq . ( 42 ) ) subsequently becomes @xmath245 \\ & & \times\,\,\left[\,\frac{1}{4}\,\,\frac{s(l)}{2l+1}\,\,\sum_{m_l } \,|\,\phi_{nlm_l}(-\vec{q_r})\,|^2\,\right].\end{aligned}\ ] ] to obtain the expression for the inclusive cross section we still need to integrate this expression over @xmath246 . following ref . @xcite we use the energy delta function to remove angle integration , and with some algebraic manipulations obtain @xmath247 \\ & & \hspace{4cm}=\,s(l)\,\,\frac{m}{2q}\,\ , \int_{q_r^{min}}^\infty q_r\,\,dq_r\,\,|\,\phi _ { nl}(q_r)\,|^2.\end{aligned}\ ] ] substituting this in eq . ( 49 ) above , and also writing @xmath248 in terms of the elementary @xmath17-@xmath17 differential cross section ( eq . ( 44 ) ) we get @xmath249 \,\,\left[\,\frac{d\sigma}{d\omega}(\epsilon)\,\right]_{cm}^{pn}\,\ , \left[\,s(l)\,\int_{q^{min}_r}^\infty q_r\,\,dq_r\,\,\left|\ , \frac{1}{(2\pi)^{3/2}}\,\,\phi _ { nl}(q_r)\,\right|^2\,\right].\ ] ] here @xmath250 is that minimum momentum which a nucleon of binding energy @xmath233 must have in the nucleus for the scattered proton to be observed at a scattering angle @xmath251 with a momentum @xmath252 . its value is given by @xmath253}{[\,p_{p^\prime}^2\,-\,p_p^\prime\,p_p\,cos\,(\theta)\,+\ , m\,b_n\,]^{1/2}}\,\,.\ ] ] eq . ( 51 ) for the knock - out cross section assumes that the incoming proton and the outgoing nucleons do not suffer any additional scattering except the hard knock - out collision . this additional scattering , however , can be incorporated in the formalism by replacing the plane wave description of the continuum nucleons by the distorted waves " , which would be solutions of the wave equation with appropriate optical potentials in it . several studies of the distortion effect in the knock - out reaction have been carried out in the literature and it has been found that in the energy range of nucleons of interest to us here , the effect of distortion is mainly absorptive . the dispersive effect is very small . the absorption factor for @xmath4c ( @xmath2 , @xmath40 ) @xmath254 reaction at 160 mev beam energy , for example , in ref . @xcite is found to be around 0.5 . finally , before closing this section we determine the extent of accuracy to which the expression in eq . ( 51 ) describes the measured inclusive proton induced single nucleon knock - out cross section . we calculate the inclusive cross sections at 160 mev beam energy for @xmath4c and compare them with the measured ( @xmath2,@xmath6 ) ones at the same beam energy @xcite . the calculated results are summed over the knocked - out nucleon ( including both neutron and proton ) from @xmath44 and @xmath69 shells . the single nucleon binding energies for them are taken from ref . the nucleon - nucleon differential cross section in the centre - of - mass required in the calculations are taken from the analytic form given in ref . @xcite , i.e. @xmath255\,(\,1\,+\,0.1\,cos^2\,\theta\,),\ ] ] where the cross section is in mb and energy ( kinetic ) , t in mev . this form is in good agreement with the energy dependence of the observed cross sections for the range 20 @xmath256 t @xmath256 200 mev . the bound nucleon wave function is described by the oscillator potential form with the length parameter , b = 1.67 fm . the spectroscopic factor , @xmath239 is taken equal to the number of nucleons ( neutrons+protons ) in the orbital @xmath235 . c after a single nucleon knock - out scattering from the nucleus along with the experimental points at the same energy @xcite . , width=415,height=340 ] with these inputs and the distortion factor equal to 0.5 ( as discussed above ) , the calculated energy spectrum for the proton going at 30@xmath257 , for example , in the lab . frame along with the experimental cross sections is shown in fig . 7 . as we see , the agreement between them , both in shape and magnitude , is very good . this validates the accuracy of eq . ( 51 ) for the description of the inclusive knock - out channel and gives confidence for its use in calculations of the ( @xmath1 , @xmath0 ) reaction . the final expression for calculating the knock - out contribution to the @xmath1 absorption probability is obtained by substituting eq . ( 51 ) for @xmath258 in eq . we get @xmath259\,\ , f(\vec { p_p^\prime})\,\times\,\frac{1}{\sigma _ t^{pb}(t_{p^\prime } ) } \,\,\left[\,\frac { 2}{qp_p^\prime}\,\right]\,\,\left[\,\frac{d\sigma } { d\omega}(\epsilon)\,\right]_{cm}^{pn}\times \\ & & \left[\,s(l)\,\int_{q^{min}_r}^\infty q_r\,\,dq_r\,\left|\ , \frac{1}{(2\pi ) ^{3/2}}\,\,\phi _ { nl}(q_r)\,\right|^2\,\right],\end{aligned}\ ] ] where @xmath260 is given by eq . ( 39 ) . as in the case of elastic scattering " we present here calculated results for the target nucleus @xmath4c . the results are the probabilities summed over two proton hole states in various pairs of ( @xmath75 ; @xmath76 ) shell model orbitals at vertex i and , for each of these pairs , summed over various one nucleon @xmath235 orbitals in nucleus @xmath80 at the knock - out vertex ii . since in the final state after knock - out we do not detect the knocked out nucleon , we use the spectroscopic factor , @xmath239 summed over both , the neutrons and protons . the radial part of the bound state wave functions , as discussed in the last section , is taken for the oscillator potential . the differential scattering cross section for @xmath2-@xmath17 is described by eq . ( 53 ) with energy taken corresponding to momentum @xmath261 of the proton incident at the vertex ii ( fig . required @xmath262 for proton on nucleus @xmath80 at energies corresponding to different proton momentum , @xmath207 is taken from ref . this cross section , of course , does not vary significantly over the @xmath263 energy range of interest here . the physical effect of knock - out scattering at vertex ii is to reduce the energy of the proton @xmath263 and deflect it from its direction of incidence . the amount of these changes , as can be seen from the experimental results on inclusive ( @xmath2 , @xmath263 ) reaction at 160 mev in ref . @xcite , are about 30 mev and above for the energy reduction and about 30@xmath257 and above for the deflection . immediate consequence of these numbers would be that the angular correlation between @xmath2 and @xmath3 shown in fig . 4 , coming from vertex i , will be widened significantly and the energy spectrum of @xmath2 and @xmath3 , shown in fig . 5 , will be shifted towards lower energies . both these effects will , therefore , spoil the agreement shown in these figures ( figs . 4 - 5 ) between the calculated results from vertex i and the corresponding finuda measurements . to get a quantitative idea about the extent of deflection the vertex ii may introduce in the proton , @xmath263 , in fig . 8 we show its angular distribution relative to the @xmath169 motion after being scattered from the vertex ii . initially the @xmath264 is taken to move at 180@xmath257 w.r.t . @xmath3 with energy as fixed at the vertex i , including the spread due to fermi motion . without the knock - out scattering , this distribution will be just a point at cos(@xmath265)=@xmath201 in this figure . due to scattering this point gets a significant spread , as we see in this figure . calculated results use eq . ( 54 ) and are integrated over the energy spread of @xmath263 . mass distribution from both the vertices along with the experimental points @xcite.,width=415,height=377 ] the consequence of above is expected to be a significant modification in the invariant mass distribution of the @xmath2 and @xmath169 . however , before we show these results we may mention that , because of various sums , integrals and constrain checks on kinematic variables , the calculations are very involved and tedious . therefore , to keep the calculations somewhat simpler and physically transparent we have put some constraints in the calculations without , of course , losing any essence of the physics of the results . we have seen earlier in the calculations on vertex i that the @xmath2 and @xmath169 from it emerge mainly back - to - back with a very small cone angle . we have , therefore , done the vertex ii calculations with @xmath265 = 180@xmath184 only . the energy variation of the @xmath2 and @xmath3 due to fermi motion of the absorbing protons in the nucleus a , however , has been kept intact . with this , in fig . 9 we show the calculated invariant mass distribution of @xmath2-@xmath3 after knock - out scattering along with the finuda measurements @xcite . it is extraordinary to see that the calculated mass distribution totally agrees with the measured ones . they agree in mass shift as well as in the shape of the distribution . this is the same observation as reported in ref . @xcite . however , to get the complete @xmath2-@xmath169 invariant mass distribution we need to add to above the contribution from the elastic scattering " fsi corresponding to vertex i ( fig . therefore , in fig . 9 we also show the @xmath2-@xmath3 invariant mass distribution due to vertex i , and the sum of the elastic " and knock - out " contributions . the summed distribution obviously has two peaks , one corresponding to elastic scattering " fsi and another to the single nucleon knock - out " fsi . this is similar to the inclusive inelastic spectrum normally seen experimentally in ( @xmath2 , @xmath263 ) or ( @xmath7 , @xmath266 ) scattering on any nucleus at intermediate energies " @xcite . unusual thing following @xmath1 absorption seen here is that the peak corresponding to elastic scattering " seems to be missing in the finuda data . this is intriguing . we summarize our observations for the vertex ii as , 1 . the calculated invariant @xmath2-@xmath3 mass distribution totally reproduces the experimentally observed distribution . this is in line with the finding in ref . though we have not shown the full calculations for the inclusive energy spectra for the proton or lambda and angular correlation between them , we believe that due to large deflection and energy shift by the single - nucleon knock - out scattering the agreement seen in figs . 4 - 5 between the observed and calculated angular correlation between the @xmath2 and @xmath169 and the inclusive @xmath2 and @xmath169 energy spectra using only the vertex i will be spoiled considerably . we have not calculated these spectra fully because , due to several integrations over various kinematic variables , they are very long and involved .

absorption is assumed to occur on two protons in the nucleus . in the formalism calculations are presented for thec nucleus , using shell model wave functions , and without any extra mass modification of the system in the nucleus . these results are compared with the corresponding experimental measurements . with only elastic scattering fsi the calculated invariant mass distribution is found to have two peaks , one corresponding to the elastic scattering fsi and another to single nucleon ko fsi .

emission in coincidence following absorption at rest in nuclei is studied using quantum mechanical scattering theory and nuclear wave functions . absorption is assumed to occur on two protons in the nucleus . in the formalism , emphasis is put on the study of the final state interaction ( fsi ) effects of and with the recoiling nucleus . we include elastic scattering and single nucleon knock - out ( ko ) channels in the fsi . calculations are presented for thec nucleus , using shell model wave functions , and without any extra mass modification of the system in the nucleus . calculated results are presented for the angular correlation distribution between and , their invariant mass distribution and the momentum spectra of and . these results are compared with the corresponding experimental measurements . with only elastic scattering fsi included , the angular correlation distribution and the momentum spectra are found to be in good accord with the corresponding measurements . with full fsi the calculated invariant mass distribution is found to have two peaks , one corresponding to the elastic scattering fsi and another to single nucleon ko fsi . the ko peak agrees fully , in position and shape , with the peak observed in ref . . the peak corresponding to elastic scattering fsi does not seem to exist in the measured distribution . considering that such a two peak structure is always seen in the inclusive ( , ) and ( , ) reactions in nuclei at intermediate energies , absence of the elastic scattering peak in the reaction is intriguing .

1003.1615

i

we have calculated the inclusive differential absorption probability for @xmath1 at rest in @xmath4c nucleus for the @xmath0 exit channel . the @xmath1 is assumed to be absorbed on a pair of protons in the nucleus . the final state interaction in the reaction includes the elastic scattering of @xmath2 and @xmath169 in the final state and the single nucleon knock - out from the recoiling nucleus . the calculated invariant @xmath2-@xmath3 mass distribution shows two peaks , one due to elastic scattering and another due to knock - out channel . the latter peak overlaps in position and width with the peak observed in the finuda measurements . the peak corresponding to elastic scattering is not seen in the experiments . measured angular correlation between @xmath2 and @xmath169 and their inclusive energy distribution agree with the corresponding calculated results including only the elastic scattering in the final state . inclusion of the knock - out channel is likely to spoil this distribution . thus , finally , we may conclude that , seen in isolation , the experimentally observed shift in the invariant @xmath2-@xmath3 mass distribution could be interpreted as due to single nucleon knock - out final state interaction . but , if we include other results , like the absence of elastic scattering peak in experiments , full agreement of the calculated @xmath2-@xmath169 angular correlation and their inclusive spectra using only vertex i with the corresponding measurements , the situation becomes quite a bit confusing . as a final comment in the present work on the @xmath200 " cluster interpretation of the observed downshift of about 100 mev in the finuda measurements of the @xmath267 invariant mass compared to its free value , the knock - out reaction in the final state seems to be a definitive alternative for this shift . only discomfiture in this conclusion comes from the absence of the elastic scattering " peak ( fig . 9 ) in the observed invariant mass distribution in the finuda measurements . this absence can not be attributed to the cut off of the @xmath3 hyperons below 300 mev / c momentum in the finuda spectrometer as these momenta in the elastic scattering " peak are above 400 mev / c . availability of absolute measurements may help to understand the ( @xmath1 , @xmath0 ) reaction better .

we include elastic scattering and single nucleon knock - out ( ko ) channels in the fsi . included , the angular correlation distribution and the momentum spectra are found to be in good accord with the corresponding measurements . with full fsi the ko peak agrees fully , in position and shape , with the peak observed in ref . the peak corresponding to elastic scattering fsi does not seem to exist in the measured distribution . considering that such a two peak structure is always seen in the inclusive ( , ) and ( , ) reactions in nuclei at intermediate energies , absence of the elastic scattering peak in the reaction is intriguing .

emission in coincidence following absorption at rest in nuclei is studied using quantum mechanical scattering theory and nuclear wave functions . absorption is assumed to occur on two protons in the nucleus . in the formalism , emphasis is put on the study of the final state interaction ( fsi ) effects of and with the recoiling nucleus . we include elastic scattering and single nucleon knock - out ( ko ) channels in the fsi . calculations are presented for thec nucleus , using shell model wave functions , and without any extra mass modification of the system in the nucleus . calculated results are presented for the angular correlation distribution between and , their invariant mass distribution and the momentum spectra of and . these results are compared with the corresponding experimental measurements . with only elastic scattering fsi included , the angular correlation distribution and the momentum spectra are found to be in good accord with the corresponding measurements . with full fsi the calculated invariant mass distribution is found to have two peaks , one corresponding to the elastic scattering fsi and another to single nucleon ko fsi . the ko peak agrees fully , in position and shape , with the peak observed in ref . . the peak corresponding to elastic scattering fsi does not seem to exist in the measured distribution . considering that such a two peak structure is always seen in the inclusive ( , ) and ( , ) reactions in nuclei at intermediate energies , absence of the elastic scattering peak in the reaction is intriguing .

0806.0045

i

laser - plasma interaction ( lpi ) @xcite is an important plasma - physics problem which poses serious challenges to theoretical modeling . lpi is the basis of several applications , including laser - based particle acceleration @xcite and the backward raman amplifier @xcite . moreover , for inertial confinement fusion ( icf)@xcite to succeed , lpi must not be so active that it prevents the desired laser energy from being delivered to the target , with the desired spatial and temporal behavior . this paper focuses on modeling the backscatter instabilities , where a laser light wave ( mode 0 ) decays into a backscattered light wave ( mode 1 ) and a plasma wave ( mode 2 ) . in stimulated raman scattering ( srs ) and stimulated brillouin scattering ( sbs ) , the plasma wave is , respectively , an electron plasma wave and an ion acoustic wave . these lpi processes pose a serious risk to indirect - drive icf @xcite . a wide array of computational tools is used to model lpi , ranging from rapid ( @xmath0secs ) calculations of linear gains along 1d profiles to massively - parallel kinetic particle - in - cell simulations . we present here a new tool , called deplete , to the less computationally expensive end of this spectrum . deplete solves for the pump intensity and scattered - wave spectral density for a set of scattered frequencies , in steady - state , along a 1d profile of plasma conditions . pump depletion is included , and the plasma waves are assumed to be in the strong damping limit ( i.e. , they do not advect ) . fully kinetic ( although linear ) formulas are used for various quantities like the coupling coefficient . bremsstrahlung noise and damping , as well as thomson scattering ( ts ) , are included . the deplete model , especially the noise sources , in some ways resembles that of ref.@xcite . other similar works which have influenced our thinking , and use 1d coupled - mode equations , are refs.@xcite-@xcite . deplete is a 1d model , but the plasma conditions are generally found by tracing 3d geometric - optics ray paths through the output of a radiation - hydrodynamics code . we therefore call this combined approach to studying lpi a ray - based one . details of this methodology , and its limits , are discussed in sec . [ s : ray ] . deplete is similar to the code newlip , which calculates linear gains for srs and sbs along 1d profiles ( newlip is discussed here in appendix a ) . both codes take seconds to analyze one profile from the laser entrance to the high - z wall in an icf ignition design . however , deplete includes substantially more physics than newlip , such as pump depletion , noise sources , and re - absorption of scattered light . deplete moreover provides pump and scattered intensities , which unlike gains can be directly compared with experiment and more sophisticated lpi codes . despite its simplicity , deplete agrees well in certain cases with results from the 3d paraxial laser propagation code pf3d . this is quite promising given deplete s much lower computing cost . there is important physics which deplete does not capture , with laser speckles or hot spots being one of the most important . recent sbs experiments @xcite at the omega laser facility @xcite show good agreement between measured reflectivity and pf3d predictions , while deplete gives a lower value . this is due to the speckle pattern of the phase plate smoothed lasers . [ s : nif ] describes one approximate way to bound the speckle enhancement by doubling the coupling coefficient ; the resulting deplete reflectivity always exceeds the experimental level . a more sophisticated idea for handling speckles is outlined in the conclusion . additional beam smoothing , like polarization smoothing ( ps ) and smoothing by spectral dispersion ( ssd ) , reduce the effective speckle intensity and can reduce the reflectivity even below the speckle - free deplete level . the paper is organized as follows . section [ s : gov ] derives the governing equations for the pump intensity and scattered - wave spectral density . our ray - based methodology and model limits are discussed in sec . [ s : ray ] . the numerical method is given in sec . [ s : num ] , including a quasi - analytic solution for the coupling - thomson step . section [ s : bench ] compares deplete with newlip linear gains and pf3d `` plane - wave '' simulations on prescribed profiles . the relationship between thomson scattering and linear gain is discussed in sec . [ s : thom ] . in sec . [ s : omsbs ] we compare deplete to the experimental and pf3d sbs reflectivities in recent omega shots . [ s : nif ] presents deplete analysis of an ignition design with a 285 ev radiation temperature for the national ignition facility ( nif ) @xcite . in particular , we show the effect of scattered light re - absorption and put a bound on speckle enhancement . we conclude and discuss future prospects in sec.[s : conc ] . a review of newlip and its linear gain is presented in appendix a. appendix b details the numerics of deplete s coupling - thomson step .

pump depletion , inverse - bremsstrahlung damping , bremsstrahlung emission , thomson scattering off density fluctuations , and whole - beam focusing are included . the numerical code deplete , which implements this model , is described . the model is compared with traditional linear gain calculations , as well as `` plane - wave '' simulations with the paraxial propagation code pf3d . comparisons with brillouin - scattering experiments at the omega laser facility [ t. r. boehly et al . , opt . commun . ( 1997 ) ] show that laser speckles greatly enhance the reflectivity over the deplete results . analysis with deplete of an ignition design for the national ignition facility ( nif ) [ j. a. paisner , e. m. campbell , and w. j. hogan , fusion technol .

a 1d , steady - state model for brillouin and raman backscatter from an inhomogeneous plasma is presented . the daughter plasma waves are treated in the strong damping limit , and have amplitudes given by the ( linear ) kinetic response to the ponderomotive drive . pump depletion , inverse - bremsstrahlung damping , bremsstrahlung emission , thomson scattering off density fluctuations , and whole - beam focusing are included . the numerical code deplete , which implements this model , is described . the model is compared with traditional linear gain calculations , as well as `` plane - wave '' simulations with the paraxial propagation code pf3d . comparisons with brillouin - scattering experiments at the omega laser facility [ t. r. boehly et al . , opt . commun . 133 , p. 495 ( 1997 ) ] show that laser speckles greatly enhance the reflectivity over the deplete results . an approximate upper bound on this enhancement , motivated by phase conjugation , is given by doubling the deplete coupling coefficient . analysis with deplete of an ignition design for the national ignition facility ( nif ) [ j. a. paisner , e. m. campbell , and w. j. hogan , fusion technol . 26 , p. 755 ( 1994 ) ] , with a peak radiation temperature of 285 ev , shows encouragingly low reflectivity . re - absorption of raman light is seen to be significant in this design .

0806.4127

i

in surface design , the user often needs to perform rounding or filleting between two intersecting surfaces . mathematically , the surface used in making the rounding is defined as the envelope of a family of spheres which are tangent to both surfaces . this envelope of spheres centered at @xmath1 with radius @xmath2 , where @xmath3 and @xmath2 are rational functions , is called a canal surface with spine curve @xmath4 . if the radius @xmath2 is constant the surface is called a pipe surface . moreover , if additionally we reduce the dimension ( take @xmath3 in a plane and consider circles instead of spheres ) we obtain the offset to the curve . canal surfaces are very popular in geometric modelling , as they can be used as a blending surface between two surfaces . for example , any two circular cones with a common inscribed sphere can be blended by a part of a dupin cyclide bounded by two circles as it was shown by @xcite ( see figure [ fig : cyclide ] ) . cyclides are envelopes of special quadratic families of spheres . for other examples of blending with canal surfaces we refer to @xcite . = 3.5 cm here we study the implicit equation of a canal surface @xmath5 and its implicit degree . the implicit equation of a canal surface can be obtained after elimination of the family variable @xmath6 from the system of two equations @xmath7 ( here @xmath8 are quadratic in the variables @xmath9 ) , i.e. by taking the resultant with respect to @xmath6 . however , this resultant can have extraneous factors . in the paper we explain how these factors appear and how we can eliminate them . by using lie and laguerre geometry , we see that the above system of equations is related to a system @xmath10 , where @xmath11 are linear in the variables @xmath12 and @xmath13 is the lie quadric ( for the exact definition see formula ( [ liequadric ] ) ) . it turns out that the variety defined by the system of equations @xmath14 is a dual variety to the curve @xmath15 , where @xmath16 is a curve on the lie quadric determined by the spine curve @xmath17 ( for the explicit definition see formula ( [ bar_e ] ) ) . for the dual variety @xmath18 we define the @xmath0-basis , which consists of two polynomials @xmath19 which are linear in @xmath20 , and of degree @xmath21 in @xmath6 are such that @xmath22 is minimal . it turns out that the resultant of @xmath23 and @xmath24 with respect to @xmath6 gives the implicit equation of the variety @xmath18 . there is a simple substitution formula ( see the algorithm at the end of section [ dualvariety ] ) to compute the implicit equation of the canal surface from the implicit equation of the variety @xmath18 . partial solutions to the problem of finding the implicit equation ( and degree ) for canal surfaces have been given in other papers . for instance , the degree of offsets to curves is studied in @xcite . in @xcite , there is a degree formula for the implicit equation of a polynomial canal surface . quadratic canal surfaces ( parametric and implicit representation ) have been studied in @xcite . we close the introduction by noting that the implicit degree of a canal surface is important for the parametric degree . our observation is that if the canal surface has the minimal parametrization of bi - degree @xmath25 then its implicit degree is close to @xmath26 . on the minimal bi - degree @xmath25 parametrizations of the canal surface we refer to @xcite . the paper is organized as follows . in the next section we develop some algebraic formalism about modules with two quasi - generators . we define the @xmath0-basis for these modules and present an algorithm for its computation . in the following section , we recall some needed facts about lie and laguerre sphere geometry . then using lie and laguerre geometry we describe the canal surface explicitly also , we introduce the @xmath27-hypersurface which contains all @xmath28-offsets to the canal surface . using the @xmath0-basis algorithm we compute the implicit equations of the dual variety @xmath18 , the @xmath27-hypersurface and the canal surface @xmath5 . next we apply the results of the previous section to the dual variety @xmath18 of the curve and explain how to compute the implicit degree of the @xmath27-hypersurface ( without computation of the implicit equation ) . finally , we give some computational examples .

a canal surface is an envelope of a one parameter family of spheres . in this paper we define the-basis for arbitrary dimension and give a simple algorithm for its computation . this is then applied to the dual variety , which allows us to deduce the implicit equations of the the dual variety , the canal surface and any offset to the canal surface . + _ key words : _ canal surface , implicit equation , resultant ,-basis , offset [ thm]lemma [ thm]corollary [ thm]conjecture [ thm]proposition [ thm]definition [ thm]remark [ thm]assumption [ thm]example

a canal surface is an envelope of a one parameter family of spheres . in this paper we present an efficient algorithm for computing the implicit equation of a canal surface generated by a rational family of spheres . by using laguerre and lie geometries , we relate the equation of the canal surface to the equation of a dual variety of a certain curve in 5-dimensional projective space . we define the-basis for arbitrary dimension and give a simple algorithm for its computation . this is then applied to the dual variety , which allows us to deduce the implicit equations of the the dual variety , the canal surface and any offset to the canal surface . + _ key words : _ canal surface , implicit equation , resultant ,-basis , offset [ thm]lemma [ thm]corollary [ thm]conjecture [ thm]proposition [ thm]definition [ thm]remark [ thm]assumption [ thm]example

1611.01782

i

a _ proper colouring _ of a graph is a labeling of its vertices with colours such that no two vertices sharing the same edge have the same colour . the smallest number of colours in a proper colouring of a graph @xmath2 is called its _ chromatic number _ , and it is denoted by @xmath3 . in this paper we are concerned with another notion of vertex colouring . a _ clique _ @xmath4 is a subset of the vertex set such that any pair of vertices in @xmath5 is connected by an edge . moreover , a clique @xmath5 is _ maximal _ if there is no vertex in @xmath6 connected by an edge to every vertex in @xmath5 ( in other words , @xmath5 is not a proper subset of another clique ) . a _ clique colouring _ of a graph @xmath7 is a colouring of the vertices so that no maximal clique is monochromatic , ignoring isolated vertices . the smallest number of colours in such a colouring is called the _ clique chromatic number _ of @xmath7 , denoted by @xmath8 . clearly , @xmath9 but it is possible that @xmath8 is much smaller than @xmath3 . for example , for any @xmath10 we have @xmath11 but @xmath12 . note that if @xmath7 is triangle - free then @xmath13 . the problem has received considerable attention for deterministic graphs : in @xcite it was shown that planar graphs satisfy @xmath14 . in @xcite a necessary and sufficient condition for @xmath15 on line graphs was given . moreover , several graph classes are known to satisfy @xmath16 : claw - free perfect graphs @xcite , co - diamond free graphs @xcite , claw - free planar graphs @xcite , powers of cycles ( other than odd cycles longer than three that need three colours ) @xcite , and also claw - free graphs with maximum degree at most @xmath17 ( except for odd cycles longer than @xmath18 ) @xcite . also , circular - arc graphs are known to have @xmath19 ( see @xcite ) . further results about other classes of graphs can also be found in @xcite , and the clique chromatic number of graphs without having long paths was studied in @xcite . on the algorithmic side , it is known that testing whether @xmath20 for a planar graph can be performed in polynomial time @xcite , but deciding whether @xmath20 is @xmath21-complete for @xmath18-chromatic perfect graphs @xcite and for graphs with maximum degree @xmath18 @xcite . the clique chromatic number for geometric graphs ( in particular , random geometric graphs ) is analysed in the accompanying paper @xcite . let us recall the classic model of random graphs that we study in this paper . the _ binomial random graph _ @xmath0 is the random graph @xmath7 with vertex set @xmath22 $ ] in which every pair @xmath23}{2}$ ] appears independently as an edge in @xmath7 with probability @xmath24 . note that @xmath1 may ( and usually does ) tend to zero as @xmath25 tends to infinity . the behaviour of many colouring problems has been investigated for @xmath0 : the classic chromatic number has been intensively studied , see @xcite and the references therein ; the list chromatic number ( known also as the choice number ) was studied among others in @xcite , and other variants were analysed recently in @xcite and @xcite . all asymptotics throughout are as @xmath26 ( we emphasize that the notations @xmath27 and @xmath28 refer to functions of @xmath25 , not necessarily positive , whose growth is bounded ) . we use the notations @xmath29 for @xmath30 and @xmath31 for @xmath32 . we also write @xmath33 if @xmath34 as @xmath35 ( that is , when @xmath36 ) . we say that events @xmath37 in a probability space hold _ with high probability _ ( or _ whp _ ) , if the probability that @xmath37 holds tends to @xmath38 as @xmath25 goes to infinity . since we aim for results that hold whp , we will always assume that @xmath25 is large enough . we often write @xmath0 when we mean a graph drawn from the distribution @xmath0 . finally , we use @xmath39 to denote natural logarithms . here is our main result . we consider the edge probability @xmath40 ranging from the sparse case when @xmath41 arbitrarily slowly , to the dense case when @xmath42 for an arbitrarily small @xmath43 , and we break this range into 8 parts . after the theorem we give a corollary which is less precise but easier to read . [ thm : gnp ] let @xmath44 be a constant ( arbitrarily small ) . let @xmath45 be a function tending to infinity with @xmath25 ( arbitrarily slowly ) , and suppose that @xmath46 . let @xmath47 for some @xmath1 . then , the following holds whp : * if @xmath48 and @xmath49 , then @xmath50 * if @xmath51 , then @xmath52 and @xmath53 * if @xmath54 , then @xmath55 * if @xmath56 , then @xmath57 * if @xmath58 , then @xmath59 and @xmath60 * if @xmath61 and @xmath62 , then @xmath63 * if @xmath64 , then @xmath59 and @xmath60 * if @xmath65 and @xmath66 , then @xmath67 note that @xmath68 in places a slightly tighter result than the one given in theorem [ thm : gnp ] can be obtained . we are interested in the ratio between the upper and lower bounds on @xmath8 for each @xmath24 , but in order to keep the statement reasonably simple , we spelled out results only for large intervals of @xmath24 . for instance , as cases ( d ) and ( f ) are concerned with @xmath69 and @xmath61 respectively , we treated these cases in the statement of theorem [ thm : gnp ] less fully . for the reader interested in more precise bounds on this ratio , we refer to the proofs in the sections below for more details . in order to understand better the behaviour of the clique chromatic number , let us introduce the following function @xmath70 : @xmath71 this function is depicted in figure [ fig1 ] . we get immediately the following corollary . related to the clique chromatic number of @xmath72,width=192,height=153 ] [ cor.main ] let @xmath70 be defined as above . let @xmath47 for some @xmath1 . if @xmath73 for some @xmath74 then whp @xmath75 . figure [ fig1 ] shows that there are dramatic transitions in corollary [ cor.main ] when @xmath24 is about @xmath76 and @xmath77 , and there is some problem at about @xmath78 in theorem [ thm : gnp ] . what is happening at these points ? here is the rough story . for @xmath79 , whp most edges are not in triangles , and @xmath8 is close to @xmath3 . for @xmath80 , whp each edge is in many triangles , and we may use triangle - free sets as colour classes ; or we may find relatively small dominating sets which yield an upper bound on @xmath8 ( see lemma [ lem : deterministic_upperbound ] ) . for @xmath81 , the first bound ( which increases with @xmath24 ) is stronger , whereas from then on the second bound ( which decreases with @xmath24 ) gives better results . for lower bounds on @xmath8 , depending on the range of @xmath24 , we find a value of @xmath82 , so that whp there are many @xmath82-cliques and most of them are not contained inside @xmath83-cliques . ( in the central range in figure [ fig1 ] , for @xmath84 , we use @xmath85 . ) for @xmath86 whp most triangles are not contained in @xmath87-cliques ; and for @xmath88 whp each triangle is in many @xmath87-cliques .

a clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic ( ignoring isolated vertices ) . the smallest number of colours in such a colouring is the clique chromatic number . in this paper we see that the typical clique chromatic number , as a function of the average degree , forms an intriguing step function . [ section ] [ theorem]lemma [ theorem]definition [ theorem]conjecture [ theorem]proposition [ theorem]algorithm [ theorem]corollary [ theorem]observation [ theorem]open problem [ theorem]remark

a clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic ( ignoring isolated vertices ) . the smallest number of colours in such a colouring is the clique chromatic number . in this paper , we study the asymptotic behaviour of the clique chromatic number of the random graph for a wide range of edge - probabilities . we see that the typical clique chromatic number , as a function of the average degree , forms an intriguing step function . [ section ] [ theorem]lemma [ theorem]definition [ theorem]conjecture [ theorem]proposition [ theorem]algorithm [ theorem]corollary [ theorem]observation [ theorem]open problem [ theorem]remark

math0603023

i

since cayley @xcite in 1857 and 100 years later merson @xcite and followed shortly after by butcher @xcite , it has been known that rooted trees are extremely useful for structuring algebras of differential operators and elementary differentials . in 1972 butcher @xcite produced the far - reaching result that runge kutta methods form a group . this was later named the butcher group in the paper by hairer and wanner @xcite , who made significant contributions to this theory . the butcher group is defined on the dual of the tree space and it was pointed out by dr @xcite that there exists a one - to - one correspondence between the butcher group and the commutative graded hopf algebra of unordered rooted trees . the hopf algebra of unordered rooted trees has had far - reaching applications in various areas of mathematics and physics . in 1998 a hopf subalgebra was discovered by connes and moscovici @xcite during work on an index theorem in non - commutative geometry and by kreimer @xcite in the renormalization method of quantum field theory . further collaborations between connes and kreimer @xcite have lead to other surprising results ; notably a connection with the riemann hilbert problem was established . brouder @xcite realized that the mathematical structure of connes kreimer was the same as that of butcher . grossman and larson @xcite also developed a cocommutative graded hopf algebra on a general class of rooted trees . it was shown by foissy @xcite and hoffman @xcite that the commutative hopf algebra of butcher and connes kreimer was the dual of the cocommutative hopf algebra of grossman larson , which corrected the original result of panaite @xcite . murua @xcite has developed series expansions of elementary differential operators and shown among other results that the logarithm of such a series is equivalent to the series expansions obtained from backward error analysis . recently a great deal of interest has been focused on developing numerical methods which preserve geometric properties of the exact flow . in particular , lie group integrators , which describe integrators that use lie group actions on manifolds , were originally proposed by crouch and grossman @xcite followed shortly after by lewis and simo @xcite . integrators of this type are now known as lie group integrators ; a survey of these methods is given in @xcite . series expansions for various classes of lie group methods have been developed ; these expansions are generally used to analyze order . munthe - kaas @xcite constructed the order conditions for a special subclass of lie group methods , where the computations are performed in a lie algebra , which is a linear space . later , @xcite it was shown that the classical order conditions could be used along with a certain transformation . owren and marthinsen @xcite developed the general order conditions for the crouch grossman methods with their analysis being based on ordered rooted trees . recently , owren @xcite derived the order conditions for the commutator free lie group methods @xcite , which were derived to overcome some of the problems associated with computing commutators . in this paper we aim to construct a commutative graded hopf algebraic structure , which can be used to analyze the order of all lie group methods . the outline of this paper is as follows : in section [ sect : derivation ] we will introduce ordered trees and forests and describe some useful operations on them . we will motivate the present hopf algebra as a universal object in a category of non - commutative derivations , and also briefly discuss lie butcher theory which will be treated in more detail in @xcite . in section [ sect : hopf ] we develop the hopf algebra of ordered trees , giving both recursive and non - recursive definitions of the coproduct and antipode , using certain cutting operations on ordered rooted trees . we show that the hopf algebra described in this paper is the dual of the grossman larson hopf algebra , thus generalizing the result of hoffman @xcite . finally , in section [ sect : homo ] we use a symmeterization operator to provide an injective hopf algebra homomorphism from the unordered trees into the ordered trees , establishing the former as a sub - algebra of the latter .

a commutative but not cocommutative graded hopf algebra , based on ordered rooted trees , is studied . this hopf algebra generalizes the hopf algebraic structure of unordered rooted trees , developed by butcher in his study of runge kutta methods and later rediscovered by connes and moscovici in the context of non - commutative geometry and by kreimer where it is used to describe renormalization in quantum field theory . recursive and non - recursive definitions of the coproduct and the antipode are derived . it is also shown that the dual of is a hopf algebra of grossman and larson .

a commutative but not cocommutative graded hopf algebra , based on ordered rooted trees , is studied . this hopf algebra generalizes the hopf algebraic structure of unordered rooted trees , developed by butcher in his study of runge kutta methods and later rediscovered by connes and moscovici in the context of non - commutative geometry and by kreimer where it is used to describe renormalization in quantum field theory . it is shown that is naturally obtained from a universal object in a category of non - commutative derivations , and in particular , it forms a foundation for the study of numerical integrators based on non - commutative lie group actions on a manifold . recursive and non - recursive definitions of the coproduct and the antipode are derived . it is also shown that the dual of is a hopf algebra of grossman and larson . contains two well - known hopf algebras as special cases : the hopf algebra of butcher connes kreimer is identified as a proper subalgebra of using the image of a tree symmetrization operator . the hopf algebra of the free associative algebra is obtained from by a quotient construction .

math0603023

l

there are two interesting commutative graded hopf sub - algebras of @xmath0 obtained by restricting from the set of ordered rooted trees to either the set of tall trees ( that is trees where each parent has one child ) or bushy trees ( that is trees where there is only one parent ) . these hopf sub - algebras are useful respectively for determining the order conditions for the problem , when @xmath304 is constant , or when the numerical scheme has high stage order . in this section we will also show that the hopf algebra @xmath1 of butcher , based on unordered trees , can be identified as a sub - algebra of @xmath0 . finally we find that the hopf algebra @xmath2 of the free associative algebra is related to @xmath0 through the operation of _ freezing _ vector fields , which can be defined as a quotient construction on @xmath0 . let @xmath305 denote all unordered trees and @xmath306 denote all unordered forests , defined as the set of all empty or non - empty unordered words over the alphabet @xmath305 . recall from @xcite the following definition . given the real vector space @xmath307 , denote the hopf algebra of unordered forests as @xmath308 . the product @xmath309 is defined as the ( commutative ) concatenation @xmath310 the unit element @xmath311 , is given by @xmath312 . the coproduct @xmath313 is defined by linearity and for any @xmath314 and @xmath315 by the recursion @xmath316 the counit @xmath317 is defined by @xmath318 and @xmath216 for @xmath319 . the antipode @xmath320 is , as usual , the two - sided inverse of the convolution in @xmath1 , see @xcite for details . the main tool used to provide the relationship between the hopf algebras of ordered and unordered forests is the symmetrization operator defined below . [ defn : symm ] the symmetrization operator @xmath321 is a map defined by linearity and the relations @xmath322 the shuffle product permutes the trees in a forest in all possible ways , and the symmetrization of a tree is a recursive splitting in sums over all permutations of the branches . the symmetrization defines an equivalence relation on @xmath5 , that is , if @xmath323 thus @xmath324 if and only if @xmath325 can be obtained from @xmath326 by permuting the order of the trees in the forest and the order of the branches of the trees . we see that an alternative characterization of @xmath327 is @xmath328 the integer @xmath329 is the classical symmetry coefficient , defined for trees and forests as @xmath330 where the integers @xmath331 count the number of equivalent trees among @xmath332 . in other words , if we consider the full group of all possible permutations of trees and branches acting on a forest @xmath281 , then @xmath329 is the size of the isotropy subgroup i.e. the number of permutations leaving @xmath20 invariant . the total number of permutations acting on a given forest @xmath20 is given by the integer @xmath333 defined as @xmath334 note that once a forest @xmath20 is symmetrized , then another application of the symmetrization yields the scaling @xmath335 let @xmath306 be the unordered forests . clearly , there is a 11 correspondence between unordered forests and equivalence classes of ordered forests , thus there is a natural isomorphism @xmath336 . through this identification , we can interpret @xmath327 as an injection @xmath337 where @xmath307 and @xmath24 . from we see that the map @xmath338 defined as @xmath339 where @xmath340 is the natural identification of an ordered forest with the corresponding unordered forest , defines a left - sided inverse @xmath341 for all @xmath342 . [ thm : hopfhomo ] the symmetrization operator @xmath337 defines an injective hopf algebra homomorphism from the hopf algebra @xmath1 of unordered forests into the hopf algebra @xmath0 of ordered forests . a hopf algebra homomorphism is a bialgebra homomorphism , which is a linear map that is both an algebra and a coalgebra homomorphism . @xmath343 is an algebra homomorphism if @xmath344 these conditions are automatically satisfied by definition [ defn : symm ] . @xmath343 is a coalgebra homomorphism if @xmath345 the first condition follows immediately given that the counits are only non - zero when the argument is the empty forest . the second relation follows using an induction argument . first we need to establish a useful relationship between @xmath327 and @xmath346 . using the _ sumless sweedler notation _ @xmath347 we find : @xmath348 now we prove by induction . for a forest we find using and @xmath349 for a tree @xmath350 and @xmath351 we find using the definition of @xmath327 and the recursion formulas and that @xmath352 the proof extends to a general @xmath27 by linearity . @xmath98 a consequence of the above theorem is a connection between the antipodes of the corresponding hopf algebras . given @xmath353 is a bialgebra homomorphism then for @xmath354 , it follows that @xmath355 note that symmetrization operator @xmath327 is invertible , so expressions for the product , coproduct and antipode of the hopf algebra of unordered forests can be directly expressed in terms of the corresponding functions in the hopf algebra of ordered forests , they are @xmath356 in the final part of this section we will elaborate on the connections between the ls - series built on ordered trees , and their commutative counterpart the s - series . these series belong respectively to the dual spaces @xmath31 and @xmath357 , are naturally associated through the dual map @xmath358 taking the series of ordered forests to unordered forests . if @xmath359 and @xmath360 , we find @xmath361 on a manifold with a commutative lie group action the elementary differential operators @xmath143 do not depend on the ordering . thus we find that the s - series of @xmath362 as defined in @xcite equals the ls - series of @xmath363 as given in , @xmath364 this shows that the normalization @xmath365 in the commutative case is compatible with our normalization in the non - commutative case . it is interesting to characterize the image of the _ logarithmic _ and _ exponential _ series under @xmath366 . if @xmath363 is logarithmic ( lemma [ lem : log_exp ] ) then @xmath367 thus @xmath362 is non - zero only on trees . if @xmath363 is exponential then @xmath368 this is a well known condition in the composition of b - series , see @xcite . in the proof of proposition [ prop : antipode_nr ] , we defined the hopf algebraic structure @xmath2 as the free associative algebra built of words over an alphabet @xmath277 , where @xmath369 is the shuffle and @xmath279 the dual of the concatenation product . we will briefly comment upon the connection between @xmath0 and @xmath2 in the context of lie group integrators . in the theory of lie group integrators , it is common to call constant sections @xmath370 _ frozen vector fields_. if @xmath106 is frozen , then the lie derivative @xmath371 = 0 $ ] for all vector fields @xmath136 . on the algebraic side , a tree @xmath372 represents a frozen vector field if the left grafting of anything non - constant to the tree is zero , @xmath373 = 0 , \qquad\mbox{for all $ \widetilde{\tau}\in \ot\backslash\{\one\}$}.\ ] ] in this case we see that the grossman larson product becomes just the concatenation product @xmath374 . the freezing of certain vector fields can be understood as the quotient @xmath375 , where @xmath376 is the linear span of any @xmath377 grafted to a frozen vector field . as a special example , we consider the case where all single - node trees are frozen , so that taller trees can not be produced . letting @xmath378 be the alphabet of all single - node trees , we find from the following well known recursion for the coproduct @xmath379

contains two well - known hopf algebras as special cases : the hopf algebra of butcher connes kreimer is identified as a proper subalgebra of using the image of a tree symmetrization operator . the hopf algebra of the free associative algebra is obtained from by a quotient construction .

a commutative but not cocommutative graded hopf algebra , based on ordered rooted trees , is studied . this hopf algebra generalizes the hopf algebraic structure of unordered rooted trees , developed by butcher in his study of runge kutta methods and later rediscovered by connes and moscovici in the context of non - commutative geometry and by kreimer where it is used to describe renormalization in quantum field theory . it is shown that is naturally obtained from a universal object in a category of non - commutative derivations , and in particular , it forms a foundation for the study of numerical integrators based on non - commutative lie group actions on a manifold . recursive and non - recursive definitions of the coproduct and the antipode are derived . it is also shown that the dual of is a hopf algebra of grossman and larson . contains two well - known hopf algebras as special cases : the hopf algebra of butcher connes kreimer is identified as a proper subalgebra of using the image of a tree symmetrization operator . the hopf algebra of the free associative algebra is obtained from by a quotient construction .

cond-mat0303513

i

manganites are currently attracting considerable attention mainly due to the presence of the colossal magnetoresistance effect in magnetotransport measurements@xcite . in addition , these materials have a complex phase diagram with a plethora of ordered phases , a typical characteristic of correlated electron systems . many experimental and theoretical investigations have unveiled the inhomogeneous character of the states of relevance to explain the cmr phenomenon , with a competition between ferromagnetic and antiferromagnetic states that induces coexistence of clusters , typically with nanometer - scale sizes@xcite . the rationalization of this phenomenon , and concomitant explanation of the cmr effect , is based on the first - order transitions that separate the metallic and insulating phases in the clean limit ( i.e. without disorder)@xcite . the first - order character of the transition is caused by the different magnetic and charge orders of the competing states . the clean - limit phase diagram is illustrated in fig . [ fig : tg3 ] ( upper panel ) . when quenched disorder is introduced in the coupling or density that is modified to change from one phase to the other , the temperature where the nel and curie temperatures meet is reduced in value and eventually reaches zero as in fig . [ fig : tg3 ] ( middle panel ) . upon further increase of the disorder strength , a spin disordered region appears at low temperatures , with a glassy behavior involving coexisting clusters ( fig . [ fig : tg3 ] ( lower panel ) ) . simulations by burgy _ @xcite have shown that the clustered state between the curie temperature and the clean - limit critical temperature ( @xmath5 ) , with preformed ferromagnetic regions of random orientations , has a huge magnetoresistance since small fields can easily align the moments of the ferromagnetic islands , leading to a percolative conductor in agreement with experiments@xcite . the quenched disorder simply triggers the stabilization of the cluster formation , but phase competition is the main driving force of the mixed state . it has been speculated@xcite that similar phenomena should occur in a variety of materials , including , for example , the high temperature superconductors where nanoclusters have been found@xcite . is some parameter needed to change from one phase to the other . _ middle _ with increasing disorder , the temperature range with first - order transitions separating the ordered states is reduced , and eventually for a _ fine - tuned _ value of the disorder the resulting phase diagram contains a quantum critical point . in this context , this should be a rare occurrence . _ bottom _ in the limit of substantial quenched disorder , a window without any long - range order opens at low temperature between the ordered phases . this disordered state has glassy characteristics and it is composed of coexisting clusters of both phases . the size of the coexisting islands can be regulated by the disorder strength and range , and by the proximity to the original first - order transition . for more details see refs . . the new scale @xmath6 discussed in the text remnant of the clean - limit transition is also shown . , width=226 ] in spite of the phenomenological success of the previous calculations discussed above , there are many issues that must be further investigated in the manganite context to reach a reasonable understanding of these materials ( for a recent discussion on open issues see ref . ) . it is the purpose of this paper to contribute to manganite studies by addressing a variety of open topics , using computational techniques applied to realistic models for mn - oxides . in particular , here the phase diagram at hole - density @xmath4=0.5 is presented for the case of _ cooperative _ jahn - teller phonons in interaction with carriers . previous studies mainly focused on non - cooperative phonons@xcite . the phase diagram reported below in fig . [ fig : figure1 ] has interesting features . for example , it contains the realistic fm charge - disordered and ce charge - ordered phases well - known from experiments . in addition , further evidence is provided that the transition from the fm to the af state is first - order at low temperatures . this first - order property allows us to obtain a huge magnetoresistance effect even working on small clusters , since small magnetic fields can unbalance the ground state from antiferro to ferro , in the vicinity of the competition region . for the calculation , techniques borrowed from mesoscopic calculations using the landauer formalism are used . the shape of the phase diagram with increasing temperature is in good qualitative agreement with recent experiments that established the bicritical form of the @xmath4=0.5 phase diagram@xcite . moreover , a novel phase is found with both charge and ferromagnetic order , in agreement with previous calculations@xcite and also with recent experiments by loudon _ et al._@xcite another feature of the results found here that is in good agreement with recent experiments ( to be discussed below ) is the value of the charge disproportionation . the widely accepted view of the extreme 3+/4 + separation needs to be revisited . a much milder charge separation appears more realistic . the @xmath4=0.5 phase diagram of the realistic model studied here is also analyzed in the presence of quenched - disorder , to test the results of phenomenological models , such as those shown in fig . [ fig : tg3 ] . overall , good agreement with previous studies is found , but surprises are also reported such as the strong sensitivity of the ce phase to quenched disorder . this behavior is also in good agreement with recent experimental results by akahoshi _ et al._@xcite and nakajima _ et al._@xcite , where an asymmetry between the behavior of the fm- and ce - phases was observed when disordered was introduced in some materials with @xmath4=0.5 . finally , since _ electron _ doping of manganites has also been experimentally actively pursued @xcite , the model studied here is analyzed for the case of electron doping as well , with an electronic density per site equal to 1.5 . a state quite similar to the ce phase is observed , which could be found in future experiments . the overall conclusion is that theoretical studies of realistic models for manganites using unbiased techniques are unveiling a remarkable qualitative agreement with experiments . while previous phenomenological descriptions of manganites are reasonable approximations for the understanding of the puzzling cmr phenomenon@xcite , many other features observed in the present studies still need further investigations , such as the strong sensitivity to disorder of the ce state .

a comprehensive analysis of half - doped manganites is presented using monte carlo simulations applied to the double - exchange model with cooperative jahn - teller lattice distortions in two dimensions . a far smaller appears more realistic , in agreement with recent experiments by garcia _ et al . _ and daoud - aladine _ et al . _ _ ( iii ) _ colossal magnetoresistance ( cmr ) effects are found in calculations of cluster resistances using the landauer formalism . this occurs near the ubiquitous first - order phase transitions between the insulating and metallic states . this is also in qualitative agreement with recent experiments by akahoshi _ _ electron doping _ regime is briefly discussed as well . a charge - ordered state is found which is the analog of the=0.5 ce phase .

a comprehensive analysis of half - doped manganites is presented using monte carlo simulations applied to the double - exchange model with cooperative jahn - teller lattice distortions in two dimensions . a variety of novel results are reported . in particular : _ ( i ) _ the phase diagram is established in the- plane , with the electron - phonon coupling and the antiferromagnetic exchange between classical spins . the results include standard phases , such as the ce - insulating and fm - metallic regimes , but they also include novel states , such as a ferromagnetic charge - ordered ( co ) orbital - ordered phase originally predicted by hotta _ et al._. this state is compatible with recent experimental results by loudon _ et al . _ _ ( ii ) _ for realistic couplings , it was observed that the charge disproportionation of the co phase is far from the widely accepted extreme limit=0.5 of a 3+/4 + charge separation . a far smaller appears more realistic , in agreement with recent experiments by garcia _ et al . _ and daoud - aladine _ et al . _ _ ( iii ) _ colossal magnetoresistance ( cmr ) effects are found in calculations of cluster resistances using the landauer formalism . this occurs near the ubiquitous first - order phase transitions between the insulating and metallic states . the present result reinforces the previous conjecture that cmr phenomenology exists in two forms : the low - temperature cmr addressed here and the more standard cmr above the curie temperature . _ ( iv ) _ the ce - state is found to be _ very sensitive to disorder _ since its long - range order rapidly disappears when quenched - disorder is introduced , contrary to the fm state which is more robust . this is also in qualitative agreement with recent experiments by akahoshi _ et al . _ and nakajima _ et al . _ _ ( v ) _ the phase diagram in the half - doped _ electron doping _ regime is briefly discussed as well . a charge - ordered state is found which is the analog of the=0.5 ce phase . it contains a 3+/2 + charge arrangement at large . numerical results suggest that an approximate symmetry exists between the hole- and electron - doped systems in the large hund coupling limit .

cond-mat0303513

c

the investigations reported in this paper have unveiled several unexpected properties of half - doped manganites . for example , the ce phase was found to be more sensitive than expected to the addition of quenched disorder . the effort in this context included the calculation of the clean - limit phase diagram in the @xmath82-@xmath1 plane , to understand the competition between the fm and ce states believed by most manganite experts to be at the heart of the cmr phenomenon . in agreement with previous calculations@xcite , a low - temperature first - order transition between the two phases was found , with a shape similar to those reported experimentally@xcite . adding disorder , the predicted@xcite depletion of critical temperatures was observed , but contrary to those expectations the reduction was not symmetrical in magnitude for the two competing phases . in fact , it was observed that the ce state rapidly transforms into a `` ce glass '' with disorder , while the ferromagnetic phase is comparatively less affected , in nice agreement with recent experiments where disorder is introduced `` by hand '' upon a clean - limit bicritical - shaped phase diagram@xcite . the zigzag chains of the ce state and its very subtle arrangement of spins , charge , and orbitals seems easily broken by imperfections , contrary to the more robust uniform fm state . the present results revealed other new properties of manganite states that also deserve further investigations . among these interesting new properties are the existence of _ novel phases _ , such as a fm / co state , and a discussion on charge disproportionation that suggests that the widely held view of charge - ordering as containing 3 + and 4 + ions needs considerable revision . a second important result presented here is the existence of a very large mr effect when the insulating state is close in energy to the fm state , as it occurs in the first - order transitions systematically found in the clean limit . as a consequence , without quenched disorder and working at low temperatures , the insulating state can be destabilized by the fm state with increasing fields . the magnitude of the field required can be very small in units of the hopping , and its actual critical value depends on how close the ce state investigated is to the fm state in the phase diagram ( i.e. how close the analyzed value of @xmath1 is to the critical value separating the phases ) . this mr effect is _ qualitatively different _ from the more standard effect found above the curie temperature , which is believed to need disorder effects to be understood@xcite . as predicted by tokura and collaborators@xcite many years ago , _ there are two types of cmr _ : a low - temperature form investigated theoretically here and a higher temperature variety with the standard profiles for resistivity vs. temperature . in carrying out these investigations , we have introduced techniques borrowed from mesoscopic physics , within the context of the landauer formalism . the study of electron - doped systems has also been initiated in the investigations reported in this paper , focusing on the half - doped limit . charge - ordered states analogous to those observed at half - hole - doping were identified . a formal particle - hole symmetry appears to exist in the system with respect to the undoped @xmath132 limit , at least for a large hund coupling . more theoretical work in this area should be pursued , since experiments are starting to investigate electron - doped manganites . it is clear that the study of models for manganites using unbiased techniques provides a plethora of interesting results . in the field of manganites the crossfertilization theory - experiments has been remarkably positive , and new surprises will likely be found in the near future . investigations of manganites and other related oxides should continue at its present rapid rate , to enhance our understanding of correlated electron systems in general .

in particular : _ ( i ) _ the phase diagram is established in the- plane , with the electron - phonon coupling and the antiferromagnetic exchange between classical spins . the results include standard phases , such as the ce - insulating and fm - metallic regimes , but they also include novel states , such as a ferromagnetic charge - ordered ( co ) orbital - ordered phase originally predicted by hotta _

a comprehensive analysis of half - doped manganites is presented using monte carlo simulations applied to the double - exchange model with cooperative jahn - teller lattice distortions in two dimensions . a variety of novel results are reported . in particular : _ ( i ) _ the phase diagram is established in the- plane , with the electron - phonon coupling and the antiferromagnetic exchange between classical spins . the results include standard phases , such as the ce - insulating and fm - metallic regimes , but they also include novel states , such as a ferromagnetic charge - ordered ( co ) orbital - ordered phase originally predicted by hotta _ et al._. this state is compatible with recent experimental results by loudon _ et al . _ _ ( ii ) _ for realistic couplings , it was observed that the charge disproportionation of the co phase is far from the widely accepted extreme limit=0.5 of a 3+/4 + charge separation . a far smaller appears more realistic , in agreement with recent experiments by garcia _ et al . _ and daoud - aladine _ et al . _ _ ( iii ) _ colossal magnetoresistance ( cmr ) effects are found in calculations of cluster resistances using the landauer formalism . this occurs near the ubiquitous first - order phase transitions between the insulating and metallic states . the present result reinforces the previous conjecture that cmr phenomenology exists in two forms : the low - temperature cmr addressed here and the more standard cmr above the curie temperature . _ ( iv ) _ the ce - state is found to be _ very sensitive to disorder _ since its long - range order rapidly disappears when quenched - disorder is introduced , contrary to the fm state which is more robust . this is also in qualitative agreement with recent experiments by akahoshi _ et al . _ and nakajima _ et al . _ _ ( v ) _ the phase diagram in the half - doped _ electron doping _ regime is briefly discussed as well . a charge - ordered state is found which is the analog of the=0.5 ce phase . it contains a 3+/2 + charge arrangement at large . numerical results suggest that an approximate symmetry exists between the hole- and electron - doped systems in the large hund coupling limit .

1508.07639

i

the use of cepheids as stellar candles requires a deep knowledge of their pulsational characteristics . the separation between fundamental ( @xmath1-mode ) and first - overtone ( @xmath2-mode ) radial pulsators @xcite is still the most relevant distinction to be taken into account to calibrate the period - luminosity relations . about the regularity of their variations , ground - based photometry of v473 lyr @xcite and polaris @xcite showed continuous changes in amplitude and phase . @xcite established that v473 lyr pulsates in the second radial overtone and shows strong amplitude and phase modulations on two long periods ( 1205@xmath33 and 5300@xmath3150 d ) . the slow changes in the light curve of polaris on a time baseline of 4.5 yr were precisely recorded by space photometry using the _ coriolis _ spacecraft . @xcite connected them to a quite fast evolutionary phase . however , @xcite proposed a different interpretation pointing out how the amplitude change is cyclic rather than monotonic and also suggesting an excess of power in the 2 - 6 d range due to the granulation . the situation was expected to change with the advent of high - precision space mission . a detailed analysis of the _ most _ observations of sz tau and rt aur revealed variations from cycle to cycle as a function of the pulsation phase @xcite . the stars were both monitored for less than seven consecutive cycles : the @xmath2-cepheid sz tau showed a pulsation less stable than the @xmath1-cepheid rt aur . the historical analysis of the times of maximum brightness proved that both stars experienced period variations on a time - scale of decades and that those of the overtone mode resulted to be more erratic than the @xmath4 ones . in the case of _ kepler _ , there was only one cepheid in the field - of - view , i.e. , v1154 cyg . @xcite provided a very detailed analysis of the data acquired in the first 600 d ( i.e. , about 120 cycles ) of the mission , detecting significant cycle - to - cycle fluctuations . a very slight correlation was also found between the fourier parameters and the o - c values , i.e. , the difference between the _ observed _ and _ calculated _ times of maximum brightness ( t@xmath5 ) . this suggested that the o - c variations ( up to 30 min ) might be due to instabilities in the light curve shape . on the basis of these observational results , @xcite suggested that convection and hot spots can explain the observed period jitter , thus supporting the @xcite interpretation of the polaris changes . the clues about the changes in the light curve of v1154 cyg were strengthened by the analysis of new _ kepler _ measurements ( up to q17 quarter ) : the analysis of the fourier parameters supplied a robust determination of a periodicity of 158.6 d @xcite . about ground - based observations , @xcite discovered significant modulations in the radial velocity curves of some cepheids . these modulations appear to be different from those reported on v1154 cyg , since they manifest as shape and amplitude variations on time - scales of years for short - period cepheids ( like v1154 cyg ) and cycle - to - cycle variations in the long - period cepheids . a common result among all these investigations is that no nonradial mode was identified in the light or radial velocity curves of galactic cepheids . hence the counterparts of the subclasses detected in the large magellanic cloud ( lmc ; for a review see * ? ? ? * ) are still missing . however , an important player has not yet contributed , i.e. , the corot data base of the light curves collected in the exoplanetary fields . such data base already proved to be a real treasure allowing the discovery of a unique triple - mode cepheid in the outskirts of the milky way @xcite . in this paper we describe the new elements brought by the corot photometry to the hot topic of the stability of the cepheid light curves .

therefore , we investigated all the candidate cepheids pointed out by the automatic classification of the corot curves . at the end the light curves were investigated to remove some instrumental effects . on the other hand , cycle - to - cycle variations of the fourier parameters were observed , but very small and always within . among the seven cepheids , i first - overtone pulsators , four pop .

a few galactic classical cepheids were observed in the programmes of space missions as _ coriolis _ , _ most _ and _ kepler_. an appealing opportunity was to detect additional nonradial modes , thus opening the possibility to perform asteroseismic studies and making the pulsational content of galactic cepheids more similar to that of magellanic clouds ones . however , only hints of cycle - to - cycle variations were found , without any strict periodicity . in this context the potential of the corot exoplanetary data base was not fully exploited despite the wide area covered on the galactic plane . therefore , we investigated all the candidate cepheids pointed out by the automatic classification of the corot curves . at the end we could identify seven bona - fide cepheids . the light curves were investigated to remove some instrumental effects . the frequency analysis was particularly delicate since these small effects can be enhanced by the large amplitude , resulting in the presence of significant , but spurious , peaks in the power spectrum . indeed , the careful evaluation of a very attracting peak in the spectra of corot 0102618121 allowed us to certify its spurious origin . once that the instrumental effects were properly removed , no additional mode was detected . on the other hand , cycle - to - cycle variations of the fourier parameters were observed , but very small and always within . among the seven cepheids , there are two pop . i first - overtone pulsators , four pop . i fundamental mode pulsators , and one pop . ii star . the corot colours allowed us to measure that times of maximum brightness occur a little earlier ( about 0.01 period ) at short wavelengths than at long ones . [ firstpage ] methods : data analysis stars : individual : v376 mon stars : individual : dp mon stars : interiors stars : oscillations stars : variables : cepheids

cond-mat0511472

i

dispersions of non - spherical colloidal particles are model systems to study a broad range of phenomena in condensed matter ranging from fluid phase separation to the emergence of liquid crystalline ordering . examples of such systems are clay suspensions @xcite , dispersed gibbsite platelets @xcite , mixtures of silica spheres and silica - coated boehmite rods @xcite , wax disks @xcite , or nonaqueous suspensions of laponite and montmorillonite @xcite . while already pure systems often possess complex liquid crystal phase behavior , binary mixtures additionally may demix into bulk phases with different chemical compositions , e.g. like in dispersions of disks and spheres @xcite . such mixtures are often viewed as composed of a primary species ( here the spheres ) that interact with an effective depletion potential that is generated by the second component referred to as the depletion agent ( here the disks ) . the depletion interaction is primarily attractive with a range of attraction similar to the size of the depletant agent and with a strength that is ruled by the concentration of the depletion agent . under appropriate conditions the depletion interaction may be sufficiently strong in order to drive a phase transition that is anaologous to the gas - liquid phase separation in simple substances , and that manifests fluid - fluid demixing when regarding the full mixture . recent theoretical work has been devoted to fluids of platelike particles near a hard wall @xcite , an interaction site model for lamellar colloids @xcite , wetting and capillary nematization of binary hard - platelet and hard - rod fluids @xcite , the lamellar zwanzig model @xcite , and colloidal hard - rod fluids near geometrically structured substrates @xcite . the phase diagram of mixtures of hard colloidal spheres and discs was obtained within a free - volume scaled particle approach @xcite . the depletion potential between two spheres immersed in a sea of platelets was studied in detail @xcite , and found to compare well with predictions from the derjaguin approximation @xcite . sedimentation was found to influence liquid crystal phase transitions of colloidal platelets @xcite , as well as multi - phase equilibria in mixtures of platelets and ideal polymer @xcite . the fundamental - measure theory ( fmt ) is an ( approximate ) density - functional theory ( dft ) @xcite , originally proposed by rosenfeld for additive hard sphere mixtures @xcite . an early extension to convex non - spherical particles has been given in refs . when applied to homogeneous and isotropic fluid states , this theory yields the correct second virial coefficient of the equation of state , but fails to recover the exact density functional up to second order in density . to remedy the latter problem , an interpolation between the hard sphere rosenfeld functional and the onsager functional for elongated rods was proposed @xcite and applied to the bulk isotropic - nematic transition . an fmt that originated from the treatment of parellel hard cubes @xcite was also used very successfully for the zwanzig model , see e.g.@xcite . see also refs.@xcite for the treatment of various hard body system mixtures . recently attention was paid to binary mixtures of hard rods and polymers @xcite . also notable is the treatment of hard body fluids based on two - point measures @xcite . the bolhuis - frenkel model of hard spheres and vanishingly thin hard needles @xcite can be considered as the simplest model hard core mixture of spheres and rods . previous work was concerned with the formulation of a dft for this model @xcite , and an extension to include rod - rod interaction on the onsager ( second virial ) level @xcite . predictions for the orientation ordering of the rods at a free interface between ( isotropic ) sphere - rich and sphere - poor phases @xcite were successfully confirmed by simulations @xcite . adding a third component to this system , the fluid demixing phase behavior of ternary mixtures of spheres , rods , and model polymers was found to be rich @xcite . quenching one of the components led to investigations of hard spheres immersed in random rod networks @xcite , and of the isotropic - nematic transition of rods in matrices of immobilized spheres @xcite . the present paper is concerned with the introduction of hard platelets to the fmt framework . central to fmt is the so - called deconvolution of the mayer bond @xmath0 between particles of species @xmath1 and @xmath2 into `` weight functions '' @xmath3 that are characteristic of the shape of particles of each species @xmath1 ; the label @xmath4 indicates the type of weight function . recall that the mayer bond is related to the pair interaction potential @xmath5 between species @xmath1 and @xmath2 via @xmath6 , where @xmath7 , @xmath8 is the boltzmann constant and @xmath9 is temperature . for hard bodies @xmath10 if the two particles overlap and @xmath11 otherwise . in the fmt framework a convolution integral is used to express @xmath0 as a sum of terms each of which is bilinear in the weight functions , i.e. possesses the structure @xmath12 , where @xmath13 label the species , @xmath14 label the type of weight function , and the asterisk indicates the ( three - dimensional ) convolution . the weight functions are characteristic of the geometry of single particles , i.e. vanish ( already ) beyond the physical extent of _ one _ particle . note that this is different from the behavior of @xmath0 , which extends to larger separations and vanishes provided _ two _ particles do not overlap . however the convolution of pairs of ( shorter ranged ) weight functions restores the correct ( longer ranged ) non - locality of the mayer bond . this concept was originally introduced for hard sphere mixtures @xcite . an extension to arbitrarily shaped convex bodies was proposed in refs.@xcite , which yields the correct second virial coefficients , but gives only an approximation for the mayer bonds . this is insufficient to describe e.g. nematic ordering ( see e.g. ref . @xcite for a discussion ) . however , obtaining exact deconvolutions is possible in principle , as the cases of the mayer bond between a sphere and a vanishingly thin needle @xcite and between a pair of thin rods @xcite demonstrate . to give an overview of the general structure of fmt , the weight functions @xmath15 are used to to build so - called weighted densities via convolution with the bare one - body density distribution ( of each species in the case of a mixture ) . the weighted densities are then input as arguments to an ( analytically given ) excess free energy density , @xmath16 . integrating @xmath16 over space ( and director space in case of anisotropic particles ) yields the excess free energy of the system , which is the only unknown contribution to the grand potential functional being the central quantity in ( classical ) dft . for any external potential ( like describing gravity or a wall ) minimizing the grand potential with respect to the one - body density distribution(s ) gives both the value of the grand potential and the structural information contained in the density profile(s ) . here we show how to treat platelet - shaped particles and mixtures with both spheres and needles within the same framework . the relevant weight functions are given and it is shown explicitly how all mayer bonds for the ternary mixture are obtained through convolutions . for two - dimensional hard disks the rosenfeld functional only yields an approximation for the exact mayer bond . despite this deficieny the two - dimensional rosenfeld functional is considered to be a useful tool to study inhomogeneous situations , see e.g. ref . @xcite for an investigation of laser - induced freezing and melting of confined colloidal particles . the treatment of the sphere - platelet case inevitably leads to the same defect . we do achieve , however , exact deconvolutions of the mayer bonds for platelet - platelet and platelet - needle interactions . furthermore we treat explicitly contributions to the excess free energy functional that are of third order in densities . while the treatment of second order terms was a crucial input for the original construction of fmt for hard spheres @xcite , this route has been only recently pursued in detail @xcite , and the authors conclude that they come `` close to the edge of fmt '' in attempting to incorporate properties of the exact expansion of the free energy functional in density on the third virial level into the approximate framework of fmt . in particular these delicate cases appear when particle touch , i.e. configurations that are close to the edge of the particles . hence we approximate the exact third virial level ( the triangle in the diagrammatic mayer @xmath0-expansion ) by terms that are non - vanishing only for cases with common triple intersection of the three particles involved . global prefactors are used to compensate for the `` lost cases '' @xcite in order to yield reasonable third virial coefficients . as a test case for the accuracy of the theory we consider the pure system of hard platelets . this is known to undergo a weak first order isotropic - nematic phase transition @xcite . in contrast to the application to thin rods , the onsager ( second virial ) functional for platelets yields only a qualitatively correct account of the phase transition : the transition density , density jump , and value of the nematic order parameter in the coexisting nematic are significantly overestimated . our theory has the same ( exact ) second order contribution to the free energy , but also sports a further third - order term . higher than third - order terms are absent due to the scaled - particle roots @xcite of the approach ; the vanishing thickness and hence vanishing volume of the platelets effectively truncates the series . our results show that fmt considerably improves on the onsager treatment , and yields a quantitatively correct picture , albeit with slightly too small transitition densities . harnau and dietrich recently considered bulk and wetting phenomena in a binary mixture of colloidal hard spheres and hard platelets @xcite , an interesting limiting case of the present ternary mixture . we postpone a detailed discussion of the relationship of the current work to ref . @xcite to sec . [ secplateconclusions ] . the paper is organized as follows . in sec . [ secplatemodel ] we define the model of a ternary hard body mixture of spheres , needles , and platelets . in sec . [ secdeconvolution ] the mayer bonds are represented as convolutions of ( weight ) functions characteristic for single particles . in sec . [ sectriangle ] we proceed to control the third virial level . readers who are interested in the structure of the resulting theory rather than its derivation might wish to skip secs.[secdeconvolution ] and [ sectriangle ] and directly go to sec.[secweighteddensities ] where the weighted densities are defined as convolutions of the weight functions with the bare density profiles . [ secdft ] presents the excess free energy functional . explicit expressions for the relevant quantities in planar and uniaxial geometry are given in sec.[secplanargeometry ] . we apply the theory to the isotropic - nematic transition of hard platelets in sec . [ secresults ] , and conclude in sec . [ secplateconclusions ] .

geometrical weight functions that are characteristic for each species are given and it is shown how convolutions of pairs of weight functions recover each mayer bond of the ternary mixture and hence ensure the correct second virial expansion of the excess free energy functional . the case of sphere - platelet overlap relies on the same approximation as does rosenfeld s functional for strictly two - dimensional hard disks . we explicitly control contributions to the excess free energy that are of third order in density . rev . e * 71 * , 011504 ( 2004 ) ] . as a test case of our approach

a geometry - based density functional theory is presented for mixtures of hard spheres , hard needles and hard platelets ; both the needles and the platelets are taken to be of vanishing thickness . geometrical weight functions that are characteristic for each species are given and it is shown how convolutions of pairs of weight functions recover each mayer bond of the ternary mixture and hence ensure the correct second virial expansion of the excess free energy functional . the case of sphere - platelet overlap relies on the same approximation as does rosenfeld s functional for strictly two - dimensional hard disks . we explicitly control contributions to the excess free energy that are of third order in density . analytic expressions relevant for the application of the theory to states with planar translational and cylindrical rotational symmetry , e.g. to describe behavior at planar smooth walls , are given . for binary sphere - platelet mixtures , in the appropriate limit of small platelet densities , the theory differs from that used in a recent treatment [ l. harnau and s. dietrich , phys . rev . e * 71 * , 011504 ( 2004 ) ] . as a test case of our approach we consider the isotropic - nematic bulk transition of pure hard platelets , which we find to be weakly first order , with values for the coexistence densities and the nematic order parameter that compare well with simulation results .

cond-mat0511472

r

already the binary subsystems with platelets ( leaving aside the full ternary mixture ) , i.e. the sphere - platelet and rod - platelet mixtures , are expected to display rich bulk and interfacial properties due to the competition between depletion and orientational order . in order to find a simple yet demanding test case for the present theory , we restrict ourselves further to the pure system of platelets . in bulk this undergoes an isotropic - nematic phase transition , where @xmath261 in the isotropic phase , and @xmath262 in the nematic phase being peaked around the nematic director that indicates the preferred direction of alignment of the particles ; @xmath263 is the angle between @xmath264 and the nematic director . the strength of the nematic order is conveniently measured via the nematic order parameter , @xmath265 where @xmath266 is the second legendre polynomial . we use the generic form of the free energy , eq.([eqfexcgeneric ] ) , as a multiple integral over a free energy density appropriate for pure platelets , @xmath224 as given in eq.([eqphid ] ) . this function depends on the weighted densities @xmath267 , @xmath268 , and @xmath269 , as defined in eqs.([eqnnud ] ) ( with @xmath270 ) , ( [ eqn1dd ] ) , and ( [ eqn2ddd ] ) , respectively . the weighted densities in turn are built as convolutions of the bare platelet one - body density , @xmath271 , with the weight functions @xmath272 , @xmath273 , and @xmath274 , as given in ( [ eqw2drosenfeld ] ) , ( [ eqw1dd ] ) , and ( [ eqw2ddd ] ) [ and summarized in eqs . ( [ eqsumw2drosenfeld ] ) , ( [ eqsumw1dd ] ) , and ( [ eqsumw2ddd ] ) ] , respectively . to carry out the actual calculations , we have started from the corresponding expressions in planar geometry ( see sec . [ secplanargeometry ] ) , that we have further ( numerically ) reduced to @xmath100-independent quantities , as appropriate for the spatially homogeneous isotropic and nematic phases . alternatively , our fmt excess free energy functional for hard platelets can be viewed as being composed of the exact second - order virial contribution and a further ( approximate ) third order term that i ) vanishes for configurations without common triple intersection of three platelets , and ii ) is constant for configurations with common triple intersection ; the constant is adjusted to give a reasonable value for the third virial coefficient . note that the theory for the full ternary mixture has a similar structure , but features higher order ( in density ) contributions that arise from the finite volume , and hence finite packing fraction , of the hard spheres . we minimize the grand potential functional @xmath275 ) = f_{\rm id}[\rho_d ] + f_{\rm exc}[\rho_d ] -\mu_d \int d{{\bf r}}\frac{d{{\bf \omega}}}{4\pi } \rho_d({{\bf r}},{{\bf \omega } } ) , \label{eqgrandpotential}\ ] ] where the ideal gas contribution is given by @xmath276 = \int d{{\bf r}}\frac{d{{\bf \omega}}}{4\pi } \rho_d({{\bf r}},{{\bf \omega}})[\ln(\rho_d({{\bf r}},{{\bf \omega}})\lambda_d^3)-1],\ ] ] where @xmath277 is the chemical potential , the dependence on temperature and volume has been suppressed in the notation , and we formally set the thermal wavelength equal to the platelet radius , @xmath278 , to fix an arbitrary additive constant to the chemical potential . for a given value of the ( scaled ) chemical potential @xmath279 , we minimized @xmath280 with respect to @xmath281 in either phase . inserting the resulting distribution into eq.([eqgrandpotential ] ) yields the grand potential of the system , and the condition for equality of the grand potential in both phases locates the phase transition . the numerical implementation uses free minimization , i.e. no parametrized form of @xmath282 is assumed a priori . we use an equidistant grid in @xmath263-space with 100 grid points in the interval @xmath283 $ ] . the integration over @xmath284 in eq.([eqw2dddplanar ] ) to obtain @xmath274 was done with 200 grid points in @xmath285 $ ] ; this only needs to be performed once at the start of the calculation . we have used alternatively fixed - point iteration or simulated annealing to minimize the density profile ; results from both approaches were consistent . up to 1000 steps were sufficient to obtain convergence . .results for the bulk isotropic - nematic phase transition from different approaches . given are values for the ( scaled ) coexisting densities in the isotropic and nematic phase , @xmath286 and @xmath287 , respectively , the ( scaled ) density jump , @xmath288 , and the value of the nematic order parameter in the coexisting nematic phase , @xmath289 , as obtained from fmt , and the simulation results by frenkel and eppenga @xcite , as well as theoretical results obtained from the onsager second virial functional for hard platelets . [ cols="<,<,<,<",options="header " , ] the resulting values of the densities in the coexisting isotropic and nematic phases , @xmath290 and @xmath291 , respectively , as well as the value of the order parameter in the coexisting nematic phase , @xmath289 , are in good agreement with frenkel and eppenga s classic simulation results @xcite , see tab . [ tabtransitionin ] . the fmt values somewhat underestimate the coexistence densities and give a too high value for @xmath289 . the onsager second virial functional , on the other hand , overestimates significantly the coexistence densities as well as @xmath289 . hence the actual phase transition , as characterized by the jump in coexistence densities and in order parameter , is weaker than predicted by the onsager treatment . to further illustrate our findings , we plot in fig . [ fig : sofmu ] the variation of @xmath17 with @xmath292 and in fig . [ fig : sofrho ] the variation of @xmath17 with @xmath293 . we also plot the behavior of the metastable nematic branch , i.e. for statepoints where the nematic phase remains locally stable against small fluctuations , although globally the grand potential for the isotropic phase is the lower one . it is apparent that the value of @xmath289 very sensitively depends on the precise location of the phase transition . taking this into account we find the overall the agreement of the fmt results with simulation data to be very reasonable , and to give confidence both into applications to interfacial phenomena of platelets , as well as into the accuracy of the fmt for the full ternary mixture . for a bulk system of pure hard platelets as a function of the ( scaled ) chemical potential , @xmath292 . shown are results from fmt in the stable ( solid line ) and metastable ( dashed line ) nematic . values in the coexisting nematic and isotropic phases , @xmath289 and @xmath294 , are shown as obtained from fmt ( squares ) and the onsager - type second virial treatment ( dots ) . ] , but as a function of the ( scaled ) density of platelets @xmath295 . note the considerable reduction of the coexistence density gap obtained from fmt as compared to the onsager - like treatment . ]

for binary sphere - platelet mixtures , in the appropriate limit of small platelet densities , the theory differs from that used in a recent treatment [ l. harnau and s. dietrich , phys . we consider the isotropic - nematic bulk transition of pure hard platelets , which we find to be weakly first order , with values for the coexistence densities and the nematic order parameter that compare well with simulation results .

a geometry - based density functional theory is presented for mixtures of hard spheres , hard needles and hard platelets ; both the needles and the platelets are taken to be of vanishing thickness . geometrical weight functions that are characteristic for each species are given and it is shown how convolutions of pairs of weight functions recover each mayer bond of the ternary mixture and hence ensure the correct second virial expansion of the excess free energy functional . the case of sphere - platelet overlap relies on the same approximation as does rosenfeld s functional for strictly two - dimensional hard disks . we explicitly control contributions to the excess free energy that are of third order in density . analytic expressions relevant for the application of the theory to states with planar translational and cylindrical rotational symmetry , e.g. to describe behavior at planar smooth walls , are given . for binary sphere - platelet mixtures , in the appropriate limit of small platelet densities , the theory differs from that used in a recent treatment [ l. harnau and s. dietrich , phys . rev . e * 71 * , 011504 ( 2004 ) ] . as a test case of our approach we consider the isotropic - nematic bulk transition of pure hard platelets , which we find to be weakly first order , with values for the coexistence densities and the nematic order parameter that compare well with simulation results .

1408.1002

i

recent advances in biotechnology are allowing us to measure cellular properties at an unprecedented level of detail @xcite . for instance , it is now possible to routinely measure various molecular entities ( e.g. dna methylation , mrna and protein expression , snps ) genome - wide in hundreds if not thousands of cellular specimens @xcite . in addition , other molecular data detailing interactions between proteins or between transcription factors and regulatory dna elements are growing at a rapid pace @xcite . all these types of data are now widely referred to collectively as `` omic '' data . the complexity and high - dimensional nature of this omic data presents a daunting challenge to those wishing to analyse and interpret the data @xcite . the difficulty of analysing omic data is further compounded by the inherent complexity of cellular systems . cells are prime examples of organized complex systems , capable of highly stable and predictable behaviour , yet an understanding of how this deterministic behaviour emerges from what is a highly complex and probabilistic pattern of dynamic interactions between numerous intra and extracellular components , still eludes us @xcite . thus , elucidating the systems - biological laws or principles dictating cellular phenotypes is also key for an improved analysis and interpretation of omic data . furthermore , important biological phenomena such as cellular differentiation are fundamentally altered in diseases like cancer @xcite . hence , an attempt to understand how cellular properties emerge at a systems level from the properties seen at the individual gene level is not only a key endeavour for the systems biology community , but also for those wanting to translate basic insights into effective medical advances @xcite . + it is now well accepted that at a fundamental level most biological systems are best modeled in terms of spatial interactions between specific entities ( e.g. neurons in the case of the brain ) , which may or may not be dynamically changing in time @xcite . it therefore seems natural to also use the mathematical and physical framework of networks to help us analyse and interpret omic data at a systems level @xcite . indeed , the cellular phenotype is determined to a large extent by the precise pattern of molecular interactions taking place in the cell , i.e. a molecular interaction network @xcite . although this network has spatial and dynamic dimensions which at present remain largely unexplored due to technological or logistical limitations , there is already a growing number of examples where network - based analysis strategies have been instrumental @xcite . for instance , a deeper understanding of why sustained treatment with egfr inhibitors can lead to dramatic sensitization of cancer cell lines to cytotoxic agents was possible thanks to a systems approach @xcite . another study used reverse engineering network approaches to identify and validate drug targets in glioblastoma multiforme , to be further tested in clinical trials @xcite . what is key to appreciate here is that these successes have been achieved in spite of noisy and incomplete data , suggesting that there are simple yet deep systems biological principles underlying cellular biology that we can already probe and exploit with current technology and data . thus , with future improvements in data quality and coverage , network - based analysis frameworks will play an ever increasing and important role in systems biology , specially at the level of systems analysis and interpretation @xcite . therefore , it is also imperative to develop novel , more powerful , network - theoretical methods for the systems analysis of omic data . + in adopting a network s perspective for the analysis and interpretation of omic data , there are in principle two different ( but not mutually exclusive ) approaches one can take . one possibility is to infer ( i.e. reverse engineer ) the networks from genome - wide data @xcite . most of these applications have done this in the context of gene expression data , with the earliest approaches using clustering or co - expression to postulate gene interdependencies @xcite . partial correlations and graphical gaussian models have proved useful as a means of further refining correlation networks by allowing one to infer the more likely direct interactions while simultaneously also filtering out those which are more likely to be indirect @xcite . these methods remain popular and continue to be studied and improved upon @xcite . other methods have drawn on advanced concepts from information theory , for instance aracne ( `` algorithm for the reconstruction of accurate cellular networks '' ) has been shown to be successful in infering regulatory networks in b - cells @xcite . + in stark contrast to reverse engineering methods , another class of algorithms have used structural biological networks from the outset , using these as scaffolds to integrate with omic data . specifically , by using a structural network one can sparsify the correlation networks inferred from reverse - engineering approaches , thus providing another means of filtering out correlations that are more likely to be indirect @xcite . besides , integration with a structural network automatically provides an improved framework for biological interpretation @xcite . the structural networks themselves are typically derived from large databases , which detail literature curated experimentally verified interactions , including interactions derived from yeast 2 hybrid screens ( y2h ) @xcite . the main example is that of protein protein interaction ( ppi ) maps , which have been generated using a number of different complementary experimental and in - silico approaches , and merging these maps from these different sources together has been shown to be an effective approach in generating more comprehensive high - confidence interaction networks @xcite . ppi networks have been used mainly as a means of integrating and analysing gene expression data ( see e.g. @xcite ) . more recently , this approach has also been successfully applied in the dna methylation context , for instance it has been shown that epigenetic changes associated with age often target specific gene modules and signalling pathways @xcite . + another class of methods that have used structural networks , ppis in particular , have integrated them with gene expression data to define an approximation to the underlying signaling dynamics on the network , thus allowing more in - depth exploration of the interplay between network topology and gene expression . typically , these studies have used the notion of random walks on weighted graphs where the weights are constructed from differential expression statistics , and where the aim is to identify nodes ( genes ) in the network which may be important in dictating the signaling flows within the pathological state . for instance , among these random walk methods is netrank , a modification of the google pagerank algorithm , which was able to identify novel , robust , network based biomarkers for survival time in various cancers @xcite . other random walk based approaches , aimed at identifying causal drivers of specific phenotypes ( e.g. expression or cancer ) , have modeled signal transduction between genes in the causal and phenotypic layers as flows in an electric circuit diagram , an elegant formulation capable of not only identifying the likely causal genes but also of tracing the key pathways of information flow or dysregulation @xcite . random walk theory has also been employed in the development of differential network methodologies . an example is netwalk @xcite , which is similar to netrank but allows differential signaling fluxes to be inferred . this approach was successful in identifying and validating the glucose metabolic pathway as a key determinant of lapatinib resistance in erbb2 positive breast cancer patients @xcite . + another important concept to have emerged recently is that of network rewiring @xcite . this refers to the changes in interaction patterns that accompany changes in the cellular phenotype . network rewiring embodies the concept that it is the changes in the interaction patterns , and not just the changes in absolute gene expression or protein activity , that are the main determinants of the cellular phenotype . that network rewiring may be key to understanding cellular phenotypes was most convincingly demonstrated in a differential epistasis mapping study conducted in yeast cells exposed to a dna damaging agent @xcite . specifically , what this study demonstrated is that responses to perturbations or cellular stresses are best understood in terms of the specific rewiring of protein complexes and functional modules . thus , this conceptual framework of network rewiring may apply equally well to the genetic perturbations and cellular stresses underlying disease pathologies like cancer . + in this article we advocate a network - theoretical framework based on statistical mechanical principles and more specifically on the notion of signalling entropy @xcite . this theoretical framework integrates gene expression ( but in principle also other functional data ) with a ppi network , merging existing concepts such as signaling dynamics ( i.e. random walks ) and network rewiring with that of signalling entropy . in previous work we have shown how signalling entropy ( i ) provides a proxy to the elevation in waddington s epigenetic landscape , correlating with a cell s differentiation potential @xcite , ( ii ) how it can be used to identify signaling pathways and nodes important in differentiation and cancer @xcite , and ( iii ) how it predicts two cancer system - omic hallmarks : ( a ) cancer is characterised by an increase in signalling entropy and ( b ) local signaling entropy changes anti - correlate with differential gene expression @xcite . here , we present and unify the different signaling entropy measures used previously and further explore a novel application of signalling entropy to understanding drug sensitivity profiles in cancer cell lines . specifically , we first use simulated data to justify the existence of an entropy - robustness theorem , and subsequently provide empirical evidence for this theorem by demonstrating that increases in local signalling entropy correlate with drug resistance ( robustness ) . we further show the importance of network topology in dictating the signalling entropy changes underlying drug response . in addition , we provide r - functions implementing the entropy rate calculation and ranking of genes according to differential entropy , all freely available from _ sourceforge.net/projects/signalentropy/files/ _ .

specifically , we propose the notion of cellular signalling entropy ( or uncertainty ) , as a novel means of analysing and interpreting omic data , and more fundamentally , as a means of elucidating systems - level principles underlying basic biology and disease . we describe the power of signalling entropy to discriminate cells according to differentiation potential and cancer status . we further argue the case for an empirical cellular entropy - robustness correlation theorem and demonstrate its existence in cancer cell line drug sensitivity data . specifically , we find that high signalling entropy correlates with drug resistance and further describe how entropy could be used to identify the achilles heels of cancer cells . in summary , signalling entropy is a deep and powerful concept , based on rigorous statistical mechanical principles , which , with improved data quality and coverage , will allow a much deeper understanding of the systems biological principles underlying normal and disease physiology .

a key challenge in systems biology is the elucidation of the underlying principles , or fundamental laws , which determine the cellular phenotype . understanding how these fundamental principles are altered in diseases like cancer is important for translating basic scientific knowledge into clinical advances . while significant progress is being made , with the identification of novel drug targets and treatments by means of systems biological methods , our fundamental systems level understanding of why certain treatments succeed and others fail is still lacking . we here advocate a novel methodological framework for systems analysis and interpretation of molecular omic data , which is based on statistical mechanical principles . specifically , we propose the notion of cellular signalling entropy ( or uncertainty ) , as a novel means of analysing and interpreting omic data , and more fundamentally , as a means of elucidating systems - level principles underlying basic biology and disease . we describe the power of signalling entropy to discriminate cells according to differentiation potential and cancer status . we further argue the case for an empirical cellular entropy - robustness correlation theorem and demonstrate its existence in cancer cell line drug sensitivity data . specifically , we find that high signalling entropy correlates with drug resistance and further describe how entropy could be used to identify the achilles heels of cancer cells . in summary , signalling entropy is a deep and powerful concept , based on rigorous statistical mechanical principles , which , with improved data quality and coverage , will allow a much deeper understanding of the systems biological principles underlying normal and disease physiology . entropy ; network ; signalling ; genomics ; drug resistance ; cancer ; differentiation ; stem cell

1408.1002

m

loosely defined , entropy of a system , refers to a measure of the disorder , randomness or uncertainty of processes underlying the system s state . in the context of a single cell , signalling entropy will reflect the amount of overall disorder , randomness or uncertainty in how information , i.e. signaling , is passed on in the molecular interaction network . at a fundamental level , all signaling in a cell is probabilistic , determined in part by the relative cellular concentrations of the interacting molecules . hence , in discussing cellular signalling entropy , it is useful to picture a molecular interaction network in which edges represent possible interactions and with the edge weights reflecting the relative probabilities of interaction ( * fig.1a * ) . thus , an interaction network in a high - entropy state is characterised by signaling interaction probabilities that are all fairly similar in value , whereas a low - entropy state will be characterised by specific signalling interactions possessing much higher weights ( * fig.1a * ) . why would this type of signalling entropy , loosely defined as the amount of uncertainty in the signaling interaction patterns , be useful to systems biology ? one way to think of signalling entropy is as representing signaling promiscuity , which has been proposed as a key systems feature underlying the pluripotent or multipotent capacity of cells ( * fig.1b * ) @xcite . indeed , it has been suggested that pluripotency is an intrinsic statistical mechanical property , best defined at the cellular population level @xcite . specifically , it has been demonstrated that pluripotent stem cells exhibit remarkable heterogeneity in gene expression levels , including well known stem cell markers such as _ nanog _ @xcite . it is also well known that a large number of genes , many encoding transcription factors , exhibit low - levels of expression in stem cells , yet simultaneously are being kept in a poised chromatin state , allowing immediate activation if this were required @xcite . thus , in a pluripotent stem cell like state , signal transduction is in a highly egalitarian and , thus , promiscuous state , i.e. a state of high signalling entropy . conversely , differentiation leads , by necessity , to activation of specific transcription factors and pathways and thus to a lowering in the uncertainty of signaling patterns , and thus to a lowering of entropy . we recently demonstrated , using gene expression profiles of over 800 samples , comprising cells of all major stages of differentiation , including human embryonic stem cells ( hescs ) , induced pluripotent stem cells ( ipscs ) , multipotent cell types ( e.g. hematopoietic stem cells ( hscs ) ) , and terminally differentiated cells within these respective lineages , that signalling entropy not only correlates with differentiation potential but that it provides a highly quantitative measure of potency @xcite . indeed , we showed that signalling entropy provides a reasonably good approximation to the energy potential in waddington s epigenetic landscape @xcite . + here we decided to explore the concept of signalling entropy in relation to cellular robustness and specifically to drug resistance in cancer . that signalling entropy may be informative of a cell s robustness is a proposal that stems from a general ( but unproven ) theorem , first proposed by manke and demetrius @xcite : namely , that a system s entropy and robustness are correlated . mathematically , this can be expressed as @xmath0 , which states that a positive change in a system s entropy ( i.e. @xmath1 ) must lead to an increase in robustness ( @xmath2 ) . now , cells are fairly robust entities , having evolved the capacity to buffer the intra - and - extracellular stresses and noise which they are constantly exposed to @xcite . much of this overall stability and robustness likely stems from the topological features of the underlying signaling and regulatory networks , for instance features such as scale - freeness and hierarchical modularity , which are thought to have emerged through natural evolutionary processes such as gene duplication @xcite . however , another key feature which contributes to cellular robustness is cross - talk and signalling pathway redundancy @xcite . pathway redundancy refers to a situation where a cell has the choice of transmitting signals via two or more possible routes . in the language of statistical mechanics , this corresponds to a state of high uncertainty ( or entropy ) in signaling . high signalling entropy could thus underpin a cell s robustness to perturbations , suggesting that a cell s entropy and robustness may indeed be correlated ( * fig.2a * ) . consistent with this , pathway redundancy is also well recognized to be a key feature underlying drug resistance of cancer cells @xcite . ) , then the system s robustness @xmath3 must also decrease , i.e. @xmath4 , so that the product @xmath0 . observe how in the low entropy state , random removal of edges through e.g. inactivating mutations , can lead to deactivation of a key signaling pathway connecting a given start and end nodes ( shown in orange ) . in the high entropy state , the same perturbations do not prevent signal transduction between the two orange nodes . * b ) * depicted are the effects of two major forms of cancer perturbation . in the upper panel , inactivation ( typically of tumour suppressors ) , leads to underexpression and a corresponding loss of correlations / interactions with neighbors in the ppi network . this is tantamount to a state of increased entropy and drug intervention is unlikely to be effective . in the lower panel , we depict the case of an oncogene , which is overexpressed in cancer . this overexpression leads to activation of a specific oncogenic pathway which results in oncogene addiction and increased sensitivity to targeted drug intervention . thus , local signalling entropy and robustness ( as determined by response to a drug ) , may also correlate locally . ] further supporting the notion that entropy and robustness may be correlated , we previously showed that ( i ) cancer is characterised by a global increase in signalling entropy compared to its respective normal tissue @xcite , in line with the observation that cancer cells are specially robust to generic perturbations , and ( ii ) that a gene s differential entropy between normal and cancer tissue is generally speaking anticorrelated with its differential expression @xcite , consistent with the view that cancer cells are specially sensitive to drug interventions that target overexpressed oncogenes , a phenomenon known as oncogene addiction ( * fig.2b * ) @xcite . interestingly , the observations that differential entropy and differential expression are anti - correlated and that cancer is characterised globally by an increase in entropy @xcite , are also consistent with the prevalent view that most driver mutations are indeed inactivating , targeting tumor suppressor genes @xcite . hence , based on all of these observations and insights , we posited that signalling entropy could also prove useful as a means of predicting drug resistance in cancer cells . briefly , we review the definitions and construction of signalling entropy as used in our previous studies @xcite . the construction relies on a comprehensive and high - confidence ppi network which is integrated with gene expression data @xcite ( see * appendix a * ) . briefly , the ppi is used as a scaffold , and edge weights are constructed from the gene expression data to approximate the interaction or signaling probabilities between the corresponding proteins in the ppi . thus , the gene expression data provides the biological context in which to modulate the ppi interaction probabilities . to compute signalling entropy requires the estimation of a stochastic matrix , reflecting these interaction probablities over the network . + the construction of the stochastic matrix can proceed in two different ways . in the earliest studies we used a construction which was done at the level of phenotypes @xcite . under this model , edge weights @xmath5 between proteins @xmath6 and @xmath7 were constructed from the correlation between the expression levels of the corresponding genes @xmath6 and @xmath7 , as assessed over independent samples all representing the same phenotype . estimating the correlations over independent samples , all within the same phenotype , can be viewed as representing a multifactorial perturbation experiment , with e.g. genetic differences between individuals mimicking specific perturbations , and thus allowing influences to be inferred . thus , this approach hinges on the assumption that highly correlated genes , whose coding proteins interact , are more likely to be interacting in the given phenotype than two genes which do not correlate . the use of a ppi as a scaffold is important to filter out significant correlations which only result from indirect influences . the correlations themselves can be defined in many different ways , for instance , using simple pearson correlations or non - linear measures such as mutual information . for example , one way to define the weights is as @xmath8 with @xmath9 describing the pearson correlation coefficient between genes @xmath6 and @xmath7 . this definition guarantees positivity , but also treats positive and negative correlations differently , which makes biological sense because activating and inhibitory interactions normally have completely distinct consequences on downstream signalling . thus , the above weight scheme treats zero or insignificant correlations as intermediate , consistent with the view that an absent interaction is neither inhibitory nor activating . however , other choices of weights are possible : e.g. @xmath10 , which treats negative and positive correlations on an equal footing . once edge weights are defined as above , these are then normalised to define the stochastic matrix @xmath11 over the network , @xmath12 with @xmath13 denoting the ppi neighbors of gene @xmath6 . thus , @xmath11 is the probability of interaction between genes @xmath6 and @xmath7 , and as required , @xmath14 . however , there is no requirement for @xmath11 to be doubly stochastic , i.e. @xmath15 is in general not a symmetric matrix . hence , edges are bi - directional with different weights labeling the probability of signal transduction from @xmath6 to @xmath7 , and that from @xmath7 to @xmath6 ( @xmath16 ) . + an alternative to the above construction of the stochastic matrix is to invoke the mass action principle , i.e. one now assumes that the probability of interaction in a given sample is proportional to the product of expression values of the corresponding genes in that sample @xcite . thus , the ppi is again used as a scaffold to only allow interactions supported by the ppi network , but the weights are defined using the mass action principle , as @xmath17 where @xmath18 denotes the normalised expression intensity value ( or normalised rna - seq read count ) of gene @xmath6 . an important advantage of this construction is that the stochastic matrix is now sample specific , as the expression values are unique to each sample . + given a stochastic matrix , @xmath11 , constructed using one of the two methods above , one can now define a local shannon entropy for each gene @xmath6 as @xmath19 where @xmath20 denotes the degree of gene @xmath6 in the ppi network . the normalisation is optional but ensures that this local shannon entropy is normalised between 0 and 1 . clearly , if only one weight is non - zero , then the entropy attains its minimal value ( 0 ) , representing a state of determinism or lowest uncertainty . conversely , if all edges emanating from @xmath6 carry the same weight , the entropy is maximal ( 1 ) , representing the case of highly promiscuous signaling . in principle , local shannon entropies can thus be compared between phenotypes to identify genes where there are changes in the uncertainty of signaling . in the case where entropies are estimated at the phenotype level , jackknife approaches can be used to derive corresponding differential entropy statistics @xcite . deriving statistics is important because node degree has a dramatic influence on the entropy variance , with high degree nodes exhibiting significantly lower variability in absolute terms , which nevertheless could be highly significant @xcite . in the case where entropies are estimated at the level of individual samples , ordinary statistical tests ( e.g. rank sum tests ) can be used to derive sensible p - values , assuming of course that enough samples exist within the phenotypes being compared . + in addition to the local entropy , it is also of interest to consider statistical properties of the distribution of local entropies , for instance their average . comparing the average of local entropies between two phenotypes would correspond to a comparison of non - equilibrium entropy rates . to see this , consider the formal definition of the entropy rate @xmath21 @xcite , i.e. @xmath22 where @xmath23 is the stationary distribution ( equivalently the left eigenvector with unit eigenvalue ) of @xmath15 ( i.e. @xmath24 ) , and where now @xmath25 note that the entropy rate @xmath21 is an equilibrium entropy since it involves the stationary distribution of the random walker . as such , the entropy rate also depends on the global topology of the network . thus , the entropy rate is a weighted average of the local unnormalized entropies , with the weights specified by the stationary distribution . it follows that comparing the unweighted averages of local entropies reflects a comparison of a non - equilibrium entropy rate since the stationary distribution is never used . the entropy rate constructed using the mass action principle is sample specific . we previously demonstrated that this entropy rate was highly discriminative of the differentiation potential of samples within a developmental lineage , as well as being highly discriminative of normal and cancer tissue @xcite ( * fig.3a * ) . since the entropy rate takes as input a ppi network and a sample s genome - wide expression profile , the significance of the resulting entropy values , as well as that of the difference between entropy rates , also needs to be assessed relative to a null distribution in which the putative information content between network and gene expression values is non - existent . in fact , since the weights in the network determine the entropy rate and these weights are dependent on both the specific network nodes and their respective gene expression profiles , it is natural to assess the significance of the entropy rate by `` destroying '' the mutual information between the network nodes and their gene expression profiles , for instance by randomising ( i.e. permuting ) the gene expression profiles over the network . thus , under this randomisation , the topological properties of the network remain fixed , but the weights are redefined . application of this randomisation procedure to the normal / cancer expression set considered previously @xcite ( * fig.3a * ) shows that the discriminatory potential is significantly reduced upon permuting the gene expression values over the network ( * fig.3b - c * ) . importantly , we observe that the entropy rate is much higher in the normal and cancer states compared to the rates obtained upon randomisation of the gene expression profiles ( * fig.3b * ) , indicating that both normal and cancer states are characterised by a higher level of signaling promiscuity compared to a network with random weights . that the discrimination between normal and cancer is significantly reduced in the randomly weighted network further demonstrates that there is substantial mutual information between the ppi network and the gene expression profiles , thus justifying the signalling entropy approach . a vignette / manual and user - friendly r - scripts that allow computation of the entropy rate is available at the following url : _ sourceforge.net / projects / signalentropy / files/_. here we briefly describe the salient aspects of this package : + _ the input : _ the main r - script provided ( _ compsr _ ) takes as input a user - specified ppi network , and a genome - wide expression vector representing the gene expression profile of a sample . it is assumed that this has been generated using either affy , illumina or rna - sequencing . in principle one ought to use as gene expression value the normalised unlogged intensity ( affy / illu ) or rna - seq count , since this is what should be proportional to the number of rna transcripts in the sample . however , in practice we advise taking the log - transformed normalised value since the log - transformation provides a better compromise between proportionality and regularisation , i.e. some regularisation is advisable since the underlying kinetic reactions are also regular . + _ the output : _ the r - functions provided in the package then allow the user to estimate the global entropy rate for a given sample , as well as the local normalised entropies for each gene in the integrated network . if a phenotype is specified then genes can be ranked according to the correlation strength of their local entropy to the phenotype . thus , the signalling entropy method allows us to assess ( i ) if the overall levels of signalling promiscuity is different between phenotypes , which could be important , for instance , to compare the pluripotent capacity of ipscs generated via different protocols or to assess the stem cell nature of putative cancer stem cells @xcite , and ( ii ) to rank genes according to differential entropy between phenotypes , allowing key signalling genes associated with differentiation , metastasis or cancer to be identified @xcite .

a key challenge in systems biology is the elucidation of the underlying principles , or fundamental laws , which determine the cellular phenotype . understanding how these fundamental principles are altered in diseases like cancer is important for translating basic scientific knowledge into clinical advances . while significant progress is being made , with the identification of novel drug targets and treatments by means of systems biological methods , our fundamental systems level understanding of why certain treatments succeed and others fail is still lacking .

a key challenge in systems biology is the elucidation of the underlying principles , or fundamental laws , which determine the cellular phenotype . understanding how these fundamental principles are altered in diseases like cancer is important for translating basic scientific knowledge into clinical advances . while significant progress is being made , with the identification of novel drug targets and treatments by means of systems biological methods , our fundamental systems level understanding of why certain treatments succeed and others fail is still lacking . we here advocate a novel methodological framework for systems analysis and interpretation of molecular omic data , which is based on statistical mechanical principles . specifically , we propose the notion of cellular signalling entropy ( or uncertainty ) , as a novel means of analysing and interpreting omic data , and more fundamentally , as a means of elucidating systems - level principles underlying basic biology and disease . we describe the power of signalling entropy to discriminate cells according to differentiation potential and cancer status . we further argue the case for an empirical cellular entropy - robustness correlation theorem and demonstrate its existence in cancer cell line drug sensitivity data . specifically , we find that high signalling entropy correlates with drug resistance and further describe how entropy could be used to identify the achilles heels of cancer cells . in summary , signalling entropy is a deep and powerful concept , based on rigorous statistical mechanical principles , which , with improved data quality and coverage , will allow a much deeper understanding of the systems biological principles underlying normal and disease physiology . entropy ; network ; signalling ; genomics ; drug resistance ; cancer ; differentiation ; stem cell

hep-ph0511098

i

the determination of the cabibbo - kobayashi - maskawa ( ckm ) matrix element @xmath2 from inclusive semileptonic @xmath6 decays requires theoretical predictions for partial decay rates , which are then compared to their experimentally measured values . because of a dominating @xmath7 background in a large portion of phase space , this procedure is adopted for a variety of restricted regions in phase space as obtained by , e.g. , accepting only events with one or more of the following features : charged - lepton energy @xmath8 , hadronic invariant mass @xmath9 , leptonic invariant mass @xmath10 , hadronic @xmath11 . here , @xmath12 , where @xmath13 denotes the energy and @xmath14 the three - momentum of the final hadronic state in the @xmath5-meson rest frame . for many of these cuts a hierarchy of energy scales exists in the decay process , for example @xmath15 , and shape - function effects become important @xcite . the theoretical expressions for differential decay rates in this region of phase space factorize into hard functions at the scale @xmath16 , and the convolution of jet functions and shape functions at the scale @xmath17 @xcite . while the jet functions are perturbatively calculable , shape functions are non - perturbative objects that capture all strong - interaction effects below the scale @xmath18 . at leading power the jet function @xmath19 and shape function @xmath20 are universal , and enter the qcd - factorization theorems for both the triple differential decay rate in @xmath1 decays and in the @xmath0 normalized photon spectrum @xcite . one strategy for the inclusive determination of @xmath2 is to use the @xmath0 photon spectrum to extract the leading shape function @xmath20 , which then allows for the calculation of arbitrary semileptonic partial decay rates @xcite . it was shown in that reference that the factorization approach can be applied to all commonly used kinematic cuts . in practice this program is realized by adopting a parameterizable model for the shape function , and fitting the parameters using information of the measured photon spectrum . it was emphasized in @xcite that such a model is only acceptable if the values for parameters are stable when fitted to different aspects of the photon spectrum , such as moments of it , or its functional form ( provided that the resolution is coarse enough to smear out hadronic resonances ) . with improving data on the photon spectrum such an approach might require further and further refinement of the models used . a different strategy is to eliminate the necessity for the extraction of the leading shape function , and to use the experimental data directly . such ideas have been investigated previously in @xcite for some specific cuts , where partial rates in semileptonic decays are expressed as weighted integrals over the @xmath21 photon spectrum , @xmath22 in the most recent analysis of this type , @xcite , the cut was chosen as @xmath11 for simplicity , and the weight function @xmath23 was calculated at complete two - loop order at the intermediate scale @xmath18 . the second term on the right - hand side of this equation denotes a residual hadronic power correction ( rhc ) , which was absorbed into the weight function in that reference . in the above relation @xmath24 denotes the partial semileptonic decay rate and @xmath25 the normalized photon spectrum in radiative decays , where @xmath26 , @xmath27 is the photon energy in the @xmath5-meson rest frame , and the total decay rate @xmath28 is defined to include all events with @xmath29 . it is beneficial to use the normalized photon spectrum instead of the absolute spectrum , because the weight function as defined in ( [ eq : sffree ] ) is independent of @xmath30 , possesses a well - behaved perturbative expansion @xcite , and because the normalized photon spectrum can be determined with better accuracy than the absolute one @xcite . in order to determine @xmath2 from relation ( [ eq : sffree ] ) both @xmath24 and the normalized photon spectrum enter as experimental input , while the weight function @xmath31 and the residual hadronic correction @xmath32 are theoretical quantities . in this paper , we extend the technology developed in @xcite to derive the weight function for an _ arbitrary _ kinematic cut that includes events with @xmath33 and @xmath34 . in general , the weight function @xmath31 ( and the correction @xmath32 ) , as well as the integration limit @xmath35 in ( [ eq : sffree ] ) depend on the particular cut that is used . for example , if we consider a cut on the charged - lepton energy @xmath36 then @xmath37 , or if a cut on hadronic invariant mass @xmath9 is considered then one needs @xmath38 . a quick way to obtain some intuition on how @xmath31 and @xmath35 depend on the specific cut qualitatively is to consider the @xmath1 phase - space depicted in figure [ fig : pspace ] . note that @xmath39 in radiative decays , so that the phase - space of @xmath40 consists only of the vertical boundary on the right - hand side of the plot . this picture explains correctly how the maximal @xmath41 depends on the nature of the cut , and where to expect some kinks in @xmath31 , for example for a combined cut on @xmath42 and @xmath43 . in the next section we will derive an expression for the weight function at leading power , which follows from exact factorization theorems for differential rates in inclusive @xmath5 decays in the shape - function region . the formula for the weight function is thereby also valid to all orders in perturbation theory . we compute its explicit form to next - to - next - to leading order ( nnlo ) at the intermediate scale @xmath18 and to next - to leading order ( nlo ) at the hard scale @xmath44 in renormalization - group ( rg ) improved perturbation theory , including three - loop running effects . this approximation is sufficiently precise for phenomenological applications since effects at subleading power become as important as higher - order perturbative effects at leading power . in section [ sec : powcorr ] we discuss first - order power corrections . kinematical power corrections enter the weight function , while hadronic power corrections give rise the quantity @xmath32 in ( [ eq : sffree ] ) . we then apply our results to a few examples of kinematic cuts in section [ sec : examples ] and perform an analysis of theoretical uncertainties on @xmath2 by using a simple model for the experimental inputs .

a leading - power factorization formula for weight functions relating the photon spectrum to arbitrary partial decay rates in is derived . we calculate the weight function in renormalization - group improved perturbation theory to complete next - to - next - to leading order at the jet scale and to next - to leading order at the hard scale .

a leading - power factorization formula for weight functions relating the photon spectrum to arbitrary partial decay rates in is derived . these weight functions are independent of the hadronic shape function and allow for the determination of in a model - independent way . we calculate the weight function in renormalization - group improved perturbation theory to complete next - to - next - to leading order at the jet scale and to next - to leading order at the hard scale . first - order power corrections are also included , where a model - dependence enters via the appearance of subleading hadronic shape functions . mit - ctp 3695 + hep - ph/0511098 + november 8 , 2005 * shape - function independent relations of charmless inclusive-decay spectra * bjrn o. lange +

astro-ph0403371

i

accurate determinations of metal abundances in planetary nebulae ( pne ) are important for understand the chemical evolution of galaxies and the nucleosynthesis and mixing processes in low- and intermediate - mass stars . a long - standing problem in nebular abundance studies has been that the heavy - element abundances derived from optical recombination lines ( orls ) are systematically higher than those derived from collisionally excited lines ( cels ; see liu 2001 , 2003 for recent reviews ) . in extreme cases , the discrepancies exceed a factor of 10 . given that cel abundances are sensitive to electron temperature and density , knowledge of temperature and density variations is an essential ingredient in understanding the abundance discrepancy problem . information on temperature and density variations can be obtained by comparing results derived from plasma diagnostics which behave differently as temperature and density vary . traditionally , nebular electron temperatures and densities are derived from cel ratios . however , for pne including high - density regions ( @xmath12e@xmath13@xmath14 ) , such as ic 4997 @xcite , mz 3 @xcite and m 2 - 24 @xcite , cels are strongly suppressed by collisional de - excitation , and so depending on their critical densities , they may become unusable in probing such high - density regions . on the other hand , hydrogen recombination lines have very high critical densities ( @xmath15@xmath14 ) . consequently , the hydrogen recombination spectrum provides a powerful tool to probe high - density regions . comparison of electron temperatures and densities derived from the hydrogen recombination spectrum ( lines plus continuum ) with those derived from cels should enable us to quantify temperature and density variations in nebulae . by measuring the balmer discontinuity , @xcite determined electron temperatures in 14 pne and found that they are systematically lower than those derived from the [ ] nebular - to - auroral forbidden line ratio . this can be explained by the presence of large temperature fluctuations since the [ ] @xmath16 auroral line of higher excitation energy tends to be emitted in higher temperature regions than the [ ] @xmath17 nebular lines of lower excitation energy , as suggested by @xcite . recent reviews on temperature fluctuations of gaseous nebulae are given by @xcite and references therein . however , no photoionization models yield such large temperature fluctuations suggested by observations . moreover , temperature fluctuations are unable to explain the large disparity between the orl and cel abundances observed in some pne @xcite . instead , it is found that the hypothesis that pne may contain clumps of different temperature , density and chemical composition , embedded in diffuse ` normal ' material , can explain many of the observed patterns @xcite . peimbert s concept of temperature fluctuations alone may not be a realistic description of the physical conditions in a nebula , although it has proven to be a convenient and useful way to characterize the complexities of real nebulae . it is expected that there are also variations of electron density inside a pn . @xcite pointed out that electron densities derived from various plasma diagnostics should correlate positively with the critical densities of the diagnostic lines involved if there are large density variations within a nebula . @xcite found that electron densities derived from the [ ] @xmath9 ratios are generally lower than those derived from the optical [ ] and [ ] doublet ratios , suggesting the presence of high density regions in pne , where the 52- and 88-@xmath18 lines are suppressed by collisional de - excitation due to their relatively low critical densities . @xcite showed that the discrepancies between @xmath0([]@xmath3 ) and @xmath0(bal ) can be explained by the presence of condensations in nebulae and gave an analytical method of estimating the filling factor of condensations . our detailed studies of the peculiar pne mz 3 and m 2 - 24 suggest that the presence of high - density regions in some pne can significantly affect abundance determinations @xcite . however , density inhomogeneities alone can not explain the large disparity between the orl and cel abundances either , unless the condensations are also h - deficient @xcite . @xcite found that the discrepancy between the orl and cel abundances correlates with the difference between the temperatures derived from the [ ] collisional lines and from the balmer jump of hydrogen recombination spectrum , thus lending strong support to the argument that some extremely cold h - deficient knots may co - exist with normal nebular material in many pne . however , it is difficult to explain the origin and evolution of such h - deficient knots with the current theories of element production in intermediate - mass stars . in this paper , we present determinations of electron temperatures and densities for a large sample of pne based on the analysis of nebular hydrogen recombination spectrum . section 2 describes the sample . in section 3 , we describe a new approach to deriving electron temperature and density simultaneously from the observed nebular hydrogen recombination spectrum ( lines plus continuum ) and present the results . in section 4 , we compare results derived from a variety of temperature and density indicators and discuss the significance of variations in electron temperature and density in pne . a discussion of he / h abundance determinations is also presented in this section . in section 5 , we present electron temperatures and densities derived from the paschen discontinuity and decrements . a summary then follows in section 6 .

a method is presented to derive electron temperatures and densities of planetary nebulae ( pne ) simultaneously , using the observed hydrogen recombination spectrum , which includes continuum and line emission . by matching theoretical spectra to observed spectra around the balmer jump at about 3646 , we determine electron temperatures and densities for 48 galactic pne .

a method is presented to derive electron temperatures and densities of planetary nebulae ( pne ) simultaneously , using the observed hydrogen recombination spectrum , which includes continuum and line emission . by matching theoretical spectra to observed spectra around the balmer jump at about 3646 , we determine electron temperatures and densities for 48 galactic pne . the electron temperatures based on this method [ hereafter(bal ) ] are found to be systematically lower than those derived from [ ] and [ ] ratios [ hereafter ,([] ) and([] ) ] . and the electron densities based on this method are found to be systematically higher than those derived from [ ] , [ ] , [ ] , [ ] and [ ] ratios . these results suggest that temperature and density fluctuations are generally present within nebulae . the comparison of([] ) and(bal ) suggests that the fractional mean - square temperature variation ( ) has a representative value of 0.031 . a majority of temperatures derived from the [ ] ratio are found to be higher than([] ) , which is attributed to the existence of dense clumps in nebulae those [ ] ir fine - structure lines are suppressed by collisional de - excitation in the clumps . by comparing([] ) ,([] ) and(bal ) and assuming a simple two - density - component model , we find that the filling factor of dense clumps has a representative value of 7 . the discrepancies between([] ) and(bal ) are found to be anti - correlated with electron densities derived from various density indicators ; high - density nebulae have the smallest temperature discrepancies . this suggests that temperature discrepancy is related to nebular evolution . in addition , he / h abundances of pne are found to be positively correlated with the difference between([] ) and(bal ) , suggesting that he / h abundances might have been overestimated generally because of the possible existence of h - deficient knots . electron temperatures and densities deduced from spectra around the paschen jump regions at 8250 are also obtained for four pne , ngc 7027 , ngc 6153 , m 1 - 42 and ngc 7009 . electron densities derived from spectra around the paschen jump regions are in good agreement with the corresponding values derived from spectra around the balmer jump , whereas temperatures deduced from the spectra around the paschen jump are found to be lower than the corresponding values derived from spectra around the balmer jump for all the four cases . the reason remains unclear .

astro-ph0403371

c

electron densities deduced from the balmer decrements , and from the [ ] @xmath5 , [ ] @xmath40 , [ ] @xmath41 and [ ] @xmath42 doublet ratios are compared in fig . [ necorr ] . [ necorr ] shows that for most pne , the hydrogen recombination spectrum yields higher densities compared to forbidden lines diagnostics . there is a trend such that the discrepancy becomes larger at higher densities . two extreme cases are ic 4997 and m 2 - 24 . for ic 3568 , [ ] @xmath41 yields an abnormally low density . the cause of it is unclear but is probably due to measurement error . in fig . [ necorrir ] we compare electron densities derived from the [ ] far - infrared fine structure line ratio and from the hydrogen recombination spectrum . evidently , densities derived from the [ ] @xmath105 ratio are significantly lower than those derived from the balmer decrements . in fact , it has been found that the far - ir lines yield systematic lower densities , lower than even those deduced from optical forbidden line density - diagnostics @xcite . a linear fit shows that the density discrepancy increases with the electron density . o@xmath64 , s@xmath64 , cl@xmath106 , o@xmath106 and ar@xmath32 have ionization potentials of 13.6 , 10.4 , 23.8 , 35.1 and 40.7 ev , respectively , and therefore embrace a wide range of regions of differing states of ionization . on the other hand , the hydrogen recombination spectrum arises from the entire ionized nebula . thus the discrepancies between densities derived from recombination spectrum and those deduced from forbidden line ratios are not likely to be caused by ionization stratifications . we notice that the [ ] , [ ] , [ ] , [ ] and [ ] density - diagnostic lines have critical densities of @xmath107 @xmath14 . therefore , if the nebulae contain regions of higher densities , these forbidden lines will be suppressed in these regions by collisional de - excitation . in contrast , the hydrogen recombination spectrum is practically unaffected by collisional de - excitation . as a result , in such cases the hydrogen recombination spectrum will yield higher average densities compared to the [ ] , [ ] , [ ] , [ ] and [ ] lines ratios . the same argument has been developed by @xcite to explain the systematically lower densities derived from the [ ] fine - structure line ratio compare to those deduced from optical forbidden line ratios . the [ ] lines have higher critical densities than the [ ] , [ ] , [ ] lines . thus , the effects are weaker for [ ] doublet ratio than for the other cases . given that the [ ] far - infrared fine - structure lines have the lowest critical densities amongst those density - diagnostic forbidden lines ( @xmath108([]@xmath109 @xmath110 ) , the effects are largest for the [ ] @xmath105 ratio . this can be seen clearly in fig . [ necorr ] and fig . [ necorrir ] . a nebular model including these high - density condensations and their effects on temperature determinations will be discussed in section 4.3 . [ tecorr ] and fig . [ tecorrir ] show that electron temperatures derived from the [ ] @xmath1 and @xmath111 ratios [ hereafter , @xmath0([]@xmath112 and @xmath0([]@xmath113 are systematically higher than those deduced from the hydrogen recombination spectrum [ hereafter @xmath0(bal ) ] , although there are a few exceptions where @xmath0(bal ) is higher than @xmath0([]@xmath3 ) . the most extreme case is he 2 - 118 , for which @xmath0(bal ) is 5000 k higher than @xmath0([]@xmath3 ) . in our sample , the four pne with the largest temperatures discrepancies , hf 2 - 2 , m 1 - 42 , m 2 - 36 and ngc 6153 , were previously studied in detail by @xcite , @xcite and @xcite . the discrepancy between @xmath0([]@xmath3 ) and @xmath0(bal ) was first discovered by @xcite and was attributed to temperature fluctuations within the nebula @xcite . he characterized the temperature structure of a nebula by the average temperature @xmath114 and a mean square temperature fluctuation @xmath10 as follows : @xmath115 and @xmath116 where @xmath50 is the ion density for an observed emission line . assuming that all the oxygen is twice ionized and that there are no density and composition variations in the nebula , @xmath114 and @xmath10 can be derived from two of the three expressions , @xmath117_{\rm na})=t_0[1+\frac{1}{2}(\frac{9.13\times10 ^ 4}{t_0}-3)t^2],\ ] ] @xmath117_{\rm fn})=t_0[1+\frac{1}{2}(\frac{2.92\times10 ^ 4}{t_0}-3)t^2],\ ] ] and @xmath118 @xcite . in fig . [ tecorr ] and fig . [ tecorrir ] , lines showing @xmath0([]@xmath3 ) as a function of @xmath119 for the case of @xmath120 , 0.02 , 0.06 and 1.00 are over plotted . fig . [ tecorr ] shows that most nebulae have @xmath121 . excluding a few extreme nebulae having @xmath122 , we obtain a mean value of @xmath123 . the value is in excellent agreement with previous result @xmath124 derived by @xcite although their analysis contained larger measurement errors . however , even excluding the extreme nebulae , a similar comparison between @xmath0([]@xmath4 ) and @xmath119 yields an average @xmath125 . as we shall show in the following subsection , it is likely that @xmath0([]@xmath4 ) have been overestimated because of density variations in the nebula , resulting in higher @xmath126 when comparing @xmath0([]@xmath4 ) and @xmath119 . another interesting point revealed by fig . [ tecorr ] is that @xmath119 covers a wider range of values than @xmath0([]@xmath3 ) . except a few extreme cases , @xmath0([]@xmath3 ) falls between 8000 and 14000 k , whereas @xmath119 varies from 900 to 19000 k. @xcite has argued that temperature fluctuations of gaseous nebulae can cause lower @xmath119 compared to @xmath0([]@xmath3 ) . however , even @xmath123 is beyond any photoionization model prediction . another explanation of higher @xmath119 compared to @xmath0([]@xmath3 ) is that there are some condensations with @xmath127 @xmath14 in pne , which then suppress the @xmath0 ( [ ] ) @xmath1034959,5007 relative to the @xmath934363 , leading to an artificially higher @xmath0([]@xmath3 ) @xcite . but no such high density condensations have been found in pne having extremely large temperature discrepancies @xcite . recently , @xcite presented an empirical nebular model containing two components , each with its own temperature , density and chemical composition to account for the temperature discrepancy . this seems to be the most plausible explanation at present . for pne having @xmath128([]@xmath3 ) , as in he 2 - 118 , the electron temperature might conceivably increase in low - ionization regions owing to heating by shock waves in the outer regions . as a result , the average temperature of the entire nebula [ @xmath0(bal ) ] appears higher than that of high - ionization regions [ @xmath0([]@xmath3 ) ] . in 4.1 , we found that electron densities derived from the recombination spectrum are systematically higher than those derived from optical forbidden lines ratios , and confirm the earlier result of @xcite that densities derived from the [ ] far - ir fine - structure lines are the lowest of all . these results show that nebular electron densities derived from various diagnostics are correlated with the critical density of the diagnostic lines , and suggest that dense clumps are generally present in pne . in such cases , the [ ] @xmath129 ratio is no longer a good temperature indicator given the relatively low critical densities of the [ ] @xmath130- and 88-@xmath131 lines ( 3500 and 1500 @xmath14 , respectively ) . when nebulae contain condensations with densities in the range between 10@xmath132 and 10@xmath133 @xmath14 , the [ ] @xmath130- and 88-@xmath131 far - infrared lines will be heavily suppressed by collisional de - excitation , leading to overestimates of @xmath0([]@xmath4 ) . on the other hand , condensations of densities in this range will not affect @xmath0([]@xmath3 ) given that the [ ] @xmath134 and @xmath16 lines have higher critical densities of @xmath135 @xmath14 and @xmath136 @xmath14 , respectively ( cf . fig [ tecorrir ] ) . another piece of evidence pointing to the presence of dense clumps comes from fig . [ oiiite ] where we compare @xmath0([]@xmath3 ) and @xmath0([]@xmath4 ) . if nebulae have temperature fluctuations but are homogeneous in density , then @xmath0([]@xmath4 ) is expected to be always lower than @xmath0([]@xmath3 ) @xcite . however , fig . [ oiiite ] shows about two - third of the objects have higher @xmath0([]@xmath4 ) compared to @xmath0([]@xmath3 ) , suggesting that density variations play an important role in @xmath0([]@xmath4 ) determinations . @xcite showed that electron temperatures derived from the [ ] @xmath137 ratio may be overestimated if nebulae contain ionized condensations of electron densities higher than 10@xmath133 @xmath14 . however , except for a few extreme pne , such as ic 4997 @xcite , mz 3 @xcite and m 2 - 24 @xcite there is no evidence that nebulae contain a substantial amount of ionized gas in condensations with densities in excess of 10@xmath133 @xmath14 , as the presence of such condensations should have been revealed by the recombination spectrum ( liu et al . 2000 , 2001a , 2001b ; fig . [ necorr ] ) . therefore , we consider @xmath138 @xmath14 as a safe upper limit for the density of these condensations . in order to estimate the filling factor of the condensations , we use an analytical method originally developed by @xcite for explaining the discrepancy of @xmath0([]@xmath3 ) and @xmath119 . assuming that some high - density regions ( @xmath127 @xmath14 ) are present , they deduced nebular - to - auroral [ ] line ratios ( see their eq . following the same deduction of their eq . 2.3 but assuming that the electron density of these condensations is about @xmath139 @xmath14 instead of @xmath45 @xmath14 , we obtain the infrared - to - auroral line ratio @xmath140 where @xmath141 is line emission coefficient , @xmath142 , f and n refer to ir fine - structure line and nebular line , l and h refer to the low- and high - density regions , @xmath143 is the density contrast and @xmath144 is the filling factor . in the following calculation , we assume that : 1 ) density variations are the only cause of the lower @xmath0([]@xmath4 ) compared to @xmath0([]@xmath3 ) and @xmath145 ; 2 ) the nebular gas has a homogeneous electron temperature , i.e. , @xmath0([]@xmath3 ) or @xmath145 ; 3 ) the condensations have densities lower than the critical density of the [ ] auroral line but higher than the critical densities of the [ ] infrared lines , whereas low - density regions have a density that is lower than the critical densities of the [ ] auroral and infrared lines . setting @xmath146_{\rm fn}})=i_{52\micron+88\micron}/i_{4959}$ ] , we have @xmath147 . for a simple assumption @xmath148 @xmath14 and @xmath149 @xmath14 , the ratio @xmath150 is about 0.04 , whereas @xmath151 is practically independent of the density when the densities are lower than the critical density of the nebular line and thus @xmath152 . therefore , @xmath153 is given by @xmath154_{\rm fn}})/f(t_{\rm bal})-1}{0.04-f(t_{[{\rm o}~{\rm iii}]_{\rm fn}})/f(t_{\rm bal})}.\ ] ] in fig . [ tecorrir ] , the lines showing the variation of @xmath0([]@xmath4 ) as a function of @xmath0(bal ) for @xmath155 , 0.1 , 0.5 , 1.0 and 2.0 are plotted . we obtain an average value of @xmath156 . however , it should be borne in mind that the @xmath153 derived by eq . ( 18 ) is a upper limit since temperature variations are actually present and contribute partly to the measured temperature discrepancies . @xmath153 can also be estimated by comparing @xmath0([]@xmath3 ) with @xmath0([]@xmath4 ) , as shown in fig . [ oiiite ] . assuming that @xmath0([]@xmath3 ) is constant across the entire nebula , @xmath153 is likewise given by @xmath154_{\rm fn}})/f(t_{[{\rm o}~{\rm iii}]_{\rm na}})-1}{0.04-f(t_{[{\rm o}~{\rm iii}]_{\rm fn}})/f(t_{[{\rm o}~{\rm iii}]_{\rm na}})}.\ ] ] in fig . [ oiiite ] , the lines showing the variation of @xmath0([]@xmath4 ) as a function of @xmath0(bal ) for @xmath155 , 0.1 , 0.5 , 1.0 and 2.0 are also plotted . an average value of @xmath157 is obtained . it is noteworthy that the presence of temperature variations will increase @xmath0([]@xmath158 ) with respect to @xmath0([]@xmath4 ) , in contrast to density variations . therefore , @xmath153 derived from eq . ( 19 ) is a lower limit . the upper- and lower - limit of @xmath153 are listed in table . [ tbl-3b ] . for a few pne having a lower limit @xmath159 , the effect of temperature variations on the temperature determination is larger than that of density variations , causing higher @xmath0([]@xmath112 compared to @xmath0([]@xmath4 ) . for all the pne , the mean value of @xmath153 is in the range from 0.4 to 1.0 . thus , we suggest that @xmath160 may be a representative value that can be used for a typical nebula . adopting the density contrast @xmath161 , a representative value of the condensations filling factor @xmath162 is about @xmath163 . the value is similar to that found by @xcite . adopting @xmath160 for all the pne , we obtain new @xmath0([]@xmath4 ) , for which the contribution of the clumps has been deduced using equation ( 17 ) . comparing the new @xmath0([]@xmath4 ) and @xmath0(bal ) , we re - evaluate the average value @xmath164 , which is in good agreement with the value yielded by the comparison of @xmath0([]@xmath3 ) and @xmath0(bal ) ( see section 4.2 ) . in fig . [ crene ] , the differences between temperatures derived from the [ ] forbidden lines and from the balmer jump are plotted against electron densities derived from the hydrogen recombination spectrum , the [ ] @xmath5 , [ ] @xmath40 , [ ] @xmath41 [ ] @xmath42 and the [ ] @xmath165m/@xmath166 m ratios , respectively . in all cases , the differences between the two temperature are anti - correlated with nebular densities derived from various density indicators . a least - squares fit yields the following relation : @xmath167 and a linear correlation coefficient of 0.51 , where @xmath168([])@xmath169 and @xmath170 is that derived from hydrogen recombination spectrum , which represents the average density of the entire nebula . @xcite found that the discrepancies of o@xmath32/h@xmath106 abundance ratios derived from orls and from cels are greater for larger , lower - surface brightness pne than for compact , dense ones . on the other hand , @xcite found that the abundance discrepancies are greater for those with larger temperature discrepancies . our results are therefore consistent with these earlier findings . @xcite found that , @xmath171 where @xmath172 . from the two relations given in equations ( 20 ) and ( 21 ) , we find that the ratio of abundances derived from orls and from cels has a power - law dependence on density , @xmath173 where @xmath174 . in general , electron densities of pne are expected to decline monotonically as nebulae expand . @xcite found an approximate relation between nebular mean electron density and nebular size , @xmath175 , where r is the nebular radius . consequently , it is suggested that the temperature discrepancy is related to the evolutionary state of pne ; more expended pne show larger temperature discrepancies . on the other hand , the actual evolution in a given nebula may depend on many factors . furthermore , the electron density measured for a given nebula may depend on the diagnostic used . for example , electron densities of low - ionization regions , as probed by the [ ] @xmath5 and [ ] @xmath40 line ratios , can be enhanced by interactions of fast stellar winds with earlier slow winds . therefore , there are slight difference in deriving the relation of @xmath176 and @xmath170 from different density diagnostics , as shown in fig . [ crene ] . we also use electron density derived from these forbidden line ratios to estimate the @xmath26-value in equation ( 22 ) . the results show that the value is within the range of 0.3 and 0.9 . to explain the relation shown in equation ( 20 ) and ( 22 ) , we apply a chemically inhomogeneous two - component model presented by @xcite . in their hypothesis , the nebula consists of two components , h - deficient clumps and diffuse material with ` normal ' abundances ( @xmath177solar ) . these h - deficient clumps are expected to have a low electron temperature due to enhanced cooling caused by high metallicity and pressure equilibrium between the two components , so that only recombination lines are emitted . as a result , orls yield overestimated heavy element abundances . in a given pn , the contribution of the two components to the balmer temperature and abundance determinations depends on their densities , i.e. , the contribution becomes weaker as density declines . if the clumps have neutral cores , as found in abell 30 @xcite and ngc 7293 @xcite , they are expected to be quite stable during the evolution of the pne . as a result , the relative contribution of the low - density and high - temperature material to the average temperature and abundance determinations decrease as nebulae expand . consequently , the discrepancies in temperature and abundance determinations become larger as the nebulae expend . a long - standing problem in abundance determinations for pne has been that the heavy element abundances derived from orls are systematically higher than those derived from cels . the ratio of abundances derived from the two types of emission lines is found to vary from target to target . @xcite analysed a pn sample and found the orl / cel discrepancy factors for these heavy element abundances span a range from about 2 to as large as 20 . for the most extreme known pne hf 2 - 2 , the orl / cel abundance ratio reaches a record value of 84 @xcite . another problem in nebular astrophysics is that @xmath0([]@xmath3 ) are systematically higher than @xmath119 , as we have discussed in 4.2 . the two problems are found to be related . a positive correlation between the ratio of o@xmath32/h@xmath31 derived from orls and from [ ] cels and the temperature discrepancy [ @xmath176=@xmath0([]@xmath3)-@xmath119 ] has been shown by @xcite . the explanations for the dichotomy in abundance and temperature determinations have focussed on temperature and density variations and chemical inhomogeneities . however , it has been shown that the temperature and density variations alone can be ruled out by current observations @xcite . at present , the most plausible explanation is that there exist some cold h - deficient clumps in nebula @xcite where only recombination lines are emitted . to fit the observed pattern of ngc 6153 , @xcite have presented several empirical models incorporating h - deficient clumps . @xcite also constructed photoionization models of ngc 6153 and m 1 - 42 , incorporating h - deficient clumps , which reproduce the observed integrated line ratio satisfactorily . in all the models , the h - deficient clumps contain only about @xmath179 of the total mass , suggesting that the h - deficient components contain only a tiny amount of nebular gas . nevertheless , they can seriously enhance the intensities of orls , causing overestimated orl abundances . therefore , the he / h abundance ratio could be overestimated owing to the existence of the h - deficient clumps can lead to since all the observable strong and lines are recombination lines . some h - deficient components have been found in some pne , such as abell 30 and abell 78 @xcite . @xcite analysed optical spectra of the h - deficient knots in abell 30 and found the he / h abundance ratios of the knots are more than ten . these pne with extremely large abundance and temperature discrepancies , such as ngc 6153 , m 1 - 42 and hf 2 - 2 , have been found to have high he / h abundances . @xcite defined the pne having high he / h abundance ( @xmath180 ) as type - i pne . according to our suggestion , these pne with high he / h abundances might result from contamination from h - deficient clumps . quantitative analysis of overestimated he / h abundances in pne is quite difficult . for this purpose , as mentioned by @xcite , one has to rely on the measurement of @xmath9310830 line which is strongly enhanced by collisional excitation from the 2s@xmath181s metastable level . if the he / h abundance is enhanced by h - deficient clumps , one expects that there is a positive correlation between the he / h abundance and the orl / cel abundance ratio of heavy element . for most pne , however , heavy element orls are quite weak and thus yield large uncertainties in the resultant abundance . instead , we can study the relation between the he / h abundance and @xmath176 since the orl / cel abundance ratio has been found to be positively correlated with @xmath176 by @xcite . fig . [ hete ] plots he / h versus @xmath176 . the measurement error of the he / h abundance ratio is negligible . in the figure , filled and open circles represent the galactic disc and bulge pne respectively and the solid line is a linear fit represented by the relation : @xmath182 which yields a linear correlation coefficient of 0.50 . although there is considerable scatter , there is a trend of increasing he / h abundance with increasing difference between the temperature from [ ] forbidden lines and balmer discontinuity for both the galactic disc and bulge pne . therefore , it seems plausible that the he / h abundances in pne have been generally overestimated due to the existence of cold h - deficient clumps . [ hete ] also shows that the most of pne have enhancement of helium with respect to the sun , which is often ascribed to the second and third dredge - up . however , the normal model of dredge - up can not explain some extreme cases , such as he 2 - 111 which has a he / h abundance of 0.219 @xcite . for these cases , the more likely reason is that pollution of the h - deficient clumps causes an overestimate of the he / h abundance ratio . on the other hand , we can not completely reject the possibility that helium in nebulae has been partly enhanced by the products of nucleosynthesis of the central star . that may be the main cause of the scatter in the derived he / h abundances . the presence of hydrogen - deficient clumps may be further proof . the orl / cel abundance discrepancies are also found in regions @xcite . and the existence of some high density clumps in regions has been indicated by previous studies ( e.g. odell et al . therefore , the h - deficient material might also be present in regions , leading to overestimated orl abundances . if this is the case , the primordial helium abundance , which is based on observations of some metal poor extragalactic regions , could have been overestimated .

the electron temperatures based on this method [ hereafter(bal ) ] are found to be systematically lower than those derived from [ ] and [ ] ratios [ hereafter ,([] ) and([] ) ] . and the electron densities based on this method are found to be systematically higher than those derived from [ ] , [ ] , [ ] , [ ] and [ ] ratios . these results suggest that temperature and density fluctuations are generally present within nebulae . a majority of temperatures derived from the [ ] ratio are found to be higher than([] ) , which is attributed to the existence of dense clumps in nebulae those [ ] ir fine - structure lines are suppressed by collisional de - excitation in the clumps . by comparing([] ) ,([] ) and(bal ) and assuming a simple two - density - component model , we find that the filling factor of dense clumps has a representative value of 7 . the discrepancies between([] ) and(bal ) are found to be anti - correlated with electron densities derived from various density indicators ; high - density nebulae have the smallest temperature discrepancies . this suggests that temperature discrepancy is related to nebular evolution .

a method is presented to derive electron temperatures and densities of planetary nebulae ( pne ) simultaneously , using the observed hydrogen recombination spectrum , which includes continuum and line emission . by matching theoretical spectra to observed spectra around the balmer jump at about 3646 , we determine electron temperatures and densities for 48 galactic pne . the electron temperatures based on this method [ hereafter(bal ) ] are found to be systematically lower than those derived from [ ] and [ ] ratios [ hereafter ,([] ) and([] ) ] . and the electron densities based on this method are found to be systematically higher than those derived from [ ] , [ ] , [ ] , [ ] and [ ] ratios . these results suggest that temperature and density fluctuations are generally present within nebulae . the comparison of([] ) and(bal ) suggests that the fractional mean - square temperature variation ( ) has a representative value of 0.031 . a majority of temperatures derived from the [ ] ratio are found to be higher than([] ) , which is attributed to the existence of dense clumps in nebulae those [ ] ir fine - structure lines are suppressed by collisional de - excitation in the clumps . by comparing([] ) ,([] ) and(bal ) and assuming a simple two - density - component model , we find that the filling factor of dense clumps has a representative value of 7 . the discrepancies between([] ) and(bal ) are found to be anti - correlated with electron densities derived from various density indicators ; high - density nebulae have the smallest temperature discrepancies . this suggests that temperature discrepancy is related to nebular evolution . in addition , he / h abundances of pne are found to be positively correlated with the difference between([] ) and(bal ) , suggesting that he / h abundances might have been overestimated generally because of the possible existence of h - deficient knots . electron temperatures and densities deduced from spectra around the paschen jump regions at 8250 are also obtained for four pne , ngc 7027 , ngc 6153 , m 1 - 42 and ngc 7009 . electron densities derived from spectra around the paschen jump regions are in good agreement with the corresponding values derived from spectra around the balmer jump , whereas temperatures deduced from the spectra around the paschen jump are found to be lower than the corresponding values derived from spectra around the balmer jump for all the four cases . the reason remains unclear .

astro-ph0403371

i

in this work , we presented a method to derive nebular electron temperature and density simultaneously by matching the theoretical with the observed hydrogen recombination spectra . using the method , we derived electron temperatures and densities of 48 galactic pne . the temperatures and densities derived by this method are compared with those derived from other indicators , enabling us to study electron temperature and density variations in pne . our main findings are as follows . the temperatures derived from the nebular - auroral line ratios of [ ] are systematically higher than those derived from hydrogen recombination spectra , suggesting that temperature variations are generally present in nebulae . we obtained the rms temperature fluctuation parameter @xmath188 as a representative value . the larger discrepancy in temperatures derived from [ ] @xmath129 line ratios and from hydrogen recombination spectra , however , can not be attributed simply to the temperature variations . the presence of clumps with @xmath138 @xmath14 can account well for the discrepancy . a comparison between electron temperatures derived from [ ] @xmath129 line ratios and those derived from hydrogen recombination spectra and [ ] @xmath1 line ratios suggests that the filling factor of the condensations has a typical value of @xmath163 . the discrepancies in temperatures derived from [ ] @xmath1 line ratios and from hydrogen recombination spectra are related to the evolution of pne . the orl / cel abundance ratio is found to have a power - law dependence on electron density ( @xmath189 , @xmath174 ) . we attribute this to different evolutionary scenario of two components in pne , i.e. expanding diffuse nebular material and stable cold h - deficient clumps . the contribution of the clumps to the temperature and abundance determinations increases as the nebulae expand . he / h abundance has a positive correlation with the temperature discrepancy , suggesting that he / h abundances might be overestimated due to the existence of h - deficient clumps . if this is the case , traditional up - dredge theory should be modified . the primordial helium abundance determination will be affected if there are significant condensations in regions . further investigations are needed . the densities derived from the spectrum near the paschen jump are in good agreement with those derived form that near the balmer jump . however , the former yield generally lower temperatures than the latter . the reason remain unknown . we conclude that the existence of h - deficient clumps in pne is the most plausible explanation for the origin of temperature and density variations and the orl / cel abundance discrepancies . however , it is a challenge to clarify the origin and evolution of these clumps . in further work , we will study the statistical relationships between the temperature and abundance discrepancies and other physical conditions of pne such as radio flux , morphology , chemical abundances and the nature of the central star . the authors are grateful to w. wang for his help with part of the data reduction . we also thank j .- s chen and s .- g . luo for their help with the preparation of this paper . constructive suggestions and comments by the referee s. m. viegas were appreciated . this work was supported in part by beijing astrophysics center ( bac ) . lcccccc hu 1 - 2 & 0.512 & 0.046 & 0.079 & @xmath190 & & 144 + ic 3568 & 0.261 & 0.095 & 0.001 & @xmath191 & @xmath192 & 132 + ngc 7027 & 1.380 & 0.058 & 0.041 & @xmath193 & @xmath194&82 + ngc 7009 & 0.130 & 0.093 & 0.011 & @xmath195 & @xmath196 & 132 + ngc 40 & 0.700 & 0.062 & 0.000 & @xmath197 & @xmath198 & 132 + ngc 6210 & 0.055 & 0.099 & 0.001 & @xmath199 & @xmath200 & 148 + ngc 6826 & 0.064 & 0.099 & 0.000 & @xmath201 & @xmath202 & 132 + ngc 6572 & 0.400 & 0.110 & 0.000 & @xmath197 & @xmath203 & 132 + ngc 6741 & 1.100 & 0.081 & 0.031 & @xmath204 & @xmath205 & 132 + ngc 6790 & 1.100 & 0.081 & 0.003 & @xmath206 & & 132 + ngc 6884 & 1.050 & 0.081 & 0.016 & @xmath207 & @xmath208 & 132 + ngc 7662 & 0.180 & 0.061 & 0.038 & @xmath209 & @xmath210 & 156 + ngc 6543 & 0.100 & 0.117 & 0.000 & @xmath211 & @xmath212 & 140 + m 2 - 24 & 0.800 & 0.099 & 0.005 & @xmath213 & & 156 + cn 2 - 1 & 1.063 & 0.104 & 0.004 & @xmath214 & & 132 + ngc 6720 & 0.200 & 0.094 & 0.017 & @xmath215 & @xmath216 & 132 + ngc 6620 & 0.760 & 0.099 & 0.021 & @xmath217 & & 132 + ngc 6567 & 0.670 & 0.105 & 0.007 & @xmath218 & @xmath219 & 132 + h 1 - 35 & 1.360 & 0.115 & 0.001 & @xmath220 & & 132 + h 1 - 50 & 0.700 & 0.100 & 0.011 & @xmath221 & & 132 + he 2 - 118 & 0.158 & 0.065 & 0.000 & @xmath222 & & 148 + m 1 - 20 & 1.200 & 0.094 & 0.000 & @xmath223 & & 123 + m 3 - 21 & 0.570 & 0.100 & 0.007 & @xmath224 & & 140 + m 3 - 32 & 0.579 & 0.120 & 0.013 & @xmath225 & & 140 + ic 1297 & 0.220 & 0.079 & 0.042 & @xmath226 & & 132 + ic 4634 & 0.454 & 0.097 & 0.000 & @xmath227 & @xmath228 & 140 + ic 4776 & 0.265 & 0.088 & 0.000 & @xmath229 & & 136 + ic 4997 & 0.438 & 0.139 & 0.000 & @xmath230 & & 127 + ngc 5873 & 0.004 & 0.063 & 0.040 & @xmath231 & & 127 + ngc 5882 & 0.380 & 0.113 & 0.003 & & & 127 + ngc 6302 & 1.460 & 0.105 & 0.067 & @xmath232 & @xmath233 & 132 + ngc 6818 & 0.400 & 0.050 & 0.051 & @xmath204 & @xmath234 & 148 + ngc 6153 & 1.300 & 0.123 & 0.011 & @xmath235 & @xmath236 & 148 + m 2 - 36 & 0.270 & 0.133 & 0.003 & @xmath237 & @xmath238 & 148 + m 1 - 42 & 0.700 & 0.139 & 0.009 & @xmath239 & @xmath240 & 140 + ngc 6778 & 0.890 & 0.155 & 0.008 & @xmath241 & & 140 + hf 2 - 2 & 0.200 & 0.131 & 0.002 & @xmath242 & & 132 + dddm 1 & 0.137 & 0.060 & 0.000 & @xmath243 & & 99 + hu 2 - 1 & 0.777 & 0.097 & 0.000 & @xmath244 & & 99 + ic 2003 & 0.347 & 0.054 & 0.049 & @xmath245 & & 99 + ic 5217 & 0.501 & 0.088 & 0.008 & @xmath246 & & 115 + me 2 - 2 & 0.343 & 0.138 & 0.000 & @xmath247 & & 90 + ngc 6803 & 0.869 & 0.112 & 0.004 & @xmath248 & & 107 + ngc 6807 & 0.642 & 0.084 & 0.000 & @xmath249 & & 107 + ngc 6833 & 0.000 & 0.078 & 0.000 & @xmath250 & & 99 + ngc 6879 & 0.401 & 0.091 & 0.003 & @xmath251 & & 99 + ngc 6891 & 0.287 & 0.094 & 0.000 & @xmath252 & & 115 + vy 1 - 1 & 0.421 & 0.094 & 0.000 & @xmath253 & & 99 + lcc hu 1 - 2 & @xmath254 & @xmath255 + ic 3568 & @xmath256 & @xmath257 + ngc 7027 & @xmath258 & @xmath259 + ngc 7009 & @xmath260 & @xmath261 + ngc 40 & @xmath262 & @xmath263 + ngc 6210 & @xmath264 & @xmath261 + ngc 6826 & @xmath265 & @xmath266 + ngc 6572 & @xmath267 & @xmath268 + ngc 6741 & @xmath269 & @xmath270 + ngc 6790 & @xmath271 & @xmath272 + ngc 6884 & @xmath273 & @xmath261 + ngc 7662 & @xmath274 & @xmath275 + ngc 6543 & @xmath276 & @xmath257 + m 2 - 24 & @xmath277 & @xmath278 + cn 2 - 1 & @xmath279 & @xmath280 + ngc 6720 & @xmath281 & @xmath282 + ngc 6620 & @xmath283 & @xmath284 + ngc 6567 & @xmath285 & @xmath286 + h 1 - 35 & @xmath287 & @xmath280 + h 1 - 50 & @xmath288 & @xmath289 + he 2 - 118 & @xmath290 & @xmath291 + m 1 - 20 & @xmath292 & @xmath293 + m 3 - 21 & @xmath294 & @xmath295 + m 3 - 32 & @xmath296 & @xmath270 + ic 1297 & @xmath297 & @xmath298 + ic 4634 & @xmath299 & @xmath300 + ic 4776 & @xmath301 & @xmath280 + ic 4997 & @xmath302 & @xmath303 + ngc 5873 & @xmath304 & @xmath305 + ngc 5882 & @xmath306 & @xmath298 + ngc 6302 & @xmath307 & @xmath308 + ngc 6818 & @xmath309 & @xmath310 + ngc 6153 & @xmath311 & @xmath312 + m 2 - 36 & @xmath311 & @xmath313 + m 1 - 42 & @xmath314 & @xmath315 + ngc 6778 & @xmath316 & @xmath317 + hf 2 - 2 & @xmath318 & @xmath319 + dddm 1 & @xmath320 & @xmath321 + hu 2 - 1 & @xmath322 & @xmath323 + ic 2003 & @xmath324 & @xmath325 + ic 5217 & @xmath326 & @xmath325 + me 2 - 2 & @xmath327 & @xmath328 + ngc 6803 & @xmath329 & @xmath330 + ngc 6807 & @xmath331 & @xmath332 + ngc 6833 & @xmath333 & @xmath334 + ngc 6879 & @xmath335 & @xmath336 + ngc 6891 & @xmath337 & @xmath338 + vy 1 - 1 & @xmath337 & @xmath325 + lccccc ic 3568 & 0.527 & 0.254 & 0.315 & 1.174 & 0.253 + ngc 7027 & 0.281 & 0.169 & 0.254 & 0.708 & 0.532 + ngc 7009 & 1.210 & 0.720 & 0.433 & 0.730 & 0 . + ngc 40 & 1.345 & 0.433 & 0.374 & 2.408 & 0 . + ngc 6210 & 0.641 & 0.392 & 0.469 & 0.681 & 0.208 + ngc 6826 & 0.641 & 0.374 & 0.520 & 0.768 & 0.415 + ngc 6572 & 0.401 & 0.151 & 0.374 & 1.857 & 1.639 + ngc 7662 & 0.271 & 0.238 & 0.228 & 0.145 & 0 . + ngc 6543 & 1.505 & 0.688 & 0.872 & 1.300 & 0.282 + ngc 6720 & 0.836 & 0.303 & 0.374 & 1.981 & 0.246 + ic 4634 & 0.688 & 0.258 & 0.411 & 1.867 & 0.634 + ngc 6818 & 0.258 & 0.171 & 0.254 & 0.543 & 0.516 + ngc 6153 & 2.543 & 0.714 & 0.254 & 2.987 & 0 . + m 2 - 36 & 2.543 & 2.056 & 0.714 & 0.249 & 0 . + m 1 - 42 & 2.543 & 1.422 & 0.543 & 0.850 & 0 . + average & & & & 1.0 & 0.4 +

the comparison of([] ) and(bal ) suggests that the fractional mean - square temperature variation ( ) has a representative value of 0.031 . in addition , he / h abundances of pne are found to be positively correlated with the difference between([] ) and(bal ) , suggesting that he / h abundances might have been overestimated generally because of the possible existence of h - deficient knots . the reason remains unclear .

a method is presented to derive electron temperatures and densities of planetary nebulae ( pne ) simultaneously , using the observed hydrogen recombination spectrum , which includes continuum and line emission . by matching theoretical spectra to observed spectra around the balmer jump at about 3646 , we determine electron temperatures and densities for 48 galactic pne . the electron temperatures based on this method [ hereafter(bal ) ] are found to be systematically lower than those derived from [ ] and [ ] ratios [ hereafter ,([] ) and([] ) ] . and the electron densities based on this method are found to be systematically higher than those derived from [ ] , [ ] , [ ] , [ ] and [ ] ratios . these results suggest that temperature and density fluctuations are generally present within nebulae . the comparison of([] ) and(bal ) suggests that the fractional mean - square temperature variation ( ) has a representative value of 0.031 . a majority of temperatures derived from the [ ] ratio are found to be higher than([] ) , which is attributed to the existence of dense clumps in nebulae those [ ] ir fine - structure lines are suppressed by collisional de - excitation in the clumps . by comparing([] ) ,([] ) and(bal ) and assuming a simple two - density - component model , we find that the filling factor of dense clumps has a representative value of 7 . the discrepancies between([] ) and(bal ) are found to be anti - correlated with electron densities derived from various density indicators ; high - density nebulae have the smallest temperature discrepancies . this suggests that temperature discrepancy is related to nebular evolution . in addition , he / h abundances of pne are found to be positively correlated with the difference between([] ) and(bal ) , suggesting that he / h abundances might have been overestimated generally because of the possible existence of h - deficient knots . electron temperatures and densities deduced from spectra around the paschen jump regions at 8250 are also obtained for four pne , ngc 7027 , ngc 6153 , m 1 - 42 and ngc 7009 . electron densities derived from spectra around the paschen jump regions are in good agreement with the corresponding values derived from spectra around the balmer jump , whereas temperatures deduced from the spectra around the paschen jump are found to be lower than the corresponding values derived from spectra around the balmer jump for all the four cases . the reason remains unclear .

astro-ph0305602

i

a model that rapidly computes the secular evolution of a gravitating disk planet system is developed . the disk is treated as a nested set of rings , with the rings/planets time evolution being governed by the lagrange planetary equations . it is shown that the solution to the dynamical equations is a modified version of the classical laplace lagrange solution for the secular evolution of the planets @xcite , with the modification being due to a ring s finite thickness @xmath409 that is a consequence of the dispersion velocity @xmath49 of that ring s constituent particles . since the ring s finite thickness @xmath0 softens its gravitational potential , this also softens the laplace coefficients appearing in the laplace lagrange solution over a scale @xmath410 . it is shown that the lagrange planetary equations admit spiral wave solutions when the tight winding approximation is applied . there are two types of spiral density ( or apsidal ) waves , long waves of wavelength @xmath411 and short waves of wavelength @xmath412 where @xmath413 is the disk s fractional thickness and @xmath232 is the angular rate at which the spiral pattern rotates . the simulations presented here show that the giant planets launch long waves at either a resonance in the disk or else at the disk s nearest edge , and that these waves propagate away until they reflect at the disk s far edge or else at a @xmath1barrier in the disk which resides where @xmath414 where @xmath415 is the maximum disk thickness that can sustain apsidal waves , with @xmath416 being the normalized disk mass . of course all of these findings may be derived from the stellar dispersion relation given in @xcite in the limit that the pattern speed @xmath232 is much smaller than the disk s mean motion @xmath51 . nonetheless , it is satisfying to see that the theory of unforced apsidal waves is readily obtained from the lagrange planetary equations ; with a little more effort the theory for forced apsidal waves [ e.g. , @xcite ] should also be recoverable . however new results are obtained for the nodal wave problem , which admits only a long wavelength solution @xmath260 to the planetary equations in the tight winding limit . in particular , it is shown that these waves can stall , that is , the waves group velocity plummets to zero as they approach a site in the disk where @xmath417 . if , however , these waves instead encounter a disk edge , they will reflect and return as long waves . in the limit that @xmath287 , the results for nodal waves propagating in a infinitesimally thin disk is recovered @xcite , but note that the wave stalling phenomenon does not appear in a @xmath118 treatment of the disk . the rings model is also used to examine the propagation of apsidal and nodal waves that are launched by the giant planets into a variety of kuiper belts having a mass @xmath418 m@xmath32 ( the estimated primordial mass ) down to @xmath337 m@xmath32 ( which is @xmath419 of the belts current mass estimate ) . in each simulation the giant planets deposit roughly the same fraction of their initial angular momentum deficits , @xmath395 and @xmath396 of the planets @xmath178 and @xmath179 , into the disk in the form of spiral waves . and since the waves angular momentum content is roughly the same in each simulation , the lower mass kuiper belts thus experience higher amplitude waves . indeed , the waves seen in the @xmath420 m@xmath32 simulations are of sufficient amplitude that they could in principle account for much of the dynamical excitation that is observed in the kuiper belt . however wave action in a @xmath421 m@xmath32 belt also requires its fractional scale height to quite thin , namely , @xmath422 . most likely , apsidal and nodal waves were shut off , due to self stirring by large kbos as well as by other external perturbations , long before the belt eroded down to its current mass , in which case the excitation by wave action would have been quite modest . the rings model developed here has many other applications . one issue of great interest is to determine whether apsidal and nodal waves may be propagating in saturn s rings . of particular interest are the short apsidal waves since their detection could yield the ring s dispersion velocity @xmath49 via a measurement of the short wavelength @xmath423 . although the ring particles dispersion velocity is of fundamental importance to ring dynamics , it is less than well constrained at saturn . of course , the differential precession due to planetary oblateness also needs to be included in the model [ c.f . , @xcite ] since this effect may actually defeat this form of wave action . the rings model can also be used to examine the forced motions of a relatively massless but much thicker circumstellar dust disk like @xmath73 pictoris . the warps and brightness asymmetries seen in this system are usually attributed to secular perturbations exerted by an unseen planetary system , and the code developed here can be used to very rapidly explore the wide range of planetary parameters . this rings model will be used to study these and other problems in greater detail in the near future . * acknowledgments * this paper is contribution xxxx from the lunar and planetary institute , which is operated by the universities space research association by cooperative agreement ncc5679 with the national aeronautics and space administration . this research was also supported by nasa via the origins of solar systems grant no . nag510946 issued through the office of space science .

a model that rapidly computes the secular evolution of a gravitating disk planet system is developed . the disk is treated as a nested set of gravitating rings , with the rings/planets time evolution being governed by the classical laplace lagrange solution for secular evolution but modified to account for the disk s finite thickness . there are two varieties of apsidal waves long waves and short waves . planets typically launch long density waves at the disk s nearer edge or else at a secular resonance in the disk , and these waves ultimately reflect downstream at a more distant disk edge or else at abarrier in the disk , whereupon they return as short density waves . planets also launch nodal bending waves , and these have the interesting property that they can stall in the disk , that is , their group velocity plummets to zero upon approaching a region in the disk that is too thick to support the continued propagation of bending waves . the rings model is used to compute the secular evolution of a kuiper belt having a variety of masses , and it is shown that the early massive belt was very susceptible to the propagation of low amplitude apsidal and nodal waves launched by the giant planets .

a model that rapidly computes the secular evolution of a gravitating disk planet system is developed . the disk is treated as a nested set of gravitating rings , with the rings/planets time evolution being governed by the classical laplace lagrange solution for secular evolution but modified to account for the disk s finite thickness . the lagrange planetary equations for this system yield a particular class of spiral wave solutions , usually denoted as apsidal density waves and nodal bending waves . there are two varieties of apsidal waves long waves and short waves . planets typically launch long density waves at the disk s nearer edge or else at a secular resonance in the disk , and these waves ultimately reflect downstream at a more distant disk edge or else at abarrier in the disk , whereupon they return as short density waves . planets also launch nodal bending waves , and these have the interesting property that they can stall in the disk , that is , their group velocity plummets to zero upon approaching a region in the disk that is too thick to support the continued propagation of bending waves . the rings model is used to compute the secular evolution of a kuiper belt having a variety of masses , and it is shown that the early massive belt was very susceptible to the propagation of low amplitude apsidal and nodal waves launched by the giant planets . for instance , these waves typically excited orbits to in a primordial kuiper belt of mass earth masses . although these orbital disturbances are quite small , the resulting fractional variations in the disk s surface density due to the short density waves is usually large , typically of order unity . this epoch of apsidal and nodal wave propagation probably lasted throughout the belt s first to years , with the waves being shut off between the time when the large km kbos first formed and when the belt was subsequently eroded and stirred up to its present configuration .

1004.5131

i

since the pioneering works @xcite , where the oscillatory behavior of the resonance charge - transfer probability for low - energy collisions was predicted , numerous publications have been devoted to the theoretical investigations of the charge - transfer , excitations and ionization in the h(1s)-h@xmath7 collisions ( see , e.g. , reviews @xcite ) . nonrelativistic two - center finite basis set calculations have been carried out in refs . nonrelativistic three - dimensional lattice methods in the position and momentum spaces have been applied for the time - dependent schrdinger equation in refs . @xcite . within the nonrelativistic approach , the probabilities and cross - sections for a homonuclear collision a@xmath8a@xmath9 for the nuclear charge @xmath10 can be easily obtained by scaling to the h(@xmath1)@xmath11h@xmath7 collision . in the straight - line trajectory approximation , the cross section @xmath12 scales exactly as @xmath13 @xcite , where @xmath14 is the projectile velocity . this scaling law is not valid , however , in the relativistic theory . collisions involving highly charged ions provide tests of relativistic and quantum electrodynamics effects in the scattering theory @xcite . the study of such processes can provide also a unique tool to probe the quantum electrodynamics ( qed ) in the supercritical coulomb field , if the total charge of the colliding ions @xmath15 is larger than the critical one @xmath16 ( see , e.g. , refs . @xcite and references therein ) . in the presence of such a field the energy of the one - electron @xmath17 state of the quasi - molecule can reach the negative - energy dirac continuum , when the distance @xmath18 between target ion @xmath19 and projectile ion @xmath20 becomes equal to the critical value @xmath21 . for the distance @xmath18 less than @xmath21 the ground state level dives into the negative - continuum spectrum . in the u@xmath5(1s)-u@xmath6 collision the critical radius for the point nucleus case was found to be @xmath22 fm @xcite . to date various approaches were developed to treat the heavy - ion collisions @xcite . in refs . @xcite , the two- and three - dimensional numerical lattice methods were employed to solve the time - dependent dirac equation at high energies . in refs . @xcite , high energy relativistic collisions of heavy ions were considered using the basis set approach , in which the time - dependent wave function was expanded in terms of the atomic eigenstates of the projectile and the target . for internuclear distances smaller than about 1000 fm some effects can also be evaluated within so - called monopole approximation , which accounts only for the spherically - symmetric part of the two - center potential @xcite . the atomic processes such as excitation , ionization and charge transfer in relativistic atomic collisions involving heavy and highly - charged projectile ions with energies ranging from 100 mev / u upward were studied in refs . @xcite and references therein . in the present work , we develop a new method for solving the two - center stationary and time - dependent dirac equations . the wave functions are expanded in terms of the dirac and dirac - sturm basis functions , which are central - field 4-component dirac bispinors centered at the ions . the radial parts of these orbitals are obtained by solving numerically the finite - difference radial one - center dirac and dirac - sturm equations . in the nonrelativistic calculations of atoms and molecules , so - called coulomb - sturmian basis set was introduced in ref . the hartree - fock calculations of atoms with this basis were considered by many authors ( see , e.g. , ref . the relativistic coulomb - sturmian basis was employed in the papers @xcite . in the present paper we use a non - coulomb relativistic sturm basis set , which is obtained by solving numerically the dirac - sturm equations with a special choice of the weight function , that was proposed in refs . this allows us to include any central - field potential in the radial equations for the large and small components of the basis functions . in particular , the coulomb potential of the other ion can be included in the radial equations within the monopole approximation . the basis set constructed in this way is described in detail in section [ subsec : basis ] . calculations of two - center integrals with the basis functions obtained require using special tools . in the nonrelativistic case , a special symmetrical procedure for such calculations , based on the lwdin reexpansion @xcite , was developed in refs . @xcite . in section [ subsec : integrals ] , we generalize this procedure to the relativistic case . to test the quality of the two - center expansion described above we perform relativistic calculations of the ground - state energy of molecular ion h@xmath23 and one - electron quasi - molecule th@xmath24 at the `` chemical '' distance @xmath25 a.u . and compare the results with high - precision calculations of refs . we also calculate the ground - state energy as a function of the internuclear distance @xmath18 and the critical radii @xmath21 for a number of one - electron quasi - molecules , including u@xmath26 . most calculations of the critical distances @xmath27 presented in the literature were performed either for the point - nucleus model @xcite or with a crude estimate of the nuclear - size effect @xcite . we calculate the critical distances for both point and extended nucleus models using the same basis set expansion . the obtained results and comparison with the calculations by other authors are presented in section [ subsec : stationary ] . the classical rutherford trajectories @xcite of the projectile and target ions are obtained by numerical solution of the newton s equations . the born - oppenheimer approximation is used to separate the motion of the electron and the nuclei . the magnetic interaction between the electron and the moving ions is neglected , because of low velocity of the projectile with respect to the target . the time - dependent dirac equation for the electron is solved using the two - center basis set expansion . the expansion coefficients can be defined employing , e.g. , the crank - nicholsen propagation scheme @xcite or the split - operator method @xcite . these methods conserve the norm of the time - dependent wave function at each time step , since the crank - nicholsen operator and the split - operator are unitary . however , in this work we use the direct evolution exponential operator method , which is more stable compared to the others . to obtain the matrix of the exponential operator in the finite basis set one has to diagonalize the generalized hamiltonian matrix at each step of time . since our basis set is not too large , the diagonalization procedure is not too time consuming . the amplitudes of the charge transfer to different bound states of the projectile ion are calculated nonperturbatively by projecting the time - dependent wave function onto the moving dirac orbitals of the projectile . in section [ subsec : charge - trnasfer ] we present the results of the relativistic calculations of the charge - transfer probabilities and cross sections for the h(1s)h@xmath7 , ne@xmath0(@xmath1)ne@xmath2 , xe@xmath28(@xmath1)xe@xmath29 , and u@xmath5(@xmath1)u@xmath6 low - energy collisions . all the calculations are performed in the laboratory frame @xmath30 , that is defined to be at rest with respect to the initial target position . the h(@xmath1)h@xmath7 collision is considered in section [ subsec : h2 ] . since the relativistic effects in this collision are negligible , the results of our calculations can be compared with nonrelativistic data obtained by other authors ( section [ subsec : h2 ] ) . the role of the relativistic effects is investigated in sections [ subsec : ne2 ] , [ subsec : xe2 ] , and [ subsec : u2 ] , where the relativistic and nonrelativistic calculations of the charge - transfer probabilities and cross sections are performed for higher-@xmath31 ions .

a new method for solving the time - dependent two - center dirac equation is developed . the time - dependent dirac wave function is represented as a sum of atomic - like dirac - sturm orbitals , localized at the ions . the atomic orbitals are obtained by solving numerically the finite - difference one - center dirac and dirac - sturm equations with the potential which is the sum of the exact reference - nucleus potential and a monopole - approximation potential from the other nucleus . an original procedure to calculate the two - center integrals with these orbitals is proposed . the approach is tested by calculations of the charge transfer and ionization cross sections for the h(1s)proton collisions at proton energies from 1 kev to 100 kev . the obtained results are compared with related experimental and other theoretical data . to investigate the role of the relativistic effects , the charge transfer cross sections for the ne()ne ( at energies from to mev / u ) and u(1s)u ( at energies from 6 to 10 mev / u ) collisions are calculated in both relativistic and nonrelativistic cases .

a new method for solving the time - dependent two - center dirac equation is developed . the time - dependent dirac wave function is represented as a sum of atomic - like dirac - sturm orbitals , localized at the ions . the atomic orbitals are obtained by solving numerically the finite - difference one - center dirac and dirac - sturm equations with the potential which is the sum of the exact reference - nucleus potential and a monopole - approximation potential from the other nucleus . an original procedure to calculate the two - center integrals with these orbitals is proposed . the approach is tested by calculations of the charge transfer and ionization cross sections for the h(1s)proton collisions at proton energies from 1 kev to 100 kev . the obtained results are compared with related experimental and other theoretical data . to investigate the role of the relativistic effects , the charge transfer cross sections for the ne()ne ( at energies from to mev / u ) and u(1s)u ( at energies from 6 to 10 mev / u ) collisions are calculated in both relativistic and nonrelativistic cases .

1205.5730

i

the last decades have inaugurated great success in precise _ ab - initio _ calculations for few - body scattering processes @xcite . these calculations made it possible to describe accurately the results of numerous recent experiments on elastic @xmath2 scattering at energies up to 350 mev and also the three - body breakup @xmath0 at low and moderate energies @xmath3 mev . however some problems remain unsettled even at such low energies . these are the so - called @xmath4-puzzle ( as well as other puzzles for various tensor and vector analyzing powers ) in elastic scattering , the problems with an adequate description of the pairwise @xmath5-channel contribution to three - body breakup at low energies @xcite and breakup cross section in some particular three - particle configurations such as the quasi - free scattering @xcite and the space - star @xcite configurations . the most plausible reason for the visible discrepancies with experimental data in this area is likely not insufficient accuracy of numerical calculations but rather some deficiency in the input @xmath6- and @xmath7-interactions . at the same time , the progress in the field of precise few - nucleon calculations , particularly in testing of new models for @xmath7-interactions , is restrained strongly by a high number of very complicated few - nucleon calculations , especially above the three - body threshold . because of these complications of traditional computational schemes for the direct solution of the faddeev yakubovsky equations , one observes a rise of interest in recent years in alternative approaches @xcite to calculate the scattering observables by simpler methods . among such alternative approaches one can note a preference for the so called @xmath8-methods . these methods are based on expansions of the scattering solution into a basis of square - integrable functions @xcite . such @xmath8-methods proved to be very well suited and quite efficient for numerous applications . one of the most successful approaches of this type is the continuum - discretized coupled - channel ( cdcc ) method developed three decades ago for treatment of breakup processes in direct nuclear reactions @xcite . the cdcc approach in its traditional form was unable to treat other channels than elastic scattering and projectile ( or target ) breakup . recently a few groups generalized the traditional cdcc approach to scattering of three - fragment projectiles by a stable target @xcite . however this generalized approach can be considered as a hybrid method : @xmath8 discretization of inner motion in the three - body projectile and the traditional treatment of a coupled - channel problem on the other hand , the present authors have developed some alternative @xmath8-technique @xcite which is based on the idea of complete continuum discretization with a special stationary wave - packet basis in momentum space ( three - body lattice basis ) . the basic distinction of such an approach from the traditional cdcc - scheme for the three - body systems is that the wave - packet approach is dealing with a full discretization of the three - body continuum . in other words , the discretization on _ both _ jacobi coordinates is used here rather than the discretization on the alone coordinate of the projectile inner motion as in the cdcc approach2 scattering and also the breakup @xmath9 process using the laguerre polynomial basis . unfortunately this prospective approach has not received any further development . ] . our approach with the global discretization over all valence coordinates leads immediately to a few important advantages in the accurate treatment of few - body scattering . among those the following are the most important : * the few - body scattering problem is consistently formulated in a hilbert space of three - body _ normalized _ states , similarly to the bound - state problem . * the approach employs the integral equation framework of scattering theory instead of the differential equation approach ( e.g. in the cdcc ) where the boundary conditions in few - body scattering channels are not easy to formulate , especially in terms of the @xmath8 basis used . contrary to this , the integral equation formulation allows to avoid any explicit account of the boundary conditions . * when working within the wave - packet formulation of the scattering problem one can derive explicit formulas for some scattering operators ( e.g. channel resolvents ) . such fully analytical finite - dimensional approximations , being substituted into integral equation kernels , lead immediately to their algebraic matrix analogues . thus , our final equations are simple matrix linear equations with regular matrices . in our previous works @xcite we have demonstrated how to find the elastic @xmath10 scattering amplitudes in lattice representation . in the present paper , we generalize the technique to the three - body breakup treatment . so we present here the complete formalism for determination of three - body breakup amplitudes . it is important to emphasize in this connection that the accurate treatment of three - body breakup within the @xmath8 type approach is much less obvious than that of elastic ones and thus requires some additional delicate theoretical studies . in particular , the matrix elements in the breakup amplitude are not truncated over all spatial coordinates ( in contrast to the elastic and rearrangement amplitudes ) , so the validity of the @xmath8 scheme in the treatment of the breakup processes should be studied carefully . as some substantiation for such approach one can consider the three - body breakup calculations within the cdcc - approach where the discretization of the continuum in the projectile inner subhamiltonian has been used for the description of the breakup amplitudes @xcite . so , the natural generalization of such a partial continuum discretization to the case of full three- and few - body continuum within the faddeev equation approach is an important next step . moreover , this fully discretized approach studied in the present work allows to simplify drastically all calculations and makes it more universal and elegant . the present work has the following structure . in the section ii , a three - body lattice - like free wave - packet basis is described in detail together with a similar basis for the channel hamiltonian . here we also discuss the properties of these bases . the complete formalism for elastic scattering and breakup , as applied to the @xmath2-system in the packet representation is presented in section iii . in the section iv , a few useful numerical illustrations and their comparison with the standard faddeev benchmark calculations are given . our results are summarized in the section v. for the sake of convenience for the reader we add three appendices . in appendix a we describe the detailed scheme for calculation of the three - body overlap matrix in the three - body lattice basis for recoupling of different jacobi coordinates . in appendix b we give the convenient wave - packet formalism for the solution of three - body scattering problem with separable pairwise interactions . in the last appendix c we discuss some features of our numerical calculations .

a novel approach is developed to find the three - body breakup amplitudes and cross sections within the modified faddeev equation framework . the method is based on the lattice - like discretization of the three - body continuum with a three - body stationary wave - packet basis in momentum space .

a novel approach is developed to find the three - body breakup amplitudes and cross sections within the modified faddeev equation framework . the method is based on the lattice - like discretization of the three - body continuum with a three - body stationary wave - packet basis in momentum space . the approach makes it possible to simplify drastically all the three- and few - body breakup calculations due to discrete wave - packet representations for the few - body continuum and simultaneous lattice representation for all the scattering operators entering the integral equation kernels . as a result , the few - body breakup can be treated as a particular case of multi - channel scattering in which part of the channels represents the true few - body continuum states . as an illustration for the novel approach , an accurate calculations for the three - body breakup process with non - local and local interactions are calculated . the results obtained reproduce nicely the benchmark calculation results using the traditional faddeev scheme which requires much more tedious and time - consuming calculations .

1703.05158

m

the discrete - element method ( dem ) is already a mature tool that is applied in conjunction with experiments both for a better understanding of the micromechanics of granular materials and as a means of @xmath2 experimentation when laboratory experiments are unavailable . in a similar vein , the inclusion of a fluid at the subgranular scale in dem simulations provides a powerful tool in the broad field of fluid - grain mixtures . obviously , the available computational power and research time restrict considerably the number of particles or the size of a physical system . in the case of dry granular materials , statistically representative samples are obtained and simulated with @xmath3 of particles in 2d @xcite . despite enhanced kinematic constraints , 2d simulations often lead to novel physical insights and realistic behaviors that can be easily generalized to 3d configurations . however , with fluid in the pore space , 2d simulations are much less reliable in the dense regime since the pore space is discontinuous with zero permeability . this two - dimensional flaw can be partially repaired by adding artificially a permeable layer on the particles . but only 3d simulations may account for a realistic behavior of particle - fluid mixtures with their natural permeability . moreover , complex geometries / boundaries relating to realistic engineering problems can not be fully captured in 2d simulations or symmetric 2d extensions ( e.g. axis - symmetry ) ; only 3d approaches can handle such problems in full generality . we developed a 3d fluid dynamics algorithm based on the lattice boltzmann method ( lbm ) . this algorithm was interfaced with a dem algorithm with a standard linear spring - dashpot - friction model of contact between particles . the combined lbm - dem method for particle - laden fluid is then further coupled to a deformable solid domain using finite elements to model a rubber - like behavior . the rubber coupling is intentionally simplified . within actual computer power , it is still a significant challenge to model the entirety of most engineering systems and problems . certain sub - scale details and complex interactions are unnecessary to capture the macroscale system response for a given loading . we utilize symmetric boundaries ( where possible ) and a variety of techniques to shrink the system size and average - up sub - scale phenomena . specifically in this work : to handle sub - scale behavior in the fluid we use a large - eddy - simulation ( les ) technique ( see sect . [ subsec : lbm ] ) , to mimic a large fluid domain outside the focus region we have created a technique we denote _ zoom - in with effective boundaries _ ( zieb ) ( see sect . [ subsec : zoominlinktech ] ) , and to reduce simulation time we developed a weak coupling to the rubber domain based on a _ neo - hookean _ model developed in abaqus . the last part is computed separately and only the result is imported into lbm - dem simulation ; the coupling and description of this part is expounded in sec . [ subsec : lbmdemrubbercoupling ] . the dem is based on the assumption of elastic solids with damping and frictional contact behavior @xcite . newton s equations of motion are integrated for all degrees of freedom with simple force laws expressing the normal and friction forces as explicit functions of the elastic deflection defined from the relative positions and displacements at contact points . we treat all quasi - rigid solids in the domain using this dem description , including grains , the valve , and solid system boundaries . correspondingly , all solid - on - solid contact forces ( e.g. grain on grain , grains on valve , grain on solid wall ) are obtained using dem contact laws . the valve and system walls are discretized as a kinematically constrained connected mesh of polyhedral solid ` particles ' . to simplify contact interactions , we assume linear elastic normal and tangential contact forces characterized by a normal stiffness @xmath4 and tangential stiffness @xmath5 . this is applied to all contact interactions , e.g. between spheres , polyhedra , or sphere - polyhedra , though the stiffnesses can vary depending on the two objects in contact . in additional to the elastic part , a dissipative part of the contact force is necessary @xcite . in our model , we use a linear visco - elastic law for normal damping and a linear visco - elasto - plastic law for tangential damping and friction forces where the plastic part uses a coulomb law . the visco - elastic law is modeled by a parallel spring - dashpot model . the contact normal force is defined as : @xmath6 where @xmath7 is the contact normal vector and @xmath8 is the relative velocity along the contact normal . @xmath9 represents a viscosity parameter with a value that depends on the normal restitution coefficient between grains . according to coulomb s law the friction force is given by : @xmath10 and @xmath11 where @xmath12 is the friction coefficient , @xmath13 is the tangential relative velocity , and @xmath14 is a viscosity parameter , which depends on the tangential restitution coefficient . the equations of motion ( both linear and angular momentum balance ) are integrated according to a velocity verlet scheme @xcite . the lbm is based on a material representation of fluids as consisting of particle distributions moving and colliding on a lattice @xcite . the partial distribution functions @xmath15 are introduced to represent the probability density of a fluid particle at the position @xmath16 with a velocity @xmath17 at time @xmath18 along discrete direction @xmath19 . the three components of @xmath20 are given in tab . [ tab : civalue ] . rrr rrr rrr rrr rrr rrr rr @xmath19 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 + @xmath21 & 0 & -1 & 0 & 0 & -1 & -1 & -1 & -1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 + @xmath22 & 0 & 0 & -1 & 0 & -1 & 1 & 0 & 0 & -1 & -1 & 0 & 1 & 0 & 1 & -1 & 0 & 0 & 1 & 1 + @xmath23 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 1 & -1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & -1 + the lattice boltzmann method is often non - dimensionalized when applied to physical problems . the governing macroscopic equations are given in terms of lattice units : @xmath24 where @xmath25 is the lattice spacing , @xmath26 is the lattice speed with @xmath27 the time step , and @xmath28 is the fluid density at zero pressure . for the following , we will describe the method in lattice units . figure [ fig : schemalattice ] shows a cartesian grid where the meshing scheme d3q19 , corresponding to 18 space directions in 3d used in our simulations , is represented . in lbm , the scheme d3q19 is defined for each node where the distribution functions evolve according to a set of rules , which are constructed so as to ensure the conservation equations of mass , momentum and energy ( with dissipation ) , so as to recover the navier - stokes equations @xcite . this holds only when the wave lengths are small compared to the lattice spacing @xcite . at each lattice node , the fluid density @xmath29 and momentum density @xmath30 are defined as @xmath31 @xmath32 and the temperature is given by @xmath33 where @xmath34 is the number of space dimensions , @xmath35 is particle mass , and @xmath36 is the boltzmann constant . the equilibrium state is assumed to be governed by the maxwell distribution : @xmath37 \label{eqn : maxw}\ ] ] where @xmath38 is the mean velocity . by expanding ( eq . [ eqn : maxw ] ) to order 2 as a function of @xmath39 , which is the local mach number with @xmath40 being the lbm sound velocity , a discretized form of the maxwell distribution is obtained and used in the lbm : @xmath41 \label{eq : equilibre_maxwell_boltzmann}\ ] ] where the factor @xmath42 , @xmath43 and the rest of @xmath44 . @xmath45 depend on the scheme with the requirement of rotational invariance @xcite . the lbm sound speed is then given by @xmath46 . the velocities evolve according to the boltzmann equation . in its discretized form , it requires an explicit expression of the collision term . we used the bhatnagar - gross - krook ( bgk ) model in which the collision term for each direction @xmath19 is simply proportional to the distance from the maxwell distribution @xcite : @xmath47 where @xmath48 is a characteristic time . hence , for the d3q19 scheme , we have a system of @xmath49 discrete equations governing the distribution functions : @xmath50 these equations are solved in two steps . in the collision step , the variations of the distribution functions are calculated from the collisions : @xmath51 where the functions @xmath52 designate the post - collision functions . in the streaming step , the new distributions are advected in the directions of their propagation velocities : @xmath53 the above equations imply an equation of state @xcite : @xmath54 the kinematic viscosity is then given by @xcite @xmath55 \label{eq : viscositelbm}\ ] ] with the requirement @xmath56 . as discussed in @xcite , the lattice boltzmann method holds only when the pressure wave lengths are small compared to the lattice spacing unit . this imposes a limitation on mach number @xmath57 and therefore fluid speeds higher than the sound speed can not be simulated . in nature , for a given fluid we have : sound speed @xmath58 , density @xmath28 and viscosity @xmath59 . from eq . [ eq : collision_step ] , we need the relaxation time @xmath48 . this is related to @xmath60 , @xmath59 and @xmath25 by : @xmath61 equation . [ eq : tauphyslbm ] shows that since @xmath60 and @xmath59 are fixed from fluid properties , only @xmath25 can be used to ensure the stability of lbm , which becomes unstable when @xmath62 . numerically , there is a limitation in computer capability regarding the smallest value of @xmath25 . to handle this , a _ sub - grid turbulent model _ based on les with a smagorinsky turbulence model is used @xcite . the viscosity formulation is : @xmath63 = c_s^2\left[\tau + \tau_{_t } - \frac12\right]\ ] ] where @xmath64 is the sub - grid les viscosity and @xmath65 is the sub - grid les lbm relaxation time . the les viscosity is calculated from the filtered strain rate tensor @xmath66 and a filter length scale @xmath67 through the relation @xmath68 where @xmath69 is the smagorinsky constant . in lbm , @xmath70 is obtained from the second momentum @xcite @xmath71 as @xmath72 , where @xmath73 . from @xmath70 and @xmath74 , @xmath65 is expressed as : @xmath75\ ] ] where the filter length scale @xmath76 is the spatial lattice discretization . there exist different techniques to model fluid - structure interaction . the most used in cfd is stress integration , however , in lbm the preferred approach is based on momentum exchange @xcite . this approach is simple in lbm , since in lbm each node already contains information about the derivatives of the hydrodynamic variables in each distribution function @xmath77 @xcite . due to the presence of a solid boundary , after the collision step but before the streaming step , we know the population of @xmath77 except those which reflect off the solid wall as shown in fig . [ fig : fluideparticule_cb ] ( @xmath78 ) . in our simulations , we use the method developed by bouzidi @xcite , which mimics a macroscopic no - slip condition . for clarity , we describe the interaction rules using lattice units . this means that the time step is @xmath79 , the space discretization is @xmath79 , etc . out of the characteristic scales , we will denote , around the fluid - solid boundary , @xmath21 as position and the subscripts @xmath80 , @xmath81 , and @xmath82 respectively will indicate either the _ fluid _ or _ solid _ domain and the _ fluid - solid boundary_. we present the method in simplified 1d form . for more clarity in the way momenta are exchanged between fluid and solid domains , let us introduce @xmath83 . according to the lbm scheme ( collide and stream sect . [ subsec : lbm ] ) in the presence of a solid wall , we have two scenarios depending on the wall position @xmath84 ( see fig . [ fig : fluideparticule_cb ] ) : 1 . @xmath85 where fluid leaves node @xmath86 , reflects off the wall , and reaches @xmath86 again in time less than @xmath27 . 2 . @xmath87 where fluid leaves node @xmath86 , reflects off the wall , and reaches @xmath86 again in time greater than @xmath27 . to handle these scenarios , we introduce a fictitious node @xmath88(see fig . [ fig : fluideparticule_cb ] ) such that after streaming , a fluid particle leaving @xmath88 arrives at @xmath86 exactly in time @xmath27 . as shown in fig . [ fig : fluideparticule_cb ] , if @xmath86 is the last fluid node before we reach the solid boundary , @xmath89 should be a solid node . let @xmath90 be the distribution function such that @xmath90 is the opposite direction of @xmath19 where @xmath19 is the direction oriented from fluid node to solid boundary . by using a linear interpolation , @xmath90 is expressed as follow : @xmath91 where @xmath92 corresponds to the distribution function of @xmath77 after the collision step but before streaming and @xmath93 . the term @xmath94 is calculated from the boundary velocity and is zero if the boundary is stationary . @xmath94 is calculated by considering that the fluid velocity @xmath95 evolves linearly between @xmath96 and @xmath97 . if @xmath98 is the boundary velocity , @xmath95 is then defined by @xmath99 at first order in @xmath95 . the equilibrium value of @xmath77 is given by @xmath100 where @xmath101 is constant and depends on the lattice discretization scheme @xcite . using a linear interpolation , @xmath94 is given by : @xmath102 hydrodynamic forces acting on the structure are calculated by using momentum exchange @xcite . let @xmath103 and @xmath104 be fluid and solid momentum calculated near the boundary . the exchanged momentum is given by : @xmath105 @xmath103 and @xmath104 are calculated as follows : @xmath106 @xmath107 where @xmath34 is the space dimension , and @xmath108 and @xmath109 are respectively the fluid and the solid distribution functions . to be clear , @xmath109 is constructed at a lattice point occupied by solid by taking the solid velocity @xmath110 and density @xmath111 and assigning a maxwell equilibrium distribution , per eq . [ eq : equilibre_maxwell_boltzmann ] . the hydrodynamic force @xmath112 and torque @xmath113 are then given by : @xmath114 @xmath115 where @xmath116 is the distance between the center - of - mass of the solid domain and the boundary node @xmath84 . unlike the coupling between lbm and dem , the lbm - rubber and dem - rubber coupling is indirect . we focus our explanation below on the case of a rubber ring component of a valve , but the idea can be generalized to other cases . a first simulation is performed using abaqus from which the deformed rubber shape and reaction force of the valve seat on the rubber are saved for many states of rubber compression . this simulation uses no fluid or particles . the abaqus simulation consists of compressing the rubber ring geometry against the bare valve seat ( see inset of fig . [ fig : force_vs_displacement ] ) . the rubber is simulated as a nearly incompressible neo - hookean elastic solid with a strain energy @xmath117 ( per unit reference volume ) given by @xmath118 where @xmath119 is the first deviatoric strain invariant defined as @xmath120 , the deviatoric stretches are given by @xmath121 , @xmath122 is the total volume ratio , and @xmath123 are the principal stretches . the ( small - strain ) shear modulus is @xmath124 and bulk modulus is @xmath125 . a frictionless contact between rubber and valve seat is used for simplicity . figure [ fig : abq_deform_snapshot ] shows several snapshots during the abaqus simulation and fig . [ fig : force_vs_displacement ] gives the seat net - force on rubber as a function of ( downward ) displacement @xmath126 of the rubber ring , @xmath127 . we index each deformed rubber configuration by the value of @xmath126 it corresponds to . the abaqus tests are performed under quasi - static conditions but we also assume damping can exist such that upward the force on the rubber satisfies the relation @xmath128 . ] then , the data from the abaqus simulation is used in a weak coupling routine to describe rubber configurations when the ring takes part in a slurry simulation . in short , the method determines which of the deformed rubber configurations from the stand - alone abaqus tests is the best representation of the actual deformed rubber state at that moment in the slurry problem . hence , the utility of this method lies in the fact that the rubber deformation that occurs in the actual slurry problem largely resembles the modes of deformation seen in a purely solid compression experiment . situations where the rubber surface becomes heavily locally deformed could be problematic for this approach . from the lbm - dem point of view , the rubber is composed of tetrahedra ; this allows us to compute contact forces for dem and the exchanged momentum for lbm as if it were a simple collection of polyhedral objects . since the abaqus simulation is performed without fluid or particles , to use its solutions we need to deduce an effective upward force from lbm - dem acting on the bottom rubber surface , which can then be referenced against the abaqus data to infer a deformed rubber shape . the effective force is needed because the rubber in the abaqus simulations has contact only with the valve seat , whereas in the slurry case , there can be additional forces from fluid and particles extending to the lateral surfaces of the rubber . key to our routine is to identify two subsets of the exposed rubber surface , denoted surface a and surface b. surface a is the part that makes contact with the valve seat and surface b remains free in the abaqus simulations ( see left fig .[fig : abq_coupling_config ] ) . in particular , surface @xmath129 and @xmath130 are geometrically defined using the the last frame of the abaqus simulation where the rubber is fully compressed . in the slurry case , we add uniform hydrostatic stress to the observed rubber loading distribution until the mean normal stress acting on surface b vanishes . this leaves us with a loading state that resembles one from abaqus . because the rubber is essentially incompressible , changing the hydrostatic stress uniformly along the surface does not affect the deformed rubber configuration . to be specific , we compute the normal stress on surface @xmath129 using @xmath131 and on surface @xmath130 using @xmath132 where @xmath133 is the stress from hydrodynamic , particle , and valve - seat forces and @xmath134 is the normal vector on section @xmath135 . since the rubber deformation is caused by the shear part of the stress , we uniformly subtract the traction @xmath136 from all normal stresses acting on the rubber . this modified loading now resembles the abaqus loading ( inset fig . [ fig : force_vs_displacement ] ) in which only an upward force on surface @xmath129 exists . therefore , we define the effective upward force on surface @xmath129 as @xmath137 . the rubber shape is updated using a four step loop , which is performed after particle positions and fluid data are updated . the goal of the iteration routine is to implicitly solve eq . [ constit ] for @xmath126 so that the effective force from particles , fluid , and the valve seat on the final rubber state matches the force from eq . [ constit ] . * step 1 : compute effective upward force @xmath138 from fluid , particle , and valve - seat interactions on rubber given the current guess for @xmath126 * step 2 : use this force and eq . [ constit ] to update to a new guess for @xmath126 . * step 3 : check if new and old @xmath126 differ by less than a tolerance . if so break , if not update the rubber shape based on the new @xmath126 . * step 4 : update applied force on surfaces @xmath129 and @xmath130 according to the rubber shape given by the new guess for @xmath126 . we assume that fluid forces do not change throughout the iteration procedure . this is true by assuming a small incremental rubber shape change between iterates , so only particle and valve - seat forces on the rubber are updated during step 4 . in step 2 we utilize the following update rule @xmath139 \frac{\delta t}{\eta } + \delta}{1 + \frac{\nu}{\eta } } \label{eqn : rubber_eqn_2}\ ] ] where @xmath140 is the actual displacement of the rubber at the beginning of the time - step . the coefficient @xmath141 is numerically selected to aid convergence . note that if the updated value of @xmath126 matches the value inputted , then eq . [ constit ] is solved . the update rule above attempts to move @xmath126 toward such a solution with each pass through the iteration loop . we check convergence of @xmath126 ( step 3 ) using a tolerance that is scaled by the error of the first iterate , @xmath142 , where @xmath143 is the value obtained for @xmath126 after the first pass of the loop . we use simple linear interpolation to compute values of @xmath127 when @xmath126 is between two neighboring values from the abaqus output . and @xmath130 , and the initial and final shape of the rubber as it proceeds through the iteration loop to solve eq . [ constit ] , with a representation of particles in contact shown . the right figure gives a 1d analog of the model to compute the rubber deformation . ] the third step consists of updating the rubber shape using the new guess for @xmath126 obtained in step 2 . again , a linear interpolation is applied as is done for @xmath127 . for the shape , we export the abaqus mesh , which allows the interpolation . for example , if the new @xmath126 lies directly between two frames of the abaqus data , the rubber shape is updated by moving the nodes of the rubber to positions halfway between those of the neighboring frames . the fourth step consists of recomputing the contact force between the tetrahedra representing the rubber , and the particles / valve - seat . the contact force is computed using eq . [ eq : normal_force ] for the normal part and eq . [ eq : friction_force ] for the tangential part . one specificity here is that we do not update @xmath144 during the coupling loop routine unless when @xmath145 where @xmath146 . a schematic model of one update through the iteration loop is presented in fig . [ fig : abq_coupling_config ] ( left ) and a 1d mechanical model of the treatment of the rubber interaction is visualized in fig . [ fig : abq_coupling_config ] ( right ) . the numerical method described in previous sections is implemented using the algorithm displayed in fig . [ fig : algo ] . note that we compute the valve acceleration by enjoining the applied force and mass of the rubber and valve components together in the verlet update ; rubber deformations are not included in the calculation of net acceleration of the valve / rubber composite as they are small compared to the movement of the valve overall . as shown in fig . [ fig : algo ] , the lbm step is computed in parallel with dem force calculation and rubber force calculation . the _ zoom - in with effective boundaries _ ( zieb ) technique replaces a fluid reservoir / domain with an analytical solution that interacts dynamically with the remainder of the domain . the challenge is to determine the correct effective dynamics at the fictitious interface , and to transfer the analytical result to lbm distribution functions . in this study we model valves , which can be positioned far from the pump that leads the slurry to the valve . the goal is to avoid having to calculate flow in the expansive pump region . from a computational point of view , one might assume a simple input velocity boundary condition should solve the problem , however , for a compressible fluid , the imposed flow and pressure may depend on the total removed volume and feedback with the dynamics within the valve system . in this section , we first detail how to obtain the analytical solution then explain how to implement this solution as an lbm boundary condition . ) and the closest virtual lattice point ( at @xmath147 ) . , width=336 ] per fig . [ virtual ] , we assume the virtual ( i.e. removed ) domain is a cylinder and piston . the cylinder is initially full of fluid and has total volume @xmath148 . as the piston moves , fluid is pushed into the simulated domain . let the movement of the piston be given by some prescribed @xmath149 , where @xmath150 measures the piston displacement . the cross - sectional area of the piston ( and cylindrical domain ) is @xmath151 . the piston velocity , @xmath152 , is simply defined from the time - derivative of @xmath150 . define as @xmath153 the mean in - flowing fluid velocity component on the interface between the domains . let @xmath29 be the average density of fluid in the virtual cylinder between the piston head and the interface . further , we make the simplifying assumption that in the cylinder region the fluid density is in fact uniform , such that it is equal to @xmath29 throughout . conservation of fluid mass in the cylinder domain can be expressed by balancing the mass rate within the cylinder against the mass flux into the simulated domain : @xmath154 \label{eq : mass_conv}\ ] ] in a fully continuum framework , the above equation would need to be augmented with momentum balance in order to provide the in - flowing velocity , @xmath153 , at the fictitious interface . however , using the lbm description , we can update @xmath153 in another way , which is consistent with momentum balance on the small scale . at a given time @xmath155 , we assume @xmath156 is given in the cylinder domain and is equal to the density at @xmath157 , where , per fig . [ virtual ] , @xmath157 is the lattice point in the simulated domain that is adjacent to the interface with the virtual domain . we suppose the velocity at @xmath157 is the interfacial velocity @xmath158 . both density and velocity at time @xmath155 at @xmath157 are defined by eq . [ dens ] and [ momentum ] through distribution functions @xmath159 . the distribution functions are updated to @xmath160 under the following procedure , which is applied after the collision step but before streaming . first we update and store the density @xmath161 at @xmath157 using explicit integration of eq . [ eq : mass_conv ] : @xmath162 \label{eq : ziebdesity8}\ ] ] next , a partial lbm streaming step is performed at @xmath157 using the distributions at time @xmath155 . during this step @xmath157 streams to and from its neighboring ` real ' lattice points within the simulated domain . however , it only streams out of the interface with the virtual domain and does not receive any distributions from the virtual domain . define @xmath163 as the density at @xmath157 after this partial streaming step . the next step is to back - solve the needed distributions to be streamed in from the virtual domain in order to guarantee the final density at @xmath157 equals @xmath161 . for example , consider a setup as shown in fig . [ virtual ] and suppose the fictitious interface is normal to the @xmath164 direction ( see fig . [ fig : schemalattice ] ) . after the partial streaming step , updated ( though not finalized ) distribution values exist for all the @xmath77 except for the values associated to @xmath165 and @xmath166 . these five distribution values are all unknown after the partial streaming step . to compute them , first we modify only the value of the @xmath167 distribution , which is the distribution that streams into the simulated domain normal to the fictitious boundary : @xmath168 this advects all the missing density at @xmath169 from a fictitious node @xmath147 ( see bottom of fig . [ virtual ] ) . with these distributions , the velocity at time @xmath160 is computed at @xmath157 according to @xmath170 . because the distributions in the @xmath171 and @xmath166 directions are still unknown at this point , the maxwell equilibrium function eq . [ eq : equilibre_maxwell_boltzmann ] is then used to redistribute all the distributions at @xmath157 to a more natural , equilibrium state . this updates all the distributions to their final values , at @xmath160 . we notice here that the initial value of @xmath29 at the beginning of the simulation should be a normalized value ( in lattice units ) otherwise an additional step of normalizing by the physical fluid reference density will be necessary before using it in eq . [ eq : mass_conv ] and eq . [ eq : ziebdesity9 ] . we first test some of the individual components of the routine . in this section , we provide separate numerical validations of the zieb technique , the rubber deformation model , and the lbm . to validate the zieb method , we performed an analysis of fluid flow in a geometry comprised of a piston driving fluid passing through a narrow restriction . this flow field is then compared to that obtained using a `` virtual piston '' in which the domain containing the moving piston is removed and in its place an effective boundary condition from zieb is used , see fig . [ fig : zoomnlink_config ] . the real piston begins positioned such that the fluid volume between the piston and input section is @xmath172 ; the same @xmath148 is used in eq . [ eq : ziebdesity8 ] for the virtual piston . we use @xmath173 and @xmath174 . as input parameters , we use a ( pressure - free ) fluid density @xmath175 , dynamic viscosity @xmath176 and a smagorinsky constant of @xmath177 for the sub - grid turbulence model @xcite . figure [ fig : velocity_density_opened ] shows the comparison between the two simulations regarding fluid velocity and the normalized input fluid density computed in the same domain ( see fig . [ fig : zoomnlink_config ] ) . the agreement is strong , even the time - dependent fluctuations , confirming the correctness of the zieb method . the test of the rubber coupling and rubber deformation is performed running a loading / unloading test without fluid . a force ( loading / unloading ) @xmath178 is directly applied on the valve which presses the rubber into contact with twelve frozen spheres ( see fig . [ fig : rubberloadunloadtest ] ) . two phases are considered : a loading phase with @xmath179 then an unloading phase with @xmath180 . we use a frictionless contact type between the rubber and spheres where normal stiffness is set to @xmath181 and no normal damping is used to insure that all dissipation comes from internal rubber damping . the rubber coupling parameters ( eq . [ eqn : rubber_eqn_2 ] ) are set to @xmath182 and @xmath183 . the valve density is set to 7850 @xmath184 and the rubber density to @xmath185 . the time step is set to @xmath186 . [ fig : rubberloadunloadtest ] shows the loading / unloading force , the reaction force @xmath138 of spheres on the rubber , the displacement @xmath126 ( right inset ) and the corresponding force @xmath127 . the agreement of @xmath178 , @xmath187 , and @xmath138 after a relaxation time verifies the coupling . the last test is focused on verifying the fluid lbm simulation by comparing flow of fluid in the rubber channel of a pressure - valve assembly against an analytical flow solution ( recall fig . [ fig : opned_closed ] ) . we can run a simulation where the fluid viscosity is large , such that the flow in the channel will be in the stokes limit . to aid in calculating an analytical solution , we treat the flow as radially directed and assert the lubrication approximation . in view of fig . [ fig : opned_closed ] for the definition of the @xmath22 direction , we obtain the following system of equations , which includes momentum and mass balance under the lubrication limit : @xmath188 this is solved by @xmath189 where @xmath129 is an undetermined constant and @xmath190 where @xmath69 is a constant and @xmath129 is the same as in the velocity field equation . from the @xmath191 equation , the pressure difference @xmath192 between @xmath193 and outer at @xmath194 ( see fig . [ fig : channel_velocity_pressure ] ) is given by : @xmath195 . using @xmath196 , @xmath197 and @xmath198 , we find @xmath199 ( see fig . [ fig : channel_velocity_pressure ] left ) from our numerical data , giving @xmath200 . hence , the predicted pressure difference is @xmath201 which is quite close to the obtained pressure difference from the simulation ( @xmath202 , see fig . [ fig : channel_velocity_pressure ] right ) . [ fig : channel_velocity_pressure_vs_r ] shows the analytical solution as a function of @xmath203 in comparison with the numerical data and agreement is found . in this test , the fluid density is @xmath175 and dynamic viscosity is @xmath204 . the valve density is set to @xmath205 and the rubber density to @xmath185 . we use zieb on the input section ( see fig . [ fig : system_cofig ] ) with a virtual piston velocity of @xmath206 , and we apply a constant pressure of @xmath207 at the output section ( see fig . [ fig : system_cofig ] ) .

a numerical method for particle - laden fluids interacting with a deformable solid domain and mobile rigid parts is proposed and implemented in a full engineering system . the main issue of this work , since separately each of these methods is a mature tool , is to develop coupling and model - reduction approaches in order to efficiently simulate coupled problems of this nature , as occur in various geological and engineering applications . the lattice boltzmann method incorporates a large - eddy simulation technique using the smagorinsky turbulence model . a neo - hookean hyperelastic model is used for the deformable solid . we provide a detailed description of how to couple the three solvers within a unified algorithm . the technique we propose for rubber modeling / coupling exploits a simplification that prevents having to solve a finite - element problem each time step . we also develop a technique to reduce the domain size of the full system by replacing certain zones with quasi - analytic solutions , which act as effective boundary conditions for the lattice boltzmann method . the major ingredients of the routine are are separately validated . to demonstrate the coupled method in full , we simulate slurry flows in two kinds of piston - valve geometries . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

a numerical method for particle - laden fluids interacting with a deformable solid domain and mobile rigid parts is proposed and implemented in a full engineering system . the fluid domain is modeled with a lattice boltzmann representation , the particles and rigid parts are modeled with a discrete element representation , and the deformable solid domain is modeled using a lagrangian mesh . the main issue of this work , since separately each of these methods is a mature tool , is to develop coupling and model - reduction approaches in order to efficiently simulate coupled problems of this nature , as occur in various geological and engineering applications . the lattice boltzmann method incorporates a large - eddy simulation technique using the smagorinsky turbulence model . the discrete element method incorporates spherical and polyhedral particles for stiff contact interactions . a neo - hookean hyperelastic model is used for the deformable solid . we provide a detailed description of how to couple the three solvers within a unified algorithm . the technique we propose for rubber modeling / coupling exploits a simplification that prevents having to solve a finite - element problem each time step . we also develop a technique to reduce the domain size of the full system by replacing certain zones with quasi - analytic solutions , which act as effective boundary conditions for the lattice boltzmann method . the major ingredients of the routine are are separately validated . to demonstrate the coupled method in full , we simulate slurry flows in two kinds of piston - valve geometries . the dynamics of the valve and slurry are studied and reported over a large range of input parameters . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

0904.1812

i

space - time ( st ) coding is a bandwidth - efficient transmission technique that can improve the reliability of data transmission in mimo wireless systems @xcite . orthogonal space - time block coding ( ostbc ) is one of the most attractive st coding approaches because the special structure of orthogonality guarantees a full diversity and a simple ( linear ) maximum - likelihood ( ml ) decoding . the first ostbc design was proposed by alamouti in @xcite for two transmit antennas and was then extended by tarokh _ et . _ in @xcite for any number of transmit antennas . a class of ostbc from complex design with the code rate of @xmath2 was also given by tarokh _ et _ in @xcite . later , systematic constructions of complex ostbc of rates @xmath3 for @xmath4 or @xmath5 transmit antennas for any positive integer @xmath6 were proposed in @xcite . however , the ostbc has a low code rate not more than @xmath7 for more than two transmit antennas @xcite . to enhance the transmission rate of the stbc , various stbc design approaches were proposed such as quasi - ostbc @xcite and algebraic number theory based stbc @xcite . the quasi - ostbc increases the code rate by relaxing the orthogonality condition on the code matrix , which was originally proposed in @xcite , @xcite , and @xcite , independently . due to the group orthogonality , the ml decoding is performed pair - wise or group - wise with an increased complexity compared to the single - symbol decoding . in @xcite , quasi - ostbc was studied in the sense of minimum decoding complexity , i.e. , a real pair - wise symbols decoding . in @xcite , the pair - wise decoding was generalized to a general group - wise decoding . the decoding for these codes is the ml decoding and their rates are basically limited by that of ostbc . the algebraic number theory based stbc are designed mainly based on the ml decoding that may have high complexity and even though some near - ml decoder , such as sphere decoder @xcite can be used , the expected decoding complexity is still dominated by polynomial terms of a number of symbols which are jointly detected @xcite . to reduce the large decoding complexity of the high rate stbc aforementioned , several fast - decodable stbc were recently proposed @xcite@xcite . the stbc proposed in @xcite achieves a high rate and a reduced decoding complexity at the cost of loss of full diversity . the fast - decodable stbc in @xcite can obtain full rate , full diversity and the reduced ml decoding complexity , but the code design is limited to @xmath8 and @xmath9 mimo transmissions only . another new perspective of reducing the decoding complexity was recently considered in @xcite and @xcite to resort to conventional linear receivers such as zero - forcing ( zf ) receiver or minimum mean square error ( mmse ) receiver instead of the ml receiver to collect the full diversity . the outage and diversity of linear receivers in flat - fading mimo channels were studied in @xcite , but no explicit code design was given to achieve the full diversity when the linear receivers are used . based on the new stbc design criterion for mimo systems with linear receivers , toeplitz stbc @xcite and overlapped - alamouti codes @xcite were proposed and shown to achieve the full diversity with the linear receivers . recently , some other new designs of stbc with linear receivers were proposed @xcite . however , the code rate of stbc achieving full diversity with linear receivers is upper bounded by one . later , guo and xia proposed a partial interference cancellation ( pic ) group decoding scheme @xcite which can be viewed as an intermediate decoding approach between the ml receiver and the zf receiver by trading a simple single - symbol decoding complexity for a high code rate larger than one symbol per channel use . moreover , in @xcite an stbc design criterion was given to achieve full diversity when the pic group decoding is applied at the receiver . the proposed pic group decoding in @xcite was also connected with the successive interference cancellation ( sic ) strategy to aid the decoding process , referred to as pic - sic group decoding . a few code design examples were presented in @xcite , but a general design of stbc achieving full diversity with the pic group decoding remains an open problem . in this paper , we propose two designs of stbc which can achieve full diversity with the pic group decoding for any number of transmit antennas . the first proposed stbc have a structure of multiple diagonal layers and for each diagonal layer there are exactly @xmath10 coded symbols embedded , being equal to the number of transmit antennas , which are obtained from a cyclotomic lattice design . indeed , each diagonal layer of the coded symbols can be viewed as the conventional rate - one diagonal stbc @xcite . the code rate of the proposed stbc can be from one to @xmath10 symbols per channel use by adjusting the codeword length , i.e. , embedding different number of layers in the code matrix . with the pic group decoding the code rate of the first proposed full - diversity stbc can be only up to @xmath0 symbols per channel use , i.e. , for two layers . for more than two layers embedded in the codeword , the code rate is increased at the cost of losing full diversity with the pic group decoding . however , with the pic - sic group decoding , the proposed stbc with arbitrary number of layers can obtain full diversity and the code rate can be up to @xmath10 . the second proposed stbc is designed with three layers of information symbols embedded in the codeword and the pic group decoding can be performed in three separate groups accordingly . without loss of decoding complexity compared to the first proposed stbc , the second proposed stbc can achieve full diversity and a code rate larger than @xmath0 . note that the code rate for the first proposed full - diversity stbc with pic group decoding can not be above @xmath0 . in the pic group decoding of the proposed stbc , every @xmath10 neighboring columns of the equivalent channel matrix are clustered into one group . this paper is organized as follows . a system model of st transmission over mimo channels with the pic group decoding is introduced in section [ sec : system ] . in section [ sec : new ] , a design of high rate stbc with the pic group decoding is proposed , which contains multiple diagonal layers of coded symbols . for a particular code design with two diagonal layers , the full diversity with the pic group decoding is proved . for the code with pic - sic group decoding , the full diversity is shown for any number of diagonal layers . several full - diversity code design examples are given in section [ sec : example ] . in section [ sec : xu ] , another design of high rate stbc with the pic group decoding is proposed , which can achieve full diversity with three layers . simulation results are presented in section [ sec : sim ] . finally , in section [ sec : conclusion ] , we draw our conclusions . _ notations _ : column vectors ( matrices ) are denoted by boldface lower ( upper ) case letters . superscripts @xmath11 and @xmath12 stand for transpose and conjugate transpose , respectively . @xmath13 denotes the field of complex numbers . @xmath14 denotes the @xmath15 identity matrix , and @xmath16 denotes the @xmath17 matrix whose elements are all @xmath18 . @xmath19 is the vectorization of matrix @xmath20 by stacking the columns of @xmath20 on top each other .

a partial interference cancellation ( pic ) group decoding based space - time block code ( stbc ) design criterion was recently proposed by guo and xia , where the decoding complexity and the code rate trade - off is dealt when the full diversity is achieved . in this paper , two designs of stbc are proposed for any number of transmit antennas that can obtain full diversity when a pic group decoding ( with a particular grouping scheme ) is applied at receiver . with the pic group decoding and an appropriate grouping scheme for the decoding , the proposed stbc are shown to obtain the same diversity gain as the ml decoding , but have a low decoding complexity . some code design examples are given and simulation results show that the newly proposed stbc can well address the rate - performance - complexity tradeoff of the mimo systems . diversity techniques , space - time block codes , linear receiver , partial interference cancellation .

a partial interference cancellation ( pic ) group decoding based space - time block code ( stbc ) design criterion was recently proposed by guo and xia , where the decoding complexity and the code rate trade - off is dealt when the full diversity is achieved . in this paper , two designs of stbc are proposed for any number of transmit antennas that can obtain full diversity when a pic group decoding ( with a particular grouping scheme ) is applied at receiver . with the pic group decoding and an appropriate grouping scheme for the decoding , the proposed stbc are shown to obtain the same diversity gain as the ml decoding , but have a low decoding complexity . the first proposed stbc is designed with multiple diagonal layers and it can obtain the full diversity for two - layer design with the pic group decoding and the rate is up to symbols per channel use . but with pic - sic group decoding , the first proposed stbc can obtain full diversity for any number of layers and the rate can be full . the second proposed stbc can obtain full diversity and a rate up to with the pic group decoding . some code design examples are given and simulation results show that the newly proposed stbc can well address the rate - performance - complexity tradeoff of the mimo systems . diversity techniques , space - time block codes , linear receiver , partial interference cancellation .

1103.1055

i

in this paper we continue the investigation of the infrared structure of quantum electrodynamics based on an algebraic model proposed earlier by one of us ( see ref . @xcite and papers cited therein ; see also @xcite ) . this model is supposed to describe asymptotic fields in the quantum maxwell - dirac system , including the gauss law constraint ( as opposed to the crossed product of free fields ) . in a recent paper @xcite this model was investigated in respect of the localization properties of fields . it was shown that one needs an extension of the localization regions : infrared / charge structure is encoded in unbounded regions . it was argued that from the point of view of scattering theory , the natural choice for extended localization regions consists of ` fattened lightcones ' , unions of intersecting : a future- and a past - lightcone . the test - functions of electromagnetic fields have well - defined asymptotes encoding the information on the long distance structure . in the present article we show that the algebra can be localized in any ` time - slice ' which is fattening under constant inclination towards infinity . in addition , the localization of electromagnetic field may be restricted to ` fattened symmetrical spacelike cones ' : the unions of a spacelike cone and its reflection with respect to a point in its inside . similar restriction seems to be ruled out , even asymptotically , for charged fields . this seems to contradict general wisdom on the expected behavior of fields in full electrodynamics , see e.g.the assumptions on which buchholz @xcite bases his selection criterion of representations in quantum electrodynamics . whether this points to some incompleteness of the model is an open question ; see the discussion at the beginning of section [ locdir ] below and in conclusions . on the other hand , we show that in the present model , in agreement with the general expectation , one can superpose two appropriately `` dressed '' dirac fields carrying opposite charges to obtain a local observable . this article should be regarded as a continuation of refs . @xcite and @xcite , and we refer the reader to these references for more detail and a wider background . however , we briefly summarize notation and the formulation of the model in the next two sections . we obtain spacelike localization of fields in sections 4 and 5 , and discuss the results in concluding section 6 .

electromagnetic fields may be localized in ` symmetrical spacelike cones ' , but there are strong indications this is not possible in the present model for charged fields , which have tails extending in all space directions . nevertheless , products of appropriately ` dressed ' fermion fields ( with compensating charges ) yield bi - localized observables .

a previously proposed algebra of asymptotic fields in quantum electrodynamics is formulated as a net of algebras localized in regions which in general have unbounded spacelike extension . electromagnetic fields may be localized in ` symmetrical spacelike cones ' , but there are strong indications this is not possible in the present model for charged fields , which have tails extending in all space directions . nevertheless , products of appropriately ` dressed ' fermion fields ( with compensating charges ) yield bi - localized observables .

quant-ph9806078

i

over the past decade there has been steady progress in the development of quantum algorithms . most attention has focused on the quantum algorithms for finding the factors of a composite integer @xcite and for finding an item in an unsorted database @xcite . these successes have inspired several researchers to look for quantum algorithms that can solve other challenging problems , such as decision problems @xcite or combinatorial search problems @xcite , more efficiently than their classical counterparts . the class of np - complete problems includes the most common computational problems encountered in practice @xcite . in particular , it includes scheduling , planning , combinatorial optimization , theorem proving , propositional satisfiability and graph coloring . in addition to their ubiquity , np - complete problems share a fortuitous kinship : any np - complete problem can be mapped into any other np - complete problem using only polynomial resources @xcite . thus , any quantum algorithm that speeds up the solution of one np - complete problem immediately leads to equally fast quantum algorithms for all np - complete problems ( up to the polynomial cost of translation ) . unfortunately , np - complete problems appear to be even harder than the integer factorization problem . whereas , classically , the best known algorithm for the latter problem scales only sub - exponentially @xcite , np - complete problems are widely believed to be exponential @xcite . thus , the demonstration that shor s quantum algorithm @xcite can factor an integer in a time that is bounded by a polynomial in the `` size '' of the integer ( i.e. , the number of bits needed to represent that integer ) , while remarkable , does not lead to a polynomial - time quantum algorithm for np - complete problems , the existence of which being considered as highly improbable @xcite . moreover , it has proven to be very difficult to adapt shor s algorithm to other computational applications . by contrast , the unstructured quantum search algorithm @xcite can be adapted quite readily to the service of solving np - complete problems . as a candidate solution to an np - complete problem can be tested for correctness in polynomial time , one simply has to create a `` database '' consisting of all possible candidate solutions and apply the unstructured quantum search algorithm . unfortunately , the speedup afforded by this algorithm is only @xmath6 where @xmath7 is the number of candidate solutions to be tested . for a typical np - complete problem in which one has to find an assignment of one of @xmath8 values to each of @xmath9 variables , the number of candidate solutions , @xmath10 , grows exponentially with @xmath9 . a classical algorithm would therefore take a time @xmath11 to find the solution whereas the unstructured quantum search algorithm would take @xmath12 . although this is an impressive speedup , one would still like to do better . while there is now good evidence that for unstructured problems , the quantum search algorithm is optimal @xcite , these results have raised the question of whether faster quantum search algorithms might be found for problems that possess _ structure _ it so happens that np - complete problems have such structure in the sense that one can often build up complete solutions ( i.e. , value assignments for all the variables ) by extending _ partial _ solutions ( i.e. , value assignments for a subset of the variables ) . thus , rather than performing an unstructured quantum search amongst _ all _ the candidate solutions , in an np - complete problem , we can perform a quantum search amongst the _ partial _ solutions in order to narrow the subsequent quantum search amongst their descendants . this is the approach presented in this paper and which allows us to find a solution to an np - complete problem in a time that grows , on average , as @xmath13 for the hardest problems , where @xmath4 is a constant depending on the problem instance considered . our improved quantum search algorithm works by _ nesting _ one quantum search within another . specifically , by performing a quantum search at a carefully selected level in the tree of partial solutions , we can narrow the effective quantum search amongst the candidate solutions so that the net computational cost is minimized . the resulting algorithm is the quantum counterpart of a _ classical _ nested search algorithm which scales as @xmath14 , giving a square root speedup overall . the nested search procedure mentioned here corresponds to a _ single _ level of ( classical or quantum ) nesting , but it can be extended easily to several nesting levels . thus , our result suggests a systematic technique for translating a nested classical search algorithm into a quantum one , giving rise a square - root speedup , which can be useful to accelerate _ efficient _ classical algorithms ( rather than a simple exhaustive search , of no practical use ) . we believe this technique is applicable in all structured quantum searches . the outline of the paper is as follows . section [ sect_nested_class ] introduces a simple classical tree search algorithm that exploits problem structure to localize the search for solutions amongst the candidates . this is not intended to be a sophisticated classical tree search algorithm , but rather is aimed at providing a baseline against which our quantum algorithm can be compared . in section [ sect_unstruc_qu ] , we outline the standard unstructured quantum search algorithm @xcite . we focus especially on the algorithm based on an arbitrary unitary search operator @xcite , as this is a key for implementing quantum nesting . finally , section [ sect_nested_quant ] describes the quantum tree search algorithm based on nesting , which is a direct quantum analog of the classical search algorithm appearing in section [ sect_nested_class ] . the quantum search algorithm with several levels of nesting is also briefly discussed . we conclude by showing that the expected time to find a solution grows as @xmath13 , that is , as the square root of the classical time for problem instances in the hard region . the constant @xmath5 , depending on the problem considered , is shown to decrease with an increasing nesting depth ( i.e. , an increasing number of nesting levels ) .

it is shown here that an improved quantum search algorithm can be devised that exploits the structure of a tree search problem by _ nesting _ this standard search algorithm . the number of iterations required to find the solution of an average instance of a constraint satisfaction problem scales as , with a constant depending on the nesting depth and the problem considered .

a quantum algorithm is known that solves an unstructured search problem in a number of iterations of order , where is the dimension of the search space , whereas any classical algorithm necessarily scales as . it is shown here that an improved quantum search algorithm can be devised that exploits the structure of a tree search problem by _ nesting _ this standard search algorithm . the number of iterations required to find the solution of an average instance of a constraint satisfaction problem scales as , with a constant depending on the nesting depth and the problem considered . when applying a single nesting level to a problem with constraints of size 2 such as the graph coloring problem , this constant is estimated to be around 0.62 for average instances of maximum difficulty . this corresponds to a square - root speedup over a classical nested search algorithm , of which our presented algorithm is the quantum counterpart .

0911.4148

i

over the last quarter of a century , _ expander _ graphs have played a vital role in a remarkable variety of areas , ranging from combinatorics to discrete geometry to theoretical computer science , while exhibiting deep connections to algebra and number theory . notable applications of expanders , to name just a few , include the design of efficient communication networks , explicit error - correcting codes with efficient encoding and decoding schemes , derandomization of randomized algorithms , compressed sensing and the study of metric embeddings . see the expository article of sarnak @xcite on these intriguing objects , as well as the comprehensive survey of hoory , linial and wigderson @xcite demonstrating their many applications . informally , an expander is a graph where every small subset of the vertices has a relatively large edge boundary ( see section [ sec : intro - exp ] for a formal definition ) . most applications utilize @xmath12-regular sparse expanders ( @xmath20 fixed ) , where it is well - known that expansion is related to the ratio between @xmath12 and @xmath17 , the second largest eigenvalue in absolute value of the adjacency matrix . the smaller @xmath17 is , the better the graph expansion becomes . as a consequence of the alon - boppana bound @xcite ( see also @xcite ) @xmath21 where the @xmath22-term tends to @xmath23 as the graph size tends to @xmath24 . graphs for which @xmath25 are in that respect optimal expanders and are called _ ramanujan _ graphs . a proof that @xmath12-regular expanders exist for any @xmath26 was given by pinsker @xcite in the early 70 s via a simple probabilistic argument . however , constructing good expanders _ explicitly _ is far more challenging and particularly important in applications ( see @xcite and the references therein ) , a task that was first achieved by margulis @xcite . thereafter ramanujan graphs were constructed explicitly in the seminal works of lubotzky - phillips - sarnak @xcite and margulis @xcite , relying on deep number theoretic facts . till this date ramanujan graphs remain mysterious : not only are there very few constructions for such graphs , but for instance it is not even known whether they exist for any @xmath26 . a striking result of friedman @xcite shows that almost every @xmath12-regular graph on @xmath0 vertices is _ nearly _ ramanujan it has @xmath27 ( the @xmath22-term tends to @xmath23 as @xmath28 ) . what proportion of these graphs satisfy @xmath25 remains an intriguing open problem . the useful connection between expanders and the topological notion of _ covering maps _ was extensively studied by many authors over the last decade . various properties of random covers of a given graph were thoroughly examined ( see e.g. @xcite ) , motivated in part by the problem of generating good ( large ) expanders from a given one . given two simple graphs @xmath1 and @xmath2 , a _ covering map _ @xmath29 is a homomorphism that for every @xmath30 induces a bijection between the edges incident to @xmath31 and those incident to @xmath32 . in the presence of such a covering map we say that @xmath2 is a _ lift _ ( or a cover ) of @xmath1 , or alternatively that @xmath1 is a _ quotient _ of @xmath2 . the _ fiber _ of @xmath33 is the set @xmath34 , and if @xmath1 is connected then all fibers are of the same cardinality , the _ covering number_. one well - known connection between covers and expansion is the fact that the universal cover of any @xmath12-regular graph is the infinite @xmath12-regular tree @xmath35 , whose spectral radius is @xmath14 , the eigenvalue threshold in ramanujan graphs . in fact , greenberg and lubotzky @xcite ( cf . @xcite*chapter 4 ) extended the alon - boppana bound to any family of general graphs in terms of the spectral radius of its universal cover ( also see @xcite*theorem 4.1 ) . it is easy to see that any lift of a @xmath12-regular base graph @xmath1 is itself @xmath12-regular and inherits all the original eigenvalues of @xmath1 . one hopes that the lift would also inherit the expansion properties of its base graph , and in particular that almost every cover of a ( small ) ramanujan graph will also be ramanujan . since our focus here is on lifts of ramanujan graphs ( regular by definition ) we restrict our attention to base graphs that are @xmath12-regular for @xmath26 . a random uniform @xmath0-lift of a base graph @xmath1 ( a uniform cover of @xmath1 with covering number @xmath0 ) has the following convenient description : it is the graph @xmath2 on the vertices @xmath3\times v(g)$ ] , where for each edge @xmath4 in @xmath1 there is an independent uniform bijection @xmath5 , and @xmath2 has all edges of the form @xmath6 . the random lift of a complicated base - graph is thus a hybrid between the complex geometry of the quotient and the randomness due to the bijections . in an important development in the study of the spectrum of random lifts friedman @xcite showed in 2003 that with high probability ( w.h.p . ) every `` new '' eigenvalue of an @xmath0-lift ( one that is not inherited from the base graph ) is at most @xmath36 , where @xmath7 is the largest eigenvalue of the base - graph and @xmath8 is the spectral - radius of its universal cover . when the base - graph @xmath1 is @xmath12-regular , @xmath14 and friedman s result implies that in its random @xmath0-lift @xmath2 the largest absolute value of all nontrivial eigenvalues is w.h.p . @xmath37 where @xmath38 denotes @xmath39 . conversely , @xmath40 and by alon - boppana it is also at least @xmath41 . this lower bound was conjectured by friedman @xcite to be tight ( for general graphs he conjectured that all new eigenvalues are at most @xmath10 as in the greenberg - lubotzky bound ) . in a recent paper @xcite , linial and puder were able to significantly improve friedman s bound and show that w.h.p . all the new eigenvalues of @xmath2 are at most @xmath42 . consequently , an @xmath0-lift @xmath2 of a @xmath12-regular @xmath1 w.h.p . satisfies @xmath43 when @xmath1 is a @xmath12-regular expander with nontrivial eigenvalues of @xmath44 as is the case for ramanujan graphs , this translates to @xmath45 . our main result in this work is the new near optimal upper bound of @xmath46 when @xmath1 is @xmath12-regular with all nontrivial eigenvalues at most @xmath17 in absolute value . for @xmath47 it substantially improves the known bounds , , and when @xmath48 as in ramanujan graphs it is tight up to a logarithmic factor , giving @xmath49 . [ thm-1 ] let @xmath1 be a @xmath12-regular graph with all nontrivial eigenvalues at most @xmath17 in absolute value and let @xmath14 be the spectral radius of its universal cover . let @xmath2 be a random @xmath0-lift of @xmath1 . for some explicit absolute constant @xmath50 , every nontrivial eigenvalue of @xmath2 is at most @xmath51 in absolute value except with probability @xmath52 . [ cor-2 ] let @xmath1 be a @xmath12-regular ramanujan graph vertices and let @xmath2 be a random @xmath0-lift of @xmath1 . with probability @xmath53 every nontrivial eigenvalue of @xmath2 is at most @xmath54 in absolute value , where @xmath50 is an explicit absolute constant . note that the above corollary implies that typical random @xmath0-lifts of ramanujan graphs are nearly ramanujan . no attempt was made to optimize the explicit constant in theorem [ thm-1 ] . finally , the statement of theorem [ thm-1 ] holds even when the size of the base - graph @xmath55 is allowed to grow with @xmath0 provided that @xmath0 is large enough in comparison ( e.g. , @xmath56 ) . the previous bounds on the spectra of random @xmath0-lifts of a fixed graph @xmath1 due to friedman @xcite and linial and puder @xcite were both obtained via wigner s trace method . the fact that the universal cover @xmath57 of a connected graph @xmath1 is the infinite tree of non - backtracking walks from an arbitrarily chosen vertex makes the trace method particularly useful for relating the new eigenvalues of the lift with @xmath8 , the spectral - radius of @xmath57 . even when the geometry of a graph is very well understood , bounding its nontrivial eigenvalues can be extremely challenging . for instance , a line of papers ( cf . @xcite ) established various bounds for the second eigenvalue of certain random regular graphs , culminating in the optimal bound @xmath58 for a uniformly chosen @xmath12-regular graph on @xmath0 vertices , proved by friedman @xcite using highly sophisticated arguments . it turns out that this model is essentially the special case of an @xmath0-lift of a graph comprising a single vertex with self - loops : it is easy to see that for @xmath12 even , the random @xmath12-regular graph obtained by @xmath59 independent uniform permutations in @xmath60 is equivalent to an @xmath0-lift of the base - graph @xmath1 that has a single vertex with @xmath59 loops ( this model is in fact contiguous to the uniform random @xmath12-regular graph for @xmath61 , cf . e.g. @xcite ) . unfortunately , when the base - graph features a complex and rich structure ( e.g. the lps - expanders , whose expansion properties hinge on a deep theorem of selberg ) it becomes significantly harder to control the spectrum of its lifts . indeed , there are many examples of geometric properties that have been pinpointed precisely for the random regular graph yet remain unknown for arbitrary expanders ( see @xcite for a recent such example ) . estimating the number of closed walks in lifts of arbitrary ramanujan graphs thus appears to be a formidable task . in this work , the bounds obtained for the spectra of lifts of arbitrary expanders rely on an approach introduced by kahn and szemerdi @xcite , which is quite different from wigner s trace method . this approach was originally used to control the spectrum of a random regular graph , and several new ideas are required to adapt it to the more complicated geometry of the lifts considered here . another related problem in the study of spectra of lifts , yet of a rather different nature , considers the @xmath62-lift of a base - graph ( rather than @xmath0-lifts of a small fixed graph ) . bilu and linial @xcite showed that for any @xmath12-regular graph @xmath1 there _ exists _ a @xmath62-lift with all new eigenvalues at most @xmath63 . this was shown by means of the lovsz local lemma , combined with the crucial observation of @xcite whereby the new eigenvalues correspond precisely to the eigenvalues of a _ signing _ of the adjacency matrix of @xmath1 ( the matrix obtained by replacing a subset of its @xmath64 entries by @xmath65 ) . in the absence of such a characterization when the covering number @xmath0 is large , different tools are needed for the problem studied here , where we seek a bound that holds for almost every @xmath0-lift with @xmath0 sufficiently large . as stated above , while a random @xmath12-regular graph @xmath1 has second eigenvalue @xmath66 w.h.p . ( the @xmath22-term tending to @xmath23 as @xmath67 ) , the probability that @xmath1 is ramanujan is unknown . see @xcite for some experimental results suggesting that this probability is bounded away from @xmath23 and @xmath64 . as this is essentially the simplest special case of a random lift ( the quotient being a single vertex with self - loops ) , it is natural to conjecture the following : [ conj-3 ] for any ramanujan graph @xmath1 there exist some @xmath68 such that its random @xmath0-lift @xmath2 satisfies @xmath69 , where the @xmath22-term tends to @xmath23 as @xmath28 . note that the limiting constant in the above conjecture depends on the base - graph @xmath1 , as it is plausible that its structure may affect the probability of being ramanujan . for instance , a random cover of a complete graph on @xmath70 vertices might behave quite differently compared to lifts of a sparse @xmath12-regular ramanujan graph . however , as we next elaborate , experimental results lead us to suspect that up to normalization this is not the case . figure [ fig : lifts : a ] shows the cumulative distribution function ( c.d.f . ) of @xmath71 where @xmath2 is the @xmath72-lift of 3 different @xmath73-regular ramanujan base - graphs : @xmath74 ( complete graph on @xmath75 vertices ) , the @xmath76-vertex petersen graph and the @xmath77-vertex dodecahedral graph . each curve was evaluated from @xmath78 random lifts . in these simulations , the probability of a random lift being ramanujan for each of these three base - graphs was bounded between @xmath79 and @xmath80 . somewhat surprisingly , aligning the number of vertices of the graph cover @xmath2 to be the same ( via @xmath72-lifts of the dodecahedral graph , @xmath81-lifts of the petersen graph and @xmath82-lifts of @xmath74 , giving @xmath83-vertex covers for each graph ) resulted in the curves of the individual c.d.f.s coinciding fairly accurately . this is demonstrated in figure [ fig : lifts : b ] , in light of which we speculate that the following stronger version of the statement of conjecture [ conj-3 ] holds . first , it seems plausible that for any integer @xmath20 the limiting distribution of the second eigenvalue of the random cover is independent of the base - graph . namely , there exists a distribution @xmath84 on @xmath85 $ ] such that for any @xmath12-regular ramanujan graph @xmath1 on @xmath55 vertices , the distribution of @xmath71 for its random @xmath0-lift @xmath2 converges to @xmath84 as @xmath28 . second , the strong fit between the curves after aligning the total graph sizes suggests that even the rate of convergence to @xmath84 depends on @xmath86 rather than on the geometry of the base - graph or even its relative size ( in case @xmath55 is allowed to depend to @xmath0 ) . of - course , one clearly needs some level of `` burn - in '' for the covering number @xmath0 compared to @xmath55 since the cover @xmath2 starts as ramanujan at @xmath87 . for example , it may be that for any @xmath88 the total - variation distance between the distribution of @xmath71 and @xmath84 decays as a function of @xmath89 alone , namely that @xmath90 where @xmath91 depends only on @xmath92 and tends to @xmath23 as @xmath93 .

a random-lift of a base graph is its cover graph on the vertices\times v(g)$ ] , where for each edge in there is an independent uniform bijection , and has all edges of the form . a main motivation for studying lifts is understanding ramanujan graphs , and namely whether typical covers of such a graph are also ramanujan . let be a graph with largest eigenvalue and let be the spectral radius of its universal cover . friedman ( 2003 ) proved that every `` new '' eigenvalue of a random lift of is with high probability , and conjectured a bound of , which would be tight by results of lubotzky and greenberg ( 1995 ) . linial and puder ( 2008 ) improved friedman s bound to . for-regular graphs , where and , this translates to a bound of , compared to the conjectured . here we analyze the spectrum of a random-lift of a-regular graph whose nontrivial eigenvalues are all at most in absolute value . , it implies that a typical-lift of a ramanujan graph is nearly ramanujan .

a random-lift of a base graph is its cover graph on the vertices\times v(g)$ ] , where for each edge in there is an independent uniform bijection , and has all edges of the form . a main motivation for studying lifts is understanding ramanujan graphs , and namely whether typical covers of such a graph are also ramanujan . let be a graph with largest eigenvalue and let be the spectral radius of its universal cover . friedman ( 2003 ) proved that every `` new '' eigenvalue of a random lift of is with high probability , and conjectured a bound of , which would be tight by results of lubotzky and greenberg ( 1995 ) . linial and puder ( 2008 ) improved friedman s bound to . for-regular graphs , where and , this translates to a bound of , compared to the conjectured . here we analyze the spectrum of a random-lift of a-regular graph whose nontrivial eigenvalues are all at most in absolute value . we show that with high probability the absolute value of every nontrivial eigenvalue of the lift is . this result is tight up to a logarithmic factor , and for it substantially improves the above upper bounds of friedman and of linial and puder . in particular , it implies that a typical-lift of a ramanujan graph is nearly ramanujan . [ multiblock footnote omitted ]

1412.4108

i

clear evidence indicates that most galactic globular clusters host multiple , distinct stellar populations ( e.g. , see reviews by gratton et al . 2004 ; 2012 ) . for mono metallic clusters exhibiting negligible spreads in [ fe / h]@xmath7log(n@xmath8/n@xmath9)@xmath10log(n@xmath8/n@xmath9)@xmath11 and log @xmath12(a)@xmath7log(n@xmath8/n@xmath13)+12.0 for elements a and b. ] , the various stellar populations are identified spectroscopically by their light element chemistry . specifically , stars within a single globular cluster are often categorized by their [ o / fe ] and/or [ na / fe ] ratios as belonging to the primordial " , intermediate " , or extreme " populations ( e.g. , carretta et al . 2009 ) . in this categorization , primordial stars ( first generation ) are similar in composition to metal poor halo field stars ( o rich ; na poor ) , and intermediate and extreme stars ( second generation ) have lower [ o / fe ] ratios and higher [ na / fe ] ratios . the extreme stars are further distinguished from the intermediate population as having the lowest oxygen abundances ( [ o / fe]@xmath50.4 ) and highest sodium abundances ( [ na / fe]@[email protected] ) . while the intermediate population tends to dominate by number over the primordial population , the extreme stars are found only in a handful of globular clusters ( carretta et al . 2009 ; their table 5 ) . the large and often ( anti)correlated abundance variations of elements ranging from carbon through aluminum is evidence that the material in globular cluster stars atmospheres has been subjected to high temperature proton capture nucleosynthesis ( e.g. , denisenkov & denisenkova 1990 ; langer et al . 1993 ; prantzos et al . 2007 ) . while changes in [ c / fe ] , [ n / fe ] , and @xmath16c/@xmath17c ratios as a function of evolutionary state on the subgiant branch ( sgb ) and red giant branch ( rgb ) are clearly linked to _ in situ _ mixing processes ( e.g. , denissenkov & vandenberg 2003 ) , the temperatures reached near the bottom of the convective envelope in more evolved low mass rgb stars are too low to significantly alter the abundances of heavier elements ( but see also dantona & ventura 2007 for a possible exception ) . observations of similar star to star abundance variations among scarcely evolved globular cluster main sequence and sgb stars ( e.g. , briley et al . 1996 ; gratton et al . 2001 ; ramrez & cohen 2002 ; ramrez & cohen 2003 ; carretta et al . 2004 ; briley et al . 2004 ; cohen & melndez 2005 ; bragaglia et al . 2010 ; dorazi et al . 2010 ; dobrovolskas et al . 2014 ) indicate that the composition differences between the various globular cluster populations were already imprinted on the gas from which the stars formed . although there is still no consensus regarding the nucleosynthesis source(s ) driving the composition differences , possible options include intermediate mass ( @xmath1858 m@xmath19 ) asymptotic giant branch ( agb ) stars ( e.g. , karakas et al . 2006 ; ventura & dantona 2009 ) , rapidly rotating massive ( @xmath1420 m@xmath19 ) main sequence stars ( e.g. , decressin et al . 2007a ) , massive binary stars ( de mink et al . 2009 ) , and super massive ( @xmath1810@xmath20 m@xmath21 ) stars ( denissenkov & hartwick 2014 ) . the physical process by which globular clusters form and evolve remains an open question as well ( e.g. , decressin et al . 2007b ; dercole et al . 2008 ; renzini 2008 ; carretta et al . 2010a ; bekki 2011 ; conroy & spergel 2011 ; valcarce & catelan 2011 ; bastian et al . 2013 ) . regardless of the pollution source(s ) in globular clusters , the nuclear processes creating o depleted and na enhanced gas may also be concurrent with he enhancements . with the exception of a few cases , such as ngc 1851 , where the c@xmath15n@xmath15o sum may be variable ( e.g. , ventura et al . 2009 ; yong et al . 2009 ; yong et al . 2014 ) , he abundance differences ranging from @[email protected] have been invoked to explain many of the multiple photometric sequences observed in recent cluster color magnitude diagrams ( e.g. , piotto et al . 2007 ; milone et al . 2012 ) . combined evidence from photometry and spectroscopy ties the traditional light element abundance patterns to varying levels of he enhancement ( e.g. , bragaglia et al . 2010 ; dupree et al . 2011 ; pasquini et al . 2011 ; marino et al . 2014 ; mucciarelli et al . in addition to causing subtle effects in absorption line formation ( bhm vitense 1979 ) , he enhancements can significantly alter a star s position on the color magnitude diagram and shorten its evolutionary timescale , relative to a he normal star . there are some indications that the extent ( or existence ) of the agb phase may be sensitive to a star s initial he abundance such that he enhanced stars may evolve off the horizontal branch to become agb manqu stars ( e.g. , greggio & renzini 1990 ; castellani et al . 2006 ; gratton et al . 2010 ; charbonnel et al . in particular , several authors have noted a peculiar feature in many clusters that cn strong and o poor / na rich stars appear with a lower frequency on the agb compared to the rgb ( mallia 1978 ; norris et al . 1981 ; suntzeff 1981 ; smith & norris 1993 ; pilachowski et al . 1996 ; ivans et al . 1999 ; sneden et al . 2000 ; campbell et al . 2006 ; campbell et al . 2010 ; smolinski et al . 2011 ; johnson & pilachowski 2012 ; campbell et al . 2012 ; campbell et al . 2013 ) . however , it is not yet clear that the lack of cn strong , o poor , and na rich stars on the agb is a ubiquitous property of all globular clusters . gratton et al . ( 2010 ) note that the agb / rgb ratio in globular clusters is correlated with the minimum mass along the horizontal branch . in other words , clusters with redder horizontal branches , and possibly also lower levels of he enhancement , tend to retain a larger fraction of stars between the rgb and agb phases . therefore , in order to investigate this phenomenon further we have obtained spectra of rgb and agb stars in the red horizontal branch , and relatively metal rich ( [ fe / h]@xmath230.7 ) , globular cluster 47 tucanae ( 47 tuc ) , and compare the [ na / fe ] distributions between stars in the two evolutionary states . 47 tuc exhibits a relatively high agb / rgb ratio ( @xmath180.10 ; gratton et al . 2010 ) , and is suspected of having a predominantly cn strong / na rich agb population ( mallia 1978 ) . if confirmed , a dominant na rich agb population in 47 tuc would strongly contrast with the completely na poor agb population in the more metal poor blue horizontal branch cluster ngc 6752 ( campbell et al .

therefore , we present radial velocities , [ fe / h ] , and [ na / fe ] abundances for 35 agb stars in the galactic globular cluster 47 tucanae ( 47 tuc ; ngc 104 ) , and compare the agb [ na / fe ] distribution with a similar rgb sample published previously .

a recent analysis comparing the [ na / fe ] distributions of red giant branch ( rgb ) and asymptotic giant branch ( agb ) stars in the galactic globular cluster ngc 6752 found that the ratio of na poor to na rich stars changes from 30:70 on the rgb to 100:0 on the agb . the surprising paucity of na rich stars on the agb in ngc 6752 warrants additional investigations to determine if the failure of a significant fraction of stars to ascend the agb is an attribute common to all globular clusters . therefore , we present radial velocities , [ fe / h ] , and [ na / fe ] abundances for 35 agb stars in the galactic globular cluster 47 tucanae ( 47 tuc ; ngc 104 ) , and compare the agb [ na / fe ] distribution with a similar rgb sample published previously . the abundances and velocities were derived from high resolution spectra obtained with the michigan / magellan fiber system ( m2fs ) and mspec spectrograph on the magellan clay 6.5 m telescope . we find the average heliocentric radial velocity and [ fe / h ] values to berv=18.56 km s (=10.21 km s ) and[fe / h]=0.68 (=0.08 ) , respectively , in agreement with previous literature estimates . the average [ na / fe ] abundance is 0.12 dex lower in the 47 tuc agb sample compared to the rgb sample , and the ratio of na poor to na rich stars is 63:37 on the agb and 45:55 on the rgb . however , in contrast to ngc 6752 , the two 47 tuc populations have nearly identical [ na / fe ] dispersion and interquartile range values . the data presented here suggest that only a small fraction ( ) of na rich stars in 47 tuc may fail to ascend the agb , which is a similar result to that observed in m13 . regardless of the cause for the lower average [ na / fe ] abundance in agb stars , we find that na poor stars and at least some na rich stars in 47 tuc evolve through the early agb phase . the contrasting behavior of na rich stars in 47 tuc and ngc 6752 suggests that the rgb [ na / fe ] abundance alone is insufficient for predicting if a star will ascend the agb .

1412.4108

r

although recent color magnitude diagrams of 47 tuc suggest the cluster has a complex star formation history ( anderson et al . 2009 ; di criscienzo et al . 2010 ; nataf et al . 2011 ; milone et al . 2012 ; monelli et al . 2013 ; li et al . 2014 ) , there is no evidence supporting a substantial spread in [ fe / h ] . instead , literature work suggests the cluster has an average [ fe / h]@xmath230.60 to 0.80 , with a star to star dispersion @xmath50.10 dex ( e.g. , carretta & gratton 1997 ; alves brito et al . 2005 ; wylie et al . 2006 ; koch & mcwilliam 2008 ; carretta et al . 2009 ; gratton et al . 2013 ; cordero et al . 2014 ; dobrovolskas et al . 2014 ; thygesen et al . 2014 ) . in agreement with past work , we find an average [ fe / h]=0.68 and a star to star dispersion of @xmath4=0.08 . this result is similar to , but more metal rich than , the rgb flames giraffe sample of cordero et al . ( 2014 ) that found @xmath0[fe / h]@xmath2=0.75 ( @xmath4=0.10 ) . the 0.07 dex difference in [ fe / h ] between the agb and rgb samples is possibly tied to a systematic offset in the temperature scales between the two data sets . a comparison of the spectroscopically determined t@xmath31 values with those derived from photometric spectral energy distribution fitting in mcdonald et al . ( 2011 ) finds systematic offsets of 20 k ( @xmath4=74 k ) for the rgb sample and @xmath1535 k ( @xmath4=71 k ) for the agb sample . if the [ fe / h ] abundances are redetermined using temperatures ( and corresponding gravities ) that are 35 k lower and 20 k higher for the agb and rgb stars , respectively , then the agb [ fe / h ] values decrease by 0.05 dex and the rgb [ fe / h ] values increase by 0.02 dex . this brings both data sets into agreement at @xmath0[fe / h]@xmath2=0.73 . in contrast to [ fe / h ] , 47 tuc exhibits a significant spread in [ na / fe ] abundance . for the agb stars analyzed here , [ na / fe ] ranges from 0.11 to @xmath150.62 with @xmath0[na / fe]@[email protected] ( @xmath4=0.17 ) . the large dispersion in [ na / fe ] is typical for globular clusters ( e.g. , see reviews by kraft 1994 ; gratton et al . 2004 ; 2012 ) , and the large fraction of stars with [ na / fe]@xmath370 matches previous observations of 47 tuc rgb , horizontal branch , and main sequence turn off stars ( e.g. , carretta et al . 2004 ; alves brito et al . 2005 ; koch & mcwilliam 2008 ; carretta et al . 2009 ; dorazi et al . 2010 ; worley & cottrell 2012 ; gratton et al . 2013 ; cordero et al . 2014 ; dobrovolskas et al . 2014 ; thygesen et al . the possible temperature scale difference between the agb and rgb samples also affects sodium . however , the change in [ na / fe ] is only an increase of 0.02 dex for the agb stars and a decrease of 0.01 dex for the rgb stars . a more detailed comparison between the agb and rgb [ na / fe ] abundances is provided in section 5 . radial velocities were also measured for each star in order to verify cluster membership . we used the cross correlation code xcsao ( kurtz & mink 1998 ) , and the heliocentric correction was determined using the iraf task _ rvcorrect_. the velocities were determined relative to a synthetic rest frame spectrum with t@xmath31=4600 k , log(g)=1.5 ( cgs ) , [ fe / h]=0.70 , and vt=1.5 km s@xmath3 , and the xcsao routine was only run on order 55 ( see table 1 ) of a single exposure for each star . order 55 was selected because it exhibits several strong lines but does not contain many problematic features ( e.g. , telluric bands ; unusually strong or broad lines ) . the average measurement uncertainty returned by xcsao for the radial velocities was 0.18 km s@xmath3 . we find an average heliocentric radial velocity of 18.56 km s@xmath3 ( @xmath4=10.21 km s@xmath3 ) . the average cluster velocity determined here is in good agreement with past work , which ranges from 16.86 km s@xmath3 to 22.43 km s@xmath3 ( dinescu et al . 1999 ; carretta et al . 2004 ; alves brito et al . 2005 ; carretta et al . 2009 ; dobrovolskas et al . 2014 ; lapenna et al . 2014 ) . all of the stars observed with m2fs have velocities consistent with cluster membership .

we find the average heliocentric radial velocity and [ fe / h ] values to berv=18.56 km s (=10.21 km s ) and[fe / h]=0.68 (=0.08 ) , respectively , in agreement with previous literature estimates . however , in contrast to ngc 6752 , the two 47 tuc populations have nearly identical [ na / fe ] dispersion and interquartile range values .

a recent analysis comparing the [ na / fe ] distributions of red giant branch ( rgb ) and asymptotic giant branch ( agb ) stars in the galactic globular cluster ngc 6752 found that the ratio of na poor to na rich stars changes from 30:70 on the rgb to 100:0 on the agb . the surprising paucity of na rich stars on the agb in ngc 6752 warrants additional investigations to determine if the failure of a significant fraction of stars to ascend the agb is an attribute common to all globular clusters . therefore , we present radial velocities , [ fe / h ] , and [ na / fe ] abundances for 35 agb stars in the galactic globular cluster 47 tucanae ( 47 tuc ; ngc 104 ) , and compare the agb [ na / fe ] distribution with a similar rgb sample published previously . the abundances and velocities were derived from high resolution spectra obtained with the michigan / magellan fiber system ( m2fs ) and mspec spectrograph on the magellan clay 6.5 m telescope . we find the average heliocentric radial velocity and [ fe / h ] values to berv=18.56 km s (=10.21 km s ) and[fe / h]=0.68 (=0.08 ) , respectively , in agreement with previous literature estimates . the average [ na / fe ] abundance is 0.12 dex lower in the 47 tuc agb sample compared to the rgb sample , and the ratio of na poor to na rich stars is 63:37 on the agb and 45:55 on the rgb . however , in contrast to ngc 6752 , the two 47 tuc populations have nearly identical [ na / fe ] dispersion and interquartile range values . the data presented here suggest that only a small fraction ( ) of na rich stars in 47 tuc may fail to ascend the agb , which is a similar result to that observed in m13 . regardless of the cause for the lower average [ na / fe ] abundance in agb stars , we find that na poor stars and at least some na rich stars in 47 tuc evolve through the early agb phase . the contrasting behavior of na rich stars in 47 tuc and ngc 6752 suggests that the rgb [ na / fe ] abundance alone is insufficient for predicting if a star will ascend the agb .

1007.1234

i

the theory of cps is a framework for design and analysis of distributed algorithms for coordination of the groups of dynamic agents . in many control problems , agents in the group need to agree upon certain quantity , whose interpretation depends on the problem at hand . the theory of cps studies the convergence to a common value ( consensus ) in its general and , therefore , abstract form . it has been a subject of intense research due to diverse applications in applied science and engineering . the latter include coordination of groups of unmanned vehicles @xcite ; synchronization of power , sensor and communication networks @xcite ; and principles underlying collective behavior in social networks @xcite and biological systems @xcite , to name a few . from the mathematical point of view , analysis of continuous time cps is a stability problem for systems of linear differential equations possibly with additional features such as stochastic perturbations or time delays . there are many effective techniques for studying stability of linear systems @xcite . the challenge of applying these methods to the analysis of cps is twofold . first , one is interested in characterizing stability under a minimal number of practically relevant assumptions on the structure of the matrix of coefficients , which may depend on time . second , it is important to identify the relation between the structure of the graph of interactions in the network to the dynamical performance of cps . a successful solution of the second problem requires a compilation of dynamical systems and graph theoretic techniques . this naturally leads to spectral methods , which play important roles in both mathematical disciplines , and are especially useful for problems on the interface between dynamics and the graph theory @xcite . a general idea for using spectral methods for analyzing cps is that , on the one hand , stability of the continuous time cp is encoded in the eigenvalues ( evs ) of the matrix of coefficients ; on the other hand , evs of the same matrix capture structural properties of the graph of the cp . the spectral graph theory offers many fine results relating the structural properties of graphs to the evs of the adjacency matrix and the graph laplacian @xcite . this provides a link between the network topology and the dynamical properties of cps . in this paper , under fairly general assumptions on cps , we study two problems : convergence of cps and their stability in the presence of stochastic perturbations . the former is the problem of asymptotic stability of the consensus subspace , a one - dimensional invariant ( center ) subspace . the latter is a special albeit representative form of stability of the consensus subspace with respect to constantly acting perturbations @xcite . the rate of convergence to the consensus subspace sets the timescale of the consensus formation ( or synchronization ) from arbitrary initial conditions or upon instantaneous perturbation . therefore , the convergence rate is important in applications where the timing of the system s responses matters ( e.g. , in decision making algorithms , neuronal networks , etc ) . stochastic stability , on the other hand , characterizes robustness of the consensus to noise . this form of stability is important when the consensus needs to be maintained in noisy environment over large periods of time ( e.g. , communication networks , control of unmanned vehicles , etc ) . we believe that our quantitative description of these two forms of stability elucidates two important aspects of the performance of cps . the questions investigated in this paper have been studied before under various hypotheses on cps : constant weights @xcite , time - dependent interactions @xcite , and cps with time - delays @xcite . optimization problems arising in the context of cp design were studied in @xcite . there is a body of related work on discrete time cps @xcite . robustness of cps to noise was studied in @xcite . in this paper , we offer a unified approach to studying convergence and stochastic stability of cps . our method applies to networks with directed information flow ; both cooperative and noncooperative interactions ; networks under weak stochastic forcing ; and those whose topology and strength of connections may vary in time . we derive sufficient conditions guaranteeing convergence of time - dependent cps and present estimates characterizing their stochastic stability . for cps on undirected graphs , we show that the rate of convergence and stability to random perturbations are captured by the generalized algebraic connectivity and the total effective resistance of the underlying graphs . previously , these results were available only for cps on graphs with positive weights @xcite . to further elucidate the role that network topology plays in shaping the dynamical properties of cps , we further develop our results for cps on simple networks ( see text for the definition of a simple network ) . our analysis of simple networks reveals the role of the geometric properties of the cycle subspace associated with the graph of the network ( such as the first betti number of the graph ; the length and the mutual position of the independent cycles ) to the stability of cps to random perturbations . in addition , we explore several implications of the results of the spectral graph theory to cp design . first , we show that expanders , sparse highly connected graphs @xcite , generate cps with the rate of convergence bounded from zero uniformly when the size of the network tends to infinity . in particular , cps based on expanders are effective for coordinating large networks . second , we point out that cps with random connections have nearly optimal convergence rate . in contrast , the convergence of cps on regular lattice - like graphs slows down rapidly as the size of the network grows . we illustrate these observations with numerical examples and refer to the relevant graph - theoretic results . the mathematical analysis of cps in this paper uses the method , which we recently developed for studying synchronization in systems of coupled nonlinear oscillators @xcite and reliability of neuronal networks @xcite . we further develop this method in several ways . first , we relate the key properties of the algebraic transformation of the coupling operator used in @xcite for studying synchronization to general properties of a certain class of pseudo - similarity transformations . second , we strengthen the graph theoretic interpretation of the stability analysis . we believe that our method will be useful for design and analysis of cps and for studying synchronization in a large class of models . the outline of the paper is as follows . in section [ algebra ] , we study the properties of a pseudo - similarity transformation , which is used in the analysis of cps in the remainder of the paper . section [ convergence ] is devoted to the convergence analysis of cps . after formulating the problem and introducing necessary terminology in [ formulation ] , we study convergence of cps with constant and time - dependent coefficients in [ stationary ] and [ nonstationary ] respectively . section [ robustness ] presents estimates characterizing stochastic stability of stationary and time - dependent cps . these results are applied to study cps protocols on undirected weighted graph in section [ undirected ] . in section [ connectivity ] , we discuss the relation between the connectivity of the graph and dynamical performance of cps . the results of this paper are summarized in section [ conclude ] .

our method applies to networks with directed information flow ; both cooperative and noncooperative interactions ; networks under weak stochastic forcing ; and those whose topology and strength of connections may vary in time . the graph theoretic interpretation of the analytical results is emphasized . we show how the spectral properties , such as algebraic connectivity and total effective resistance , as well as the geometric properties , such the dimension and the structure of the cycle subspace of the underlying graph , shape stability of the corresponding cps . in addition , we explore certain implications of the spectral graph theory to cp design . in particular , we point out that expanders , sparse highly connected graphs , generate cps whose performance remains uniformly high when the size of the network grows unboundedly . similarly , we highlight the benefits of using random versus regular network topologies for cp design . we illustrate these observations with numerical examples and refer to the relevant graph - theoretic results . + + * key words . * consensus protocol , dynamical network , synchronization , robustness to noise , algebraic connectivity , effective resistance , expander , random graph + + * ams subject classifications . * 34d06 , 93e15 , 94c15

a unified approach to studying convergence and stochastic stability of continuous time consensus protocols ( cps ) is presented in this work . our method applies to networks with directed information flow ; both cooperative and noncooperative interactions ; networks under weak stochastic forcing ; and those whose topology and strength of connections may vary in time . the graph theoretic interpretation of the analytical results is emphasized . we show how the spectral properties , such as algebraic connectivity and total effective resistance , as well as the geometric properties , such the dimension and the structure of the cycle subspace of the underlying graph , shape stability of the corresponding cps . in addition , we explore certain implications of the spectral graph theory to cp design . in particular , we point out that expanders , sparse highly connected graphs , generate cps whose performance remains uniformly high when the size of the network grows unboundedly . similarly , we highlight the benefits of using random versus regular network topologies for cp design . we illustrate these observations with numerical examples and refer to the relevant graph - theoretic results . + + * key words . * consensus protocol , dynamical network , synchronization , robustness to noise , algebraic connectivity , effective resistance , expander , random graph + + * ams subject classifications . * 34d06 , 93e15 , 94c15

1307.1838

i

we have examined the local symmetry distribution ( positions of locally symmetric domains as a function of their range ) and dynamics ( spatial evolution of the distances between local symmetry axes for given range ) of one - dimensional aperiodic lattices generated by representative binary sequences , including ( generalized ) fibonacci , period - doubling , thue - morse and cantor . for each case , explicit scaling behaviors and correlation properties were found numerically for the sequences of occurring maximal local symmetry ranges , and for their spacings along the lattice , in relation to the original generating inflation rules or recursive schemes . in particular , the generalized fibonacci lattices were shown to incorporate multiplet symmetry range subsequences with different structural features determined by the asymptotic length ratio of the corresponding binary word generations , with a dominant subsequence containing all complete local symmetries ( i.e. , palindromic word prefixes ) . moreover , for the fibonacci , pd and tm lattices , the number of different local symmetry spacings is restricted to @xmath212 or @xmath238 for any symmetry range . in terms of the subsequences of local symmetry spacing return maps for given ranges , the ( asymptotic ) copper fibonacci lattice was shown to be structurally equivalent to the period - doubling lattice , of which in turn the thue - morse case constitutes a reduced version ( with two sub - subsequences discarded ) . remarkably , the aperiodic order of each type of lattice was found to be encoded exactly in its local symmetry dynamics : for each occurring local symmetry range , the renormalized sequence of axis spacings is generated by the inflation rule of the original aperiodic sequence . the local symmetry distribution of the above lattices was , finally , contrasted with the cantor lattice , whose inflation rule is inherently palindromic with gapped maximal symmetry distribution and unbounded symmetry spacing return maps . in conclusion , we have shown that a unified view of different classes of aperiodic lattices is provided by the distribution , and , in particular , by the fixed range spacing dynamics , of their ( maximal ) local symmetries . they yield a compellingly simple analysis tool to distinguish structural similarities but also lower level differences between given lattices . the encoding of the corresponding inflation rule within any given symmetry spacing trajectory further demonstrates that local symmetries completely characterize deterministic aperiodic order at any scale . since we perform a numerical investigation of the considered aperiodic sequences , throughout the paper the notion of asymptotic convergence is not used in a mathematically strict sense , but refers to the observed behavior for large number of corresponding iterations . the first and second range subsequences would be @xmath333 and @xmath334 , respectively . thus , in order to consider the very _ first _ @xmath196-tet bundle @xmath335 , one has to include , apart from the single - letter maximal palindromes with length @xmath173 , also the _ empty _ maximal palindromes @xmath7 of length @xmath336 . the latter are the interfaces between two different letters @xmath21 and @xmath22 in a word . since @xmath337 and @xmath115 are not plotted throughout our analysis , we describe the @xmath196-tets starting from the second bundle @xmath338 .

a unifying description of lattice potentials generated by aperiodic one - dimensional sequences is proposed in terms of their local reflection or parity symmetry properties . we demonstrate that the ranges and axes of local reflection symmetry possess characteristic distributional and dynamical properties which can be determined for every aperiodic binary lattice . a striking aspect of such a property is given by the return maps of sequential spacings of local symmetry axes , which typically traverse few - point symmetry orbits . this local symmetry dynamics allows for a classification of inherently different aperiodic lattices according to fundamental symmetry principles . illustrating the local symmetry distributional and dynamical properties for several representative binary lattices ,

a unifying description of lattice potentials generated by aperiodic one - dimensional sequences is proposed in terms of their local reflection or parity symmetry properties . we demonstrate that the ranges and axes of local reflection symmetry possess characteristic distributional and dynamical properties which can be determined for every aperiodic binary lattice . a striking aspect of such a property is given by the return maps of sequential spacings of local symmetry axes , which typically traverse few - point symmetry orbits . this local symmetry dynamics allows for a classification of inherently different aperiodic lattices according to fundamental symmetry principles . illustrating the local symmetry distributional and dynamical properties for several representative binary lattices , we further show that the renormalized axis spacing sequences follow precisely the particular type of underlying aperiodic order . our analysis thus reveals that the long - range order of aperiodic lattices is characterized in a compellingly simple way by its local symmetry dynamics .

astro-ph0302066

r

using our data supplemented with that of blc02 , 49 of the 54 candidate cluster radio galaxies have measured velocities . of these , 40 are confirmed cluster members . the list of these galaxies and their attributes ( velocities , magnitudes , radio fluxes , and classifications ) may be found in table [ tbl - clusrg ] . prior papers ( bf76 ; * ? ? ? * ; * ? ? ? * ) applied alphabetical designations to the prominent radio features , and we have included these designations in table [ tbl - clusrg ] when they are associated with specific galaxies . in addition , the alphabetical designations have been noted in figure [ fig - bigmap ] . to avoid confusion we will adhere to the following terms and definitions . the radio _ halo _ in abell 2256 is the diffuse emission around the cluster center . its surface brightness is too low for it to be present in our images , although it may be associated with one cluster radio galaxy ( `` d '' in the prior studies ) . the _ relic _ is the diffuse emission to the north of the cluster center , identified as `` g '' and `` h. '' it is often referred to as a halo , although under current usage of the term this is incorrect as it has a high polarization ( up to @xmath37 ; * ? ? ? * ) and is located off the cluster center . lastly , the diffuse emission labeled `` f '' will be referred to as the ultrasteep spectrum source @xcite . it is most likely another relic source . our spectra include 28 of the 40 confirmed cluster radio galaxies , an even split of which are powered by each agn and star formation . among the agn , we associate four galaxies with head - tail radio sources . @xcite identified these four galaxies as definite head - tails ( features a , b , c , and i ; see table [ tbl - clusrg ] and figures [ fig - koalatad ] , [ fig - flagpole ] , and [ fig - flag ] ) , and noted four additional potential head - tails ( f , g , h , and m ) . the resolution of our maps excludes source m as a cluster radio galaxy , and source f ( the ultrasteep spectrum source ; see figure [ fig - f ] ) is unlikely to be a true head - tail source . it is resolved into three components by our observations ( potentially attached with lower surface brightness emission than our observations detect ) , the easternmost of which ( f3 ) is associated with a cluster galaxy ( and probably unassociated with the remainder of the emission of source f , * ? ? ? sources g and h comprise the relic source , with source g apparently associated with a cluster elliptical , as the peak in the radio emission fell only 3.1 from the galaxy ( see figure [ fig - relic1 ] ) . this galaxy may have once been a head - tail radio galaxy which was disrupted by the merger environment ( as suggested by * ? ? ? * and further explored in section [ sec : discuss ] ) . even if only four of the galaxies are associated with head - tail radio sources , this is an unusually high number for a cluster . typical clusters only possess from 02 such sources @xcite . one further cluster agn may be associated with extended emission . the galaxy located at j170448 + 783829 is one of the brighter ellipticals in abell 2256 and is associated with an unresolved 11.4 mjy radio source ( radio feature d in bf76 ; see figure [ fig - center ] ) . it sits in the center of the diffuse cluster halo emission @xcite , and about 1 to the east of the brightest cluster galaxy ( which is not a radio source ) . of the remaining cluster agn for which we have optical spectra , all but one are low - luminosity radio sources with absorption line spectra . the spectrum of j170150 + 790216 is that of a strong liner . the radio emission is unresolved , and the optical image of the galaxy suggests a recent merger or interaction ( see figure [ fig - fig8 ] ) . fourteen cluster radio galaxies are confirmed to be star - forming on the basis of their spectra . the wavelength coverage adopted for the multi - fiber observations did not include cluster [ oii ] emission , preventing us from classifying the galaxies in the morphs scheme . however , we can compensate for this somewhat by assuming that the equivalent width of [ oii ] is @xmath38 that of the h@xmath39-[nii ] equivalent width @xcite . making this assumption , we find that one of the galaxies ( j170534 + 785116 ) is a starburst galaxy , or `` e(b ) '' in the morphs classification scheme . not coincidentally , this galaxy has the largest 1.4 ghz flux of all the cluster star - forming galaxies ( @xmath14.2 mjy ) and was even identified in the early radio study of bf76 ( source `` e '' ) . three of the star - forming galaxies have h@xmath40 absorption sufficient enough to place them in the `` e(a ) '' class ( j165253 + 784629 , j170335 + 785055 , and j171016 + 780207 ) . galaxies with such spectra are likely associated with starbursts in which selective extinction has diminished their [ oii ] emission in dusty hii regions , yet longer - lived a stars have exited ( or dispersed ) these hii regions and produce a strong h@xmath40 absorption feature @xcite . one further galaxy has h@xmath40 absorption which places it close to the e(a ) category ( j170512 + 784814 ) . the remainder of the star - forming galaxies appear to be relatively normal , given the limits of our data . six of the remaining galaxies with measured velocities are background radio galaxies seen in projection on the cluster , and the remaining three are foreground radio galaxies . these galaxies , along with the five candidate cluster radio galaxies which lack velocities , are listed in table [ tbl - morerg ] . relaxing the formal limits of @xmath41 and the radial limit of 3 mpc produces a further four radio galaxies with measured velocities which are also included in table [ tbl - morerg ] . three of these are cluster members , while the fourth is a background galaxy . blc02 note that most of the non - cluster galaxies in their velocity list come from a background group at a velocity of 52,413 km s@xmath4 . three of our background radio galaxies appear to belong to this system , including an apparent fr i source ( j170454 + 782954 , see figure [ fig - backfr1 ] ) . this source has a radio luminosity of @xmath42 w hz@xmath4 and optical magnitude of @xmath43 , typical of fr i sources with wide - angle tail morphology . another obvious radio galaxy can be seen in the figure ( j170515 + 783018 ) , although the velocity of this galaxy is unknown . should it belong to the same system , it would have a radio luminosity of @xmath44 w hz@xmath4 and an absolute magnitude of @xmath45 . lastly , there is faint , diffuse emission just to the north of the fr i source . some of this may be associated with the faint galaxy at the western edge of the emission ( j170450 + 783011 ) , although it is unlikely that this represents a head - tail galaxy at the redshift of the fr i source . if the galaxy were in the same system , its absolute magnitude of @xmath46 would be unusually faint for a head - tail source . how does the radio galaxy population of abell 2256 compare to other nearby clusters ? in mo03 this question was investigated for 20 nearby clusters , including abell 2256 . the analysis used a chi - square test to evaluate the fraction of galaxies with radio emission in a given cluster compared to the same fraction for the composite sample formed by the other 19 clusters . in that paper , the dynamically active cluster abell 2255 was demonstrated to have a highly significant excess of radio galaxies , but abell 2256 appeared relatively normal . this result is not affected by the lack of velocities for five potential members of abell 2256 ; three of these have radio luminosities lower than the limit adopted for the other clusters , and the remaining two do not change any of the statistical results . we note that the lack of an excess of radio galaxies in abell 2256 is consistent with the low blue fraction ( @xmath47 ) for the cluster , reported in @xcite . there is one specific case in which the radio galaxy population of abell 2256 differs from the other clusters , and that is the fraction of optically brightest galaxies ( @xmath48 ) harboring radio sources . there are seven such radio galaxies in abell 2256 , six of which have spectra indicative of agn ( five have absorption line spectra , the sixth is a strong liner ) . while we do not have a spectrum for the seventh galaxy , it is probably also an agn . it is the middle galaxy in a triple galaxy system , the western galaxy of which is a very strong tailed radio galaxy ( see figure [ fig - koalatad ] ) . the excess of these optically - bright radio galaxies is not unexpected , as many prior studies have noted the unusually large number of head - tail radio galaxies in abell 2256 . in addition , the _ chandra _ observations of smmv02 noted at least three x - ray point sources corresponding to cluster elliptical galaxies . abell 2256 is known to contain substructure . blc02 used the kmm algorithm ( kaye s mixture model , see * ? ? ? * ) to decompose the cluster into three separate substructures based on both velocity and position . this is a maximum likelihood algorithm , which fits a specified number of gaussians to the data and determines to which of these gaussians each individual galaxy most likely belongs . roughly 70@xmath49 of the galaxies belonged to the primary cluster ( gaussian # 2 in blc02 , referred to hereafter as the `` primary '' ) , and about 20@xmath49 belonged a second subcluster ( gaussian # 1 , hereafter the `` subcluster '' ) , presumably the infalling subcluster identified in the x - ray observations . the remaining @xmath5 belonged to a third group ( gaussian # 3 , hereafter the `` group '' ) concentrated to the north of the primary center and at higher radial velocity . we have repeated the kmm procedure on our combined velocity list in order to place all galaxies ( including the radio galaxies ) into their respective substructures . our kmm code differs slightly from that of @xcite , in that we assume the covariances among the three attributes ( two position , one velocity ) are zero . of all cluster galaxies . ] thus , we calculate the distance from each galaxy to each specified gaussian , @xmath50 : @xmath51.\ ] ] each galaxy is assigned to the gaussian which produces the smallest distance , and once all galaxies have been assigned the new means and dispersions are calculated . these values are then used in successive iterations until the assignments are stable . in particular , the log likelihood of the fit is described by : @xmath52\ ] ] where the summation over @xmath50 refers to the number of groups to be fit to the data and the summation over @xmath53 refers to the individual galaxies . the factor @xmath54 refers to the probability that a given galaxy @xmath53 belongs to group @xmath50 , which is evaluated based on the values of @xmath55 for that galaxy . the factor @xmath56 is the `` mixing fraction , '' or the fraction of the total population which are members of group @xmath50 . convergence of the algorithm is assumed once @xmath57 has converged . as discussed in @xcite , the significance of a multi - component fit is not easily surmised . consequently , our code also calculates @xmath57 for monte carlo shuffles of the data in order to determine the frequency with which the algorithm can fit a randomized data set as well as it does the real data . in this case , we only shuffled the velocity data so as to avoid issues related to positional sampling of cluster velocities ( our mc test is thus directly analogous to that used to quantify the significance of the dressler - shectman test ) . the results are presented in table [ tbl - kmm ] ; the derived velocity means and dispersions are consistent with those presented in table 8 of blc02 . in total , 186 galaxies were assigned to the primary , 78 galaxies to the large infalling subcluster , and 30 galaxies to the group identified primarily on the basis of its higher radial velocity . only @xmath58 of the mc shuffles had 3-gaussian fits as good as or better than the actual data . note that for the two - component and four - component fits the results were @xmath59 and @xmath60 , respectively , arguing for the three - component model ( when attempting a fifth component , the algorithm failed to converge without removing nearly all of the galaxies from one of the substructures ) . a relatively large number of the radio sources were assigned to the group ( 7 of 30 galaxies with velocities placing them in this structure are in table [ tbl - clusrg ] ) . spatially , this may be seen in figure [ fig - dist ] as an apparent excess of radio galaxies to the north of the cluster center . quantitatively , a chi - square test places this excess of radio sources at only marginal significance , about @xmath36 . individually , there is no statistical difference in the radio galaxy fractions of the primary and the subcluster . if we include the three additional confirmed cluster radio galaxies which were fainter than our @xmath61 limit ( table [ tbl - morerg ] ) , the excess of radio sources in the group is accentuated . one of these sources belongs to the group , increasing the significance of the radio galaxy excess to over @xmath62 . interestingly , the radio galaxies in the group are almost exclusively associated with star formation . our spectra include six of the eight identified radio galaxies in the group , all of which exhibit emission lines . one source , j170150 + 790216 , is a strong liner whose optical morphology suggests a recent galaxy - galaxy interaction ( see figure [ fig - fig8 ] ) . this galaxy is also separated from the other radio detections of the group , lying at a higher declination . the remaining five galaxies have spectra of star - forming galaxies . the propensity for the group members to be radio - detected star - forming galaxies is significant at nearly @xmath63 ( 5 of 30 compared to 9 of 264 ) , although this claim is tempered by our incomplete spectroscopy . however , should all of the group galaxies lacking spectroscopic classification be agn and all the non - group galaxies lacking spectroscopic classification be star forming ( i.e. , a worst case scenario ) , the excess of star - forming galaxies in the group would still be significant at over @xmath64 . star formation therefore seems more prevalent in the group than in the remainder of abell 2256 . additionally , we note that the two group member radio galaxies for which we do not possess spectra appear to be spirals and their radio - to - optical flux ratios are suggestive of star formation rather than agn . conversely , only one of the radio galaxies assigned to the subcluster is associated with star formation . our kmm code suggests that the probability that this galaxy belongs to the subcluster is only slightly greater than the probability that it belongs to the primary . thus , the activity level of the infalling subcluster appears to be quite low .

over 40 cluster radio galaxies are identified , with optical spectroscopy indicating the emission source ( star formation or agn ) for most of them . while the overall fraction of galaxies exhibiting radio emission is consistent with a large sample of other nearby clusters , we find an increase in the activity level of galaxies belonging to the third system ( hereafter , the `` group '' ) .

abell 2256 is a nearby ( ) , rich cluster of galaxies with fascinating observed properties across a range of wavelengths . long believed to represent a cluster merger , recent x - ray and optical results have suggested that in addition to the primary cluster and subcluster there is evidence for a third , poorer system . we present wide - field , high sensitivity 1.4 ghz vla radio observations of abell 2256 in conjunction with optical imaging and additional spectroscopy . over 40 cluster radio galaxies are identified , with optical spectroscopy indicating the emission source ( star formation or agn ) for most of them . while the overall fraction of galaxies exhibiting radio emission is consistent with a large sample of other nearby clusters , we find an increase in the activity level of galaxies belonging to the third system ( hereafter , the `` group '' ) . specifically , the group has relatively more star formation than both the primary cluster and main subcluster . the position of the group is also coincident with the observed cluster radio relic . we suggest that the group recently (.3 gyr ) merged with the primary cluster and that this merger , not the ongoing merger of the primary and the main subcluster , might be responsible for many of the unusual radio properties of abell 2256 . furthermore , the greater star formation activity of the group suggests that the infall of groups is an important driver of galaxy evolution in clusters .

astro-ph0302066

c

we have presented a multiwavelength observational study of abell 2256 . our deep radio continuum imaging reached an rms sensitivity as low as @xmath69 @xmath13jy beam@xmath4 and encompassed an area from the center of the cluster out to radii of @xmath150 , or about 3 mpc . potential cluster radio galaxies were identifed using optical imaging of the same area , with membership being evaluated on the basis of both new and existing optical spectroscopy . in total , 43 cluster radio galaxies were identified and many of these were characterized as agn or star - forming galaxies on the basis of their spectra . paralleling the study of blc02 , the application of a maximum likelihood algorithm assigned the galaxies to their probable substructures , which correspond to the primary , the infalling subcluster , and a group . taken collectively , the radio galaxy population of abell 2256 is not unusual when compared to a sample of nineteen other nearby clusters . we confirm a statistical excess of the optically brightest radio galaxies , which might have been guessed from the previously noted unusual number of galaxies with extended radio emission in the cluster . outside of this specific population , the radio galaxy fraction of abell 2256 is relatively normal . we do , however , note that the galaxies belonging to the group are more frequently associated with radio emission and nearly all of these galaxies are powered by star formation . the radial velocity of the group is @xmath70 km s@xmath4 greater than that of the primary , and its position is coincident with the cluster radio relic source . we therefore argue that the group has recently merged with the cluster and is located behind the cluster relative to the observer , roughly 0.3 gyr after core passage . under this scenario , the radio relic would owe its existence to the cluster - group merger rather than the ongoing cluster - subcluster merger , which appears to be about 0.2 gyr prior to core passage and entering the primary from behind and to the west @xcite . in the broader picture , if abell 2256 is a representative example then galaxy activity in clusters may be most influenced by the merger history of the parent cluster with outlying groups . these minor mergers would provide a relatively large influx of gas - rich spirals which might undergo starbursts followed by truncation of star formation . mergers of larger partners , while having more profound effects on the icm , may not prove overly important toward understanding star formation in clusters simply because the partners are less likely to include the potential starbursting galaxies . thus , active clusters such as those identified in butcher - oemler type studies would simply correspond to dynamically young clusters in larger - scale structures rich in peripheral groups . nam thanks the national radio astronomy observatory for a predoctoral fellowship , and acknowledges the support of a national research council associateship award at nasa s goddard space flight center . this research was supported in part by nasa through the american astronomical society s small research grant program . we thank keith ashman for providing the original kmm code , and eric greisen for assistance with aips and the multi - resolution clean . markevitch , m. , ponman , t.j . , nulsen , p.e.j . , bautz , m.w . , burke , d.j . , david , l.p . , davis , d. , donnelly , r.h . , forman , w.r . , jones , c. , kaastra , j. , kellogg , e. , kim , d .- w . , kolodziejczak , j. , mazzotta , p. , pagliaro , a. , patel , s. , van speybroeck , l. , vikhlinin , a. , vrtilek , j. , wise , m. , & zhao , p. 2000 , , 541 , 542 l l r r c r r 16:47:57.9 & 78:52:07 & 17784 & 31 & emi & & + 16:50:39.6 & 78:39:07 & 16756 & 42 & abs & & + 16:51:11.5 & 79:06:05 & 17671 & 58 & abs & & + 16:52:32.0 & 78:57:03 & 30106 & 78 & emi & & + 16:52:52.8 & 78:46:29 & 16836 & 19 & emi & & + 16:53:05.9 & 78:30:13 & 16663 & 69 & emi & & + 16:53:13.6 & 78:44:13 & 16397 & 41 & abs & & + 16:53:33.3 & 79:00:00 & 18742 & 37 & emi & & + 16:53:34.3 & 79:06:21 & 18500 & 40 & emi & & + 16:54:06.1 & 78:37:42 & 16674 & 39 & abs & 16710 & 36 + 16:54:16.6 & 79:11:20 & 53488 & 53 & abs & & + 16:54:29.4 & 78:54:34 & 16228 & 39 & abs & & + 16:54:36.1 & 79:14:56 & 39573 & 60 & emi & & + 16:54:58.9 & 79:08:45 & 17433 & 61 & abs & & + 16:55:32.2 & 78:32:02 & 15969 & 57 & abs & 16180 & 35 + 16:55:36.9 & 79:05:10 & 17808 & 30 & abs & & + 16:56:06.9 & 79:23:28 & 26352 & 60 & emi & & + 16:56:09.6 & 78:56:36 & 17815 & 46 & abs & 17851 & 46 + 16:56:16.1 & 78:55:09 & 18288 & 32 & emi & 18309 & 48 + 16:56:18.3 & 79:09:29 & 17598 & 60 & emi & & + 16:57:05.5 & 78:32:48 & 16350 & 56 & abs & 16495 & 31 + 16:57:58.5 & 78:36:15 & 4260 & 22 & emi & 4253 & 45 + 16:58:21.1 & 78:43:25 & 15791 & 67 & abs & 15909 & 52 + 16:58:46.8 & 78:25:21 & 15632 & 62 & abs & 15635 & 49 + 17:00:07.5 & 78:49:52 & 16514 & 68 & abs & & + 17:00:11.1 & 79:00:16 & 17383 & 70 & abs & 17415 & 70 + 17:00:11.5 & 79:03:33 & 17597 & 58 & abs & 17668 & 34 + 17:00:16.0 & 78:34:25 & 18362 & 47 & abs & 18355 & 62 + 17:00:20.4 & 78:33:55 & 16454 & 62 & abs & 16372 & 55 + 17:00:32.6 & 78:56:10 & 15021 & 65 & abs & 15237 & 47 + 17:00:52.3 & 78:41:21 & 17157 & 37 & abs & 17325 & 39 + 17:00:59.2 & 78:44:59 & 15966 & 42 & abs & 16137 & 53 + 17:01:00.2 & 79:05:09 & 18377 & 42 & emi & 18234 & 48 + 17:01:34.6 & 78:50:46 & 19531 & 30 & emi & 19518 & 15 + 17:01:49.5 & 79:02:16 & 19307 & 90 & emi & 19303 & 60 + 17:02:07.5 & 78:17:29 & 17760 & 32 & emi & 18001 & 47 + 17:02:12.3 & 78:51:33 & 19325 & 16 & emi & 19421 & 59 + 17:02:17.8 & 78:45:52 & 20086 & 99 & abs & 19907 & 66 + 17:02:18.6 & 78:46:03 & 19581 & 38 & emi & 19643 & 44 + 17:02:46.8 & 78:51:46 & 16638 & 30 & abs & 16574 & 53 + 17:03:02.9 & 78:35:56 & 16417 & 39 & abs & 16303 & 100 + 17:03:10.9 & 78:14:40 & 18028 & 34 & emi & 17973 & 48 + 17:03:25.4 & 78:20:38 & 18922 & 43 & emi & 18886 & 48 + 17:03:29.5 & 78:37:55 & 17565 & 54 & abs & 17796 & 44 + 17:03:34.5 & 78:50:55 & 18989 & 72 & emi & 18943 & 49 + 17:03:56.5 & 78:44:44 & 17082 & 44 & abs & 17115 & 36 + 17:04:04.2 & 78:33:23 & 17316 & 53 & abs & 17355 & 53 + 17:04:14.5 & 79:23:13 & 26291 & 32 & emi & & + 17:04:15.4 & 78:06:07 & 18107 & 90 & abs & & + 17:04:41.0 & 78:13:49 & 17956 & 58 & abs & 18008 & 56 + 17:04:48.2 & 78:38:29 & 19013 & 38 & abs & 19094 & 49 + 17:04:54.0 & 78:29:54 & 52284 & 60 & abs & 52295 & 114 + 17:05:11.9 & 78:48:14 & 19921 & 32 & emi & 19942 & 48 + 17:05:21.8 & 79:08:05 & 18825 & 42 & emi & 18807 & 48 + 17:05:24.2 & 78:22:00 & 17189 & 55 & abs & 17322 & 36 + 17:05:33.6 & 78:51:16 & 19511 & 46 & emi & 19479 & 54 + 17:06:00.8 & 79:06:43 & 17908 & 68 & abs & 17927 & 47 + 17:06:28.4 & 78:49:15 & 18994 & 22 & emi & 19093 & 45 + 17:07:00.6 & 78:41:23 & 16956 & 62 & abs & 17161 & 79 + 17:07:02.4 & 79:10:19 & 11408 & 31 & emi & 11436 & 49 + 17:07:14.1 & 78:53:18 & 16563 & 36 & abs & 16524 & 42 + 17:07:15.9 & 78:30:08 & 18379 & 30 & emi & 18582 & 48 + 17:07:21.4 & 78:53:38 & 16520 & 56 & abs & 16564 & 35 + 17:08:33.5 & 78:29:54 & 18663 & 39 & abs & 18684 & 46 + 17:08:49.0 & 79:14:31 & 16668 & 44 & abs & & + 17:08:58.2 & 78:50:15 & 16249 & 99 & abs & 16064 & 51 + 17:09:10.0 & 78:52:32 & 18624 & 60 & emi & 18812 & 46 + 17:09:44.3 & 78:06:06 & 18039 & 32 & emi & & + 17:10:14.4 & 78:39:25 & 16922 & 47 & abs & 17043 & 53 + 17:10:15.9 & 78:02:07 & 17801 & 38 & emi & & + 17:10:31.5 & 78:29:10 & 15783 & 65 & abs & 15792 & 66 + 17:10:48.8 & 78:32:18 & 18682 & 36 & abs & 18767 & 31 + 17:13:57.2 & 78:59:33 & 19696 & 30 & abs & 19614 & 62 + l l l r l l r l l c c 16:52:52.8 & 78:46:29 & 16805 & 27 & ke & 16.39 & 0.572 & 0.107 & 2.2 & & sf + 16:53:05.8 & 78:30:13 & 16663 & 69 & me & 16.51 & 0.534 & 0.151 & 2.7 & & sf + 16:54:58.9 & 79:08:45 & 17433 & 61 & mf & 15.38 & 0.338 & 0.127 & 1.3 & & agno + 16:55:36.9 & 79:05:10 & 17808 & 30 & kf & 14.76 & 2.897 & 0.100 & 0.2 & & agno + 16:56:18.3 & 79:09:29 & 17616 & 36 & ke & 15.86 & 0.641 & 0.155 & 0.8 & & sf + 16:58:18.5 & 78:29:34 & 17882 & 47 & b & 14.99 & 3.397 & 0.340 & 0.8 & ad & + 16:58:46.8 & 78:25:21 & 15632 & 62 & mf & 15.23 & 1.000 & 0.218 & 5.0 & & agno + 17:00:25.8 & 78:49:26 & 15376 & 52 & b & 16.58 & 0.225 & 0.071 & 1.5 & & + 17:00:52.3 & 78:41:21 & 17157 & 37 & mf & 15.40 & 7.8 & 0.3 & & i & agno + 17:00:59.2 & 78:44:59 & 15966 & 42 & kf mf & 15.29 & 0.247 & 0.060 & 0.4 & & agno + 17:01:00.2 & 79:05:09 & 18232 & 48 & ke & 16.20 & 0.865 & 0.124 & 0.5 & & sf + 17:01:23.4 & 78:41:16 & 16217 & 67 & b & 15.55 & 0.425 & 0.126 & 2.2 & & + 17:01:34.6 & 78:50:46 & 19531 & 30 & me & 16.64 & 0.549 & 0.083 & 0.4 & & sf + 17:01:49.5 & 79:02:16 & 19342 & 60 & ke & 14.65 & 2.962 & 0.100 & 1.2 & & agnl + 17:02:12.3 & 78:51:33 & 19325 & 16 & ke me & 15.76 & 1.438 & 0.100 & 0.7 & & sf + 17:02:15.0 & 78:35:47 & 18757 & 39 & b & 15.36 & 0.642 & 0.185 & 1.7 & & + 17:02:18.6 & 78:46:03 & 19612 & 23 & ke & 15.52 & 1.687 & 0.104 & 1.2 & k & sf + 17:02:46.8 & 78:51:46 & 16638 & 30 & kf & 15.12 & 1.840 & 0.100 & 0.7 & ac & agno + 17:03:02.9 & 78:35:56 & 16417 & 39 & mf & 14.90 & 50.3 & 0.7 & & b & agno + 17:03:22.4 & 78:46:55 & 19723 & 45 & b & 16.33 & 0.454 & 0.087 & 1.2 & & + 17:03:25.4 & 78:20:38 & 18922 & 43 & me & 16.43 & 0.714 & 0.155 & 0.6 & & sf + 17:03:29.5 & 78:37:55 & 17565 & 54 & mf & 14.73 & 120.6 & 0.3 & & a & agno + 17:03:30.1 & 78:39:55 & 17558 & 36 & b & 15.01 & 40.7 & 0.5 & & c & + 17:03:34.1 & 78:37:48 & 16929 & 47 & b & 14.70 & 1.261 & 0.148 & 2.3 & & + 17:03:34.5 & 78:50:55 & 18989 & 72 & me & 16.02 & 0.697 & 0.115 & 1.4 & & sf + 17:03:56.5 & 78:44:44 & 17082 & 44 & mf & 15.51 & 5.816 & 1.000 & 3.1 & g & agno + 17:04:15.4 & 78:06:07 & 18107 & 90 & kf & 15.74 & 3.583 & 0.200 & 0.4 & & agno + 17:04:48.2 & 78:38:29 & 19103 & 38 & mf & 14.64 & 11.4 & 0.1 & 1.9 & d & agno + 17:05:11.9 & 78:48:14 & 19921 & 32 & me & 16.20 & 0.576 & 0.112 & 0.3 & & sf + 17:05:33.6 & 78:51:16 & 19511 & 46 & me & 15.75 & 4.178 & 0.100 & 0.3 & e & sf + 17:06:43.4 & 78:35:57 & 14910 & 77 & b & 16.17 & 1.025 & 0.076 & 0.9 & & + 17:06:56.4 & 78:41:09 & 16867 & 38 & b & 15.30 & 0.883 & 0.104 & 1.7 & f3 & + 17:07:15.9 & 78:30:08 & 18391 & 36 & ke & 16.41 & 1.329 & 0.086 & 1.2 & & sf + 17:08:33.5 & 78:29:54 & 18663 & 39 & kf mf & 14.78 & 2.895 & 0.100 & 1.0 & aa & agno + 17:08:47.7 & 78:25:56 & 16860 & 7 & b & 16.60 & 0.283 & 0.107 & 2.3 & & + 17:08:58.1 & 78:50:15 & 16249 & 99 & kf & 15.21 & 0.600 & 0.083 & 1.2 & & agno + 17:09:10.0 & 78:52:32 & 18624 & 60 & me & 15.31 & 1.211 & 0.134 & 0.9 & & sf + 17:09:19.1 & 78:30:16 & 16700 & 44 & b & 16.78 & 0.647 & 0.165 & 4.1 & & + 17:09:39.9 & 78:33:07 & 16120 & 33 & b & 15.44 & 0.500 & 0.132 & 2.9 & & + 17:10:15.9 & 78:02:07 & 17801 & 38 & me & 16.58 & 0.405 & 0.105 & 1.7 & & sf + & & & & & & & & & & + & & & & & & & & & & + & & & & & & & & & & + & & & & & & & & & & + & & & & & & & & & & + & & & & & & & & & & + l l r r l l r l l c c c 16:53:19.4 & 78:30:40 & & & & 15.08 & 0.261 & 0.091 & 1.0 & & & + 16:54:16.6 & 79:11:20 & 53488 & 53 & mf & 16.72 & 0.535 & 0.162 & 2.4 & & agno & + 16:54:36.0 & 79:14:56 & 39573 & 60 & ke & 16.72 & 0.520 & 0.145 & 1.4 & & sf & + 16:56:06.9 & 79:23:28 & 26352 & 60 & ke & 15.75 & 2.105 & 0.2 & 1.1 & & sf & + 16:57:58.5 & 78:36:15 & 4260 & 22 & ke me & 15.22 & 1.922 & 0.1 & 0.3 & af & sf & + 17:02:57.9 & 78:03:18 & & & & 16.80 & 0.656 & 0.110 & 0.8 & & & + 17:03:06.8 & 78:11:43 & & & & 16.79 & 0.429 & 0.141 & 3.0 & & & + 17:04:14.5 & 79:23:13 & 26291 & 32 & me & 16.67 & 0.853 & 0.141 & 0.8 & & sf & + 17:04:54.0 & 78:29:54 & 52284 & 60 & kf & 16.64 & 1.898 & 0.154 & 1.5 & & agno & + 17:04:55.4 & 78:51:08 & 19980 & 45 & b & 17.15 & 0.438 & 0.094 & 0.9 & & & + 17:04:56.9 & 78:43:06 & 18682 & 45 & b & 17.05 & 0.419 & 0.096 & 1.2 & & & + 17:06:28.4 & 78:49:15 & 18994 & 22 & me & 16.96 & 1.135 & 0.098 & 1.4 & & sf & + 17:07:02.4 & 79:10:19 & 11408 & 31 & ke me & 16.06 & 0.678 & 0.132 & 0.8 & & sf & + 17:07:40.1 & 78:56:57 & 53835 & 61 & b & 17.57 & 0.276 & 0.078 & 1.6 & & & + 17:09:01.1 & 78:46:18 & 11844 & 26 & b & 15.24 & 2.180 & 0.1 & 0.9 & ae & & + 17:11:20.7 & 78:49:47 & 39493 & 84 & b & 16.64 & 1.231 & 0.1 & 1.0 & & & + 17:14:18.9 & 78:05:20 & & & & 15.10 & 2.390 & 0.270 & 0.8 & & & + 17:18:38.8 & 78:13:38 & & & & 15.90 & 1.299 & 0.100 & 1.0 & & & +

abell 2256 is a nearby ( ) , rich cluster of galaxies with fascinating observed properties across a range of wavelengths . long believed to represent a cluster merger , recent x - ray and optical results have suggested that in addition to the primary cluster and subcluster there is evidence for a third , poorer system . specifically , the group has relatively more star formation than both the primary cluster and main subcluster . the position of the group is also coincident with the observed cluster radio relic . we suggest that the group recently (.3 gyr ) merged with the primary cluster and that this merger , not the ongoing merger of the primary and the main subcluster , might be responsible for many of the unusual radio properties of abell 2256 . furthermore , the greater star formation activity of the group suggests that the infall of groups is an important driver of galaxy evolution in clusters .

abell 2256 is a nearby ( ) , rich cluster of galaxies with fascinating observed properties across a range of wavelengths . long believed to represent a cluster merger , recent x - ray and optical results have suggested that in addition to the primary cluster and subcluster there is evidence for a third , poorer system . we present wide - field , high sensitivity 1.4 ghz vla radio observations of abell 2256 in conjunction with optical imaging and additional spectroscopy . over 40 cluster radio galaxies are identified , with optical spectroscopy indicating the emission source ( star formation or agn ) for most of them . while the overall fraction of galaxies exhibiting radio emission is consistent with a large sample of other nearby clusters , we find an increase in the activity level of galaxies belonging to the third system ( hereafter , the `` group '' ) . specifically , the group has relatively more star formation than both the primary cluster and main subcluster . the position of the group is also coincident with the observed cluster radio relic . we suggest that the group recently (.3 gyr ) merged with the primary cluster and that this merger , not the ongoing merger of the primary and the main subcluster , might be responsible for many of the unusual radio properties of abell 2256 . furthermore , the greater star formation activity of the group suggests that the infall of groups is an important driver of galaxy evolution in clusters .

1403.1605

i

over the last decades , cosmological observations have provided us with a wealth of information about the structure and evolution of the universe . dedicated studies of anisotropies in the cosmic microwave background ( cmb ) by the _ cobe _ , _ wmap _ and _ planck _ satellites as well as ground - based telescopes have yielded increasingly precise information on the spectrum of cosmic density fluctuations on the largest scales @xcite . extensive surveys of galaxies and clusters of galaxies and their gravitational lensing effects have clarified the power - spectrum of large - scale structure ( lss ) on somewhat smaller scales @xcite . finally , investigations of inhomogeneities in intergalactic hydrogen through the lyman alpha forest have led to valuable constraints on the distribution of cosmic structure on the smallest scales to date @xcite . most current observations can be accommodated consistently by the concordance @xmath7cdm cosmological model , based on cold dark matter ( cdm ) , a cosmological constant , and a power - law spectrum of adiabatic primordial density fluctuations that is nearly scale - invariant @xcite . however , the @xmath7cdm model has yet to be sufficiently tested against observations on scales much smaller than those corresponding to massive galaxies . the small - scale power spectrum ( ssps ) , that is , the power spectrum of large - scale structure on such scales , is of great interest from different perspectives , not only for cosmology but also for fundamental physics . on the premise that the concordance @xmath7cdm cosmology is basically valid , one aspect of the ssps that has received great attention is its role in constraining the mass of neutrinos @xcite . the existence of finite rest mass of neutrinos has been demonstrated by experiments probing solar and atmospheric neutrino oscillations , which are sensitive to the relative mass differences between different neutrino families . an important cosmological effect caused by light , massive neutrinos is the suppression of the matter power spectrum on scales smaller than the free streaming scale , which becomes stronger when the total mass of neutrinos is larger . in turn , various cosmological observations including the ssps can provide valuable upper limits on the total mass of neutrinos . current such limits on the sum of neutrino masses of all families is conservatively of order @xmath8 . it is of great interest whether further observations can improve on them or even provide measurements of the neutrino mass . going somewhat beyond the simplest @xmath7cdm model , an interesting possibility is that the spectrum of primordial fluctuations is not a pure power - law but has a running spectral index ( rsi ) , that is , a spectral index @xmath9 that is scale - dependent . standard , single - field , slow - roll models of cosmological inflation in the early universe predict a nearly scale - invariant spectrum with @xmath10 , where @xmath11 and @xmath3 are expected to be first and second order respectively in the small , slow - roll parameters . however , in some inflation models , relatively large rsi can be realized @xcite . current cosmological observations indicate @xmath12 , consistent with standard , slow - roll inflation models . more extensive observations over a wider range of scales including the ssps should lead to more precise constraints and help to discriminate among inflation models that are indistinguishable in terms of @xmath9 alone . finally , as a more drastic alternative to cdm , warm dark matter ( wdm ) with particles masses in the kev range has been proposed on various grounds @xcite . the cosmological effect of wdm is characterized by its free streaming scale that depends on its mass @xmath4 . whereas above this scale , it behaves in a way similar to cdm and is indistinguishable from it , below this scale , it dramatically suppresses the power spectrum , resulting in much fewer dark matter halos on small scales compared to cdm . one specific particle physics candidate for wdm is sterile neutrinos , which are currently constrained to have a mass somewhere in the range @xmath13 kev @xcite . furthermore , wdm has also been motivated from an astrophysical viewpoint . current observations indicate that the abundance of satellite galaxies around the milky way and in the local group is much lower than compared to the abundance of cdm subhalos on corresponding scales , the so - called `` missing satellites problem '' @xcite . although the resolution may lie in astrophysical feedback effects that preferentially quench star formation in smaller systems , wdm can provide an intriguing alternative explanation by attributing the lack of satellite galaxies to the absence of the relevant dark halos in such cosmologies . such an interpretation may favor a wdm mass in the range of a few kev @xcite ( see however @xcite ) . further observations of the ssps can offer a critical test of wdm as a viable dark matter candidate . in order to investigate the ssps in greater depth and test @xmath7cdm and its alternatives , the most direct approach would be to observationally probe dark matter halos on scales much smaller than those of galaxies in the present universe . however , this is made difficult by the fact that the bulk of the intergalactic medium ( igm ) has been fully ionized after completion of cosmic reionization at @xmath14 . in such circumstances , the gas in sufficiently small halos , in particular `` minihalos '' with masses @xmath15 and virial temperature @xmath16 k , is expected to have been substantially photoevaporated @xcite . thus the only way to probe such halos in the present day may be via gravitational lensing effects @xcite or possibly dm annihilation gamma rays @xcite , which are quite challenging prospects . on the other hand , an alternative avenue may open up by focusing on redshifts @xmath17 , before cosmic reionization and minihalo photoevaporation have proceeded significantly . at such epochs , the cold , neutral gas associated with collapsed systems may be observable as a series of redshifted absorption features due to the 21 cm transition in the continuum spectrum of luminous background radio sources such as radio quasars or possibly gamma - ray bursts ( grbs ) . dubbed the `` 21 cm forest '' in analogy with the lyman alpha forest , previous work has shown that the gas in minihalos can give rise to numerous , narrow absorption features that may be observable with future facilities such as the square kilometer array ( ska ) , as long as sufficiently bright radio sources exist at the relevant redshifts @xcite . the mass function of minihalos is dependent on the ssps on scales @xmath0 mpc@xmath1 , much smaller than the smallest scales that are at present observationally accessible via the lyman alpha forest . therefore , future observations of the 21 cm forest can potentially provide a very sensitive probe of the ssps , which in turn can provide valuable constraints or measurements of fundamental physics parameters such as the neutrino mass , the wdm particle mass or rsi of primordial fluctuations , the prospects of which are the main topic of this paper . note that minihalos themselves are unlikely to harbor appreciable star formation , as their virial temperatures are below the threshold for efficient gas cooling via atomic transitions . however , the 21 cm forest signal can also be significantly affected by external astrophysical effects , such as a background of uv photons or heating of the gas via x - rays or shocks , which are expected to become progressively more important as cosmic reionization proceeds . indeed , the implications of such astrophysical feedback processes have been the main focus of studies on the 21 cm forest so far @xcite ( see however @xcite ) . since the consequences for the 21 cm forest of the ssps beyond the standard @xmath7cdm cosmology is being discussed here for the first time , as an initial step , here we choose not to account for the complicating effects of a uv background or heating caused by astrophysical sources . thus we are able to isolate and clarify the effects of modifications to the ssps itself . the neglect of feedback effects would be more justifiable at higher redshifts , @xmath18 , where the formation and evolution of stars and other objects are expected to be more limited . we mention that the global signal of 21 cm emission from minihalos and/or the igm at high redshifts ( for reviews , see @xcite ) has been previously discussed as a potentially powerful probe of the ssps @xcite . however , the major obstacle to such prospects is the huge level of foreground emission , several orders of magnitude brighter than the expected signal , that must be removed very efficiently in order to observe such emission @xcite . in contrast , foregrounds are not a concern for observing the 21 cm forest , as long as sufficiently bright background radio sources exist . this paper is organized as follows . in section ii , we describe our basic assumptions and formulation regarding the halo gas profile , spin temperature , optical depth and abundance of absorbers . section iii continues with how we formulate the modifications to the ssps caused by massive neutrinos , rsi and wdm , and also presents the corresponding expectations for the 21 cm forest , in comparison with the standard @xmath7cdm case . we end with discussions on the observability , various caveats , and a summary in section iv . for our baseline cosmological model , we adopt @xmath7cdm with the following parameters from the latest planck data : @xmath19 , @xmath20 , @xmath21=0.68 , @xmath22 $ ] , @xmath23 , @xmath24 , where @xmath25 are the densities of cold dark matter , baryons , and cosmological constant , respectively , in units of the critical density , @xmath26 is the hubble constant , and @xmath27 is the variance of mass fluctuations averaged over a sphere with radius 8@xmath28 @xcite .

although the cosmological paradigm based on cold dark matter and adiabatic , nearly scale - invariant primordial fluctuations is consistent with a wide variety of existing observations , it has yet to be sufficiently tested on scales smaller than those of massive galaxies , and various alternatives have been proposed that differ significantly in the consequent small - scale power spectrum ( ssps ) of large - scale structure . here we show that a powerful probe of the ssps at mpc can be provided by the 21 cm forest , that is , systems of narrow absorption lines due to intervening , cold neutral hydrogen in the spectra of high - redshift background radio sources in the cosmic reionization epoch . such features are expected to be caused predominantly by collapsed gas in starless minihalos , whose mass function can be very sensitive to the ssps . as specific examples , we consider the effects of neutrino mass , running spectral index ( rsi ) and warm dark matter ( wdm ) on the ssps , and evaluate the expected distribution in optical depth of 21 cm absorbers out to different redshifts . within the current constraints on quantities such as the sum of neutrino masses , running of the primordial spectral index and wdm particle mass , the statistics of the 21 cm forest manifest observationally significant differences that become larger at higher redshifts . in particular , it may be possible to probe the range of kev that may otherwise be difficult to access . future observations of the 21 cm forest by the square kilometer array may offer a unique and valuable probe of the ssps , as long as radio sources such as quasars or population iii gamma - ray bursts with sufficient brightness and number exist at redshifts of 10 - 20 , and the astrophysical effects of reionization and heating can be discriminated .

although the cosmological paradigm based on cold dark matter and adiabatic , nearly scale - invariant primordial fluctuations is consistent with a wide variety of existing observations , it has yet to be sufficiently tested on scales smaller than those of massive galaxies , and various alternatives have been proposed that differ significantly in the consequent small - scale power spectrum ( ssps ) of large - scale structure . here we show that a powerful probe of the ssps at mpc can be provided by the 21 cm forest , that is , systems of narrow absorption lines due to intervening , cold neutral hydrogen in the spectra of high - redshift background radio sources in the cosmic reionization epoch . such features are expected to be caused predominantly by collapsed gas in starless minihalos , whose mass function can be very sensitive to the ssps . as specific examples , we consider the effects of neutrino mass , running spectral index ( rsi ) and warm dark matter ( wdm ) on the ssps , and evaluate the expected distribution in optical depth of 21 cm absorbers out to different redshifts . within the current constraints on quantities such as the sum of neutrino masses , running of the primordial spectral index and wdm particle mass , the statistics of the 21 cm forest manifest observationally significant differences that become larger at higher redshifts . in particular , it may be possible to probe the range of kev that may otherwise be difficult to access . future observations of the 21 cm forest by the square kilometer array may offer a unique and valuable probe of the ssps , as long as radio sources such as quasars or population iii gamma - ray bursts with sufficient brightness and number exist at redshifts of 10 - 20 , and the astrophysical effects of reionization and heating can be discriminated .

1004.4916

r

to survey both low and intermediate mass core collapse supernovae , neutrino emission histories were taken from two core collapse calculations , one from a @xmath69 @xcite supernova calculation and the other from a @xmath70 @xcite supernova calculation . since the pnss studied have significantly different masses and neutrino emission characteristics , one is able to get a rough picture of how integrated nucleosynthesis in the ndw varies with progenitor mass . , the dashed line corresponds to @xmath71 , the dot - dashed line corresponds to @xmath72 . ] the first set of neutrino luminosities and temperatures are taken from @xcite . this calculation began with a @xmath69 progenitor meant to model the progenitor of 1987a @xcite . the resulting neutron star had a gravitational mass of @xmath0 and the neutrino sphere was taken to be at 10 km . the neutrino luminosities and average energies as a function of time from this model are shown in figure [ fig : woos94_neut ] . after about 4 seconds , the neutrino energies become constant and the large difference between the electron neutrino and anti - neutrino energies implies that the wind will be neutron rich . this supernova model had some numerical deficiencies ( sam dalhed , private communication ) . the entropy calculated for the wind in @xcite ( @xmath73 ) were unrealistically large due to some problems with the equation of state . here , that is not so important because the ndw is being calculated separately , but this study did rely on older neutrino interaction rates and did not include weak magnetism corrections ( see [ sec : neutrino_rates ] ) . therefore , the results obtained using these neutrino histories are only suggestive of what might happen in a more massive star . if weak magnetism were taken into account , the calculated electron and anti - electron neutrino temperatures would probably be somewhat further apart . ( solid line ) and the radial velocity in units of @xmath74 ( dot - dashed line ) . the middle panel shows the net energy deposition rate from weak and strong interactions in units of @xmath75 ( dot - dashed line ) and the entropy ( solid line ) . the bottom panel shows the temperature in units of @xmath76 ( solid line ) , the electron fraction ( dot - dahsed line ) , and the fraction of material contained in nuclei ( dotted line ) . ] the calculation was run for a total of 18 seconds . during this time , the mass loss rate decreased by almost three orders of magnitude while a total mass of @xmath77 was lost in the wind . a snapshot of the wind structure two seconds after bounce is shown in figure [ fig : wnd_struct ] . note that the wind velocity stays very sub - luminal throughout the calculation . therefore , the neglect of special relativistic effects is reasonable . the secondary bump in the energy deposition rate occurs at the same radius where nucleons and alpha - particles assemble into heavy nuclei . this increases the entropy by about 10 units . clearly , the electron fraction is set interior to were nuclei form . the radius where nuclei form is at a large enough value that the alpha effect @xcite is not significant at early times in the wind . however , as the neutrino luminosity decreases with time , nucleon recombination occurs at a smaller radius , and the alpha effect becomes increasingly important . the time evolution of the wind as calculated by kepler is shown in figure [ fig : woos94_wndprop ] . the increase in asymptotic entropy is mainly driven by the decrease in neutrino luminosity , since the average neutrino energies do not vary greatly . the analytic approximation ( calculated using equation [ eq : ent ] and the neutrino interaction rates given in [ sec : neutrino_rates ] ) to the entropy tracks the entropy calculated in kepler fairly well . this implies that the variation in the neutrino luminosity with time does not significantly alter the dynamics from a steady state wind . in contrast to the high entropies reported in @xcite , the entropy here never exceeds 130 . for the time scales and electron fractions also obtained , such a low value of entropy is not sufficient to give a strong r - process ( see below ) . the electron neutrino and anti - neutrino energies do move further apart as a function of time though , which causes the wind to evolve from proton - rich conditions at early times to neutron - rich conditions later . a transition occurs from the synthesis of proton - rich isotopes via the @xmath78-process at early times to the @xmath79process mediated by the reaction sequence @xmath24(@xmath24n,@xmath28)@xmath29be(@xmath24,n)@xmath25c later . the slight difference between the analytic approximation and the kepler calculation of @xmath6 is due to the alpha effect @xcite . integrated production factors for the wind are shown in figure [ fig : wwsolopf ] . the production factor for the species @xmath80 is defined as @xmath81 where @xmath82 is the mass fraction of species @xmath80 in the wind after all material has decayed to stable isotopes , @xmath83 is the mass ejected in the wind , and @xmath84 is the amount of mass ejected by the entire supernova . @xmath85 is the mass fraction of isotope @xmath80 in the sun for which the values of @xcite were used . the only isotopes that are co - produced in the wind alone are @xmath1rb , @xmath2sr , @xmath3y , and @xmath4zr , with production factor of @xmath2sr about a factor of 3 higher than the other two n = 50 closed shell isotopes . before eight seconds , the production factors had been much closer . after eight seconds though , the wind is dominated by @xmath2sr because @xmath86 and only 53% of alpha particles are free after freeze out which puts @xmath87 of the heavy nuclei just below the range given in equation [ eq : n50_prdod ] . there are not enough free neutrons to make any significant amount of heavier nuclei , and this results in significant production of the stable n = 50 closed shell isotope with the lowest @xmath88 . of material was ejected in the supernova in addition to the wind . the top dashed line corresponds to the greatest production factor in the wind , the solid line is a factor of two below that , and the bottom dashed line is a factor of two below the solid line . these lines specify an approximate coproduction band for the wind alone . ] . the red line is the neutrino temperatures from @xcite . ] . the production factors are calculated assuming that @xmath89 of material was ejected in the supernova in addition to the wind . the horizontal lines are similar to those in figure [ fig : wwsolopf ] . ] of material was ejected in the supernova in addition to the wind . the horizontal lines are similar to those in figure [ fig : wwsolopf ] . ] of material was ejected in the supernova in addition to the wind . the horizontal lines are similar to those in figure [ fig : wwsolopf ] . ] during the first four seconds , the wind is proton rich and the isotopes @xmath90ga , @xmath91ge , @xmath92se , and @xmath93kr are produced by proton captures on seed nuclei produced by the triple - alpha reaction and subsequent ( @xmath24,p ) reactions . although the mass loss rate is much higher when the wind is proton rich , the alpha - fraction freezes out at 98% of its initial value , which results in significantly decreased production of heavy nuclei . the difference in final alpha fraction between the neutron- and proton - rich phases of the wind is due mainly to the difference in speed of the reaction chains @xmath24(@xmath27,@xmath28)@xmath25c and @xmath24(@xmath24n,@xmath28)@xmath29be(@xmath24,n)@xmath25c , but also to the decreased entropy at early times . we can compare this with the analytic predictions for nucleosynthesis by plotting the neutrino temperature evolution from this model on a neutrino `` two - color plot '' ( figure [ fig : tcp_woos ] ) . here we have set @xmath94 which is approximately correct at late times in the calculation of @xcite . the wind never reaches a region in which r - process nucleosynthesis is expected , but spends a significant amount of time making nuclei in the n = 50 closed shell isotones . since the neutrino temperatures from the original model were uncertain , several other models were calculated . one had a reduced ( by @xmath95 ) electron antineutrino temperature ; another had the weak magnetism corrections to the neutrino interaction rates turned off . a smaller antineutrino temperature is more in line with recent calculations of pns cooling @xcite . because the model of @xcite did not include weak magnetism corrections , our model with weak magnetism corrections turned off is more consistent with the original supernova model . the production factors for the model with a reduced electron antineutrino temperature are shown in figure [ fig : wwredpf ] . the yield of @xmath2sr is reduced by almost a factor of ten from the base case , while the production factors of @xmath3y and @xmath4zr are reduced by a factor of three . in this case , the wind also produces the proton - rich isotopes @xmath35se , @xmath36kr , and @xmath37sr . the coproduction line for lighter elements like oxygen in a @xmath69 supernova at solar metallicity is around 18 , so the wind could contribute to the total nucleosynthesis if the antineutrino temperature was reduced , but its contribution would be small . the yields when weak magnetism corrections are ignored are shown in figure [ fig : wwnowmpf ] . without weak magnetism , the electron fraction drops below 0.4 at late times when the entropy is fairly high . equation [ eq : n50_prdod ] is no longer satisfied and material moves past the n = 50 closed shell towards a @xmath96 110 . some r - process isotopes are produced , such as @xmath97zr and @xmath98mo , but not anywhere near solar ratios , and no material reaches the first r - process peak . to investigate the possible effect of a wind termination shock on nucleosynthesis , another model was run with a boundary pressure and temperature determined by equation [ eq : pbound ] . an explosion energy of @xmath99 was assumed and the shock velocity was taken as @xmath100 . this resulted in a wind termination shock that was always at a radius greater than @xmath101 . the production factors from this model are shown in figure [ fig : wwpbd ] . similar to the simulation without a wind termination shock , the n=50 closed shell elements dominate the wind s nucleosynthesis . the main difference between the case with and without a wind termination shock is a shift in the mass of isotopes produced during the proton - rich phase . this can be seen in the increased production of mo . during this phase , the post shock temperature varied from 2.5 gk down to 0.8 gk and the density varied from @xmath102 to @xmath103 . these conditions are very favorable for continued proton capture once the long lived waiting point isotopes @xmath104ni and @xmath105ge are bypassed by ( n , p ) reactions . because these conditions persist for at least a second after a fluid element passes through the wind termination shock , significantly more proton captures can occur on seed nuclei that have moved past mass @xmath106 relative to the case with no termination shock . still , not many more neutrons are produced per seed nucleus relative to the base run . therefore , the net number of seeds that get past the long lived waiting points remains small and the proton - rich wind does not contribute to the integrated nucleosynthesis . it should also be noted that a different treatment of the wind s interaction with the supernova shock might result in a breeze solution which may supply more favorable conditions for @xmath23p - process nucleosynthesis @xcite . in figure [ fig : comb_pf ] , the production factors from a @xmath107 supernova model from @xcite have been combined with the production factors we calculated in the ndw with the unaltered neutrino histories of @xcite with weak magnetism corrections included . the wind could be responsible for synthesizing the isotopes @xmath1rb , @xmath2sr , @xmath3y , and @xmath4zr . @xmath2sr production is above the co - production band , but the rest are in agreement with the stellar yields . this overproduction of @xmath2sr is similar to the result of @xcite . for the model with a reduced anti - electron neutrino temperature combined with the yields from the @xmath69 supernova model , the wind contributes 28% , 42% , 35% , 75% , 75% , and 80% of the total @xmath35se , @xmath36kr , @xmath37sr , @xmath2sr , @xmath3y , and @xmath4zr abundances in the supernova , respectively . this wind model does not result in any isotopes being overproduced relative to the rest of the yields of the supernova . for the case with weak magnetism turned off , the nuclei produced by the wind are overproduced relative to those made in the rest of the star by factor of nearly 100 , hence this would need to be a very rare event if this model were realistic . clearly , weak magnetism corrections and variations in the neutrino temperatures have a very significant effect on nucleosynthesis in the wind . aside from the effects of an extra source of energy ( [ modifications ] ) , the neutrino spectra are the largest current theoretical uncertainty in models of the ndw . . ] the second pns model is a more modern one - dimensional calculation of an electron - capture supernova @xcite that started from an @xmath70 progenitor model @xcite . this resulted in a pns with a gravitational mass of @xmath5 and a radius of 15 km . together the lower mass and increased radius imply a lower gravitational potential at the neutrinosphere . this work employed neutrino interaction rates which took weak magnetism and `` in - medium '' effects into account . the neutrino luminosities and average energies as a function of time are shown in figure [ fig : jnk_neut ] . the maximum difference between the electron and anti - electron neutrino average energies is significantly less than in the model of @xcite . this is likely due in part to both the decreased gravitational potential of the pns and the more accurate neutrino interaction rates in the newer model . the calculation was run for a total of nine seconds , at which point the mass loss rate had dropped by two orders of magnitude . the total amount of mass ejected in the wind was @xmath108 . in figure [ fig : jnk_wndprop ] , the properties of the ndw calculated using kepler are plotted as a function of time . notice that the entropy never reaches above 100 in this model , which diminishes the likelihood of significant nucleosynthesis . for comparison , we also include the analytic estimates detailed above . there is reasonable agreement between the analytic and the numerical calculations , but not nearly as good as in the @xmath69 model . . ] in contrast to the simulation run with the neutrino luminosities of @xcite , the electron fraction continues to increase with time . the difference between the average electron neutrino energy and electron anti - neutrino energy is , at most , about 3 mev , compared to a maximum of 8 mev in the @xcite calculations . also , the difference between the average neutrino energies decreases as a function of time , compared to an increase with time in @xcite . finally , the energies of all kinds of neutrinos are lower in the @xcite calculation , so that the proton - neutron rest mass difference significantly suppresses the anti - neutrino capture rate relative to the neutrino capture rate . these differences are presumably due to both the different neutron star masses and neutrino interaction rates employed . of material was ejected in the supernova in addition to the wind . the horizontal lines are similar to those in figure [ fig : wwsolopf ] . notice that none of the production factors are significantly greater than one . ] the conditions in this model thus preclude _ any _ r - process nucleosynthesis , but they are potentially favorable for production of some low mass p - process isotopes by the @xmath23p - process . the integrated isotopic production factors are shown in figure [ fig : jnkpf ] . the total ejected mass was take as 7.4 @xmath60 , as 1.4 @xmath60 neutron star is left behind in the calculation of @xcite . during the calculation a maximum network size of 988 isotopes is reached . the p - process elements @xmath35se and @xmath36kr are co - produced with @xmath109cu , @xmath110zn , and @xmath90ga , but the maximum production factor for any isotope is 1 when weighted with the total mass ejected in the supernova . therefore , in this simple model , the proton - rich wind from low mass neutron stars will not contribute significantly to galactic chemical evolution . the entropies encountered when the mass loss rate is high are low ( @xmath111 ) , so that there is more production of @xmath104ni by triple - alpha and a subsequent @xmath24p - process . as the neutron abundance available for the @xmath23p - process is given by @xmath112 increased seed production reduces the available neutron abundance and therefore hinders production of the p - process elements @xmath35se , @xmath36kr , @xmath37sr , and @xmath38mo . additionally , at early times , the dynamical time scale is short which implies a smaller integrated neutron to seed ratio , @xmath113 ( see the appendix ) . the yields of from this model can not be combined with the yields from the rest of the supernova because they are not published . as @xcite has discussed , the mass inside the helium burning shell was close to the mass of the neutron star that was left after the explosion . therefore the ejecta of the supernova is expected to have small production factors . this implies that , even when the yields of the ndw are combined with the rest of the supernova , it is unlikely that these low mass core collapse supernovae will contribute significantly to galactic chemical evolution . of material was ejected in the supernova in addition to the wind . the horizontal lines are similar to those in figure [ fig : wwsolopf ] . notice that the production factors are almost unchanged when an external boundary pressure is added . ] as was mentioned above , it is very possible that a transonic wind solution may not be appropriate this early in the supernovas evolution . @xcite have found that a wind termination shock is not present in a one - dimensional supernova model using the progenitor from @xcite . still , it is interesting to consider the effect of a reverse shock on the wind nucleosynthesis . a second simulation was run with a time dependent boundary pressure given by equation [ eq : pbound ] , with @xmath114 and @xmath115 . this results in a wind termination shock at a radius of approximately @xmath116 throughout the simulation . inside the wind termination shock the wind dynamics are very similar to those in the run with no boundary pressure . the production factors from this model are shown in figure [ fig : jnkpfpbd ] . clearly , there is almost no difference in the nucleosynthesis in the runs with and without a wind termination shock . after @xmath117 , the post shock temperature drops below 1 gk and the wind termination shock has little effect on subsequent nucleosynthesis . because the post shock temperature is high for less than one second , the wind termination shock has very little effect on the integrated nucleosynthesis . a larger explosion energy would likely result in a larger effect on the nucleosynthesis , but there are still very few neutrons available to bypass the long lived waiting points and it seems unlikely that the production factors would be increased by more than a factor of a few .

although they are but a small fraction of the mass ejected in core - collapse supernovae , neutrino - driven winds ( ndws ) from nascent proto - neutron stars ( pnss ) have the potential to contribute significantly to supernova nucleosynthesis . in previous works , , it contributes to the production of the n = 50 closed shell elements and some light p - nuclei . in doing so

although they are but a small fraction of the mass ejected in core - collapse supernovae , neutrino - driven winds ( ndws ) from nascent proto - neutron stars ( pnss ) have the potential to contribute significantly to supernova nucleosynthesis . in previous works , the ndw has been implicated as a possible source of r - process and light p - process isotopes . in this paper we present time - dependent hydrodynamic calculations of nucleosynthesis in the ndw which include accurate weak interaction physics coupled to a full nuclear reaction network . using two published models of pns neutrino luminosities , we predict the contribution of the ndw to the integrated nucleosynthetic yield of the entire supernova . for the neutrino luminosity histories considered , no true r - process occurs in the most basic scenario . the wind driven from an older model for a pns is moderately neutron - rich at late times however , and producesrb ,sr ,y , andzr in near solar proportions relative to oxygen . the wind from a more recently studied pns is proton - rich throughout its entire evolution and does not contribute significantly to the abundance of any element . it thus seems very unlikely that the simplest model of the ndw can produce the r - process . at most , it contributes to the production of the n = 50 closed shell elements and some light p - nuclei . in doing so , it may have left a distinctive signature on the abundances in metal poor stars , but the results are sensitive to both uncertain models for the explosion and the masses of the neutron stars involved .

1004.4916

c

we have performed calculations of the dynamics and nucleosynthesis in time dependent neutrino driven winds . this was done for two sets of neutrino spectra calculated in one - dimensional supernova models taken from the literature . the nucleosynthesis in these models was compared with supernova yields to determine if these models were consistent with observations . additionally , we compared the results of these numerical models to analytic models of the neutrino driven wind and found good agreement . similar to most of the work on the ndw after @xcite , we find that it is unlikely that the r - process occurs in the neutrino driven wind unless there is something that causes significant deviation from a purely neutrino driven wind . additionally , in the simplest case , there is little production of p - process elements at early times in the wind . in our calculation that used spectra from a more massive neutron star , the wind only produces the n=50 closed shell elements @xmath1rb , @xmath2sr , @xmath3y , and @xmath4zr . this result is sensitive to small changes in the neutrino interaction rates ( i.e. the inclusion of weak magnetism ) and changes to the neutrino temperature of order 10% . comparing our models with the over abundance of strontium seen in sn 1987a suggests that the difference between the electron and anti - electron neutrino temperatures in the model of @xcite may have been to large . we also find that the effect of a wind termination shock on the wind nucleosynthesis is small . using neutrino spectra from an @xmath70 supernova that drives a wind which is proton rich throughout its duration @xcite , we find that no significant @xmath23p - process occurs and the wind does not contribute to the yields of the supernova . the neutrino spectra from this model are probably more accurate than the spectra from the model of @xcite . we also investigated the effect of an outer boundary pressure which resulted in a wind termination shock . this had a negligible effect on the nucleosynthesis . however , one also expects that the nucleosynthesis in the ndw will vary considerably from event to event , especially with the mass and possibly the rotation rate of the pns . the winds from more massive pns have greater entropy and might , in general , be expected to produce heavier elements and more of them . the neutrino spectral histories of pns as a function of mass have yet to be determined over a wide range of parameter space . currently , the neutrino luminosities and temperatures are the largest uncertainties in models of the ndw . we would like to thank alex heger , david lai , enrico ramirez - ruiz , sanjay reddy , and yong - zhong qian for useful discussions about issues relating to this work . l. r. was supported by an nnsa / doe stewardship science graduate fellowship ( de - fc52 - 08na28752 ) and the university of california office of the president ( 09-ir-07 - 117968-woos ) . s. w. was supported by the us nsf ( ast-0909129 ) , the university of california office of the president ( 09-ir-07 - 117968-woos ) , and the doe scidac program ( defc-02 - 06er41438 ) . r. h. was supported by the doe scidac program ( defc-02 - 06er41438 ) and under the auspices of the department of energy at lawrence livermore national laboratory under contract de - ac52 - 07na27344 .

the wind driven from an older model for a pns is moderately neutron - rich at late times however , and producesrb ,sr ,y , andzr in near solar proportions relative to oxygen . the wind from a more recently studied pns is proton - rich throughout its entire evolution and does not contribute significantly to the abundance of any element .

although they are but a small fraction of the mass ejected in core - collapse supernovae , neutrino - driven winds ( ndws ) from nascent proto - neutron stars ( pnss ) have the potential to contribute significantly to supernova nucleosynthesis . in previous works , the ndw has been implicated as a possible source of r - process and light p - process isotopes . in this paper we present time - dependent hydrodynamic calculations of nucleosynthesis in the ndw which include accurate weak interaction physics coupled to a full nuclear reaction network . using two published models of pns neutrino luminosities , we predict the contribution of the ndw to the integrated nucleosynthetic yield of the entire supernova . for the neutrino luminosity histories considered , no true r - process occurs in the most basic scenario . the wind driven from an older model for a pns is moderately neutron - rich at late times however , and producesrb ,sr ,y , andzr in near solar proportions relative to oxygen . the wind from a more recently studied pns is proton - rich throughout its entire evolution and does not contribute significantly to the abundance of any element . it thus seems very unlikely that the simplest model of the ndw can produce the r - process . at most , it contributes to the production of the n = 50 closed shell elements and some light p - nuclei . in doing so , it may have left a distinctive signature on the abundances in metal poor stars , but the results are sensitive to both uncertain models for the explosion and the masses of the neutron stars involved .

1612.09511

i

weak gravitational field of a spatially compact source is identified as a perturbation about a background space - time which is a solution of the einstein equation in the source free region . in the presence of a positive cosmological constant , the background space - time is the de sitter space - time . unlike the minkowski background for the zero cosmological constant , de sitter space - time has different patches eg the global patch , a poincare patch and a static patch . in the cosmological context , a poincare patch is appropriate which is what we focus on . a solution at the _ linearized level _ , valid throughout the poincare patch and extending to the future null infinity @xmath2 , is available in @xcite . however the space - like character of the @xmath0 poses challenges for defining energy , momentum and their fluxes . let us recall that the cleanest articulation of ` infinity ' arises in the conformal completion of physical space - times . conformal completion preserves the light cone structure of the physical space - time and captures the notion of approaching infinity along many directions as well as with different speeds . it naturally identifies boundary components , @xmath3 where time - like and null geodesics ` terminate ' . the causal nature of these boundary components is determined by the asymptotic form of ` source - free ' equations : @xmath4 are null when @xmath5 and space - like for @xmath6 ( time - like for @xmath7 ) . these boundary components serve to define out - going ( in - coming ) fields as those solutions of the asymptotic equations that have suitably finite limiting values on @xmath8 . it is then a result that the weyl tensor of out - going fields evaluated along out - going null geodesics , has a definite pattern of fall - off in inverse powers of an affine parameter along the geodesics ( the peeling - off theorem ) @xcite . this enables one to identify the leading term as representing gravitational radiation ( far field of a source ) , in a coordinate invariant manner . it is conveniently described in terms of the weyl scalars which are defined with respect to a suitable null tetrad . when @xmath2 is null , a null tetrad at a point @xmath9 is uniquely determined ( modulo real scaling and rotation ) by the tangent vector @xmath10 of an outgoing null geodesic reaching @xmath11 , _ and _ the null normal , say @xmath12 at @xmath11 normalized so that with @xmath13 . clearly as the null geodesic changes its direction , @xmath14 changes but not @xmath15 and hence the weyl scalar @xmath16 remains unchanged . its non - zero value can be taken as showing the presence of gravitational radiation . this feature is lost when the @xmath2 is space - like . now the null vector @xmath12 , with @xmath13 , is chosen to be in the plane defined by @xmath10 and the ( time - like ) normal @xmath17 . clearly , as @xmath10 changes , so does @xmath12 and then _ no _ weyl scalar is invariant . an invariant characterization of gravitational radiation is no longer available @xcite . to test if a notion of radiation based on asymptotic behaviour is physically useful , one still needs suitable definitions of flux of energy - momentum , angular momentum in terms of the fields which will then correlate physical quantities being radiated away from a spatially compact source . fortunately , in the poincare patch , there are seven globally defined killing vectors , all space - like in the vicinity of @xmath0 , and corresponding conserved quantities can be expected . one of the definition of such conserved quantities is based on the covariant phase space framework @xcite . in the context of the linearised theory , it exploits the phase space structure of the space of solutions and defines a manifestly gauge invariant and conserved ` hamiltonian ' corresponding to every _ isometry _ of the background space - time . although defined on each space - like hypersurface of the poincare patch , the simplest expressions results for evaluation at @xmath2 . being defined as an integral over a hypersurface , it is also referred to as a _ flux_. thus we see that the null infinity provides a characterization of the radiation ( up to origin dependence for @xmath18 ) which can transport energy - momentum - angular momentum away from a spatially compact gravitating source . one other implication of this characterization at infinity is that the energy / momentum / angular momentum which _ is _ received at the future null infinity is _ necessarily lost _ from the source under the assumption of _ no incoming radiation _ boundary condition . it is this last implication that follows also for the cosmological horizon of a spatially compact source : energy / momentum / angular momentum that crosses the cosmological horizon is also lost forever from the source . note that the worldlines of different components of a spatially extended ( but compact ) source must reach the same point on @xmath0 to maintain a _ finite _ physical separation among them . it follows that that the cosmological horizon for a spatially compact source is well defined : it is the past light cone of the common point on @xmath0 where the source world tube converges . equally well , _ any observer _ who remains at a finite physical distance from the compact source for all times , must necessarily lie within the cosmological horizon i.e. within the static patch . we would like to explore to what extent and under what conditions may we regard the cosmological horizon as a `` substitute '' for the _ future null infinity_. it is obvious at the outset that the out - going null geodesics emanating from the source intersect the cosmological horizon at a _ finite _ value of any affine parameter and it can be chosen to be 1 by a suitable normalization . such a normalized affine parameter equals the ratio of the physical distance from the source to @xmath19 . all spatially compact sources may be taken to be within a sphere of radius @xmath20 . furthermore , only sources varying over cosmological time scales , will have comparable wavelengths . in other words , most sources producing gravitational waves would have wavelengths far smaller than @xmath21 and any wave crossing the horizon may be taken to be a ` far zone field ' . cosmological horizon being a null hypersurface , we do have the weyl scalar @xmath22 defined on it , which is independent of the _ direction _ of null geodesics meeting the horizon . for sources which are sufficiently rapidly varying ( relative to the scale set by the cosmological constant ) , there is an alternative identification of gravitational waves as _ ripples on a background _ within the so called _ short wave approximation _ @xcite . furthermore , it is possible to define an _ effective gravitational stress tensor _ , @xmath23 for the ripples . for vanishing @xmath24 , it is gauge invariant , conserved and symmetric . for non - zero @xmath24 it is _ not _ gauge invariant but the gauge violations are suppressed by powers of @xmath25 . it is very convenient to have such a stress tensor to define and compute fluxes of energy and momenta carried by the ripples across any hypersurface . we will use this to show that for the retarded solution given in @xcite , the flux of energy - momentum across the cosmological horizon exactly equals the corresponding flux across the @xmath0 and also equals the flux computed in @xcite . like - wise , the instantaneous power received at infinity matches with that crossing the horizon . this is our main result . the paper is organised as follow . in section [ solution ] , we recall the solution at the linearised level for which the fluxes will be evaluated . section [ energy ] is divided into five subsections . in subsection [ covariant ] we summarise the covariant phase space framework and in subsection [ effectivestresstensor ] , we discuss the isaacson prescription adapted to the presence of the cosmological constant . since this constructs a symmetric , conserved and gauge invariant ( to _ leading order _ ) stress tensor , we get conserved currents for each killing vector of the background space - time . in the subsection [ fluxcomputations ] , we present computations of the energy flux across various hypersurfaces . in particular we show that the fluxes across the out - going null hypersurfaces are zero , implying for example , that the energy propagation is sharp . subsection [ momentumfluxes ] contains the fluxes for the momentum and the angular momentum . in the subsection [ cosmologicalhorizon ] , we discuss its applications and show that the quadrupole power can be computed at the cosmological horizon . the final section [ finalsection ] concludes with a discussion . 0.3 cm

an important class of observables for gravitational waves consists of the fluxes of energy , momentum and angular momentum carried away by them and are well understood for weak gravitational waves in minkowski background . in de sitter background , the future null infinity , , is space - like which makes the meaning of these observables subtle . a spatially compact source in de sitter background also provides a distinguished null hypersurface , its _ cosmological horizon _ , . for sources supporting the short wavelength approximation , we adopt the isaacson prescription to define an effective gravitational stress tensor .

an important class of observables for gravitational waves consists of the fluxes of energy , momentum and angular momentum carried away by them and are well understood for weak gravitational waves in minkowski background . in de sitter background , the future null infinity , , is space - like which makes the meaning of these observables subtle . a spatially compact source in de sitter background also provides a distinguished null hypersurface , its _ cosmological horizon _ , . for sources supporting the short wavelength approximation , we adopt the isaacson prescription to define an effective gravitational stress tensor . we show that the fluxes computed using this effective stress tensor can be evaluated at and match with those given by ashtekar et al at .

1603.00113

c

directed self - assembly of charged particles along line segments has been considered . in the assembly process , a number of particles move in one dimension along a line segment under the repulsive forces experienced from interactions with other particles , and the process is directed towards formation of a desired pattern by external forces applied from charged electrodes located at fixed points in the line segment . the potentials of these electrodes are precisely controlled in time so that the formation of a desired pattern is secured with the highest probability despite the inherent uncertainty in the initial position and the dynamical behaviors of the particles . a challenging aspect of such a control is that the actual positions of the particles are not measurable during the assembly process . two models have been proposed to describe the uncertain dynamics of the particles . the first model which is mathematically more tractable consists of a set of nonlinear stochastic differential equations and is suitable for larger particles of micrometer scale . the second model is a discrete ising model consisting of a continuous - time markov chain and is more accurate for nanometer scale particles . a class of piecewise constant controls has been proposed for these models and the optimal values of the constant pieces have been determined as the solutions to certain optimization problems . consider @xmath126 with @xmath119 . the goal is to show that the algebraic equation @xmath352 has exactly one solution @xmath93 in @xmath126 and that the jacobian matrix of the vector field @xmath353 is negative definite over the set @xmath126 . to that end , denote the @xmath18th component of @xmath354 by @xmath355 and let @xmath356 be the @xmath319th component of @xmath94 , where by hypothesis @xmath357 . then using , @xmath355 can be written as @xmath358 where @xmath359 denotes the signum function . further , for @xmath360 , the partial derivatives of @xmath361 with respect to @xmath362 and @xmath363 , @xmath364 exist and are given by given @xmath360 , for each @xmath68 , let @xmath366 denote the smallest integer satisfying @xmath367 and define @xmath368 fix all components of @xmath360 except for @xmath362 which is allowed to vary in the segment @xmath369 . the negative derivative implies that @xmath361 is strictly decreasing in @xmath370 with all other variables fixed . in addition , implies that this function tends to @xmath198 and @xmath371 as @xmath362 tends to @xmath372 and @xmath373 , respectively . thus , it is concluded that @xmath374 has one and only one solution for @xmath362 in the interval @xmath370 , with all other variables fixed . this solution depends on @xmath375 and is denoted by @xmath376 let @xmath377 be a vector - valued function with the components @xmath378 , @xmath68 . then , @xmath93 is a solution to , if and only if it is a fixed point of the mapping @xmath379 , i.e. , if it solves @xmath380 it is proven next that has exactly one solution in @xmath126 by showing that @xmath379 is a contraction map on @xmath126 , and thereby it has exactly one fixed point in this set . the gradient @xmath381 of the scalar function @xmath378 is obtained as follows . based on the definition of @xmath378 , the identity @xmath382 holds true for any @xmath383 , where the components of @xmath384 are similar to @xmath122 except for @xmath385 which is equal to @xmath386 . differentiating this identity with respect to @xmath387 leads to @xmath388 solving this equation for @xmath389 results in @xmath390 since @xmath378 does not depend on @xmath362 , its partial derivative with respect to @xmath362 is identically @xmath57 so that @xmath391 substituting the explicit expressions into and noting that @xmath357 , it is straightforward to verify that @xmath392 let @xmath393 and consider the line segment @xmath394.\ ] ] since @xmath126 is a convex set , all points on this line segment are inside the set . applying the mean value theorem @xcite to the scalar function @xmath395 implies that there exists @xmath396 such that @xmath397 substituting @xmath398 , @xmath399 , and @xmath400 into this equality and taking absolute values of its sides lead to @xmath401 because this inequality holds for every @xmath68 , it is concluded that @xmath402 where @xmath403 this verifies that @xmath379 is a contraction map . finally , it is shown that @xmath99 is strictly convex on @xmath126 , and thereby the solution @xmath93 to is a stable equilibrium . the gergorin circle theorem @xcite applied to the jacobian matrix of the vector field @xmath353 implies that the eigenvalues @xmath404 of this matrix are inside the circles of the form @xmath405 since @xmath406 according to and the right - hand side of the inequality is smaller than @xmath407 , the eigenvalues of the jacobian matrix have negative values . s. ouk kim , h. h. solak , m. p. stoykovich , n. j. ferrier , j. j. de pablo , and p. f. nealey , `` epitaxial self - assembly of block copolymers on lithographically defined nanopatterned substrates , '' _ nature _ , vol . 424 , no . 6947 , pp . 411414 , 2003 . j. c. love , l. a. estroff , j. k. kriebel , r. g. nuzzo , and g. m. whitesides , `` self - assembled monolayers of thiolates on metals as a form of nanotechnology , '' _ chem . _ , vol . 105 , no . 4 , pp . 11031170 , 2005 . a. khaled , s. guo , f. li , and p. guo , `` controllable self - assembly of nanoparticles for specific delivery of multiple therapeutic molecules to cancer cells using rna nanotechnology , '' _ nano letters _ , vol . 5 , no . 9 , pp . 17971808 , 2005 . l. adleman , q. cheng , a. goel , m. huang , and h. wasserman , `` linear self - assemblies : equilibria , entropy , and convergence rates , '' in _ proc . of the 6th international conference on difference equations and applications , icdea 2001 _ , ( augsburg , germany ) , jun . 2001 . n. stephanopoulos , e. o. p. solis , and g. stephanopoulos , `` nanoscale process systems engineering : toward molecular factories , synthetic cells , and adaptive devices , '' _ aiche journal _ , vol . 51 , no . 7 , pp . 18581869 , 2005 . a. winkleman , b. d. gates , l. s. mccarty , and g. m. whitesides , `` directed self - assembly of spherical particles on patterned electrodes by an applied electric field , '' _ advanced materials _ , vol . 17 , no . 12 , pp . 15071511 , 2005 . m. p. stoykovich , h. kang , k. c. daoulas , g. liu , c .- c . liu , j. j. de pablo , m. mller , and p. f. nealey , `` directed self - assembly of block copolymers for nanolithography : fabrication of isolated features and essential integrated circuit geometries , '' _ acs nano _ , vol . 1 , no . 3 , pp . 168175 , 2007 . c. h. lalander , y. zheng , s. dhuey , s. cabrini , and u. bach , `` dna - directed self - assembly of gold nanoparticles onto nanopatterned surfaces : controlled placement of individual nanoparticles into regular arrays , '' _ acs nano _ , vol . 4 , no . 10 , pp . 61536161 , 2010 . e. o. p. solis , p. i. barton , and g. stephanopoulos , `` controlled formation of nanostructures with desired geometries . 1 . robust static structures , '' _ ind . eng . _ , vol . 49 , no . 17 , pp . 77287745 , 2010 . e. o. p. solis , p. i. barton , and g. stephanopoulos , `` controlled formation of nanostructures with desired geometries . 2 . robust dynamic paths , '' _ ind . _ , vol . 49 , no . 17 , pp . 77467757 , 2010 . roux , `` brownian particles at different times scales : a new derivation of the smoluchowski equation , '' _ physica a : statistical mechanics and its applications _ , vol . 188 , no . 4 , pp . 526552 , 1992 . d. liberzon and r. w. brockett , `` spectral analysis of fokker - planck and related operators arising from linear stochastic differential equations , '' _ siam journal on control and optimization _ , vol . 38 , no . 5 , pp . 14531467 , 2000 . p. kesavan , r. j. allgor , e. p. gatzke , and p. i. barton , `` outer approximation algorithms for separable nonconvex mixed - integer nonlinear programs , '' _ mathematical programming _ , 100 , no . 3 , pp . 517535 , 2004 . m. tawarmalani and n. v. sahinidis , _ convexification and global optimization in continuous and mixed - integer nonlinear programming : theory , algorithms , software , and applications_. dordrecht ; boston : kluwer academic publishers , 2002 .

an optimal control strategy is developed to construct nanostructures of desired geometry along line segments by means of directed self - assembly of charged particles . such a control strategy determines the electric potentials of a set of electrodes located at fixed points in the line segment . such an open - loop control optimally changes the electrode potentials in time in order to create a desired pattern with the highest probability , despite the inherent uncertainty in the initial positions and the dynamical behaviors of the particles . two models are proposed to describe the uncertain dynamics of the particles : a continuous model relying on a set of nonlinear stochastic differential equations , and a discrete ising model consisting of a large dimensional continuous - time markov chain . while the first model is more mathematically tractable , the second one more precisely describes particles at the nanometer scale .

an optimal control strategy is developed to construct nanostructures of desired geometry along line segments by means of directed self - assembly of charged particles . such a control strategy determines the electric potentials of a set of electrodes located at fixed points in the line segment . the particles move under the electric forces generated by these electrodes and by the interactions between the particles themselves to form a desired pattern eventually . due to technology limitations , the particle positions can not be measured during the course of control , so that the control is open - loop in nature . such an open - loop control optimally changes the electrode potentials in time in order to create a desired pattern with the highest probability , despite the inherent uncertainty in the initial positions and the dynamical behaviors of the particles . two models are proposed to describe the uncertain dynamics of the particles : a continuous model relying on a set of nonlinear stochastic differential equations , and a discrete ising model consisting of a large dimensional continuous - time markov chain . while the first model is more mathematically tractable , the second one more precisely describes particles at the nanometer scale . the control design procedure begins with the continuous model and identifies the structure of its stable equilibria , which is used later to propose a piecewise constant structure for the control and to demonstrate that the optimal value of each piece is independently obtained from a certain static optimization problem . it is shown next that the design procedure can be applied to the discrete model with only minor modifications . a numerical example of control design is presented . directed self - assembly , fokker - planck equation , ising model , nanostructure , optimal control .

nucl-ex0407002

i

experimental studies , carried out during last two decades ( for review see @xcite and for update of the recent progress see e.g. @xcite ) , demonstrate that , in nuclear reactions such as the nucleus - nucleus collisions in the fermi energy domain or the collisions of a high - energy light particle with a heavy target , the amount of kinetic energy dissipated into the internal excitation energy is large enough to produce highly excited ( hot ) nuclei which may further de - excite via multifragmentation , a decay mode where the hot nucleus disintegrates into many fragments . in the nature , nuclear processes of such type can be observed in the interaction of cosmic rays with terrestrial and interplanetary matter . such spallation reactions contribute to the observed isotopic composition of the interstellar matter which can be used to study the properties of the cosmic rays and of their astrophysical sources ( for review see e.g. @xcite ) . with the advent of powerful accelerators of high - energy particles and , more recently , of heavy ions of intermediate and high energy , it became possible to study the process of multifragmentation in the laboratory . extensive experimental studies of multifragmentation are conducted since eighties of the last century . as can be deduced from observed experimental data @xcite , during the process of multifragmentation the nuclear medium becomes hot , expands and is supposed to undergo a phase transition where two phases are created , a dense isospin - symmetric phase ( liquid ) and a dilute isospin - asymmetric phase ( vapor or gas ) . such a phase transition was predicted by theory ( for review see @xcite ) . the most common theoretical description of multifragmentation is based on the statistical model of multifragmentation @xcite , typically assuming a thermally equilibrated freeze - out configuration with hot ( excited ) or cold fragments . the probabilities of various fragment partitions can be evaluated analytically using the grand - canonical approach . while the statistical model of multifragmentation is quite successful in describing a wide range of fragment observables , it does not provide a specific information on the mechanism of the underlying process of fragment formation , apart from the assumption that at the freeze - out configuration a thermal equilibrium among the fragments is reached . the underlying process of cluster formation can be related to the equation of state of asymmetric nuclear matter at sub - saturation densities , where the phase transition is supposed to occur . the coexistence line , typical for the one - component system such as symmetric nuclear matter , develops with introduction of isospin asymmetry into a complex coexistence region @xcite , typically suggesting a coexistence of the dilute isospin - asymmetric phase and isospin - symmetric dense phase . further theoretical investigations @xcite suggest that instability modes responsible for the phase transition are typically a combinations of mechanical and chemical instability modes . in this article , we present an overview of the recent progress in the studies of nuclear multifragmentation . we place a special emphasis on the exploration of isotopic trends in nuclear multifragmentation and the possibilities to extract physical information related to the nuclear equation of state . experimental studies of isotopic trends in nuclear multifragmentation have been made possible by use of the state - of - the - art detection techniques , which allow to identify individual isotopes of light charged particles and intermediate mass fragments in wide angular ranges . the isotopic composition of fragments can be used to extract the values of thermodynamical observables of the system undergoing multifragmentation such as temperature and chemical potentials ( free nucleon densities ) . various methods for extraction of temperature from isotopic composition are analyzed . an exponential scaling of the yields of specific isotopes from different reactions with the isotope neutron and proton numbers ( isoscaling ) is used to study the evolution of the isospin degrees of freedom of the system . finally , the nuclear equation of state and the isospin - asymmetric liquid - gas phase transition in the nuclear matter are discussed .

an overview of the recent progress in the studies of nuclear multifragmentation is presented . special emphasis is put on the exploration of isotopic trends in nuclear multifragmentation and the possibilities to extract physical information related to the nuclear equation of state . the isotopic composition of fragments is used to extract the values of thermodynamical observables of the system undergoing multifragmentation such as temperature and chemical potentials . various methods for extraction of thermodynamical variables are analyzed . an exponential scaling of relative isotopic yields from reactions with different neutron content , called isoscaling , is used to explore the evolution of the isospin degrees of freedom of the system . finally , the nuclear equation of state and the isospin - asymmetric liquid - gas phase transition in the nuclear matter are discussed .

an overview of the recent progress in the studies of nuclear multifragmentation is presented . special emphasis is put on the exploration of isotopic trends in nuclear multifragmentation and the possibilities to extract physical information related to the nuclear equation of state . relevant experimental methods of isotope identification are described . the isotopic composition of fragments is used to extract the values of thermodynamical observables of the system undergoing multifragmentation such as temperature and chemical potentials . various methods for extraction of thermodynamical variables are analyzed . an overview of methods of isotope thermometry , exploring the sensitivity of various yield ratios to temperature , is presented . an exponential scaling of relative isotopic yields from reactions with different neutron content , called isoscaling , is used to explore the evolution of the isospin degrees of freedom of the system . finally , the nuclear equation of state and the isospin - asymmetric liquid - gas phase transition in the nuclear matter are discussed .

1511.04731

i

an _ rna sequence _ is a string composed of four types of nucleotides , namely @xmath0 , and @xmath1 . given an rna sequence , the goal of the _ rna folding _ problem is to find a maximum cardinality set of crossing - free pairs of nucleotides , where all the pairs are either @xmath2 or @xmath3 . the problem is central in bioinformatics and has found applications in many areas of molecular biology . for a more comprehensive exposition of the topic , the reader is referred to e.g. @xcite . it is well - known that the problem can be solved in cubic time using a simple dynamic programming method @xcite . due to the importance of rna folding in practice , there has been a long line of research on improving the cubic time algorithm ( see e.g. @xcite ) . currently the best upper bound is @xmath4 @xcite , and this can be obtained via four - russian method or fast min - plus multiplication ( based on ideas from valiant s cfg parser @xcite ) . whether the rna folding problem can be solved in @xmath6 time for some @xmath8 is still a major open problem . other than attempting to improve the upper bound , we should also approach the problem in the opposite direction , i.e. showing a lower bound or arguing why the problem is hard . a popular way to show hardness of a problem is to demonstrate a lower bound conditioned on some widely accepted hypothesis . [ strongly exponential time hypothesis ( seth ) ] [ c-1 ] there exists no @xmath9 such that @xmath7-sat with @xmath10 variables can be solved in time @xmath11 for all @xmath12 . [ c-2 ] there exists no @xmath13 such that @xmath7-clique on graphs with @xmath10 nodes can be solved in time @xmath14 for all @xmath12 , where @xmath15 is the matrix multiplication exponent . assuming that seth ( conjecture [ c-1 ] ) holds , the following bounds are unattainable for any @xmath8 : * an @xmath16 algorithm for @xmath7-dominating set problem @xcite , * an @xmath17 algorithm for dynamic time warping , longest common subsequence , and edit distance @xcite , * an @xmath18 algorithm for ( @xmath19)-approximating the diameter of a graph with @xmath20 edges @xcite . as remarked in @xcite , it is easy to reduce the longest common subsequence problem on binary strings to the rna folding problem as following : given two binary strings @xmath21 , we let @xmath22 be the string such that @xmath23 = a$ ] if @xmath24 = 0 $ ] , @xmath23 = c$ ] if @xmath24 = 1 $ ] , and we let @xmath25 be the string such that @xmath26 = u$ ] if @xmath27 = 0 $ ] , @xmath26 = g$ ] if @xmath27 = 1 $ ] . then we have a 1 - 1 correspondence between rna foldings of @xmath28 ( i.e. concatenation of @xmath29 and the reversal of @xmath30 ) and common subsequences of @xmath31 and @xmath32 . it has been shown in @xcite that there is no @xmath17 algorithm for longest common subsequence problem on binary strings conditioned on seth , and we immediately get the same conditional lower bound for rna folding from the simple reduction ! very recently , based on a conjectured hardness of @xmath7-clique problem ( conjecture [ c-2 ] ) , a higher conditional lower bound was proved for a generalized version of the rna folding problem ( which coincides with the rna folding problem when the alphabet size is 4 ) @xcite : [ @xcite ] [ thm-1 ] if the generalized rna folding problem on sequences of length @xmath10 with alphabet size 36 can be solved in @xmath33 time , then @xmath34-clique on graphs with @xmath35 can be solved in @xmath36 time . therefore , a @xmath37 time algorithm for the generalized rna folding with alphabet size at least 36 will disprove conjecture [ c-2 ] , yielding a breakthrough to the parameterized complexity of clique problem . however , the above theorem is irrelevant to the rna folding problem in real life ( which has alphabet size 4 ) . it is unknown whether the generalized rna folding for alphabet size @xmath38 admits a faster algorithm than the case for alphabet size @xmath39 . in fact , there are examples of string algorithms whose running time scales with alphabet size ( e.g. string matching with mismatched @xcite and jumbled indexing @xcite ) . we also note that when the alphabet size is 2 , the generalized rna folding can be trivially solved in linear time . in this paper , we improve upon theorem [ thm-1 ] by showing the same conditional lower bound for the rna folding problem : [ thm-2 ] if the rna folding problem on sequences in @xmath40 can be solved in @xmath33 time , then @xmath34-clique on graphs with @xmath35 can be solved in @xmath41 time . note that we also get an @xmath42 factor improvement inside @xmath43 , though it does not affect the conditional lower bound . the current state - of - art algorithm for @xmath44clique , which takes @xmath45 time , requires the use of fast matrix multiplication @xcite which does not perform very efficiently in practice . for combinatorial , non - algebraic algorithm for @xmath44clique , the current best one runs in @xmath46 time @xcite , which is only slightly better than the trivial approach . as a result , by theorem [ thm-2 ] , even a @xmath47 time combinatorial algorithm for rna folding will lead to an improvement for combinatorial algorithms for @xmath44clique ! in the proof of theorem [ thm-1 ] in @xcite , given a graph @xmath48 , a sequence of length @xmath49 is constructed in such a way that we can decide whether @xmath50 has a @xmath34-clique according to the number of pairs in an optimal generalized rna folding of @xmath51 . such a construction requires many different types of letters in order to build various `` walls '' which prevent undesired pairings between different parts of the sequence . hence extending their approach to handle the case where the alphabet size is 4 may not be easy without aid from other techniques and ideas . * overview of our approach . * at a high level , our reduction ( from @xmath34-clique problem to rna folding problem ) follows the approach in @xcite : we enumerate all @xmath7-cliques , and each of them is encoded as some gadgets . all the gadgets are then put together to form an rna sequence . the goal is to ensure that an optimal rna folding corresponds to choosing three @xmath7-cliques that form a @xmath34-clique , given that the underlying graph admits a @xmath34-clique . to achieve this goal without using extra types of letters that force the gadgets to match in a desired manner , we construct the gadgets via a key lemma in @xcite , whose original purpose is to prove that longest common subsequence and other edit distance problems are seth - hard even on binary strings . we will treat it as a black box and apply it multiple times during the construction . this powerful tool will allow us to test whether two @xmath7-cliques form a @xmath52-clique by the longest common subsequence between the two strings representing the two @xmath7-cliques . in the final rna sequence , all clique gadgets are well - separated by some carefully designed sequences whose purpose is to `` trap '' all the clique gadgets except three of them . since we know that these three clique gadgets are guaranteed to match well if the graph has a @xmath34-clique , we can infer whether the graph has a @xmath34-clique from the optimal rna folding of the rna sequence . * dyck edit distance . * one other way to formulate the rna folding problem is as follows : deleting the minimum number of letters in a given string to transform the string into a string in the language defined by the grammar @xmath53 ( empty string ) . dyck edit distance problem _ @xcite , which asks for the minimum number of edits to transform a given string to a well - balanced parentheses of @xmath54 different types , has a similar formulation . due to the similarity , the same conditional lower bound as theorem [ thm-1 ] was also shown for the dyck edit distance problem ( with alphabet size @xmath55 ) in @xcite . in this paper , we improve and simplify their result by demonstrating a simple reduction from rna folding to dyck edit distance problem : [ thm-3 ] if dyck edit distance problem on sequences of length @xmath10 with alphabet size 10 can be solved in @xmath33 time , then the rna folding problem on sequences in @xmath40 can be solved in @xmath56 time . combining theorem [ thm-2 ] , [ thm-3 ] , we get the following corollary : [ cor-1 ] if the dyck edit distance problem on sequences of length @xmath10 with alphabet size 10 can be solved in @xmath33 time , then @xmath34-clique on graphs with @xmath35 can be solved in @xmath41 time .

an rna sequence is a string composed of four types of nucleotides , , and . the problem is central in bioinformatics and has received much attention over the years . however , the current best algorithm for the problem still takes time , which is only a slight improvement over the classic dynamic programming algorithm . whether the rna folding problem can be solved in time remains an open problem . recently , abboud , backurs , and williams ( focs15 ) made the first progress by showing a conditional lower bound for a generalized version of the rna folding problem based on a conjectured hardness of the-clique problem . a drawback of their work is that they require the rna sequence to have at least 36 types of letters , making their result biologically irrelevant . in this paper , we show that by constructing the gadgets using a lemma of bringmann and knnemann ( focs15 ) and surrounding them with some carefully designed sequences , the framework of abboud et al . can be improved upon to work for the case where the alphabet size is 4 , yielding a conditional lower bound for the rna folding problem . we also investigate the dyck edit distance problem . we demonstrate a reduction from rna folding problem to dyck edit distance problem of alphabet size 10 , establishing a connection between the two fundamental string problems . keywords : rna folding , dyck edit distance , longest common subsequence , conditional lower bound , clique

an rna sequence is a string composed of four types of nucleotides , , and . given an rna sequence , the goal of the rna folding problem is to find a maximum cardinality set of crossing - free pairs of the form or . the problem is central in bioinformatics and has received much attention over the years . however , the current best algorithm for the problem still takes time , which is only a slight improvement over the classic dynamic programming algorithm . whether the rna folding problem can be solved in time remains an open problem . recently , abboud , backurs , and williams ( focs15 ) made the first progress by showing a conditional lower bound for a generalized version of the rna folding problem based on a conjectured hardness of the-clique problem . a drawback of their work is that they require the rna sequence to have at least 36 types of letters , making their result biologically irrelevant . in this paper , we show that by constructing the gadgets using a lemma of bringmann and knnemann ( focs15 ) and surrounding them with some carefully designed sequences , the framework of abboud et al . can be improved upon to work for the case where the alphabet size is 4 , yielding a conditional lower bound for the rna folding problem . we also investigate the dyck edit distance problem . we demonstrate a reduction from rna folding problem to dyck edit distance problem of alphabet size 10 , establishing a connection between the two fundamental string problems . this leads to a much simpler proof of the conditional lower bound for dyck edit distance problem given by abboud et al . and lowers the required alphabet size for the lower bound to work . keywords : rna folding , dyck edit distance , longest common subsequence , conditional lower bound , clique

1305.5564

i

ever since the discovery of the @xmath1 by the belle collaboration in 2003 @xcite , one of the leading interpretations has been a charm meson molecule whose constituents are a superposition of @xmath13 and @xmath16 @xcite . this identification would require the @xmath2 quantum numbers of @xmath1 to be @xmath3 . the observation of its decay into @xmath17 determined the charge conjugation @xmath18 to be @xmath19 @xcite . in 2006 , the cdf collaboration reduced the options for @xmath2 to @xmath3 and @xmath6 by analyzing decays into @xmath0 @xcite . the lhcb collaboration recently ruled out @xmath6 , finally establishing the quantum numbers of @xmath1 as @xmath3 @xcite . the option @xmath6 had been disfavored on various theoretical grounds , especially if the @xmath1 is identified with the @xmath20 charmonium state @xmath21 . the prediction of the mass of @xmath21 in most potentials models is lower than 3872 mev by 40 to 100 mev @xcite . the decay of @xmath21 into @xmath17 should have strong multipole suppression @xcite . the expected production rate for d - wave charmonium in a hadron collider is much smaller than the observed production rate of the @xmath1 at the tevatron @xcite . the decay of d - wave charmonium into @xmath22 should have angular momentum suppression @xcite . finally , the degree of isospin violation required by the observed branching fraction into @xmath0 is difficult to accommodate for d - wave charmonium @xcite . all of these problems are solved , or at least ameliorated , if the quantum numbers are @xmath3 . back in 2010 , the babar collaboration analyzed decays of @xmath1 into @xmath12 and concluded that @xmath6 was preferred over @xmath3 @xcite . they quantified this preference in terms of a probability that was 7.1% for @xmath3 and 61.9% for @xmath6 . we will point out that a proper quantification of the likelihood for the observed result increases the probability for @xmath3 to 18.7% . with the properly calculated probabilities , the preference for @xmath6 over @xmath3 is no longer so significant . however it is still worth considering whether a more accurate description of the resonance in the @xmath12 channel would be important in the babar analysis or in future analyses . since it has quantum numbers @xmath3 , the @xmath8 has an s - wave coupling to @xmath9 . the proximity of the @xmath9 threshold to the @xmath13 and @xmath14 thresholds and the narrow width of the @xmath11 suggest that the effects of scattering between @xmath9 and charm meson pairs could be significant . we therefore study the effects of scattering between these coupled channels on the @xmath1 resonance in the @xmath12 channel . we also analyze the effects of scattering through the @xmath15 charmonium resonance , which has quantum numbers @xmath3 . in section [ sec : widths ] , we introduce our notation for the three coupled channels and for the many masses that are relevant to this problem . in section [ sec : ddscat ] , we derive the scattering amplitudes due to s - wave scattering between the coupled channels . we use them in section [ sec : lineshape ] to determine the inclusive line shape of the @xmath1 resonance and its line shape in the @xmath12 channel . we also determine the effect of the @xmath15 resonance on the line shape . in section [ sec:3pionmass ] , we derive a simple expression for the @xmath4 invariant mass distribution . we examine the babar results in ref . @xcite and point out that a proper evaluation of the @xmath7 significantly increases the probability for the quantum numbers @xmath3 . we show that the experimental resolution , which was ignored in previous theoretical analyses , has a significant effect on the @xmath4 invariant mass distribution . finally we study the effect on that invariant mass distribution of scattering between the three coupled channels and scattering through the @xmath15 charmonium resonance .

we point out that a proper evaluation of the in that analysis increases the probability for from 7.1% to about 18.7% . in the case of quantum numbers , where the has an s - wave coupling to , the proximity of the threshold to thresholds and the narrow width of the suggest that the effects of scattering between and charm meson pairs could be significant . we derive invariant mass distributions for and that take into account s - wave scattering between the , , and channels . we also analyze the effects of scattering through the charmonium resonance .

analyses of the decay channel of the resonance by the cdf , belle , and lhcb collaborations have established its quantum numbers as . an analysis of the invariant mass distribution in the decay channel by the babar collaboration indicated a preference for over . we point out that a proper evaluation of the in that analysis increases the probability for from 7.1% to about 18.7% . in the case of quantum numbers , where the has an s - wave coupling to , the proximity of the threshold to thresholds and the narrow width of the suggest that the effects of scattering between and charm meson pairs could be significant . we derive invariant mass distributions for and that take into account s - wave scattering between the , , and channels . we also analyze the effects of scattering through the charmonium resonance . we find that scattering effects are unable to produce significant changes in the shape of the invariant mass distribution .

cond-mat0604665

i

the dressing of the charge carriers in high - t@xmath0 superconductors ( htscs ) is still one of the most exciting topics in solid state physics . the htscs are a paradigm for the transition of a correlated system from an insulating to a metallic state . the dressing of the charge carriers in htscs is likely caused by the same interaction that gives rise to the superconductivity , hence the understanding of the quasiparticle self - energy may help to understand the origin of the mechanism of high - t@xmath0 superconductivity . the dressing can be studied by various experimental methods but angle - resolved photoemission spectroscopy is the only method which gives a quantitative information on the momentum dependence of those renormalization effects . in htscs there are two important regions on the fermi surface : the nodal region , where the diagonal of the brillouin zone cuts the fermi surface and where the @xmath3wave superconducting order parameter changes sign . this region mostly contributes to the transport properties , particularly in the underdoped region , where a pseudogap opens up , squeezing the fermi surface to a region near the nodes . the other ( antinodal ) region is one where the edge of the brillouin zone cuts the fermi surface . in this region , the order parameter in hole doped superconductors has its maximum . this region is therefore mostly relevant for the studies of the superconducting properties . there are numerous arpes studies on the renormalization effects near the nodal point,@xcite but only a few studies are concentrated at the antinodal point @xcite . in the bilayer systems the study of the antinodal point is complicated by the bilayer splitting , which could not be resolved for 15 years . on the other hand , only in the bilayer system of the bi - htsc family the entire superconducting region from underdoped ( ud ) via optimally doped ( op ) to overdoped ( od ) can be studied . in the superconducting state a well pronounced peak - dip - hump structure has been detected @xcite . this structure was originally explained @xcite solely in terms of a coupling to a bosonic mode , similar to the mcmillan - rowell explanation of the tunnelling spectra in conventional superconductors@xcite . later on , it was established that this peak - dip - hump structure is partially caused by the bilayer splitting @xcite . by varying the photon energy h@xmath4 in the arpes experiments , and exploiting the different energy dependence of the matrix elements for the excitations from the bonding and the antibonding bands , it became possible to separate the two bands @xcite and to extract the full energy- and momentum - dependent spectral weight separately in each of the bands . this procedure allowed the authors of refs . to find the intrinsic peak - dip - hump structure , and to demonstrate that the strength of this intrinsic effect is doping - dependent , and decreases in going from ud to od materials . an important characteristic of the interaction between fermionic and bosonic excitations is the energy - dependent , dimensionless coupling @xmath5 . in theories where the fermionic self - energy depends on energy , @xmath6 , much stronger than on the momentum @xmath7 , this dimensionless coupling is related to the self - energy via @xmath8 . it is also relevant whether the measurements are performed in the normal or in the superconducting state . we will label the corresponding couplings as @xmath9 and @xmath10 , respectively . if the normal state is a fermi liquid , @xmath11 is finite , and is often called a dimensionless coupling constant . it determines the mass renormalization of the fermionic quasiparticles via @xmath12 . the coupling constant can , in principle , be extracted from arpes measurements of the quasiparticle dispersion in the normal state at the lowest energies , however this procedure requires one to know both @xmath13 and the _ bare _ mass , @xmath14 . in previous analysis @xcite , the mass , @xmath14 , was extracted from a tight - binding model with parameters derived from a fit of the fermi surface and from the quasiparticle dispersion measured along the nodal direction @xcite . the analysis of the experimental data in the antinodal region yielded @xmath15 both in ud and od materials . this result should be contrasted with values @xcite of @xmath16 at the nodal point . it is consistent with expectations as for non rotationally - invariant systems the coupling @xmath17 depends on the position on the fermi surface . in the superconducting state , the measured quasiparticle energy in the antinodal region is bounded by the superconducting gap , @xmath18 , and it becomes an issue at which energy one extracts the coupling @xmath10 from the data . in previous analysis , the coupling was extracted from the self - energy measured at @xmath19 . this coupling @xmath20 turns out to be larger than @xmath21 , and it also rapidly increases from od to ud samples ( @xmath22 for dopant concentration @xmath23 ) . in this communication we extend our previous analysis of the antinodal self - energy in the superconducting state @xcite and show how one can extract the coupling at zero frequency @xmath24 from the arpes data . we find that @xmath25 is smaller than @xmath20 and within a certain model is also smaller than the normal state coupling @xmath21 , in agreement with earlier calculations @xcite . we show that the large value of @xmath20 and its strong doping dependence are at least partially due to the fact that the fermionic self - energy in a superconductor actually diverges at @xmath26 , where @xmath27 is the energy of the bosonic mode . if the bosonic mode is the spin resonance peak , its energy decreases with decreasing doping . then @xmath28 and @xmath29 come closer to each other in the ud regime , and @xmath20 strongly increases . this is consistent with the analysis in ref . . our present analysis is based on the measurements of the quasiparticle spectral function in the antinodal region of the high - t@xmath0 superconductor ( bi , pb)@xmath1sr@xmath1cacu@xmath1o@xmath30 ( bipb2212 ) in the superconducting state . we go beyond a previous arpes study which has analyzed the energy dependence of the spectral weight just at the @xmath31 point @xcite , and study the whole antinodal region . we interpret our results in the superconducting state in terms of model self - energy function which is composed of two terms . the first and dominant term is due to a strong coupling of the charge carriers to a single bosonic mode . the second term describes a band renormalization at higher energies and is assumed to have a fermi - liquid form . we extract both couplings from the fits to the data . we used two models for electron - boson coupling . the first model is a one - mode model for an interaction with an einstein boson , which is assumed to be independent on fermions . second is a collective mode model , in which the bosonic spectrum in the normal state is rather flat and incoherent , but splits into a mode and into gapped continuum in the superconducting state due to the feedback effect from the pairing . this second model is appropriate if the boson is a spin collective mode of fermions . we obtain a rather good agreement between the parameters derived from the analysis of the experimental data using the model self - energy function and the calculated values using the collective mode model . this yields a strong indication that the dominant part of the renormalization of the fermionic dispersion is due to a coupling of collective spin excitations . the paper is organized as follows . in sec . ii we review the two fermion - boson models in the normal and the superconducting state . the experimental setup is discussed in sec . iii . in sec . iv we present the experimental results together with the data analysis . in sec . v we discuss the results and compare them with other renormalization effects studied by arpes in solid state physics . the conclusions of our study are presented in sec .

angle - resolved photoemission spectroscopy ( arpes ) is used to study the spectral function of the optimally doped high - t superconductor ( bi , pb)srcacuo in the vicinity of the antinodal point in the superconducting state . using a parameterized self - energy function , it was possible to describe both the coherent and the incoherent spectral weight of the bonding and the antibonding band .

angle - resolved photoemission spectroscopy ( arpes ) is used to study the spectral function of the optimally doped high - t superconductor ( bi , pb)srcacuo in the vicinity of the antinodal point in the superconducting state . using a parameterized self - energy function , it was possible to describe both the coherent and the incoherent spectral weight of the bonding and the antibonding band . the renormalization effects can be assigned to a very strong coupling to the magnetic resonance mode and at higher energies to a bandwidth renormalization by a factor of two , probably caused by a coupling to a continuum . the present reevaluation of the arpes data allows to come to a more reliable determination of the value of the coupling strength of the charge carriers to the mode . the experimental results for the dressing of the charge carriers are compared to theoretical models .

1312.4835

c

in this paper we have discussed extragalactic anisotropies in the electromagnetic emission produced by dm annihilation or decay as a promising tool to search for a dm signal . we have first reviewed the formalism needed to compute a generic 2-point angular power spectrum by following the halo - model description of clustering of structures in the universe . this formalism was then applied to realize estimates involving relevant emissions induced by particle dm annihilations or decays . we have discussed the features and the relative size of the various auto- and cross - correlation angular power spectra that can be envisaged for anisotropy studies . from the side of dm signals we have considered the full multi - wavelength spectrum , including the synchrotron emission at radio frequencies , the ic radiation in the x - ray and gamma - ray bands , as well as the prompt emission of gamma - rays . the angular power spectra of auto - correlation of each of these signals and of the cross - correlation between any pair of them is presented . as a way to enhance the capability of detection of such non - gravitational signals of dm ( and to improve their disentanglement from other astrophysical backgrounds ) we introduce their cross - correlation with maps tracing the gravitational potential . we have analyzed this possibility studying specific gravitational tracers of dm distribution in the universe : weak - lensing cosmic shear , large scale structure matter distribution and cmb - lensing . we have shown that cross - correlating a multi - wavelength dm signal ( which is a direct manifestation of its particle physics nature ) with a gravitational tracer ( which is a manifestation of the presence of large amounts of unseen matter in the universe ) may offer a prime tool to demonstrate that what we call dm is indeed formed by an elementary particle .

anisotropies in the electromagnetic emission produced by dark matter annihilation or decay in the extragalactic sky are a recent tool in the quest for a particle dark matter evidence . we review the formalism to compute the two - point angular power spectrum in the halo - model approach and discuss the features and the relative size of the various auto- and cross - correlation signals that can be envisaged for anisotropy studies . from the side of particle dark matter signals , we consider the full multi - wavelength spectrum , from the radio emission to x - ray and gamma - ray productions . we discuss the angular power spectra of the auto - correlation of each of these signals and of the cross - correlation between any pair of them . we then extend the search to comprise specific gravitational tracers of dark matter distribution in the universe : weak - lensing cosmic shear , large - scale - structure matter distribution and cmb - lensing . we have shown that cross - correlating a multi - wavelength dark matter signal ( which is a direct manifestation of its particle physics nature ) with a gravitational tracer ( which is a manifestation of the presence of large amounts of unseen matter in the universe ) may offer a promising tool to demonstrate that what we call dm is indeed formed by elementary particles .

anisotropies in the electromagnetic emission produced by dark matter annihilation or decay in the extragalactic sky are a recent tool in the quest for a particle dark matter evidence . we review the formalism to compute the two - point angular power spectrum in the halo - model approach and discuss the features and the relative size of the various auto- and cross - correlation signals that can be envisaged for anisotropy studies . from the side of particle dark matter signals , we consider the full multi - wavelength spectrum , from the radio emission to x - ray and gamma - ray productions . we discuss the angular power spectra of the auto - correlation of each of these signals and of the cross - correlation between any pair of them . we then extend the search to comprise specific gravitational tracers of dark matter distribution in the universe : weak - lensing cosmic shear , large - scale - structure matter distribution and cmb - lensing . we have shown that cross - correlating a multi - wavelength dark matter signal ( which is a direct manifestation of its particle physics nature ) with a gravitational tracer ( which is a manifestation of the presence of large amounts of unseen matter in the universe ) may offer a promising tool to demonstrate that what we call dm is indeed formed by elementary particles .

1607.03582

i

a large amount of experimental and theoretical studies have been devoted to studying the unusual properties of high temperature cuprate superconductors in the past thirty years @xcite . although some consensuses have been reached , many fundamental problems are still in debate , including the microscopic pairing mechanism @xcite , the origin of pseudogap @xcite , and the description of non - fermi liquid behaviors of the normal state @xcite . in the past decade , there have been accumulating experimental evidences for the existence of a strong anisotropy in many of the physical properties of yba@xmath5cu@xmath6o@xmath7 ( ybco ) @xcite and bi@xmath5sr@xmath5cacu@xmath5o@xmath8 ( bscco ) @xcite . such an anisotropy is widely believed to be driven by the formation of a novel electronic nematic order @xcite , which spontaneously breaks the @xmath9 symmetry down to a @xmath10 symmetry . in case the nematic transition line goes across the superconducting transition line and penetrates into the superconducting dome , there exists a zero - temperature nematic quantum critical point ( qcp ) . the nematic quantum phase transition and the associated quantum critical behaviors have been investigated extensively in recent years @xcite . from a theoretical perspective , there are two widely studied scenarios to induce an electronic nematic order . first , the nematic order can be generated by melting a stripe order that spontaneously breaks both translational and rotational symmetry @xcite . the other way is related to pomeranchuk instability which refers to the deformation of the shape of the fermi surface of a metal due to coulomb interaction @xcite . in the simplest case , pomeranchuk instability occurs when the circular fermi surface of a two - dimensional metal becomes ellipse - like via quadrupolar distortion . @xcite . the hubbard model defined on square lattices @xcite provides a pertinent platform to investigate the electronic nematic order . halboth and metzner @xcite studied a two - dimensional hubbard model using functional renormalization group ( rg ) method , and revealed pomeranchuk instability and nematic order . more recently , the square lattice hubbard model is studied by variational cluster approximation @xcite and found to display a local nematic phase under certain circumstances , which might be applied to understand the intra - unit - cell electronic nematicity observed in the scanning tunneling spectroscopy ( sts ) measurements by lawler _ et al _ @xcite . another context of studying nematic order is provided by the superconducting dome of cuprate superconductors . for a pure @xmath0-wave superconductor , the discrete @xmath11 symmetry is preserved . however , when superconductivity coexists with a nematic order , the gap nodes are shifted from their original positions and the @xmath11 symmetry is broken down to @xmath12 @xcite . such an anisotropic superconducting state is physically equivalent to a @xmath13-wave superdconducting state @xcite . at the nematic qcp , the massless fermions excited from the @xmath14-wave gap nodes couple strongly to the quantum critical fluctuation of nematic order parameter , which can be effectively described by a ( 2 + 1)-dimensional field theory @xcite . this model was first analyzed by vojta _ @xcite , who made an @xmath15-expansion and argued that the nematic phase transition is turned to first order . @xcite later tackled the same model by means of @xmath16-expansion , where @xmath17 is a large fermion flavor , and concluded that the transition remains continuous . huh and sachdev @xcite performed a renormalization group ( rg ) analysis , and showed that the fermion velocity ratio @xmath18 in the lowest energy limit , where @xmath19 is the fermi velocity of nodal fermions and @xmath20 the gap velocity @xcite . the unusual velocity renormalization leads to significant changes of a number of quantities , including density of states ( dos ) @xcite , specific heat @xcite , low-@xmath21 thermal conductivity @xcite , superfluid density @xcite , and london penetration depth @xcite . in this article , we revisit the issue of unusual physical properties caused by the strong interaction between massless nodal fermions and critical nematic fluctuation . we will apply the powerful rg approach to calculate the fermion damping rate @xmath1 , where @xmath22 is the imaginary part of retarded fermion self - energy , and the corresponding quasiparticle residue @xmath23 . _ @xcite have previously computed the fermion self - energy and spectral function . interestingly , it will become clear below that the rg analysis give rise to different results once the singular renormalization of fermion velocities is taken into account . in particular , we will show that the fermion damping rate vanishes upon approaching the fermi level more rapidly than the energy @xmath2 , namely @xmath24 . according to the conventional notion of quantum many - particle physics , one would expect the system to behave as a normal fermi liquid . however , by analyzing the rg results , we find that the quasiparticle residue vanishes , i.e. , @xmath25 , in the limit @xmath4 . therefore , the system is actually a non - fermi liquid that represents an even weaker violation of fermi liquid theory comparing to a marginal fermi liquid ( mfl ) . to the best of our knowledge , this type of unconventional non - fermi liquid behavior has not been reported previously . in realistic materials , there are always certain amount of disorders , which may play an important role . as demonstrated earlier by nersesyan _ @xcite , the most important disorder in cuprates behaves like a randomly distributed gauge potential . thus , we will mainly study the influence of random gauge potential on the physical properties of nodal fermions and also the stability of nematic qcp . in the absence of nematic order , the coupling between nodal fermions and random gauge potential has attracted considerable interest @xcite . here , we consider the case in which nodal fermions couple to both the nematic order and random gauge potential , and then study the interplay of these two interactions by means of rg method . we find that the effective strength of gauge potential disorders tends to diverge at the lowest energy . this behavior signals the emergence of a finite zero - energy dos @xmath26 and the happening of quantum phase transition from an unconventional non - fermi liquid to a disorder dominated diffusive state . the nodal fermions acquire a finite scattering rate @xmath27 , which in turn affects the thermodynamic and spectral behaviors of nodal fermions . the rg results for the self - energy of nodal fermions can be used to understand a number of experimental facts observed in cuprate superconductors . we will show that the rg results are qualitatively consistent with some recent measurements of specific heat , fermion damping rate , and temperature dependence of fermion velocities . the rest of the paper will be organized as follows . we first present the effective low - energy field theory for the interaction between nematic order and nodal fermions in section [ sec : modelderivation ] . the random gauge potential is also introduced in this section . in section [ sec : rganalysisclean ] , we make detailed theoretical analysis for the self - consistent rg equations of the fermion velocities in the clean case . based on the rg solutions , we proceed to compute the fermion damping rate , quasiparticle residue , and other physical quantities . we will show that the quantum critical fluctuation of nematic order leads to unconventional non - fermi liquid behaviors of nodal fermions . we consider the influence of random gauge potential in section [ sec : rganalysisdisorder ] and find that the effective disorder strength flows to the strong coupling regime at low energies . therefore , even weak disorders play a significant role and drive the system to enter into a disorder controlled diffusive state . in section [ sec : compaisionexperiments ] , we discuss the possible application of the rg results to some experimental findings of cuprates . in section [ sec : summary ] , we briefly summarize our main results .

the nodal fermions thus constitute an unconventional non - fermi liquid that represents an even weaker violation of fermi liquid theory than a marginal fermi liquid . we also investigate the interplay of quantum nematic critical fluctuation and gauge - potential - like disorder , and find that the effective disorder strength flows to the strong coupling regime at low energies . therefore , even an arbitrarily weak disorder can drive the system to become a disorder controlled diffusive state . based on these theoretical results , we are able to understand a number of interesting experimental facts observed in curpate superconductors .

at the nematic quantum critical point that exists in the-wave superconducting dome of cuprates , the massless nodal fermions interact strongly with the quantum critical fluctuation of nematic order . we study this problem by means of renormalization group approach and show that , the fermion damping rate vanishes more rapidly than the energy and the quasiparticle residue in the limit . the nodal fermions thus constitute an unconventional non - fermi liquid that represents an even weaker violation of fermi liquid theory than a marginal fermi liquid . we also investigate the interplay of quantum nematic critical fluctuation and gauge - potential - like disorder , and find that the effective disorder strength flows to the strong coupling regime at low energies . therefore , even an arbitrarily weak disorder can drive the system to become a disorder controlled diffusive state . based on these theoretical results , we are able to understand a number of interesting experimental facts observed in curpate superconductors .

1511.04270

i

the swimming of many microorganisms is powered by periodic bending motion of cilia and flagella . the flagellar beat results from a feedback : dynein motors generate sliding forces that bend the flagellum ; and bending leads to deformations and stresses , which feed back and regulate motors . three alternative feedback mechanisms have been proposed : regulation by the sliding forces , regulation by the curvature of the axoneme , and regulation by the normal forces that tend to separate adjacent doublets . in this work we combine theoretical and experimental techniques to test whether any of these mechanisms can account for the waveforms of the short flagella of the unicellular alga _ chlamydomonas_. we show that the sliding control mechanism can not produce bend propagation for short flagella , which results in a poor fit to the data . comparison of the waveforms of wild type _ chlamydomonas _ with those of a mutant that has a symmetric beat argues against normal force regulation . by contrast , the waveforms predicted by the curvature control model accord with the experimental data . importantly , we make the surprising prediction that the motors respond to the time derivative of curvature , rather than curvature itself , hinting at an adaptive mechanism within the cilium .

these motors generate sliding forces between adjacent microtubule doublets within the axoneme , the motile cytoskeletal structure inside the flagellum . , we used isolated , reactivate axonemes of the unicellular alga _ chlamydomonas _ as a model system . we found that regulation by sliding forces fails to account for the measured beat , due to the short lengths of _ chlamydomonas _ cilia .

axonemal dyneins are the molecular motors responsible for the beating of cilia and flagella . these motors generate sliding forces between adjacent microtubule doublets within the axoneme , the motile cytoskeletal structure inside the flagellum . to create regular , oscillatory beating patterns , the activities of the axonemal dyneins must be coordinated both spatially and temporally . it is thought that coordination is mediated by stresses or strains that build up within the moving axoneme , but it is not known which components of stress or strain are involved , nor how they feed back on the dyneins . to answer this question , we used isolated , reactivate axonemes of the unicellular alga _ chlamydomonas _ as a model system . we derived a theory for beat regulation in a two - dimensional model of the axoneme . we then tested the theory by measuring the beat waveforms of wild type axonemes , which have asymmetric beats , and mutant axonemes , in which the beat is nearly symmetric , using high - precision spatial and temporal imaging . we found that regulation by sliding forces fails to account for the measured beat , due to the short lengths of _ chlamydomonas _ cilia . we found that regulation by normal forces ( which tend to separate adjacent doublets ) can not satisfactorily account for the symmetric waveforms of the _ mbo2 _ mutants . this is due to the model s failure to produce reciprocal inhibition across the axes of the symmetrically beating axonemes . finally , we show that regulation by curvature accords with the measurements . unexpectedly , we found that the phase of the curvature feedback indicates that the dyneins are regulated by the dynamic ( i.e. time - varying ) component of axonemal curvature , but not by the static one . we conclude that a high - pass filtered curvature signal is a good candidate for the signal that feeds back to coordinate motor activity in the axoneme .

1511.04270

i

cilia and flagella are long , thin organelles whose regular oscillatory bending waves propel cells through fluids and drive fluid flows across the surfaces of cells . the internal motile structure , the axoneme , contains nine doublet microtubules , a central pair of single microtubules , motor proteins in the axonemal dynein family and a large number of additional structural and regulatory proteins @xcite . the axonemal dyneins power the beat by generating sliding forces between adjacent doublets . the sliding is then converted to bending by constraints at the base of the axoneme ( e.g. provided by the basal body ) and/or along the length of the axoneme ( e.g. nexin links ) @xcite . while the constrained - sliding mechanism of bend formation is well established , it is not known how the activities of the dyneins are coordinated in space and time to produce the periodic beating pattern . it is thought that the beat is the result of feedback . the axonemal dyneins generate forces that deform the axoneme ; the deformations , in turn , regulate the dyneins . because of the geometry of the axoneme , deformation leads to stresses and strains that have components in various directions ( e.g. axial and radial ) . however , which component ( or components ) regulates the dyneins is not known . . the two filaments are constrained to have a spacing @xmath0 . the dyneins step towards the base of the doublets . dyneins sitting at the bottom filament with their head ( blue circle ) on the top filament produce a tensile force density @xmath1 on the top filament , which tends to slide it towards the distal end ; and a compressive force density @xmath2 on the bottom filament . the dyneins sitting on the opposite filament create antagonistic forces . the local sliding displacement is given by @xmath3 , and the sliding at the base is @xmath4 . the spring and dashpot at the base correspond to the compliance of the base with stiffness @xmath5 and friction coefficient @xmath6 . the green springs indicate the normal compliance which supports a normal force @xmath7 . the position of the point at the arc length @xmath8 from the origin is @xmath9 , characterized by a tangent vector @xmath10 , a normal vector @xmath11 , and a tangent angle @xmath12 with respect to the horizontal axis of the lab - frame @xmath13 . ( * b * ) schematic of dynein regulation mechanisms . under _ curvature control _ the dynein head detaches due to an increase in curvature . in _ sliding control _ detachment is enhanced by a tangential loading force , and in _ normal force control _ it is the normal force that enhances detachment . signs indicate doublets polarity . [ fig : axoscheme ] ] three different , but not mutually exclusive , models for dynein regulation have been suggested in the literature , see fig . [ fig : axoscheme ] . according to the sliding control model , dyneins are regulated by tangential forces acting parallel to the long axis of the microtubule doublets @xcite . according to the curvature control model , dyneins are regulated by doublet curvature @xcite . according to the normal force control model , dyneins are regulated by transverse forces that act to separate adjacent doublets when they are curved @xcite . which of these mechanisms regulates the beat of the axoneme is not known . in this work , we isolated _ chlamydomonas _ axonemes , and tracked them with high spatial and temporal resolution . performing a fourier decomposition of the data , we observed that the beat of wild type and _ mbo2 _ mutant axonemes have similar dynamics . the primary difference lies in the static asymmetry characteristic of the wild type beat , which is lacking in the mutant @xcite . by developing a two - dimensional theory for the axonemal beat in the presence of a static asymmetry , we show that sliding control , curvature control , and normal force control can all , in principle , give raise to wave propagation in asymmetric axonemes . however , the observed beats of wild type cells and _ mbo2 _ mutants are not consistent with sliding and normal force control . by contrast , curvature control accorded with the wild type and _ mbo2 _ beats provided that the curvature signal adapts ( or is insensitive to ) the static component of curvature .

to create regular , oscillatory beating patterns , the activities of the axonemal dyneins must be coordinated both spatially and temporally . it is thought that coordination is mediated by stresses or strains that build up within the moving axoneme , but it is not known which components of stress or strain are involved , nor how they feed back on the dyneins . to answer this question we derived a theory for beat regulation in a two - dimensional model of the axoneme .

axonemal dyneins are the molecular motors responsible for the beating of cilia and flagella . these motors generate sliding forces between adjacent microtubule doublets within the axoneme , the motile cytoskeletal structure inside the flagellum . to create regular , oscillatory beating patterns , the activities of the axonemal dyneins must be coordinated both spatially and temporally . it is thought that coordination is mediated by stresses or strains that build up within the moving axoneme , but it is not known which components of stress or strain are involved , nor how they feed back on the dyneins . to answer this question , we used isolated , reactivate axonemes of the unicellular alga _ chlamydomonas _ as a model system . we derived a theory for beat regulation in a two - dimensional model of the axoneme . we then tested the theory by measuring the beat waveforms of wild type axonemes , which have asymmetric beats , and mutant axonemes , in which the beat is nearly symmetric , using high - precision spatial and temporal imaging . we found that regulation by sliding forces fails to account for the measured beat , due to the short lengths of _ chlamydomonas _ cilia . we found that regulation by normal forces ( which tend to separate adjacent doublets ) can not satisfactorily account for the symmetric waveforms of the _ mbo2 _ mutants . this is due to the model s failure to produce reciprocal inhibition across the axes of the symmetrically beating axonemes . finally , we show that regulation by curvature accords with the measurements . unexpectedly , we found that the phase of the curvature feedback indicates that the dyneins are regulated by the dynamic ( i.e. time - varying ) component of axonemal curvature , but not by the static one . we conclude that a high - pass filtered curvature signal is a good candidate for the signal that feeds back to coordinate motor activity in the axoneme .

1511.04270

r

to test the different mechanisms of beat regulation we performed high precision measurements of the flagellar waveform , see _ experimental procedures _ and fig . [ fig : tracking ] . using the tracking software developed in @xcite we calculated the temporal trajectories of 20 points along the arc length of the axoneme . the uncertainty of the shape in the @xmath13 space was @xmath104 and the uncertainty in the tangent angle was @xmath105 ( see panels @xmath106 and @xmath107 in fig . [ fig : tracking ] ) . the latter corresponds to a sliding displacement between adjacent doublet microtubules of only @xmath108 . ) and three higher harmonics ( @xmath109 ) are labeled . ( _ ii _ ) angular representation of the static ( @xmath35 ) mode as a function of arc length . the constant slope indicates an arc length independent static curvature @xmath110 . ( _ iii _ ) the amplitude and phase ( argument ) of the fundamental mode are shown in _ iii _ and _ iv _ , respectively . the linear decrease in phase indicates steady wave propagation . the data of a representative axoneme is highlighted in the panels _ ii__iv _ , with error bars indicating the standard error of the mean calculated by hexadecimation . ( * b * ) equivalent plots to ( a ) for _ mbo2 _ axonemes . ( * c * ) beat shapes of one representative beat cycle of the wild type axoneme highlighted in panel a ( left panel , data ) and shapes reconstructed from the superposition of the static and fundamental modes , neglecting all higher harmonics . the progression of shapes through the beat cycle is represented by the rainbow color code ( see inset ) . ( * d * ) same as ( c ) for an _ mbo2 _ mutant axoneme . [ fig : tracked ] ] because the beat of _ chlamydomonas _ is periodic in time , it is convenient to decompose the tangent angle @xmath111 into fourier modes @xmath33 , see eq . [ eq : modes ] . before doing so , we note that wild type _ chlamydomonas _ axonemes swim in circles at a slow angular rotation speed @xmath112 , see @xmath113 and @xmath114 in fig . [ fig : tracking]a . while the effect of this rotation is small for a single beat it becomes large for a long time series . before performing the fourier decomposition we subtracted @xmath115 from the tangent angle @xmath111 . and @xmath116 . ] the power spectrum of the tangent angle ( averaged over the flagellar length ) shows clear peaks at harmonics of its fundamental frequency , see fig . [ fig : tracked]a__i__. note that the peak of the fundamental mode @xmath36 accounts for @xmath117 of the total power spectrum , and so we neglect the higher harmonics @xmath118 for reconstructing the flagellar shape . the amplitude of the static mode ( @xmath35 ) and the amplitude and phase of the fundamental mode ( @xmath36 ) are shown in fig . [ fig : tracked]_ii__iv_. the main difference between the wild type and mutant axonemes comes from the static angular mode @xmath119 . for wild type axonemes , @xmath119 decreases with an approximately constant slope over arc length . this corresponds to a constant static curvature @xmath120 , and indicates that the static mode of the shape is a circular arc of radius @xmath121 . the static curvature of wild type axonemes leads to a highly asymmetric waveform . in contrast , _ mbo2 _ mutant axonemes have a very small static mode with a curvature @xmath122 , and the resulting beat is approximately symmetric . in comparison to the large difference in the static mode between wild type and mutant axonemes , the fundamental modes are similar . the amplitude of @xmath65 is roughly constant and has a characteristic dip in the middle . the argument of @xmath65 , which determines the phase profile of the wave , decreases at a roughly constant rate in both cases . since the total decay is about @xmath123 , the wavelength of the beat is approximately equal to the length of the axoneme . in summary , both wild type and mutant cells have an approximately sinusoidal dynamic beat whose amplitudes drop in the middle of the axoneme and whose phases decrease monotonically , consistent with a beat wavelength of about @xmath124 . the response of the motor force to strains and stresses of the axoneme described by eq . [ eq : fullmotmod ] allows for regulation via sliding , curvature and normal forces . while it is possible that all three mechanisms of motor control are involved in regulation of the axonemal beat , we now show each individual mechanism alone is capable of producing dynamic bending patterns . is plotted for beats resulting from sliding control , curvature control , and normal force control . ( * b * ) analogous to ( a ) for _ mbo2_. note that here also curvature control and normal force control provide good agreement , but not sliding control . ( * c * ) and ( * d * ) theoretical and experimental shape reconstruction in position space for the wild type and _ mbo2 _ beats under curvature control . [ fig : fits ] ] because the axonemal beat is dominated by its static mode ( @xmath35 ) and fundamental dynamic mode ( @xmath36 ) , see fig . [ fig : tracked ] , we constrain our physical model to one dynamic mode , which corresponds to the critical point of an oscillatory hopf bifurcation @xcite . within our theoretical description , _ sliding control _ corresponds to the case in which @xmath125 and @xmath126 , and the motor force only responds to sliding changes through @xmath127 , with single and double primes denoting real and imaginary parts respectively and @xmath113 the imaginary unit . additionally , because the active response must dominate for oscillations to occur , we have @xmath128 and @xmath129 @xcite . in _ curvature control _ , the motors have a passive response to sliding ( corresponding to the slope of their force - velocity curves ) , so that @xmath130 ; the motors are actively regulated by curvature , thus @xmath131 ; and they are not regulated by normal force , that is @xmath126 . finally , in _ normal force control _ there is a passive response to sliding , @xmath130 , an active response to normal forces , with @xmath132 and @xmath133 , and no response to curvature , @xmath125 . and @xmath134 change , but not those of @xmath135 . ] -2.25in0 in .*parameters for beat generation in wild type axonemes . * [ cols="<,<,<,<,<",options="header " , ] [ tab : mbo2fits ] as for wild type beats curvature control and normal force control provide very good fits , and sliding control does not . values indicated are averages and standard deviations for @xmath136 axonemes ( when the standard deviation was larger than the mean it was replaced by the mean itself ) . the static curvature was @xmath137 . note that the values of @xmath80 in normal force control are very spread and different relative to those obtained for wild type fits . in fact , in one case we obtained @xmath138 , indicating that motors must amplify the normal force they sense by almost two orders of magnitude . the values for curvature control are very similar to those of wild type fits . [ tab : mbo2fits ] . the beating pattern obtained by curvature control is compared to the experimental reconstruction in fig . [ fig : fits]c . the good agreement reinforces the conclusion from figure 4a that the curvature control model accords with the experimental data for wild type cells . similar good agreement for wild type cells was found with the normal - force model . table [ tab : wtfits ] summarizes average parameters resulting from the fits of 9 different axonemes . we compared theory and experiments for the symmetric beat of the mutant _ mbo2 _ , where the static curvature is reduced by at least one order of magnitude compared to wild type beats , see fig . [ fig : tracked ] . in this case the results were similar to those of wild type , see fig . [ fig : fits]b : sliding control can not produce bend propagation , while curvature and normal force control are in good agreement with the experimental data . the parameters obtained from the fit of _ mbo2 _ beats are given in table [ tab : mbo2fits ] . there are two related reasons why the sliding control mechanism provides poor fits to the observed beating patterns . first , under sliding control the equations describing the flagellar beat become symmetric with respect to a change in sign of the arc length . as a consequence , the only possible solutions are standing waves . this can be most easily seen by using the plane - wave approximation , @xmath54 with @xmath55 real , for a symmetric beat ( @xmath51 ) regulated by sliding ( @xmath125 and @xmath126 ) . in that case , eq . [ eq : asym ] becomes an equation for the wave - length @xmath139 where @xmath140 is the dimensionless complex sliding response coefficient . note that this equation is symmetric with respect to the change @xmath141 , and thus admits simultaneously waves traveling in both directions . in the absence of boundary asymmetries , these opposing waves interfere to form a standing wave . thus to the extent that the asymmetry of the boundary can be neglected , the sliding control mechanism can not account for the bend propagation . ] the second reason for the poor fits of the sliding control mechanism is that short flagella can not exhibit bend propagation , as already noted in @xcite . this is true even in the presence of a boundary asymmetry ( from the basal compliance ) . to understand this , we note that the dimensionless parameter @xmath142 can be written as @xmath143 where @xmath144 is the characteristic length at which oscillations decay in a boundary driven axoneme @xcite . for the case of flagella short with respect to @xmath144 , we can approximate @xmath145 in eq . [ eq : plane ] . the two allowed wavelengths are @xmath146 and @xmath147 , where one can show that @xmath148 in order to satisfy the boundary condition of no torques on the distal end @xcite . these two modes correspond again to opposing waves , and imposing no torques on the basal end it can be shown that they must have equal amplitudes . the result is again a standing wave , irrespective of the basal asymmetry . to verify this argument we studied the speed of bend propagation , which is defined as @xmath149 , for two different flagellar lengths @xmath150 , see fig . [ fig : slilen]a . for axonemes with lengths @xmath150 similar or smaller than @xmath144 , the resulting beats exhibit no bend propagation , figs . [ fig : slilen ] b and c bottom . for axonemes significantly longer than @xmath144 wave propagation can be strong , figs . [ fig : slilen ] b and c top . while in bull sperm we have @xmath151 , which is enough to produce wave propagation @xcite ; for _ chlamydomonas _ we have @xmath152 , which results in almost no bend propagation . indeed , we estimated the wave speed of the sliding control waveform in fig . [ fig : fits ] to be two orders of magnitude smaller than the observed value . despite the good fits obtained for wild type and _ mbo2 _ axonemes , there are two observations that argue against normal force control . the first is that unlike the curvature response coefficient @xmath79 , the mean value of the normal - force response coefficient @xmath80 is much greater in mutants cells than in wild type cells ( table [ tab : mbo2fits ] ) . the reason is that in _ mbo2 _ the small static curvature results in a small normal force , which requires a correspondingly larger response coefficient @xmath80 . in other words , to obtain agreement between the observed and theoretical waveforms , the sensitivity of the motors to normal force needs to be much greater in the _ mbo2 _ mutant than in the wild type . such a big difference in the response coefficient is not expected , given that the dynamic components of the mutant and wild type waveforms are very similar ( compare fig . [ fig : tracked]c and [ fig : tracked]g ) . is proportional to the static curvature @xmath42 but the fundamental mode stays unchanged , the normal force response coefficient @xmath80 is inversely proportional to the curvature . in red values for _ mbo2 _ and in green for wild type . ( * b * ) the curvature control response coefficient @xmath153 is independent of the asymmetry , and stays constant even for a change in asymmetry of several orders of magnitude . [ fig : normalasym ] ] the second argument against the normal force control is that @xmath80 varies greatly from cell to cell for the case of _ mbo2_. this is due to the observation that , while the static curvature is small in _ mbo2 _ axonemes , it is variable . this leads to a large variability in @xmath154 from axoneme to axoneme , which correlates strongly with the inverse of the static curvature ( see fig . [ fig : normalasym]a ) , i.e. @xmath155 . thus , even though the amplitude and phase of the first mode in _ mbo2 _ axonemes is very similar from axoneme to axoneme ( fig . [ fig : tracked]b ( @xmath114 ) and ( @xmath106 ) ) , the response coefficients vary widely in amplitude . the reason for this correlation between @xmath154 and @xmath42 is that , according to eq . [ eq : fndyn ] , the dynamic component of the normal force is linearly proportional to the asymmetry @xmath42 . this means that to preserve a similar fundamental dynamic mode , axonemes with a smaller static asymmetry @xmath42 require motors to have a higher response coefficient @xmath154 . by contrast , the curvature control response coefficient is highly consistent for wild type and mutant axonemes . in summary , despite the similarities in the dynamics of the beats of wild type and _ mbo2 _ axonemes , the normal force model requires very different values for the response coefficient @xmath80 ( and also the basal response coefficient @xmath156 ) for wild type and _ mbo2 _ axonemes . furthermore , the normal force model requires very large differences in @xmath80 from axoneme to axoneme , despite the similarity in the dynamics between axonemes . thus , we conclude that normal force is not a plausible parameter for controlling the ciliary beat . the curvature control model provides a good fit to the experimental data for both wild type and _ mbo2 _ axonemes ( figure [ fig : fits]a and b , middle panel ) . however , the strategy of curvature regulation that we used is strikingly different from those previously studied @xcite . while in previous work it was assumed that the static and dynamic response to curvature are of equal importance , here we found that when @xmath157 and @xmath158 were unconstrained , their best fit values were not significantly different from zero . we therefore set them both to zero ( see table [ tab : wtfits ] ) . we can understand this by writing eq . [ eq : asym ] for symmetric plane waves regulated by curvature , which , after separating real and imaginary part , becomes @xmath159 where @xmath160 is the dimensionless real curvature response coefficient , @xmath161 is the complex one , and we assume a passive response to sliding , @xmath130 and @xmath162 . these equations show that for a wave traveling forward ( @xmath163 ) , @xmath164 produces the active force to counter viscous effects of fluid and filament sliding , while @xmath131 counters the elastic forces of filament bending and sliding . for the case in which the contribution of the response to sliding is smaller than that of the fluid and filament bending , we can divide the equations above to obtain @xmath165 . since for @xmath166 @xmath167 and @xmath58 , we find that @xmath168 . in other words , for _ chlamydomonas _ , elastic forces dominate over viscous forces , and thus @xmath169 , as we obtained from the fits . furthermore , in the limit of @xmath170 ( short lengths , low viscosity ) , we can set @xmath171 and still obtain plane waves . the opposite however is not true : for @xmath172 the balance of elastic forces can not be satisfied . the molecular implications of this finding will be expanded upon in the discussion section . between the theoretical and a reference experimental beat as a function of the sliding response coefficient @xmath173 and the curvature response coefficient @xmath153 . the ellipsoid delimits the region with @xmath174 . black lines delimit the region with a passive base . moving along the long axis ( green circles ) affects the amplitude dip in the midpoint of the axonemem , see left panel in ( * b * ) . moving along the small axis towards the region of active base results in waveforms with a large amplitude at the base ( blue and red circles ) , see central and right panels in b. the axis in a are normalized by the reference fit , such that @xmath175 corresponds to the highest value of @xmath176 . [ fig : curspace ] ] we therefore set @xmath177 in the subsequent analysis . by doing so , we significantly simplify the model , because the number of free parameters is now only two , @xmath173 and @xmath153 . note that the two parameters , @xmath178 and @xmath179 , which characterize the stiffness and viscosity at the base respectively , are determined once @xmath173 and @xmath153 are specified as we are looking for oscillating solutions to the boundary value problem ( see _ appendix _ ) . thus , the curvature control model is specified by just two free parameters , @xmath173 and @xmath153 , which are specified by the sliding elasticity between doublet microtubules and the rate of change of axonemal curvature . the average values of @xmath173 and @xmath153 varied little between wild type and _ mbo2 _ mutant axonemes ( compare the third column of table [ tab : wtfits ] with that of table [ tab : mbo2fits ] ) . this accords with the observation that there is little difference in the dynamical properties of the beat between wild type and _ mbo2 _ axonemes . furthermore , the standard deviation of @xmath173 and @xmath153 are small , indicating that there is little variation from axoneme to axoneme . thus , the tight distribution of values of the parameters in the model reflects the similarity in the observed shapes in different axonemes . in other words , @xmath173 and @xmath153 are well constrained by the experimental data . to understand what aspects of the experimental data specify these two parameters , we performed a sensitivity analysis on @xmath173 and @xmath153 . in fig . [ fig : curspace]a we show a density map of the mean square distance @xmath176 between the theoretical waveforms and a reference experimental beating pattern as a function of @xmath173 and @xmath153 . a red ellipse delimits the region with @xmath180 , which very closely coincides with the region where @xmath178 and @xmath179 are both positive , delimited by black lines . this is very important because negative values of the basal parameters imply an active process at the base . such an active process would drive a whiplike motion of the cilium as discussed in @xcite . evidently , the observed shapes of the beats rule out such a whiplike motion . to investigate how the beat pattern is affected by variations in @xmath173 and @xmath153 , as well as the existence of active processes in the base , we systematically varied @xmath173 and @xmath153 parallel and perpendicular to the long axis of the ellipse . moving parallel affects the amplitude of the beat , with the middle - dip becoming more or less prominent , see green shapes in fig . [ fig : curspace]b . moving perpendicular in the region of active base indeed results in whiplike beats , with a large amplitude at the base , see blue and red shapes in figs . [ fig : curspace]b . . note that _ points lie away from wild type circles in the direction of the short axis of the ellipse . all values are normalized by those of the reference fit used also to normalize the heat map axis . ( * b * ) the distance of the circles to the long axis of all fits shows a clear correlation with the axonemal length . note also that _ mbo2 _ axonemes are systematically shorter than wild type axonemes . ( * c * ) the basal response coefficient also correlates with the length , resulting in a high value for wild type axonemes , which are longer . the values are normalized by the value for a reference axoneme . [ fig : sensitivity ] ] to better understand the cell to cell variability we placed all the axonemes recorded in the @xmath181 space , see fig . [ fig : sensitivity]a . points scatter mainly along the long axis of the ellipse , where there is a large region of small shape variations . importantly , we consistently see a shift perpendicular to the long axis between the wild type and _ mbo2 _ mutant axonemes . this variation mainly comes from the difference in length between wild type and mutant axonemes , as can be seen in fig . [ fig : sensitivity]b . the implication of this length variation is a stiffening of the base , see fig . [ fig : sensitivity]c , which explains the variability in basal compliance between _ mbo2 _ and wild type axonemes for curvature control in tables [ tab : wtfits ] and [ tab : mbo2fits ] .

axonemal dyneins are the molecular motors responsible for the beating of cilia and flagella . we then tested the theory by measuring the beat waveforms of wild type axonemes , which have asymmetric beats , and mutant axonemes , in which the beat is nearly symmetric , using high - precision spatial and temporal imaging . we found that regulation by normal forces ( which tend to separate adjacent doublets ) can not satisfactorily account for the symmetric waveforms of the _ mbo2 _ mutants .

axonemal dyneins are the molecular motors responsible for the beating of cilia and flagella . these motors generate sliding forces between adjacent microtubule doublets within the axoneme , the motile cytoskeletal structure inside the flagellum . to create regular , oscillatory beating patterns , the activities of the axonemal dyneins must be coordinated both spatially and temporally . it is thought that coordination is mediated by stresses or strains that build up within the moving axoneme , but it is not known which components of stress or strain are involved , nor how they feed back on the dyneins . to answer this question , we used isolated , reactivate axonemes of the unicellular alga _ chlamydomonas _ as a model system . we derived a theory for beat regulation in a two - dimensional model of the axoneme . we then tested the theory by measuring the beat waveforms of wild type axonemes , which have asymmetric beats , and mutant axonemes , in which the beat is nearly symmetric , using high - precision spatial and temporal imaging . we found that regulation by sliding forces fails to account for the measured beat , due to the short lengths of _ chlamydomonas _ cilia . we found that regulation by normal forces ( which tend to separate adjacent doublets ) can not satisfactorily account for the symmetric waveforms of the _ mbo2 _ mutants . this is due to the model s failure to produce reciprocal inhibition across the axes of the symmetrically beating axonemes . finally , we show that regulation by curvature accords with the measurements . unexpectedly , we found that the phase of the curvature feedback indicates that the dyneins are regulated by the dynamic ( i.e. time - varying ) component of axonemal curvature , but not by the static one . we conclude that a high - pass filtered curvature signal is a good candidate for the signal that feeds back to coordinate motor activity in the axoneme .

1703.04823

i

graphs provide a mathematical structure for describing relationships between objects in a system . owing to their intuitive representation , well - understood theoretical properties , the wealth of the algorithmic methodology and available code base , graphs have also become a major framework for modeling biological systems . protein - protein interaction networks , protein 3d structures , drug - target interaction networks , metabolic networks and gene regulatory networks are some of the major representations of biological systems . unfortunately , molecular and cellular systems are only partially observable and may contain significant amount of noise due to their inherent stochastic nature as well as the limitations of both low - throughput and high - throughput experimental techniques . this highlights the need for the development and application of computational approaches for predictive modeling ( e.g. , inferring novel interactions ) and identifying interesting patterns in such data . learning on graphs can be generally seen as supervised or unsupervised . under a supervised setting , typical tasks involve _ graph classification _ ; i.e. , the assignment of class labels to entire graphs @xcite , _ vertex or edge classification _ ; i.e. , the assignment class labels to vertices or edges in a single graph @xcite , or _ link prediction _ ; i.e. , the prediction of the existence of edges in graphs @xcite . alternatively , frequent subgraph mining @xcite , motif finding @xcite , clustering @xcite , and community detection @xcite are traditional unsupervised approaches . regardless of the category , the development of techniques that capture local / global network structure , measure graph similarity and incorporate domain - specific knowledge in a principled manner lie at the core of all these problems . the focus of this study is on classification problems across various biological networks . a straightforward approach to this problem is the use of topological and other descriptors ( e.g. , vertex degree , clustering coefficient , betweenness centrality ) that summarize graph neighborhoods . these descriptors straightforwardly lead to vector - space representations of vertices or edges in the graph , after which standard machine learning algorithms can be applied to learn a target function @xcite . another approach involves the use of kernel functions on graphs @xcite . kernels are mappings of pairs of objects from an input space @xmath0 to an output space @xmath1 with special properties , such as symmetry and positive semi - definiteness , that lead to efficient learning . graph kernels often exploit similar ideas as traditional vector - space approaches . finally , classification on graphs can be approached using probabilistic graphical models such as markov random fields @xcite and related label - propagation @xcite or flow - based @xcite methods . these `` global '' formulations are generally well adjusted to learning smooth functions over neighboring nodes . despite the success and wide adoption of these methods in machine learning and computational biology , it is well - understood that graph representations suffer from information loss since every edge can only encode pairwise relationships @xcite . a protein complex , for instance , can not be distinguished from a set of proteins that interact only pairwise . such disambiguation , however , is important in order to understand the biological activity of these molecules . hypergraphs , a generalization of graphs , naturally capture these higher - order relationships @xcite . as we show later , they also provide a representation that can be used to unify several conventional classification problems on ( hyper)graphs as a single vertex classification approach on hypergraphs . in this paper , we present and evaluate a kernel - based framework for the problems of vertex classification , edge classification and link prediction in graphs and hypergraphs . we first use the concepts of hypergraph duality to demonstrate that all such classification problems can be unified through the use of hypergraphs . we then describe the development of edit - distance hypergraphlet kernels for vertex classification in hypergraphs and combine them with support vector machines into a semi - supervised predictive methodology . finally , we use sixteen biological network data sets , eleven assembled specifically for this work , to provide evidence that the proposed approaches compare favorably to the previously established methods .

this approach has been highly successful owing to the theory , methodology and software that support analysis and learning on graphs . graphs , however , often suffer from information loss when modeling physical systems due to their inability to accurately represent multiobject relationships . we introduce a novel kernel method on vertex- and edge - labeled ( colored ) hypergraphs for analysis and learning .

biological and cellular systems are often modeled as graphs in which vertices represent objects of interest ( genes , proteins , drugs ) and edges represent relational ties among these objects ( binds - to , interacts - with , regulates ) . this approach has been highly successful owing to the theory , methodology and software that support analysis and learning on graphs . graphs , however , often suffer from information loss when modeling physical systems due to their inability to accurately represent multiobject relationships . hypergraphs , a generalization of graphs , provide a framework to mitigate information loss and unify disparate graph - based methodologies . in this paper , we present a hypergraph - based approach for modeling physical systems and formulate vertex classification , edge classification and link prediction problems on ( hyper)graphs as instances of vertex classification on ( extended , dual ) hypergraphs in a semi - supervised setting . we introduce a novel kernel method on vertex- and edge - labeled ( colored ) hypergraphs for analysis and learning . the method is based on exact and inexact ( via hypergraph edit distances ) enumeration of small simple hypergraphs , referred to as hypergraphlets , rooted at a vertex of interest . we extensively evaluate this method and show its potential use in a positive - unlabeled setting to estimate the number of missing and false positive links in protein - protein interaction networks .

1703.04823

m

we consider binary classification problems on graphs and hypergraphs and propose to unify all such learning problems through semi - supervised vertex classification on hypergraphs . first , vertex classification falls trivially into this framework . second , the problems of edge classification in graphs and hyperedge classification in hypergraphs are equivalent to the problem of vertex classification on dual hypergraphs . as discussed in section [ duality ] , both graphs and hypergraphs give rise to dual hypergraph representations and , thus , ( hyper)edge classification on a graph @xmath2 straightforwardly translates into vertex classification on its dual hypergraph @xmath58 . we note here that vertices with the degree of one in @xmath2 give rise to self - loops in the dual hypergraph @xmath58 . to account for them , we add one dummy node per self - loop with the same vertex label as the original vertex and connect them with an appropriately labeled edge . third , one can similarly see link prediction as vertex classification on dual hypergraphs , where the set of existing links is treated as positive data , the set of known non - existing links is treated as negative data , and the remaining set of missing links is treated as unlabeled data . this formulation further requires an extension of dual hypergraph representations as follows . consider a particular negative or missing link @xmath84 in the original graph @xmath2 with its dual hypergraph @xmath58 ( fig . [ duality : example]c ) . to make a prediction on this edge @xmath61 , we must first introduce a new vertex @xmath61 in the dual hypergraph as well as modify those hyperedges in @xmath58 that correspond to the vertices @xmath85 in @xmath2 ( fig . [ duality : example]c ) . we denote this extended hypergraph as @xmath86 . it now easily follows that the sets of negative and unlabeled examples can be created by considering a collection of extended graphs @xmath86 , one at a time , for all non - existing vertices @xmath84 or a subset thereof . since most graph data in biological networks lack large sets of representative negative examples , we approach vertex classification , ( hyper)edge classification and link prediction as instances of vertex classification on ( extended , dual ) hypergraphs in a positive - unlabeled setting . we believe this is a novel and useful attempt at generalizing three distinct graph classification problems in a common kernel - based semi - supervised setting . the following sections introduce hypergraphlet kernels that are the core of our classification approach . to @xmath87 vertices , with the root node of each hypergraphlet inscribed in a square . all hypergraphlets are presented in a compressed notation ; e.g. , the two non - isomorphic hypergraphlets @xmath88 and @xmath89 are shown in one drawing . ] * hypergraphlets . * inspired by graphlets @xcite , we define _ hypergraphlets _ as small , simple , connected , rooted hypergraphs . a hypergraphlet with @xmath15 vertices is called an @xmath15-hypergraphlet ; and the @xmath90-th hypergraphlet of order @xmath15 is denoted as @xmath91 . we consider hypergraphlets up to isomorphism and will refer to these isomorphisms as root- and label - preserving isomorphisms when hypergraphs are rooted and labeled . figure [ hypergraphlets ] displays all non - isomorphic unlabeled @xmath15-hypergraphlets with up to three vertices . there is only one hypergraphlet of order @xmath92 ( @xmath93 ; fig . [ hypergraphlets]a ) and one hypergraphlet of order @xmath94 ( @xmath95 ; fig . [ hypergraphlets]b ) . on the other hand , there are nine hypergraphlets of order @xmath87 ( @xmath96 ; fig . [ hypergraphlets]c ) and @xmath97 hypergraphlets of order @xmath98 ( not shown ) . we refer to all these hypergraphlets as _ base hypergraphlets _ since they correspond to the case when @xmath99 . consider now a vertex- and hyperedge - labeled ( or _ fully labeled _ for short ) hypergraphlet with @xmath15 vertices and @xmath100 hyperedges , where @xmath8 and @xmath11 denote the vertex - label and hyperedge - label alphabets , respectively . if @xmath101 and/or @xmath102 , then automorphic structures with respect to the same base hypergraphlet may exist ; hence , the number of fully labeled hypergraphlets per base structure is generally smaller than @xmath103 . for example , if one only considers vertex - labeled @xmath87-hypergraphlets , then there are @xmath104 vertex - labeled hypergraphlets corresponding to the asymmetric base hypergraphlets @xmath88 , @xmath105 and @xmath106 but only @xmath107 corresponding to the base hypergraphlets @xmath108 , @xmath89 , @xmath109 , @xmath110 , @xmath111 , @xmath112 ; see table [ hypergraphs : polya ] . this is a result of symmetries in the base hypergraphlets that give rise to automorphisms among vertex - labeled structures . similarly , if @xmath102 , then new symmetries may exist with respect to the base hypergraphlets that give rise to different automorphisms among hyperedge - labeled structures . in section [ growth : pattern ] , we provide a more detailed discussion on these symmetries . the relevance of these symmetries and enumeration steps relates to the dimensionality of the hilbert space in which the prediction is carried out . motivated by the case for graphs @xcite , we introduce _ hypergraphlet kernels_. let @xmath113 , be a fully labeled hypergraph where @xmath6 is a vertex - labeling function @xmath7 , @xmath9 is a hyperedge - labeling function @xmath10 , and @xmath114 . the vertex- and hyperedge - labeled @xmath15-hypergraphlet count vector for any vertex @xmath13 is defined as @xmath115 where @xmath116 is the count of the @xmath90-th fully labeled @xmath15-hypergraphlet and @xmath117 is the total number of vertex- and hyperedge - labeled @xmath15-hypergraphlets . a kernel function between the @xmath15-hypergraphlet counts for vertices @xmath118 and @xmath16 is defined as an inner product between @xmath119 and @xmath120 ; i.e. , @xmath121 the hypergraphlet kernel function incorporating all hypergraphlets up to the size @xmath122 is given by @xmath123 where @xmath122 is a small integer . in this work we use @xmath124 due to the exponential growth of the number of base hypergraphlets . consider a fully labeled hypergraph @xmath113 . given a vertex @xmath13 , we define the vector of counts for a @xmath125-generalized edit - distance hypergraphlet representation as @xmath126 where @xmath127 here , @xmath128 is the set of all @xmath15-hypergraphlets such that for each @xmath129 there exists an edit path of total cost at most @xmath125 that transforms @xmath130 into @xmath131 and @xmath132 is a user - defined constant . in words , the counts for each hypergraphlet @xmath91 are updated by also counting all other hypergraphlets @xmath133 that are in the @xmath125 vicinity of @xmath91 . the function @xmath134 can be used to adjust the weights of these pseudocounts . we set @xmath135 for all @xmath90 and @xmath136 and the cost of all edit operations was also set to @xmath92 . this restricts @xmath125 to nonnegative integers . the length-@xmath125 edit - distance @xmath15-hypergraphlet kernel @xmath137 between vertices @xmath118 and @xmath16 can be computed as an inner product between the respective count vectors @xmath138 and @xmath139 ; i.e. , @xmath140 finally , the length-@xmath125 edit - distance hypergraphlet kernel function is given as @xmath141 the edit operations considered here incorporate substitutions of vertex labels , substitutions of hyperedge labels , and insertions / deletions ( indels ) of hyperedges . given these edit operations , we also define three subclasses of edit - distance hypergraphlet kernels referred to as vertex label - substitution @xmath142 , hyperedge label - substitution @xmath143 and hyperedge - indel kernels @xmath144 . although the functions from equations ( [ eq : kernel ] ) and ( [ eq : kernel2 ] ) are defined as inner products , other formulations such as radial basis functions can be similarly considered @xcite . we also note that the combined kernels from equations ( [ eq : combined : kernel ] ) and ( [ eq : combined : kernel2 ] ) can be generalized beyond linear combinations @xcite . for the simplicity of this work , however , we only explore equal - weight linear combinations and normalize the functions from equations ( [ eq : kernel ] ) and ( [ eq : kernel2 ] ) using a cosine transformation . the implementation and the analysis of hypergraphlet kernels is an extension of the available solutions for string kernels @xcite . let @xmath145 be a neighborhood hypergraph , as defined in section [ sec : graphs ] and suppose it is significantly smaller than the original hypergraph @xmath2 . the hypergraphlet counting algorithm takes @xmath146 steps , where @xmath147 is the maximum degree of a vertex . similarly , the generation of the minimum cost edit path takes @xmath148 per single hypergraphlet edit operation . therefore , for each vertex @xmath16 an order of @xmath149 operations are necessary , where the @xmath150 term enumerates possible @xmath15-hypergraphlets in @xmath151 . note that the possible number of edges in a hypergraph @xmath152 can be significantly larger than the possible number of edges in a standard graph . hence , in a practical setting , the edit distance hypergraphlet kernels could greatly benefit from effective sampling techniques or exploitation of special types of hypergraphlets . the proposed implementation for computing hypergraphlet kernel functions is computed in time linear in the number of non - zero elements .

hypergraphs , a generalization of graphs , provide a framework to mitigate information loss and unify disparate graph - based methodologies . in this paper , we present a hypergraph - based approach for modeling physical systems and formulate vertex classification , edge classification and link prediction problems on ( hyper)graphs as instances of vertex classification on ( extended , dual ) hypergraphs in a semi - supervised setting .

biological and cellular systems are often modeled as graphs in which vertices represent objects of interest ( genes , proteins , drugs ) and edges represent relational ties among these objects ( binds - to , interacts - with , regulates ) . this approach has been highly successful owing to the theory , methodology and software that support analysis and learning on graphs . graphs , however , often suffer from information loss when modeling physical systems due to their inability to accurately represent multiobject relationships . hypergraphs , a generalization of graphs , provide a framework to mitigate information loss and unify disparate graph - based methodologies . in this paper , we present a hypergraph - based approach for modeling physical systems and formulate vertex classification , edge classification and link prediction problems on ( hyper)graphs as instances of vertex classification on ( extended , dual ) hypergraphs in a semi - supervised setting . we introduce a novel kernel method on vertex- and edge - labeled ( colored ) hypergraphs for analysis and learning . the method is based on exact and inexact ( via hypergraph edit distances ) enumeration of small simple hypergraphs , referred to as hypergraphlets , rooted at a vertex of interest . we extensively evaluate this method and show its potential use in a positive - unlabeled setting to estimate the number of missing and false positive links in protein - protein interaction networks .

1202.1815

i

walker first noticed @xcite that the endomorphism induced in a @xmath1-tqft ( defined over a field ) by the exterior of a closed off seifert surface of a knot in zero - framed surgery along the knot can be used to give lower bounds for the genus of the knot . he did this by showing the number of non - zero eigenvalues of this endomorphism counted with multiplicity is an invariant @xcite , i.e. it does not depend on the choice of the seifert surface . thus the number of such eigenvalues must be less than or equal to the dimension of the vector space that the tqft assigns to a closed surface of this minimal genus . next turaev and viro @xcite , again assuming the tqft is defined over a field , saw that the similarity class of the induced map on the vector space associated to a seifert surface modulo the generalized @xmath2-eigenspace was a stronger invariant . if the tqft is defined over a more general commutative ring , the second author observed that the strong shift equivalence class of the endomorphism is an invariant of the knot @xcite . strong shift equivalence ( abbreviated sse ) is a notion from symbolic dynamics which we will discuss in [ sse ] below . for a tqft defined over a field @xmath3 , the similarity class considered by turaev - viro is a complete invariant of sse . in this case , the vector space modulo the generalized @xmath2-eigenspace together with the induced automorphism , considered as a module over @xmath4 $ ] , is called the turaev - viro module . it should be considered as somewhat analogous to the alexander module . the order of the turaev - viro module is called the turaev - viro polynomial and lies in @xmath4 $ ] . we will refer to the endomorphisms constructed as above ( and those in the same sse class ) as turaev - viro endomorphisms . in @xcite , turaev - viro endomorphisms were studied and methods for computing the endomorphism explicitly were given . these methods adapted rolfsen s surgery technique of studying infinite cyclic covers of knots . this method requires finding a surgery description of the knot ; that is a framed link in the complement of the unknot such that the framed link describes @xmath5 and the unknot represents the original knot . moreover each of the components of the framed link should have linking number zero with the unknot . for this method to work , it is important that the surgery presentation have a nice form . in this paper , we will show that all knots have a surgery presentations of this form ( in fact an even nicer form that we will call standard . ) another explicit method of computation was given by achir , and blanchet @xcite . this method starts with any seifert surface . the second author also considered the further invariant obtained by decorating a knot with a colored meridian ( this was needed to give formulas for the turaev - viro endomorphism of a connected sum , and to use the turaev - viro endomorphism to compute the quantum invariants of branched cyclic covers of the knot ) . ohtsuki @xcite arrived at the same invariant as the turaev - viro polynomial but from a very different point of view . ohtsuki extracts this invariant from a surgery description of a knot ( alternatively of a closed 3-manifold with a primitive one dimensional cohomology class ) and the data of a modular category . his method starts from any surgery description standard or not . this is a significant advantage of his approach . ohtsuki s proof of the invariance of the polynomial in @xcite is only sketched . he stated that his invariant is the same as the turaev - viro polynomial , but does not give an explanation . recently viro has returned to these ideas @xcite . he has studied the turaev - viro endomorphism of a knot after coloring both the meridian and the longitude of the knot . viro observed that a weighted sum of the traces of these endomorphisms is the colored jones polynomial evaluated at a root of unity . in @xcite , turaev - viro endomorphisms were defined more generally for infinite cyclic cover of 3-manifolds . suppose @xmath6 is a closed connected oriented @xmath7-manifold @xmath8 with @xmath9 such that @xmath10 is onto . let @xmath11 be the infinite cyclic cover of @xmath8 corresponding to @xmath12 . choose a surface @xmath13 in @xmath8 dual to @xmath12 . by lifting @xmath13 to @xmath11 , we obtain a fundamental domain @xmath14 with respect to the action of @xmath15 on @xmath11 . @xmath14 is a cobordism from a surface @xmath13 to itself . let @xmath16 be a @xmath1-tqft on the cobordism category of extended @xmath7-manifolds and extended surfaces . applying @xmath16 to @xmath14 and @xmath13 , we can construct an endomorphism @xmath17 . in @xcite , it is proved that the strong shift equivalent class of @xmath17 is an invariant of the pair @xmath6 , i.e. it does not depend on the choice of @xmath13 . we denote this sse class by @xmath18 . we will sometimes refer to a pair @xmath19 as above , informally , as a 3-manifold with an infinite cyclic covering . the knot invariants discussed above can be obtained as special cases of the above invariants of 3-manifolds with an infinite cyclic covering . for any _ oriented _ knot @xmath0 in @xmath5 , we obtain an extended @xmath7-manifold @xmath20 by doing @xmath2-surgery along @xmath0 . we choose @xmath12 to be the integral cohomology class that evaluates to @xmath21 on a positive meridian of @xmath0 . then it is easy to see that the invariant @xmath22 corresponding to @xmath23 only depends on @xmath0 . if our tqft is defined for 3-manifolds with colored links , one may obtain further invariants by coloring the meridian and the longitude ( a little further away ) of the knot . for the knot invariants discussed above , it is required , in general , that @xmath0 be oriented . ] this is so that the exterior of a seifert surface acquires a direction as a cobordism from the seifert surface to itself . however , we decided to delay mentioning this technicality . to avoid issues that arise from phase anomalies in tqft , in this paper , we work with extended manifolds as in walker @xcite and turaev @xcite . in this introduction , we omit mention of the integer weights and lagrangian subspaces of extended manifolds . we discuss extended manifolds carefully in the main text . inspired by ohtsuki , we construct a sse class @xmath24 from a framed ( or banded ) tangle in @xmath25 that arises in a surgery presentation of @xmath6 . we call this the tangle endomorphism . moreover we show that the endomorphism ( or square matrix ) that ohtsuki considers in this situation is well defined up to sse . by relating the definition of the turaev - viro endomorphism to ohtsuki s matrix , we give a different proof of the invariance of ohtsuki s invariant . in fact , we show that ohtsuki s matrix has the same sse class as the turaev - viro endomorphism , i.e. @xmath26 . we do not prove these results in the general case of a tqft arising from a modular category . we only work in the context of the skein approach for tqfts associated to @xmath27 and @xmath28 . we work with a modified blanchet - habegger - masbaum - vogel approach @xcite as outlined in @xcite . this theory is defined over a slightly localized cyclotomic ring of integers . it is worthwhile studying endomorphisms defined up to strong shift equivalence over this ring rather than passing to a field . we show that the traces of the turaev - viro polynomial of knots with the meridian and longitude colored turns out to encode exactly the same information as the colored jones polynomial evaluated at a root of unity . in section [ preliminary ] , we discuss extended manifolds , a variant of the tqft constructed in @xcite , surgery presentations and the definition of sse . in section [ thetanglemorphism ] , we construct an endomorphism for each framed tangle in @xmath25 and apply it to the tangle obtained from a surgery presentation of an infinite cyclic cover of a 3-manifold . we call it the tangle endomorphism . then we state theorem [ main ] which states that the sse class of a tangle endomorphism constructed from a surgery presentation of @xmath19 is an of invariant @xmath19 . in section [ tvm ] , we discuss technical details concerning the turaev - viro endomorphism for @xmath6 , and the method of calculating @xmath29 introduced in @xcite . in section [ relation ] , we relate the tangle endomorphism associated to a nice surgery presentation to the corresponding turaev - viro endomorphism . in section [ pmt ] , we prove theorem [ main ] . in section [ cj ] , we give formulas relating the colored jones polynomial to the traces of turaev - viro endomorphism of a knot whose meridian and longitude are colored . in section [ example ] , we compute two examples to illustrate these ideas . all surfaces and 3-mainifolds are assumed to be oriented .

following viro , we consider the endomorphisms that one obtains after coloring the meridian and longitude of the knot . we show that the traces of these endomorphisms encode the same information as the colored jones polynomials of at a root of unity . most of the discussion is carried out in the more general setting of infinite cyclic covers of 3-manifolds .

by applying a variant of the tqft constructed by blanchet , habegger , masbaum , and vogel , and using a construction of ohtsuki , we define a module endomorphism for each knot by using a tangle obtained from a surgery presentation of . we show that it is strong shift equivalent to the turaev - viro endomorphism associated to . following viro , we consider the endomorphisms that one obtains after coloring the meridian and longitude of the knot . we show that the traces of these endomorphisms encode the same information as the colored jones polynomials of at a root of unity . most of the discussion is carried out in the more general setting of infinite cyclic covers of 3-manifolds .

1309.6219

i

by embedding four - rod resonators inside double - split ring resonators superlattice , a composite metamaterial system is introduced , where a common subradiant oscillator in four - rod resonator is coupled with two superradiant oscillators in double - split ring resonator to give rise to double fano resonances . as a classical analogue of four - level tripod atomic system , the extinction spectrum of the composite metamaterial exhibits a double fano - based coherent effect . double fano resonances are correlated owing to the conjugation of two coupling strengths @xmath4 and @xmath5 , which shows up as a transfer of absorbed power from one superradiant dipole to the other superradiant dipole in double - split ring resonators . double fano resonances in a tripod metamaterial system can be utilized in designing plasmonic resonances of metamaterial to engineer coherent effects in metamaterials .

by embedding four - rod resonators inside double - split ring resonators superlattice , a planar composite metamaterial possessing tripod plasmonic resonances is fabricated . double fano resonances are observed where a common subradiant driven oscillator is coupled with two superradiant oscillators . as a classical analogue of four - level tripod atomic system , the extinction spectrum of the composite metamaterial exhibits a coherent effect based on double fano resonances .

by embedding four - rod resonators inside double - split ring resonators superlattice , a planar composite metamaterial possessing tripod plasmonic resonances is fabricated . double fano resonances are observed where a common subradiant driven oscillator is coupled with two superradiant oscillators . as a classical analogue of four - level tripod atomic system , the extinction spectrum of the composite metamaterial exhibits a coherent effect based on double fano resonances . transfer of the absorbed power between two orthogonal superradiant oscillators is shown to be mediated by the common subradiant oscillator .

math0508269

i

classically , a _ chord diagram _ is a collection of chords of a circle ; the intersection graph for these chords is called a _ circle graph_. circle graphs have been intensively studied in graph theory , with classifications given by even and itai @xcite and bouchet @xcite , among others . in recent years , interest in chord diagrams has spread to topologists as part of the theory of finite type knot and link invariants @xcite , as well as the theory of virtual knots @xcite . a natural extension of knot theory is to look at spatial embeddings of more complex graphs , and it is natural to ask whether some analogue of chord diagrams could be equally useful in this context . the goal of this paper is to define a reasonable notion of chord diagram for general graphs , and to construct some tools to study them . in particular , we will look at a variety of intersection graphs for these chord diagrams . we will use these intersection graphs to determine when an embedding of a planar graph can be extended to an embedding of a chord diagram on that graph . as an application , we will define _ gauss codes _ for immersions of arbitrary graphs and give algorithms for determining whether a crossing sequence is realizable as a gauss code . acknowledgement : the authors would like to acknowledge the hospitality of waseda university , tokyo , and professor kouki taniyama during the international workshop on knots and links in a spatial graph in july , 2004 , where the idea for this project was conceived . the second author was supported by an lmu faculty research grant .

we will define chord diagrams for planar embeddings of planar graphs and their intersection graphs , and prove some basic results . then , as an application , we will introduce gauss codes for immersions of graphs in the plane and give algorithms to determine whether a particular crossing sequence is realizable as the gauss code of an immersed graph .

chord diagrams on circles and their intersection graphs ( also known as _ circle graphs _ ) have been intensively studied , and have many applications to the study of knots and knot invariants , among others . however , chord diagrams on more general graphs have not been studied , and are potentially equally valuable in the study of spatial graphs . we will define chord diagrams for planar embeddings of planar graphs and their intersection graphs , and prove some basic results . then , as an application , we will introduce gauss codes for immersions of graphs in the plane and give algorithms to determine whether a particular crossing sequence is realizable as the gauss code of an immersed graph .

gr-qc9810003

i

consider einstein s non - covariant first - order action@xcite , the 4-integral of a `` bulk '' lagrangian which is quadratic in the christoffel symbols and thus often called the `` @xmath12 action . '' starting with the einstein action , one applies standard techniques associated with noether s theorem in order to derive , among other things , an energy definition in general relativity : namely , the 2-integral of an einstein superpotential over some generic 2-surface @xmath13 in spacetime @xmath14 . the einstein energy is well known to be ambiguously defined because it depends on the choice of background coordinates . nevertheless , one may use the einstein construction to define sensible notions of total gravitational energy . indeed , consider the scenario of asymptotic flatness , say , towards future null infinity @xmath15 . in this case , @xmath13 tends to a round , infinite - radius , 2-sphere cut of @xmath15 , and the ( now suitably unique ) choice of asymptotically cartesian coordinates ensures that the einstein energy agrees with the accepted trautman - bondi - sachs ( tbs ) notion of total energy.@xcite however , were we to offer the einstein definition as the energy contained within some _ quasilocal _ ( that is , finite ) 2-surface @xmath13 , we would still be confronted with the task of choosing a physically meaningful set of background coordinates . the only natural choice would be coordinates which are partially adapted to the embedding of @xmath16 . however , such a choice wrecks the agreement between the einstein and tbs energies as @xmath13 tends towards @xmath15 . in fact , choosing such @xmath13-adapted coordinates , one finds that the einstein energy blows up in the said limit . similar statements can be made regarding other approaches for defining energy , which trade coordinate ( that is , holonomic - frame ) ambiguity for ambiguity of a different stripe , e. g. tetrad ( or rigid ) frame , spin frame , or auxiliary vector ( or spinor ) fields . the traditional party line regarding these issues is the following : _ there is no over - arching rule , applicable for all quasilocal 2-surfaces , for selecting a ( suitably unique ) background frame _ ; whence gravitational quasilocal energy is not well - defined . to what extent does the stubborn presence of frame ambiguity in the quasilocal context point to a gap in our understanding of gravitational energy ? to address this question and sharpen our thoughts on these issues , let us consider a covariant version of the einstein construction . employing a straightforward field - theoretic generalization of the hamilton - jacobi ( hj ) method@xcite , one may derive from a covariant action functional an expression for canonical quasilocal energy ( qle ) in general relativity.@xcite we call this definition of qle _ canonical _ , because , owing to its intimate connection with hj theory , this qle is also the on - shell value of the gravitational hamiltonian for the choice of unit lapse function and vanishing shift vector at the system boundary.@xcite we also note that the canonical qle is the thermodynamic internal energy in a thermodynamical description of a ( relativistic ) self - gravitating system.@xcite the analysis that leads to the canonical qle runs along a somewhat different line than the one followed in a noether - type analysis , but it also leads to a concept of energy which is not unique . indeed , _ as is always the case with energy _ , the canonical qle is defined only up to the choice of a zero - point . the zero - point ambiguity may be traced to a freedom present in the action principle . namely , one may always add to the action any functional of the fixed boundary data without affecting the variational principle . as with the situation above , if the goal is to obtain agreement with the accepted notions of gravitational energy at spacelike or null infinity , then there is a suitably unique choice of energy zero - point@xcite , whereas at the quasilocal level there seems to be no preferred choice .- adapted coordinates for the einstein definition . like before , such a choice , leading to an infinite energy , wrecks the agreement between the canonical quasilocal and tbs energies in the large - sphere null limit . ] while at first sight this seems no better or worse than the situation encountered above , notice that now the ambiguity in the energy has a physical interpretation , and , moreover , is a field - theoretic generalization of the standard ambiguity present in the hj definition of energy in ordinary mechanics . we may now restate the emphasized portion of the party line above as follows : _ there is no over - arching rule , applicable for all quasilocal 2-surfaces , for selecting a ( suitably unique ) zero - point . _ taking this statement at face value , we claim that it is the physicist s job to select an appropriate energy zero - point , guided by the principle that the selection should be appropriate for the physics of the problem at hand . we would like to point out that this is a common enough state of affairs in general relativity , a many - faceted theory known for its wealth of possible boundary conditions . indeed , by way of analogy consider the search for solutions of the einstein field equations . in practice , relativists certainly do not attempt to find _ the _ general solution , rather they attempt to find solutions given some additional physical input ( boundary conditions , symmetries , etc . ) . in practice , the same such additional input is needed to associate a meaningful qle with a particular quasilocal 2-surface . bearing these points in mind , we recall the form of the canonical qle : @xmath17 where we adopt geometrical units ( in which both newton s constant and the speed of light are set to unity ) , @xmath13 is a closed 2-surface , @xmath18 is the proper area element on @xmath13 , and @xmath19 is the mean curvature of @xmath13 as embedded in a spanning 3-surface @xmath5 . it is important to realize that this @xmath0 , while obtained as a proper surface integral over @xmath13 , is the energy of the gravitational and matter fields which live on @xmath5 , that is to say , @xmath0 is a functional of the initial data of @xmath5 . this concept of energy is rooted in the @xmath20 view of spacetime geometry , and for a fixed @xmath13 it is slightly sensitive to the choice of spanning @xmath5 . [ more precisely , @xmath0 depends on the equivalence class of spanning 3-surfaces determined by a unit timelike vector on ( and pointing orthogonal to ) @xmath13 . ] this sensitivity is quite analogous to the observer dependence of energy in special relativity , and _ a priori _ we expect its presence.@xcite in general , the term @xmath21 represents an arbitrary ( local ) function of the intrinsic metric @xmath22 of @xmath13 and corresponds to the freedom to assign the qle zero - point . notice that this freedom corresponds to a proper surface integral of what is effectively a free function of two variables . this freedom , stemming from the field - theoretic character of gravity , is rather more subtle than the freedom in simple mechanics of simply adding a constant to the energy . for our analysis here we only consider two energy zero - points , one determined by _ lightcone reference _ and the other by _ euclidean reference_. we make these concepts precise below . in this paper we examine the canonical qle in the standard `` small - sphere limit , '' first considered by horowitz and schmidt in their classic examination of hawking s quasilocal mass . by the term _ small sphere _ we mean a cut @xmath1 , level in an affine radius @xmath2 , of the lightcone @xmath3 belonging to a generic spacetime point @xmath4 . as a power series in @xmath2 , we compute the energy @xmath0 of the gravitational and matter fields on a spacelike hypersurface @xmath5 spanning @xmath1 . much of our analysis concerns conceptual and technical issues associated with assigning the zero - point of the energy , and , therefore , particularly elucidates the points raised in the first two paragraphs above . for the small - sphere limit , we argue that the correct zero - point is obtained via the aforementioned lightcone reference , which stems from a certain isometric embedding of @xmath1 into a genuine lightcone of minkowski spacetime and amounts to fixing @xmath23\!{}^{-1}\right\ } \label{lcmeancurvature}\ ] ] in the above qle definition . here @xmath24 is the laplacian operator and @xmath25 is twice the gaussian curvature of @xmath1 ; therefore , as advertised , this choice for @xmath26 depends solely on the intrinsic geometry of @xmath1 . notice that , due to the presence of the square root in this expression , one expects this choice for @xmath26 to be valid only for ( topologically spherical ) 2-surfaces possessing everywhere positive gaussian curvature ( as is the case both for the small spheres we study here ) . choosing the proper surface integral of this choice for @xmath27 as the energy zero - point , we find the following results for small spheres : in the presence of matter @xmath28|_{p } + { \rm o}(r^{4 } ) \label{result_one}\ ] ] and in vacuo @xmath29|_{p } + { \rm o}(r^{6 } ) { \ , } \label{result_two}\ ] ] here , @xmath8 is a unit , future - pointing , timelike vector in the tangent space at @xmath4 ( which defines the choice of affine radius ) ; @xmath9 is the matter stress - energy - momentum tensor ; @xmath10 is the bel - robinson gravitational super stress - energy - momentum tensor ; ] where @xmath30 is the weyl tensor , @xmath31 is the left - dual of the weyl tensor , and @xmath32 is the self - dual @xmath33 [ anti - self - dual @xmath34 part of the weyl tensor . further curvature conventions are discussed in the appendix . ] and @xmath11 denotes `` restriction to @xmath4 . '' it is interesting to note that , when integrated , the bel - robinson `` energy '' in ( [ result_two ] ) has been proven to be very useful already in the sense of a mathematical `` energy '' in the study of existence of solutions of hyperbolic equations in einsteinian gravity.@xcite it is noteworthy that it shows up also in the physical limit given in ( [ result_two ] ) . although the full physical significance of the physical limit is not known to us , we think that these mathematical and physical properties go beyond mere coincidences . for both the vacuum and non - vacuum cases , hawking s quasilocal mass expression agrees with the canonical qle results ( [ result_one ] ) and ( [ result_two ] ) up to and including the first non - trivial order in the affine radius . we find this result rather striking in light of the fact that the hawking mass has no apparent connection with the gravitational action or hamiltonian . we show that the non - vacuum result ( [ result_one ] ) has the expected form based on the results of newtonian potential theory . compare the small - sphere limit considered here with the large - sphere limit when @xmath1 tends to a cut of @xmath15 , in which case we know that the choice of euclidean reference yields agreement between @xmath0 and the tbs energy.@xcite in both limiting cases , the 2-surface @xmath1 of interest is a cut , level in an affine radius @xmath2 , of an outgoing congruence of null geodesics . there is , however , a crucial distinction to be made . in the small - sphere case , @xmath1 arises as the cut of a genuine lightcone , while in the large - sphere case this is generally not true . we find it remarkable that this distinction can be mirrored in the choice of zero - points . to grasp this point , consider first the euclidean reference for either limit . this reference involves an _ embedding of @xmath1 into a flat euclidean 3-space @xmath35 . now , in both limiting scenarios @xmath1 is , in general , slightly distorted from perfect roundness . therefore , with @xmath35 viewed in turn as an inertial slice of minkowski space @xmath36 , the enveloped @xmath37 can not be the cut of a genuine lightcone of @xmath36 . ( technically , in this case the outward null congruence associated with @xmath1 is not shear - free , but the lightcones of @xmath36 are shear - free . ) therefore , in the general large - sphere scenario , neither the @xmath1 embedding into the physical spacetime @xmath14 nor its embedding into the reference spacetime @xmath36 corresponds to a lightcone embedding [ that is , in neither case is @xmath1 the cut of a lightcone ] . of course , for the small - sphere limit we may employ either the euclidean reference or the lightcone reference . of these two choices , the lightcone employs flat _ spacetime _ to put the reference space on an equal footing with the small sphere construction in the physical spacetime . to define the lightcone reference , we first isometrically embed @xmath1 into the lightcone @xmath38 of a point @xmath39 , and then select a certain 3-surface @xmath40 spanning @xmath1 [ so that @xmath41 . the details of this construction , eventually leading to the expression ( [ lcmeancurvature ] ) , are found in subsection 2.a . now , having defined the lightcone reference ( tailored to the small sphere limit ) and found the resulting closed - form expression ( [ lcmeancurvature ] ) , we may now invert the question , asking whether or not a lightcone - reference @xmath26 defines a qle ( [ brownyorkqle ] ) possessing a correct large - sphere limit towards @xmath15 . we discuss this issue in appendix a. the organization of this paper is as follows . in section 1 we lay the foundations for our examination of the small - sphere limit . we describe the geometry of the limit in subsection 1.a , fix some general conventions in subsection 1.b , and make some general observations concerning the embedding of a @xmath42-surface in minkowski spacetime in subsection 1.c . no choice of energy zero - point is made in section 1 , although the results of subsection 1.c are used in the subsequent sections to construct zero - points . in section 2 we study the small - sphere limit , subject to the choice of lightcone reference , and derive the results ( [ result_one ] ) and ( [ result_two ] ) . in section 3 we discuss the relationships between the main results ( [ result_one ] ) , ( [ result_two ] ) and results from newtonian potential theory . appendix a contains discussions of the small - sphere limit subject to the choice of euclidean reference , and the large - sphere limit towards @xmath15 subject to the choice of lightcone reference . throughout our analysis , we use the newman - penrose ( np ) formalism@xcite , with which we assume the reader is familiar . in appendix b we collect various conventions and results associated with the np formalism which are used in the main parts of the paper .

consider the definition of quasilocal energy stemming from the hamilton - jacobi method as applied to the canonical form of the gravitational action . we examine in the standard `` small - sphere limit , '' first considered by horowitz and schmidt in their examination of hawking s quasilocal mass . by the term _ small sphere _ we mean a cut , level in an affine radius , of the lightcone belonging to a generic spacetime point . as a power series in , we compute the energy of the gravitational and matter fields on a spacelike hypersurface spanning . much of our analysis concerns conceptual and technical issues associated with assigning the zero - point of the energy . for the small - sphere limit , we argue that the correct zero - point is obtained via a `` lightcone reference , '' which stems from a certain isometric embedding of into a genuine lightcone of minkowski spacetime . choosing this zero - point , we find the following results : ( i ) in the presence of matter|_{p } + { \rm o}(r^{4})$ ] and ( ii ) in vacuo|_{p } + { \rm o}(r^{6})$ ] . here , is a unit , future - pointing , timelike vector in the tangent space at ( which defines the choice of affine radius ) ; is the matter stress - energy - momentum tensor ; is the bel - robinson gravitational super stress - energy - momentum tensor ; and denotes `` restriction to . '' hawking s quasilocal mass expression agrees with the results ( i ) and ( ii ) up to and including the first non - trivial order in the affine radius . the non - vacuum result ( i ) has the expected form based on the results of newtonian potential theory .

consider the definition of quasilocal energy stemming from the hamilton - jacobi method as applied to the canonical form of the gravitational action . we examine in the standard `` small - sphere limit , '' first considered by horowitz and schmidt in their examination of hawking s quasilocal mass . by the term _ small sphere _ we mean a cut , level in an affine radius , of the lightcone belonging to a generic spacetime point . as a power series in , we compute the energy of the gravitational and matter fields on a spacelike hypersurface spanning . much of our analysis concerns conceptual and technical issues associated with assigning the zero - point of the energy . for the small - sphere limit , we argue that the correct zero - point is obtained via a `` lightcone reference , '' which stems from a certain isometric embedding of into a genuine lightcone of minkowski spacetime . choosing this zero - point , we find the following results : ( i ) in the presence of matter|_{p } + { \rm o}(r^{4})$ ] and ( ii ) in vacuo|_{p } + { \rm o}(r^{6})$ ] . here , is a unit , future - pointing , timelike vector in the tangent space at ( which defines the choice of affine radius ) ; is the matter stress - energy - momentum tensor ; is the bel - robinson gravitational super stress - energy - momentum tensor ; and denotes `` restriction to . '' hawking s quasilocal mass expression agrees with the results ( i ) and ( ii ) up to and including the first non - trivial order in the affine radius . the non - vacuum result ( i ) has the expected form based on the results of newtonian potential theory .

1210.3449

i

consider a minimal - delay space - time coded rayleigh quasi - static flat fading mimo channel with full channel state information at the receiver ( csir ) . the input output relation for such a system is given by @xmath0 where @xmath1 is the channel matrix and @xmath2 is the additive noise . both @xmath3 and @xmath4 have entries that are i.i.d . complex - gaussian with zero mean and variance 1 and @xmath5 respectively . the transmitted codeword is @xmath6 and @xmath7 is the received matrix . the ml decoding metric to minimize over all possible values of the codeword @xmath8 is @xmath9 [ ld_stbc_def ] @xcite : a linear stbc @xmath10 over a real ( 1-dimensional ) signal set @xmath11 , is a finite set of @xmath12 matrices , where any codeword matrix belonging to the code @xmath10 is obtained from , @xmath13 by letting the real variables @xmath14 take values from a real signal set @xmath15 where @xmath16 are fixed @xmath12 complex matrices defining the code , known as the weight matrices . the rate of this code is @xmath17 complex symbols per channel use . we are interested in linear stbcs , since they admit sphere decoding ( sd ) @xcite and other qr decomposition based decoding techniques such as the qrdm decoder @xcite which are fast ways of decoding for the variables . designing stbcs with low decoding complexity has been studied widely in the literature . orthogonal designs with single symbol decodability were proposed in @xcite , @xcite , @xcite . for stbcs with more than two transmit antennas , these came at a cost of reduced transmission rates . to increase the rate at the cost of higher decoding complexity , multi - group decodable stbcs were introduced in @xcite , @xcite , @xcite . another set of low decoding complexity codes known as the fast decodable codes were studied in @xcite . fast decodable codes have reduced sd complexity owing to the fact that a few of the variables can be decoded as single symbols or in groups if we condition them with respect to the other variables . fast decodable codes for asymmetric systems using division algebras have been reported @xcite . the properties of fast decodable codes and multi - group decodable codes were combined and a new class of codes called fast group decodable codes were studied in @xcite . a new code property called the _ block - orthogonal _ property was studied in @xcite which can be exploited by the qr - decomposition based decoders to achieve significant decoding complexity reduction without performance loss . this property was exploited in @xcite to reduce to the average ml decoding complexity of the golden code @xcite and also in @xcite to reduce the worst - case complexity of the golden code with a small performance loss . while the other low decoding complexity stbcs use the zero entries in the upper left portion of the upper triangular matrix after the qr decomposition , these decoders utilize the zeroes in the lower right portion to reduce the complexity further . the contributions of this paper are as follows : * we generalize the set of sufficient conditions for an stbc to be block orthogonal provided in @xcite for sub - block sizes greater than 1 . * we provide analytical proofs that the codes obtained from the sum of clifford unitary weight designs ( cuwds ) @xcite exhibit the block orthogonal property when we choose the right ordering and the right number of matrices . * we provide new methods of construction of bostbcs using coordinate interleaved orthogonal designs ( ciods ) @xcite , cyclic division algebras ( cdas ) @xcite and crossed product algebras ( cpas ) @xcite along with the analytical proofs of their block orthogonality . * we show that the ordering of variables of the stbc used for the qr decomposition dictates the block orthogonal structure and its parameters . * we show how the block orthogonal property of the stbcs can be exploited to reduce the decoding complexity of a sphere decoder which uses a depth first search approach . * we provide bounds on the maximum possible reduction in the euclidean metrics ( em ) calculation during sphere decoding of bostbcs . * simulation results show that we can reduce the decoding complexity of existing stbcs by upto 30% by utilizing the block orthogonal property . the remaining part of the paper is organized as follows : in section [ sec2 ] the system model and some known classes of low decoding complexity codes are reviewed . in section [ sec3 ] , we derive a set of sufficient conditions for an stbc to be block orthogonal and also the effect of ordering of matrices on it . in section [ sec4 ] , we present proofs of block orthogonal structure of various existing codes and also discuss some new methods of constructions of the same . in section [ sec5 ] , we discuss a method to reduce the number of em calculations while decoding a bostbc using a depth first search based sphere decoder and also derive bounds for the same . simulation results for the decoding complexity of various bostbcs are presented in section [ sec6 ] . concluding remarks constitute section [ sec7 ] . _ notations : _ throughout the paper , bold lower - case letters are used to denote vectors and bold upper - case letters to denote matrices . for a complex variable @xmath18 , denote the real and imaginary part of @xmath18 by @xmath19 and @xmath20 respectively . the sets of all integers , all real and complex numbers are denoted by @xmath21 and @xmath22 , respectively . the operation of stacking the columns of @xmath23 one below the other is denoted by @xmath24 . the kronecker product is denoted by @xmath25 , @xmath26 and @xmath27 denote the @xmath28 identity matrix and the null matrix , respectively . for a complex variable @xmath18 , the @xmath29 operator acting on @xmath18 is defined as follows @xmath30.\ ] ] the @xmath29 operator can similarly be applied to any matrix @xmath31 by replacing each entry @xmath32 by @xmath33 , @xmath34 , @xmath35 , resulting in a matrix denoted by @xmath36 . given a complex vector @xmath37^{t}$ ] , @xmath38 is defined as @xmath39^{t}$ ] .

construction of high rate space time block codes ( stbcs ) with low decoding complexity has been studied widely using techniques such as sphere decoding and non maximum - likelihood ( ml ) decoders such as the qr decomposition decoder with m paths ( qrdm decoder ) . recently ren et al . , presented a new class of stbcs known as the block orthogonal stbcs ( bostbcs ) , which could be exploited by the qrdm decoders to achieve significant decoding complexity reduction without performance loss . we show that codes formed as the sum of clifford unitary weight designs ( cuwds ) or coordinate interleaved orthogonal designs ( ciods ) exhibit block orthogonal structure . we also provide new construction of block orthogonal codes from cyclic division algebras ( cdas ) and crossed - product algebras ( cpas ) . in addition , we show how the block orthogonal property of the stbcs can be exploited to reduce the decoding complexity of a sphere decoder using a depth first search approach .

construction of high rate space time block codes ( stbcs ) with low decoding complexity has been studied widely using techniques such as sphere decoding and non maximum - likelihood ( ml ) decoders such as the qr decomposition decoder with m paths ( qrdm decoder ) . recently ren et al . , presented a new class of stbcs known as the block orthogonal stbcs ( bostbcs ) , which could be exploited by the qrdm decoders to achieve significant decoding complexity reduction without performance loss . the block orthogonal property of the codes constructed was however only shown via simulations . in this paper , we give analytical proofs for the block orthogonal structure of various existing codes in literature including the codes constructed in the paper by ren et al . we show that codes formed as the sum of clifford unitary weight designs ( cuwds ) or coordinate interleaved orthogonal designs ( ciods ) exhibit block orthogonal structure . we also provide new construction of block orthogonal codes from cyclic division algebras ( cdas ) and crossed - product algebras ( cpas ) . in addition , we show how the block orthogonal property of the stbcs can be exploited to reduce the decoding complexity of a sphere decoder using a depth first search approach . simulation results of the decoding complexity show a 30% reduction in the number of floating point operations ( flops ) of bostbcs as compared to stbcs without the block orthogonal structure .

hep-th0306028

i

the importance of understanding quantum field theory in de sitter space , the space - time associated with a positive cosmological constant , has been heightened by recent observations of both the early and late universe . the dramatic results of wilkinson microwave anisotropy probe @xcite have provided further strong evidence that the universe underwent a rapid inflationary expansion . both the large - angle anti - correlation in the temperature - polarization cross - power spectrum and the nearly flat spectral index are consistent with the predictions of inflation . more surprisingly , the dimming of the type ia supernovae seen by the supernova search team @xcite and the supernova cosmology project @xcite , combined with other observations , is yielding a new standard picture for the contents of the universe , the largest component of which is a dark energy whose properties are consistent with a positive cosmological constant . a striking difference between de sitter and flat space - time is the richer vacuum structure of the former . for a free scalar field in a minkowski space , there exists an essentially unique poincar invariant vacuum state . in contrast , for a de sitter background , mottola @xcite and allen @xcite discovered an infinite family of vacua for the quantum theory of a free massive scalar field that are invariant under the isometries of de sitter space . these vacua can be parameterized by a single complex number , @xmath0 , and are usually called the @xmath0-vacua . most of these @xmath0-vacua have a host of peculiar features , such as a mixture of positive and negative frequency modes at short - distances and a non - thermal behavior that violates the principle of detailed balance . only one of these states , the euclidean or bunch - davies @xcite vacuum , behaves thermally when viewed by an unruh detector @xcite and reduces to the minkowski vacuum as we take the cosmological constant to zero . the assumption that the universe was at least approximately in the euclidean vacuum underlies the successful predictions of inflation for the calculation of the density fluctuations which produced the temperature anisotropies in the cosmic microwave background radiation . despite their unappealing features , the @xmath0-vacua are perfectly valid vacua for a free scalar field . if they can not be shown to be unphysical , then their existence would undercut some of the robustness of the inflationary paradigm we would need to explain how the epoch prior to inflation managed to place the universe in the euclidean vacuum rather than one of the other infinite family of @xmath0-vacua . for example , the regularization needed by the energy - momentum tensor even for the free theory in the @xmath0-vacuum is not generally compatible with that needed after inflation @xcite . a complication in formulating quantum field theory in de sitter space is its lack of a well - defined @xmath1-matrix . in an interacting theory we have two sources of time dependence for matrix elements one induced by any inherent time dependence of the background geometry and another introduced by the interactions . in such a system , it is therefore appropriate only to ask time dependent questions to study how a matrix element evolves from a given initial state . schwinger @xcite and keldysh @xcite developed a formalism to solve for this finite time evolution . in their approach , we specify the state of the system at an initial time and then evolve to a finite time later . here , both the ` in ' and ` out ' states correspond to the same state and are evolved together when we evaluate the expectation value of an operator in effect this formalism evaluates matrix elements between two ` in ' states . the schwinger - keldysh formalism is thus ideally suited for studying the behavior of the @xmath0-vacua in the presence of interactions . we place the system initially in an @xmath0-vacuum and then study whether a sensible evolution results . since the quantum field theory only is evolved over a finite interval , our results are relevant not only for the more formal question of the @xmath0-vacua in an eternal de sitter background but also for the phenomenological problem of a finite epoch of inflation . the methods established here can also be applied to any initial state , such as the ` truncated @xmath0-vacua ' of @xcite . in these vacua , the short - distance behavior of the @xmath0-vacua is modified either in accord with some specific theory , such as the stringy uncertainty relation of @xcite , or simply by truncating the @xmath0 mode functions above some energy scale to reflect our ignorance of the new physics @xcite . we address the formal case of a pure @xmath0-vacuum and shall study the truncated case later in @xcite . in this article , we show that an interacting scalar field theory in a general @xmath0-vacua contains linear divergences which can not be removed with a de sitter invariant renormalization prescription . these divergences appear in the one loop corrections and are present for arbitrarily weak interactions . the specific example we study is the expectation value of the number of euclidean particles in an @xmath0-vacuum . the divergences appear in the high momentum region of the loop integral . we show that they only vanish for the euclidean vacuum , which is completely renormalizable . the subject of the @xmath0-vacua for an interacting theory has also been recently investigated in related work @xcite . both of these works essentially studied the corrections to the two - point correlation function obtained between an ` in ' @xmath0 state and an ` out ' state given by the @xmath0 state at a later time . banks and mannelli @xcite found that the interacting theory in the @xmath0-vacuum required non - local counterterms while einhorn and larsen @xcite found pinched singularities in the loop corrections . these features provided highly suggestive evidence that the @xmath0-vacua are pathological in the presence of interactions . some attempts to modify the theory to avoid these problems appear in @xcite . we begin with a review of the de sitter invariant vacua for a free scalar field in sec . [ greens ] . this section also shows the form of the wightman functions in conformally flat coordinates . section [ sk ] derives the expectation value of an operator in an interacting theory based on the schwinger - keldysh formalism . in sec . [ number ] we calculate the change in the number of euclidean particles in an @xmath0-vacuum due to a cubic interaction and show that in the presence of this interaction , the expectation value is renormalizable for the euclidean vacuum while an unrenormalizable divergence appears for the @xmath0-vacuum . section [ discuss ] explores the origin of these divergences in the @xmath0-vacua in a more general setting . we derive the necessary conditions for these divergences to arise and show how they can appear in a general interacting scalar field theory . section [ conclude ] summarizes our results and suggests future applications for this formalism .

de sitter space - time has a one complex parameter family of invariant vacua for the theory of a free , massive scalar field . for most of these vacua , in an interacting scalar theory these divergences are not of a form that can be removed by a de sitter invariant counterterm , except in the case of the euclidean , or bunch - davies , vacuum .

de sitter space - time has a one complex parameter family of invariant vacua for the theory of a free , massive scalar field . for most of these vacua , in an interacting scalar theory the one loop corrections diverge linearly for large values of the loop momentum . these divergences are not of a form that can be removed by a de sitter invariant counterterm , except in the case of the euclidean , or bunch - davies , vacuum .

1212.6497

i

hard - particle systems in general offer a great opportunity to understand the mechanism of a variety of physical phenomena , including crystal nucleation @xcite , glass transition @xcite and jamming @xcite . the spherical particles , such as hard spheres and hard disks , have been studied intensively during the past decades . in particular , for a system of hard spheres , an entropy - driven liquid - crystal transition occurs at sufficiently high density @xcite and a hard - sphere glass can also be formed above the glass transition point if crystallization is avoided @xcite . on the other hand , non - spherical particles play a fundamental role in the physics of molecular liquids and attract growing interest in recent years because of their usefulness in forming functional structures . from the theoretical point of view , it is interesting to include the rotational degrees of freedom for the particles and investigate the physical behavior of a system composed of anisotropic particles . such study uncovers new types of behavior and holds potential for providing new insights that can not be obtained from studies of spherical objects @xcite . from the practical point of view , the constituent particles often have non - spherical shapes in real materials such as laponite @xcite , and the building blocks with anisotropic shape can provide a powerful candidate for the assembly of particular targeted structures @xcite . thus , exploration of systems composed of anisotropic particles is also valuable for materials design . in this paper , we present an event - driven molecular dynamics ( edmd ) study of hard ellipses . a system of hard ellipses can be regraded as one of the simplest models of two - dimensional ( 2d ) anisotropic particles and its phase behavior is very rich . yet , the thermodynamic , structural and dynamical properties of hard ellipses remain largely unknown . the phase transitions , in particular , the isotropic - nematic ( i - n ) transition , of hard ellipses have been studied by computer simulations and various theoretical approaches . in vieillard - baron s pioneering monte carlo ( mc ) simulations in @xmath7 ensemble ( i.e. , the particle number @xmath8 , the volume @xmath9 and the termperature @xmath10 are constant ) @xcite , three different phases , including isotropic liquid , nematic liquid crystal and solid , have been identified in a hard - ellipse system with an aspect ratio of @xmath11 , where @xmath12 and @xmath13 denote semi - major and semi - minor axes . vieillard - baron observed that both i - n and nematic - solid ( n - s ) transitions are of first order , as evidenced by a small discontinuity shown in the equation of state ( eos , i.e. , density - dependence of the reduced pressure @xmath3 ) . however , computer simulations show that the i - n transition in a 2d fluid of hard needles @xcite and long hard rods @xcite , which in principle share the similar physics with elongated hard ellipses , is continuous rather than first - order . moreover , the @xmath14 ( i.e. , the particle number @xmath8 , the pressure @xmath15 and the termperature @xmath10 are constant ) mc simulations of cuesta and frenkel @xcite indicate that while there is no stable nematic phase in hard ellipses for @xmath16 , the i - n transition is of first order for @xmath17 and continuous via disclination unbinding for @xmath18 . hence , it appears that the nature of the i - n transition depends on the aspect ratio and that the phase diagram possesses a tricritical point at which the transition changes from first - order to continuous . thus , on the simulation side , a definite conclusion has not been reached on the phase transitions of hard ellipses . this issue is also contentious on the theoretical side . ward and lado @xcite found based on the percus - yevick ( py ) integral equation theory that the i - n transition does not take place in a fluid of hard ellipses although orientational ordering is allowed . while , the density - functional theory ( dft ) @xcite and the effective - liquid approach @xcite indicate that the i - n transition is continuous for hard ellipses with arbitrary aspect ratios . the other forms of dft @xcite , the onsager theory - based approach @xcite , and the scaled particle theory ( spt ) @xcite predict the existence of the i - n transition in hard ellipses , but its location differs from these theoretical predictions . therefore , a clear understanding of the phase transitions in hard ellipses is still lacking . here , it should also be mentioned that the phase behavior of binary hard ellipses was recently investigated by a scaled particle dft @xcite and a weighted dft @xcite . another important issue concerns the orientational order in the 2d nematic liquid crystals . straley has shown @xcite that true long - range order ( lro ) can not exist in a 2d nematic phase if the particles interact via a separable potential , but the existence of true lro can not be excluded if the potential is not separable into positional and orientational parts . in fact , it has been demonstrated by computer simulations @xcite that a general property of a 2d nematic phase is the lack of true lro , although earlier simulations @xcite also show that the system with the nonseparable potential exhibits true lro . for this reason , the systems that lack true lro are usually referred to as having quasi - lro . in the elastic continuum theory @xcite , quasi - lro is expected to appear in a 2d nematic phase if the free energy associated with collective fluctuations in the particle orientations can be expressed as @xmath19 where @xmath20 characterizes the orientation at position @xmath21 with respect to a fixed axis and @xmath22 is the 2d frank s elastic constant @xcite . based on eq . ( 1 ) , one can derive that the amplitude of the orientational fluctuations diverges with the system size in a logarithmic manner , @xmath23 where @xmath24 denotes the ensemble average , @xmath25 is boltzmann s constant , @xmath10 is the absolute temperature and @xmath8 is the particle number . as a consequence , both the 2d nematic order parameter @xmath26 and the angular correlation function @xmath27 decay algebraically , i.e. , @xmath28 @xmath29)>\sim r^{-2l^{2}k_{b}t/\pi k},\ ] ] where @xmath30 is a positive integer . equations ( 3 ) and ( 4 ) imply that a 2d nematic phase with quasi - lro can be characterized by a vanishing orientational order parameter in the thermodynamic limit and a power - law decay of the angular correlation function . if the transition from a 2d nematic phase with quasi - lro to an isotropic phase proceeds via the kosterlitz - thouless ( kt ) disclination unbinding mechanism @xcite , it should occur at a universal value of the renormalized frank s constant , @xmath31 @xcite . as pointed out in ref . @xcite , a different mechanism is also possible , but there is no stable nematic phase at values of the frank s constant below @xmath32 . thus , one can quantitatively identify the location where the stable nematic phase starts to form by monitoring the system size dependence of the nematic order parameter or from the spatial correlation of the particle orientations . we note that the algebraic decay of the angular correlation function has been evidenced by simulations of 2d hard needles @xcite , 2d hard rods @xcite , hard ellipses @xcite , and recently by experiments of quasi-2d suspensions of hard ellipsoids @xcite . in contrast to the efforts made on understanding the phase behavior and the structure of hard ellipses , little attention has been paid on their dynamics despite its fundamental and technological importance . the dynamical properties of hard needles have been studied by molecular dynamics ( md ) simulations about three decades ago @xcite . the diffusive process of a single hard ellipsoid confined in two dimensions was measured just several years ago in experiments @xcite . in particular , the diffusion of quasi-2d suspensions of hard ellipsoids with @xmath33 has been explored very recently by video - microscopy experiments @xcite and the enhancement of the translational diffusion with respect to the rotational diffusion has been rationalized in terms of the formation of unstable nematic - like regions with an average lifetime that exceeds the characteristic time of diffusion in a recent mc study @xcite . in spite of the aforementioned progress , our knowledge of the dynamics of hard ellipses is far from complete . to our knowledge , there are no available md results concerning the self - diffusion of hard ellipses so far . in this work , we analyze in detail the thermodynamic , structural and self - diffusive properties in the various phases of hard ellipses , with the aim of examining the effects of aspect ratio and area fraction and exploring connections between different properties . to this end , we perform edmd simulations for hard ellipses over a wide range of aspect ratios and area fractions , covering state points including the isotropic , plastic and nematic phases . we present the eos for a series of @xmath4 values and compare the simulation results with a recent prediction of the spt in the isotropic phase . we find that spt well describes the isotropic branch of the eos in hard ellipses . dependence of @xmath4 on the reduced pressure @xmath3 is explored in the various phases . at large fixed area fractions , @xmath3 increases with @xmath4 in both the isotropic and the plastic phases , and interestingly , its dependence on @xmath4 is rather weak in the nematic phase . we rationalize the thermodynamics of hard ellipses in terms of particle motions , which show different features in different phases . the static structures , both positionally and orientationally , are then investigated in the various phases . our simulations show that the plastic crystal forms for aspect ratios up to @xmath5 , while appearance of the stable nematic phase starts approximately at @xmath6 for the studied density range . the locations of the isotropic - plastic ( i - p ) transition and the i - n transition are determined by analyzing the bond - orientation correlation functions and the angular correlation functions , respectively . as expected , the i - p transition point is found to increase with @xmath4 , while a larger @xmath4 leads to a smaller area fraction where the i - n transition takes place . in addition , our results strongly support that the 2d nematic phase in hard ellipses has only quasi - lro . we further explore the self - diffusion of hard ellipses . we find that a phase transition in the translational degrees of freedom is indeed reflected by an enhancement in the rotational diffusion and vice versa . hence , a clear connection is revealed between the structure and the self - diffusion . we discuss the relevance of our results to the glass transition in hard ellipses , which is a subject of recent experimental investigation @xcite . we finally provide the results of the isodiffusivity lines for hard ellipses and discuss the effect of spatial dimension on the diffusive dynamics of hard ellipsoidal particles by comparing results of hard ellipses and uniaxial hard ellipsoids @xcite . the paper is organized as follows . in sec . ii , we specify the details of the simulation and the model system used in this work . thermodynamic , structural and self - diffusive properties of hard ellipses are presented and discussed in sec . a summary of our findings is given in sec . iv . in the appendix , we add the main ingredients for implementing an edmd simulation for hard ellipses and a brief discussion on the effect of the system size .

despite their fundamental and practical interest , the physical properties of hard ellipses remain largely unknown . in this paper we identify three different phases , including isotropic , plastic and nematic states . we analyze in detail the thermodynamic , structural and self - diffusive properties in the formed various phases of hard ellipses . the reduced pressure increases with in both the isotropic and the plastic phases and , interestingly , its dependence on is rather weak in the nematic phase . we rationalize the thermodynamics of hard ellipses in terms of particle motions . the plastic crystal is shown to form for aspect ratios up to , while appearance of the stable nematic phase starts approximately at . we quantitatively determine the locations of the isotropic - plastic ( i - p ) transition and the isotropic - nematic ( i - n ) transition by analyzing the bond - orientation correlations and the angular correlations , respectively . as expected , the i - p transition point is found to increase with , while a larger leads to a smaller area fraction where the i - n transition takes place . moreover , our simulations strongly support that the two - dimensional nematic phase in hard ellipses has only quasi - long - range orientational order . we discuss the relevance of our results to the glass transition in hard ellipses . finally , the results of the isodiffusivity lines are evaluated for hard ellipses and we discuss the effect of spatial dimension on the diffusive dynamics of hard ellipsoidal particles .

despite their fundamental and practical interest , the physical properties of hard ellipses remain largely unknown . in this paper , we present an event - driven molecular dynamics study for hard ellipses and assess the effects of aspect ratio and area fraction on their physical properties . for state points in the plane of aspect ratio ( ) and area fraction ( ) , we identify three different phases , including isotropic , plastic and nematic states . we analyze in detail the thermodynamic , structural and self - diffusive properties in the formed various phases of hard ellipses . the equation of state ( eos ) is shown for a wide range of aspect ratios and is compared with the scaled particle theory ( spt ) for the isotropic states . we find that spt provides a good description of the eos for the isotropic phase of hard ellipses . at large fixed , the reduced pressure increases with in both the isotropic and the plastic phases and , interestingly , its dependence on is rather weak in the nematic phase . we rationalize the thermodynamics of hard ellipses in terms of particle motions . the static structures of hard ellipses are then investigated both positionally and orientationally in the different phases . the plastic crystal is shown to form for aspect ratios up to , while appearance of the stable nematic phase starts approximately at . we quantitatively determine the locations of the isotropic - plastic ( i - p ) transition and the isotropic - nematic ( i - n ) transition by analyzing the bond - orientation correlations and the angular correlations , respectively . as expected , the i - p transition point is found to increase with , while a larger leads to a smaller area fraction where the i - n transition takes place . moreover , our simulations strongly support that the two - dimensional nematic phase in hard ellipses has only quasi - long - range orientational order . the self - diffusion of hard ellipses is further explored and connections are revealed between the structure and the self - diffusion . we discuss the relevance of our results to the glass transition in hard ellipses . finally , the results of the isodiffusivity lines are evaluated for hard ellipses and we discuss the effect of spatial dimension on the diffusive dynamics of hard ellipsoidal particles .

1212.6497

r

in this section , we present and discuss our results of the thermodynamic , structural and self - diffusive properties in hard ellipses . first of all , the eos is shown for a wide range of aspect ratios and the reduced pressure ( i.e. , compressibility factor ) in the isotropic phase is compared with a recent result of the spt . we also explore the dependence of the reduced pressure on the aspect ratio in the different phases and discuss the influence of the particle motions on the thermodynamics . in subsection iii b , we analyze in detail the positional and orientational order of hard ellipses and compare our results with previous simulations when possible . finally , we investigate the self - diffusion of hard ellipses and demonstrate connections between the structure and the self - diffusion . the relevance of our results to the glass transition in hard ellipses is briefly discussed . we also show the results of the isodiffusivity lines in both the translational and the rotational degrees of freedom and discuss the effect of spatial dimension on the diffusive dynamics of hard ellipsoidal particles . as a function of area fraction @xmath2 for ( a ) hard disks and hard ellipses with @xmath59 and ( b ) hard ellipses with @xmath38 . the results are shifted up by @xmath60 from the preceding small @xmath4 for clarity . the dotted lines are simulation results and the solid lines are the results of the spt [ eq . the green dashed line in ( a ) is the spt prediction for the equation of state of hard disks in the isotropic state , i.e. , eq . ( 10 ) in ref . the circles indicate the area fractions where the system starts to form stable plastic crystals in ( a ) and stable nematic crystals in ( b ) , respectively ( see the structural analysis in subsection iii b ) . the semi - log plot in the inset of fig . 1(b ) highlights the change of dependence of @xmath3 on @xmath2 with increasing @xmath4.,scaledwidth=45.0% ] as a function of @xmath61 at fixed densities . the solid lines are the results of eq . ( 5).,scaledwidth=45.0% ] -dependence of @xmath3 at the highest densities studied . the dotted lines are a guide to the eye . for these densities , the system has a plastic phase for @xmath62 , a nematic phase for @xmath63 and an isotropic phase in between.,scaledwidth=45.0% ] knowledge of an accurate eos for a model system is of fundamental importance and particularly useful for providing information on what type of an underlying phase transition is . we present the results of eos for hard ellipses for various aspect ratios in fig . 1 . the eos for a system of hard disks is also included in fig . 1(a ) . here , the reduced pressure @xmath64 is computed by the momentum exchange from particle collisions in edmd ( see appendix a ) . we checked that our results for @xmath16 , @xmath65 and @xmath66 are consistent with the compression runs in previous mc simulations @xcite . let us first focus on hard ellipses with small @xmath4 values . in this situation , the orientational order is not appreciable in the system but the positional order can develop upon compression , leading to the formation of plastic crystals at sufficiently high densities . the appearance of the positional order can significantly affect the thermodynamics of the system and it is indeed reflected by a discontinuity in the eos , as shown in fig . 1(a ) . the plastic crystal was already observed for a system of hard ellipses with @xmath67 about four decades ago by vieillard - baron @xcite . our results demonstrate that the plastic crystal can be formed in hard ellipses with aspect ratios up to @xmath5 . the observation of the discontinuity in the eos is compatible with the scenario of a first - order transition , but it should be viewed with some caution since the nature of the formation of a 2d crystalline phase is unclear even in the case of hard disks ; e.g. , although the kosterlitz , thouless , halperin , nelson and young ( kthny ) theory may not rule out other melting scenarios @xcite , a famous one is that the melting of hard disks proceeds via two continuous phase transitions with an intermediate phase called hexatic phase @xcite , while the recent large - scale computer simulations @xcite reveal that the liquid - hexatic transition is of first order . we will further discuss this issue in subsection iii b by analyzing the structure of the system . moreover , we find in fig . 1(a ) that the discontinuity shown in the eos becomes less apparent as @xmath4 gets larger and nearly undetectable within the studied density range for @xmath68 . this can be easily understood since the particle s anisotropy tends to destroy the positional order of the system , and thus will be in favor of the isotropic phase . for hard ellipses with sufficiently large elongations , the orientational order is expected to appear and the positional order will be highly suppressed , resulting in the formation of 2d nematic liquid crystals in hard ellipses . as already found in other 2d models @xcite , the thermodynamic properties of the system are indeed not very sensitive to the occurrence of the orientational order . this is also confirmed by our results of the eos for hard ellipses with @xmath63 , as shown in fig . for these @xmath4 values , the system does have a nematic phase above some area fraction , but the corresponding change in the eos is rather mild . thus , we can not unambiguously determine the nature of the i - n transition based on solely the thermodynamics . however , by a close inspection of the results in the semi - log plot [ see the inset of fig . 1(b ) ] , some difference in the eos can be seen with varying @xmath4 . it seems that some inflection points exhibit in the eos for large aspect ratios , while the slope @xmath69 monotonically increases with @xmath2 for small aspect ratios , i.e. , the @xmath2-evolution of @xmath3 differs qualitatively from relatively small to very large aspect ratio . this suggests that the nature of the i - n transition changes with aspect ratio . in fact , previous mc simulations imply the existence of a tricritical point between @xmath17 and @xmath66 in the phase diagram of hard ellipses @xcite . the i - n transition is first - order below the tricritical point and becomes continuous above it . interestingly , we find that the appearance of the inflection points in the eos starts within the same @xmath4 range . thus , our results appear to support the existence of a tricritical point in the phase diagram of hard ellipses . however , since the location of such a point is very hard to determine even if it exists , we will focus on characterizing the properties of different phases in hard ellipses . next , we compare the reduced pressure in the isotropic phase of hard ellipses with a recent result of the spt . as mentioned in section i , the eos of hard ellipses can also be obtained by various theories . hence , it is meaningful to test these theoretical predictions as our results cover a wide window of @xmath4 and @xmath2 and contain data in the various phases . spt in general provides a simple form for the eos of a model system and its usefulness has been confirmed by the study of hard disks @xcite . other existing theories also provide relevant results , but the solution of the eos needs to be solved numerically @xcite . thus , we concentrate here on a very recent result of the spt @xcite . on the basis of the spt , boulk derived an eos for the isotropic fluid of hard ellipses @xcite , which has the following form @xmath70}{(1-\phi)^{2}},\ ] ] where the non - circularity parameter @xmath61 has been introduced . @xmath61 is calculated according to the formula @xmath71 with @xmath72 the perimeter of the ellipse . the value of @xmath72 can be expressed accurately by the complete elliptic integral of the second kind , and a rather good approximation can be obtained from the following formula : @xmath73 where @xmath74 . we use eq . ( 6 ) to calculate @xmath61 in this work and the results of the eq . ( 5 ) are shown as solid lines in fig . 1 . as a reference , the area fractions where the system starts to form stable plastic crystals and stable nematic crystals are also shown in fig . 1 . as can be seen in fig . 1(a ) , the simulation results and the spt prediction are almost indistinguishable in the isotropic phase for @xmath75 , indicating a fairly good description of eq . ( 5 ) for the eos of hard ellipses with small @xmath4 values . meanwhile , the predicted eos deviates from the simulation results above some area fraction for @xmath63 , as shown in fig . 1(b ) . as explained in ref . @xcite , the system is not isotropic but exhibits orientational order at high densities for large @xmath4 values . in fig . 1(b ) , one can find that the area fraction where eq . ( 5 ) starts to deviate from the simulations is much smaller than the i - n transition point [ see circles in the main plot of fig . however , it should be emphasized that the short - range orientational order already appears in the system at smaller area fractions ( see subsection iii b ) . taking this into account , we find that eq . ( 5 ) well describes the isotropic branch of the eos for hard ellipses even with large @xmath4 values . we can also use @xmath61 as a variable to test eq . the results for representative area fractions are shown in fig . we find that the theoretical and simulation results agree well for low densities . at high densities , deviations appear for large @xmath61 values because of the occurrence of short - range or some long - range orientational order , as evidenced by the non - monotonic dependence of @xmath3 on @xmath61 . therefore , we conclude that spt provides a good description of the eos for the isotropic phase of hard ellipses . it is interesting to further explore dependence of the reduced pressure on the aspect ratio in the various phases of hard ellipses . to this end , we plot @xmath3 as a function of @xmath4 for @xmath76 , @xmath77 and @xmath41 in fig . 3 . for these densities , the system has a plastic phase for @xmath62 , a nematic phase for @xmath63 and an isotropic phase in between ( see subsection iii b ) . we find that @xmath3 monotonically increases with @xmath4 at fixed densities in both the isotropic and the plastic phases ( i.e. , @xmath75 at @xmath78 ) , suggesting that the thermodynamics of the system can be significantly influenced by the particle shape when the ellipses are oriented at random . note that this conclusion is still valid for even larger @xmath4 values ( see the results for @xmath79 in fig . 2 ) . interestingly enough , in the nematic phase ( i.e. , @xmath63 at @xmath78 ) , when the particles show some long - range orientational order but have their centers of mass at random , @xmath3 depends on @xmath4 very weakly . although not pointed out before , the same conclusion can be drawn from previous @xmath14 mc simulations @xcite , which indicate that the area fractions are almost identical at the same pressure in the nematic phase of hard ellipse at least for @xmath17 and @xmath66 . to understand the above findings , we recall that the reduced pressure is calculated from the momentum exchange during collisions of two ellipses , which indeed contains translational and rotational parts ( although they are coupled with each other ) . it is expected that the contribution from the rotational motions becomes more apparent as the ellipses get more elongated . this is the reason why @xmath3 increases with @xmath4 in the plastic and the isotropic phases , where the particles can freely move both translationally and rotationally ( see fig . 14 ) . in the nematic phase , however , particles on average line up preferentially along a common direction . in this case , the rotational dynamics becomes extremely slow , but the motions of the particles have little hindrance in the translational degrees of freedom ( see fig . 14 ) and thus the translational part becomes dominant in determining the reduced pressure of the system . we then speculate that the translational part must be very similar for all aspect ratios at the same density in the nematic phase so that @xmath3 has only weak dependence on @xmath4 . therefore , the thermodynamics of hard ellipses could be explained in terms of the particle motions . we now turn to the structural properties of hard ellipses . as a system of hard ellipses with sufficiently large aspect ratios has a transition from the isotropic liquid to the 2d liquid crystal , much attention has been paid on the structure of the nematic phase in hard ellipses , i.e. , on characterizing the orientational order of elongated hard ellipses . instead of focusing solely on the orientational correlations , we explore both positional and orientational order in the various phases of hard ellipses in this subsection . of hard ellipses at @xmath80 for @xmath68 . the results are shifted up by @xmath81 from the preceding small @xmath4 for clarity . as there are two basic length scales in a system of hard ellipses , @xmath82 also peaks at @xmath83 , as indicated by the arrows for @xmath84.,scaledwidth=45.0% ] of hard ellipses with @xmath85 in the vicinity of the isotropic - plastic transition . the solid lines are the results of the oz fittings and the dash dotted lines are the results of the power - law fittings . the green dashed line indicates @xmath86 . the results are similar for other aspect ratios with @xmath62.,scaledwidth=45.0% ] as a function of @xmath4 . the solid line is a guide to the eye.,scaledwidth=45.0% ] as a function of nematic director @xmath87 for hard ellipses with @xmath18 for various @xmath2 . note that the result is shown for a single sample because @xmath88 can not be averaged among different samples since the nematic direction differs from one sample to another.,scaledwidth=45.0% ] as a function of @xmath2 for various @xmath4 . lower : contour plot of @xmath89 in the plane of @xmath2 and @xmath4.,scaledwidth=45.0% ] for hard ellipses with @xmath75 at @xmath80 . the green dashed line indicates @xmath90.,scaledwidth=45.0% ] for a system of hard ellipses with @xmath18 in the vicinity of the isotropic - nematic transition . the solid lines are the results of the power - law fittings . the green dashed line indicates @xmath90 . the results are similar for other aspect ratios with @xmath63 . , scaledwidth=45.0% ] as a function of @xmath4 . the triangle and the squares are the mc results taken from ref . @xcite and ref . @xcite , respectively . the dashed and solid lines are the fits to the edmd data by @xmath91 with @xmath92 and @xmath93 with @xmath94 and @xmath95 , respectively.,scaledwidth=45.0% ] for hard ellipses with @xmath63 at @xmath80.,scaledwidth=45.0% ] and area fraction @xmath2 . ( b)-(e ) representative snapshots for a high - density isotropic phase with @xmath96 and @xmath80 , a low - density isotropic phase with @xmath18 and @xmath97 , a plastic phase with @xmath85 and @xmath80 , and a nematic phase with @xmath84 and @xmath80 . note that there is visually no distinction for the orientation of an ellipse with @xmath98 and @xmath99 so that particles with @xmath98 are shown as the same color as those with @xmath99 in the snapshots.,scaledwidth=45.0% ] we first consider the pair correlation function of hard ellipses , which is defined as @xmath100 where @xmath101 is the distance between the centers of mass of the particles . in general , characteristic peaks will appear in @xmath82 at large distances if the system possesses some long - range positional order . this is indeed observed in systems of hard ellipses with @xmath62 at high densities ( data not shown ) , where the centers of mass of the particles form a triangular ( or hexagonal ) lattice . in hard ellipses with larger aspect ratios , an ordered structure in principle can also form translationally although it does not need to be a perfect triangular one . we present the results of @xmath82 at @xmath80 for @xmath68 in fig . 4 . note that there are two basic length scales ( i.e. , the semi - major axis @xmath12 and the semi - minor axis @xmath13 ) in a system of hard ellipses so that @xmath82 also peaks at the distance of @xmath102 . beyond the distance of @xmath102 , we find from fig . 4 that @xmath82 rapidly decays to the value of @xmath39 , indicating the absence of any long - range translational order even at the largest density studied . the reason , why we did not observe the formation of a translationally ordered structure for large aspect ratios , has been explained in section ii . moreover , it is clear that several peaks are also shown in @xmath82 at between @xmath45 and @xmath102 , suggesting the existence of some short - range translational order . as we shall see later , this is indeed due to the appearance of the orientational order . the pair correlation function only provides a qualitative way to see whether a system possesses long - range positional order or not . meanwhile , we can study the bond - orientation correlation functions @xcite , in order to quantitatively identify the location of the transition from the isotropic liquid to the hexagonal crystal . we first define the sixfold bond - orientation order parameter @xmath103 where @xmath104 , @xmath105 is the number of the neighbors of the @xmath106th particle and @xmath107 is the angle between the vector @xmath108 and the @xmath109 axis . here , we define two ellipses to be neighbors if they overlap when uniformly expanding their sizes by @xmath110 . then , the spatial correlation of @xmath111 can be calculated by @xmath112 the quasi - long - range bond - orientation order is evidenced by an algebraic decay of @xmath113 , i.e. , @xmath114 in particular , according to the kthny theory @xcite , @xmath115 has a value of @xmath116 at the boundary between the liquid phase and the hexatic phase . we thus use the criterion @xmath86 to separate the plastic states from the isotropic states for hard ellipses with @xmath62 . in the liquid phase , the system only exhibits short - range bond - orientation order and @xmath113 should decay exponentially . however , for high densities and especially in vicinity of the formation of the hexatic phase , we find that the bond - orientation correlation function can indeed be better described by the ornstein - zernike ( oz ) function @xmath117 with @xmath118 the static correlation length . in fact , it is observed that the oz function can well characterize the behavior of @xmath113 in 2d supercooled liquids @xcite . we confirm the above claims by presenting the results of @xmath113 for a system of hard ellipses with @xmath85 in fig . as can be seen , @xmath113 decays slowly with increasing @xmath2 . below @xmath119 , the envelopes of @xmath113 can be well fitted by the oz function . with further increasing @xmath2 , a power - law decay with an exponent of @xmath120 holds for the behavior of @xmath113 , indicating the emergence of quasi - long - range bond - orientation order . we thus estimate the location of the isotropic - plastic transition to be @xmath121 for @xmath85 . note that @xmath122 obtained in this way is actually an upper bound to the exact value . the values of @xmath122 for other aspect ratios are shown in fig . and we confirmed that the obtained value for hard disks ( i.e. , @xmath123 ) is in agreement with other studies within the simulation accuracy @xcite . the above analysis indicates that hard ellipses for @xmath62 share the similar behavior with hard disks on the structure as well as the thermodynamics ( see discussion in subsection iii a ) . thus , the nature of the isotropic - plastic transition of hard ellipses should be the same as that of hard disks . turning to the orientational order of hard ellipses , we first focus on the nematic order parameter as a function of @xmath2 and @xmath4 . the 2d nematic order parameter is defined as @xmath124 where @xmath125 is the angle characterizing the orientation of the @xmath106th ellipse with respect to the @xmath109 axis and @xmath87 the orientation of the nematic director . note that this definition is equivalent to eq . ( 3 ) . as the nematic director is not known _ a priori _ , @xmath126 and @xmath87 are usually determined by finding the eigenvalues and the eigenvectors of a tensor order parameter @xcite . as pointed out and observed in previous work @xcite , the nematic order parameter obtained from the above method has a clearly nonzero value even in the isotropic phase due to the finite system size used in simulations . instead of using the tensor order parameter , we set @xmath87 as a variable and calculate @xmath126 as a function of @xmath87 in this work . we show the results of @xmath88 for various area fractions for @xmath18 in fig . 7 as an illustration . as can be seen , @xmath88 shows extrema , and the absolute values of these extrema are the same ( we denote it by @xmath89 ) because of the periodicity of the cosine function in the definition of @xmath126 . we then use @xmath89 to quantify the nematic order of hard ellipses . in fig . 8 , @xmath89 is presented as a function of @xmath2 for various aspect ratios and also shown as a contour plot in the plane of @xmath2 and @xmath4 . we find from fig . 8 that @xmath89 is nearly zero at low densities and has clearly nonzero values at high densities for @xmath63 . thus , it appears that @xmath89 is more sensitive to the onset of the orientational order than @xmath26 . for @xmath75 , a clearly nonzero but still very small value of @xmath89 sets in above @xmath127 , implying the absence of any long - range orientational order . the quasi - long - range orientational order is expected to appear for systems of hard ellipses with larger aspect ratios , as confirmed by the results of @xmath89 for @xmath63 . moreover , the area fraction where the system starts to orientate becomes small as @xmath4 increases , revealing the fact that a system of more elongated hard ellipses has a stronger tendency to form nematic liquid crystals . it is interesting to explore whether @xmath89 is dependent on the system size in the nematic phase . unlike the order parameter @xmath26 computed from the tensor order parameter ( dependence of the system size on @xmath26 has been confirmed by mc studies @xcite ) , we find that the system size does nt strongly influence the value of @xmath89 for both the isotropic and the nematic phases studied ( see appendix b ) . hence , it is not proper to estimate the point of the i - n transition from the dependence of the system size on @xmath89 . we can quantitatively determine the location of the i - n transition @xmath128 from the angular correlation function defined by eq . we mainly concentrate on the case of @xmath129 , i.e. , @xmath130)>\sim r^{-\eta_{2}},\ ] ] where @xmath131 and the average is performed over all pairs with the distance @xmath101 . as introduced in section i , the stable nematic phase is then identified by a power - law decay with @xmath132 for @xmath133 . the results of @xmath133 are shown for @xmath75 at @xmath80 in fig . consistent with the results of @xmath89 , we find that @xmath133 decays much faster than @xmath90 even at the largest studied density for these aspect ratios , indicating again there is no quasi - long - range orientational order in the system . the algebraic decay appears at high densities for @xmath63 , as evidenced in fig . 10 for a system of hard ellipses with @xmath18 . we then use the power law @xmath134 to fit @xmath133 at @xmath135 and estimate the location of the i - n transition @xmath128 to be the area fraction when the value of @xmath136 is smaller than @xmath116 . again , @xmath128 obtained in this way is actually an upper bound to the exact value . it should also be stressed that the i - n transition in a finite system tends to occur at a lower density than in an infinite system because the frank s constant obtained from simulations on small systems is larger than that in the infinite system size @xcite . we show the results of @xmath128 as a function of @xmath4 in fig . as can be expected , @xmath128 decreases with increasing @xmath4 . comparing with previous mc results , we find that @xmath128 for @xmath18 obtained in this work is larger than that in ref . @xcite but close to the result in ref . the reason , why the i - n transition was observed at a smaller area fraction by vieillard - baron , has been hinted by frenkel and eppenga @xcite . similar to the case of @xmath18 , the value of @xmath128 for @xmath17 in this work is also a little smaller than that of ref . the slight difference between the results of ref . @xcite and ours is possibly due to the different particle number and different method used ( @xmath137 and @xmath14 mc in ref . @xcite versus @xmath48 and @xmath7 md in this work ) . we also note a recent mc study @xcite , where @xmath128 for @xmath84 is roughly estimated to be @xmath138 , which is much larger than ours . our results also provide an opportunity to test theoretical predictions on the @xmath4-dependence of @xmath128 for hard ellipses . several theories @xcite have predicted the occurrence of the i - n transition in hard ellipses . in particular , the onsager theory - based approach @xcite and spt @xcite predict that the relation @xmath139 emerges for sufficiently elongated hard ellipses . since the values of @xmath4 studied in this work are not large enough and the hard - needle regime has probably not been reached yet , such a relation can not well describe our results ( see the dashed line in fig . 11 ) . instead , we find that a different form , @xmath93 with @xmath140 and @xmath141 adjustable parameters , provides a fairly good description of our data , as shown by a solid line in fig . 11 with @xmath94 and @xmath95 . as mentioned in section i , the nature of the orientational order in the 2d nematic phase is also a central theme in the study of liquid crystals . the algebraic decay of @xmath133 already indicates the lack of the true long - range orientational order in the nematic phase of hard ellipses . we further confirm that the algebraic orientational order holds even for the largest studied density , as shown in fig . 12 , although @xmath133 may decay very slowly in this case . thus , our simulations strongly support that the nematic phase in hard ellipses has quasi - lro . based on the above structural analysis , we present a phase diagram of hard ellipses and several representative snapshots in fig . 13 . our simulations show three different phases within the investigated state points . for @xmath62 , the system has a transition from a low - density isotropic liquid to a plastic crystal at high densities [ fig . 13(d ) ] . with increasing the aspect ratio up to @xmath16 , we did not observe any phase transition within the whole density range studied in this work [ fig . 13(b ) ] . for hard ellipses with @xmath63 , an isotropic phase [ fig . 13(c ) ] will transform into a nematic phase [ fig . 13(e ) ] at sufficiently high densities . again , we confirm from the snapshot in fig . 13(c ) that some short - range orientational order exists even in the isotropic phase of elongated hard ellipses . on the other hand , although the simulated phase diagram does not include any solid phase , we should stress again that hard ellipses do have the transition from an isotropic or a nematic phase to a solid phase , as revealed in previous mc simulations @xcite . we even expect that a transition from a plastic crystal to a solid phase will occur for small @xmath4 values at even larger densities than @xmath80 . the reason why such a transition was not observed in our work , as explained in section ii , is due to the fact that a solid is hard to spontaneously form from a liquid in @xmath7 md simulations . on the other hand , it will be very interesting to explore the high - density part of the phase diagram for hard ellipses , where the stable crystalline phase will exist . in fact , our knowledge is very limited so far about the equilibrium structure of the solid in hard ellipsoidal particles , although it can be readily recognized that the densest packing of hard ellipses has the same density as the densest packing of hard disks and has a stretched triangular lattice structure @xcite . however , such explorations are hindered by the present method . more elegant methods will be needed to clarify this issue , which remains a challenge from both theoretical and computational points of view . self - diffusion is of great importance in many real processes , but it is still poorly understood in hard ellipses . the diffusive property of hard ellipses with @xmath84 has been studied recently by experiments @xcite and mc simulations @xcite . yet , the effects of aspect ratio and area fraction on the self - diffusion of hard ellipses are unclear . in this subsection , we investigate in detail the self - diffusion of hard ellipses over a wide range of aspect ratios and area fractions , including the isotropic , plastic and nematic phases , and demonstrate the connections between the structure and the self - diffusion . we further discuss the relevance of our results to the glass transition in hard ellipses and provide the results of the isodiffusivity lines . for three aspect ratios . the system has a plastic phase for @xmath142 , an isotropic phase for @xmath16 and a nematic phase for @xmath18.,scaledwidth=45.0% ] the self - diffusion is measured here by the translational and rotational mean squared displacements ( msd ) @xmath143 @xmath144 and the corresponding diffusion constants obtained from the long - time data of the msd @xmath145 @xmath146 we first focus on the behavior of the msd in the different phases of hard ellipses . the results of @xmath147 and @xmath148 are shown at @xmath80 for the plastic ( @xmath142 ) , isotropic ( @xmath16 ) and nematic ( @xmath18 ) phases of hard ellipses in fig . as can be seen , both translational and rotational motions of particles are ballistic at short times , and cross over into diffusive behavior at sufficiently long times in both the plastic and the isotropic phases . moreover , an apparent subdiffusive behavior is seen in the intermediate time regime of the translational msd due to the cage effect in the plastic phase [ see the solid line in fig . by contrast , the rotational diffusion of particles becomes extremely slow in the nematic phase , while the translational msd can still retain a diffusive regime at long times within the simulation time window ( see the dotted lines in fig . we have seen in subsection a that the above features shown in the particle motions may be responsible for the thermodynamics in the different phases of hard ellipses . and ( b ) rotational diffusion constant @xmath149 as a function of area fraction @xmath2 for various @xmath4.,scaledwidth=45.0% ] as a function of translational diffusion constant @xmath150 for various @xmath4 . the green dashed line indicates @xmath151.,scaledwidth=45.0% ] to better illustrate the effects of aspect ratio and area fraction on the self - diffusion of hard ellipses , we extract the translational and rotational diffusion constants from those of the corresponding msds retaining a diffusive regime . the results of @xmath150 and @xmath149 are shown in fig . 15 . at fixed area fractions , we observe that increasing @xmath4 leads to a slowing down of the rotational diffusion , while @xmath150 increases with aspect ratio . moreover , the occurrence of the phase transition seems not to affect these trends at least within the studied density range . then , a natural question is how self - diffusion responds to the phase transition . in a recent md study @xcite , it was found for a system of hard ellipses with @xmath84 that if the diffusion constants are normalized by the values in the infinite dilution limit , then both diffusion constants still monotonically decrease with @xmath2 but the normalized @xmath150 will exceed the normalized @xmath149 at the area fraction where the i - n transition takes place , i.e. , the translational diffusion is enhanced with respect to the rotational diffusion as a result of the appearance of the quasi - long - range orientational order . we did nt observe crossing of the two normalized curves for the same aspect ratio by performing the same analysis ( data not shown ) . instead , our simulations reveal that the self - diffusion responds to the phase transition in a rather straight manner , which can be detected already in each curve ; i.e. , a sudden increase of @xmath149 is seen when a plastic crystal forms , as evidenced by a small peak in @xmath149 in fig . 15(b ) , while the emergence of a nematic phase leads to an enhancement of @xmath150 although this observation seems to become less evident as @xmath4 gets smaller . the small peaks in fig . 15 immediately indicate that the formation of the orientational or positional order will increase the translational or rotational mobilities of particles . hence , a phase transition in the translational degrees of freedom is reflected by a corresponding change in the rotational diffusion and vice versa . we thus uncover a clear connection between the structure and the self - diffusion in hard ellipses . it is interesting to further explore the effect of the particle s anisotropy on the self - diffusion in each degrees of freedom by plotting @xmath149 as a function of @xmath150 at varying @xmath4 . as shown in fig . 16 , @xmath149 is larger than @xmath150 within the full density range for @xmath75 and the particles diffuse slower in the rotational degrees of freedom than in the translational degrees of freedom for @xmath152 . in addition , the curves show an upturn at high densities for @xmath75 , while they turn down for @xmath63 . this means that the translational mobility of hard ellipses will be similar with the rotational mobility at some aspect ratio between @xmath153 and @xmath154 . if a glass is allowed to form in hard ellipses , we then expect that the rotational glass transition sets in at a lower density than the translational glass transition for @xmath155 , which was indeed observed recently in experiments of quasi-2d hard ellipsoids with @xmath156 @xcite . instead , the translational glass transition density will be smaller than the rotational one for @xmath157 . this immediately suggests that there is a point in the phase diagram of glassy hard ellipses , where the translational and the rotational glass transition lines will intersect . therefore , glasses in hard ellipses can be classified into three categories : a plastic glass in which dynamic arrest occurs in the translational degrees of freedom but the system behaviors as a liquid in the rotational degrees of freedom , a liquid glass in which the system becomes glass in the rotational degrees of freedom but the translational motions of particles remains ergodic , and a glass in which motions of the particles are dynamically arrested in both the translational and the rotational degrees of freedom . and area fraction @xmath2 . the solid lines are isodiffusivity lines from translational diffusion coefficients @xmath150 and the dashed lines are isodiffusivity lines from rotational diffusion coefficients @xmath149 . the green squares and the red circles indicate the locations of the isotropic - plastic transition and the isotropic - nematic transition , respectively.,scaledwidth=45.0% ] another relevant property is the isodiffusivity of hard ellipses . as there are related results for hard ellipsoids of revolution @xcite , we can evaluate such isodiffusivity lines for hard ellipses by proper interpolation and thus explore the qualitative effect of spatial dimension . we present the results of the isodiffusivity lines in fig . we find that an almost perpendicular crossing of the translational and the rotational isodiffusivity lines is shown in the plane of @xmath2 and @xmath4 . moreover , the rotational isodiffusivity lines of hard ellipses reproduce qualitatively the shape of the i - n transition line at least for sufficiently small @xmath149 values , which is similar as the result in hard ellipsoids of revolution @xcite . the latter observation indicates that the rotational motions of the ellipses is mostly controlled by the particle s anisotropy . on the other hand , the translational isodiffusivity lines in uniaxial hard ellipsoids mimic the swallowlike shape of the coexistence line between the isotropic liquid and the crystalline phase @xcite , suggesting a non - monotonic change of @xmath150 with @xmath4 at fixed density ( see fig . 2 in ref . note that the swallowlike shape appears because there is a distinction between prolate ( rodlike ) and oblate ( disklike ) particles for hard ellipsoids , while such a distinction does not exist in hard ellipses since there are only two symmetry axes . our results , already apparent from fig . 15 , show that @xmath150 monotonically increases with @xmath4 at fixed @xmath2 , which remains valid even when a phase transition intervenes . thus , it appears that the spatial dimension does influence the translational dynamics . however , it should be noted that our result does not rule out a non - monotonic dependence of @xmath150 on @xmath4 at even larger fixed densities than those presented in fig . in fact , our preliminary study for polydisperse hard ellipses ( which prevent the formation of crystal phases ) does show that the translational glassy dynamics depends on aspect ratio non - monotonically at large fixed densities . therefore , a common mechanism will be responsible for the translational glassy dynamics of hard ellipsoidal particles although different origins may exist at low densities .

, we present an event - driven molecular dynamics study for hard ellipses and assess the effects of aspect ratio and area fraction on their physical properties . for state points in the plane of aspect ratio ( ) and area fraction ( ) , the equation of state ( eos ) is shown for a wide range of aspect ratios and is compared with the scaled particle theory ( spt ) for the isotropic states . we find that spt provides a good description of the eos for the isotropic phase of hard ellipses . at large fixed , the static structures of hard ellipses are then investigated both positionally and orientationally in the different phases . the self - diffusion of hard ellipses is further explored and connections are revealed between the structure and the self - diffusion .

despite their fundamental and practical interest , the physical properties of hard ellipses remain largely unknown . in this paper , we present an event - driven molecular dynamics study for hard ellipses and assess the effects of aspect ratio and area fraction on their physical properties . for state points in the plane of aspect ratio ( ) and area fraction ( ) , we identify three different phases , including isotropic , plastic and nematic states . we analyze in detail the thermodynamic , structural and self - diffusive properties in the formed various phases of hard ellipses . the equation of state ( eos ) is shown for a wide range of aspect ratios and is compared with the scaled particle theory ( spt ) for the isotropic states . we find that spt provides a good description of the eos for the isotropic phase of hard ellipses . at large fixed , the reduced pressure increases with in both the isotropic and the plastic phases and , interestingly , its dependence on is rather weak in the nematic phase . we rationalize the thermodynamics of hard ellipses in terms of particle motions . the static structures of hard ellipses are then investigated both positionally and orientationally in the different phases . the plastic crystal is shown to form for aspect ratios up to , while appearance of the stable nematic phase starts approximately at . we quantitatively determine the locations of the isotropic - plastic ( i - p ) transition and the isotropic - nematic ( i - n ) transition by analyzing the bond - orientation correlations and the angular correlations , respectively . as expected , the i - p transition point is found to increase with , while a larger leads to a smaller area fraction where the i - n transition takes place . moreover , our simulations strongly support that the two - dimensional nematic phase in hard ellipses has only quasi - long - range orientational order . the self - diffusion of hard ellipses is further explored and connections are revealed between the structure and the self - diffusion . we discuss the relevance of our results to the glass transition in hard ellipses . finally , the results of the isodiffusivity lines are evaluated for hard ellipses and we discuss the effect of spatial dimension on the diffusive dynamics of hard ellipsoidal particles .

1507.03966

i

the exoplanet revolution is well underway . the last decade has seen order - of - magnitude increases in the number of known planets beyond the solar system . surveys reveal that while hot jupiters on few - day orbits occur around @xmath01% of sunlike stars @xcite ; smaller , cooler sub - neptunes and super - earths occur far more frequently @xcite , and are 23@xmath1 more common still around low - mass stars @xcite . at larger orbital separations planets with super - jovian masses are rare , occurring around @xmath25% of stars @xcite . even greater than the advances made in planetary demographics is the dramatic progress made toward understanding the atmospheres of these distant worlds . detailed characterization of a planet s atmosphere provides the best hope for distinguishing the makeup of its outer layers , and the _ only _ hope for understanding the interplay between initial composition @xcite , chemistry @xcite , dynamics & circulation @xcite , and disequilibrium processes @xcite . mass and radius measurements alone can not uniquely identify the h@xmath3 , ice , and rock content of many planets @xcite . atmospheric measurements are essential . given the rapid expansion of the field , a steady stream of updated reviews is required to keep up to date . a number of excellent review articles have been published on the topic of exoplanet atmospheres , and much in these remains accurate and relevant @xcite . since the latest of these do not discuss the most recent exciting developments , while others discuss results now seen to have been illusory , the time is ripe for another comprehensive review . this article gives one observer s perspective of our current understanding of extrasolar planet atmospheres . the document is organized as follows : observing methods and the modeling approaches used to infer atmospheric properties are described in secs . [ sec : obs ] and [ sec : retrieval ] , respectively . [ sec : composition ] summarizes the status of measurements of atmospheric composition and chemistry , leading in to a discussion of clouds and hazes in sec . [ sec : clouds ] and circulation , energy budgets , and variability in sec . [ sec : energy ] . finally , sec . [ sec : future ] concludes with a look toward the exciting future of the field .

detailed characterization of an extrasolar planet s atmosphere provides the best hope for distinguishing the makeup of its outer layers , and the only hope for understanding the interplay between initial composition , chemistry , dynamics & circulation , and disequilibrium processes . in recent years this article gives an observer s perspective on the current understanding of extrasolar planet atmospheres prior to the considerable advances expected from the next generation of observing facilities .

detailed characterization of an extrasolar planet s atmosphere provides the best hope for distinguishing the makeup of its outer layers , and the only hope for understanding the interplay between initial composition , chemistry , dynamics & circulation , and disequilibrium processes . in recent years , some areas have seen rapid progress while developments in others have come more slowly and/or have been hotly contested . this article gives an observer s perspective on the current understanding of extrasolar planet atmospheres prior to the considerable advances expected from the next generation of observing facilities . atmospheric processes of both transiting and directly - imaged planets are discussed , including molecular and atomic abundances , cloud properties , thermal structure , and planetary energy budgets . in the future we can expect a continuing and accelerating stream of new discoveries , which will fuel the ongoing exoplanet revolution for many years to come .

astro-ph0001096

i

recent detections of distant dusty galaxies with the scuba camera ( the submillimeter common user bolometer array ; holland et al . 1999 ) on the 15 m james clerk maxwell telescope constitute a substantial fraction of the cosmic fir background detected by the _ firas _ and _ dirbe _ experiments on the _ cobe _ satellite ( puget et al . 1996 ; guiderdoni et al . 1997 ; schlegel , finkbeiner , & davis 1998 ; fixsen et al . 1998 ; hauser et al . 1998 ; lagache et al . 1999 ) . since the observed fir background is comparable to the total unobscured emission at ultraviolet / optical wavelengths , a full determination of the global star formation history of the universe requires a comprehensive understanding of this dust - enshrouded galaxy population . the 850 @xmath0 m scuba surveys to date have reported galaxy number counts that are in general agreement ( smail , ivison & blain 1997 ; barger et al . 1998 ; hughes et al . 1998 ; blain et al . 1999a ; eales et al . 1999 ; barger , cowie , & sanders 1999 ) ; the cumulative surface density above 2 mjy is about @xmath9 . the discrete sources have bolometric luminosities that are characteristically @xmath10 if they lie at @xmath11 . moreover , the sources for which measurements exist at multiple wavelengths ( e.g. ivison et al . 1998 ) show thermal spectral energy distributions ( seds ) . thus , the scuba sources are inferred to be the distant analogs of the local ultraluminous infrared galaxy ( ulig ; sanders & mirabel 1996 ) population . an essential observational goal is to determine the redshift distribution of the submillimeter population in order to trace the extent and evolution of obscured emission in the distant universe . however , identifying the optical / nir counterparts to the submillimeter sources is difficult due to the uncertainty in the scuba positions . barger et al . ( 1999b ) presented a spectroscopic survey of possible optical counterparts to a flux - limited sample of galaxies selected from the 850 @xmath0 m survey of massive lensing clusters by smail et al . candidate optical counterparts in the scuba error - boxes were identified using moderately deep ground - based and _ hst _ exposures ( @xmath12 and @xmath13 , respectively ) . one - quarter of the sources could be reliably identified , and those had redshifts in the range @xmath8 . a lower limit of 20 per cent of the full sample showed signs of agn activity . however , for the majority of the submillimeter sources there were either no optical counterparts or the optical associations were not secure . such sources could either be at very high redshift or be so highly obscured that they emit their energy almost entirely in the submillimeter . high resolution radio continuum maps with subarcsecond positional accuracy and resolution offer new opportunities for locating submillimeter sources and determining their physical properties . the unique advantage of centimeter and fir observations is that galaxies and the intergalactic medium are transparent at these wavelengths , so observed flux densities are proportional to intrinsic luminosities . in galaxies without a powerful agn , the radio luminosity is dominated by diffuse synchrotron emission from relativistic electrons accelerated in supernovae remnants from stars more massive than @xmath14 . these massive stars live @xmath15 yr ; the relativistic electrons probably live @xmath16 yr ( condon 1992 ) . thus , radio observations probe very recent star formation . fir observations of starburst galaxies are also a direct measure of massive star formation . as summarized by condon ( 1992 ) , radio continuum emission and thermal dust emission are empirically observed to be tightly correlated due to both being linearly related to the massive star formation rate . if the fir - radio correlation applies to high redshift , as is plausibly the case ( though at the very highest redshifts compton cooling of the relativistic electrons by the microwave background may suppress the radio emission ) , then very sensitive radio observations can be used to pinpoint distant submillimeter sources . in this paper we investigate the feasibility of using radio data to identify and characterize the bright submillimeter source population . richards ( 1999b ) recently obtained an extremely deep very large array ( vla ) 1.4 ghz image centered on the hubble deep field ( hdf ) . richards et al . ( 1999 ) matched ground - based optical data from barger et al . ( 1999a ) to the radio image and found that @xmath17 per cent of the galaxies in the sample could not be identified to optical magnitude limits of @xmath18 . in other respects , such as radio size and spectral index , the optically - faint objects were not any different from the remaining population . richards et al . ( 1999 ) proposed four possible scenarios to explain this population , including @xmath19 obscured starbursts ( beyond @xmath20 the sensitivity to star forming galaxies cuts off due to the flux density limits of the radio data ) , extreme redshift ( @xmath21 ) agn , @xmath22 obscured agn , or one - sided radio jets . in the first phase of our program we observed with lris on the keck ii 10 m telescope a complete subsample of the radio sources . our primary objective was to determine the redshifts of the optical / nir - faint ( @xmath23 ) radio sources . although we were able to spectroscopically identify nearly all the objects in our subsample to @xmath24 ( all had @xmath25 ) , we were unable to obtain redshifts for the fainter objects . either these sources are distant ( @xmath26 ) with spectral features lying outside the optical wavelength range , or the visibility of remarkable features is strongly affected by dust in the galaxies . if the optical / nir - faint radio sources are highly dust obscured systems , then it is possible that they will be detectable in the submillimeter . in the second phase of our program we observed with scuba 15 of the 22 optical / nir - faint radio sources in the central @xmath27 square arcminute region of the radio map ; another 4 were observed by hughes et al . ( 1998 ; hereafter h98 ) in the hdf - proper . the jiggle map mode enabled simultaneous observations of a large fraction ( 31/48 ) of the optical / nir - bright radio sources . even with relatively shallow scuba observations ( a @xmath2 detection limit of 6 mjy at 850 @xmath0 m ) , we detected 5 of the optical / nir - faint radio sources ; a sixth with submillimeter flux @xmath28 mjy was detected in the deep hdf - proper submillimeter map of h98 . in contrast , none of the optical / nir - bright radio sources were detected . we additionally detected two @xmath5 mjy sources that did not have radio counterparts . thus , our targeted scuba survey of optical / nir - faint radio sources turned up @xmath29 per cent of the bright submillimeter sources in our surveyed areas . in the final phase of our program , we explored the feasibility of obtaining redshift estimates from the submillimeter - to - radio flux ratios , as recently suggested by carilli & yun ( 1999 ) . we find that the redshifted arp 220 sed reasonably describes both redshifted local ulig data and known high redshift submillimeter sources and hence can be used as a rough redshift estimator . we estimate that all of our bright submillimeter sources fall in the redshift range @xmath6 , consistent with the redshifts for the lensed submillimeter sources of barger et al . ( 1999b ) . once we have redshifts for the distant submillimeter source population , we can determine the global evolution of star formation in dust - obscured galaxies . previous studies of the star formation rate density ( sfrd ) have primarily used rest - frame ultraviolet data ( e.g. madau et al . however , the ultraviolet emission from a galaxy is heavily affected by the presence of even small amounts of dust , and the extinction corrections are highly uncertain ; for example , corrections for dust obscuration at @xmath30 range from factors of @xmath31 ( pettini et al . 1997 ) to factors of @xmath32 ( meurer et al . 1997 ) , though the more recent estimate of meurer , heckman , & calzetti ( 1999 ) lowers the latter to a factor of @xmath33 . only direct measurements of the reradiated light at submillimeter wavelengths can securely address the sfrd at high redshifts . the true sfrd will receive contributions from both the ultraviolet / optical and the submillimeter . we find that the submillimeter contribution to the sfrd in the @xmath6 range from our @xmath5 mjy sources is comparable to the observed ultraviolet / optical sfrd contribution . the ultraviolet lyman - break galaxies are on the average undetected in the submillimeter at a @xmath34 level of @xmath35 mjy , and thus the lyman - break galaxy population is largely distinct from the bright submillimeter population ( chapman et al . 1999 ) . if we assume that fainter submillimeter sources have the same redshift distribution and properties as the @xmath5 mjy sample , then the sfrd of the entire population contributing to the submillimeter background is about an order of magnitude higher than the observed ultraviolet / optical sfrd . the contribution from uligs to the sfrd increases by more than two orders of magnitude from @xmath7 to @xmath6 , which supports a scenario in which the distant submillimeter sources are the progenitors of massive spheroidal systems . this rapid evolution in the ulig population sampled by scuba can be reproduced in models ( e.g. , blain et al . 1999b , c ) . 2 we present our radio sample and optical / nir imaging , along with our new scuba and lris observations . in 3 we determine our radio - selected submillimeter source counts . in 4 we introduce the predicted high redshift submillimeter - radio flux correlation and obtain millimetric redshift estimates for our sources . in 5 we use the complementary information from radio and submillimeter fluxes to gain insights into the characteristics of our radio - selected submillimeter source population . in 6 we compare the rest - frame colors of the radio sources that have spectroscopic redshifts with the rest - frame colors of our submillimeter sources and find that the latter are likely to fall in the extremely red object category . in 7 we calculate the luminosities , number densities , and sfrds of our submillimeter sources . we compare the submillimeter contribution to the sfrd with contributions from radio and ultraviolet / optical wavebands over a range of redshifts . in 8 we summarize our main conclusions . we take @xmath36 km s@xmath37 mpc@xmath37 and consider both @xmath38 , @xmath39 and @xmath40 , @xmath41 , which should cover the full range of possible cosmologies .

even with relatively shallow scuba observations ( a detection limit of 6 mjy at 850 m ) , we were successful at making submillimeter detections of optical / nir - faint ( and ) radio sources , and our counts closely match the bright counts from submillimeter surveys . redshift estimation , places the bright submillimeter population at where it forms the high redshift tail of the faint radio population . the star formation rate density ( sfrd ) due to ultraluminous infrared galaxies increases by more than two orders of magnitude from to .

direct submillimeter imaging has recently revealed the 850 m background to be mostly composed of a population of distant ultraluminous infrared galaxies , but identifying the optical / near - infrared ( nir ) counterparts to these sources has proved difficult due to the poor submillimeter spatial resolution . however , the proportionality of both centimeter and submillimeter data to the star formation rate suggests that high resolution radio continuum maps with subarcsecond positional accuracy can be exploited to locate submillimeter sources . in this paper we present results from a targeted scuba survey of micro - jansky radio sources in the flanking fields of the hubble deep field . the sources were selected from the uniform ( 8 jy at 1 ) 1.4 ghz vla image of richards ( 1999b ) . even with relatively shallow scuba observations ( a detection limit of 6 mjy at 850 m ) , we were successful at making submillimeter detections of optical / nir - faint ( and ) radio sources , and our counts closely match the bright counts from submillimeter surveys . an important corollary is that a large fraction of the bright ( mjy ) submillimeter sources in untargeted submillimeter surveys have extremely faint optical / nir counterparts and hence are inaccessible to optical imaging and spectroscopy . however , redshift estimates can be made from the ratio of the submillimeter flux to the radio flux across the 100 ghz break in the spectral energy distribution . this procedure , which we refer to as _ millimetric _ redshift estimation , places the bright submillimeter population at where it forms the high redshift tail of the faint radio population . the star formation rate density ( sfrd ) due to ultraluminous infrared galaxies increases by more than two orders of magnitude from to . the sfrd at high redshift inferred from our mjy submillimeter observations is comparable to that observed in the ultraviolet / optical .

1312.7695

i

farfield narrowband source localization based on observed snapshots of a sensor array is a major problem in array signal processing @xcite , known also as direction of arrival ( doa ) estimation . the difficulty of the problem arises from the fact that the observed snapshots are nonlinear functions of the directions of interest . according to estimation schemes adopted , existing methods for the doa estimation can be classified into three categories : parametric , nonparametric and semiparametric , which are described as follows . _ parametric _ methods explicitly carry out parameter estimation using optimization or other methods . a prominent example is the nonlinear least squares ( nls ) method ( see , e.g. , @xcite ) , which adopts the least squares criterion and has a strong statistical motivation . however , nls admits the following two shortcomings : 1 ) it requires the knowledge of the source number which is typically unavailable , and 2 ) its global optimum can not be guaranteed with a practically efficient algorithm due to nonconvexity caused by the nonlinearity nature . either of them may greatly degrade the parameter estimation performance . music @xcite does the parameter estimation by studying the subspace of the data covariance matrix and is a large - snapshot realization of the maximum likelihood ( ml ) method in the case of uncorrelated sources . but it also requires the source number and does not have reliable performance when the number of snapshots is small or correlation exists between the sources . other subspace - based methods include esprit and their variants ( see @xcite for a complete review ) . in contrast , a _ nonparametric _ method typically produces a _ dense _ spectrum whose peaks are interpreted as source directions . so an alternative name _ dense _ is also used . examples include conventional beamformer and mvdr ( or capon s method ) ( see e.g. , @xcite ) and recently introduced iterative adaptive approach ( iaa ) @xcite which eliminates to a large extent leakage of the beamformer and is robust to correlation of the sources . but it will be shown in this paper that iaa suffers from resolution limit , especially in the moderate / low snr regime . while the classification above is consistent with @xcite , music is sometimes categorized in the literature as a nonparametric / dense method by interpreting the plot of its objective function as its power spectrum whose peaks are the optima of the directions . _ semiparametric _ methods have been popular in the past decade which remove the direction variables in the observation model by parameter discretization and transform the nonlinear parameter estimation problem into a sparse signal recovery problem under a linear model , followed by some sparse signal recovery technique and support detection of the sparse solution . in particular , the continuous direction range is approximated by a set of discrete grid points under the assumption that the grid is fine enough such that any of the true sources lies on ( practically , close to ) some grid point . after that , the knowledge is exploited that each of the expanded vector of source signals [ composed of ( virtual ) source signals from candidate directions on the grid ] is a sparse signal since the grid size is greatly larger than the source number , and a sparse solution is sought after . finally , the directions are retrieved from the support of the sparse solution . consequently , the semiparametric methods are also named as _ sparse _ methods in contrast to _ dense_. in principle , a source will be detected once the estimated source signal / source power from some direction is nonzero . following from the literature of sparse signal representation ( ssr ) and later developed compressed sensing ( cs ) @xcite , @xmath0 norm minimization and other sparse signal recovery techniques have been widely used in the semiparametric methods ( see , e.g. , @xcite ) . however , their theoretical support based on ssr and cs can not be applied due to dense discretization . moreover , the sparse recovery techniques usually require one or more practically unknown parameters , e.g. , the noise statistics , the source number or regularization parameters , etc . the semiparametric iterative covariance - based estimation ( spice ) method @xcite is a breakthrough of the semiparametric methods , in which covariance fitting criteria are adopted with sound statistical motivation , the source power and noise variance(s ) are estimated in a natural manner , no user - parameters are required and a connection to the @xmath0 norm minimization is shown . in principle , the semiparametric methods are approximation methods due to the discretization scheme adopted . furthermore , grid selection remains a major problem since 1 ) a coarse grid leads to a high modeling error and 2 ) too dense a grid is computationally prohibitive and might result in computational instability ( see , e.g. , @xcite ) . to alleviate the drawbacks of the discretization , preliminary results have been obtained in @xcite . overall , because of the inherent discretization scheme , the performance of semiparametric methods is dependent on the trade off between the discretization grid size and the computational workload . this paper aims at developing discretization - free techniques for doa estimation with an affordable computational workload in a common scenario of linear arrays , more specifically , uniform linear arrays ( ulas ) and sparse linear arrays ( slas ) . we consider stochastic source signals and the same covariance fitting criteria as in spice . by exploiting the hermitian toeplitz structure in the data covariance matrix , the covariance fitting problem is cast as semidefinite programming ( sdp ) and solved using off - the - shelf sdp solvers , e.g. , sdpt3 @xcite , in a polynomial time . a postprocessing technique is presented in this paper to retrieve , from the data covariance estimate , the parameters of interest including source locations , source powers and noise variance(s ) . the proposed method is a _ sparse _ method because it utilizes the same covariance fitting criteria of the semiparametric spice method and guarantees to produce a sparse parameter estimate . at the same time , it is a _ parametric _ method since it is proven to be equivalently solving a covariance fitting problem parameterized by the aforementioned parameters . therefore , the proposed method is named as sparse and parametric approach ( spa ) . the spa method differs from the existing sparse / semiparametric methods by no need of discretization . unlike the existing parametric methods , spa is based on convex optimization and does not require the source number . theoretical analysis shows that spa is a large - snapshot realization of the ml estimation and is statistically consistent ( in the number of snapshots ) under uncorrelated sources . other merits of spa include improved resolution , applicability to arbitrary number of snapshots , robustness to correlation of the sources and no requirement of user - parameters . numerical simulations are carried out to verify our theoretical results and demonstrate the superior performance of spa compared to existing methods . though most , if not all , of the existing sparse methods can be applied to sensor arrays with arbitrary geometry , problems may occur when applied to the ula and sla cases as studied in this paper . in particular , their parameter estimates might suffer from some identifiability problem and not be truly sparse as demonstrated in this paper , which conflicts with the name _ sparse _ and decreases their resolution considerably . as a byproduct , a modified spice method , named as spice - pp , is presented to rectify the problem by incorporating the postprocessing technique presented in this paper . in the single - snapshot case , the parameter estimation problem studied in this paper is mathematically equivalent to spectral analysis @xcite , for which discretization - free methods have been developed recently in @xcite . it is noted that the results of @xcite are based on very different techniques . furthermore , @xcite and @xcite are mainly focused on the noise - free case ( the former on the ula case and the latter on the sla case in the language of this paper ) . @xcite studies the ula case in the presence of i.i.d . gaussian noise with known noise statistics . the spa method proposed in this paper can deal with all of the above scenarios without the knowledge of noise statistics . while this paper is focused on the array processing applications in which the multisnapshot case is of the main interest , it is still not clear whether the techniques in @xcite specialized for the single - snapshot case can be extended to other cases . notations used in this paper are as follows . @xmath1 , @xmath2 and @xmath3 denote the sets of real numbers , nonnegative real numbers and complex numbers , respectively . boldface letters are reserved for vectors and matrices . for an integer @xmath4 , @xmath5 $ ] is defined as the set @xmath6 . @xmath7 denotes the absolute value of a scalar or cardinality of a set . @xmath8 , @xmath9 and @xmath10 denote the @xmath0 , @xmath11 and frobenius norms , respectively . @xmath12 , @xmath13 and @xmath14 are the matrix transpose , conjugate transpose and complex conjugate of @xmath15 , respectively . @xmath16 is the @xmath17th entry of a vector @xmath15 and @xmath18 is the @xmath19th entry of a matrix @xmath20 . unless otherwise stated , @xmath21 is a subvector of @xmath15 with the index set @xmath22 . for a vector @xmath15 , @xmath23 is a diagonal matrix with @xmath15 being its diagonal . @xmath24 means @xmath25 for all @xmath17 . @xmath26 denotes the trace of a matrix @xmath20 . for positive semidefinite matrices @xmath20 and @xmath27 , @xmath28 means that @xmath29 is positive semidefinite . @xmath30 denotes expectation and @xmath31 is an estimator of @xmath32 . for notational simplicity , a random variable and its numerical value will not be distinguished . the rest of this paper is organized as follows . section [ sec : problem ] describes the problem of array processing with linear arrays . section [ sec : spa_ula ] presents the proposed spa method in the ula case followed by section [ sec : spa_sla ] on the sla case . section [ sec : property ] introduces theoretical properties of spa . section [ sec : connectionprior ] discusses its connections to existing methods . section [ sec : simulation ] presents our numerical simulations . section [ sec : conclusion ] concludes this paper .

direction of arrival ( doa ) estimation in array processing using uniform / sparse linear arrays is concerned in this paper . while sparse methods via _ approximate _ parameter discretization have been popular in the past decade , the discretization may cause problems , e.g. , modeling error and increased computations due to dense sampling . in this paper , an _ exact _ discretization - free method , named as sparse and parametric approach ( spa ) , is proposed for uniform and sparse linear arrays . spa carries out parameter estimation in the continuous range based on well - established covariance fitting criteria and convex optimization . it guarantees to produce a sparse parameter estimate without discretization required by existing sparse methods . theoretical analysis shows that the spa parameter estimator is a large - snapshot realization of the maximum likelihood estimator and is statistically consistent ( in the number of snapshots ) under uncorrelated sources . other merits of spa include improved resolution , applicability to arbitrary number of snapshots , robustness to correlation of the sources and no requirement of user - parameters . numerical simulations are carried out to verify our analysis and demonstrate advantages of spa compared to existing methods . array processing , doa estimation , sparse and parametric approach ( spa ) , continuous parameter estimation , compressed sensing .

direction of arrival ( doa ) estimation in array processing using uniform / sparse linear arrays is concerned in this paper . while sparse methods via _ approximate _ parameter discretization have been popular in the past decade , the discretization may cause problems , e.g. , modeling error and increased computations due to dense sampling . in this paper , an _ exact _ discretization - free method , named as sparse and parametric approach ( spa ) , is proposed for uniform and sparse linear arrays . spa carries out parameter estimation in the continuous range based on well - established covariance fitting criteria and convex optimization . it guarantees to produce a sparse parameter estimate without discretization required by existing sparse methods . theoretical analysis shows that the spa parameter estimator is a large - snapshot realization of the maximum likelihood estimator and is statistically consistent ( in the number of snapshots ) under uncorrelated sources . other merits of spa include improved resolution , applicability to arbitrary number of snapshots , robustness to correlation of the sources and no requirement of user - parameters . numerical simulations are carried out to verify our analysis and demonstrate advantages of spa compared to existing methods . array processing , doa estimation , sparse and parametric approach ( spa ) , continuous parameter estimation , compressed sensing .

1311.6785

i

the increasing availability of large - scale relational datasets in a variety of fields has led to the widespread analysis of complex networks . in particular , the current interest in quantitative social sciences has been fuelled by the importance of social networks and by the wealth of socio - economic datasets widely available today @xcite . due to the sheer complexity of these networks , it has become crucial to develop tools for network analysis that can increase our insight into such data . a key direction in this area is that of _ community detection _ , which aims at extracting a simplified , yet meaningful , coarse - grained representation of a complex network in terms of ` communities ' of nodes at different levels of resolution @xcite . a common characteristic of many social , engineering and biological networks is the importance of directionality . clearly , it is not the same to ` follow ' a widely known personality in twitter than to be followed by one . directionality is also key in food - webs @xcite , brain networks @xcite , economics datasets @xcite , protein interaction networks @xcite , and trade networks @xcite , to name but a few . failure to consider directionality when present in the data , as is commonly done in numerous network analyses , entails ignoring the true nature of the asymmetric relationships and information propagation . from a methodological perspective , however , the analysis of directed networks presents unique challenges that put them beyond standard methodologies . in particular , it is difficult to extend the structural notion of community ( i.e. , a group of nodes with strong connectivity within and with weaker connectivity to the outside ) to the case of directed networks . here we show how the analysis of flow patterns on a network can be integrated to provide a framework for community @xcite and role @xcite detection . this framework is naturally applicable to directed networks where flow is an intrinsic feature of the system they represent . our analysis is able to reveal a layered view of the data from four complementary perspectives : interest communities of nodes at different levels of resolution ; a personalised view of interest in the network from any vantage point ; the identification of user roles in the network based on directed flows ; and a classification of the interest communities into distinctive information organigrams . our framework is applicable to generic directed networks , but we showcase our approach through the analysis of the twitter network of influential twitter users during the 2011 riots in england , compiled from the list published by the british newspaper the guardian . the riots of august 6 - 10 2011 in england were followed by an intense public debate about the role and influence of social media during the unrest . politicians , journalists , pundits and bloggers alike weighed in on the issue but few arguments were based on data @xcite . a few months after the riots , the guardian made available to the public a list of the 1000 ` most influential ' ( i.e. , most _ re - tweeted _ ) twitter users during the riots @xcite . the guardian s list comprised a diverse set of twitter users including newspapers , broadcasting services , news agencies , as well as individual accounts of journalists , politicians , entertainers , global and local activists , and members of the public . to enable a quantitative analysis of the guardian s list , we mined twitter in february 2012 and recovered the _ directed _ network of followers within the list ( see si ) . henceforth we study the largest connected component of this network consisting of @xmath0 nodes ( twitter users ) . the remaining 86 users were either disconnected ( i.e. , they did not follow nor were followed by anyone on the list ) or their accounts had been since deleted . in our network , an edge indicates that the source node is subscribed to the _ tweets _ of the target node , i.e. , the direction of the edge indicates the declared interest whereas information and content travel in the opposite direction ( fig . s1 in the si ) . ) . the inset shows the network with nodes and edges coloured according to their community at four illustrative markov times . two of these partitions at different resolutions are shown in more detail : * b. * at relatively short markov times ( @xmath1 ) we find 149 communities ( coarse - grained network view in the centre ) . three examples of communities in this partition are ` police and crime reporting ' ( top ) , ` hackney ' ( bottom ) , ` the daily telegraph ' ( left ) shown with their members and their self - description word clouds ; * c. * at longer markov times ( @xmath2 ) we find 4 communities ( coarse - grained view in the centre ) : three large communities broadly corresponding to ` uk ' ( bottom - right ) , ` international ' ( top ) , ` celebrities / entertainment ' ( bottom - left ) , and a small one corresponding to the ` bbc ' ( right ) . a detailed view of the partitions can be found in the si.,scaledwidth=80.0% ]

directionality is a crucial ingredient in many complex networks in which information , energy or influence are transmitted . in such directed networks , we showcase here a flow - based approach for community detection in networks through the study of the network of the most influential twitter users during the 2011 riots in england . firstly , we use directed markov stability to extract descriptions of the network at different levels of coarseness in terms of interest communities , i.e. , groups of nodes within which flows of information are contained and reinforced . the study of flows also allows us to generate an interest distance , which affords a personalised view of the attention in the network as viewed from the vantage point of any given user .

directionality is a crucial ingredient in many complex networks in which information , energy or influence are transmitted . in such directed networks , analysing flows ( and not only the strength of connections ) is crucial to reveal important features of the network that might go undetected if the orientation of connections is ignored . we showcase here a flow - based approach for community detection in networks through the study of the network of the most influential twitter users during the 2011 riots in england . firstly , we use directed markov stability to extract descriptions of the network at different levels of coarseness in terms of interest communities , i.e. , groups of nodes within which flows of information are contained and reinforced . such interest communities reveal user groupings according to location , profession , employer , and topic . the study of flows also allows us to generate an interest distance , which affords a personalised view of the attention in the network as viewed from the vantage point of any given user . secondly , we analyse the profiles of incoming and outgoing long - range flows with a combined approach of role - based similarity and the novel relaxed minimum spanning tree algorithm to reveal that the users in the network can be classified into five roles . these flow roles go beyond the standard leader / follower dichotomy and differ from classifications based on regular / structural equivalence . we then show that the interest communities fall into distinct informational organigrams characterised by a different mix of user roles reflecting the quality of dialogue within them . our generic framework can be used to provide insight into how flows are generated , distributed , preserved and consumed in directed networks .

1311.6785

r

to gain a structured view of the communities in the network at different levels of resolution , we use markov stability community detection @xcite which has been extended to deal with directed networks ( see methods , si and ref . a key advantage of markov stability is that it is based on a quantitative criterion that relies on flow propagation and containment , and thus identifies _ flow communities_. the communities so found correspond to ` interest communities ' , insomuch as information , interest and influence are propagated , retained and reinforced within them following the edges . if edge directionality is ignored , the community structure is blurred and the analysis severely hindered , as shown below . a second advantage of our method is that the network is scanned for structure at all scales , and flow communities are found to be relevant at different levels of resolution . figure [ fig : communities]a illustrates how , as the network is swept by a continuous - time diffusion process , the method goes from detecting many small , granular communities ( at short markov times ) to fewer and coarser communities ( at longer markov times ) . as a visual aid to interpret the theme of the communities , we create ` word clouds ' from the most frequently - used words in the twitter self - biographies of the users in each community . it is important to remark that the biographies are not used in the network analysis , i.e. , the word clouds serve as an independent , _ a posteriori _ annotation or ` self - description ' of the communities found ( see si ) . an example of a highly granular partition ( 149 communities ) at short markov times is shown in fig . [ fig : communities]b ( and figs . s3 and s4 in the si ) . at this resolution , some communities are defined by the geographic origin of the twitter accounts ( e.g. , midlands , manchester , liverpool , even croydon and hackney within london ) ; others are determined by employer or institution ( e.g. , media such as the independent , itv , channel 4 , or the daily telegraph ) ; while others correspond to interest groups ( e.g. , a community grouping together police forces and fire departments of riot areas with crime reporters and civil organisations highlights the police s use of twitter during the riots @xcite ) . as the markov time increases , we find coarser partitions with larger communities . at @xmath3 we find 48 communities , including a football / sports community ( clubs , athletes , sports journalists and supporters ) , a politics / labour community , and a relatively small bbc community ( fig . s5 in the si ) . at a longer markov time ( @xmath4 ) , we find a partition into 15 communities including the bbc community , a sky community , a community of guardian journalists , a community of international and alternative media / journalists / activists ( including wikileaks , al jazeera , and anonymous - related accounts ) , among other topical communities ( see fig . [ fig : organigram ] ) . at even longer markov times , we show in fig . [ fig : communities]c a coarse partition with four communities corresponding broadly to uk media / activism , international media / activism , entertainment / sports , and the bbc , which remains as a distinct community across a large span of markov times . we provide a supplemental spreadsheet with all partitions of the network at all markov times so that interested parties can explore the all - scale structure of interest communities in the network . furthermore , we have carried a similar analysis using the well - known information - theoretic infomap community detection algorithm @xcite , which in this case leads to an overpartitioned description with non - optimal compression ( i.e. , a large compression gap ) _ and _ unbalanced partitions ( see si for a discussion ) @xcite . in systems that are represented as directed networks , such as twitter , the directionality of the edges is central to their function . the full consideration of edge directionality , which is naturally incorporated in our analysis , is crucial for the community structure detected . to illustrate this phenomenon , we compare the communities found in the original , directed twitter network with those obtained when edge orientation is ignored . we have analysed both versions of the network ( directed and undirected ) using the extended markov stability method which can deal with both types of networks . see the si ( fig . s6 ) for a discussion of the differences in community structure between the directed and undirected versions of this twitter network . importantly , relevant communities can go undetected if one uses standard approaches for community detection based on undirected structural notions ( typically density of connections @xcite ) . , @xmath4 , and @xmath5 , along with their word clouds . in the middle we show the size of the communities of the today show in both versions of the network for markov times between @xmath6 and @xmath7 . , scaledwidth=80.0% ] as stated above , the bbc is an example of a flow community that stands out in its persistency . in fig . [ fig : bbcr4today ] , we show how the community of bbc s today show ( a morning news broadcast with a broad audience ) remains consistently grouped across many levels of resolution in the analysis of the directed network : from an early markov time , bbc - related accounts are grouped together and remain so all the way up to the top levels of resolution , with consistent word clouds throughout . this phenomenon depends strongly on the directionality of the flows : the nodes in the bbc community are among the most important in the network ( high in - degree and pagerank ) , attracting flow ( attention ) from elsewhere in the network and retaining it for long periods of markov time . in a symmetrised network , such communities can go undetected , as shown in fig . [ fig : bbcr4today ] , where the corresponding undirected community of the bbc s today show is quickly blurred across markov times and gets mixed with a variety of users with little in common , consisting mainly of politicians from the labour party and journalists . interestingly , this drastic difference between directed and undirected communities is not observed for all communities in the network . there are communities , such as the one including guardian columnist george monbiot , which behave in an essentially similar fashion in both cases across levels of resolution ( see fig . [ fig : georgemonbiot_size_comparison ] ) . this difference between communities that are sensitive or insensitive to directionality persists across time scales , signalling the fact that some groupings ( such as the bbc community ) are fundamentally based on retention of directed flows , while others ( such as the monbiot community ) follow from a balanced flow and , thus , can be captured by standard undirected measures . we note that the directed markov stabilty method is able to detect both types of communities simultaneously . , @xmath8 , and @xmath5 , along with their word clouds . compare these results to those obtained in fig . [ fig : bbcr4today ] bbc radio 4 s today show.,scaledwidth=80.0% ] as the markov diffusion explores the network , it can provides us with information of how interesting the members of the network are to a given node or groups of nodes ( denoted the ` vantage point ' ) . using our flow - based communities , we establish the _ interest distance _ from the vantage point to any node in the network as the earliest markov time at which the node belongs to the same community as the vantage point ( i.e. , we compute how ` near ' they are in an ultrametric space @xcite ) . in fig . [ fig : interest_distance]a , we show the computed interest distance from the vantage point of the anonymous community ( from @xmath9 onwards ) . consistent with other studies @xcite , the closest nodes to anonymous include wikileaks , human rights watch , al jazeera and amnesty international , followed by a mix of activists and writers , mainstream international media , and the uk media . of least interest to anonymous are celebrities , uk politicians and footballers . unsurprisingly , the picture is starkly different from the vantage point of the nodes that are of least interest to anonymous . figure [ fig : interest_distance]b shows the interest distance from the vantage point of footballer wayne rooney ( of little interest to anonymous ) whose neighbourhood of interest is dominated by football , sports and tv celebrities , with news and activists as distant interests . the computed interest distance is able to capture the nuanced information provided by all the directed paths in the network . this is shown by the fact that stephen fry ( english actor , tv personality and writer ) is distant from _ both _ wayne rooney and anonymous ( fig . [ fig : interest_distance]b ) , while rio ferdinand ( rooney s ex - teammate at manchester united ) is always close to rooney . these examples highlight the sensitivity of our markov exploration and how the use of vantage points can be used to provide personalised information about the system . a flow - based analysis of directed networks also provides a different angle for the classification of nodes according to their role in generating and disseminating information . conceptually , it is clear that an account with millions of followers , such as bbc news , acts as a source of information ( i.e. , a reference ) while a personal account with only a handful of followers yet with subscriptions to media outlets acts mostly as a sink of information ( i.e. , a listener ) . to go beyond this source / sink dichotomy , or the traditional leader / follower and hub / authority @xcite categories , we use here the full structure of flows in the network to develop a quantitative methodology that reveals ` flow roles ' in the network without imposing the number of roles _ a priori_. our algorithm starts by building the _ role - based similarity _ ( rbs ) matrix ( see methods below ) @xcite . a feature vector for each node @xmath10 is constructed from the scaled pattern of incoming and outgoing paths of _ all lengths _ and the pairwise cosine similarities ( @xmath11 $ ] ) between all such vectors ( see methods ) are stored in the @xmath12 similarity matrix @xmath13 . nodes with similar profiles of incoming and outgoing flows of all lengths are classified as having similar _ flow roles _ in the network ( i.e. , when @xmath14 is close to 1 ) . the extreme cases correspond to the standard sources and sinks , but an assortment of nuanced roles spanning these two extremes emerges in our results . this analysis provides a complementary use of flows to infer different properties of nodes : instead of grouping nodes according to flow persistence ( as in the detection of interest communities described above ) , rbs provides a grouping of nodes according to their function in information propagation . we have extended the rbs method by using the relaxed minimum spanning tree ( rmst ) algorithm to extract a _ role similarity graph _ from the matrix @xmath13 ( fig . [ fig : riot1000-rbs - t97]a ) . this novel algorithm creates a new graph by emphasising strong similarities between nodes and downplaying weaker , redundant similarities based on local continuity and global geometric properties of the data similarity @xmath13 ( see methods ) . note that in this rmst - rbs role similarity graph ( which is generated from the twitter graph but is distinct from it ) , nodes with similar connectivity patterns lie close to each other regardless of how close they are ( in a geodesic way ) in the original network . we then apply graph - theoretical community detection algorithms ( such as markov stability ) to the rmst - rbs graph and , in doing so , we reveal groups of nodes ( the communities in the role similarity graph ) with similar in- and out - flow patterns corresponding to _ flow - based roles_. the number of communities on the role similarity graph correspond to the number of roles in the network . note that this procedure does not impose an _ a priori _ number of roles to be detected ( see si ) . our rmst - rbs analysis finds that there are five flow - based roles for the nodes in this twitter network . examination of their incoming and outgoing flow patterns reveals that some of the groups identified correspond to traditional roles such as _ listeners _ ( ` followers ' ) or _ references _ ( ` leaders ' ) , but also distinguishes between different types of leaders , followers and intermediate roles ( fig . [ fig : riot1000-rbs - t97]a - b ) . the description of the five flow role categories we obtained is as follows : 1 . * references : * typically , institutional accounts , important sources of content , or well - known personalities with many followers who follow few accounts , e.g. , bbc breaking news , al jazeera , stephen fry , or the new york times . * engaged leaders : * accounts with large number of followers who , unlike references , also follow other users . this category includes institutional and personal accounts often meant to interact with the public , e.g. , sky news , the british prime minister s office , tom watson ( a british mp ) , or paul lewis ( guardian editor ) . * mediators : * users who interact with the two leader categories ( i.e. , they follow and are followed by high profile accounts ) , as well as with nodes in the listener categories below . many such accounts belong to journalists and reporters . examples of mediators include ross chainey ( reuters employee ) , bbc - have - your - say and the london fire brigade . 4 . * diversified listeners : * accounts with few followers that follow many nodes from all categories , suggesting diversity in their interests and sources of information . 5 . * listeners : * accounts with few followers ( within this network , not necessarily over the whole of twitter ) who follow mostly reference nodes . within this particular network , they can be considered as passive recipients of mainstream content . in the supplemental spreadsheet we give the roles of all nodes in the network . we remark that this classification of nodes into roles is pertinent _ only _ in the context of the network within the guardian s list ; it is possible that the role of certain users will be different if considered embedded in the wider twitter network , since the pattern of paths of different lengths attached to each node is likely to change . figure [ fig : riot1000-rbs - t97]c illustrates the mathematical basis for the classification of nodes into roles by our rmst - rbs algorithm : the patterns of incoming and outgoing flow at all path lengths are combined to reveal the different flow roles . because rmst - rbs takes into account the whole spectrum of short to long paths ( from length 1 to @xmath15 in this case , and everything in between ) our classification goes beyond similarity scores that only use single features , such as in- and out - degrees of the nodes ( which appear as the paths of length 1 in columns 1 and @xmath16 of the matrix @xmath17 in equation ( [ eq : rbs ] ) ) or eigencentrality - type stationary flow metrics ( columns @xmath18 and @xmath19 ) . therefore our method obtains information which is not apparent if we just rank the nodes according to in / out degree or centrality and then split them into groups . for example , ranking the nodes according to pagerank is not enough to distinguish the ` reference ' and ` engaged leader ' categories , or to separate mediator from engaged leader or diversified listener ( see fig . s8 in the si and supplemental spreadsheet ) . to confirm the relevance of our findings , we examine the cumulative distribution of retweets attained by each node class ( fig . [ fig : riot1000-rbs - t97]d ) , where we see a clear separation between the leader ( reference and engaged leader nodes ) and follower ( diversified listeners and listeners ) categories , with the mediators lying in between both groups . it is important to remark that the retweet data in fig . [ fig : riot1000-rbs - t97]d is not part of our role detection and is only used _ a posteriori _ to inform our understanding of the flow roles obtained ( see also fig . s8 in the si ) . the flow roles we find here are conceptually and practically different from those obtained using well established theories in social network analysis , e.g. structural equivalence ( se ) @xcite and regular equivalence ( re ) @xcite . se bases node similarity on overlapping sets of neighbours ( i.e. , two nodes are similar if many of their neighbours are the same ) , while re - based methods rely on node colorations and neighbourhoods ( i.e. , two nodes have the same role if the colours of their neighbours are the same , regardless of the number of common neighbours ) . hence se and re are essentially short - path methods and not suitable for networks like the one studied here where the full structure of flow is essential ( see si for a detailed description of re and se roles and their lack of information content in this network ) . furthermore , re methods are not robust to small random perturbations in network connectivity due to their combinatorial nature . . the communities reflect a diverse set of topical groupings ( see word clouds with the top 50 nontrivial words in the user biographies ) and are characterised by different mixes of the five flow roles , as shown by the pie charts . the organigrams range from reference - listener schemes ( ` broadcast ' and ` monologue ' ) to more balanced dialogue communities ( ` engaged dialogue ' and ` dialogue in public ' ) in which engaged leaders , mediators and diversified listeners dominate . , scaledwidth=80.0% ] heretofore , our two - pronged flow - based analysis has led to groupings of the nodes according to two criteria : interest communities ( at different resolutions ) and flow roles . both perspectives present complementary views of the information in the network and can be combined to characterise the internal organisation of interest communities in terms of the mix of roles of their members . figure [ fig : organigram ] presents this integrated view for the 15 interest communities at medium resolution ( markov time @xmath4 ) , and the five node roles found through rbs - rmst . using a simple @xmath20-means clustering of their role - mixes , we find that the 15 communities fall into four types of informational organigrams ( see si ) . two of these organigrams broadly conform to communities formed mostly by leaders ( ` references ' ) and their followers ( ` listeners ' ) , though with some important differences : _ `` monologue communities '' _ are predominantly composed of references with a set of loyal ( non diversified ) listeners with information flowing mostly in one direction ( e.g. , ` celebrities / entertainers ' , ` sport ' , parody accounts ) , while in _ `` broadcast communities '' _ most members are references delivering content broadly to a wide variety of users in the network ( e.g ` bbc ' and ` international media ' ) . in addition , there exist two organigrams with a more balanced dialogue structure : _ `` dialogue in public '' _ , which involves many diversified listeners ( e.g. , ` panel show celebrities ' , ` london ' , or groups heavily based on internet interaction such as ` uk journalists & activists ' ) ; and _ `` engaged dialogue '' _ , which is dominated by engaged leaders and mediators ( e.g. , ` politics ' and ` the guardian ' ) . these two dialogue organigrams reflect the importance of online interaction in information networks , where bottom - up grassroots associations , bloggers and commentators from the public interact directly with accounts linked to news outlets and official political organisations .

analysing flows ( and not only the strength of connections ) is crucial to reveal important features of the network that might go undetected if the orientation of connections is ignored . such interest communities reveal user groupings according to location , profession , employer , and topic . secondly , we analyse the profiles of incoming and outgoing long - range flows with a combined approach of role - based similarity and the novel relaxed minimum spanning tree algorithm to reveal that the users in the network can be classified into five roles . these flow roles go beyond the standard leader / follower dichotomy and differ from classifications based on regular / structural equivalence .

directionality is a crucial ingredient in many complex networks in which information , energy or influence are transmitted . in such directed networks , analysing flows ( and not only the strength of connections ) is crucial to reveal important features of the network that might go undetected if the orientation of connections is ignored . we showcase here a flow - based approach for community detection in networks through the study of the network of the most influential twitter users during the 2011 riots in england . firstly , we use directed markov stability to extract descriptions of the network at different levels of coarseness in terms of interest communities , i.e. , groups of nodes within which flows of information are contained and reinforced . such interest communities reveal user groupings according to location , profession , employer , and topic . the study of flows also allows us to generate an interest distance , which affords a personalised view of the attention in the network as viewed from the vantage point of any given user . secondly , we analyse the profiles of incoming and outgoing long - range flows with a combined approach of role - based similarity and the novel relaxed minimum spanning tree algorithm to reveal that the users in the network can be classified into five roles . these flow roles go beyond the standard leader / follower dichotomy and differ from classifications based on regular / structural equivalence . we then show that the interest communities fall into distinct informational organigrams characterised by a different mix of user roles reflecting the quality of dialogue within them . our generic framework can be used to provide insight into how flows are generated , distributed , preserved and consumed in directed networks .

1311.6785

c

in this work , we have used the twitter network constructed from the list of influential users during the uk riots of 2011 collected by the guardian to showcase how flow - based methods in directed networks can lead to enhanced insight into the structure of data . our analysis reveals interest communities into which users fall at different levels of resolution . the interest communities found confirm the relevance of news organisations and media , yet providing a layered view in terms of their focus ( uk / international , mainstream / alternative ) and of relationships to each other and to the overall network . the enhanced sensitivity of our multi - resolution analysis allows us to uncover small but significant groups related to local organisations or clean - up groups in riot areas which appear close to police and law enforcement groupings . in addition , our analysis reveals groupings that have an unexpected relevance in a network that was selected on the basis of ` retweeting ' importance during an event of civil unrest . in particular , a significant grouping of celebrities , sport personalities and pop musicians act as the center of a significant interest community . also intriguing is the role of interest groups based on humour in such situations , as represented by communities of comedians and parody accounts . our work points at future studies on how to use this type of analyses to improve and tailor communication strategies during times of unrest , in particular with regards to providing a personalised view of the network from any given vantage point ( i.e. , from any node or group of nodes ) based on the interest distance of information flow . these results can be a starting point to examine textual information and analyse the influence of groups of interest on observed behaviors in this and similar datasets . using flow transfer in the network , we find that the twitter users in this network fall into a palette of five flow roles , whereas interest communities exhibit distinct mixes of such roles reflecting diverse communication patterns within them . some communities contain one - way communication patterns ( e.g. , celebrities and their followers ) , whereas other communities harbour more balanced dialogue patterns . in particular , our analysis highlights the differences between media organisations and their distinct patterns of interaction with the influential users in this network . for instance , international mainstream media tend to fall into the broadcast and monologue categories , as would be expected in a network of uk - based events . on the other hand , the uk and specialised media exhibit a more diverse pattern of interactions with their followers : some of them are highly engaged with mediators and diversified listeners whereas others largely maintain the more traditional role of publishing content . this work also highlights the use of multiscale network analyses , which go beyond local information of individual users towards aggregate global metrics , to deliver an enriched view of information dissemination in social networks , thus uncovering relationships and roles of nodes and providing concise coarse - grained descriptions of the network . we hope that our results ( all available in the supplemental material ) could be a helpful resource to aid in the study of online interactions during the uk riots of 2011 . more generally , our work highlights the importance of directionality in network analysis . when the notion of flows ( e.g. , of people , information , energy , goods ) is central to a network , ignoring directionality destroys information , ` blurring ' the structure , especially at the finer levels of resolution , so that key communities ( e.g. , the bbc , sky , and geographical communities in our analysis ) will go undetected . the formulation of community and role detection in terms of flow dynamics thus provides an integrated methodology for the analysis of systems ( natural or man - made ) with directed network representations .

we then show that the interest communities fall into distinct informational organigrams characterised by a different mix of user roles reflecting the quality of dialogue within them . our generic framework can be used to provide insight into how flows are generated , distributed , preserved and consumed in directed networks .

directionality is a crucial ingredient in many complex networks in which information , energy or influence are transmitted . in such directed networks , analysing flows ( and not only the strength of connections ) is crucial to reveal important features of the network that might go undetected if the orientation of connections is ignored . we showcase here a flow - based approach for community detection in networks through the study of the network of the most influential twitter users during the 2011 riots in england . firstly , we use directed markov stability to extract descriptions of the network at different levels of coarseness in terms of interest communities , i.e. , groups of nodes within which flows of information are contained and reinforced . such interest communities reveal user groupings according to location , profession , employer , and topic . the study of flows also allows us to generate an interest distance , which affords a personalised view of the attention in the network as viewed from the vantage point of any given user . secondly , we analyse the profiles of incoming and outgoing long - range flows with a combined approach of role - based similarity and the novel relaxed minimum spanning tree algorithm to reveal that the users in the network can be classified into five roles . these flow roles go beyond the standard leader / follower dichotomy and differ from classifications based on regular / structural equivalence . we then show that the interest communities fall into distinct informational organigrams characterised by a different mix of user roles reflecting the quality of dialogue within them . our generic framework can be used to provide insight into how flows are generated , distributed , preserved and consumed in directed networks .

1411.7936

c

distillation of entanglement from shared quantum states is a useful technique for several quantum information protocols . it is important , for both fundamental and practical reasons , to consider this protocol in a scenario of limited resources . we have considered distillation of entanglement in bipartite and multipartite quantum states in the situation where there is a limited amount of energy that is accessible for the process to be accomplished . in particular , we consider constraints on average energy : a canonical energy constraint and a weak canonical energy constraint , that naturally lead to the concepts of canonical distillability and special canonical distillability . we have shown that for a bipartite system described by a non - interacting hamiltonian , almost no states are special canonically distillable . this , in turn , implies that almost no states are canonically distillable , as canonically distillable states form a subset of the set of special canonically distillable states . significant understanding about the set of special canonically distillable states can be obtained by looking at the probability distributions of the average energies of the shared states . the concept has been applied to a number of bipartite and multipartite systems described by well - known spin hamiltonians , namely , the spin-@xmath1 @xmath2 model in transverse and longitudinal fields , the spin-@xmath1 @xmath3 model in an external field , and the bilinear - biquadratic hamiltonian of a spin-@xmath4 system in the presence of an external field . we find that the probability that a randomly chosen state is special canonically distillable can be manipulated by altering the direction of the external field . the canonical distillability of a number of mixed states such as the thermal state , the bell - diagonal states , and mixed bipartite states with fixed magnetizations are also investigated . it has been shown that for a fixed external field value , the thermal state can be special canonical distillable only above a critical temperature , which we have called the special canonically distillable temperature . the concept of canonical distillability of three - qubit ghz and w class states have also been introduced . the results are expected to be of importance in realization of quantum communication channels . 100 r. horodecki , p. horodecki , m. horodecki , and k. horodecki , rev . phys . * 81 * , 865 ( 2009 ) , and the references therein . h. j. briegel , d. e. browne , w. dr , r. raussendorf , and m. van den nest , nature physics * 5 * , 19 - 26 ( 2009 ) . a. sen(de ) and u. sen , physics news * 40 * , 17 - 32 ( 2010 ) ( arxiv:1105.2412 [ quant - ph ] ) . a. acn , n. brunner , n. gisin , s. massar , s. pironio , and v. scarani , phys . 98 , 230501 ( 2007 ) ; s. pironio , a. acn , n. brunner , n. gisin , s. massar , and v. scarani , new j. phys . * 11 * , 045021 ( 2009 ) ; n. gisin , s. pironio , and n. sangouard , phys . 105 , 070501 ( 2010 ) ; l. masanes , s. pironio , and a. acn , nature commun . * 2 * , 238 ( 2011 ) ; h. k. lo , m. curty , and b. qi , phys . 108 , 130503 ( 2012 ) ; c. h. bennett , g. brassard , s. popescu , b. schumacher , j. a. smolin , and w. k. wootters , phys . rev . lett . * 76 * , 722 ( 1996 ) ; m. horodecki , p. horodecki , and r. horodecki , phys . rev . lett . * 78 * , 574 ( 1997 ) ; m. murao , m. b. plenio , s. popescu , v. vedral , and p. l. knight , phys . rev . a * 57 * , r4075 ( 1998 ) ; w. dr , j. i. cirac , and r. tarrach , phys . . lett . * 83 * , 3562 ( 1999 ) ; m. horodecki and p. horodecki , phys . rev . a * 59 * , 4206 ( 1999 ) ; w. dr and j. i. cirac , phys . rev . a * 61 * , 042314 ( 2000 ) ; w. dr , j. i. cirac , m. lewenstein , and d. bru , phys . rev . a * 61 * , 062313 ( 2000 ) ; g. alber , a. delgado , n. gisin , and i. jex , j. phys . a. * 34 * , 8821 ( 2001 ) ; j. dehaene , m. van den nest , b. d. moor , and f. vestraete , phys . rev . a * 67 * , 022310 ( 2003 ) ; w. dr , h. aschaur , and h. j. briegel , phys . rev . lett . * 91 * , 107903 ( 2003 ) ; e. hostens , j. dehaene , and b. d. moor , arxiv : quant - ph/0406017 ( 2004 ) ; k. g. h. vollbrecht and f. vestraete , phys . a * 71 * , 062325 ( 2005 ) ; a. miyake and h. j. briegel , phys . rev . 95 * , 220501 ( 2005 ) ; h. aschaur , w. dr , and h. j. briegel , phys . rev . a * 71 * , 012319 ( 2005 ) ; e. hostens , j. dehaene , and b. d. moor , phys . a * 73 * , 042316 ( 2006 ) ; e. hostens , j. dehaene , and b. d. moor , phys . a * 74 * , 062318 ( 2006 ) ; a. kay , j. k. pachos , w. dr , and h. j. briegel , new . j. phys . * 8 * , 147 ( 2006 ) ; c. kruszynska , a. miyake , h. j. briegel , and w. dr , phys . a * 74 * , 052316 ( 2006 ) ; s. glancy , e. knill , and h. m. vasconcelos , phys . a * 74 * , 032319 ( 2006 ) . p. horodecki , phys . a * 232 * , 333 ( 1997 ) ; m. horodecki , p. horodecki , and r. horodecki , phys . rev . lett . * 80 * , 5239 ( 1998 ) ; w. dr and j. i. cirac , phys . a * 62 * , 022302 ( 2000 ) ; p. horodecki , j. a. smolin , b. m. terhal , and a. v. thapliyal , theor . comput . sc . * 292 * , 589 ( 2003 ) ; j. lavoie , r. kaltenbaek , m. piani , and k. j. resch , nature physics * 6 * , 827 ( 2010 ) ; j. lavoie , r. kaltenbaek , m. piani , and k. j. resch , phys . lett . * 105 * , 130501 ( 2010 ) ; j. diguglielmo , a. samblowski , b. hage , c. pineda , j. eisert , and r. schnabel , phys . rev . 107 , 240503 ( 2011 ) ; e. amselem , m. sadiq , and m. bourennane , scientific reports * 3 * , 1966 ( 2013 ) . d. p. divincenzo , p. w. shor , j. a. smolin , b. m. terhal , and a. v. thapliyal , phys . a * 61 * , 062312 ( 2000 ) ; b. kraus , m. lewenstein , and j. i. cirac , phys . a * 65 * , 042327 ( 2002 ) ; j. watrous , phys . lett . * 93 * , 010502 ( 2004 ) . a. ekert , phys . lett . * 67 * , 661 ( 1991 ) ; t. jennewein , c. simon , g. weihs , h. weinfurter , and a. zeilinger , phys . lett . * 84 * , 4729 ( 2000 ) ; d. s. naik , c. g. peterson , a. g. white , a. j. berglund , and p. g. kwiat , phys . rev . * 84 * , 4733 ( 2000 ) ; w. tittel , t. brendel , h. zbinden , and n. gisin , phys . rev . lett . * 84 * , 4737 ( 2000 ) ; n. gisin , g. ribordy , w. tittel , and h. zbinden , rev . * 74 * , 145 ( 2002 ) . c. h. bennet and s. j. wiesner , phys * 69 * , 2881 ( 1992 ) ; k. mattle , h. weinfurter , p. g. kwiat , and a. zeilinger , phys . * 76 * , 4656 ( 1996 ) . c. h. bennett , g. brassard , c. crpeau , r. jozsa , a. peres , and w. k. wootters , phys . lett . * 70 * , 1895 ( 1993 ) ; d. bouwmeester , j. w. pan , k. mattle , m. eibl , h. weinfurter , and a. zeilinger , nature * 390 * , 575 ( 1997 ) ; j. w. pan , d. bouwmeester , h. weinfurter , and a. zeilinger , phys . lett . * 80 * , 3891 ( 1998 ) ; d. bouwmeester , j. w. pan , h. weinfurter , and a. zeilinger , j. mod . opt . * 47 * , 279 ( 2000 ) . j. briegel , w. dr , j. i. cirac , and p. zoller , phys . * 81 * , 5932 ( 1998 ) . h. bombin and m. a. martin - delgado , phys . rev . lett . * 97 * , 180501 ( 2006 ) . p. g. kwiat , s. barraza - lopez , a. stefanov , and n. gisin , nature * 409 * , 1014 ( 2001 ) ; j. w. pan , c. simon , c. brukner , and a. zeilinger , nature * 410 * , 1067 ( 2001 ) ; j. w. pan , s. gasparoni , r. ursin , g. weihs , and a. zeilinger , nature * 423 * , 417 ( 2003 ) . p. horodecki , r. horodecki , and m. horodecki , acta physica slovaca * 48 * , 141 ( 1998 ) . p. m. hayden , m. horodecki , and b. m. terhal , j. phys . a * 34 * , 6891 ( 2001 ) ; m. b. plenio and s. virmani , quant . * 7 * , 1 ( 2006 ) . s. hill and w. k. wootters , phys . lett . * 78 * , 5022 ( 1997 ) ; w. k. wootters , phys . lett . * 80 * , 2245 ( 1998 ) ; v. coffman , j. kundu , and w. k. wootters , phys . a * 61 * , 052306 ( 2000 ) ; w. k. wootters , quant . * 1 * , 27 ( 2001 ) . j. p. gordon , in proceedings of the international school of physics , `` enrico fermi , course xxxi , '' edited by p. a. miles ( academic press , new york , 1964 ) , p. 156 ; l. b. levitin , in proceedings of the vi national conference on inf . theory , p. 111 ( tashkent 1969 ) . h. p. yuen , in quantum communication , computing , and measurement , edited by o. hirota et al . , ( plenum , new york , 1997 ) . a. s. holevo , probl . peredachi inf . * 9 * , 3 ( 1973)[probl . inf . transm . 9 , 110 ( 1973 ) ] ; b. schumacher , m. westmoreland , and w. k. wootters , phys . * 76 * , 3452 ( 1996 ) ; p. badziag , m. horodecki , a. sen(de ) , and u. sen , phys . rev.lett . * 91 * , 117901 ( 2003 ) ; m. horodecki , j. oppenheim , a. sen(de ) , and u. sen , _ ibid_. * 93 * , 170503 ( 2004 ) . h. p. yuen and m. ozawa , phys . lett . * 70 * , 363 ( 1993 ) ; c. m. caves and p. d. drummond , rev 66 * , 481 ( 1994 ) ; a. s. holevo , m. sohma , and o. hirota , phys . rev . a * 59 * , 1820 ( 1999 ) ; a. s. holevo and r. f. werner , ibid . * 63 * , 032312 ( 2001 ) ; s. lloyd , phys . rev . lett . * 90 * , 167902 ( 2003 ) ; v. giovannetti , s. lloyd , l. maccone , and p. w. shor , ibid . * 91 * , 047901 ( 2003 ) ; v. giovannetti , s. lloyd , and l. maccone , ibid . * 70 * , 012307 ( 2004 ) ; v. giovannetti , s. guha , s. lloyd , l. maccone , and j. h. shapiro , ibid . * 70 * , 032315 ( 2004 ) ; v. giovannetti , s. guha , s. lloyd , l. maccone , j. h. shapiro , and h.p . yuen , ibid . * 92 * , 027902 ( 2004 ) ; a. sen(de ) , u. sen , b. gromek , d. bru , and m. lewenstein , phys . lett . * 95 * , 260503 ( 2005 ) ; a. sen(de ) , u. sen , b. gromek , d. bru , and m. lewenstein , phys . a * 75 * , 022331 ( 2007 ) ; m. m. wolf , d. prez - garca , and g. giedke , phys . 98 , 130501 ( 2007 ) ; c. lupo , v. giovannetti , and s. mancini , phys . rev . lett . 104 , 030501 ( 2010 ) ; v. giovannetti , r. garca - patrn , n. j. cerf , and a. s. holevo , nature photonics * 8 * , 796 ( 2014 ) . e. lieb , t. schultz , and d. mattis , ann . * 16 * , 407 ( 1961 ) ; e. barouch , b.m . mccoy , and m. dresden , phys . rev . * 2 * , 1075 ( 1970 ) ; e. barouch and b.m . mccoy , phys . rev . * 3 * , 786 ( 1971 ) . b. k. chakrabarti , a. dutta , and p. sen , quantum ising phases and transitions in transverse ising models ( springer , heidelberg , 1996 ) ; s. sachdev , quantum phase transitions ( cambridge university press , cambridge , 2011 ) ; s. suzuki , j. -i . inou , b. k. chakrabarti , quantum ising phases and transitions in transverse ising models ( springer , heidelberg , 2013 ) . c. n. yang and c. p. yang , phys . rev . * 150 * , 321 ( 1966 ) ; c. n. yang and c. p. yang , _ ibid . _ * 150 * , 327 ( 1966 ) . e. demler and f. zhou , phys . lett . * 88 * , 163001 ( 2002 ) ; s. k. yip , phys . . lett . * 90 * , 250402 ( 2003 ) ; o. romero - isart , k. eckert , and a. sanpera , phys . rev . a * 75 * , 050303(r ) ( 2007 ) . d. m. greenberger , m. a. horne , and a. zeilinger , in bell s theorem , quantum theory , and conceptions of the universe , ed . m. kafatos ( kluwer academic , dordrecht , the netherlands , 1989 ) . a. zeilinger , m. a. horne , and d. m. greenberger , in proceedings of squeezed states and quantum uncertainty , edited by d. han , y. s. kim , and w. w. zachary , nasa conf . publ . * 3135 * , 73 ( 1992 ) . w. dr , g. vidal , j. i. cirac , phys . a * 62 * , 062314 ( 2000 ) . p. w. shor , j. a. smolin , and b. m. terhal , phys . lett . * 86 * , 2681 ( 2001 ) . a. peres , phys . lett . * 77 * , 1413 ( 1996 ) ; m. horodecki , p. horodecki , and r. horodecki , phys . a * 223 * , 1 ( 1996 ) . a. sen(de ) , u. sen , m. wiesniak , d. kaszlikowski , and m. ukowski , phys . a , * 68 * , 062306 ( 2003 ) .

distilling highly entangled quantum states from weaker ones is a process that is crucial for efficient and long - distance quantum communication , and has implications for several other quantum information protocols . we show that for non - interacting hamiltonians , almost no states are canonically distillable , while the situation can be drastically different for interacting ones .

distilling highly entangled quantum states from weaker ones is a process that is crucial for efficient and long - distance quantum communication , and has implications for several other quantum information protocols . we introduce the notion of distillation under limited resources , and specifically focus on the energy constraint . the corresponding protocol , which we call the canonical distillation of entanglement , naturally leads to the set of canonically distillable states . we show that for non - interacting hamiltonians , almost no states are canonically distillable , while the situation can be drastically different for interacting ones . several paradigmatic hamiltonians are considered for bipartite as well as multipartite canonical distillability . the results have potential applications for practical quantum communication devices .

1406.4270

i

elastic manifolds in random media are one of the most important model systems in the statistical physics of disordered systems , which exhibit disorder - dominated pinned phases with many features common to glassy systems @xcite . likewise , the depinning of an elastic manifold from a disorder potential under the action of a driving force is a paradigm for the non - equilibrium dynamical behavior of disordered systems capturing the avalanche dynamics of many complex systems if they are driven through a complex energy landscape @xcite . in particular , the problem of a directed line ( dl ) or directed polymer ( i.e. , an elastic manifold in @xmath0 dimensions ) in a random potential and driven by a force has been subject of extensive study @xcite . at zero temperature , there is a threshold force , at which the manifold changes from a localized state with vanishing mean velocity to a moving state with a non - zero mean velocity . the depinning transition has been treated within the framework of classical critical phenomena by functional renormalization group techniques starting from the more general problem of depinning of @xmath1-dimensional elastic interfaces ( with @xmath2 corresponding to lines ) . in @xmath3 dimensions , `` critical '' exponents at depinning can be calculated by functional renormalization using dimensional regularization in an @xmath4-expansion @xcite . at finite temperature , there is experimental evidence for a creep motion at any non - vanishing driving forces which can be understood qualitatively as thermally activated crossing of energy barriers which result from an interplay of both elastic energies of the line and the disorder potential . the energy of dls such as flux lines , domain walls , wetting fronts is proportional to their length ; therefore , the elastic properties of directed lines are governed by their line tension , which favors the straight configuration of shortest length . here , we concentrate on stiff directed lines ( sdls ) , whose elastic energy is given by the curvature of the line and , thus , represents a bending energy . this gives rise to configurations which are locally curvature - free and straight but , in contrast to the dl , the straight segments of sdls can assume any orientation even if this increases the total length of the line . there are a number of applications and interesting general theoretical issues concerning sdls in random media . the overdamped equation of motion of a sdl is the ( fourth - order ) herring - mullins linear diffusion equation @xcite , which also describes surface growth governed by surface diffusion . the depinning dynamics of the herring - mullins equation in quenched disorder has been subject of a number of prior studies @xcite , whose findings ( e.g. , an unphysically small roughness exponent ) differ in part significantly from ours as we will point out below ( see sec . [ sec : prev ] ) . sdls also describe semiflexible polymers with contour lengths smaller than their persistence length for bending fluctuations , such that the assumption of a directed line is not violated @xcite . our results can be applied to the depinning dynamics of semiflexible polymers such as dna or cytoskeletal filaments like f - actin in a random environment , such as a porous medium , as long as the correlation length of the depinning transition is smaller than the persistence length . as in other semiflexible polymer phase transitions ( such as adsorption ) non - universal quantities such as the value of the depinning threshold itself will be governed by the bending elasticity . at the depinning transition , where the correlation length diverges , semiflexible polymers will exhibit a crossover to critical properties of effectively flexible lines ( dls ) with a segment length set by the persistence length . moreover , static sdls in a random potential feature a disorder - driven localization transition at finite temperatures already in @xmath0 dimensions @xcite . due to an interesting dimensional shift in the problem , an analogous transition occurs for dls only in higher dimensions . in principle , this offers an opportunity to observe new phenomena arising from an interplay of depinning and delocalization for sdls ( the disorder used in refs . @xcite does not feature such a transition in the static problem ) . the localization transition at a finite temperature also offers the opportunity to test the usage of static quantities in the treatment of creep motion , because the sdl is not pinned by disorder above the critical temperature . a lot of the progress for the depinning theory of dls has been based on two basic theorems due to middleton @xcite , which essentially state that a forward moving dl can only move forward and will stop in a localized configuration , if such exists . this allows for an unambiguous time - ordering of a sequence of states . we will show that these theorems do not hold for the sdl , which can be seen as a consequence of the next - to - neighbor terms that are introduced by the bending elasticity . the paper is organized as follows . we present the model and the relevant equations of motion in sec . [ sec : model ] , where we also comment on some equilibrium properties and previous work on the sdl . in sec.[sec : ana ] we present analytical results based on scaling arguments and functional renormalization group calculations . in sec . [ sec : num ] we present numerical methods and results and , in sec.[sec : middle ] , we comment on the non - applicability of middleton s theorems for sdls . we conclude with a summary of our results in sec.[sec : concl ] .

driven elastic manifolds in random media exhibit a depinning transition to a state with non - vanishing velocity at a critical driving force . we study the depinning of stiff directed lines , which are governed by a bending rigidity rather than line tension . their equation of motion is the ( quenched ) herring - mullins equation , which also describes surface growth governed by surface diffusion . rev.lett . * 68 * , 670 ( 1992 ) ) is not applicable . we employ analytical arguments and numerical simulations to determine the critical exponents and compare our findings with previous works and functional renormalization group results , which we extend to the different line elasticity .

driven elastic manifolds in random media exhibit a depinning transition to a state with non - vanishing velocity at a critical driving force . we study the depinning of stiff directed lines , which are governed by a bending rigidity rather than line tension . their equation of motion is the ( quenched ) herring - mullins equation , which also describes surface growth governed by surface diffusion . stiff directed lines are particularly interesting as there is a localization transition in the static problem at a finite temperature and the commonly exploited time ordering of states by means of middleton s theorems ( a. middleton , phys . rev.lett . * 68 * , 670 ( 1992 ) ) is not applicable . we employ analytical arguments and numerical simulations to determine the critical exponents and compare our findings with previous works and functional renormalization group results , which we extend to the different line elasticity . we see evidence for two distinct correlation length exponents .

1312.7712

i

through remarkable developments in solid earth science since the late 1960s , our understanding of earthquakes has increased significantly . the availability of relevant data has steadily increased as the study of earthquakes has progressed remarkably in geophysics . after every major earthquake , researchers have elucidated important seismic mechanisms associated with it . however , even though detailed analysis and discussions have been conducted , large uncertainties remain because of diversity and complexity of the earthquake phenomenon . this leads to unachievable challenges in deterministic earthquake prediction because all diverse and complex scenarios must faithfully reflect the processes of earthquakes to be considered for effective earthquake prediction . on the other hand , several techniques for predicting earthquakes have been proposed on the basis of anomalies of various types ; however , the effectiveness of these techniques is controversial ( @xcite ) . therefore , objectivity is required for such evaluation ; otherwise , arguments presented may lack merit . new prediction models that claim to incorporate potentially useful information over those used in standard seismicity models should be evaluated to determine whether predictive power is improved . earthquake forecasting models should evolve in this manner . recently , there has been growing momentum for seismologists to develop an organized research program on earthquake predictability . an international cooperative study known as _ collaboratory for the study of earthquake predictability _ ( csep ; http://www.cseptesting.org/ ) is currently under way among countries prone to major earthquakes for exploring possibilities in earthquake prediction ( e.g. , @xcite ) . an immediate objective of the project is to encourage the development of statistical models of seismicity , such as those subsequently discussed in section [ sec2 ] , and to evaluate their predictive performances in terms of probability . in addition , the csep study aims to develop a scientific infrastructure to evaluate statistical significance and _ probability _ _ gain _ ( @xcite ) of various methods used to predict large earthquakes by using observed abnormalities such as seismicity anomaly , transient crustal movements and electromagnetic anomaly . here _ probability gain _ is defined as the ratio of probability of a large earthquake estimated based on an anomaly to the underlying probability without anomaly . section [ sec3 ] describes this important concept , and then discusses statistical point - process models to examine the significance of causality of anomalies and also to evaluate the probability gains conditional on the anomalous events . for prediction of large earthquakes with a higher probability gain , comprehensive studies of anomalous phenomena and observations of earthquake mechanisms are essential . several such studies are summarized in sections [ sec4][sec6 ] . particularly , i have been interested in elucidating abnormal seismic activities and geodetic anomalies to apply them for promoting forecasting abilities , as described in these sections .

earthquakes occur because of abrupt slips on faults due to accumulated stress in the earth s crust . because most of these faults and their mechanisms are not readily apparent , deterministic earthquake prediction is difficult . for effective prediction , complex conditions and uncertain elements must be considered , which necessitates stochastic prediction .

earthquakes occur because of abrupt slips on faults due to accumulated stress in the earth s crust . because most of these faults and their mechanisms are not readily apparent , deterministic earthquake prediction is difficult . for effective prediction , complex conditions and uncertain elements must be considered , which necessitates stochastic prediction . in particular , a large amount of uncertainty lies in identifying whether abnormal phenomena are precursors to large earthquakes , as well as in assigning urgency to the earthquake . any discovery of potentially useful information for earthquake prediction is incomplete unless quantitative modeling of risk is considered . therefore , this manuscript describes the prospect of earthquake predictability research to realize practical operational forecasting in the near future .

0908.2527

i

the large hadron collider ( lhc ) is presently being commissioned at cern @xcite . its main beam parameters are given in table [ tab : lhc - beams ] . in its first phase of operation it will collide protons but later also heavy nuclei , starting with @xmath0 at center - of - mass energies up to 1.15 pev @xcite . this will open up a new energy regime in experimental nuclear physics , extending the study of the hadronic matter ( `` quark - gluon plasma '' ) that existed in the early universe about @xmath1 after the big bang @xcite . at the same time , colliding heavy - ion beams at this unprecedented energy will present beam physics challenges not encountered in previous colliders . when two fully stripped ions collide at an interaction point ( ip ) , a variety of processes leading to fragmentation and particle production can occur . hadronic nuclear interactions , which are usually the main object of study of the experiments , occur only when the impact parameter @xmath2 is smaller than about twice the nuclear radius @xmath3 . _ ultraperipheral collisions _ are those in which two colliding ions pass close to each other with @xmath4 . in such events , the intense lorentz - contracted fields of the nuclei can be represented as pulses of virtual photons in the equivalent photon picture of fermi , weizscker and williams @xcite . these extremely energetic photons collide and cause electromagnetic interactions that are particularly strong in heavy - ion collisions because the density of virtual photons around the nuclei is proportional to @xmath5 . reviews of this field can be found in refs . @xcite . in comparison with the cross - section for inelastic hadronic interactions , @xmath6 , the rates of ultraperipheral interactions are enormous : for example , the cross - section for @xmath7 pair production given by the racah formula @xcite is of order @xmath8 . from the point of view of collider operation , the most important electromagnetic processes are the sub - classes of these interactions which remove ions from the beam , namely bound - free pair production ( bfpp ) and electromagnetic dissociation ( emd ) @xcite . .design parameters for the lhc s proton and @xmath0 beams in collision conditions @xcite . the values of @xmath9 and @xmath10 refer to ip2 for @xmath0ions and ip1 and ip5 for protons ( see fig . [ fig : lhc - layout ] ) . [ cols="<,>,>",options="header " , ] in bfpp , the virtual photons surrounding relativistic ions collide and produce an @xmath7 pair where , in contrast to the much more frequent free pair production , the electron is created in an atomic shell of one of the ions . schematically , the reaction between two bare nuclei with atomic numbers @xmath11 , @xmath12 can be written as @xmath13 in emd one nucleus absorbs a photon and undergoes a transition into an excited state , typically by excitation of the giant dipole resonance . when this decays , it emits one or several nucleons . because of the nuclear coulomb barrier , the most common processes are the emission of one or two neutrons . for the case of @xmath0 ions , the 1-neutron reaction ( called emd1 hereafter ) can be written as @xmath14 both bfpp and emd change the magnetic rigidity of at least one of the colliding nuclei . ( as usual , magnetic rigidity is defined as a particle s momentum @xmath15 per unit charge @xmath16 , or @xmath17 , where @xmath18 is the bending radius in a magnetic field @xmath19 . ) if the charge of an ion changes by @xmath20 , and the number of nucleons by @xmath21 , the resulting rigidity can be written as @xmath22 , where the fractional deviation @xmath23 from the main beam with atomic number @xmath24 and mass number @xmath25 is given to a very good approximation ( neglecting increments of the mass excess ) by @xmath26 here @xmath27 is the fractional momentum deviation per nucleon with respect to an ion circulating on the central orbit of the storage ring . during the interactions , the transverse momentum recoil is very small , so the modified ions emerge at a small angle to the main beam . in emd , the recoil changes @xmath28 ( see sec . [ sec : emd - quench ] ) , while this is negligible for bfpp . however , ions from both processes follow dispersive orbits according to their magnetic rigidity and may be lost at the first point in the ring where the horizontal aperture @xmath29 and dispersion @xmath30 generated locally since the ip satisfy @xmath31 the beam losses caused by these processes contribute significantly to the decay of intensity and luminosity in an ion collider @xcite . this has been evaluated for the lhc @xcite , where ions might collide at three ips : the atlas experiment at ip1 , alice at ip2 and cms at ip5 . the general layout of the lhc is shown in fig . [ fig : lhc - layout ] . it consists of two concentric 27 km rings with counter rotating beams . each ring is made of eight long straight sections matched to eight regular arcs through dispersion suppressor regions . the two rings overlap in four of the eight long straight sections housing colliding - beam experiments . the optics design continues to evolve through various _ versions _ , usually corresponding to a given layout of the collider elements , with the latest one called v6.503 . the lhc uses superconducting magnets with the two apertures within the same cryostat , where some are cooled down to 1.9 k by liquid helium . further details of the lhc optics and layout can be found in ref . . + the rate of removal of particles from the beam at the ips is directly proportional to the interaction cross section , which in the case of bfpp takes the approximate form @xcite @xmath32 where the electron is captured by nucleus 1 and the sum is taken over atomic shells @xmath33 . here @xmath34 is the relativistic factor of the ions in the center - of - mass frame , and @xmath35 depend only weakly on @xmath36 . for 2.76 tev / nucleon @xmath0collisions in the lhc , the total cross section for electron capture to _ one _ of the nuclei is @xcite @xmath37 using tab . [ tab : lhc - beams ] and eq . ( [ eq : sigbfpp - lhc ] ) , the bfpp event rate is @xmath38 ( @xmath9 is the collider luminosity ) . predictions of the cross - sections for bfpp have varied substantially over the years but have converged on values close to those found using the plane - wave born approximation of ref . @xcite ; as these authors point out , it was shown earlier @xcite , that higher order coulomb corrections may reduce the values by a few percent . we estimate the uncertainty in the bfpp cross section to about 20% @xcite . the corresponding cross section for emd1 was calculated with the monte carlo program fluka @xcite , by simulating a large number of the events through direct calls to the event generator : @xmath39 this relies on a recent improved implementation of emd effects which has been benchmarked against the reldis code @xcite ( e.g. , in this case reldis gives 104 b @xcite ) . fluka was also used to estimate the cross section for 2-neutron emd ( called emd2 ) to @xmath40 while the total emd cross section , including all decay channels , is @xmath41 thus , the total cross section for particle loss by electromagnetic processes is around 507 b , compared to the 8 b for nuclear inelastic interactions , and is the dominant limit on usable luminosity lifetime for the experiments . furthermore , if eq . ( [ eq : loss - condition ] ) is satisfied somewhere , then the lost particles hit the vacuum chamber in a well - defined location which may lie in a superconducting magnetic element @xcite . with the energy @xmath42 per ion in tab . [ tab : lhc - beams ] , the total power in the bfpp beam is @xmath43 . this induced heating may raise the temperature enough to bring the superconducting cable irreversibly over the critical surface in its phase diagram , the space spanned by magnetic field , current , and temperature . the resulting departure from the superconducting state is called a quench . the induced joule heating also quenches neighboring volumes so that the quench propagates . in case of a quench in the lhc , the beam will be dumped within one machine turn and the quench protection system will fire heaters to quickly quench a number of magnets and dissipate the stored magnetic energy over a larger volume , avoiding damage to the magnets . nevertheless , quenches have to be avoided by all means during collider operation since recovery involves the lengthy process of cooling the magnets down again , followed by refilling , ramping and `` squeezing '' of the beams . downtime of the lhc is costly . bfpp has been measured in fixed target experiments @xcite . the first measurement in a collider @xcite occurred during 100 gev / nucleon @xmath44cu@xmath45 operation of the relativistic heavy ion collider ( rhic ) at brookhaven national laboratory , where the detected induced showers agreed within a factor 2 with fluka simulations and the location of the losses was predicted within 2 m. although neither bfpp nor emd pose any risk of quenches at rhic , mutual emd between ions of the colliding beams has also been measured at rhic @xcite and is routinely used to monitor the luminosity @xcite . + + + the risk of inducing quenches means that it is vital to study , quantify and possibly alleviate the impact of bfpp and emd in the lhc , in order to ensure safe operation uninterrupted by lengthy quench - recovery procedures . to study the loss processes , we use a three - step simulation method , consisting of optical tracking of the ions with modified rigidity , a monte carlo simulation of the hadronic and electromagnetic shower created as the ions fragment following their initial impact on the vacuum envelope , and a thermal network simulation of the heat flow inside the surrounding cryo - magnet . to illustrate the method we apply it to bfpp in the alice experiment at ip2 of the lhc , with brief comparisons with the other ips in sec . [ sec : tracking]-[sec : operating - cond ] . in sec . [ sec : emd - quench ] we treat losses from emd and in sec . [ sec : monitoring ] we describe a system of beam loss monitors ( blms ) to survey the losses . finally , in sec . [ sec : alleviation ] , we discuss possible methods of alleviation .

electromagnetic interactions between colliding heavy ions at the large hadron collider ( lhc ) at cern will give rise to localized beam losses that may quench superconducting magnets , apart from contributing significantly to the luminosity decay . to quantify their impact on the operation of the collider , we have used a three - step simulation approach , which consists of optical tracking , a monte - carlo shower simulation and a thermal network model of the heat flow inside a magnet . we present simulation results for the case of ion operation in the lhc , with focus on the alice interaction region , and show that the expected heat load during nominal operation is 40% above the quench level . furthermore , we discuss methods of monitoring the losses and possible ways to alleviate their effect .

electromagnetic interactions between colliding heavy ions at the large hadron collider ( lhc ) at cern will give rise to localized beam losses that may quench superconducting magnets , apart from contributing significantly to the luminosity decay . to quantify their impact on the operation of the collider , we have used a three - step simulation approach , which consists of optical tracking , a monte - carlo shower simulation and a thermal network model of the heat flow inside a magnet . we present simulation results for the case of ion operation in the lhc , with focus on the alice interaction region , and show that the expected heat load during nominal operation is 40% above the quench level . this limits the maximum achievable luminosity . furthermore , we discuss methods of monitoring the losses and possible ways to alleviate their effect .

cond-mat0205350

i

quantum dots are man - made sub - micron structures in a solid , typically consisting of 10@xmath0 - 10@xmath1 atoms and a comparable number of electrons @xcite . in semiconductor quantum dots all electrons are tightly bound , except for a small number of free electrons , which can range from zero to several thousands . for the quantum dot devices considered in this review , the starting point for fabrication is formed by a heterostructure consisting of different semiconductor materials ( gaas / algaas ) . the free electrons are strongly confined to the interface between gaas and algaas , forming a 2-dimensional electron gas ( 2deg ) . confinement in the other two dimensions is accomplished by locally depleting the 2deg , via etching techniques or metal gate electrodes . the resulting structure is weakly coupled to source and drain electrical contacts by tunnel barriers , connecting it to the outside world . + an important element of electronic transport through quantum dots is coulomb blockade @xcite . an extra electron can only be added to the dot , if enough energy is provided to overcome the coulomb repulsion between the electrons . next to this purely classical effect , the confinement in all three directions leads to quantum effects that strongly influence electronic transport at low temperature . in particular it leads to the formation of a discrete ( 0d ) energy spectrum , resembling that of an atom . this and other similarities have therefore lead to the name artificial atoms for quantum dots @xcite . + the next logical step after studying individual quantum dots is to study systems of more than one dot . where single quantum dots are regarded as ` artificial atoms ' , two quantum dots can be coupled to form an ` artificial molecule ' . depending on the strength of the inter - dot coupling , the two dots can form ionic - like ( weak tunnel coupling ) or covalent - like bonds ( strong tunnel coupling ) . in the case of ionic bonding the electrons are localized on the individual dots . the binding occurs , because a static redistribution of electrons leads to an attractive coulomb interaction . weakly , electrostatically coupled quantum dots with negligible inter - dot tunnel conductance are covered by orthodox coulomb blockade theory @xcite . in the case of covalent bonding , two electron states are quantum - mechanically coupled . the main requirement for covalent binding is that an electron can tunnel many times between the two dots in a phase - coherent way . here the electron can not be regarded as a particle that resides in one particular dot , but it must be thought of as a coherent wave that is delocalized over the two dots . the bonding state of a strongly coupled artificial molecule has a lower energy than the energies of the original states of the individual dots . this energy gain forms the binding force between the two dots . + the theoretical possibility to perform certain tasks in a much more efficient way using a ` quantum computer ' instead of a ` classical computer ' , has stimulated the search for physical realizations of the basic building block of such a computer : the quantum bit . in principle , any quantum two - level system can be used as such a qubit . in particular , recent studies have put forward double quantum dots as interesting candidates for realizing qubits @xcite . the possible application of double quantum dot devices in quantum logic forms an important motivation for this work . + in this review we concentrate on electron transport through lateral double quantum dots coupled in series . all devices have been fabricated and all experiments have been performed at delft university of technology and ntt basic research laboratories . by now there exists an extensive literature on experimental studies of electron transport through double lateral quantum dots coupled in series , and lateral double dots coupled in parallel . vertical double quantum dot structures fall outside the focus of this review . in vertical structures , the characteristics of the tunnel barriers are set by the growth parameters of the heterostructure , limiting the experimental tunability . besides that , the gate geometry used in these devices , makes it difficult to address dots independently . + as a first step to understanding double dot systems we introduce the stability diagram @xcite , or honeycomb diagram , in section [ stability ] . it is a convenient tool in the analysis of double dot transport properties . resonant tunneling experiments discussed in section [ restun ] , show that the resonant widths are only determined by the lifetime of the discrete energy states , independent of the electron temperature . in section [ magnetiz ] we discuss level spectroscopy in a magnetic field . the double dot geometry offers sufficient energy resolution to probe intra - dot level repulsion . in section [ spectro ] we present microwave spectroscopy measurements on a quantum dot molecule . we illustrate the transition from a weakly coupled double dot to a strongly coupled double dot , by discussing a two - level system in section [ 2level ] . although being a clear simplification , the mapping of the double dot on a two - level system grasps much of the physics of the experiments presented in section [ spectro ] . irradiation with microwaves leads to photon assisted tunneling ( pat ) ( sections [ weakcpat ] and [ strongcpat ] ) , which turns out to be a powerful tool not only to reveal the character of the inter - dot coupling , but also to quantitatively determine the bonding strength .

electron transport experiments on two lateral quantum dots coupled in series are reviewed . resonant tunneling experiments show that the double dot geometry allows for an accurate determination of the intrinsic lifetime of discrete energy states in quantum dots . the resolution allows to resolve avoided crossings in the spectrum of a quantum dot . with microwave spectroscopy it is possible to probe the transition from ionic bonding ( for weak inter - dot tunnel coupling ) to covalent bonding ( for strong inter - dot tunnel coupling ) in a double dot artificial molecule

electron transport experiments on two lateral quantum dots coupled in series are reviewed . an introduction to the charge stability diagram is given in terms of the electrochemical potentials of both dots . resonant tunneling experiments show that the double dot geometry allows for an accurate determination of the intrinsic lifetime of discrete energy states in quantum dots . the evolution of discrete energy levels in magnetic field is studied . the resolution allows to resolve avoided crossings in the spectrum of a quantum dot . with microwave spectroscopy it is possible to probe the transition from ionic bonding ( for weak inter - dot tunnel coupling ) to covalent bonding ( for strong inter - dot tunnel coupling ) in a double dot artificial molecule this review on the present experimental status of double quantum dot studies is motivated by their relevance for realizing solid state quantum bits .

1503.07895

i

a rotation is an example of an isometry , a map that moves points without changing the distances between them . a rotation is a linear transformation that describes the motion of a rigid body around a fixed point or an axis and can be expressed with an orthonormal matrix which is called a rotation matrix . @xmath0 rotation matrices form a special orthogonal group , denoted by @xmath1 , which , for @xmath2 is non - abelian . the group of @xmath0 rotation matrices is isomorphic to the group of rotations in an @xmath3 dimensional space . this means that multiplication of rotation matrices corresponds to composition of rotations . rotation matrices are used extensively for computations in geometry , kinematics , physics , computer graphics , animations , and optimization problems involving the estimation of rigid body transformations . for this reason , the generation of a rotation matrix is considered to be an important problem in mathematics . in the two dimensional euclidean space , a rotation matrix can easily be generated using basic linear algebra or complex numbers . similarly , in the lorentzian plane , a rotation matrix can be generated by double ( hyperbolic ) numbers . in higher dimensional spaces , obtaining a rotation matrix using the inner product is impractical since each column and row of a rotation matrix must be a unit vector perpendicular to all other columns and rows , respectively . these constraints make it difficult to construct a rotation matrix using the inner product . instead , in higher dimensional spaces , rotation matrices can be generated using various other methods such as unit quaternions , the rodrigues formula , the cayley formula , and the householder transformation . we will give a brief review of these methods and use elliptical versions of these methods later in the paper . * 1 . a unit quaternion :* each unit quaternion represents a rotation in the euclidean 3-space . that is , only four numbers are enough to construct a rotation matrix , the only constraint being that the norm of the quaternion is equal to 1 . also , in this method , the rotation angle and the rotation axis can be determined easily . however , this method is only valid in the three dimensional spaces ( @xcite , @xcite ) . in the lorentzian space , timelike split quaternions are used instead of ordinary usual quaternions ( @xcite , @xcite ) . * rodrigues formula :* an orthonormal matrix can be obtained using the matrix exponential @xmath4 where @xmath5 is a skew symmetric matrix and @xmath6 is the rotation angle . in this method , only three numbers are needed to construct a rotation matrix in the euclidean 3-space ( @xcite , @xcite , @xcite and , @xcite).the vector set up with these three numbers gives the rotation axis . this method can be extended to the @xmath3 dimensional euclidean and lorentzian spaces ( @xcite , @xcite , @xcite , @xcite and , @xcite ) . * cayley formula :* the formula @xmath7 gives a rotation matrix , where @xmath5 is a skew symmetric matrix . rotation matrices can be given by the cayley formula without using trigonometric functions . the cayley formula is an easy method but it does nt give the rotation angle directly ( @xcite , @xcite , @xcite , @xcite and , @xcite ) . householder transformation :* the householder transformation gives us a reflection matrix . we can obtain a rotation matrix using two householder transformations . this method is an elegant method but it can be long and tedious . also , the rotation angle has to be calculated separately . this transformation can be used in several scalar product spaces ( @xcite , @xcite , @xcite , @xcite and , @xcite ) . details about generating rotation matrices , particularly in the euclidean and lorentzian spaces , using these methods can be found in various papers , some of which are given in the reference section . those authors mostly studied the rotation matrices in the positive definite scalar product space whose associated matrices are diag@xmath8 and interpreted the results geometrically . for example , quaternions and timelike split quaternions were used to generate rotation matrices in the three dimensional euclidean and lorentzian spaces where the associated matrices were @xmath9 and @xmath10 respectively . in these spaces , rotations occur on the sphere @xmath11 or the hyperboloids @xmath12 . that is , euclidean and lorentzian rotation matrices help us to understand spherical and hyperbolic rotations . in the euclidean space , a rotation matrix rotates a point or a rigid body through a circular angle about an axis . that is , the motion happens on a circle . similarly , in the lorentzian space , a rotation matrix rotates a point through an angle about an axis circularly or hyperbolically depending on whether the rotation axis is timelike or spacelike , respectively . in this paper , we investigate elliptical rotation matrices , which are orthogonal matrices in the scalar product space , whose associated matrix is diag@xmath13 with @xmath14 . first , we choose a proper scalar product to the given ellipse ( or ellipsoid ) such that this ellipse ( or ellipsoid ) is equivalent to a circle ( or sphere ) for the scalar product space . that is , the scalar product does nt change the distance between any point on the ellipse ( or ellipsoid ) and origin . interpreting a motion on an ellipsoid is an important concept since planets usually have ellipsoidal shapes and elliptical orbits . the geometry of ellipsoid can be examined using affine transformations , because of an ellipsoid can be considered as an affine map of the unit sphere . for example , for the ellipsoid @xmath15 and the unit sphere @xmath16 we can write @xmath17 using the affine transformation @xmath18 , @xmath19 then we have , @xmath20 where @xmath21 the aim of this study is to explain the motion on the ellipsoid @xmath22 as a rotation , using the proper inner product , vector product and elliptical orthogonal matrices . in this method , the elliptical inner product , the vector product and the angles are compatible with the parameters @xmath6 and @xmath23 of the parametrization @xmath24 . we use the classical methods to generate elliptical rotation matrices . in the preliminaries section , first we explain how to define a suitable scalar product and a vector product for a given ellipsoid . then we introduce the symmetric , skew symmetric and orthogonal matrices in this elliptical scalar product space . finally , we examine the motion on an ellipsoid using elliptical rotation matrices . in section 3 , we generate the elliptical rotation matrices using various classical methods ( such as , cayley formula , rodrigues formula and householder transformation ) compatible with the defined scalar product . furthermore , we defined the elliptic quaternions and generate elliptical rotations using unit elliptic quaternions .

elliptical rotation is the motion of a point on an ellipse through some angle about a vector . the purpose of this paper is to examine the generation of elliptical rotations and to interpret the motion of a point on an elipsoid using elliptic inner product and elliptic vector product . to generate an elliptical rotation matrix , finally , we define elliptic quaternions and generate an elliptical rotation matrix using those quaternions . each method is proven and is provided with several numerical examples . * keywords :* elliptic quaternion , rotation matrices , rodrigues formula , cayley transformation , householder transformation . * msc 2010 : * 15a63 , 15a66 , 70b05 , 70b10 , 70e17 , 53a17 .

elliptical rotation is the motion of a point on an ellipse through some angle about a vector . the purpose of this paper is to examine the generation of elliptical rotations and to interpret the motion of a point on an elipsoid using elliptic inner product and elliptic vector product . to generate an elliptical rotation matrix , first we define an elliptical ortogonal matrix and an elliptical skew symmetric matrix using the associated inner product . then we use elliptic versions of the famous rodrigues , cayley , and householder methods to construct an elliptical rotation matrix . finally , we define elliptic quaternions and generate an elliptical rotation matrix using those quaternions . each method is proven and is provided with several numerical examples . * keywords :* elliptic quaternion , rotation matrices , rodrigues formula , cayley transformation , householder transformation . * msc 2010 : * 15a63 , 15a66 , 70b05 , 70b10 , 70e17 , 53a17 .

1504.04160

i

versus the excitation frequency , of a microwave billiard with the shape of a quarter stadium measured at room temperature ( upper panel ) and at 2 k ( lower panel ) , where it is normal conducting and superconducting , respectively . the shape of the resonator and the positions of the antennas are illustrated in the inset . reprinted from phys . rev . lett . * 69 * , 1296 ( 1992 ) . ] this article provides a brief survey on groundbreaking experiments that were performed in the last two decades in our group with superconducting microwave billiards . they focused on the understanding of the generic properties of quantum systems like nuclei and of condensed matter structures like graphene or fullerene . in the first case , we exploited the equivalence of the scalar helmholtz equation governing flat , metallic microwave resonators , so - called microwave billiards , and the non - relativistic schrdinger equation of the corresponding quantum billiard , to address problems from the fields of quantum chaos and compound nucleus reaction theory @xcite . in the second case , the analogy between the scalar helmholtz equation describing microwave photonic crystals squeezed between two metal plates or enclosed in a microwave billiard , so - called dirac billiards , and the dirac equation was used to experimentally investigate relativistic phenomena occurring in graphene . some details on the experimental setups for the measurements of resonance spectra at superconducting conditions will be provided in . a classical billiard @xcite is a bounded domain , in which a pointlike particle moves freely and is reflected elastically on impact with the boundary . since its dynamics only depends on its shape , such systems are widely used to investigate the effects of the classical dynamics on properties of the associated quantum system @xcite . the correspondence between the latter , i.e. , a quantum billiard and a microwave billiard of the same shape , holds below a certain excitation frequency @xmath2 of the microwaves that are sent into the resonator , @xmath3 , where the electric field vector is parallel to the cylinder axis . here , @xmath4 is the velocity of light and @xmath5 is the height of the resonator . originally , this analogy was used to study universal fluctuation properties of the eigenvalues of a quantum billiard with microwave billiards at room temperature @xcite . the resolution of the resonances in the spectra , however , was very limited due to the relatively low quality factor @xmath6 for normal conducting microwave billiards . we obtained the eigenvalues of ( non ) relativistic quantum billiards by measuring resonance spectra to a very high accuracy with superconducting microwave billiards with quality factors of up to @xmath7 . this is an indispensable requirement for the determination of complete sequences of several hundreds and even thousands of eigenvalues , as illustrated in fig . [ spektren2dstadium ] . shown are transmission spectra of a microwave billiard made out of niobium metal with the shape of a quarter stadium ( see ) measured at room temperature ( upper panel ) , where the resonator is normal conducting and at @xmath8 k , where it is superconducting . its eigenfrequencies , and thus the eigenvalues of the corresponding quantum billiard , are obtained from the positions of the resonances . in the upper spectrum in fig . [ spektren2dstadium ] many resonances overlap , whereas all resonances are well resolved in the lower one , so that the determination of the complete sequence of eigenvalues is achievable only in the superconducting case . because of this capability we were able to tackle a wide range of problems that are far from feasible with normal conducting microwave billiards . since the resonance spectra are measured by emitting microwave power into the resonator via one antenna and receiving it at the same or another one , microwave billiards can also be viewed as scattering systems . we utilized this capability and studied the universal properties of quantum scattering systems with an unprecedented accuracy . due to limitation in space we focus here only on a few of our experiments . in secs . [ 2dstadium ] and [ triangular ] we present results on the spectral properties of classically chaotic systems . in we review our studies concerning the effect of symmetry breaking on the properties of the eigenvalues and the wave functions of chaotic systems . we also investigated the spectral properties of three - dimensional resonators that do not have a quantum analogue . these experiments provided additional important information to previously performed ones @xcite of , due to the superconductivity , hitherto never achieved precision . one example will be discussed in . the analysis of the spectral properties addresses the manifestation of the gross structure of the poincar surface of section ( psos ) , namely the relative sizes of the regions of regular and chaotic dynamics , in the associated quantum system . the phenomenon of quantum dynamical tunneling @xcite , that is , the classically forbidden transition between states localized in separated parts of the classical phase space has been studied theoretically intensively , but only in a few experiments @xcite , because the measurements require an exceptionally high precision . in sec . [ tunneling ] we will review the main results of our investigations of chaos - assisted tunneling @xcite and of dynamical tunneling @xcite . only lately we started experimental investigations of relativistic phenomena associated with graphene and will present some results of our most recent experiments in .

we will present the results of these and subsequent experiments and also concerning our latest studies of dynamical tunneling . most of the works on quantum chaos concentrate on quantum systems described by the non - relativistic schrdinger equation . we started lately with their experimental investigation and will present the system and recent results on the statistical properties of the associated eigenvalues in this article . *

experiments with superconducting microwave cavities have been performed in our laboratory for more than two decades . the purpose of the present article is to recapitulate some of the highlights achieved . we briefly review ( i ) results obtained with flat , cylindrical microwave resonators , so - called microwave billiards , concerning the universal fluctuation properties of the eigenvalues of classically chaotic systems with no , a threefold and a broken symmetry ; ( ii ) summarize our findings concerning the wave - dynamical chaos in three - dimensional microwave cavities ; ( iii ) present a new approach for the understanding of the phenomenon of dynamical tunneling which was developed on the basis of experiments that were performed recently with unprecedented precision , and finally , ( iv ) give an insight into an ongoing project , where we investigate universal properties of ( artificial ) graphene with superconducting microwave photonic crystals that are enclosed in a microwave resonator , i.e. , so - called dirac billiards . * quantum chaos , a field that focuses on the quantum manifestations of classical chaos , attracted a lot of attention during the last years . the signatures of classical chaos in universal properties of the corresponding quantum system are indeed of relevance in nuclear physics , atomic physics , condensed matter physics , optics and in acoustics . they were observed in the statistical properties of the eigenvalues and the wave functions , in the fluctuation properties of the scattering matrix elements of chaotic scattering processes , in the transport properties of quantum dots and also in systems , where time - reversal invariance is broken , e.g. , by a magnetic field . billiards provide a most appropriate system for the study of features from the field of quantum chaos , because the degree of chaoticity of their classical dynamics only depends on their shape . the eigenvalues and wave functions of two - dimensional quantum billiards are measured since two decades in our laboratory using flat cylindrical , superconducting and normal conducting microwave resonators , respectively . thereby large sequences of approximately 1000 eigenvalues were determined in the first experiments with a hitherto never achieved high precision yielding new insight into quantum chaos phenomena , because at that time their computation was either not possible or only feasible with a large numerical effort . we will present the results of these and subsequent experiments and also concerning our latest studies of dynamical tunneling . the latter demonstrate that even nowadays the measurements with superconducting microwave resonators may provide valuable new information that is indispensable for the development of a theoretical description of quantum phenomena like dynamical tunneling , even though eigenvalues may now be computed with high accuracy . most of the works on quantum chaos concentrate on quantum systems described by the non - relativistic schrdinger equation . only a few studies focus on the properties of the eigenvalues and the wave functions of relativistic billiards that are governed by the dirac equation of a massless spin- particle . we started lately with their experimental investigation and will present the system and recent results on the statistical properties of the associated eigenvalues in this article . *

1409.1333

i

owing to the increasing availability of high - dimensional datasets , regression models for multivariate response and high - dimensional predictors have become important tools . in this article , we describe two procedures where a random target variable @xmath0 depends on explanatory variables or covariates within a cluster - specific regression model . each cluster is represented by a parametric conditional distribution , the entire dataset being modeled by a mixture of these distributions . it provides a rigorous statistical framework . the model assumes that each observation @xmath1 originates from one of @xmath2 disjoint classes and that the data @xmath3 are independent and identically distributed such that if @xmath4 belongs to class @xmath5 , the target variable @xmath6 results from a conditional regression model @xmath7 with an unknown matrix of coefficients @xmath8 and independent errors @xmath9 with an unknown diagonal covariance matrix @xmath10 . we work with high - dimensional dataset , in other words the number of parameters to estimate @xmath11 is larger than the number of observed target values @xmath12 . two ways are considered in this paper , coefficients sparsity and ranks sparsity . the first approach consists in estimating the matrix @xmath8 by a matrix with few nonzero coefficients . the well - known lasso estimator , introduced by @xcite for linear models , is the solution chosen here . we refer to @xcite for an overview of the lasso estimator and to @xcite for stability selection results . in the second approach , we consider the rank sparsity in @xmath8 . this approach dates back to the @xmath13 s and was initiated by @xcite for the linear model . @xcite introduced the term of reduced - rank regression for this class of models . for more recent works , we refer to @xcite and to @xcite . nevertheless , the linear model is appropriate for homogeneous observations , which is not always the case . we extend in this paper those methods to mixture regression models . an important example of high - dimensional dataset is functional dataset . we refer to ramsay and silverman s book @xcite for an overview . moreover , a lot of recent works have been done on regression models for functional datasets : for example , we refer to @xcite for a study with scalar response and functional regressors . in this paper , we focus on the projection of functions into a well - suited wavelet basis . indeed , they handle many types of functional data , because they represent global and local attributes of functions and can deal for example with discontinuities . moreover , a large class of functions can be well represented with few nonzero coefficients for a suitable wavelet , which leads to sparse matrix of coefficients and then to sparse regression matrix . we propose here two procedures for clustering high - dimensional or functional datasets , where the high - dimension or functional random target variable @xmath0 depends on high - dimensional or functional predictors @xmath14 with a cluster - specific regression model . remark that we estimate the number of components , model parameters and cluster proportions . in the case of a large number of regressor variables , we use variable selection tools in order to detect relevant regressors . since the structure of interest may often be contained into a subset of available variables and many attributes may be useless or even harmful to detect a reasonable clustering structure , it is important to select the relevant variables . moreover , removing irrelevant variables enables us to get an easier model and can largely enhance comprehension . our two procedures are mainly based on three recent works . firstly , we refer to the article of @xcite , which studies finite mixture regression model . even if we work on a multivariate version of it , the model considered in the article of @xcite is adopted here . the second , the article of @xcite , deals with model - based clustering in density estimation . they propose a procedure , called lasso - mle procedure , which determines the number of clusters , the set of relevant variables for the clustering and a clustering of the observations , with high - dimensional data . we extend this procedure to regression models . finally , @xcite suggests a low rank estimator for the linear model . to consider the matrix structure , we develop this last approach to mixture models . we consider a finite mixture of gaussian regression models . the two procedures we propose follow the same sequence of steps . firstly , a penalized likelihood approach is considered to determine potential sets of relevant variables . introduced by @xcite , the lasso is used to select variables . varying the regularization parameter , it constructs efficiently a data - driven model collection where each model has a reasonable complexity . the second step of the procedures consists in refitting parameters by a less biased estimator , focusing on selected variables . then , we select a model among the collection using the slope heuristic , which was developed by @xcite . the difference between the two procedures is the refitting step . in the first procedure , later called lasso - mle procedure , the maximum likelihood estimator is used/ the second procedure , called lasso - rank procedure , deals with low rank estimation . for each model in the collection , a subcollection of models with conditional means estimated by various low rank matrices is constructed . it leads to sparsity for the coefficients and for the rank , and considers the conditional mean within its matrix structure . the article is organized as follows . section [ gaussianmixtureregression ] deals with gaussian mixture regression models . the model collection that we consider is described . in section [ twoprocedures ] , the two procedures that we propose to solve the problem of high - dimensional regression data clustering are described . section [ illustrativeexample ] presents an illustrative example , to highlight each choice done in the two procedures . section [ functionaldatasets ] states the functional data case , with a description of the projection proposed to convert these functions into coefficients data . we end this section by studying simulated and benchmark data . finally , a conclusion section ends this article .

finite mixture regression models are useful for modeling the relationship between response and predictors arising from different subpopulations . in this article , we study high - dimensional predictors and high - dimensional response and propose two procedures to cluster those observations according to the link between predictors and the response . we extend these procedures to functional data , where predictors and responses are functions . for this purpose , we use a wavelet - based approach .

finite mixture regression models are useful for modeling the relationship between response and predictors arising from different subpopulations . in this article , we study high - dimensional predictors and high - dimensional response and propose two procedures to cluster those observations according to the link between predictors and the response . to reduce the dimension , we propose to use the lasso estimator , which takes into account the sparsity and a maximum likelihood estimator penalized by the rank , to take into account the matrix structure . to choose the number of components and the sparsity level , we construct a model collection , varying those two parameters and select a model among this collection with a non - asymptotic criterion . we extend these procedures to functional data , where predictors and responses are functions . for this purpose , we use a wavelet - based approach . for each situation , we provide algorithms and apply and evaluate our methods both on simulated and real dataset , to understand how it works in practice .

0810.3497

i

emulsions are mixtures of two immiscible fluids consisting of droplets of one phase dispersed into the other , stabilized against coalescence by surfactant . at low droplets volume fraction , the emulsions have basically a newtonian behavior @xcite . when the volume fraction of the dispersed phase is increased , droplets come into contact . if a small stress is applied to a dense emulsion , the interfaces between the droplets can be strained to store surface energy ; it results in an elastic response @xcite . above a yield stress , they flow as a result of droplets rearrangements @xcite . the flow behavior of dense emulsions can be measured in classical rheological experiments , and seems to be well represented by a herschel - bulkley model @xmath1 ; in monodisperse emulsions , the exponent @xmath2 has been found to vary between @xmath3 and @xmath4 for volume fractions varying between 0.58 and 0.65 @xcite . at first glance , dense emulsions behavior thus seems to be well understood and modelled . however , it has been shown recently that the behavior of dense emulsions @xcite , and more generally of pasty materials @xcite , may be much more complex than what can be inferred from simple rheometric experiments , particularly at the approach of the yield stress . the reason is that the flow behavior of materials is usually studied in viscosimetric flows @xcite , in which one measures macroscopic quantities ( a torque @xmath5 vs. a rotational velocity @xmath6 ) . a constitutive law relating the shear stress @xmath7 to the shear rate @xmath8 in any flow can then be derived easily from these macroscopic measurements provided that the flow is homogeneous . however , in complex fluids , this last requirement may not be fulfilled as the flow may be localized , as observed in many pasty materials @xcite , even in the cone - and - plate geometry @xcite . another problem of importance , that can complicate the analysis of the rheological data , is the existence of slippage of dense emulsions at the walls @xcite even with slightly roughened walls @xcite . these problems can be got round by coupling macroscopic torque measurements and local measurements of the shear rate , e.g. through mri techniques @xcite , optical methods @xcite or sound methods @xcite , in order to account for these heterogeneities and to get a proper constitutive law . local measurements in dense emulsions have recently yielded surprising results @xcite : in some emulsions , it is impossible to find a single constitutive law compatible with all flows . the constitutive law of the material then seems to depend on the velocity at the inner cylinder in a couette cell @xcite or the pressure gradient in a poiseuille cell @xcite . three main reasons have been evoked to explain the apparent absence of a constitutive law : thixotropy , non local effects , and shear - induced migration of the droplets . thixotropy may be a cause from the apparent absence of a constitutive law . long times may indeed be needed to reach a steady state . when the measurements are not performed in this steady state , it may then leave the idea that different laws are required to describe the flow of the sample . such long times to reach steady state have been evidenced in some experiments dealing with highly adhesive emulsion @xcite . importantly , @xcite , although they claim to have reached a steady state , were unable to account for the flows of their adhesive emulsion with a single constitutive law . note that we will study again the emulsion of @xcite in this paper . the apparent absence of a single local constitutive law may also be the signature of non - local phenomena . @xcite have actually proposed a non local model to describe the flows of jammed materials ( and thus dense emulsions ) , based upon the cooperativity of the flow . in these systems , flow occurs via a succession of elastic deformation and irreversible plastic events . these localized plastic events induce a relaxation of the stress on the whole system . their zone of influence is quantified by a length @xmath9 . in presence of high stress gradient or in the vicinity of a surface , this process creates non local effects : indeed , in these situations , the rate of rearrangements of the neighboring fluid differs from what would be the bulk rearrangement rate . as a consequence , these effects affect drastically the rheological behavior when @xmath9 is comparable to the size of the confinement . such behavior has been recently pointed out by @xcite . from experimental data , the authors show that the cooperativity length @xmath9 is zero below the jamming concentration @xmath10 and is typically a few oil droplet diameters above @xmath10 . using a wide gap should then prevent us from being sensitive to these non local effects . we will check this feature by studying the flows of the @xcite emulsion in this paper . particle migration may also be a cause for the apparent lack of a local constitutive law @xcite . in multiphase materials , wide gap couette flows are indeed known to lead to concentration heterogeneities due to shear - induced migration of the dispersed elements toward the low shear zones . the migration of deformable particles , and particularly drops , has been much less studied than the one of rigid particles . single droplets have been found to migrate away from the rigid walls in any shear flow , due to asymmetric flows around deformed droplets @xcite . this leads to an equilibrium position somewhere between the walls , exactly at the center of the gap in the case of a narrow gap couette geometry , nearer from the inner cylinder in the case of a wide gap couette geometry @xcite . when dealing with a dilute emulsion , it has been observed , in contrast with dilute suspensions of rigid particles , that the equilibrium distribution of droplets is parabolic around a position that is near the equilibrium position of single droplets @xcite . the theoretical explanation is that all particles would tend to accumulate at the same equilibrium position between the walls , but their distribution is broadened by their binary collisions ( this second process may be modelled as it is for rigid particles ) . only rather dilute emulsions ( up to 10% of droplets ) have been studied so far , and nothing is known about what happens for higher droplet concentrations . in this case , one would expect the wall effect on the droplet migration to be negligible compared to the effect of the interactions between droplets ; then , one would expect migration to product the same effects as in suspensions , with the same kinetics as it has basically the same physical origin . we can thus expect that relevant information for migration in dense emulsions can be found in the literature dealing with suspensions of rigid particles . in suspensions of noncolloidal rigid particles , migration has been observed in many situations : wide - gap couette flows @xcite , parallel - plate flows @xcite , pipe flows @xcite . in wide - gap couette flows , the consequence of migration is an excess of particles near the outer cylinder . migration is related to the shear - induced diffusion of particles @xcite : the gradients in shear rate that exist in all but the cone and plate geometry generate a particle flux towards the low shear zones ( the outer cylinder in the case of the couette geometry ) , which is counterbalanced by a particle flux due to viscosity gradients . in an alternative model @xcite , particle fluxes counterbalance the gradients in normal stresses . most experiments @xcite observe that migration in suspensions of volume fraction up to 55% is rather slow , in accordance with its diffusive origin . however , it was recently found in a couette geometry that in the case of very dense suspensions ( up to 60% ) , which exhibit an apparent yield stress , migration is almost instantaneous @xcite ( it lasts for a few revolutions ) so that it may be unavoidable . a reason for this very fast kinetics may be that particles are closely packed together : any shear may then push the particles towards the outer cylinder with an instantaneous long range effect on the particle concentration . migration may then be expected to be very fast for dense emulsions exhibiting a yield stress , i.e. in which droplets are in contact as the particles in the very dense suspensions of @xcite . in this paper , we address the questions of the existence and the determination of a local constitutive law for the flows of dense emulsions . in this aim , we use a wide gap couette cell to avoid non local effects and we study in detail the question of shear - induced droplet migration in these systems . we also take care of performing our experiments in a steady state . we propose to couple macroscopic measurements and local measurements of concentration and velocity through mri techniques during the wide gap couette flows of dense emulsions . a new method designed to measure the local droplet concentration in emulsions is developed and we seek for the occurrence of migration on many formulations . local measurements of the constitutive law are then compared to purely macroscopic measurements . in sec . [ section_display ] , we present the materials and the experimental setup . we present the concentration profiles obtained on all materials after long time experiments in sec . [ section_concentration ] . the velocity profiles are shown in sec . [ section_velocity ] . the macroscopic rheometrical measurements are displayed in sec . [ section_torque ] , and a purely macroscopic determination of the constitutive law is presented . the results are analyzed in sec . [ section_local ] : we determine locally the constitutive law of the material from the velocity profiles and we compare macroscopic and local measurements of the constitutive law .

are expected in the wide gap couette flows of multiphase materials , we also designed a new method to measure the local droplet concentration in emulsions with a mri device . the existence of a constitutive law accounting for all flows contrasts with previous results obtained within a microchannel by goyon _ et al . _ [ phys . lett . * 96 * , 138302 ( 2006 ) ] .

flows of dense emulsions show many complex features among which long range nonlocal effects pose problem for macroscopic characterization . in order to get round this problem , we study the flows of several dense emulsions , with droplet size ranging from 0.3 to 40 m , in a wide gap couette geometry . we couple macroscopic rheometric experiments and local velocity measurements through mri techniques . as concentration heterogeneities are expected in the wide gap couette flows of multiphase materials , we also designed a new method to measure the local droplet concentration in emulsions with a mri device . in contrast with dense suspensions of rigid particles where very fast migration occurs under shear in wide gap couette flows , we show for the first time that no migration takes place in dense emulsions even for strain as large as 100000 in our systems . as a result of the absence of migration and of finite size effect , we are able to determine very precisely the local rheological behavior of several dense emulsions . as the materials are homogeneous , this behavior can also be inferred from purely macroscopic measurements . we thus suggest that properly analyzed purely macroscopic measurements in a wide gap couette geometry can be used as a tool to study the local constitutive laws of dense emulsions . all behaviors are basically consistent with herschel - bulkley laws of index 0.5 . the existence of a constitutive law accounting for all flows contrasts with previous results obtained within a microchannel by goyon _ et al . _ [ nature * 454 * , 84 ( 2008 ) ] : the use of a wide gap couette geometry is likely to prevent here from nonlocal finite size effects ; it also contrasts with the observations of bcu _ et al . _ [ phys . rev . lett . * 96 * , 138302 ( 2006 ) ] . we also evidence the existence of discrepancies between a perfect herschel - bulkley behavior and the observed local behavior at the approach of the yield stress due to slow shear flows below the apparent yield stress in the case of a strongly adhesive emulsion .

0810.3497

r

in order to search for the occurrence of shear - induced migration in dense emulsions , we sheared all materials for long times . we chose to apply velocities ranging between 20 and 100 rpm ; this ensures that in all cases the whole gap is sheared ( see sec . [ section_velocity ] ) . we also checked on the 6.5@xmath0 m emulsions that the same results are obtained when the flow is localized during the experiment ( i.e. when the material is sheared at 1 rpm ) . the 0.3@xmath0 m emulsion was sheared for 10000 revs , the 1@xmath0 m emulsion was sheared for 90000 revs , both 6.5@xmath0 m emulsions were sheared for 35000 revs , and the 40@xmath0 m emulsion was sheared for 20000 revs . the corresponding local strains , extracted from the velocity profiles ( see sec . [ section_velocity ] ) range between 5 times the number of revolutions near the inner cylinder , and 0.25 times near the outer cylinder . m adhesive emulsion for 10000 revs ( squares ) , and a 40@xmath0 m non - adhesive emulsion for 20000 revs ( open circles).,width=340 ] in all the experiments , within the experimental uncertainty , we observe that the materials remain homogeneous after shear : there is no observable shear - induced migration . results for the 0.3@xmath0 m and the 40@xmath0 m emulsion are depicted in fig . [ fig_emul_concentration ] . note also that the relaxation times @xmath28 of the oil and water phases were not observed to change during all the experiments : as pointed out in sec . [ section_display ] , this is consistent with the fact that the droplet size is unchanged by shear and that all the measurements deal with the same homogeneous suspension . consistently , optical observations of the droplets after shear did not reveal size changes . as noted in sec . [ section_introduction ] , up to now only rather dilute emulsions have been studied : we can not compare our results with any result from the literature dealing with emulsions . however , the relevant mechanism involved in migration of droplets in dense emulsions should be basically the same as in suspensions of rigid particles , namely a shear - induced diffusion of particles . from this model and results from the literature of migration in dense suspension , we can thus infer whether or not migration would have been expected in our systems . in the diffusive model of migration @xcite , the particles undergo a shear - induced diffusion characterized by a self - diffusion coefficient @xmath51 @xcite , where @xmath52 is the particle volume fraction , @xmath53 the shear rate , @xmath54 the particle size , and @xmath55 is a dimensionless coefficient whose dependence on @xmath52 may theoretically be @xmath56 for small @xmath52 . the gradients in shear rate that exist in a couette geometry then generate a particle flux towards the outer cylinder , which is counterbalanced by a particle flux due to viscosity gradients . as qualitatively confirmed experimentally @xcite , one would then expect the migration phenomenon to last for a number of revolutions @xmath57 until a stationary heterogeneous profile is established , where @xmath58 and @xmath59 are respectively the outer and inner radius , and @xmath60 . within the frame of this model , we can now evaluate the expected @xmath61 for complete migration in our experiment from experimental results from literature . actually , as shown by @xcite two inconsistent set of data exist , depending on the concentration : migration has been shown to occur much more rapidly for concentrations above 55% . e.g , for moderately dense suspensions , @xcite find @xmath62 revs for 140@xmath0 m particles , at @xmath63 , with @xmath64 cm and @xmath65 cm . in the case of our materials , this would imply a value of @xmath66 revs for the 0.3@xmath0 m emulsion , @xmath67 revs for the 1@xmath0 m emulsion , @xmath68 revs for the 6.5@xmath0 m emulsions , and @xmath69 revs for the 40@xmath0 m emulsion . on the other hand for very dense suspensions , @xcite found that migration occurs during the first 50 revs within the same couette geometry as our . this would imply an expected value of @xmath70 revs for the 0.3@xmath0 m emulsion , @xmath71 revs for the 1@xmath0 m emulsion , @xmath72 revs for the 6.5@xmath0 m emulsions , and @xmath73 revs for the 40@xmath0 m emulsion in our experiments . let us recall that @xmath61 is the number of revolutions expected for _ complete _ migration , and that heterogeneities should be observable for a value of @xmath74 revolutions ( see @xcite and @xcite for measurements of transient concentration profiles in suspensions ) . if a shear - induced migration mechanism similar to the one observed in suspensions of rigid particles was acting in dense emulsions , we should therefore have observed migration in the 40@xmath0 m emulsion and in the 6.5@xmath0 m emulsions . on the other hand , our experiments can not be used to conclude directly for the failure of a diffusive model of migration in the case of the smaller droplets ( 0.3 and 1@xmath0 m ) , but we can expect to predict what happens in this case from the @xmath75 scaling of the diffusion process if this process is relevant . even if it impossible to conclude definitely for the absence of migration for any strain , we can conclude that if a diffusive process acts in dense emulsions , it is much slower than the process involved in dense suspensions of rigid particles . importantly , our results provide reliable lower bounds for the occurrence of migration in dense emulsions . for practical purposes , in most flows and particularly in most rheometrical studies of dense emulsions , it can therefore be claimed that the emulsions remain homogeneous . surprisingly , our results contrast with what was observed by @xcite in dense suspensions of hard spheres where very fast migration was found to occur . a first reason could be that jamming prevents from migration in dense emulsions . however , it was found that migration in dense suspensions can lead to regions having a concentration higher than the maximum packing fraction @xmath10 above which there is no more flow @xcite : this shows that jamming does not necessarily prevents from migration . another reason could be that deformability of the particles play a central role . the shear stresses involved in the flow are of the order of 100 to 200 pa in all materials , whereas the typical stress due to surface tension range between 1000pa for the 40@xmath0 m droplets and 120000pa for the 0.3@xmath0 m particles . the droplets may then be poorly deformable ( except for the 40@xmath0 m emulsion ) under shear , but they must be deformed due to confinement and to their high concentration . in particular they should have flat contacts which may allow the droplets to slip easily past each other ; this is a major difference with the suspensions of hard spheres in which any shear may push the particles towards the outer cylinder with an instantaneous long range effect on the particle concentration thanks to the force chains originating from direct frictional contact forces . note that consistency of our observations with the modelling of shear - induced migration based on normal stresses is difficult to check as normal stresses in dense emulsions are poorly known . it should be noted however that the normal stresses due to surface tension should be of the order of the yield stress @xcite in dense emulsions , while normal stresses tend to diverge at the approach of the maximum packing fraction in dense suspensions of rigid particles ( this difference is linked to the deformability of the droplets leading to flat contacts as pointed out above ) . it is thus not unexpected from this point of view that the kinetics of shear induced migration in a couette flow , which depends on the spatial variation of the first normal stress difference and of the radial normal stress @xcite , is very fast near the maximum packing fraction in suspensions of hard spheres while it would have to remain slow in dense emulsions . anyway , it is not possible to be more quantitative at this stage and to state if migration would have been expected to occur within this picture in our experimental conditions . in these experiments , after a 5 min preshear at 100 rpm , we measured the velocity profiles at various rotational velocities . in all systems , the stationary velocity profiles were found to develop within a few seconds and to remain stable for hours . this means that these systems are not thixotropic @xcite . in the case of the 0.3 @xmath0 m adhesive emulsion , this contrasts with the long time evolution of the velocity profiles observed by @xcite ; we will comment on this point below . the stability of the velocity profiles is also consistent with the absence of shear - induced droplet migration in all systems : as the behavior of dense emulsions depends on their concentration , changes in the concentration profiles in time would have led to changes in the velocity profiles in time . this feature is important : in the absence of a method to measure the concentration profiles , the stability of the velocity profiles can be used as a proof of the absence of migration as long as the migration process is not nearly instantaneous . in fig . [ fig_emul_velocity1 ] , we plot the dimensionless velocity profiles for the steady flows of all the emulsions for various rotational velocities ranging from 0.3 to 100rpm . m adhesive emulsion , at various rotational velocities ranging from 5 to 100rpm ; the line is the theoretical dimensionless velocity profile for a newtonian fluid . b ) same plot for a 1@xmath0 m non - adhesive emulsion . c ) same plot for a 6.5@xmath0 m adhesive emulsion . d ) same plot for a 6.5@xmath0 m non - adhesive emulsion . e ) same plot for a 40@xmath0 m non - adhesive emulsion.,title="fig:",width=302 ] m adhesive emulsion , at various rotational velocities ranging from 5 to 100rpm ; the line is the theoretical dimensionless velocity profile for a newtonian fluid . b ) same plot for a 1@xmath0 m non - adhesive emulsion . c ) same plot for a 6.5@xmath0 m adhesive emulsion . d ) same plot for a 6.5@xmath0 m non - adhesive emulsion . e ) same plot for a 40@xmath0 m non - adhesive emulsion.,title="fig:",width=302 ] m adhesive emulsion , at various rotational velocities ranging from 5 to 100rpm ; the line is the theoretical dimensionless velocity profile for a newtonian fluid . b ) same plot for a 1@xmath0 m non - adhesive emulsion . c ) same plot for a 6.5@xmath0 m adhesive emulsion . d ) same plot for a 6.5@xmath0 m non - adhesive emulsion . e ) same plot for a 40@xmath0 m non - adhesive emulsion.,title="fig:",width=302 ] m adhesive emulsion , at various rotational velocities ranging from 5 to 100rpm ; the line is the theoretical dimensionless velocity profile for a newtonian fluid . b ) same plot for a 1@xmath0 m non - adhesive emulsion . c ) same plot for a 6.5@xmath0 m adhesive emulsion . d ) same plot for a 6.5@xmath0 m non - adhesive emulsion . e ) same plot for a 40@xmath0 m non - adhesive emulsion.,title="fig:",width=302 ] m adhesive emulsion , at various rotational velocities ranging from 5 to 100rpm ; the line is the theoretical dimensionless velocity profile for a newtonian fluid . b ) same plot for a 1@xmath0 m non - adhesive emulsion . c ) same plot for a 6.5@xmath0 m adhesive emulsion . d ) same plot for a 6.5@xmath0 m non - adhesive emulsion . e ) same plot for a 40@xmath0 m non - adhesive emulsion.,title="fig:",width=302 ] the 1@xmath0 m non adhesive emulsion , the 6.5@xmath0 m adhesive and non adhesive emulsions , and the 40 @xmath0 m non adhesive emulsion share a common behavior . mri measurements show that the velocity profiles are curved , and that they occupy only a small fraction of the gap at low rotational velocities ( fig . [ fig_emul_velocity1 ] ) : in this case , the velocity tends to zero within the measurement uncertainty at some radius before the outer cylinder . the fraction of the material that is sheared increases with the rotational velocity . beyond a critical velocity @xmath76 , that depends on the emulsion and is of order 3rpm in the case of the 6.5@xmath0 m adhesive emulsion ( fig . [ fig_emul_velocity1]c ) , or of the order 50 rpm in the case of the 1@xmath0 m non - adhesive emulsion ( fig . [ fig_emul_velocity1]b ) , the whole sample is sheared . this shear localization is a classical feature of yield stress fluids flows in couette geometry where the shear stress is a decreasing function of the radius : the flow must stop at a radius @xmath11 such that the shear stress equals the material yield stress at this place . note that this behavior is not evidenced for the non adhesive 6.5@xmath0 m emulsion and for the non adhesive 40@xmath0 m emulsion : this is simply due to the fact that the applied rotational velocities are too high to probe this behavior . when all the gap is sheared , the velocity profiles are clearly different from those of a newtonian fluid ( fig . [ fig_emul_velocity1 ] ) . the curvature we observe is due to the use of a wide - gap couette geometry and is actually typical of shear - thinning fluid : the shear rate decreases more rapidly within the gap ( i.e when the shear stress decreases ) than for a newtonian fluid . the shear rate may vary here of a factor 20 within the gap while it would vary of a factor 2.1 for a newtonian fluid in this geometry ( this factor 2.1 is simply the ratio between the shear stress at the inner cylinder and the shear stress at the outer cylinder , see eq . [ eq_localstress ] in sec . [ section_torque ] ) . this implies that in the analysis of macroscopic experiments , one can not simply compute a mean shear rate within the gap to characterize the flow . finally , note that no data can be recorded close to the inner cylinder . as a consequence , the expected absence of wall slip ( due to the use of rough surfaces of roughness larger than the droplet size ) can not be proved at this stage . it can only be observed that the extrapolation of the dimensionless velocity profiles to a value of 1 at the inner cylinder is reasonable , and that wall slip if any should be of a limited amount of order 5% or less . we will show in sec . [ section_local ] that it is possible to infer the emulsion velocity at the walls from its local constitutive law , and that wall slip if any is actually less than 1% of the inner cylinder velocity . in log scale ( for readability , not all profiles are plotted ) . the solid lines of fig . [ fig_emul_velocity1log]a are the velocity profiles predicted by a herschel - bulkley law.,title="fig:",width=302 ] in log scale ( for readability , not all profiles are plotted ) . the solid lines of fig . [ fig_emul_velocity1log]a are the velocity profiles predicted by a herschel - bulkley law.,title="fig:",width=302 ] in log scale ( for readability , not all profiles are plotted ) . the solid lines of fig . [ fig_emul_velocity1log]a are the velocity profiles predicted by a herschel - bulkley law.,title="fig:",width=302 ] in log scale ( for readability , not all profiles are plotted ) . the solid lines of fig . [ fig_emul_velocity1log]a are the velocity profiles predicted by a herschel - bulkley law.,title="fig:",width=302 ] in log scale ( for readability , not all profiles are plotted ) . the solid lines of fig . [ fig_emul_velocity1log]a are the velocity profiles predicted by a herschel - bulkley law.,title="fig:",width=302 ] at first sight , the 0.3@xmath0 m adhesive emulsion presents the same behavior as the other samples . the velocity profiles are curved , and they occupy only a small fraction of the gap at low rotational velocities . however , focusing on the low velocities part and thus very low shear rates part of the profiles , a striking behavior is evidenced in the semi - logarithmic scale figure ( fig . [ fig_emul_velocity1log]a ) . a slope break in the profile is evidenced at a given radius inside the gap . the velocity then starts to decrease very slowly with the radius . there is then a large zone of the material in which there remains a very slow flow . this behavior is reminiscent of the one observed previously by @xcite and by @xcite in adhesive emulsions ( this behavior will be shortly commented in sec . [ section_local ] ) . these slowly varying slow flows occur here for velocities lower than a few [email protected]@xmath77 . this behavior seems to be a specific feature of the strongly adhesive 0.3 @xmath0 m emulsion . we measured velocity as low as [email protected]@xmath77 at the approach of flow stoppage in the 1@xmath0 m non adhesive and 6.5@xmath0 m adhesive emulsion ( fig . [ fig_emul_velocity1log]b , c ) but did not observe any slope break in these cases : there is no evidence of any slow flow in the apparently jammed zones in these emulsions . in the following , we build the constitutive laws of the emulsions accounting for their flow properties . to get this information , we use two different methods . first , focusing on the macroscopic measurements , we present an experimental procedure allowing us to get a relationship between local values of the shear stress and the shear rate by measuring only macroscopic quantities ( sec . [ section_torque ] ) . second , we analyze in detail the velocity profiles and extract a local constitutive law by differentiating the velocity field to get the local shear rate ( sec . [ section_local ] ) . in this section , we present the macroscopic rheometric measurements ( torque vs. rotational velocity ) . we then show that it is possible to extract the constitutive law of the material from a proper analysis of this set of purely macroscopic data only provided that the material is homogeneous ( absence of migration ) and that there is no wall slip , which is what we have ensured . for each rotational velocity @xmath6 , we have measured the torque exerted on the inner cylinder vs. time until a stationary state is reached ( see inset of fig . [ fig_emul_rheomacro]a for the 0.3 @xmath0 m adhesive emulsion ) . we observe that for each rotational velocity , a stationary state is reached for a strain of a few unities , consistently with the absence of evolution of the velocity profiles in time . in fig . [ fig_emul_rheomacro]a we plot the stationary torque vs. the rotational velocity for the steady flows of the 0.3 @xmath0 m adhesive emulsion . m adhesive emulsion ; inset : torque vs. angular displacement for various rotational velocities ranging between 0.01 and 50 rpm ( bottom to top ) . b ) constitutive law inferred from the purely macroscopic measurements thanks to eq.[eq_localstress ] and eq . [ eq_localrate ] ; the solid line is a herschel - bulkley fit @xmath78 of the data with @xmath79pa , @xmath80pa.s and @xmath81.,title="fig:",width=302 ] m adhesive emulsion ; inset : torque vs. angular displacement for various rotational velocities ranging between 0.01 and 50 rpm ( bottom to top ) . b ) constitutive law inferred from the purely macroscopic measurements thanks to eq.[eq_localstress ] and eq . [ eq_localrate ] ; the solid line is a herschel - bulkley fit @xmath78 of the data with @xmath79pa , @xmath80pa.s and @xmath81.,title="fig:",width=302 ] we observe that the apparent flow curve of the material is that of a yield stress shear - thinning fluid . as we have shown in sec . [ section_concentration ] that the material is homogeneous , we are allowed to use these macroscopic data to infer the constitutive law of the material . because of the strong stress and shear rate heterogeneities , in particular shear localization , standard formula relating simply the torque and rotational velocity values to a mean shear stress and a mean shear rate can not be used . however , it is possible to analyze the macroscopic data to account for these heterogeneities and to obtain the unique constitutive law consistent with this set of purely macroscopic data . the method used to analyze the wide gap couette rheometric data in order to obtain the constitutive law of the material is detailed e.g. in @xcite , we give here the main steps . first , it has to be noted that the shear stress distribution @xmath82 within the gap at a radius @xmath11 is known whatever the constitutive law of the material ; it reads ( r)=t/(2h r^2)[eq_localstress]where @xmath25 is the torque and h is the height of the couette geometry . then , the local shear rate @xmath83 is related to the rotational velocity @xmath6 through = ^r_o_r_i ( r)/r dr[eq_omega0]where @xmath59 is the inner radius and @xmath58 is the outer radius . from eqs . [ eq_localstress ] and [ eq_omega0 ] , we see that from the knowledge of the torque @xmath25 and the rotational velocity @xmath6 , we obtain a non straightforward relationship between the local shear stress and the local shear rate . in order to go one step further , eqs . [ eq_localstress ] and [ eq_omega0 ] may be combined into = -^(r_o)_(r_i ) ( ) / d[eq_omega]note that the possibility of shear localization is naturally taken into account in this equation , where @xmath84 when @xmath85 with @xmath86 the material yield stress . the constitutive law may then be derived from the whole macroscopic rheometric curve @xmath5 thanks to the differentiation of eq . [ eq_omega ] relative to @xmath25 : 2t / t=((r_i))-((r_o))[eq_localrate]finally , in order to get the shear rate @xmath87 at the inner cylinder as a function of the shear stress at the inner cylinder , one needs to eliminate the shear rate at the outer cylinder @xmath88 from this equation . this can be done by summing this relationship with a series of successive decreasing torques ( that differ by a factor @xmath89 at each step ) chosen to ensure that the shear rate at the outer cylinder is eliminated in the summation at each step , until a zero shear rate is reached at the outer cylinder : the summation stops when the shear stress computed at the outer cylinder at a given step falls below the yield stress @xcite . while this summation is in principle infinite for simple fluids , only a few steps are needed in the case of yield stress fluids , and the number of steps is more reduced for wider gaps . in the case of our couette geometry , as the ratio between the shear stress at the inner cylinder and the shear stress at the outer cylinder ( @xmath89 ) is equal to 2.1 , we see that the data of fig . [ fig_emul_rheomacro]a can be easily processed thanks to a maximum number of two summations of eq . [ eq_localrate ] . from eq . [ eq_omega ] , note that such analysis is based on the assumption that there is a local constitutive law characterizing the material and that there is negligible wall slip . the constitutive law obtained with this method is depicted in fig . [ fig_emul_rheomacro]b . we observe that the behavior of the material obtained with this procedure is basically well fitted to a herschel - bulkley law . in sec . [ section_local ] we compare this law to the constitutive law built from the local shear rates extracted from the velocity profiles . the constitutive laws of the materials accounting for their velocity profiles can be built from our experimental data , using both the velocity profiles and the torque measurements . the stress distribution @xmath82 within the gap is obtained from the macroscopic torque measurements thanks to eq . [ eq_localstress ] . the local shear rate @xmath83 in the gap is inferred from the velocity profiles @xmath90 through ( r)=v / r - v / rboth measurements performed at a given radius @xmath11 for a given rotational velocity @xmath6 thus provide a local data point of the constitutive law @xmath91 . this analysis provides a fair local measurement of the constitutive law since we measure the true local shear rate within the bulk of the material and the shear stress distribution is known from the momentum balance independently of any hypothesis ; in particular , it is independent of what happens at the interface so that a possible wall slip does no affect this analysis . we are finally allowed to combine the data measured at various radiuses because the materials are homogeneous as shown in sec . [ section_concentration ] . m emulsion , a non - adhesive 6.5@xmath0 m emulsion and an adhesive 6.5@xmath0 m emulsion ; the solid line is a herschel - bulkley fit @xmath78 of the data with @xmath81 for all materials , @xmath92pa and @xmath93pa.s for the non - adhesive 1@xmath0 m emulsion , @xmath94pa and @xmath93pa.s for the non - adhesive 6.5@xmath0 m emulsion , and @xmath95pa and @xmath96pa.s for the adhesive 6.5@xmath0 m emulsion . b)constitutive law measured locally in the gap of a couette cell for the flows of a 0.3@xmath0 m adhesive emulsion ( empty symbols ) ; the solid line is a herschel - bulkley fit @xmath78 of the data with @xmath79pa , @xmath80pa.s and @xmath81 . the crosses are the data extracted from the analysis of the purely macroscopic measurements ( see fig . [ fig_emul_rheomacro]b ) , title="fig:",width=302 ] m emulsion , a non - adhesive 6.5@xmath0 m emulsion and an adhesive 6.5@xmath0 m emulsion ; the solid line is a herschel - bulkley fit @xmath78 of the data with @xmath81 for all materials , @xmath92pa and @xmath93pa.s for the non - adhesive 1@xmath0 m emulsion , @xmath94pa and @xmath93pa.s for the non - adhesive 6.5@xmath0 m emulsion , and @xmath95pa and @xmath96pa.s for the adhesive 6.5@xmath0 m emulsion . b)constitutive law measured locally in the gap of a couette cell for the flows of a 0.3@xmath0 m adhesive emulsion ( empty symbols ) ; the solid line is a herschel - bulkley fit @xmath78 of the data with @xmath79pa , @xmath80pa.s and @xmath81 . the crosses are the data extracted from the analysis of the purely macroscopic measurements ( see fig . [ fig_emul_rheomacro]b ) , title="fig:",width=302 ] the local constitutive law extracted from the experiments performed at various rotational velocities @xmath6 on all the emulsions are plotted in fig . [ fig_emul_local1 ] ( we could not analyze the data for the 40@xmath0 m emulsion as torque data could not be measured in this case ) . it should first be noted that for each emulsion , all the shear stress vs. shear rate data fall along a single curve . this means that for a given material , our different data obtained under different boundary conditions effectively reflect the behavior of a single material with a given intrinsic constitutive law in simple shear . note also that this is consistent with the absence of migration : any heterogeneity would have apparently yielded different constitutive laws for different rotational velocities @xcite . our observation of a single local constitutive law accounting for all flows of the non - adhesive 6.5@xmath0 m emulsion ( fig . [ fig_emul_local1]a ) contrasts with the observations of @xcite on the same system in microchannel flows of up to 250@xmath0 m width : the velocity profiles observed in the channel can not be accounted for by a single constitutive law . this apparent paradox has been solved by @xcite : they showed that a rather simple non - local flow rule accounts for all the velocity profiles . they concluded that non - local effects should be observable in a zone involving up to 100 particles ; beyond this zone , this non - local law reduces to a local law . this explains why our measurements are unsensitive to such effects and why we measure a single local constitutive law : using a wide gap geometry prevents from being sensitive to these non - local effects . our observation of a single local constitutive law accounting for all flows of the adhesive 0.3@xmath0 m emulsion ( fig . [ fig_emul_local1]b ) also contrasts with the observations of @xcite . on exactly the same 0.3@xmath0 m adhesive emulsion sheared in a thin gap couette geometry , @xcite were actually unable to fit all their velocity profiles with a single constitutive law . in the @xcite study , the gap being 1 mm for 0.3@xmath0 m droplets , the non - local effects evidenced by @xcite can not be the reason for the apparent absence of a single constitutive law accounting for all flows . an important point should be noted that may explain the differences between both studies of the same material : the experimental procedures are slightly different , and the time evolutions of the emulsion properties are very different . in our experiments , the emulsion was first presheared with a mixer and then at 100 rpm for 5 minutes in a couette cell with rough surfaces . the stationary velocity profiles then measured at various rotational velocities were found to develop within a few seconds and to remain stable for hours . in the experimental procedure of @xcite , each experiment is conducted on a fresh sample which is directly sheared at the studied shear rate during at least 3 hours . a key point may then be that @xcite found their system to evolve for at least two hours , in contrast with our observations : whatever the rotational velocity , they found the apparent viscosity of the system to decrease slowly in time . this would mean that their material was in an initially structured state , and was destructured by shear , whatever the magnitude of the shear rate . then , while the authors claim that the velocity profiles no longer change significantly after 2 h , one may think that some slow relaxation still occurs and that the steady state is not really reached . this would mean that the measurements performed at different rotational velocities have been performed on different structural states of the emulsion ( in contrast with our measurements ) , naturally leading to different constitutive laws for the different velocities studied . this explanation based on the influence of the initial state of the material would be consistent with the observations of @xcite who showed that simple emulsions whose flow properties show no apparent thixotropic effects when first presheared at high shear rate are actually subject to significant aging at rest : this imply in particular that such systems have to be strongly presheared before any study . this preshear was performed in our study , it was not in the @xcite study . two other points may be noted : first , although the composition of the two emulsions is the same , the samples are different and chemical impurities may play an important role in the adhesion process ; second , the couette cell used by @xcite is smooth and huge wall slip is evidenced . focusing on the non adhesive 1@xmath0 m , the adhesive 6.5 @xmath0 m , and the non adhesive 6.5 @xmath0 m emulsions , we observe that the behavior of all materials is well fitted to a herschel - bulkley behavior @xmath78 of index @xmath81 in all the range of measured shear rates ; this is consistent with previous observations of the behavior of dense emulsions @xcite . coming back to the 0.3@xmath0 m adhesive emulsion , at moderate and high shear rates ( above 0.1s@xmath77 ) , the behavior is the same as the one encountered for the large droplets emulsions : the constitutive law is well fitted to an herschel - bulkley law on index n=0.5 . at low shear rate ( in a 0.01 - 0.1s@xmath77 range ) , the behavior of the 0.3@xmath0 m adhesive emulsion differs from the previous ones . all the points still fall on the same curve but a slight slope break is noticed in the flow curve . while a stress plateau should be reached for a simple yield stress fluid , the shear stress continues decreasing when the shear rate decreases . this is also observed on the law inferred from the macroscopic measurement . as far as we can say , this behavior is specific to the 0.3@xmath0 m adhesive emulsion : we took care of measuring the local behavior at low shear rates in the 6.5@xmath0 m adhesive emulsion but observed no slope break for shear rates as low as 10@xmath97s@xmath77 . a probable reason for the differences observed between these systems is that the 0.3 micron emulsion is more adhesive than the others , in particular because of their larger surface / volume ratio . this low shear rate behavior corresponds to the slow flows observed below a first apparent yield stress in fig . [ fig_emul_velocity1log]a . at low rotational velocity , when there is apparently shear localization , the velocity profiles are well fitted to the herschel - bulkley law predictions only for velocities larger than a few [email protected]@xmath77 . for lower velocities , at the approach of the radius where the yield stress should be reached and the velocity should tend to zero , the velocity starts to decrease very slowly with the radius ; on the other hand , at high rotational velocity , when all the gap is sheared , the velocity profile is very well fitted to the herschel - bulkley law prediction . the analysis of this behavior is difficult . one may think that the emulsion flows homogeneously and really follows a single continuous local constitutive law that corresponds to a herschel - bulkley equation at high shear rate and differs from this equation at low shear rate . one may also suggest that the law at low shear rates does not correspond to steady flows . indeed , it has to be noted that the measurements of the very low velocities in fig . [ fig_emul_velocity1log]a are averages over 1min : we can not know if they correspond to steady - flows ( the measurements may e.g. reflect a stick - slip behavior ) . these flows may thus reflect the existence of a jammed phase perturbated by some unsteady local rearrangements . in this case , this behavior would be reminiscent of shear banding , which has already been observed for strongly adhesive emulsions @xcite and is explained by a competition between structuration due to the adhesion process and destructuration by shear @xcite . the local measurements obtained for the 0.3@xmath0 m adhesive emulsion are compared in fig.[fig_emul_local1]a with the law inferred from the purely macroscopic measurements ( see sec.[section_torque ] ) . we observe that both laws are very well matched . in particular , the macroscopic measurements also allow us to evidence the slope break in the constitutive law at low shear rate . this validate the use of a wide gap couette geometry as a tool to obtain the constitutive law of dense emulsions from purely macroscopic rheometrical measurements , provided that a proper analysis of the macroscopic data is performed . we recall that this agreement is due to four important features ( i ) we have shown that all dense emulsions seem to be free from migration and thus remain homogeneous during the experiments , ( ii ) the use of a wide gap prevents from the nonlocal effects observed by @xcite , ( iii ) the material is initially destructured and a steady state is studied ( this may have not be the case in the @xcite study ) , ( iv ) wall slip has to be negligible to allow for a proper analysis of the macroscopic data : this is ensured here by the use of rough walls of roughness larger than the droplet size , as shown below . as pointed out in sec . [ section_velocity ] , our velocity measurement method does not provide the velocity at the walls : the first reliable data is obtained at around 0.5 to 1 mm from the inner cylinder . however , the local measurements of the constitutive law we have performed make it possible to study in more detail a possible wall slip effect . for a given rotational velocity of the inner cylinder , the velocity of the emulsion expected at the walls can actually be computed thanks to the knowledge of the shear rate in the emulsion corresponding to the shear stress value at the wall . the same shear stress level was actually reached somewhere in the bulk of the material for a higher rotational velocity of the inner cylinder , and the shear rate corresponding to this shear stress was then measured locally thanks to the velocity profile measurement . the velocity profile expected in the zone near the inner cylinder can then be reconstructed from the velocity profile measured at some distance from the walls and from the expected shear rate near the wall . of course , this analysis is based on the hypothesis that the material is the same as in the bulk in this 0.5 to 1 mm zone near the walls , which may be no more true in a zone where there are nonlocal effects @xcite . note also that this method can be used only when the reproducibility of the experiments is very good ( allowing data measured at a given rotational velocity to be used to predict what happens at another velocity ) . an example of this method applied to the flows of the 6.5@xmath0 m adhesive emulsion is depicted in fig . [ fig_emul_velocity2_slippage ] . m adhesive emulsion , for a 10 and a 20 rpm rotational velocity . b ) dimensionless velocity profiles expected near the walls at 10 and 20 rpm ; the velocities are computed thanks to the local shear rate measurements performed at 50 rpm.,title="fig:",width=302 ] m adhesive emulsion , for a 10 and a 20 rpm rotational velocity . b ) dimensionless velocity profiles expected near the walls at 10 and 20 rpm ; the velocities are computed thanks to the local shear rate measurements performed at 50 rpm.,title="fig:",width=302 ] we observe in fig . [ fig_emul_velocity2_slippage]b that the dimensionless velocity profiles expected near the walls have a value that is very close to 1 . this analysis was performed on several velocity profiles , and in all cases the dimensionless velocity values predicted at the inner cylinder were found to be equal to 1 within 1% . we finally conclude that , thanks to the use of rough surfaces of roughness larger than the droplet size , there is probably no wall slip in our experiments ; if any , it is of order 1% or less . this feature shows that the macroscopic measurements performed with a wide gap couette rheometer with rough walls can be trusted : this explains the very good agreement observed in fig . [ fig_emul_local1]a between the constitutive law measured locally and the constitutive law inferred from the purely macroscopic measurements .

flows of dense emulsions show many complex features among which long range nonlocal effects pose problem for macroscopic characterization . in order to get round this problem , we are able to determine very precisely the local rheological behavior of several dense emulsions . as the materials are homogeneous , this behavior can also be inferred from purely macroscopic measurements . [ nature * 454 * , 84 ( 2008 ) ] : the use of a wide gap couette geometry is likely to prevent here from nonlocal finite size effects ; it also contrasts with the observations of bcu _ et al . _ rev .

flows of dense emulsions show many complex features among which long range nonlocal effects pose problem for macroscopic characterization . in order to get round this problem , we study the flows of several dense emulsions , with droplet size ranging from 0.3 to 40 m , in a wide gap couette geometry . we couple macroscopic rheometric experiments and local velocity measurements through mri techniques . as concentration heterogeneities are expected in the wide gap couette flows of multiphase materials , we also designed a new method to measure the local droplet concentration in emulsions with a mri device . in contrast with dense suspensions of rigid particles where very fast migration occurs under shear in wide gap couette flows , we show for the first time that no migration takes place in dense emulsions even for strain as large as 100000 in our systems . as a result of the absence of migration and of finite size effect , we are able to determine very precisely the local rheological behavior of several dense emulsions . as the materials are homogeneous , this behavior can also be inferred from purely macroscopic measurements . we thus suggest that properly analyzed purely macroscopic measurements in a wide gap couette geometry can be used as a tool to study the local constitutive laws of dense emulsions . all behaviors are basically consistent with herschel - bulkley laws of index 0.5 . the existence of a constitutive law accounting for all flows contrasts with previous results obtained within a microchannel by goyon _ et al . _ [ nature * 454 * , 84 ( 2008 ) ] : the use of a wide gap couette geometry is likely to prevent here from nonlocal finite size effects ; it also contrasts with the observations of bcu _ et al . _ [ phys . rev . lett . * 96 * , 138302 ( 2006 ) ] . we also evidence the existence of discrepancies between a perfect herschel - bulkley behavior and the observed local behavior at the approach of the yield stress due to slow shear flows below the apparent yield stress in the case of a strongly adhesive emulsion .

0810.3497

c

we have studied the flows of several dense emulsions in a wide gap couette geometry . we have coupled macroscopic rheometric experiments and local velocity and concentration measurements through mri techniques . the method devoted to measure the local droplet concentration was developed specifically for this study . in contrast with dense suspensions of rigid particles where very fast migration occurs under shear , we showed for the first time that no migration takes place in dense emulsions even for strain as large as 100000 in our systems . this may imply that another mechanism is involved in dense emulsions than in dense suspensions . the homogeneity of our materials under shear allows to infer their constitutive law from purely macroscopic measurements . this constitutive law is consistent with the one inferred from the velocity profiles . this contrasts with previous results obtained by @xcite within a microchannel where nonlocal finite size effects are likely to have prevented from obtaining a constitutive law . it also differs from the results of @xcite , where the loading procedure probably prevented the authors to reach a steady state . we thus suggest that properly analyzed purely macroscopic measurements in a wide gap couette geometry can be used as a tool to study dense emulsions . all behaviors we observed are basically consistent with herschel - bulkley laws of index 0.5 . however , we also evidence the existence of discrepancies with this law at the approach of the yield stress due to slow shear flows in the case of strongly adhesive emulsions , whose physical origin is unclear . finally , we have shown that there is probably no wall slip .

, we study the flows of several dense emulsions , with droplet size ranging from 0.3 to 40 m , in a wide gap couette geometry . we couple macroscopic rheometric experiments and local velocity measurements through mri techniques . as concentration heterogeneities in contrast with dense suspensions of rigid particles where very fast migration occurs under shear in wide gap couette flows , we show for the first time that no migration takes place in dense emulsions even for strain as large as 100000 in our systems . as a result of the absence of migration and of finite size effect we thus suggest that properly analyzed purely macroscopic measurements in a wide gap couette geometry can be used as a tool to study the local constitutive laws of dense emulsions . all behaviors are basically consistent with herschel - bulkley laws of index 0.5 . we also evidence the existence of discrepancies between a perfect herschel - bulkley behavior and the observed local behavior at the approach of the yield stress due to slow shear flows below the apparent yield stress in the case of a strongly adhesive emulsion .

flows of dense emulsions show many complex features among which long range nonlocal effects pose problem for macroscopic characterization . in order to get round this problem , we study the flows of several dense emulsions , with droplet size ranging from 0.3 to 40 m , in a wide gap couette geometry . we couple macroscopic rheometric experiments and local velocity measurements through mri techniques . as concentration heterogeneities are expected in the wide gap couette flows of multiphase materials , we also designed a new method to measure the local droplet concentration in emulsions with a mri device . in contrast with dense suspensions of rigid particles where very fast migration occurs under shear in wide gap couette flows , we show for the first time that no migration takes place in dense emulsions even for strain as large as 100000 in our systems . as a result of the absence of migration and of finite size effect , we are able to determine very precisely the local rheological behavior of several dense emulsions . as the materials are homogeneous , this behavior can also be inferred from purely macroscopic measurements . we thus suggest that properly analyzed purely macroscopic measurements in a wide gap couette geometry can be used as a tool to study the local constitutive laws of dense emulsions . all behaviors are basically consistent with herschel - bulkley laws of index 0.5 . the existence of a constitutive law accounting for all flows contrasts with previous results obtained within a microchannel by goyon _ et al . _ [ nature * 454 * , 84 ( 2008 ) ] : the use of a wide gap couette geometry is likely to prevent here from nonlocal finite size effects ; it also contrasts with the observations of bcu _ et al . _ [ phys . rev . lett . * 96 * , 138302 ( 2006 ) ] . we also evidence the existence of discrepancies between a perfect herschel - bulkley behavior and the observed local behavior at the approach of the yield stress due to slow shear flows below the apparent yield stress in the case of a strongly adhesive emulsion .