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cond-mat9711302

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it is known that quenched disorder can strongly affect the large - scale , long - time behaviour of nonequilibrium driven systems with interacting constituents . the interplay of disorder , interactions and drive opens up the possibility of new regimes of complex and interesting behaviour arising in these systems @xcite . in the theoretical effort to delineate and explore regimes of new behaviour , an important role is played by simple models which capture some features of more complex physical systems . in this paper , we study disordered driven diffusive systems by analysing stochastically evolving lattice gas models , with quenched disordered hopping rates @xcite . driven diffusive systems in the absence of disorder have been studied extensively and are reviewed in @xcite . also , systems with disorder and drive but no interactions between particles are well studied and understood @xcite . but there have been only sporadic studies of disordered driven diffusive systems of interacting particles . it has been argued that strong enough random site dilution can substantially affect the transport properties of particles with hard - core interactions , and can make the system respond nonmonotonically to the driving field @xcite . on the other hand , a low concentration of blocked sites was found numerically not to affect the critical behaviour of a driven lattice gas with additional attractive inter - particle interactions @xcite . finally , a driven lattice gas with a quenched noise distribution was studied using field - theoretic techniques in @xcite , but the connection of this study with particle - conserving disordered lattice gas models is not clear . in this paper , we study disordered lattice gas models with a view towards identifying different sorts of generic behaviour that can arise on large scales as a consequence of disorder . the only interaction included is the hard - core constraint which limits the allowed occupancy of each site . our results pertain mostly , but not exclusively , to one dimension . in the remainder of this introduction , we discuss the different types of behaviour displayed by the lattice gas models under study . we find three distinct regimes in disordered driven diffusive systems in one dimension : in the _ homogeneous _ regime , the state of the system is characterized by a single density and a nonzero current . quenched disorder induces variations of the density on the microscopic scale , of the order of a few lattice spacings . however , the system has a macroscopically homogeneous density . in the thermodynamic limit , the current approaches a finite value . in the _ segregated - density _ regime , the state of the system is characterized by two distinct values of density , and a nonzero current . besides microscopic - scale variations of the density , there are macroscopic regions with differing high and low densities . the state is thus characterized by phase separation of the density , and a spatially constant time - averaged current which remains finite in the thermodynamic limit . in the _ vanishing - current _ regime , the state of the system is characterized by two distinct values of the density , and an essentially zero current . the hallmark of this regime is that the current decreases as the system size increases , and vanishes in the thermodynamic limit . this is a consequence of rare but rate - limiting backbends , or stretches of bonds which disfavour the forward flow of current . the density is inhomogeneous on a macroscopic scale . the density profiles in typical states in each of the three regimes are depicted in figure 1 , while figure 2 shows the variation of the current with system size in the three cases . examples of these behaviours are discussed in this paper for two types of lattice - gas models , namely the disordered drop - push process ( ddpp ) and the disordered asymmetric simple exclusion process ( dasep ) . the models are defined in detail in sections [ sec : ddpp ] and [ sec : dasep ] respectively , but for the purpose of discussion here , it suffices to note that the models are similar in that there is a maximum occupancy of each site in both , and are different in the dynamical moves attempted nearest neighbour jumps in the dasep , and slightly longer - ranged leapfrogging moves in the ddpp . the absence of detailed balance , together with the breaking of translational invariance , in disordered off - equilibrium systems makes the characterization of even the stationary state difficult in general . it is shown that the steady state of the disordered drop - push process can be found explicitly the first such instance we are aware of , in a system with disorder , interactions and drive @xcite . this determination which is based on the condition of pairwise balance @xcite shows that a product measure form is valid in all dimensions . the form reflects the microscopic inhomogeneities coming from the underlying disorder , and results in a macroscopically homogeneous state . for the disordered asymmetric exclusion process , the steady state measure is not analytically characterizable , and we study the problem within a site - wise inhomogeneous mean - field theory and by numerical simulation . the result depends crucially on whether or not the system has backbends , which are stretches of the lattice where the local bias is against the particle flow . in the no - backbend case , when the average particle density is sufficiently away from 1/2 , the spatial profile of the density has microscopic shocks , but is uniform on macroscopic scales ( fig . [ fig : regimes1]a ) . however , in a finite region around half - filling , disorder induces phase separation into macroscopic regions of high and low density ( fig . [ fig : regimes1]b ) . we give approximate arguments to understand the origin and nature of this phase separation , and to obtain the form of the phase diagram in the current - density plane . this sort of behaviour has also been seen earlier in a model with a single weak bond @xcite . we argue that disorder - induced phase separation is a generic feature of systems in which the current @xmath1 versus density @xmath2 shows a maximum at some intermediate density , in the absence of disorder . in the version of the dasep in which the easy direction of hopping is itself a quenched random variable , the model represents a system of hard - core particles in a random potential with an overall downward tilt , but with backbends of arbitrary length . long backbends severely limit the maximum current that can flow through the system , and in fact the current decreases to zero as the system size increases ( fig . [ fig : regimes2 ] ) ; the system is in the vanishing - current regime . although our emphasis in this paper is on the analysis of lattice models , we comment briefly on certain constraints that are important in a continuum description . such a description is expected to be valid for the large - scale , long - time behaviour , and is based on stochastic differential equations involving appropriate coarse - grained variables . it is argued that quenched randomness is manifest in random multiplicative coefficients in a gradient expansion . conservation of particle number which implies spatial constancy of the current in the steady state imposes strong constraints on these terms . in one dimension , using a well known mapping @xcite , the particle models are equivalent to stochastic growth models of a 1-@xmath0 interface moving in a 2-@xmath0 medium . the interface moves with a speed proportional to the current in the particle model . the disordered jump rates now become local growth rates which are disordered in a columnar fashion for the moving interface @xcite . the three principal regimes of behaviour discussed above for the particle models translate into distinct regimes for interface motion , namely ( i ) a moving interface with normal roughness , ( ii ) a moving interface with large segments with different mean slopes , and ( iii ) an interface with different - slope segments , which is stationary in the thermodynamic limit . the paper is organized as follows . in section [ sec : ddpp ] we define and discuss the steady state properties of the disordered drop - push process in arbitrary dimensionality . the disordered asymmetric exclusion process with only forward - easy - direction of hopping , but quenched random rates , is discussed in section [ sec : dasep ] ; the case in which there are some backward - easy - direction bonds is discussed in section [ sec : sinai ] . in section [ sec : continuum ] we discuss the constraints on a continuum description , while section [ sec : height ] discusses the implications of our results for models interface growth in the presence of columnar disorder . section [ sec : conclusion ] is the conclusion .

we study the effect of quenched spatial disorder on the steady states of driven systems of interacting particles . two sorts of models are studied : disordered drop - push processes and their generalizations , and the disordered asymmetric simple exclusion process . we find that three qualitatively different regimes of behaviour are possible in 1- disordered driven systems . in the vanishing - current regime , the steady - state current approaches zero in the thermodynamic limit we comment on certain important constraints to be taken care of in any field theory of disordered systems .

we study the effect of quenched spatial disorder on the steady states of driven systems of interacting particles . two sorts of models are studied : disordered drop - push processes and their generalizations , and the disordered asymmetric simple exclusion process . we write down the exact steady - state measure , and consequently a number of physical quantities explicitly , for the drop - push dynamics in any dimensions for arbitrary disorder . we find that three qualitatively different regimes of behaviour are possible in 1- disordered driven systems . in the vanishing - current regime , the steady - state current approaches zero in the thermodynamic limit . a system with a non - zero current can either be in the homogeneous regime , chracterized by a single macroscopic density , or the segregated - density regime , with macroscopic regions of different densities . we comment on certain important constraints to be taken care of in any field theory of disordered systems . 0.5 cm pacs numbers : 05.60.+w , 47.55.mh , 64.60.-i , 05.50.+q 2

cond-mat9711302

c

in this paper , we have studied the stationary current - carrying states of driven lattice gas models with quenched disordered hopping rates . the principal results are of two types first the exact determination of the steady states for a class of disordered models , and second the demarcation of distinct regimes of behaviour on macroscopic length scales , as a result of disorder . in this section , we briefly review these results , and discuss some related open problems . the steady states of a family of disordered models the disordered drop - push process ( ddpp ) , and related models have been found in all dimensions by an application of the condition of pairwise balance . the result is a product measure state , with site - dependent weights , reflecting the microscopic disorder in the model . the current has been computed as well . the system is characterized by a strictly uniform current density , and a coarse - gained particle density that is approximately uniform . on a macroscopic scale , the state is homogeneous . disorder can lead to macroscopically non - homogeneous states , as in the 1-@xmath0 disordered asymmetric simple exclusion process ( dasep ) . our numerical and mean - field results show that a macroscopically density - segregated state occurs in the dasep model with no backbends , for densities in a finite range around half - filling . the origin of density separation is traceable to the existence of a largest current that can be carried by a stretch of weak bonds . this low current can be sustained in the rest of the lattice only by separating the density into distinct large and small values in macroscopic regions of the lattice . backbends introduce a third type of possible behaviour in one dimension . like the stretch of weak forward bonds in the density - segregated regime , a backbend rate - limits the current , leading to density segregation . however , there is an important difference : the longer the stretch of weak forward bonds , the closer is the current to a finite asymptotic value ; by contrast , the longer the stretch of reverse - biased bonds , the smaller the current it decreases exponentially fast with backbend length . since the probability of occurrence of a backbend decreases exponentially with its length , the result is a current that decreases as a bias - dependent power of the overall size of the lattice . the crucial physical point which underlies the behaviour in each of these regimes is the requirement that the steady - state current be constant at all points in the system . continuum field - theoretical approaches must ensure that this local constraint is respected ; while this is automatically assured for translationally invariant systems , it may need special care to guarantee in disordered systems . it would be of interest to generalize our results to higher dimensions and also to include interactions between particles at different sites . a few scattered results along these lines are available . ( i ) for the drop - push class of problems , we have seen in section [ sec : ddpp ] that the exact steady state even in higher dimensions is characterized by inhomogeneous product measure . on large scales , this leads to a homogeneous state . ( ii ) the transport of particles with hard - core interactions through the infinite cluster of a randomly diluted lattice above the percolation concentration has been studied @xcite . in a certain regime of dilution , backbends act as local traps , but unlike the one - dimensional case considered in section [ sec : sinai ] , there exist infinitely long paths on which the length of every backbend is less than a fixed value @xcite . the sub - network of such paths is expected to carry a current which then remains finite in the thermodynamic limit . there is thus no vanishing - current regime in this system . ( iii ) with attractive interactions between particles , the driven lattice gas system with nearest - neighbour hopping is known to undergo phase separation below a certain temperature @xcite . a numerical simulation showed that the addition of a low concentration of blocked sites did not alter the critical behaviour of this system @xcite . more systematic studies of higher - dimensional systems are called for . in particular , it would be interesting to know whether disorder - induced large - scale inhomogeneities , akin to the phase separation found in one dimension , persist in higher dimensions as well . acknowledgements : we thank r. e. amritkar , d. dhar , s. krishnamurthy , r. lahiri , g. i. menon and n. trivedi for useful discussions .

we write down the exact steady - state measure , and consequently a number of physical quantities explicitly , for the drop - push dynamics in any dimensions for arbitrary disorder . . a system with a non - zero current can either be in the homogeneous regime , chracterized by a single macroscopic density , or the segregated - density regime , with macroscopic regions of different densities .

we study the effect of quenched spatial disorder on the steady states of driven systems of interacting particles . two sorts of models are studied : disordered drop - push processes and their generalizations , and the disordered asymmetric simple exclusion process . we write down the exact steady - state measure , and consequently a number of physical quantities explicitly , for the drop - push dynamics in any dimensions for arbitrary disorder . we find that three qualitatively different regimes of behaviour are possible in 1- disordered driven systems . in the vanishing - current regime , the steady - state current approaches zero in the thermodynamic limit . a system with a non - zero current can either be in the homogeneous regime , chracterized by a single macroscopic density , or the segregated - density regime , with macroscopic regions of different densities . we comment on certain important constraints to be taken care of in any field theory of disordered systems . 0.5 cm pacs numbers : 05.60.+w , 47.55.mh , 64.60.-i , 05.50.+q 2

1702.03782

i

this paper concerns one - dimensional quasilinear reaction - diffusion pdes . we study the existence of monotone heteroclinic connections of traveling - wave type for reaction - diffusion equations with saturating and density - dependent diffusion , in presence of a convective term ; precisely , we deal with the equation @xmath0 ) - { \partial}_x h(u ) + f(u ) , \quad ( u = u(t , x)),\ ] ] under a suitable set of assumptions ( see section [ sez2 ] ) . not to overload the notation and the readability of the paper , our main results will focus on the special case when @xmath1 is of mean curvature type , namely @xmath2 ( see , for instance , @xcite ) . however , the presented results may hold true for a larger class of operators , as it will be pointed out later on in the paper . reaction - convection - diffusion type models like , with a special focus on traveling waves , are very popular and can be used to describe different mathematical and physical phenomena : the main areas of application , for which we quote a few references , are , for instance , neurophysics and biophysics @xcite , chemical physics @xcite , population genetics @xcite , tumor growth @xcite and mathematical ecology @xcite . as describing quite complicated realistic models , most of the above applications concern systems of equations in higher dimensional spaces . inside this vast class , the fisher - kolmogorov - petrovsky - piskounov ( fkpp ) equation @xmath3 introduced by fisher @xcite to model the spread of an advantageous gene inside a @xmath4-dimensional population , and the viscous burgers equation @xcite @xmath5 finding application , for instance , in problems related to gas dynamics and traffic flows , are probably among the most popular . in the recent years , an increasing interest has been devoted , as well , to a `` combination '' of and , called _ fisher - burgers _ ( or _ burgers - fisher _ ) equation , which is obtained by setting @xmath6 , @xmath7 , @xmath8 and @xmath9 in equation ; this is perhaps the easiest example of pde taking into account both the reaction - diffusion and the convection effects . such an equation has been widely studied by means of both theoretical and numerical methods ( see , e.g. , @xcite ) , and represents the starting point of our investigation . in order to better set up our scheme and introduce our results for generalized fisher - burgers equations displaying saturating effects , we thus now start briefly reviewing the results which are available , in our perspective , for equations and . after its introduction @xcite ( as pointed out , for instance , in @xcite , actually the investigation and analysis of traveling waves in chemical reactions is probably first due to luther @xcite ) , the fisher equation was widely studied , from a mathematical point of view , starting with the paper @xcite . as mentioned above , it was meant to describe the evolution of the relative concentration of a gene inside a population ; under the natural assumption @xmath10 ( meaning that no reaction can be present if the gene is completely spread or not spread at all ) , it was already observed in these first articles the natural appearance of heteroclinic solutions connecting the equilibria @xmath11 and @xmath4 - we will call them _ @xmath12connections _ - of traveling wave type , embodying the transition from the situation where the gene is not present to the one where every individual possesses it . by standard phase - plane arguments , it is immediate to see ( see , for instance , @xcite ) that such solutions are necessarily monotone ; it was then proved in @xcite that they also have some stability properties for a certain class of initial data . for a better comprehension of the model , it is thus natural to search first for solutions having the form @xmath13 , with @xmath14 an increasing function such that @xmath15 , @xmath16 . the values of @xmath17 yielding such solutions are called _ admissible speeds _ , and it turns out that the set of the admissible speeds for equation is an unbounded interval @xmath18 ; @xmath19 takes the name of _ critical speed_. for example , in the fisher case it is @xmath20 . later on , reaction terms other than the fisher one were studied : the most popular are the so called _ type b _ and _ type c _ ( according to the terminology in @xcite ) . the former has been considered in some combustion problems , while the latter appears naturally in biology ( see @xcite ) , in relation with the death rates of populations presenting different genotypes with respect to a particular gene , or in aggregative - diffusive models for population dynamics ( see , e.g. , @xcite for the case of a sign - changing diffusion ) . by some monotonicity arguments , for both of these situations ( see section [ sez4 ] below ) it may be seen that the admissible speed exists and is unique . on the other hand , the viscous burgers equation often appears as a simplification of more complicated models : a well known example is the one given by the navier - stokes equations , describing the dynamics of compressible or incompressible fluids . if we consider without the second order viscous term and with general flux function describing the convection phenomenon , it is well known that solutions with discontinuities ( called internal shocks ) are allowed in the class of the so called _ entropy solutions _ ; in particular , in order to provide uniqueness , it is common to assume the flux function to be convex ( for more details , see @xcite ) . on the other hand , when the viscosity is switched on " , all the discontinuities that appear at the hyperbolic level turn out into smooth transition layers . in this setting , smooth traveling wave solutions to the cauchy problem associated with have been extensively studied over the past years ; indeed , when looking for a solution of the form @xmath21 , equation can be explicitly solved by integration , leading to @xmath22 where @xmath23 ( @xmath24 being an integration constant ) , and @xmath25 . in particular , @xmath26 by definition , so that @xmath27 and represents a decreasing function connecting the values @xmath28 and @xmath29 . finally , it holds that @xmath30 meaning that , once the values @xmath31 and @xmath32 are chosen , there exists a unique @xmath33connection of traveling wave type , having speed @xmath17 . what we have described so far holds true when the diffusion is _ linear_. however , assuming that a diffusive process in nature takes place in a homogeneous way , disregarding local biological and physical conditions ( both environmental - like light , humidity , temperature - and intrinsic in the behavior of the described population - e.g. , its density or concentration , its gradient , cf . , for instance , @xcite ) , seems to be quite a strong restriction . for instance , it may be more likely to imagine that individuals move from richly to scarcely populated areas , or , on the contrary , that under some conditions they tend to aggregate ( for instance under the risk of extinction ) , in opposition to a diffusion - type phenomenon . similarly , it may be expected that the diffusion relents once the gradient of the density becomes large , in view of a kind of blockage effect at too high gradients . of course , there are many ways of introducing such effects in a model , leading to different pictures for the considered problem ; many authors worked in this direction , part of the associated bibliography being available inside the references we mention at the end of the paper . our choice in the present article is to take into account a usual diffusion process ( towards less `` populated '' areas ) which depends on the population density and displays a saturation effect for large gradients ; explicitly , motivated , among the others , by the papers @xcite and @xcite , we consider a diffusion depending on @xmath34 through the mean - curvature type operator ( as the one considered in @xcite ) and possibly density - dependent . in any case , we warn the reader that modifying the assumptions in the diffusion makes possible , in principle , the appearance of other kinds of solutions which are not regular any more , since the functional space to appropriately set the problem into may include discontinuous functions ( for instance , in the case one is led to work in the space of bounded variation functions ) . however , we here disregard this issue , focusing since the beginning on regular traveling waves . as for the admissible speeds , and referring to the previous discussion about the fisher equation , introducing a density - dependent and saturating diffusion may lead to some different outcomes with respect to the linear case . if , on one hand , the case of fisher - type nonlinearities like @xmath8 essentially displays the same features , on the other hand it may be seen that the situation changes when the reaction term is of type c : in the recent paper @xcite this was highlighted when dealing with the model @xmath35 namely equation with @xmath36 and @xmath37 . let us also notice that , wishing to deal instead with a similar issue for the quasilinear counterpart of the burgers equation @xmath38 the situation is slightly different from the one for , since equation admits all the constants as equilibria . fixed two equilibria @xmath39 to be connected , in order to find monotone @xmath33-connections of traveling wave type @xmath21 one thus has to solve the boundary value problem @xmath40 asking @xmath41 for every @xmath42 ( or @xmath43 for every @xmath42 ) . integrating the differential equation on the real line and noticing that @xmath44 in view of the monotonicity , it is immediate to see that it has to hold @xmath45 so that there exists at most a unique @xmath46 giving a monotone @xmath33connection ; we stress that , in the burgers case @xmath9 , the previous condition is exactly ( and in this case @xmath43 for every @xmath42 ) . hence , here the picture is a bit different - actually , as we will see in section 3 , the situation is much more restrictive - and we can explicitly solve the problem , due to the particular form of the equation . it is thus natural to wonder which effect on the admissible speeds prevails when combining a nonlinear advection @xmath47 and a nonlinear forcing term @xmath48 in equation , together with a ( realistic ) saturating and density - dependent diffusion . this will be our main concern ; precisely , we study the problem @xmath49 ) ' - ( h'(v)+c)v ' + f(v ) = 0 \\ v(-\infty)=u_- , \ ; v(+\infty)=u_+ , \quad v'(s ) > 0 , \end{array } \right.\ ] ] derived from the search for monotone heteroclinic traveling waves for , and we determine the set of the admissible speeds for different choices of @xmath48 , providing some numerical evidence , as well . complementarily , we also start investigating the admissible speeds in the small viscosity limit , analyzing the dependence of the critical speed on a small viscosity parameter braking the diffusion ( see section [ sez6 ] ) . in this respect , let us mention that the case without reaction was taken into account , e.g. , in @xcite ; on the other hand , the linear diffusion case with both a reaction and a convective term was analyzed , for instance , in @xcite , where it was shown that one could in general provide only an estimate of @xmath19 . in this framework , the nonlinear ( but not saturating ) diffusion case has been studied in @xcite , as well as in @xcite for the case of porous media - type equations . finally , the specific case of the linear fisher - burgers equation , with particular emphasis on traveling waves , has attracted the interest of several authors in these last years ( we refer , as examples , to the previously mentioned papers @xcite ) . for completeness , let us also cite the paper @xcite , the monography @xcite , as well as the references @xcite for reviews and detailed explanation about density - dependent processes , and , as an example , the papers @xcite for the study of different boundary value problems for the mean - curvature operator . on the other hand , traveling waves for the fkpp equation have been largely studied , respectively , in association with porous media - type diffusions or when the diffusion process takes place on a riemannian manifold . far from being complete in the references , we quote , respectively , @xcite and @xcite , referring the reader to the references in these papers , as well . we close this introduction with a plan of the paper . in section [ sez2 ] we present the general procedure to reduce equation to the first order ; we here adapt to the presence of the nonlinear term @xmath50 the technique already exploited , for instance , in the papers @xcite . essentially , it consists in a suitable change of variables heavily relying on the fact that we search for _ monotone _ solutions . for our purposes , it is here crucial that the shape of the nonlinearity @xmath48 is invariant under `` rescalement '' by @xmath50 . even more , this may happen in presence of a wider class of operators ( see assumption @xmath51 in section [ sez2 ] ) . section [ sez3 ] is the core of the paper and it is entirely devoted to the study of the admissible speeds for equation , assuming the reaction term to be positive on @xmath52 . in particular , the main result of the section is the following . [ ilprincipale ] let @xmath53 be a continuous and positive function and let @xmath1 be as in . moreover , let @xmath54 be a @xmath55-function such that @xmath56 , and @xmath57 be a continuous function such that @xmath58 exists , @xmath59 for every @xmath60 and @xmath61 , @xmath62 for every @xmath63 $ ] , for suitable constants @xmath64 ( notice that this implies @xmath10 ) . then , there exists @xmath65 , with @xmath66 } \frac{d(u)f(u)}{u } } - \min_{u \in [ 0 , 1 ] } h'(u),\ ] ] such that has a solution if and only if @xmath67 . the proof is carried out by means of a shooting technique and lower - upper solutions , after having performed the change of variables described in section [ sez2 ] . we underline that theorem [ ilprincipale ] gives the same bound for the critical speed as in the linear case , and finds immediate application to the fisher - burgers equation . of course , such a bound may be negative , in principle , but for sure we have an unbounded set of positive admissible speeds . incidentally , we observe that the analysis of section [ sez3 ] can be easily extended to the case of suitable general convective terms of the form @xmath68 ( see , for instance , @xcite ) . on the other hand , in section [ sez4 ] we start taking into account reaction terms @xmath48 which are not positive in a right neighborhood of zero . in propositions [ possibileb ] and [ possibilec ] , we first give some sufficient conditions for the existence of a _ positive _ admissible speed . due to the presence of the convection , we observe as well the possible appearance of _ a unique negative _ admissible speed ( cf . @xcite ) , differently from @xcite ; at the same time , we are able to show examples where no admissible speeds exist . the numerical description of these two situations is the concern of section [ sez5 ] . finally , section [ sez6 ] is devoted to the study of equation in the small viscosity limit , meaning that we consider our problem with a small parameter @xmath69 in front of the diffusive term . using the results obtained in the previous sections , we show that the positive critical speeds are proportional to @xmath69 . this result is meaningful in the spirit of the possible appearance of a phenomenon know as _ metastable dynamics _ , as explained in details at the end of the section . in particular , we expect the speed rate of convergence of the time dependent solutions towards their asymptotic limits to be influenced by the size " of the viscosity , as already proved for the linear - diffusion case ( see , for instance , the pioneering article @xcite ) . these features are currently the object of our study , and may represent a possibly interesting follow - up of the present work .

we study the existence of monotone heteroclinic traveling waves for a general fisher - burgers equation with nonlinear and possibly density - dependent diffusion . we give an estimate for the critical speed ( namely , the first speed for which a monotone heteroclinic traveling wave exists ) for some different shapes of the reaction term , and we analyze its dependence on a small real parameter when this brakes the diffusion , complementing our study with some numerical simulations . maurizio garrione , marta strani + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ saturating diffusion , fisher - burgers equation , density - dependent diffusion , traveling waves , admissible speeds . .15 cm _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 34c37 , 35k55 , 35k57 . .15 cm _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

we study the existence of monotone heteroclinic traveling waves for a general fisher - burgers equation with nonlinear and possibly density - dependent diffusion . such a model arises , for instance , in physical phenomena where a saturation effect appears for large values of the gradient . we give an estimate for the critical speed ( namely , the first speed for which a monotone heteroclinic traveling wave exists ) for some different shapes of the reaction term , and we analyze its dependence on a small real parameter when this brakes the diffusion , complementing our study with some numerical simulations . maurizio garrione , marta strani + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ saturating diffusion , fisher - burgers equation , density - dependent diffusion , traveling waves , admissible speeds . .15 cm _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 34c37 , 35k55 , 35k57 . .15 cm _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

1611.05620

i

the togliatti surface in @xmath6 is the rational surface parametrized by the cubic monomials in three variables @xmath7 . it was introduced and studied by eugenio togliatti in his two articles @xcite , @xcite about rational surfaces satisfying laplace equations . the apolar system of cubics , i.e. the ideal @xmath8 generated by the cubic monomials @xmath9 , has the remarkable property of being the only homogeneous ideal , generated by four cubics of the form @xmath10 , failing the weak lefschetz property ( @xcite ) . this twofold example has lead the authors of this note , together with g. ottaviani , to establish the connection between the geometric notion of variety satisfying laplace equations and the algebraic notion of artinian ideal failing the weak lefschetz property , thanks to the notion of togliatti system , introduced in @xcite , see definition [ togliattisystem ] . togliatti systems of cubics have been completely described in @xcite and @xcite . for degree @xmath11 , the picture quickly becomes much more complicated and only partial results are known so far , see for instance @xcite containing a characterization of togliatti systems with low number of generators . an interesting property of the ideal @xmath8 is that the associated morphism @xmath12 is a cyclic galois cover of degree @xmath13 of the image surface , such that for a general line @xmath14 the inverse image of @xmath15 is a union of three lines in general position . this observation has been exploited to give a new beautiful proof of the theorem of togliatti ( see @xcite , theorem 2.2.1 ) . in this article we start from this observation to construct a new class of examples of togliatti systems in three variables of any degree @xmath1 , and by consequence of rational surfaces parametrized by polynomials of degree @xmath1 satisfying a laplace equation of order @xmath16 . we call them gt - systems in honour to galois and togliatti . precisely , we consider the galois cyclic covers with domain @xmath17 , with cyclic group @xmath2 , for any @xmath3 and any representation of @xmath2 on @xmath18 . we prove in theorem [ upper ] that they are all defined by monomial togliatti systems of degree @xmath1 . in other words , the ideal generated by the invariant homogeneous polynomials of degree @xmath1 for any action of this type is a monomial togliatti system . moreover we prove in theorem [ minimal ] that the gt - systems are all minimal in the case of an action represented by a matrix of the form @xmath19 , where @xmath5 denotes a primitive @xmath1-th root of the unity and @xmath20 . in particular this is always the case when @xmath1 is prime or a power of a prime . we then perform a detailed study of the togliatti systems associated to the representation @xmath21 , for any coprime @xmath22 and @xmath1 , giving a complete description of the classes and number of generators of the ideals of invariant degree @xmath1 polynomials @xmath23 . we also describe for any degree @xmath1 the geometry of the toric surface @xmath24 , image of the morphism defined by the monomials invariant for the action of @xmath25 . these we call generalized classical togliatti systems . finally , we also explain the interesting relations between gt - systems and linear configurations in @xmath17 , in particular ceva configurations and their dual : fermat arrangements . let us briefly explain how this paper is organized . in section [ prelim ] , we fix the notation and basic facts needed later on . in particular , we recall the definition of smooth monomial minimal togliatti system and we introduce new families of togliatti systems , the so - called gt - systems , which will be our objects of study . in section [ galois ] , we establish the basic properties of a gt - system @xmath26 generated by all forms of degree @xmath1 invariants under the action of the matrix @xmath27 , where @xmath5 denotes a primitive @xmath1-th root of the unity and @xmath28 . namely , @xmath26 is generated by monomials , its minimal number of generators is bounded by @xmath29 , it fails wlp from degree @xmath16 to degree @xmath1 and it defines a galois cover @xmath30 with cyclic galois group @xmath2 . section [ minimality ] is entirely devoted to study the minimality of gt - systems . as a main tool we use circulant matrices . in section [ d prime ] , we completely classify gt - systems in the case @xmath1 prime or a power of a prime , while in section [ general ] we give the complete classification for all @xmath1 of the actions represented by matrices of the form @xmath31 in section [ geomproperties ] , we study from a geometric point of view the rational surfaces @xmath24 associated to generalized classical gt - systems . we prove that the homogeneous ideal @xmath32 is a cohen - macaulay ideal generated by quadrics and cubics if @xmath1 is odd and only by quadrics if @xmath1 is even . finally , we describe the singular locus of @xmath24 : the 3 fundamental points of @xmath33 are sent to the singular points of @xmath24 which are cyclic quotient singularities . finally , in section [ config ] , we establish the link between gt - systems and ceva linear configurations @xmath34 ; and we study the freeness of the arrangement @xmath35 of lines associated to @xmath34 . _ acknowledgement . _ many of the ideas for this paper were developed during a stay at birs ( banff international research station ) and the authors are very grateful to birs for the financial support . we also thank j. valls , f. perroni , r. pardini , a. logar , m. reid and a. iarrobino for interesting comments and conversations .

we prove that they are all monomial togliatti systems , and that they are minimal if the action is defined by a diagonal matrix having on the diagonal , where is a primitive-th root of the unity . we get a complete description when is prime or a power of a prime . we also establish the relation of these systems with linear ceva configurations .

we study the homogeneous artinian ideals of the polynomial ring $ ] generated by the homogenous polynomials of degree which are invariant under an action of the cyclic group , for any . we prove that they are all monomial togliatti systems , and that they are minimal if the action is defined by a diagonal matrix having on the diagonal , where is a primitive-th root of the unity . we get a complete description when is prime or a power of a prime . we also establish the relation of these systems with linear ceva configurations .

hep-th0408071

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over the latest years a great deal of work and effort has been done in the direction of understanding canonical noncommutative field theories and quantum mechanics ( see @xcite and references therein ) . the recent interest in these theories is motivated by works that establish a connection between noncommutative geometry and string theory @xcite . intensive research is carried out to investigate their interesting properties , such as the ir - uv mixing and nonlocality @xcite , lorentz violation @xcite , new physics at very short distances @xcite , and the equivalence between translations in the noncommutative directions and gauge transformations ( see , e.g. @xcite ) . several investigations have been pursued to verify the possible role of noncommutativity in a great deal of cosmological scenarios . among them we quote newtonian cosmology @xcite , cosmological perturbation theory and inflationary cosmology @xcite , noncommutative gravity @xcite , and quantum cosmology @xcite . in a previous work @xcite , an investigation into the influence of noncommutativity of the minisuperspace variables in the early universe scenario was carried out for the kantowski - sachs universe . although noncommutativity effects proved to be relevant to the universe history at intermediate times , they were shown not to be capable of removing the future and past cosmological singularities of that model in the classical context . in the quantum context , on the other hand , non - singular universe solutions were shown to be present . however , since they exist for the commutative quantum kantowski - sachs universe , their presence in the ensemble of solutions of the noncommutative quantum model can not be attributed to the noncommutativity effects . although the investigation carried out in @xcite was restricted to a particular model , one expected that some of the results obtained there could be of general validity . indeed , in this work we shall show that noncommutativity can appreciably modify the evolution of the friedman - robertson - walker ( frw ) universe with a conformally coupled scalar field @xcite . as in reference @xcite , our investigation is carried out by means of a comparative study of the universe evolution in four different scenarios : classical commutative , classical noncommutative , quantum commutative and quantum noncommutative . the main motivation for the choice of the conformally coupled scalar field is that it admits exact solutions in the simpler cases discussed along this work and it is rich enough to be useful as a probe for the significant modifications noncommutative geometry introduces in classical and quantum cosmologies . the analytical treatment renders easy the study of the singular behavior of the model in its four versions . as we shall show later , even in the classical context noncommutative geometry can remove singularities . moreover , depending on the value of the noncommutative parameter , noncommutative classical models can mimic quantum effects . as an interpretation for quantum theory , we are adopting the bohmian one , which we briefly review in section 4 . originally proposed by bohm in 1952 @xcite , and further developed by him in a collaboration with hiley @xcite , such an interpretation of quantum theory has acquired an increasing number of adepts along the years @xcite . decisive contributions for the development of the bohmian quantum physics were made by bell @xcite , who was its arduous defender for three decades , by holland @xcite , and by d. drr , s. goldstein and n. zangh ( see , e.g. , @xcite-@xcite and ref . therein ) . due to the capability it has to reproduce the experimental results of quantum mechanics and quantum field theory providing an intuitive interpretation of the underlying dynamics , bohmian quantum physics is presently an issue of interest for broad community ( see , e.g. , @xcite ) . in this work the main reason that compels us to adopt the bohmian interpretation in quantum cosmology is the absence of external observers in the primordial quantum universe , which renders the standard copenhagen interpretation inapplicable in its description . as an alternative , the bohmian interpretation has been employed in several works of quantum cosmology and quantum gravity ( see , e.g. , @xcite ) . other interesting aspect of the bohmian approach to quantum theory , which motivates us for its adoption , is the efficient framework it provides for comparison between the classical and quantum counterparts of a physical model in the common language of trajectories . we shall benefit from this facility in our comparative study of the four versions of the frw universe . since our preference for bohmian interpretation is justified for technical , rather than philosophical reasons , we shall not focus our discussion on fundamental questions regarding the interpretation of quantum theory ( for references on this subject see @xcite ) . instead , whenever possible , we shall give preference for the interpretation - independent aspects . this work is organized as follows . sections 2 and 3 are devoted to a comparative study of the classical frw with a conformally coupled scalar field and its noncommutative counterpart . in section 4 we present the commutative quantum version of the model , and analyze it by using the bohmian formalism of quantum trajectories . a similar study is carried out in sections 5 and 6 , which are concerned with the noncommutative quantum version of the model . in section 7 we end up with a general discussion and summary of the main results .

the investigation is carried out by means of a comparative study of the universe evolution in four different scenarios : classical commutative , classical noncommutative , quantum commutative , and quantum noncommutative , the last two employing the bohmian formalism of quantum trajectories . , we find that noncommutative geometry can remove singularities in the classical context for sufficiently large values of .

we study the implications of a noncommutative geometry of the minisuperspace variables for the frw universe with a conformally coupled scalar field . the investigation is carried out by means of a comparative study of the universe evolution in four different scenarios : classical commutative , classical noncommutative , quantum commutative , and quantum noncommutative , the last two employing the bohmian formalism of quantum trajectories . the role of noncommutativity is discussed by drawing a parallel between its realizations in two possible frameworks for physical interpretation : the nc - frame , where it is manifest in the universe degrees of freedom , and in the c - frame , where it is manifest through-dependent terms in the hamiltonian . as a result of our comparative analysis , we find that noncommutative geometry can remove singularities in the classical context for sufficiently large values of . moreover , under special conditions , the classical noncommutative model can admit bouncing solutions characteristic of the commutative quantum frw universe . in the quantum context , we find non - singular universe solutions containing bounces or being periodic in the quantum commutative model . when noncommutativity effects are turned on in the quantum scenario , they can introduce significant modifications that change the singular behavior of the universe solutions or that render them dynamical whenever they are static in the commutative case . the effects of noncommutativity are completely specified only when one of the frames for its realization is adopted as the physical one . non - singular solutions in the nc - frame can be mapped into singular ones in the c - frame .

hep-th0408071

c

in this work , we carried out an investigation into the role of noncommutative geometry in the cosmological scenario by introducing a noncommutative deformation in the algebra of the minisuperspace variables along the same lines proposed in @xcite and followed in @xcite . as a cosmological model to carry out such an investigation , we chose a friedman - robertson - walker universe with conformally coupled scalar field . a parallel was drawn between the realizations of noncommutativity in two possible frameworks for physical interpretation : the c - frame , where it manifest through @xmath0-dependent terms the hamiltonian , and in the nc - frame , where it is manifest directly in the universe degrees of freedom . the influence of noncommutativity in the universe evolution and its capability to remove cosmological singularities was investigated by means of a comparative study of the frw model in four different versions : classical commutative , classical noncommutative , quantum commutative and quantum noncommutative . the confrontation between the classical and quantum versions was rendered easy by the bohmian interpretation of quantum theory , which provided a common language for comparison through the quantum trajectory formalism . an extension of the bohmian formulation to comprise noncommutative effects was previously proposed for the kantowski - sachs model in @xcite . in our comparative study we have dealt with the noncommutative quantum model along the same lines . the beable mapping commonly employed in bohmian quantum mechanics was extended to noncommutative quantum cosmology . in the commutative context , our formulation is reduced to the one proposed by holland @xcite in the minisuperspace approximation . in the classical context , the main result of our investigation is that , contrary to the noncommutative kantowski - sachs model , for @xmath0 sufficiently large the noncommutative frw can be non - singular . when @xmath265 , noncommutativity can give rise to bouncing universes in the nc- and c - frame realizations . the @xmath59 case is of particular interest since it reveals the capability noncommutativity has to mimic quantum effects under special conditions . the bouncing solutions that appear in both nc- and c - frame realizations [ eqs . ( [ 29 ] ) and ( [ 30 ] ) ] can be mapped into the commutative quantum solution ( [ 78 ] ) with an appropriate identification of the integration constants . therefore noncommutative classical quantum universe can be indistinguishable from a commutative quantum universe . similar correspondences involving the noncommutative classical universe and other bouncing solutions of noncommutative quantum cosmology were described in section 6 . while in the classical context non - singular universe solutions can exist only in the noncommutative universe model and for @xmath266 , in the quantum context one may find non - singular universe solutions even in the commutative case . the qualitative behavior of the universe solutions in noncommutative quantum cosmology was discussed in section 4 , where examples were presented that contain non - singular periodic solutions , non - singular solutions presenting one bounce , as well as non - singular solutions containing an infinite sequence of bounces enveloped by a larger bounce . when noncommutativity effects are turned on in the quantum scenario , they give rise to dynamical universes in situations where bohmian quantum cosmology admits only static universes . one example is that of real wavefunctions , which in the noncommutative frw model with a conformally coupled scalar field represent always non - singular bouncing universes . noncommutativity effects can also induce a dynamical behavior yielding non - singular periodic universes in cases where the commutative counterpart is static and the noncommutative wavefunction is complex . an example was presented in section 6 . the investigation into the nc - c - frame correspondence revealed that , even in the classical context , the description of the universe evolution provided by these two possible scenarios for the realization of noncommutativity can differ radically . in section 3 we showed that for some values of the integration constants the universe can be non - singular in the nc - frame , while its c - frame counterpart is singular . an example of the drastic difference that can occur between the nc- and c - frame realizations was exhibited in that section for the case where @xmath59 . it consists of a universe that is non - singular in the nc - frame and that has no correspondent in the c - frame , where it is at all times singular . in the quantum context the distinction between the nc- and c - frame descriptions was shown to be as relevant as it is in the classical one . an example was worked out that discusses how a universe with large @xmath267 in the nc - frame can correspond to two universes in the c - frame , one with large @xmath268 @xmath267 , and other with very small @xmath269 . such an interplay between small and large scale distances was previously reported in @xcite , where it was interpreted as a sort of ir - uv mixing , in analogy with noncommutative field theory . other examples involving the nc - c - frame correspondence were presented and solved numerically for the case @xmath191 and analytically for the case @xmath73 . again the case where @xmath59 provided an interesting example . when @xmath59 noncommutativity effects act to drop the kinetic term from the wheeler - dewitt equation . this justifies the poor and highly constrained dynamics found in this case . no matter what is the wavefunction in this case , if the initial conditions are non - singular the universe is non - singular and experiments a single bounce in both nc- and c - frame descriptions ( fig . @xmath270 ) . in case noncommutativity of the minisuperspace variables has in fact played a role in the evolution of the primordial universe , as proposed in @xcite , the study carried out in this work renders evident the need of an ontology in order to have a clear picture of the essential features of the noncommutative universe models . the correspondence between degrees of freedom in two different frames of realization is not sufficient to define the theory completely , which is only fixed by assuming one of them as the physical frame . this necessity seems not to be an exclusive feature of the cosmological model considered here , where the dramatic difference in the universe evolution can be attributed , in part , to the fact that the noncommutativity in question is that of the system s degrees of freedom - the minisuperspace variables- . in the models where noncommutativity does not involve directly the system s degrees of freedom , as the canonical noncommutative field theories that come from string theory @xcite , the study of the correspondence between the nc- and c - frame descriptions is also a relevant subject . in the context of gauge theories , where the connection between the nc- and c- frames is via the seiberg - witten map , an investigation into the properties of the theory that have resemblance with gravity was carried out , e.g. , in @xcite , where the equivalence between spacetime translations and gauge transformations is shown to occur in the nc - frame . in the c - frame , on the other hand , where such an equivalence seems to be lost , noncommutative fields can be interpreted as ordinary theories immersed in a gravitational background generated by the gauge field , as shown in the interesting work by rivelles @xcite , and further in @xcite .

we study the implications of a noncommutative geometry of the minisuperspace variables for the frw universe with a conformally coupled scalar field . the role of noncommutativity is discussed by drawing a parallel between its realizations in two possible frameworks for physical interpretation : the nc - frame , where it is manifest in the universe degrees of freedom , and in the c - frame , where it is manifest through-dependent terms in the hamiltonian . as a result of our comparative analysis moreover , under special conditions , the classical noncommutative model can admit bouncing solutions characteristic of the commutative quantum frw universe . in the quantum context , we find non - singular universe solutions containing bounces or being periodic in the quantum commutative model . when noncommutativity effects are turned on in the quantum scenario , they can introduce significant modifications that change the singular behavior of the universe solutions or that render them dynamical whenever they are static in the commutative case . the effects of noncommutativity are completely specified only when one of the frames for its realization is adopted as the physical one .

we study the implications of a noncommutative geometry of the minisuperspace variables for the frw universe with a conformally coupled scalar field . the investigation is carried out by means of a comparative study of the universe evolution in four different scenarios : classical commutative , classical noncommutative , quantum commutative , and quantum noncommutative , the last two employing the bohmian formalism of quantum trajectories . the role of noncommutativity is discussed by drawing a parallel between its realizations in two possible frameworks for physical interpretation : the nc - frame , where it is manifest in the universe degrees of freedom , and in the c - frame , where it is manifest through-dependent terms in the hamiltonian . as a result of our comparative analysis , we find that noncommutative geometry can remove singularities in the classical context for sufficiently large values of . moreover , under special conditions , the classical noncommutative model can admit bouncing solutions characteristic of the commutative quantum frw universe . in the quantum context , we find non - singular universe solutions containing bounces or being periodic in the quantum commutative model . when noncommutativity effects are turned on in the quantum scenario , they can introduce significant modifications that change the singular behavior of the universe solutions or that render them dynamical whenever they are static in the commutative case . the effects of noncommutativity are completely specified only when one of the frames for its realization is adopted as the physical one . non - singular solutions in the nc - frame can be mapped into singular ones in the c - frame .

cond-mat0605165

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in monte carlo studies of statistical systems undergoing phase transitions , critical slowing down is one of the prominent problems . consider a monte carlo algorithm with dynamic exponent @xmath6 . in order to generate a given number of effectively independent samples , one has to spend computing effort @xmath7 , where @xmath8 is the linear sytem size and @xmath9 accounts for the volume of the sytem of interest . for the local metropolis algorithm for the potts model , the dynamic exponent is around @xmath10 . thus , in two dimensions , the required computing effort grows like @xmath11 . therefore , a central task of computational statistical physics is to develop algorithms such that @xmath12 vanishes or is significantly suppressed . for a discussion of monte carlo methods , see ref . @xcite . for the potts model , a significant breakthrough was the invention of the swendsen - wang cluster method @xcite and its single - cluster version the wolff cluster method @xcite . for the two- and three - dimensional ising model , the dynamic exponent of the sw algorithm is about @xmath13 and @xmath14 @xcite , respectively . in comparison with the metropolis simulations , the critical slowing down is significantly suppressed . in addition to the potts model , another important class of models in statistical physics is the @xmath0 model . the @xmath0 model is defined in terms of @xmath2-component spins on a lattice , with an isotropic pair coupling of the form @xmath15 , where @xmath16 and @xmath17 are a pair of neighboring lattice sites and @xmath18 is a function . a particularly interesting case is the honeycomb @xmath0 model , where function @xmath18 is @xmath19 , with @xmath20 a measure of the inverse temperature . it turns out that the @xmath0 model has a nice graph representation @xcite ; the graph consists of a number of nointeracting and nonoverlapping loops on the honeycomb lattice . however , in contrast to the potts model , an efficient cluster algorithm is still lacking for the @xmath0 model . when simulating the @xmath0 model , one has to apply a metropolis - like local algorithm , except for some special cases such as @xmath21 or @xmath22 . even worse is that local updates of loop configurations require some global connectivity information . therefore , the computing effort grows like @xmath23 as @xmath8 increases , where the exponent @xmath24 accounts for the effective critical slowing down due to the global - connectivity - checking procedure . the value of @xmath24 is close to @xmath4 in two dimensions , unless some complicated data structure is applied . apart from the above computational considerations , developing efficient algorithms for the @xmath0 loop model is also highly desirable from physical point of view . this is because , while much exact information about the critical properties of the @xmath0 loop model has been accumulated in the past decades , many open questions still exist ; for some recent publications , see e.g. , refs . @xcite . as a generalization , dilution can be introduced into the @xmath0 loop model and the potts model . for a sufficient number of diluted sites , a different universality class can arise at the so - called tricritical point . in contrast to the tricritical potts model , exact results for the tricritical @xmath0 model are scarce . therefore , a high - precision numerical study of the @xmath0 model is of great importance . in this work , we solved the long - standing problem of developing an efficient algorithm for the @xmath0 loop model for @xmath1 . we apply the newly developed cluster algorithm , an embedding swendsen - wang - type cluster method , to the honeycomb @xmath0 loop model without dilution , for which many exact predictions are available . the numerical data confirm the exact predictions by coulomb gas theory and by conformal field theory . further , the dynamical data imply that , for @xmath25 , the embedding cluster algorithm _ hardly _ suffers from critical slowing down .

can be simulated by means of an embedding swendsen - wang - type cluster method . by taking into account the symmetry among loops of different colors , the numerical data strongly suggest that our cluster algorithm _ hardly _ suffers from critical slowing down . s h p [ section ] [ theorem]definition [ theorem]proposition [ theorem]lemma [ theorem]corollary [ theorem]conjecture [ theorem]question

we study the loop model on the honeycomb lattice with real value by means of a cluster algorithm . the formulation of the algorithm is based on the equivalence of the loop model and the low - temperature graphical representation of a-color ashkin - teller model on the triangular lattice . the latter model with integer can be simulated by means of an embedding swendsen - wang - type cluster method . by taking into account the symmetry among loops of different colors , we develop another version of the swendsen - wang - type method . this version allows the number of colors to take any real value . as an application , we investigate the , and loop model at criticality . the determined values of various critical exponents are in excellent agreement with theoretical predictions . in particular , from quantities associated with half of the loops , we determine some critical exponents that corresponds to those for the tricritical potts model but have not been observed yet . dynamic scaling behavior of the algorithm is also analyzed . the numerical data strongly suggest that our cluster algorithm _ hardly _ suffers from critical slowing down . # 1([#1 ] ) # 1to 0pt#1 # 1proof of # 1 . # 1#2 # 1 # 1 s h p [ section ] [ theorem]definition [ theorem]proposition [ theorem]lemma [ theorem]corollary [ theorem]conjecture [ theorem]question

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by making use of the equivalence of the loop configurations of the @xmath0 loop model and the low - temperature graph of the ashkin - teller model in the infinite - coupling limit ( iat ) , we formulated an embedding swendsen - wang - type algorithm for the @xmath0 loop model with real value @xmath1 . for @xmath22 , this algorithm reduces to the conventional swendsen - wang method for the ising model . with some modifications , an embedding wolff - type method ( single - cluster version ) is readily available . we then applied our cluster algorithm to the @xmath0 loop model on the honeycomb lattice . the numerical data reveal the finite - size scaling behavior of several quantities . the associated exponents are confirmed to be those exact values predicted by the coulomb gas theory and by conformal field theory . the dynamical data strongly imply that the embedding cluster algorithm suffers little from critical slowing down . this is somewhat impressive in the sense that the dynamic exponent @xmath12 of the swendsen - wang type algorithm must satisfy the li - sokal bound @xcite : @xmath442 , unless it can be proved that the amplitude of terms with exponent @xmath443 vanishes . the value of @xmath444 can be easily calculated from eq . ( [ critical_exponent ] ) , which yields @xmath445 , @xmath446 , @xmath447 , and @xmath64 for for @xmath387 , @xmath226 , @xmath233 , and @xmath4 , respectively . the absence of a dynamic exponent @xmath448 in the embedding sw simulations of the @xmath0 loop model implies that the amplitude for terms with exponent @xmath449 indeed vanishes . we argue that this is because our embedding sw cluster algorithm has made use of the symmetry among the @xmath2 colors of ising spins . similar scenarios exist elsewhere . for instance , it can be proved that , in metropolis simulations of the ising model , the dynamic exponent @xmath12 must satisfy @xmath450 . however , the dynamic exponent @xmath12 of the swendsen - wang algorithm is much smaller than @xmath451 . this is because the symmetry between the up- and down - pointing ising spins is fully taken into account in the swendsen - wang algorithm . to have a better understanding of our above argument , let us consider another version of swendsen - wang - type cluster method for the iat or the @xmath0 model with integer @xmath2 , as described in sec . this algorithm directly simulates each color of the spin variables @xmath44 in the iat model for @xmath452 . in the language of loop configurations , the algorithm only updates loops in the same color ; it does not interchange or reassign colors of loops . in other words , the symmetry among loops in different colors is not taken into account . such a cluster algorithm has been applied to the critical @xmath56 iat model on the triangular lattice , which is equivalent to the critical @xmath102 model on the honeycomb lattice or the baxter - wu model . it was found that the dynamic exponent @xmath12 for the integrated correlation time is @xmath453 ; indeed , @xmath12 satisfies the li - sokal bound : @xmath454 . instead of the individual spin variables @xmath44 , the final version of the embedding sw algorithm simulates the product variable @xmath455 . after each update of spin variable @xmath105 , the colors of loops are reassigned randomly , irrespective of the existing colors of loops . in this sense , it is natural that dynamic exponent @xmath456 vanishes . the @xmath22 loop model is just the ising model . from universality , one has @xmath458 for the @xmath459 model on any planar graph . on the honeycomb lattice , eqs . ( [ critical_exponent ] ) and ( [ relation_gq ] ) tell that @xmath460 is a monotonically decreasing function of @xmath5 . therefore , one has @xmath461 for @xmath25 . in this sense , our cluster algorithm still satisfies the li - sokal bound . since the value of @xmath460 decreases as a function of @xmath2 , it is also expected that , as @xmath2 increases , the value of @xmath381 decrease . this is consistent with @xmath395 in table [ table_fit_tau ] . it is clear from sec . iii that the embedding cluster algorithm described in this work can not be applied to the @xmath462 case . nevertheless , since all the loop configurations are the low - temperature graphs of the @xmath105 spin variable , and vice versa . it seems that , for @xmath463 , a reweighting modification of the swendsen - wang simulation of the ising model can still be useful . such a procedure can be described as * * step 1 . * for a spin configuration @xmath464 , in which the number of loops is @xmath38 , generate a new spin configuration @xmath465 by using the swendsen - wang algorithm . * * step 2 . * derive the loop information for the new configuration @xmath465 , and calculate the loop number @xmath466 . accept the new configuration with probability @xmath467 , namely , set @xmath468 . otherwise , keep the old spin configuration @xmath105 . repeating of these two steps forms a valid ` cluster ' algorithm for @xmath469 . for the @xmath0 loop model on the honeycomb lattice , it turns out that , for @xmath470 and small system sizes @xmath471 , this algorithm works pretty well . 100 k. binder and d. stauffer , _ applications of the monte carlo method in statistical physics _ ( springer - verlad , berlin , 1984 ) . swendsen and j .- s . wang , phys . lett . * 58 * , 86 ( 1987 ) . u. wolff , phys . lett . * 62 * , 361 ( 1989 ) . g. ossola and a.d . sokal , nucl . b * 691 * , 259 ( 2004 ) . stanley , phys . * 20 * , 589 ( 1968 ) . e. domany , d. mukamel , b. nienhuis , and a. schwimmer , nucl . b * 190 * , 279 ( 1981 ) . b. nienhuis , phys . * 49 * , 1062 ( 1982 ) . b. nienhuis , j. stat . phys . * 34 * , 731 ( 1984 ) . l. chayes , l.p . pryadko , and k. shtengel , nucl . b * 570 * , 590 ( 2000 ) . w. guo , h.w.j . blte , and f.y . wu , phys . lett . * 85 * , 3874 ( 2000 ) . jacobsen , n. read , and h. saleur , phys . lett . * 90 * , 090601 ( 2003 ) . w. janke and a.m.j . schakel , phys . lett . * 95 * , 135792 ( 2005 ) . w. guo , b. nienhuis , and h.w.j . blte , phys . lett . * 96 * , 045704 ( 2006 ) . j. ashkin and e. teller , phys . rev . * 64 * , 178 ( 1943 ) . c. fan , phys . a * 39 * , 136 ( 1972 ) . grest and m. widom , phys . b * 24 * , 6508 ( 1981 ) . e. fradkin , phys . lett . * 53 * , 1967 ( 1984 ) . baxter , _ exactly solved models in statistical mechanics _ ( academic press , london new york , 1982 ) . g. musial , l. debski , and g. kamieniarz , phys . rev . b * 66 * , 012407 ( 2002 ) . j. salas , and a.d . sokal , j. stat . phys . * 87 * , 1 ( 1997 ) . y. deng , j. salas , t. garoni , a. sportielle , and a.d . sokal , unpublished . l. chayes and j. machta , physica a * 254 * , 477 ( 1998 ) . cardy , _ phase transitions and critical phenomena _ , edited by c. domb and j.l . ( academic press , london , 1987 ) , vol . 11 , p. 55 , and references therein . sokal , _ monte carlo methods in statistical mechanics : foundations and new algorithms _ , lecture notes at the cargse summer school , 1996 . x .- li and a.d . sokal , phys . lett . * 63 * , 827 ( 1989 ) .

the formulation of the algorithm is based on the equivalence of the loop model and the low - temperature graphical representation of a-color ashkin - teller model on the triangular lattice . the latter model with integer in particular , from quantities associated with half of the loops , we determine some critical exponents that corresponds to those for the tricritical potts model but have not been observed yet .

we study the loop model on the honeycomb lattice with real value by means of a cluster algorithm . the formulation of the algorithm is based on the equivalence of the loop model and the low - temperature graphical representation of a-color ashkin - teller model on the triangular lattice . the latter model with integer can be simulated by means of an embedding swendsen - wang - type cluster method . by taking into account the symmetry among loops of different colors , we develop another version of the swendsen - wang - type method . this version allows the number of colors to take any real value . as an application , we investigate the , and loop model at criticality . the determined values of various critical exponents are in excellent agreement with theoretical predictions . in particular , from quantities associated with half of the loops , we determine some critical exponents that corresponds to those for the tricritical potts model but have not been observed yet . dynamic scaling behavior of the algorithm is also analyzed . the numerical data strongly suggest that our cluster algorithm _ hardly _ suffers from critical slowing down . # 1([#1 ] ) # 1to 0pt#1 # 1proof of # 1 . # 1#2 # 1 # 1 s h p [ section ] [ theorem]definition [ theorem]proposition [ theorem]lemma [ theorem]corollary [ theorem]conjecture [ theorem]question

1204.5738

i

the physics of nano - sized metallic grains has attracted much attention following a series of experiments by ralph , black and tinkham , @xcite in which individual energy levels of ultra - small aluminum grains were resolved by single - electron - tunneling spectroscopy . recent advances have made it possible to achieve better control over the size of the grain , which is important for quantitative measurements . in the experiments of ref . , very high - quality spectra of chemically synthesized gold nano - particles were obtained . typical grains used in the spectroscopic experiments are in the ballistic regime , i.e. , their size is smaller than the mean free path , and electron transport is determined by scattering from the boundaries of the grain rather than from impurities . when the boundaries are sufficiently irregular , the single - particle dynamics are chaotic . this induces sample - specific fluctuations of observables , and the meaningful quantities are the statistical distributions of these observables ; see ref . and references therein . the single - particle energies and wave functions follow the statistics of the random - matrix theory ( rmt ) @xcite in a thouless energy window @xmath0 around the fermi energy , where @xmath0 is determined by the time it takes for an electron to move across the grain . when @xmath0 is much larger than the single - particle mean level spacing @xmath1 , the grain is described by the so - called universal hamiltonian . @xcite this hamiltonian contains three interaction terms : a `` classical '' charging energy term , a pairing term that is characterized by a bulk pairing gap @xmath2 , and exchange term that depends on the total spin of the grain and is characterized by a coupling constant @xmath3 . these three interaction terms are universal , i.e. , they are independent of the particular realization of the single - particle hamiltonian . here we assume @xmath4 so that we are below stoner instability of macroscopic polarization . when the pairing term is suppressed , ( i.e. , when only charging and exchange terms contribute ) , thermodynamic observables of the universal hamiltonian can be calculated in closed form using a spin - projection method . @xcite in refs . and , a hubbard - stratonovich transformation @xcite was employed to calculate in closed form observables such as the tunneling density of states and spin susceptibility . in the absence of the exchange term , the universal hamiltonian has the form of the bardeen - cooper - schrieffer ( bcs ) @xcite hamiltonian . in the bulk limit @xmath5 , an attractive pairing interaction leads to superconductivity . effects of the bcs interaction in nano - sized metallic grains were studied extensively ; see ref . and references therein . anderson argued @xcite that the smallest possible size of a system that can be a superconductor is determined by the condition @xmath6 . in the experiments of ref . , a pairing gap was clearly observed in the excitation spectra of the largest aluminum grains containing an even number of electrons , while it was impossible to resolve such a gap in the smaller grains . this , however , does not necessarily mean that pairing correlations disappear in the smaller grains . it was proposed that thermodynamic properties could be a more suitable tool to probe this fluctuation - dominated regime , in which @xmath7 . @xcite signatures of pairing correlations in this regime are the dependence of observables on the number parity of electrons in the grain . a good example is the re - entrant behavior ( i.e. , a local minimum ) of the spin susceptibility with decreasing temperature in an odd grain . @xcite odd - even effects in the heat capacity and magnetic susceptibility were experimentally observed in small palladium clusters . @xcite bcs theory breaks down when @xmath7 and fluctuations of the gap order parameter ( beyond its mean - field bcs value ) are important . in the static - path approximation ( spa ) , @xcite only static fluctuations of the gap are taken into account . a better approximation , the spa plus random - phase approximation ( rpa ) , takes into account small - amplitude time - dependent quantal fluctuations of the order parameter around each static field . @xcite number - parity effects can be studied by using an exact number - parity projection.@xcite the heat capacity and spin susceptibility of a metallic grain ( without exchange correlations ) as functions of temperature were studied in the spa+rpa method together with a number - parity projection in ref . as well as by quantum monte carlo methods @xcite and by richardson s solution . @xcite in all of those calculations , number - parity effects were clearly identified in both the heat capacity and spin susceptibility of the grain . signatures of pairing correlations were also found in the spin susceptibility as a function of magnetic field . @xcite the exchange interaction competes with the bcs - like pairing interaction . exchange tends to maximize spin polarization , while pairing correlations tend to minimize the spin . it is known that , depending on the values of @xmath8 and @xmath9 , the ground state of a system can be superconducting , ferromagnetic , or one in which pairing and ferromagnetic correlations coexist . @xcite the effects of mesoscopic fluctuations on this competition were studied in ref . . the effect of both pairing and exchange correlations on the thermodynamic properties of the grain ( heat capacity and spin susceptibility ) was studied in ref . for the case of an equally spaced single - particle spectrum by using a quantum monte carlo method . these thermodynamic quantities can also be calculated directly from the eigenvalues of the universal hamiltonian using richardson s solution , modified to take into account the exchange interaction . @xcite the combined effect of exchange and pairing interactions on the spin susceptibility as a function of magnetic field at zero temperature was studied in ref . . in this work , we study the general problem of mesoscopic fluctuations of thermodynamic properties of the grain in the presence of both pairing and exchange correlations assuming spin - orbit coupling is negligible . the quantum monte carlo method and richardson s solution mentioned above are computationally intensive , and are less practical in calculating the mesoscopic fluctuations of thermodynamic properties for which many realizations of the grain must be studied . richardson s solution also becomes less tractable at larger values of the pairing gap or at higher temperatures , where a very large number of energy eigenvalues is required . here we use a more efficient method to calculate the heat capacity and spin susceptibility of the grain at finite temperature . the exchange interaction is treated exactly using a spin - projection method , @xcite and the corresponding spin - projected partition functions are calculated in the spa+rpa approach . number - parity effects are captured by a number - parity projection . this approach is particularly suitable for studying the mesoscopic fluctuations . the outline of the paper is as follows . in sec . [ section_model ] , we discuss the universal hamiltonian and briefly review the conditions of its validity . in sec . [ section_theory ] , we discuss the calculation of the canonical partition function , and use it to evaluate the heat capacity and spin susceptibility of a system described by the universal hamiltonian . we also discuss the stability of the rpa , which is unstable below a certain critical value of the temperature . in sec . [ section_results ] , we present our results and discuss their physical significance . we conclude in sec . [ section_conclusions ] .

we study the mesoscopic fluctuations of thermodynamic observables in a nanosized metallic grain in which the single - particle dynamics are chaotic and the dimensionless thouless conductance is large . such a grain is modeled by the universal hamiltonian describing the competition between exchange and pairing correlations . odd - even particle - number effects induced by pairing correlations are included using a number - parity projection .

we study the mesoscopic fluctuations of thermodynamic observables in a nanosized metallic grain in which the single - particle dynamics are chaotic and the dimensionless thouless conductance is large . such a grain is modeled by the universal hamiltonian describing the competition between exchange and pairing correlations . the exchange term is taken into account exactly by a spin - projection method , and the pairing term is treated in the static - path approximation together with small - amplitude quantal fluctuations around each static fluctuation of the pairing field . odd - even particle - number effects induced by pairing correlations are included using a number - parity projection . we find that the exchange interaction shifts the number - parity effects in the heat capacity and spin susceptibility to lower temperatures . in the regime where the pairing gap is similar to or smaller than the single - particle mean level spacing , these number - parity effects are suppressed by exchange correlations , and the fluctuations of the spin susceptibility may be particularly large . however , for larger values of the pairing gap , the number - parity effects may be enhanced by exchange correlations .

1204.5738

c

in conclusion , we have studied the thermodynamic properties of ultra - small chaotic metallic grains with a large dimensionless thouless conductance in the presence of both superconducting and ferromagnetic correlations . we have used the so - called universal hamiltonian ( [ universal_hamiltonian ] ) as our model , in which the one - body part is sample - specific and modeled by rmt , while the dominating interaction terms are universal . sample - to - sample fluctuations of the interaction are suppressed and ignored in the limit of large dimensionless thouless conductance . the exchange interaction has been treated exactly by means of a spin - projection method , while the pairing interaction has been treated in a path - integral approach in which all static fluctuations of the pairing gap and small - amplitude time - dependent fluctuations around each static value of the gap are included ( spa+rpa method ) . particle - number projection is approximated in the saddle - point approximation while number - parity effects are preserved using an exact number - parity projection . the method is efficient and very accurate when compared to exact canonical calculations . however , it can not be used at very low temperatures , when the rpa correction becomes unstable . this limitation can potentially be overcome using the method developed in ref . . we have found that the exchange interaction shifts the number - parity - dependent signatures of pairing correlations ( such as the enhancement of heat capacity in the even grain and the re - entrant behavior of the spin susceptibility in the odd grain ) to lower temperatures . in the fluctuation - dominated regime @xmath7 , these signatures are suppressed by exchange correlations . however , at sufficiently large values of @xmath8 , exchange correlations have the opposite effect , i.e. , the heat capacity of the even grain develops a peak , and the re - entrant behavior of the spin susceptibility in the odd grain is enhanced . mesoscopic fluctuations of thermodynamic observables can further hinder the odd - even effects for sufficiently small @xmath8 and large @xmath9 . the mesoscopic fluctuations of the spin susceptibility are especially large in the fluctuation - dominated regime @xmath7 for values of @xmath9 above @xmath211 because of the large dispersion of the magnetization . it would be interesting to extend our work to the study of granular metals , @xcite i.e. , arrays of metallic nanoparticles that are coupled via tunnel junctions . for weakly coupled grains and when the charging energy @xmath212 satisfies @xmath213 , the majority of grains are in the coulomb - blockade regime @xmath214 with suppressed inter - grain tunneling . these coulomb - blockaded grains provide the dominant contribution to the thermodynamic properties of the granular metal at low temperatures . therefore , the values of thermodynamic observables of a granular metal ( per grain ) at @xmath215 can be effectively calculated by averaging the observables of individual grains over different random - matrix realizations and the number parity of electrons . we note that number - parity effects must still be taken into account since they lead to effects that could be missed in grand - canonical calculations . an example is the curie - like divergence in the average spin susceptibility . we thank sebastian schmidt for useful discussions . this work was supported in part by the u.s . doe grant no . de - fg02 - 91er40608 , and by the facilities and staff of the yale university faculty of arts and sciences high performance computing center .

the exchange term is taken into account exactly by a spin - projection method , and the pairing term is treated in the static - path approximation together with small - amplitude quantal fluctuations around each static fluctuation of the pairing field . we find that the exchange interaction shifts the number - parity effects in the heat capacity and spin susceptibility to lower temperatures . in the regime where the pairing gap is similar to or smaller than the single - particle mean level spacing , these number - parity effects are suppressed by exchange correlations , and the fluctuations of the spin susceptibility may be particularly large . however , for larger values of the pairing gap , the number - parity effects may be enhanced by exchange correlations .

we study the mesoscopic fluctuations of thermodynamic observables in a nanosized metallic grain in which the single - particle dynamics are chaotic and the dimensionless thouless conductance is large . such a grain is modeled by the universal hamiltonian describing the competition between exchange and pairing correlations . the exchange term is taken into account exactly by a spin - projection method , and the pairing term is treated in the static - path approximation together with small - amplitude quantal fluctuations around each static fluctuation of the pairing field . odd - even particle - number effects induced by pairing correlations are included using a number - parity projection . we find that the exchange interaction shifts the number - parity effects in the heat capacity and spin susceptibility to lower temperatures . in the regime where the pairing gap is similar to or smaller than the single - particle mean level spacing , these number - parity effects are suppressed by exchange correlations , and the fluctuations of the spin susceptibility may be particularly large . however , for larger values of the pairing gap , the number - parity effects may be enhanced by exchange correlations .

1412.4839

i

the optimization of trading strategies has long been an important goal for investors in financial markets . as demonstrated in the context of a linear equilibrium model by kyle almost thirty years ago @xcite , the optimal strategy for an investor with insider information on the fundamental price of an asset is to trade incrementally through time . this strategy allows the trader to minimize costs whilst also minimizing the revelation of information to the rest of the market . the precise way in which it is optimal to split the large order ( herein called _ metaorder _ ) @xcite depends on the objective function and on the market impact model , _ i.e. _ the change in price conditioned on signed trade size . in part due to the increasing tendency toward a full automation of exchanges and in part due to the discovery of new statistical regularities of the microstructure of financial markets , the problem of optimal execution is receiving growing attention from the academic and practitioner communities @xcite . as pointed out in gatheral et al . @xcite , a first generation of market impact models @xcite distinguishes between two impact components . the first component is temporary and only affects the individual trade that has triggered it . the second component is permanent and affects all current and future trades equally . these models can be either in discrete or in continuous time and can assume either linear or nonlinear market impact for individual trades . the second generation of market impact models focusses on the _ transient _ nature of market impact @xcite . in such models , market impact is typically assumed to factorize into two components : instantaneous market impact and a decay component . the instantaneous component models the reaction of price to traded volume . the decay component describes how the market price relaxes on average after the execution of an order . in such models , each trade affects future price dynamics with an intensity that decays with time . the problem of optimal execution in the presence of transient impact has been considered in a series of recent studies . in the case of linear instantaneous market impact @xcite , the problem has been completely solved by showing that the cost minimization problem is equivalent to solving an integral equation . in particular gatheral et al . @xcite proved that optimal strategies can be characterized as measure - valued solutions of a fredholm integral equation of the first kind . they show that optimal strategies always exist and are nonalternating between buy and sell trades when price impact decays as a convex function of time . this extends the result of alfonsi et al . @xcite on the non existence of transaction triggered price manipulation , _ i.e _ strategies where the expected execution costs of a sell ( buy ) program can be decreased by intermediate buy ( sell ) trades . however , a series of empirical studies @xcite has clearly shown that the instantaneous market impact is a strongly concave function of the volume , well approximated by a power law function . the resulting optimal execution problem in the presence of nonlinear and transient impact is mathematically much more complicated than the linear case . in this paper we consider this optimization problem and propose several methods to find optimal solutions . some important results in the nonlinear transient case were established by gatheral @xcite who showed that under certain conditions , the model admits price manipulation , _ i.e. _ the existence of round trip strategies with positive expected revenues . this money machine should of course be avoided in the modeling of market impact . in particular gatheral set some necessary conditions for the absence of price manipulation ( see below for details ) . a step toward the solution of the optimal execution problem under nonlinear transient impact has been made recently by dang @xcite . in his paper , dang suggests a way to convert the cost minimization problem into a nonlinear integral equation and proposes a numerical fixed point method on a discretization of the trading time interval to solve this equation . as we discuss in detail below , we find that dang s fixed point method has convergence problems when the degree of nonlinearity of impact is significant and/or when the discretization grid is fine enough . in this paper we propose two methods to solve the optimal execution under nonlinear transient impact . the first is based on the homotopy analysis method ( ham ) @xcite applied to the discretized version of the integral equation proposed by dang @xcite . the method starts from an initial guess and deforms it continuously in order to find better and better approximations of the solution of the integral equation . in doing this , we are implicitly restricting the space of solutions to continuous nonvanishing functions of the trading rate . we find that the optimal solution is a non time - symmetric u - shape ; in the case of concave ( convex ) instantaneous impact , it is optimal to trade more at the beginning ( end ) of the metaorder . a comparative cost analysis shows that our solution outperforms conventional strategies . once again , the ham method explores only a restricted subspace of possible solutions . for this reason , in the second part of the paper we consider a fully numerical cost optimization method on a discrete grid . by using sequential quadratic programming ( sqp ) we minimize directly the cost functional ( _ i.e. _ we do not try to solve the integral equation ) . we find that the cost landscape is rugged , _ i.e. _ composed by a very large number of local minima separated by peaks . a significant number of these minima correspond to strategies with similar costs ; for a buy program the corresponding strategy is an alternation of intense and short bursts of buying periods and long periods of weak selling . in other words , the model admits transaction triggered price manipulation . more important , when the nonlinearity is strong and/or the partition is fine , some such strategies have a negative expected cost , indicating that the model admits price manipulation . we then further extend our analysis by minimizing the cost functional with the extra constraint that all trades should have the same sign , so that for example selling is disallowed during execution of a buy metaorder . this case requires a derivative - free optimization method . by using a direct - search method , namely the generating set search ( gss ) method , we find positive expected execution costs and _ sparse _ optimal strategies , _ i.e. _ it is optimal to trade with a few bursts at a high trading rate interspersed with long periods of no - trading . in order to eliminate negative cost solutions , we propose two ways of regularizing the model , one based on the addition of a spread cost and one based on a modification of the instantaneous impact function . in the latter case the function becomes convex for sufficiently high trading rates . both methods succeed in avoiding solution with negative costs and obviously reflect features of real markets . the paper is organized as follows . in section [ gatheral s model and fredholm integral equations ] , we state the problem and explain why it is difficult to solve . we also briefly summarize our results on the convergence of the dang fixed point method . in section [ sec : homotopy ] , we present the ham approach to the solution of the cost minimization problem and in section [ sec : generalpiecewise ] we present our results on the sqp and direct - search brute force minimization of the cost function . section [ regularize ] presents two proposed regularization methods . in section [ conclusions ] , we summarize and conclude .

we find that the optimal solution is front loaded for concave impact and that its expected cost is significantly lower than that of conventional strategies . we then consider brute force numerical optimization of the cost functional ; we find that the optimal solution for a buy program typically features a few short intense buying periods separated by long periods of weak selling . indeed , in some cases we find negative expected cost . we show that this undesirable characteristic of the nonlinear transient impact model may be mitigated either by introducing a bid - ask spread cost or by imposing convexity of the instantaneous market impact function for large trading rates .

we study the problem of the optimal execution of a large trade in the presence of nonlinear transient impact . we propose an approach based on homotopy analysis , whereby a well behaved initial strategy is continuously deformed to lower the expected execution cost . we find that the optimal solution is front loaded for concave impact and that its expected cost is significantly lower than that of conventional strategies . we then consider brute force numerical optimization of the cost functional ; we find that the optimal solution for a buy program typically features a few short intense buying periods separated by long periods of weak selling . indeed , in some cases we find negative expected cost . we show that this undesirable characteristic of the nonlinear transient impact model may be mitigated either by introducing a bid - ask spread cost or by imposing convexity of the instantaneous market impact function for large trading rates .

1612.03098

c

in this work we developed a theory for the melting of stripe phases in two dimensional quantum systems with competing interactions of variable range , considering both quantum and thermal fluctuations . our main conclusion is that the nature of the phase transitions as a consequence of the melting process can be very different depending on the range of the competing repulsive interactions , @xmath5 , extending considerably the known results which , almost exclusively , were restricted to short range interactions . a mapping of the problem to a model of quantum rotors in the plane with generalized dipolar interactions allowed us to obtain several interesting properties of the phase transitions and universality classes of the models . at @xmath9 we showed that the melting of stripes proceeds through a two step process , which can produce two quantum critical points for sufficiently long ranged repulsive interactions , @xmath230 , while when @xmath3 the smectic - nematic transition turns out to be of first order . at finite temperatures only some kind of orientational order is possible . when @xmath7 the well known critical phase with algebraic orientational correlations is present , ending at a kt line . but for sufficiently long range repulsive interactions , @xmath16 , a phase with long range nematic order is possible , ending at a second order critical line . at @xmath9 , critical exponents can be computed due the equivalence of the quantum @xmath18 problem at @xmath9 with the classical model of rotors in @xmath286 dimensions at finite temperature , which properties for short range interactions are well known . an approximate treatment for finite temperatures allowed us to compute the behavior of thermodynamic quantities near the qcp , specially the temperature dependence of the uniform susceptibility , the correlation length and the critical line , summarized in table [ tb : exponents ] . in this work the melting of stripes defined by a scalar density order parameter was studied . our results could be tested , e.g. in ultra - cold dipolar fermi gases in the case where the dipoles point perpendicular to the plane of the system , in which case the system of dipoles recovers space rotational invariance . important extensions for future work are the consideration of a vector order parameter , e.g. inclusion of different spin components , and also an interaction among different degrees of freedom , which is important for the physics of high @xmath193 superconductors and other strongly correlated electronic systems in which electron and spin density waves are intertwinned , as considered , e.g. in [ ] . lattice anisotropies can be also important perturbations in real situations which deserve to be studied in future work . the brazilian agencies _ conselho nacional de desenvolvimento cientfico e tecnolgico _ ( cnpq ) , _ fundao carlos chagas filho de amparo pesquisa do estado do rio de janeiro _ ( faperj ) , and _ coordenao de aperfeioamento de pessoal de nvel superior _ ( capes ) are acknowledged for partial financial support . d.g.b also acknowledges partial financial support by the associate program of the abdus salam international centre for theoretical physics , ictp , trieste , italy . 56ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/revmodphys.75.1201 [ * * , ( ) ] link:\doibase 10.1080/00018730903122242 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.87.457 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.82.394 [ * * , ( ) ] link:\doibase 10.1103/physrevb.59.8065 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.113.076803 [ * * , ( ) ] http://dx.doi.org/10.1038/nature09597 [ * * , ( ) ] http://dx.doi.org/10.1038/nature08716 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.117.127001 [ * * , ( ) ] link:\doibase 10.1126/science.1063539 [ * * , ( ) ] , link:\doibase 10.1126/science.1134796 [ * * , ( ) ] , link:\doibase 10.1146/annurev - conmatphys-070909 - 104117 [ * * , ( ) ] , link:\doibase 10.1103/physrevb.23.316 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.64.115312 [ * * , ( ) ] link:\doibase 10.1103/physrevb.55.142 [ * * , ( ) ] http://gen.lib.rus.ec/book/index.php?md5=d95c8ba24f70b2291c7a02cdf2039e91[__ ] , ed . , world scientific lecture notes in physics 67 ( , ) @noop _ _ ( , ) link:\doibase 10.1103/physrevb.86.115138 [ * * , ( ) ] link:\doibase 10.1103/physrevb.91.035122 [ * * , ( ) ] @noop * * ( ) \doibase http://dx.doi.org/10.1016/j.physc.2012.04.014 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.215301 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.114.205302 [ * * , ( ) ] link:\doibase 10.1103/physreva.82.013643 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.115301 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.235124 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.145304 [ * * , ( ) ] @noop * * , link:\doibase 10.1103/physrevb.79.075437 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.114.116101 [ * * , ( ) ] link:\doibase 10.1103/physrevb.73.085101 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.075127 [ * * , ( ) ] link:\doibase 10.1016/0921 - 4534(93)90581-a [ * * , ( ) ] @noop * * , ( ) @noop * * , @noop * * , ( ) link:\doibase 10.1103/physrevb.89.094112 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.98.200604 [ * * , ( ) ] link:\doibase 10.1103/physreve.88.062140 [ * * , ( ) ] link:\doibase 10.1103/physrevb.70.134430 [ * * , ( ) ] link:\doibase 10.1103/physrevb.14.3110 [ * * , ( ) ] @noop _ _ , vol . ( , ) link:\doibase 10.1103/physrevb.64.184106 [ * * , ( ) ] @noop _ _ ( , , ) \doibase http://dx.doi.org/10.7566/jpscp.3.016018 [ * * ( ) , http://dx.doi.org/10.7566/jpscp.3.016018 ] link:\doibase 10.1103/physrevb.63.214503 [ * * , ( ) ] link:\doibase 10.1103/physrevd.87.114504 [ * * , ( ) ] link:\doibase 10.1103/physreva.4.1238 [ * * , ( ) ] @noop _ _ ( , ) link:\doibase 10.1103/physrevb.78.085124 [ * * , ( ) ] link:\doibase 10.1103/physrevb.80.039901 [ * * , ( ) ] link:\doibase 10.1103/physrevb.83.100509 [ * * , ( ) ] link:\doibase 10.1103/physreva.6.936 [ * * , ( ) ] link:\doibase 10.1103/physreva.7.1673 [ * * , ( ) ]

we find a two step disordering of the stripe phases with the growth of quantum fluctuations . critical exponents as a function of are computed at one loop order in an expansion and , whenever available , compared with known results . for finite temperatures it is found that for orientational order decays algebraically with distance until a critical kosterlitz - thouless line . nevertheless , for it is found that long range orientational order can exist at finite temperatures until a critical line which terminates at the quantum critical point at .

we study the quantum melting of stripe phases in models with competing short range and long range interactions decaying with distance as in two space dimensions . at zero temperature we find a two step disordering of the stripe phases with the growth of quantum fluctuations . a quantum critical point separating a phase with long range positional order from a phase with long range orientational order is found when , which includes the coulomb interaction case . for the transition is first order , which includes the dipolar case . another quantum critical point separates the orientationally ordered ( nematic ) phase from a quantum disordered phase for any value of . critical exponents as a function of are computed at one loop order in an expansion and , whenever available , compared with known results . for finite temperatures it is found that for orientational order decays algebraically with distance until a critical kosterlitz - thouless line . nevertheless , for it is found that long range orientational order can exist at finite temperatures until a critical line which terminates at the quantum critical point at . the temperature dependence of the critical line near the quantum critical point is determined as a function of .

1508.02049

c

we have studied implications of the presence of the positive and negative correlation with the radio flux for the soft and hard x - ray fluxes , respectively , in the hard spectral state of cyg x-3 . we have found that the radio flux is then completely uncorrelated with the observed , absorbed , broad - band , 1.3100 kev x - ray flux , which appears to be a good approximation to the absorbed bolometric flux . an intrinsic lack of correlation is unlikely on theoretical grounds , as well as it is unlikely given the presence of the strong narrow x - ray band correlations . however , we can recover a clear positive correlation if the absorption by the stellar wind of the donor in cyg x-3 is strong enough . this implies that the intrinsic soft x - ray spectrum in the hard state is relatively soft , similar to those of intermediate states of bh binaries . this nature of the apparent hard state of cyg x-3 is also consistent with the luminosity of the hard state being much higher in this object than in the hard state of the high - mass bh binary , cyg x-1 . we have found the radio spectra in the hard state to be hard on average , with @xmath67 . such spectra are characteristic to bh x - ray binaries in the hard state . we have compared cyg x-3 with other x - ray binaries . on the x - ray colour - colour and colour - eddington ratio diagrams , cyg x-3 , after correcting its spectra for absorption and for the bh mass of @xmath116 , appears similar to grs 1915 + 105 , with similar colours and @xmath1171 . on the other hand , cyg x-3 , at @xmath118 is also similar on such diagrams to ns z - type binaries . however , the radio emission of those sources is much weaker than that of cyg x-3 , which is both the most radio - luminous and , in its hard state , the most radio - loud ( i.e. , with the highest ratio of the radio to bolometric luminosities ) source among all known x - ray binaries . still , its radio loudness is much closer to those of bh binaries than to those of ns ones . this represents a strong argument for the presence of a bh in cyg x-3 . we interpret the radio loudness and radio fluxes of cyg x-3 being much higher than those of grs 1915 + 105 as due to an enhancement of the jet emission in the former due to interaction of the jet with the very strong stellar wind from the donor . we have found ( appendix [ lc ] ) that the flux distributions of the radio emission and soft x - rays can be described by two log - normal functions , one corresponding to the hard state and one to soft state . this indicates a multiplicative and correlated character of variability of cyg x-3 in both the hard and soft state and in both the accretion flow and the jet . finally , we have presented ( appendix [ lc ] ) the long - time light curves of cyg x-3 using the available data , and compared them to the occurrences of high - energy @xmath1-ray detection . we have found that a low hard x - ray flux , @xmath119 , is both a necessary and sufficient criterion for detectable @xmath1-rays . we have found that this criterion has not been satisfied since mjd 55650 ( 2011 march 30 ) until now , mjd 57333 .

we study the radio / x - ray correlation in cyg x-3 . we show that this implies that the observed kev flux ( which is a fair approximation to the bolometric flux ) is completely uncorrelated with the radio flux . we can recover a positive correlation ( seen in other sources and expected theoretically ) if the soft x - rays are strongly absorbed by a local medium . then , however , the intrinsic x - ray spectrum of cyg x-3 in its hard state becomes relatively soft , similar to that of an intermediate spectral state of black - hole binaries , but not to their true hard state . we also find the radio spectra in the hard state of cyg x-3 are hard on average , and the flux distributions of the radio emission and soft x - rays can be described by sums of two log - normal functions . we compare cyg x-3 with other x - ray binaries using colour - colour , colour - eddington ratio and eddington ratio - radio flux diagrams . we find cyg x-3 to be spectrally most similar to grs 1915 + 105 , except that cyg x-3 is substantially more radio loud , which appears to be due to its jet emission enhanced by interaction with the powerful stellar wind from the wolf - rayet donor . [ firstpage ] radiation mechanisms : non - thermal radio continuum : stars stars : individual : cyg x-3 stars : winds , outflows

we study the radio / x - ray correlation in cyg x-3 . it has been known that the soft and hard x - ray fluxes in the hard spectral state are correlated positively and negatively , respectively , with the radio flux . we show that this implies that the observed kev flux ( which is a fair approximation to the bolometric flux ) is completely uncorrelated with the radio flux . we can recover a positive correlation ( seen in other sources and expected theoretically ) if the soft x - rays are strongly absorbed by a local medium . then , however , the intrinsic x - ray spectrum of cyg x-3 in its hard state becomes relatively soft , similar to that of an intermediate spectral state of black - hole binaries , but not to their true hard state . we also find the radio spectra in the hard state of cyg x-3 are hard on average , and the flux distributions of the radio emission and soft x - rays can be described by sums of two log - normal functions . we compare cyg x-3 with other x - ray binaries using colour - colour , colour - eddington ratio and eddington ratio - radio flux diagrams . we find cyg x-3 to be spectrally most similar to grs 1915 + 105 , except that cyg x-3 is substantially more radio loud , which appears to be due to its jet emission enhanced by interaction with the powerful stellar wind from the wolf - rayet donor . [ firstpage ] radiation mechanisms : non - thermal radio continuum : stars stars : individual : cyg x-3 stars : winds , outflows x - rays : binaries .

astro-ph0303098

i

the nature of the sources responsible for ionising the neutral hydrogen in the intergalactic medium ( igm ) has been the subject of a long - standing debate . there are both observational and theoretical indications that some or perhaps all of this ionising background comes from stars and not hard sources , such as quasars . while it seems clear that quasars provide a significant contribution to the ionising background at low redshifts , @xmath14 , the steep drop in the abundance of bright quasars at earlier times brings into question whether they are capable of maintaining the ionised state of the igm at @xmath15 . in an earlier paper ( sokasian et al . 2003 ) , we showed that the observed opacities in both h i and he ii indicate that quasars alone can not produce the required emissivity to match observations at @xmath16 . the situation is exacerbated at higher redshifts where observations by fan et al . ( 2000 ) , becker et al . ( 2001 ) , and djorgovski et al . ( 2001 ) imply that the igm was highly ionised at @xmath7 even though the emissivity from bright quasars was likely unimportant . given this discrepancy , it appears that a substantial portion of the ultraviolet photons at @xmath17 were produced by an additional class of sources . it has been argued that star forming galaxies may yield the required ionising emissivities ( e.g. haehnelt et al . 2001 , bianchi et al . 2001 , steidel et al . 2001 ) , although it is possible that harder low - luminosity active galactic nuclei ( agn ) may also make up the difference ( haiman & loeb 1998 ) . early star formation also provides a possible explanation for the widespread abundance of metals in the igm ( cowie et al . 1995 ) while reionisation by faint agns may provide a more plausible solution if the escape fraction for stellar radiation from galaxies is restrictively small ( schirber & bullock 2002 , wood & loeb 2000 ) . there have been numerous studies of cosmological reionisation using semi - analytical methods ( e.g. haardt & madau 1996 , madau et al . 1999 , chiu & ostriker 2000 , valageas & silk 1999 ) . in these models , the impact of the uv background is treated using a spatially and directionally averaged radiative transfer equation which incorporates the mean luminosity function per unit volume for the source term and the statistical properties of absorbing clouds for the recombination term . inevitably though , to accurately address questions related to the time - dependent propagation of radiation fronts , especially during the early stages of the reionisation process where the radiation field is highly inhomogeneous , a detailed radiative transfer approach is required . this has proved to be an extremely challenging endeavour owing to the high dimensionality of the problem . a brief review of the problem and the various techniques which have been employed in recent years is presented in razoumov et al . ( 2002 ) . one of the major challenges involved with such exercises is determining how to efficiently process the large number of sources that may be responsible for reionisation . in the case of bright quasars , the associated number density is small enough that straightforward approaches can be employed ( e.g. sokasian et al . 2001 , 2002 ) . the situation becomes much more difficult when galaxies are the primary sources . for typical cosmologies , a @xmath18 mpc@xmath19 comoving volume can easily harbour @xmath20 star forming galaxies by @xmath21 . tracking the ionising flux from such a large pool of sources over many time steps inevitably requires the implementation of sophisticated algorithms designed to efficiently process the cumulative effects from each source . by combining a photon conserving algorithm with two independent hierarchies of trees , one for rays ( abel & wandelt 2002 ) and one for the sources themselves , razoumov et al . ( 2002 ) were able to improve the efficiency of the calculations to study the inhomogeneous process of reionisation from star forming sources down to a redshift of @xmath22 . in particular , they were able to carefully follow time - dependent radiative transfer calculations on a uniform @xmath23 cartesian grid representing a volume with a comoving length of @xmath24 mpc on each side which stored the results of pre - computed hydrodynamical simulations of galaxy formation . as a result of their calculations , the authors were able to address a number of questions related to the morphological evolution of the reionisation process . in particular , they showed a picture of stellar reionisation in which photons initially do not travel far from ionising sources , in contrast to images from simulations by gnedin ( 2000 ) . more recently , ciardi et al . ( 2003b ) combined high - resolution n - body simulations and a semi - analytic model of galaxy formation with a monte carlo radiative transfer code to study reionisation in a volume with a comoving length of @xmath25 mpc . in their analysis they assessed the effect of the environment by performing a subsequent simulation on a smaller @xmath26 mpc comoving box centered on a cluster . they found that the environment where the radiation is produced modifies reionisation through the influence of the density dependence of the recombination rates . this conclusion raises the question of whether the exclusion of the copious number of dim low - mass sources which fall below the resolution limit of these simulations may play a more important role in determining the morphological evolution of the reionisation process . the fact that most models of galaxy formation predict that the bulk of the resultant ionising radiation is produced in relatively massive systems has provided some motivation for ignoring the low - mass sources altogether , especially in lieu of the high computational cost of radiative transfer calculations . however , since dim low - mass sources reside in relatively less clumpy environments compared to their brighter counterparts , their overall contribution to the ionising emissivity in the igm may be significant in spite of their intrinsic faintness . we examine the impact of low - mass sources by conducting very fast radiative transfer calculations capable of processing a large number of sources from high resolution hydrodynamic simulations . in particular , by supplementing our previous _ one - step _ radiative transfer code ( see sokasian et al . 2001 ) with an adaptive ray tracing algorithm we are able to significantly speed up the calculations to the point where handling more than @xmath27 sources becomes technically feasible . our cosmological simulation is taken from a high resolution run in a series of calculations performed by springel & hernquist ( 2003a ) , and utilises @xmath28 particles in a box of comoving length @xmath29 mpc . radiative transfer calculations are then performed on a 200@xmath19 grid where we store the corresponding density fields at each redshift . this allows us to simulate reionisation with a source mass resolution that is about an order of magnitude below the limit of the ciardi et al . ( 2003b ) simulations . furthermore , the use of a 200@xmath19 grid offers a factor of @xmath30 improvement in spatial resolution for the same box length . it is also important to point out that by using hydrodynamics to describe the gas physics , rather than an n - body simulation , we are able to estimate the clumping on unresolved scales in our grid by computing the spatial distribution of particles within a given cell ( see sokasian et al . 2001 ) . this represents an important advantage over the density interpolation schemes associated with n - body codes which smooth out density variations on sub - grid scales . as a result we expect to more properly account for the recombination rates in high density regions where clumping is important . in the present paper , which is part of a series examining reionisation , we focus on the redshift interval @xmath31 to @xmath7 with specific interest in how the level of cosmic star formation predicted in the comprehensive set of hydrodynamic simulations conducted by springel & hernquist ( 2003a ) performs in the context of observational constraints on the opacity of the igm at @xmath7 . our method also treats the simultaneous reionisation of helium in the igm , allowing us to explore the plausibility of the scenario where agns are primarily responsible for reionisation . here , we restrict our attention to sources of radiation similar to normal , star - forming galaxies at the present day . in particular , we do not account for massive , population iii stars that may have been prevalent at @xmath32 . tentative measurements of polarisation in the cmb by the wmap satellite ( kogut et al . 2003 ) suggest that much of the igm was ionised earlier than indicated by the sdss quasars ( e.g. , spergel et al . we explore the implications of this result in the context of our present model by allowing the escape fraction to evolve with redshift . in the future , we will explore the combined effect of reionisation by both population ii and population iii stars . in the next section we review some of the technical aspects related to our method , including the approximations inherent to our radiative transfer calculations . we then follow with a description of the cosmological simulation that will be used in our analysis . next , we discuss our methodology for source selection and present an in - depth analysis regarding resolution effects and how to correct for them . finally , we discuss the results of our reionisation simulation and review the properties of the inhomogeneously ionised igm and make comparisons with observational constraints .

we study the reionisation of the universe by stellar sources using a numerical approach that combines fast 3d radiative transfer calculations with high resolution hydrodynamical simulations . by supplementing a _ this allows us to study how dim low - mass sources , excluded in previous investigations owing to computational limitations , affect the morphological evolution of the reionisation process . ionising fluxes for the sources are derived from intrinsic star formation rates computed in the underlying hydrodynamical simulations . we present the results of our reionisation simulation assuming a range of escape fractions for ionising photons and make statistical comparisons with observational constraints on the neutral fraction of hydrogen at derived from the sdss quasar of becker and coworkers . our method is also capable of handling the simultaneous reionisation of the helium component in the igm , allowing us to explore the plausibility of the scenario where sources with harder spectra are primarily responsible for reionisation . in this case

we study the reionisation of the universe by stellar sources using a numerical approach that combines fast 3d radiative transfer calculations with high resolution hydrodynamical simulations . by supplementing a _ one - step _ radiative transfer code specifically designed for following ionisation processes with an adaptive ray - tracing algorithm , we are able to significantly speed up the calculations to the point where handling a vast number of sources becomes technically feasible . this allows us to study how dim low - mass sources , excluded in previous investigations owing to computational limitations , affect the morphological evolution of the reionisation process . ionising fluxes for the sources are derived from intrinsic star formation rates computed in the underlying hydrodynamical simulations . analysis of numerically converged results for star formation rates and halo mass functions allows us to assess the consequences of not including low - mass objects and enables us to correct for resolution effects . with these corrections , we are able to reduce the effective mass resolution limit for sources to m , which is roughly an order of magnitude smaller than in previous studies of this kind . our calculations reveal that the process by which ionised regions in the igm percolate is complex and is especially sensitive to the inclusion of dim sources . moreover , we find that given the same level of cosmic star formation , the number of ionising photons required to reionise the universe is significantly overestimated if sources with masses below m are excluded . this result stems from the fact that low - mass sources preferentially reside in less clumpy environments than their massive counterparts . consequently , their exclusion has the net effect of concentrating more of the cosmic ionising radiation in regions which have higher recombination rates . we present the results of our reionisation simulation assuming a range of escape fractions for ionising photons and make statistical comparisons with observational constraints on the neutral fraction of hydrogen at derived from the sdss quasar of becker and coworkers . we find that given the amplitude and form of the underlying star formation predictions , an escape fraction near is most consistent with the observational results . in these models , reionisation is expected to have occurred between , although the igm remains fairly opaque until . our method is also capable of handling the simultaneous reionisation of the helium component in the igm , allowing us to explore the plausibility of the scenario where sources with harder spectra are primarily responsible for reionisation . in this case we find that if the sources responsible for reionising hydrogen by had spectra similar to agns , then the helium component of the igm should have been reionised by . we find that such an early reionisation epoch for helium does not necessarily conflict with observational constraints obtained at , but may be challenged by future observations at higher redshifts . the recent wmap measurements of the electron scattering optical depth ( according to the `` model independent '' analysis of kogut et al . ) appear to be inconsistent with the relatively late onset of reionisation by the normal population ii type stars that we consider . in order to simultaneously match the observations from the sdss quasar and the optical depth measurement from wmap with the sources modeled here , we require an evolving escape fraction that rises from near to at . such a steep enhancement in the stellar production rate of ionising photons would be consistent with an imf that becomes more and more top heavy with increasing redshift .

astro-ph0303098

r

in our analysis , we carefully follow the reionisation process from a set of models parameterised by the escape fraction . in this context , the escape fraction is defined to be the fraction of ionising photons intrinsic to each source that escape and enter into the radiative transfer calculations . it must be noted here that this parameter inevitably also carries with it any uncertainties associated with the underlying amplitude of the star formation rates computed for the sources . nevertheless , our approach allows us to predict the epoch of reionisation as a function of this parameter . by comparing these predictions with recent observational constraints on the escape fractions and neutral densities near @xmath3 , we are able to test the overall consistency of the theoretical predictions of the computed star formation rates . along the way we are also able to make general statements regarding the morphological aspects of reionisation . the evolution of the global ionisation fraction in the simulation volume provides a useful way of characterising the overall ionisation state of the universe as a function of redshift . in figures 5a and 5b we show how the ionised mass - weighted and volume - weighted fraction , respectively , evolve with redshift . the rate of the evolution to a completely ionised state is quite similar for all the models , although there is a subtle trend towards a swallower rise of ionisation fractions with decreasing escape fractions . nevertheless , in all cases the reionisation epoch , which we define as the redshift when the global volume - weighted ionisation fraction reaches @xmath124 , occurs by @xmath125 . later in this section we show how a more detailed analysis involving statistical comparisons between simulated spectra and observational results for flux transmission at @xmath7 can lead to more rigid constraints as to how early reionisation could have taken place . figures 5a and 5b also show that the ionised mass - weighted fraction is consistently larger than the volume - weighted fraction at redshifts before reionisation . this result is seen more clearly in figure 5c where we have plotted the corresponding ratio of the two fractions . the larger values for the ionised mass - weighted fraction at early redshifts is consistent with the general picture of the pre - overlap epoch where sources are preferentially ionising their relatively dense surroundings before affecting the more tenuous igm . the fact that the inequality between the mass and volume fraction is amplified with decreasing escape fraction highlights an interesting morphological feature related to the number and intensity of the sources responsible for reionisation . namely , when escape fractions are low , a relatively larger number of sources are directly involved in the process of ionising a neutral density field . since the ionisation zones associated with these sources are also smaller , the net effect is that massive regions are more likely to be ionised from the ionising radiation emanating from within rather than radiation that has first traversed the less dense igm . this predilection naturally manifests itself in relatively larger ratios for the mass - to - volume ionisation fractions during the early stages of reionisation in these cases . eventually , intervening patches of neutral material become ionised and there is a sharp rise in the intensity of the ionising background as contributions from different sources combine . as this diffuse component comes to dominate the radiation field , all regions become uniformly ionised and the ratio of mass - to - volume ionisation fraction converges to unity . a visual illustration of the reionisation process is shown in figure 6 where we plot a series of projected slices through the simulation volume . in each panel , a @xmath126mpc slice from the @xmath127 run is projected in both density and ionisation fraction . source locations in each slice are denoted by white crosses making it easier to follow how the ionised regions ( _ blue _ ) percolate to turn a neutral igm ( _ yellow _ ) into one that is highly ionised ( _ red _ ) . compared with earlier studies of reionisation , figure 6 shows that morphologically the ionised regions in our analysis trace the large scale distribution of sources . in particular , we do not find that reionisation occurs mainly from low- to high - density regions , as would be the case if photons escaped preferentially into the igm before ionising the high density regions near the sources . the panels in figure 6 graphically illustrate the large number of sources that are involved in the highly inhomogeneous reionisation process . a more dramatic illustration of this point is shown in figure 7 where we have plotted a 3d image of the same simulation showing iso - surfaces around the ionised regions at @xmath128 . this highly inhomogeneous morphology represents a departure from the general picture of reionisation proposed by gnedin ( 2000 ) where only a handful of bright sources are responsible for reionising the universe . in the following section we discuss how the morphology of the reionisation process can affect the number of ionising photons required to achieve full reionisation . the number of ionising photons per hydrogen atom required for the ionised regions around sources to completely overlap has developed into an interesting topic that is commonly addressed in studies of this kind ( ciardi et al . 2003b , razoumov et al . 2002 , haiman et al . 2001 , miralda - escud et al . 2000 , gnedin 2000 ) . to reliably estimate this quantity with simulations requires a high level of resolution capable of properly estimating gas clumping factors which govern the overall recombination rate . this requirement is especially important in light of the results presented in haiman et al . ( 2001 ) which show that mini - halos with temperatures below @xmath129k can dominate the average clumping in the early stages of reionisation and significantly increase the required photon budget necessary to achieve complete overlap . given the high resolution inherent to our hydrodynamic simulations and the corresponding radiative transfer grids , coupled with our ability to probe clumping factors on sub - cell scales based on the underlying distribution of sph particles , we are able for the first time to probe the scales needed to account for recombination effects arising from such low - mass objects . this represents a significant improvement upon previous attempts which have relied on conservative assumptions related to the escape fraction to accommodate unresolved clumping . in figure 8 , we plot the cumulative number of escaping ionising photons per hydrogen atom as a function of redshift . the diamond symbol on each curve corresponding to different values for @xmath130 represents the point at which the volume ionisation fraction exceeds @xmath124 . for @xmath131 , between 2 and 3 ionising photons are required to reionise the universe while 4 - 5 ionising photons are needed in the case where @xmath132 . the difference in the number of required ionising photons resulting from variations in the escape fraction arises from the interesting interplay between the evolution of clumping factors and the mean density of the universe . in particular , the rapid evolution of the mean density plays a slightly more important role than the evolution of clumping factors in the redshift range of interest . therefore , a delay in reionisation to lower redshifts requires fewer ionising photons since the decline in gas densities outweighs the corresponding increase in clumping factors , resulting in comparatively fewer recombinations occurring than at higher redshifts . environmental factors influence the number of photons required to produce reionisation . a related issue is whether the distribution of ionising intensities among the sources themselves can also have an important impact on this process . in particular , given the same amplitude and evolution of the cosmic star formation rate , it is interesting to explore how a redistribution of the ionising flux from dim low - mass sources to more massive sources would affect reionisation . in figure 9 , we examine this scenario by way of comparisons involving the volume - weighted ionisation fraction and the cumulative number of ionising photons that are released . specifically , we examine two situations : 1 ) the case where we have retained the full source list which includes sources down to our resolution limit of @xmath133 m@xmath1 , and 2 ) the case where we have systematically transferred ionising fluxes from sources with @xmath134 m@xmath1 to the nearest neighbouring sources with masses above this limit . the figure shows that the affect can be significant , namely that reionisation takes place later and requires more photons ( by roughly @xmath135 ) for the same escape fraction ( @xmath11 ) when low - mass sources are excluded . this result stems from the fact that low - mass sources preferentially reside in less clumpy environments than their massive counterparts . consequently , the net effect of transferring ionising flux from these sources to more massive systems amounts to increasing the overall ionising intensity in regions with higher recombination rates . this leads to recombinations playing a more dominant role and thus delaying the onset of reionisation . this finding therefore supports the requirement that dim low - mass sources be properly resolved and included in future simulations of this kind . in particular , inaccuracies related to the inability to resolve low - mass sources and their corresponding contribution to the cosmic star formation rate may be amplified due to their preferential location in less clumpy environments . recent discoveries of quasars at redshifts 5.8 and greater ( becker et al . 2001 , fan et al . 2001 , fan et al . 2000 ) are finally making possible quantitative studies of the status of the igm at high redshifts . in fan et al . ( 2000 ) the spectrum of a @xmath136 quasar ( sdss j1010 - 0125 ) was examined and revealed no evidence for a gunn - peterson trough , indicating that the igm was highly ionised near @xmath137 . in subsequent observations ( fan et . al 2001 ) the authors studied three new quasars at @xmath138 and found a significant increase in ly@xmath44 absorption from redshift 5.5 to 6.0 . in particular , they found a @xmath139 region in the ly@xmath44 forest portion of the spectrum of a @xmath4 quasar ( sdss 1030.10 + 0524 ) which had a measured flux transmittance consistent with zero , indicating a flux decrement of @xmath140 , and suggesting the possible detection of a gunn - peterson trough . to more accurately quantify these findings becker et al . ( 2001 ) obtained higher resolution spectra of the three new @xmath138 quasars , allowing them to place better constraints on the status of the high redshift igm . to relate our results to these observations , we extract artificial ly@xmath44 absorption spectra from the simulation outputs and then derive statistical probabilities for the resultant flux transmittance which can readily be compared to the observational results . we extract artificial spectra using the particle information in the sph simulation coupled with the ionisation information from our radiative transfer calculations . for each model , we generate 500 spectra along randomly selected lines of sight ( loss ) between @xmath141 . our procedure is similar to that in the tipsy software package ( katz & quinn 1995 ) , but does not require that a los be perpendicular to a box face in the simulation volume . each los has a unique and arbitrary direction relative to the box coordinate system and wraps through the simulation volume repeatedly via periodic boundaries . using the smoothing kernels of the sph particles , gas densities and temperatures are computed along a los at appropriately sampled intervals . the component of the peculiar velocity of the gas in the direction of the los is also computed at each point . once all physical quantities have been gathered , voigt profiles are fitted to each spectrum by interpolating between the corresponding line - absorption coefficients provided in harris ( 1948 ) . it is important to point out that in computing the line profiles we use a minimum gas temperature of @xmath142 k as a correction to the sph temperatures which exclude the extra heating introduced by radiative transfer effects ( see abel and haehnelt , 1999 ) . our simulations were stopped at @xmath136 , allowing us to make comparisons with the observational results from the spectrum of the @xmath4 quasar . the presence of an apparent gunn - peterson trough in the spectrum of this quasar makes it an especially interesting case . by using the results of our simulated spectra to make statistical comparisons with this region and another region with a non - zero transmittance , we can constrain the evolution of the reionisation process . our analysis is based on probability distribution functions involving transmission levels that have been averaged over the same redshift range as the observational measurements . specifically , we focus on the redshift ranges @xmath143 and @xmath144 , corresponding to the regions over which the average transmittance was measured by becker et al . ( 2001 ) in the spectrum of the @xmath4 quasar . in figure 10 , we plot the probability distribution functions of simulated transmissions averaged over the two redshift ranges and overlay the corresponding @xmath145 measurements from becker et al . ( 2001 ) ( _ vertical - dashed lines _ ) . the distributions were obtained using 500 los , although we have only plotted a restricted range of transmissions for the purposes of observational comparisons . it is clear from figure 10 that the ionising emissivities predicted in the models with @xmath5 provide the most plausible match to the observational results in terms of transmission probabilities for the two redshift ranges . for this choice , reionisation is expected to have occurred near @xmath146 , although as the analysis here demonstrates , the igm remains fairly opaque until @xmath3 . this spread in redshift between the time when the universe first became ionised and the time when opacities declined enough to allow non - zero transmission in ly@xmath44 is important for the proper interpretation of observed quasar spectra . in particular , it is incorrect to assume that a volume - weighted ionisation fraction above @xmath124 will necessarily produce non - zero transmission levels consistent with current observations of quasar spectra at @xmath7 . a much more careful analysis involving extraction of spectra from simulations which include high precision information regarding ionisation fractions is necessary in order to make meaningful comparisons with observational results . interestingly , an escape fraction of @xmath147 is reasonably close to the range of values ( @xmath148 ) that have been observationally deduced from @xmath149 starburst galaxies ( see heckman et al 2001 ; hurwitz et al . 1997 ; leitherer 1995 ) . it is important to note that these observational results may be underestimating the true values of the escape fraction owing to undetected absorption from interstellar components . also unclear , in context of our analysis , is the possible evolution of the escape fraction with redshift which may lead to significantly different values at @xmath97 . in any case , the fact that we require a reasonable escape fraction in order to match the observations lends credence to the overall consistency of the theoretical predictions of the computed star formation rates . our radiative transfer calculations have been designed to handle the simultaneous reionisation of both the hydrogen and helium components of the igm . this allows us to explore the scenario where objects with spectra harder than stellar sources were primarily responsible for the reionisation of the universe . such a scenario has been proposed by e.g. haiman & loeb ( 1998 ) . in their analysis , a theoretical extrapolation of the quasar luminosity function at fainter luminosities and higher redshifts than currently detected appears to favour reionisation by low - luminosity mini - quasars ( or agns ) . reionisation by agn type sources has also been supported on the grounds that they may be the only type of sources capable of having large enough escape fractions to allow enough ionising photons to enter the igm ( see wood & loeb 2000 ) . to explore the implications of sources with hard spectra reionising the universe , we have run a model in which the sources are capable of ionising he ii in addition to h i. specifically , he ii ionising rates are computed for the sources by assuming a single power - law form for the spectrum , @xmath150 , normalised to deliver the same number of h i ionising photons below 4 ryd predicted from soft sources with @xmath11 . this ensures that the hydrogen component of the igm will reach roughly the same level of ionisation by @xmath3 as in the @xmath11 stellar model and therefore will also provide a reasonable match to the observational results of becker et al . ( 2001 ; see 5.3 ) . extrapolated results for the far - uv synthetic spectrum of a stellar population with continuous star formation reveals a typical quasar spectrum with @xmath151 up to 4 ryd . adopting the same spectral index above 4 ryd for the hard sources allows us to relate the rate of ionising photons , @xmath68 , in he ii and h i directly with the expression @xmath152 where @xmath153 is the ratio of ionisation threshold frequencies for the two species . this leads to a conversion factor of @xmath154 he ii ionising photons for every h i ionising photon . in figure 11 , we show the resultant evolution in the volume - weighted ionisation fraction for both soft and hard sources . note how the hydrogen component in the igm becomes reionised earlier in the case where hard sources are invoked . this is due to the fact that the he iii regions now surrounding the sources act as an additional source of h i ionising photons ( see 2.1 ) . the bottom panel of figure 11 shows that the helium component becomes reionised by @xmath155 in this case . such an early reionisation epoch for helium by high redshift sources such as agns should be testable in the coming decade with the next generation space telescope . nevertheless , an interesting question we can explore currently is whether such a scenario may also leave an imprint on he ii ionisation levels at lower redshifts where observations currently exist . in particular , we would like to test whether a hypothetical population of agns which suddenly turn off at @xmath156 will have lingering effects in terms of subsequent ionisation levels when another known population of quasars turn on . for such an analysis , we re - ran our most realistic quasar model ( model 5 ) from sokasian et al . ( 2002 ) with the initial conditions that the helium component in the igm is doubly ionised everywhere in the box . we present the results showing the subsequent evolution in the volume - weighted ionisation fraction in figure 12 . here the solid line represents the scenario described above while the dotted line indicates the original simulation where quasars turn on in a medium that is only singly ionised in helium . in the case where the igm is initially fully ionised , we see that he iii recombinations quickly cause the ionisation fraction to decrease shortly after @xmath156 when the ionising emissivity from quasars is still low . eventually , the emissivity from quasars grows and the ionisation fraction starts to rise again . however by @xmath9 enough recombinations have already taken place since @xmath156 to effectively remove all traces of an early reionisation epoch , leading to ionisation levels that are governed only by the quasar population . it thus appears as if the present observations can not rule out the simple agn scenario we have considered here . the recent tentative measurement of polarisation in the cmb by the wmap satellite ( kogut et al . 2003 ) suggests that reionisation occurred in a more complex manner than indicated by the sdss quasars alone ( e.g. , spergel et al . the measurements exhibit an excess in the cmb te cross - section spectrum on large angular scales ( @xmath157 ) yielding an electron optical depth to the cmb surface of last scattering of @xmath158 . the uncertainty associated with this measurement depends on fitting all parameters concerned with the tt power spectrum and the te cross power spectrum . kogut et al . ( 2003 ) obtain a 68% confidence range of @xmath159 corresponding to an instantaneous reionisation epoch which occurred between @xmath160 . none of the models we have considered in this paper is able reionise the universe by such early redshifts . the connection between the implications of the wmap results and those of the sdss quasars for the evolution of the igm is unclear . in an attempt to determine the extent to which the sources examined in this paper would require additional ionising emissivity to provide a consistent match with the wmap measurements we consider an evolving escape fraction of the form @xmath161 . here , a positive value for the parameter @xmath162 has the effect of producing ionisation levels consistent with the observations from the @xmath4 sdss quasar , as in the model with @xmath11 , while leading to relatively larger ionising emissivities at higher redshifts . we note that with this choice for @xmath130 , the `` escape fraction '' can exceed unity . in the context of our present analysis , this simply means that the rate of production of ionising photons is higher than would be implied for our choice of the star formation history and the imf . for simplicity , we absorb these uncertainties in what follows into our definition of the `` escape fraction . '' in figure 13 we compare three different cases for this form : 1 ) a case with a constant escape fraction of @xmath11 ( @xmath163 ; _ solid - line _ ) , 2 ) a case with @xmath164 resulting in an evolving escape fraction that rises to unity at @xmath165 around when the first sources turn on ( _ dotted - line _ ) , and 3 ) a case with @xmath166 resulting in an escape fraction that exceeds unity beyond @xmath167(_dashed - line _ ) . in panels b and c of figure 13 we plot the resultant evolution of the volume weighted h ii ionisation fraction and the evolution of the electron optical depth to electron scattering , @xmath168 , given by : @xmath169 where @xmath170 @xmath76 is the thompson cross section and @xmath171 is the mean electron number density at @xmath172 . in the calculation of @xmath168 we assume that helium is singly ionised to the same degree as the hydrogen component for @xmath173 and doubly ionised everywhere at @xmath149 . it is evident from figure 13 that matching the the optical depth measurement from wmap with the sources studied here would require substantially larger ionising emissivities at high redshifts . in particular , only the case with @xmath166 , which reaches an escape fraction greater that 10 near @xmath165 , is capable of producing the measured electron optical depth . such a large increase in the stellar production rate of ionising photons would require either a greatly enhanced star formation rate at high @xmath174 relative to that obtained by hernquist & springel ( 2003 ) or an imf which becomes more top - heavy with increasing redshift , as might be expected for population iii type stars . in a recent paper ciardi et al . ( 2003a ) argue that a separate population is not necessary if one adopts a top - heavy imf for population ii type stars with an optimistically large value for the escape fraction . however , it is unclear whether the escape fraction adopted in their analysis is meaningful because of their inability to resolve either clumping on sub - grid scales or objects below @xmath175 m@xmath1 . furthermore , in their best - fit model , reionisation is complete by @xmath176 . such an early reionisation seems difficult to reconcile with observations of the gunn - peterson effect measured in the @xmath4 sdss quasar ( becker et al 2001 ) .

this result stems from the fact that low - mass sources preferentially reside in less clumpy environments than their massive counterparts . , reionisation is expected to have occurred between , although the igm remains fairly opaque until . the recent wmap measurements of the electron scattering optical depth ( according to the `` model independent '' analysis of kogut et al . ) such a steep enhancement in the stellar production rate of ionising photons would be consistent with an imf that becomes more and more top heavy with increasing redshift .

we study the reionisation of the universe by stellar sources using a numerical approach that combines fast 3d radiative transfer calculations with high resolution hydrodynamical simulations . by supplementing a _ one - step _ radiative transfer code specifically designed for following ionisation processes with an adaptive ray - tracing algorithm , we are able to significantly speed up the calculations to the point where handling a vast number of sources becomes technically feasible . this allows us to study how dim low - mass sources , excluded in previous investigations owing to computational limitations , affect the morphological evolution of the reionisation process . ionising fluxes for the sources are derived from intrinsic star formation rates computed in the underlying hydrodynamical simulations . analysis of numerically converged results for star formation rates and halo mass functions allows us to assess the consequences of not including low - mass objects and enables us to correct for resolution effects . with these corrections , we are able to reduce the effective mass resolution limit for sources to m , which is roughly an order of magnitude smaller than in previous studies of this kind . our calculations reveal that the process by which ionised regions in the igm percolate is complex and is especially sensitive to the inclusion of dim sources . moreover , we find that given the same level of cosmic star formation , the number of ionising photons required to reionise the universe is significantly overestimated if sources with masses below m are excluded . this result stems from the fact that low - mass sources preferentially reside in less clumpy environments than their massive counterparts . consequently , their exclusion has the net effect of concentrating more of the cosmic ionising radiation in regions which have higher recombination rates . we present the results of our reionisation simulation assuming a range of escape fractions for ionising photons and make statistical comparisons with observational constraints on the neutral fraction of hydrogen at derived from the sdss quasar of becker and coworkers . we find that given the amplitude and form of the underlying star formation predictions , an escape fraction near is most consistent with the observational results . in these models , reionisation is expected to have occurred between , although the igm remains fairly opaque until . our method is also capable of handling the simultaneous reionisation of the helium component in the igm , allowing us to explore the plausibility of the scenario where sources with harder spectra are primarily responsible for reionisation . in this case we find that if the sources responsible for reionising hydrogen by had spectra similar to agns , then the helium component of the igm should have been reionised by . we find that such an early reionisation epoch for helium does not necessarily conflict with observational constraints obtained at , but may be challenged by future observations at higher redshifts . the recent wmap measurements of the electron scattering optical depth ( according to the `` model independent '' analysis of kogut et al . ) appear to be inconsistent with the relatively late onset of reionisation by the normal population ii type stars that we consider . in order to simultaneously match the observations from the sdss quasar and the optical depth measurement from wmap with the sources modeled here , we require an evolving escape fraction that rises from near to at . such a steep enhancement in the stellar production rate of ionising photons would be consistent with an imf that becomes more and more top heavy with increasing redshift .

astro-ph0303098

c

we have utilised a fast radiative transfer code to study the reionisation of the universe by stellar sources . our method has been applied to a high resolution cosmological hydrodynamical simulation which enables us to probe scales previously unresolved in similar type of studies . by supplementing a _ one - step _ radiative transfer code specifically designed for ionisation processes with an adaptive ray - tracing algorithm , we are able to significantly speed up the calculations and are able to handle a large number sources . one of the main goals of this analysis was to study the effect of low - mass sources residing in mini - halos on the overall morphology of the reionisation process . analysis of numerically converged results for star formation rates and halo mass functions in the underlying cosmological simulation allowed us to assess the resolution effects associated with low - mass objects and apply well motivated corrections . with these corrections , we were able to reduce the effective mass resolution limit for source objects to @xmath177 m@xmath1 , which is roughly an order of magnitude smaller than previous studies of this kind . our calculations reveal that the process by which ionised regions in the igm percolate is highly inhomogeneous and especially sensitive to the inclusion of dim sources . more specifically , we find that given the same level of cosmic star formation , the number of ionising photons required to reionise the universe is overestimated by @xmath178 when only objects with masses @xmath179 m@xmath1 are used as source locations . this result stems from the fact that low - mass sources preferentially reside in less clumpy environments than their massive counterparts . consequently , their exclusion has the net effect of concentrating more of the cosmic ionising radiation in regions which have higher recombination rates . our results from the full source list that includes objects down to @xmath0 m@xmath1 appear to show a similar evolution to reionisation for various values of the escape fraction . in all cases , the epoch of reionisation , which we define as occurring when the global volume - weighted ionisation fraction reaches @xmath124 , happens by @xmath125 . this result is consistent with the absence of a gunn - peterson trough in observations of quasars below @xmath3 . to relate our results to these observations , we extract artificial ly@xmath44 absorption spectra from simulation outputs and then derive statistical probabilities for the resultant flux transmittance . our comparisons are based on probability distribution functions involving transmission levels that have been averaged over the same redshift range as the observational measurements . specifically , we focus on the redshift ranges @xmath143 and @xmath144 , corresponding to the regions over which the average transmittance was measured by becker et al . ( 2001 ) in the spectrum of their @xmath4 quasar . we find that given the amplitude and form of the underlying star formation predictions , an escape fraction near @xmath11 is most consistent with the observational results . this leads to a reionisation epoch that is complete by @xmath180 and requires roughly @xmath181 ionising photons per hydrogen atom . in an attempt to explore the implications arising from a scenario where sources with harder spectra such as agns are responsible for hydrogen reionisation , we have analysed a model in which the spectrum of sources is allowed to extend beyond 4 ryd . as a result , sources are able to reionise he ii simultaneously with h i. our analysis shows that helium reionisation occurs near @xmath182 when the spectrum is normalised to deliver the same number of h i ionising photons below 4 ryd as predicted from soft sources with @xmath11 . in a hypothetical case where these hard sources abruptly turn off before the onset of quasars below @xmath156 , we find that an early reionisation of the helium component is not necessarily in conflict with observational constraints of he ii opacities obtained at @xmath9 . this is due to the fact that by @xmath9 , enough he iii recombinations haven taken place to cause ionisation levels to converge to the values governed by quasar emissivities alone . in this paper , we have focused on reionisation by stellar sources similar to those seen in local galaxies ; i.e. population ii type stars . our results show that the star - formation history inferred by springel & hernquist ( 2003a ) based on detailed hydrodynamic simulations can account for the properties of the igm measured from quasars at @xmath183 . to the extent that these comparisons are confirmed by future theoretical and observational studies , our results support the evolutionary history of `` ordinary '' star formation derived analytically by hernquist & springel ( 2003 ) . the recent measurements by the wmap satellite ( kogut et al . 2003 ) indicate that a large fraction of the igm may have been ionised early , by @xmath184 . taken together with the observations of the sdss quasars at @xmath185 , the implications of this result for the evolution of the igm are unclear . cen ( 2002 ) has proposed a scenario in which the universe was reionised early by population iii stars , but then much of the igm recombined once these stars were no longer able to form , and the universe was reionised a second time by the next generation of stars . our results in this paper with a constant escape fraction would describe the second stage of reionisation in this model , provided that the igm mostly recombined at intermediate redshifts . our models with a constant escape fraction predict a relatively late reionisation epoch and can not account for the optical depth inferred by wmap . in an effort to study how this difference could be reconciled in the context of population ii type stars , we employed models with evolving escape fractions and found that if @xmath130 were to rise from @xmath11 near @xmath7 to @xmath12 near @xmath13 , the wmap and sdss constraints can both be satisfied . in this picture , an `` escape fraction '' larger than unity implies that the stellar production rate of ionising photons is much higher than that which would be produced according to our model assumptions . one mechanism that would boost the production rate of ionising photons would be if the imf evolved so that it became more and more top - heavy with increasing redshift . in the future , we will combine simulations of early structure formation with a detailed treatment of radiative transfer to investigate the combined effects of reionisation by both population iii and population ii stars . preliminary work on related issues by yoshida et al . ( 2003 ) indicates that dynamical heating by ongoing accretion limits the rate at which cold , dense clouds of molecular material can form in halos . in principle , this effect will severely reduce the rate of formation of massive stars through molecular cooling , perhaps rendering them unimportant as sources of ionising radiation . dynamical simulations will be required to understand these processes in detail , and to reliably calibrate semi - analytical treatments , in a manner similar to the approach described here . we thank matias zaldarriaga for discussions regarding the statistical significance of observational results in the context of our analysis . we also thank brant robertson for his services as a system administrator for the computer cluster used in this analysis . thanks daniel harvey for many constructive discussions related to this study . this work was supported in part by nsf grants aci 96 - 19019 , ast 98 - 02568 , ast 99 - 00877 , and ast 00 - 71019 and nasa atp grant nag5 - 12140 . the simulations were performed at the center for parallel astrophysical computing at the harvard - smithsonian center for astrophysics .

one - step _ radiative transfer code specifically designed for following ionisation processes with an adaptive ray - tracing algorithm , we are able to significantly speed up the calculations to the point where handling a vast number of sources becomes technically feasible . analysis of numerically converged results for star formation rates and halo mass functions allows us to assess the consequences of not including low - mass objects and enables us to correct for resolution effects . with these corrections , we are able to reduce the effective mass resolution limit for sources to m , which is roughly an order of magnitude smaller than in previous studies of this kind . our calculations reveal that the process by which ionised regions in the igm percolate is complex and is especially sensitive to the inclusion of dim sources . moreover , we find that given the same level of cosmic star formation , the number of ionising photons required to reionise the universe is significantly overestimated if sources with masses below m are excluded . consequently , their exclusion has the net effect of concentrating more of the cosmic ionising radiation in regions which have higher recombination rates . we find that given the amplitude and form of the underlying star formation predictions , an escape fraction near is most consistent with the observational results . in these models we find that such an early reionisation epoch for helium does not necessarily conflict with observational constraints obtained at , but may be challenged by future observations at higher redshifts .

we study the reionisation of the universe by stellar sources using a numerical approach that combines fast 3d radiative transfer calculations with high resolution hydrodynamical simulations . by supplementing a _ one - step _ radiative transfer code specifically designed for following ionisation processes with an adaptive ray - tracing algorithm , we are able to significantly speed up the calculations to the point where handling a vast number of sources becomes technically feasible . this allows us to study how dim low - mass sources , excluded in previous investigations owing to computational limitations , affect the morphological evolution of the reionisation process . ionising fluxes for the sources are derived from intrinsic star formation rates computed in the underlying hydrodynamical simulations . analysis of numerically converged results for star formation rates and halo mass functions allows us to assess the consequences of not including low - mass objects and enables us to correct for resolution effects . with these corrections , we are able to reduce the effective mass resolution limit for sources to m , which is roughly an order of magnitude smaller than in previous studies of this kind . our calculations reveal that the process by which ionised regions in the igm percolate is complex and is especially sensitive to the inclusion of dim sources . moreover , we find that given the same level of cosmic star formation , the number of ionising photons required to reionise the universe is significantly overestimated if sources with masses below m are excluded . this result stems from the fact that low - mass sources preferentially reside in less clumpy environments than their massive counterparts . consequently , their exclusion has the net effect of concentrating more of the cosmic ionising radiation in regions which have higher recombination rates . we present the results of our reionisation simulation assuming a range of escape fractions for ionising photons and make statistical comparisons with observational constraints on the neutral fraction of hydrogen at derived from the sdss quasar of becker and coworkers . we find that given the amplitude and form of the underlying star formation predictions , an escape fraction near is most consistent with the observational results . in these models , reionisation is expected to have occurred between , although the igm remains fairly opaque until . our method is also capable of handling the simultaneous reionisation of the helium component in the igm , allowing us to explore the plausibility of the scenario where sources with harder spectra are primarily responsible for reionisation . in this case we find that if the sources responsible for reionising hydrogen by had spectra similar to agns , then the helium component of the igm should have been reionised by . we find that such an early reionisation epoch for helium does not necessarily conflict with observational constraints obtained at , but may be challenged by future observations at higher redshifts . the recent wmap measurements of the electron scattering optical depth ( according to the `` model independent '' analysis of kogut et al . ) appear to be inconsistent with the relatively late onset of reionisation by the normal population ii type stars that we consider . in order to simultaneously match the observations from the sdss quasar and the optical depth measurement from wmap with the sources modeled here , we require an evolving escape fraction that rises from near to at . such a steep enhancement in the stellar production rate of ionising photons would be consistent with an imf that becomes more and more top heavy with increasing redshift .

1506.05548

i

in this paper we consider a directed random polymer model in random media in two ( one discrete and one continuous ) dimension introduced by oconnell and yor @xcite . for @xmath0 independent one - dimensional standard brownian motions @xmath1 and the parameter @xmath2 representing the inverse temperature , the polymer partition function is defined by @xmath3 here @xmath4 for @xmath5 and @xmath6 represents the energy of the polymer . in the last fifteen years much progress has been made on this oconnell - yor polymer model , by which we can access some explicit information about @xmath7 and the polymer free energy @xmath8 @xcite . the first breakthrough was made in the zero temperature @xmath9 case . in this limit , @xmath10 becomes @xmath11 where @xmath12 is the ground state energy . for @xmath13 , the following relation was established @xcite : @xmath14^n}\prod_{j=1}^ndx_j \cdot p_{\text{gue}}(x_1,\cdots , x_n;t ) , \label{t0result } \\ & p_{\text{gue}}(x_1,\cdots , x_n;t)= \prod_{j=1}^n\frac{e^{-x_j^2/2t}}{j!t^{j-1}\sqrt{2\pi t}}\cdot \prod_{1\le j < k\le n}(x_k - x_j)^2 , \label{t0result0}\end{aligned}\ ] ] where @xmath15 is the probability density function of the eigenvalues in the gaussian unitary ensemble ( gue ) in random matrix theory @xcite . this type of connection of the ground state energy of a directed polymer in random media with random matrix theory was first obtained for a directed random polymer model on a discrete space @xmath16 @xcite by using the robinson - schensted - knuth(rsk ) correspondence . . can be regarded as its continuous analogue . note that is written in the form of a product of the vandermonde determinant @xmath17 . this feature implies that the @xmath18-point correlation function is described by an @xmath19 determinant , i.e. the eigenvalues of the gue are a typical example of the determinantal point processes @xcite . in addition based on this fact and explicit expression of the correlation kernel , we can study the asymptotic behavior of @xmath13 in the limit @xmath20 . in @xcite , it has been shown that under a proper scaling , the limiting distribution of @xmath13 becomes the gue tracy - widom distribution @xcite . in this paper , we provide a representation for a moment generating function of the polymer partition function which holds for arbitrary @xmath21 : @xmath22 = \int_{\r^n}\prod_{j=1}^ndx_j \ , f_f(x_j - u ) \cdot w(x_1,\cdots , x_n;t ) , \label{result } \\ w(x_1,\cdots , x_n;t ) = \prod_{j=1}^n\frac{1}{j!}\cdot \prod_{1\le j < k\le n } ( x_k - x_j)\cdot \det\left(\psi_{k-1}(x_j;t)\right)_{j , k=1}^n , \label{w1}\end{gathered}\ ] ] where @xmath23 is the fermi distribution function and @xmath24 for more details see definition [ defdgue ] and theorem [ thmmain ] below . this is a simple generalization of to the case of finite temperature . we easily find that it recovers in the zero - temperature limit ( @xmath25 ) . note that the function @xmath26 is also written as a product of two determinants and thus retains the determinantal structure in . in most cases , to find a finite temperature generalization of results for zero - temperature case is highly nontrivial and in fact often impossible . but for the oconnell - yor polymer model and a few related models , rich mathematical structures have been discovered for finite temperature and the studies on this topic entered a new stage @xcite . in @xcite , oconnell found a connection to the quantum toda lattice , and based on the developments in its studies and the geometric rsk correspondence , it was revealed that the law of the free energy @xmath27 is expressed as @xmath28}dx_1\int_{\r^{n-1}}\prod_{j=2}^{n } dx_{j}\cdot m(x_1,\cdots , x_n;t ) . \label{whit0}\end{aligned}\ ] ] here the probability measure @xmath29 , which is called the _ whittaker measure _ , is defined by the density function @xmath30 in terms of the whittaker function @xmath31 ( for the definition , see @xcite ) and the sklyanin measure @xmath32 ( see below ) as follows , @xmath33 where @xmath34 represents @xmath35 . in contrast to , the density function is not known to be expressed as a product of determinants and the process associated with does not seem to be determinantal . nevertheless some determinantal formulas for the oconnell - yor polymer have been found : first in @xcite , oconnell showed a determinantal representation for the moment generating function ( lhs of ) in terms of the sklyanin measure . ( see below . ) next in @xcite , borodin and corwin obtained a fredholm determinant representation for the same moment generating function ( see below ) . a direct proof of the equivalence between the two determinantal expressions was given in @xcite . in @xcite , by considering its continuous limit , the authors also obtained an explicit representation of the free energy distribution for the directed random polymer in two continuous dimension described by stochastic heat equation ( she ) @xcite . the distribution in this limit , which describes the universal crossover between the kardar - parisi - zhang ( kpz ) and the edwards - wilkinson universality class , was first obtained in @xcite and can be interpreted also as the height distribution for the kpz equation @xcite . furthermore in @xcite , they consider not only the oconnell - yor model but a class of stochastic processes having the similar fredholm determinant expressions , the macdonald processes , the probability measures on a sequence of partitions which are written in terms of the macdonald symmetric functions and include the whittaker measure defined by as a limiting case . the purpose of this paper is to investigate further the mechanism of appearance of such determinantal structures and is the central formula in our study . although @xmath36 defined by is not a probability measure but a signed measure except when @xmath25 , a remarkable feature of this measure is that it is _ determinantal _ for arbitrary @xmath37 contrary to the whittaker measure . this determinantal structure allows us to use the conventional techniques developed in the random matrix theory and thus from the relation we readily get a fredholm determinant representation with a kernel using biorthogonal functions which is regarded as a generalization of the kernel with the hermite polynomials for the gue . in , the parameter @xmath37 , which originally represents the inverse temperature in the polymer model appears in the fermi distribution function @xmath38 with the chemical potential @xmath39 as well as @xmath40 in rhs . this fact with the determinantal structure suggests that the rhs might have something to do with the free fermions at a finite temperature . related to this , a curious relation of the height of the kpz equation with fermions has been discussed in @xcite . for establishing the relation , we introduce a measure on a larger space @xmath41 . by integrating the measure in two different ways , we get its two marginal weights . in one formula appears a determinant which solves the @xmath0 dimensional diffusion equation with some condition ( see , , and ) and the other one with a symmetrization is exactly the rhs of . the relation follows immediately from the equivalence of these two expressions . our approach is similar to the one by warren @xcite for getting the relation . actually in the zero - temperature limit @xmath25 , we see that the integration of the measure is written in terms of the probability measure introduced in @xcite , which describes the positions of the reflected brownian particles on the gelfand - tsetlin cone . note that the macdonald processes ( especially the whittaker process in our case ) @xcite are also another generalizations of @xcite . although the whittaker process has rich integrable properties , they do not inherit the determinantal structure of @xcite . on the other hand , our measure is described without using the whittaker functions and keeps the determinantal structure . furthermore combining with the fact that the quantity can be rewritten as the fredholm determinant found in @xcite ( corollary [ corfred ] and proposition [ propfred2 ] below ) , our approach can be considered as another proof of the equivalence between and in @xcite . one feature of our proof is to bring to light the larger determinantal structure behind the two relations . this paper is organized as follows . in the next section , after stating the definition of a determinantal measure , we give our main result , theorem [ thmmain ] and its proof . the proof consists of two major steps : we first introduce in lemma [ lemoc ] a determinantal representation for the moment generating function which is a deformed version of the representation in @xcite . next we introduce another determinantal measure on larger space @xmath41 and then we find two relations about its integrations which play a key role in deriving our main result . in sec . 3 we show that this approach can be considered as an extension of the one in @xcite . in sec.4 . , we consider the fredholm determinant formula with biorthogonal kernel obtained by applying conventional random matrix techniques to our main result . the scaling limit to the kpz equation is discussed in sec.5 . we check that our kernel goes to the one obtained in the studies of the kpz equation . a concluding remark is given in the last section .

we study the semi - discrete directed random polymer model introduced by oconnell and yor . this measure is an extension of the probability measure of the eigenvalues for the gaussian unitary ensemble ( gue ) in random matrix theory . to establish the relation , we introduce another determinantal measure on larger degrees of freedom and consider its few properties , from which the representation above follows immediately

we study the semi - discrete directed random polymer model introduced by oconnell and yor . we obtain a representation for the moment generating function of the polymer partition function in terms of a determinantal measure . this measure is an extension of the probability measure of the eigenvalues for the gaussian unitary ensemble ( gue ) in random matrix theory . to establish the relation , we introduce another determinantal measure on larger degrees of freedom and consider its few properties , from which the representation above follows immediately

1003.0056

c

we have studied the statistical nature of primordial fluctuations from an anisotropic inflation which is realized by a vector field coupled to an inflaton . first , we have classified metric fluctuations according to the 2-dimensional rotational symmetry . to choose a convenient gauge in an anisotropic universe , we have started from the flat slicing gauge in an isotropic universe and made an appropriate gauge transformation to get a canonical gauge in an anisotropic universe . this gauge choice has made the subsequent analysis and the interpretation of the variables easier . using the canonical gauge , we have revealed the structure of the couplings between curvature perturbations , vector waves , and gravitational waves . we found that there are two sources for anisotropy , i.e. the anisotropy due to the anisotropic expansion of the universe and that due to the anisotropic couplings among variables . it turned out that the latter effect is dominant . we have numerically obtained power spectra . we also presented analytical formula using in - in formalism . since the coupling between the curvature perturbations and vector waves is the strongest one , the anisotropy in the curvature perturbations is larger than that in gravitational waves . more interestingly , we found the cross correlation between curvature perturbations and gravitational waves which is peculiar to anisotropic inflation . we also found the linear polarization of gravitational waves . although there are several mechanism to produce circular polarization in the primordial gravitational waves @xcite , this is the first example which realized the linear polarization in the primordial gravitational waves . we have only considered power spectrum for simplicity . however , as is pointed out in the paper @xcite , the statistical anisotropy could appear in the non - gaussianity strongly and modify the shape of the bispectrum and trispectrum . hence , it is interesting to study non - gaussianity in anisotropic inflation models . we can extend anisotropic inflation in various ways . although we have investigated a chaotic inflation in this paper , it is easy to extend the analysis to other inflation models . it is possible to incorporate multi - vector fields . from the string theory point of view , it is intriguing to consider anti - symmetric tensor field . it is also interesting to consider other bianchi type models @xcite in the context of anisotropic inflation . sk would like to thank the yitp members in kyoto for warm hospitality . a part of this work was done while sk was visiting yitp supported by jsps grant - in - aid for scientific research ( a ) 21244033 . sk is supported by an stfc rolling grant . js is supported by the japan - u.k . research cooperative program , grant - in - aid for scientific research fund of the ministry of education , science and culture of japan no.18540262 , grant - in - aid for scientific research on innovative area no.21111006 and the grant - in - aid for the global coe program the next generation of physics , spun from universality and emergence " .

we study the statistical nature of primordial fluctuations from an anisotropic inflation which is realized by a vector field coupled to an inflaton . using the canonical gauge , we reveal the structure of the couplings between curvature perturbations , vector waves , and gravitational waves . we identify two sources of anisotropy , i.e. the anisotropy due to the anisotropic expansion of the universe and that due to the anisotropic couplings among variables . it turns out that the latter effect is dominant . since the coupling between the curvature perturbations and vector waves is the strongest one , the statistical anisotropy in the curvature perturbations is larger than that in gravitational waves .

we study the statistical nature of primordial fluctuations from an anisotropic inflation which is realized by a vector field coupled to an inflaton . we find a suitable gauge , which we call the canonical gauge , for anisotropic inflation by generalizing the flat slicing gauge in conventional isotropic inflation . using the canonical gauge , we reveal the structure of the couplings between curvature perturbations , vector waves , and gravitational waves . we identify two sources of anisotropy , i.e. the anisotropy due to the anisotropic expansion of the universe and that due to the anisotropic couplings among variables . it turns out that the latter effect is dominant . since the coupling between the curvature perturbations and vector waves is the strongest one , the statistical anisotropy in the curvature perturbations is larger than that in gravitational waves . we find the cross correlation between the curvature perturbations and gravitational waves which never occurs in conventional inflation . we also find the linear polarization of gravitational waves . finally , we discuss cosmological implication of our results .

1411.5222

i

surface tension of ionic solutions strongly depends on their salt composition and , in general , increases with ionic strength for low salt concentrations @xcite . wagner @xcite was the first to connect this finding with the dielectric discontinuity at the air / water interface , suggesting that dielectric image charge interactions could explain the increase in surface tension . this suggestion was implemented in the pioneering work of onsager and samaras ( os ) , who used the debye hckel theory for electrolytes @xcite . the os result presents a universal limiting law for the excess surface tension . it depends on the dielectric mismatch at the interface and on the bulk salt concentration @xcite , and implies that the increase in the surface tension would be independent of the ion type . this simplified observation , however , turned out to be violated in many experimental situations @xcite , and led over the years to numerous investigations of non - electrostatic _ ion - specific _ interactions between ions and surfaces @xcite , and their role in modifying the surface tension of ionic solutions @xcite . in fact , ion - specific effects date back to the late 19th century , when hofmeister @xcite measured the amount of protein precipitation from solution in presence of various salts , and found a universal series of ionic activity . the same _ hofmeister series _ emerges in a large variety of experiments in chemical and biological systems @xcite . among others , they include forces between mica or silica surfaces @xcite , osmotic pressure in the presence of ( bio)macromolecules @xcite , and more specifically , measurements of surface tension at the air / water and oil / water interfaces @xcite . for simple monovalent salts , the surface tension ( in particular at the air / water interface ) was experimentally found @xcite to depend strongly on the type of anion , while the dependence on the cation type is much weaker . this finding is consistent with the fact that anion concentration at the air / water interface exceeds that of cations . furthermore , for halides @xcite , the lighter ions lead to a larger excess in surface tension in a sequence that precisely corresponds to the _ reverse _ of the hofmeister series . discrepancies between the os predictions and the observed ion - specific surface tension motivated numerous attempts to modify the original os model . here , we limit ourselves to briefly review some of the more recent works on surface tension of ionic solutions @xcite , which are directly related to the present study . dean and horgan @xcite calculated the surface tension of ionic solutions to first order in a systematic cumulant expansion , where the zeroth order is equivalent to the debye - hckel ( dh ) linear theory @xcite . the specific ion - surface interactions are modeled via an ionic exclusion ( stern ) layer of finite thickness . thus , the interaction of ions with the interface contains only a length scale without any energy scale . this model is solved via a formal field - theoretic representation of the partition function . the os result is reproduced exactly , and ion - specific effects are described in terms of the thickness of the salt - exclusion layer . in yet a separate study by the same authors @xcite , a system of two interacting surfaces including a cation - specific short - range surface interaction was addressed . the consequences of this cation - specific interaction on determining the effective charge , as well as the disjoining pressure as a function of the separation between the surfaces were studied in detail . in a series of papers levin and coworkers @xcite calculated the solvation free - energy of polarizable ions at air / water and oil / water interfaces . their mean - field theory ( mft ) modifies the poisson - boltzmann ( pb ) theory by adding an ion - surface interaction potential . the modified potential is based on several ion - surface interaction terms that are added in an _ ad hoc _ way to the pb equation . these interactions include the image charge interaction , stern exclusion layer , ionic cavitation energy and ionic polarizability . while the additional interaction terms may represent some physical mechanisms for ion - specific interaction with the surface , one can not , in general , simply add such terms to the mft potential in a self - consistent way . these terms , which are sometimes mutually exclusive , are neither completely independent nor can they be obtained from a mft formulation @xcite . computing numerically the surface tension for an homologous series of sodium salts , levin and co - workers fitted their predictions to the hofmeister series . they used the hydrated size of the sodium ion as a single fit parameter . furthermore , the surface tension at the oil / water interface was fitted for another series of potassium salts @xcite . in order to apply their theory to the different interface , a second fit parameter was used to account for the dispersion forces at the oil / water interface . a different line of reasoning was initiated by netz and coworkers @xcite , who calculated the surface tension for both charged and neutral surfaces using a two - scale ( atomistic and continuum ) modeling approach . explicit solvent - atomistic molecular - dynamics ( md ) simulations furnished non - electrostatic ion - specific potentials of mean force . these interaction potentials were then added to the pb theory that provides the electrostatic part of the potential of mean force . within this framework , netz and coworkers were able to show that the polarity of the surface may reverse the order of the hofmeister series . the fitted results agree well with experiments performed on hydrophobic and hydrophilic surfaces . although many works @xcite tried to generalize the seminal os theory , it nevertheless remains largely misunderstood what is the accurate theoretical framework of the os theory . the os theory makes use of the ( linearized ) pb equation in the presence of a planar dielectric boundary , to obtain the one - dimensional image charge potential of mean force , [ see eq . ( 4 ) in ref . ] . it should be stated that the latter is _ not _ a solution of the one - dimensional pb equation . in fact , the pb solution can not describe any image charge effects on its own . this subtle , yet essential point , gets often irreparably lost when generalizations of the os theory are attempted based on elaborate decorations of the pb equation itself . if the os theory is not simply a redressed version of the pb theory , then what exactly is the relation between them ? while the os theory can not be obtained from the mean - field theory , it is deduced from the thermodynamic fluctuations of the instantaneous electric fields around the pb solution @xcite . the free energy is expanded in a loop expansion and only the one - loop correction to the mft result is retained . while the detailed formal derivation ( as shown below ) is complex , we believe that its physical basis is quite simple and straightforward . the os result does not generalize the pb equation to include image charge effects at an interface between two dielectric media , but rather it solves the problem on a higher level of approximation by going beyond the mft level . we consider this conceptual clarification to have large importance on generalizations of the os theory itself . in this paper we introduce two important modifications , relevant to the calculations of the surface tension of electrolytes . first , we demonstrate that the os contribution is effectively fluctuational in nature , and follows from the one - loop expansion of the coulomb partition function around the mean - field solution . second , we propose a phenomenological approach that consistently describes ion - interface interactions in the form of a coupling term in the free energy . this new formulation allows us to obtain a simple analytical theory that reunites the os pioneering result , which does not agree with experimental data , with the ionic specificity of the hofmeister series . we take ionic specificity into account through the ionic size and an ion - surface interaction . each ionic species is characterized by a phenomenological adhesivity parameter @xcite . specifically , short - range non - electrostatic effects such as the ion chemical nature , size and polarizability , as well as the preferential ion - solvent interaction @xcite , are introduced by adding one phenomenological parameter to the free energy . this allows us to obtain a modified pb mean - field theory and to evaluate the contribution of fluctuations ( beyond mean - field ) to the surface tension . the latter includes the dielectric image charge effects ( os ) , as well as the couplings between image charge effects and surface - specific interactions . our analytical surface tension prediction fits well a variety of interfacial tension data at the air / water and oil / water interfaces . using one fit parameter per electrolyte it reproduces the reversed hofmeister series for several types of monovalent anions . the outline of this paper is as follows . in sec . ii , we present the model and a general derivation of the grand - potential to the one - loop order . in sec . iii we find the free energy and treat the spurious divergencies of our model , while in sec . iv we derive an analytical expression for the surface tension . finally , a comparison of our results with experimental data ( sec . v ) and some concluding remarks ( sec . vi ) are presented .

the free energy is expanded to first - order in a loop expansion beyond the mean - field result . we calculate the excess surface tension and obtain analytical predictions that reunite the onsager - samaras pioneering result ( which does not agree with experimental data ) , with the ionic specificity of the hofmeister series .

we study the surface tension of ionic solutions at air / water and oil / water interfaces . by using field - theoretical methods and including a finite proximal surface - region with ionic - specific interactions . the free energy is expanded to first - order in a loop expansion beyond the mean - field result . we calculate the excess surface tension and obtain analytical predictions that reunite the onsager - samaras pioneering result ( which does not agree with experimental data ) , with the ionic specificity of the hofmeister series . we derive analytically the surface - tension dependence on the ionic strength , ionic size and ion - surface interaction , and show consequently that the onsager - samaras result is consistent with the one - loop correction beyond the mean - field result . our theory fits well a wide range of salt concentrations for different monovalent ions using one fit parameter per electrolyte , and reproduces the reverse hofmeister series for anions at the air / water and oil / water interfaces .

1411.5222

c

our work presents a general self - consistent theory for calculating free energies up to one - loop order for ionic solutions with a surface proximal layer of finite width . in this layer , we consider a slowly varying ion - specific surface potential . the loop expansion we use can be computed systematically to higher orders , and re - summing the loop terms is actually equivalent to the cumulant expansion done in refs . . the calculation of the excess surface tension , @xmath195 , is based on the free - energy difference between a system with an air / water interface on one hand , and two semi - infinite systems ( electrolyte and air ) with no interface on the other hand . the same calculation can be applied to other liquid / liquid interfaces such as oil / water , simply by using @xmath8 as the dielectric constant of the oil instead of air . this calculation method is equivalent to the gibbs absorbtion isotherm method @xcite , but it is mathematically more accessible as it avoids the explicit calculation of the ionic densities . it is also possible to use the grand potential instead of the free - energy , but the latter does not simplify the calculation , because one has to consider the one - loop correction to the fugacities ( appendix c ) . we have computed the linearized mft electrostatic potential as well as the mft and one - loop free energies , utilizing the _ secular determinant _ method , where we have extended the secular determinant method of ref . for the three boundaries as in our system . the one - loop excess surface tension , @xmath212 , is calculated and the os surface tension result was naturally recovered and extended . in fact , we showed that the os result is obtained by considering the thermodynamic fluctuations of the electrostatic potential around its mft solution , while the volume fluctuations recovered the known dh correction to the mft free energy . our surface - tension result is analytical and interpolates between several known limits : the result of ref . for @xmath268 , the os one for @xmath221 or @xmath269 , and a stern layer for @xmath270 . a wide variety of monovalent ions at the air / water and oil / water interfaces were fitted by our model , all taken on a common and unified ground . within some approximations we obtain an analytical dependence of the excess surface tension on the salt concentration . the fits for @xmath195 agree well with experiments and show clearly the reversed hofmeister series ( f@xmath271cl@xmath271br@xmath271i@xmath272 ) both at the air / water and oil / water interfaces . the various fits reveal an even more extended ( hofmeister ) series : @xmath263 . in ref . a different series was obtained for anions : @xmath273 . at present , it is not clear which of these two predictions is more accurate . other experimental measurements , such as surface potential , might shed light on this discrepancy . we intend to further investigate this issue in the future by calculating the surface potential to one - loop order . in the weak - coupling linear regime , where our theory is valid , fluctuations dominate over the mft contribution , and the cation and anion adhesivities are roughly equal , @xmath274 . this then leads to a small mft contribution , while the one - loop contribution has terms proportional to @xmath275 and independent on @xmath58 . these terms arise from the image charge interaction and , indeed , give the os result with an ionic - size correction ( minimal distance of approach between ions ) . two important limitations of our theory are related to the ion finite size , and to the linearization of the mft electrostatic equations . for large adhesivities , where the ion density on the surface is high and does not correspond to the dilute solution limit , our theory is expected to fail . instead , other theories , such as the modified pb ones @xcite , can be utilized as they take into account explicitly the ion finite size . considering , for example , the surface tension of acids such as hcl and hclo@xmath276 , the fitted @xmath258 s from our theory are found to be rather large ( see table ii ) and are not expected to be as reliable , for the reasons mentioned above . furthermore , when @xmath277 , the linear approximation fails and one should solve the full mft electrostatic equation . note that the image charge term ( or other ion - interface interactions ) can not be simply added in the boltzmann weight factor ( potential of mean force ) . since the pb equation is a mft equation that follows from a certain free - energy minimization , a consistent way to generalize it should be based on an augmented free - energy functional , which then gives a generalized pb equation . in this way , double - counting of different electrostatic contributions is prevented , and remedy a common ambiguity where the image charge or other ion - interface interactions are added to the pb equation in an _ ad hoc _ fashion . our model provides a self - consistent way to calculate the fluctuations around the mft free energy . the image charge contribution naturally arises from the fluctuations hence , @xmath58 originate only from solvent ( short - range ) interactions and the problem of double counting is avoided . an augmented free energy ( `` action '' ) , eq . ( [ m9 ] ) , is obtained and its minimization indeed leads to a modified pb equation . the microscopic origin of the adhesivity is still not very well understood . nevertheless , several possibilities have been suggested recently . for the special case of silica / water interface @xcite the orientation of water molecules in the vicinity of the interface was proposed to lead to changes in the hydrogen bond strength at the interface . this surface effect can be identified as a possible microscopic source of @xmath58 , whose value is proportional to the difference in free energy between a single ion solvated in the bulk as compared to its partially solvated state at the surface . another possible origin for the adhesivity is the cavitational energy @xcite . the ion interferes with the water structure by breaking hydrogen bonds . this costs energy , leading to an effective attraction of the ion to the interface . other effects such as the ion polarizability @xcite or dispersion forces @xcite might also play a role in the ion - surface interaction . since all proposed origins of the adhesivity are short range ( their range is comparable to the ionic radius ) and our theory is a coarse - grained one , it is appropriate to average over their degrees of freedom and obtain @xmath58 , whose magnitude is reasonably obtained from the fits to the experimental data . our model can be applicable for many systems which exhibit a spatial region with ionic - specific slowly varying interactions . examples of such systems are , polymer - brushes densely attached to a surface or polyelectrolyte gels . our model is ionic - specific and can be used to obtain simple and analytical predictions for such complex systems . our model is applicable to many systems that exhibit a spatial region characterized by ionic - specific and slowly varying interactions . two examples of such systems are a densely layer of polymer brushes that are attached to a solid surface , or a polyelectrolyte gel occupying a finite volume of solution . our model is ionic - specific and can be used to obtain simple and analytical predictions also for such complex inhomogeneous systems . this work was supported in part by the israel science foundation ( isf ) under grant no . 438/12 and the us - israel binational science foundation ( bsf ) under grant no . we thank m. biesheuvel , y. ben - yakar , a. cohen , h. diamant , r. netz , h. orland and y. tsori for numerous suggestions , and d. frydel , y. levin and w. kunz for useful discussions . one of us ( rp ) would like to thank tel aviv university for its hospitality during his stay there , and acknowledges the support of the arrs through grant p1 - 0055 .

we study the surface tension of ionic solutions at air / water and oil / water interfaces . by using field - theoretical methods and including a finite proximal surface - region with ionic - specific interactions . our theory fits well a wide range of salt concentrations for different monovalent ions using one fit parameter per electrolyte , and reproduces the reverse hofmeister series for anions at the air / water and oil / water interfaces .

we study the surface tension of ionic solutions at air / water and oil / water interfaces . by using field - theoretical methods and including a finite proximal surface - region with ionic - specific interactions . the free energy is expanded to first - order in a loop expansion beyond the mean - field result . we calculate the excess surface tension and obtain analytical predictions that reunite the onsager - samaras pioneering result ( which does not agree with experimental data ) , with the ionic specificity of the hofmeister series . we derive analytically the surface - tension dependence on the ionic strength , ionic size and ion - surface interaction , and show consequently that the onsager - samaras result is consistent with the one - loop correction beyond the mean - field result . our theory fits well a wide range of salt concentrations for different monovalent ions using one fit parameter per electrolyte , and reproduces the reverse hofmeister series for anions at the air / water and oil / water interfaces .

1104.3148

c

we have studied the complexity of the two - variable fragments of dependence logic and independence - friendly logic . we have shown ( theorem [ dtwo nexptime ] ) that both the satisfiablity and finite satisfiability problems for @xmath34are decidable , @xmath2-complete to be exact . we have also proved ( theorems [ iftwo sat complexity ] and [ iftwo finsat complexity ] ) that both problems are undecidable for @xmath35 ; the satisfiability and finite satisfiabity problems for @xmath35are @xmath3-complete and @xmath4-complete , respectively . while the full logics @xmath18and @xmath17are equivalent over sentences , we have shown that the finite variable variants @xmath34and @xmath35are not , the latter being more expressive . this was obtained as a by - product of the deeper result concerning the decidability barrier between these two logics . there are many open questions related to these logics . we conclude with two of them : 1 . what is the complexity of the validity problems of @xmath34and @xmath35 ? is it possible to define np - complete problems in @xmath34or in @xmath35 ?

we study the two - variable fragments and of dependence logic and independence - friendly logic . we consider the satisfiability and finite satisfiability problems of these logics and show that for , both problems are-complete , whereas for , the problems areand-complete , respectively . we also show that is strictly less expressive than and that already in , equicardinality of two unary predicates and infinity can be expressed ( the latter in the presence of a constant symbol ) .

we study the two - variable fragments and of dependence logic and independence - friendly logic . we consider the satisfiability and finite satisfiability problems of these logics and show that for , both problems are-complete , whereas for , the problems areand-complete , respectively . we also show that is strictly less expressive than and that already in , equicardinality of two unary predicates and infinity can be expressed ( the latter in the presence of a constant symbol ) . this is an extended version of a publication in the proceedings of the 26th annual ieee symposium on logic in computer science ( lics 2011 ) . dependence logic , independence - friendly logic , two - variable logic , decidability , complexity , satisfiability , expressivity f.4.1 computability theory , model theory ; f.1.3 reducibility and completeness

1112.5807

i

nanoscale electromechanical systems ( nems ) convert small amplitude mechanical motion into measurable electrical currents @xcite . devices based on nems have found applications as sensitive detectors of mass @xcite , force @xcite and electrical charge @xcite . considerable research efforts have been dedicated to improving detection sensitivity by fabricating devices with higher resonance frequencies , lower damping rates ( high quality factors ) and larger coupling between electrical and mechanical degrees of freedom . the problem of detecting the quantum state of a macroscopic mechanical resonator gave rise to several measuring schemes , proposed @xcite as well as realized @xcite . continuous improvements in device fabrication in combination with new techniques for cooling mechanical motion @xcite have pushed the sensitivity threshold to the quantum limit @xcite . the use of superconducting devices , in particular , superconducting qubits to detect and control the mechanical motion is in focus of modern research @xcite . it gives rise to a growing interest in techniques of coupling nems to superconducting circuits . superconducting nano - devices frequently use coulomb blockade that makes their transport properties sensitive to the gate voltages . the same gate voltage can be used to excite the mechanical motion which is detected from the change of d.c . transport properties of the device @xcite . without superconductors , this scheme has been successfully realized for a metallic single - electron transistor @xcite and for a coulomb - blockaded quantum dot in an ultra - clean carbon nanotube ( cnt ) @xcite . the results revealed high resonance frequencies , reaching gigahertzs , and unprecedented quality factors of the order of @xmath0 . these devices can be made superconducting by connecting them to superconducting leads and providing sufficiently large coupling between the states of the lead and device . we have successfully realized josephson junctions based on the ultra - clean cnt . the supercurrent observed demonstrates a pronounced gate - voltage sensitivity that indicates a well - developed coulomb blockade @xcite . a very interesting proposal that combines josephson dynamics and mechanical resonator has been recently put forward by gothenburg collaboration @xcite . the authors consider an ideal ballistic cnt between two superconducting leads biased at voltage @xmath1 . owing to josephson relation , the current in the nanotube oscillates at frequency @xmath2 . the authors notice that in external magnetic field this gives rise to an oscillating lorentz force . if the frequency matches the frequency of the mechanical resonator , the force excites mechanical motion which rectifies the josephson current enabling the observation of the effect in d.c . electric response of the junction . one would observe a narrow current peak in @xmath3 characteristics of the device . the same mechanism is responsible for fiske steps @xcite : the difference is that in fiske experiments the resonance is electrical rather than mechanical . this provides us motivation for the present theoretical study where we address superconducting nems where a mechanical resonator is integrated with a superconducting circuit element , a josephson junction . the focus of this work is to describe the techniques of driving and detecting mechanical motion using superconducting current and to explore the coupled dynamics of the oscillator displacement and superconducting phase difference . while the study is general for any type of oscillating nanowire , for the purpose of illustration and for concrete estimations we refer to one of the more successful nems devices : a suspended metallic cnt resonator connected to the metallic leads . we have fabricated and studied such devices . studies of cnt josephson junctions have shown that their josephson energy can be modulated by the gate - induced charge on the cnt . in this paper , we have shown that this gives rise to a different and generally more important mechanism of mechanical driving than that considered in @xcite . we consider all possible non - linearities in coupled josephson - mechanical dynamics . we have shown that in our situation the most important one is the mechanical non - linearity . we provide detailed estimations of the displacement , force and electrical current scales involved . it is our conclusion that the mechanical response in our devices should be small modifying the current on the scale of @xmath4 of the critical current ( in this case , the critical current is estimated from the product of the junction conductance and superconducting energy gap and can exceed the experimentally measured switching current by two orders of magnitude ) . in the present paper , we have investigated in detail the mechanical response under conditions of phase bias and d.c . voltage bias . we dedicated special attention to the dynamics in the presence of external a.c . drive , in the regime where josephson junction exhibits well - developed shapiro steps @xcite . one of the motivations of the research on shapiro steps was the better synchronization conditions in comparison with d.c . voltage bias . the reason for this is that the big quality factor @xmath5 of the nanomechanical resonance results in a narrow fiske - like current peak . its width in voltage can be estimated as @xmath6 . this imposes a severe limitation on voltage noise @xmath7 : to resolve the peak one must achieve @xmath8 . this may be challenging under realistic experimental circumstances . there is a way out : the voltage can be synchronized with the frequency of external irradiation . this effectively reduces the noise . from the other hand , the external irradiation can excite the mechanical motion by itself . the resulting complicated dynamics should be augmented with non - linear effects . in this paper we present our theoretical results concerning the mechanical response manifested in d.c . electric response of the junction , in particular , the modification of the width and position of shapiro steps , and analyze a variety of distinct regimes that differ in relative and absolute magnitudes of driving forces and resonant conditions , and are manifested in distinct lineshapes of the resonant response . our preliminary experimental results show corresponding features . they will be presented elsewhere @xcite upon completion of detailed analysis and comparison with our theoretical findings . the paper is organized as follows . in section [ prop ] we describe the setup concentrating separately on the electrical and mechanical properties in the corresponding subsections and introduce the notations . in section [ fe ] we analyze in detail the mechanisms of coupling between mechanical and electrical degrees of freedom . we consider electrostatic energy , derive and explicate the concept of _ josephson mechanical force_. we present a rather involved analysis of competing non - linearities and conclude that for our devices the mechanical non - linearities dominate and the josephson force can be strong enough to induce the non - linear mechanical response . we give the workflow we use to compute the mechanical response . in the end of the section , we specify a set of concrete parameters based on experimentally measured values . in section [ sec : phasebias ] we address the phase bias conditions . we reveal that the _ phase - dependent shift _ of the resonant frequency can be quite noticeable and discuss lorentz - like and fano - like frequency dependences of the mechanical response . in section [ sec : dcvoltage ] we discuss the fiske - type mechanical response at josephson frequency matching the resonance frequency of the mechanical resonator , or an integer fraction of this frequency by higher harmonics , and give the estimations of the effect . we shortly discuss the parametric excitation . in section [ sec : shapiro - res ] we study the mechanical response at the shapiro steps in the regime where the a.c . driving frequency matches the resonant frequency and present the mechanisms and peculiarities of this response . in section [ sec : shapiro - non - res ] we consider the non - resonant driving that appears to efficiently excite the mechanical motion in the regime of shapiro steps . our concluding remarks are presented in section [ concl ] .

we study theoretically dynamics in a josephson junction coupled to a mechanical resonator looking at the signatures of the resonance in d.c . electrical response of the junction . a nearby gate electrode can be used to tune the junction parameters and to excite mechanical motion . we augment theoretical estimations with the values of setup parameters measured in the samples fabricated . the force can excite the resonant mode provided the superconducting current in the junction has oscillating components with a frequency matching the resonant frequency of the mechanical resonator . we compute the mechanical response ( the effect of mechanical motion ) in the regime of phase bias and d.c . voltage bias . we thoroughly investigate the regime of combined a.c . and d.c . bias where shapiro steps are developed and reveal several distinct regimes characteristic for this effect . our results can be immediately applied in the context of experimental detection of the mechanical motion in realistic superconducting nano - mechanical devices .

we study theoretically dynamics in a josephson junction coupled to a mechanical resonator looking at the signatures of the resonance in d.c . electrical response of the junction . such a system can be realized experimentally as a suspended ultra - clean carbon nanotube brought in contact with two superconducting leads . a nearby gate electrode can be used to tune the junction parameters and to excite mechanical motion . we augment theoretical estimations with the values of setup parameters measured in the samples fabricated . we show that charging effects in the junction give rise to a mechanical force that depends on the superconducting phase difference . the force can excite the resonant mode provided the superconducting current in the junction has oscillating components with a frequency matching the resonant frequency of the mechanical resonator . we develop a model that encompasses the coupling of electrical and mechanical dynamics . we compute the mechanical response ( the effect of mechanical motion ) in the regime of phase bias and d.c . voltage bias . we thoroughly investigate the regime of combined a.c . and d.c . bias where shapiro steps are developed and reveal several distinct regimes characteristic for this effect . our results can be immediately applied in the context of experimental detection of the mechanical motion in realistic superconducting nano - mechanical devices .

1201.2256

i

let @xmath0 be a stationary stochastic process of @xmath4-valued random variables with marginal distribution @xmath5 . we denote the empirical measure of order @xmath6 by @xmath7 . the classical empirical process is defined by @xmath8)-\mu((-\infty , t]))$ ] , @xmath9 . in the case of i.i.d . processes , the limit behavior of the empirical process was first investigated by @xcite , who proved that @xmath10 converges weakly to a brownian bridge process . this result , known as donsker s empirical process central limit theorem , confirmed a conjecture of @xcite who had observed that certain functionals of the empirical process converge in distribution towards the corresponding functionals of a brownian bridge . donsker s empirical process clt has been generalized to dependent data by many authors . one of the earliest results is @xcite , who considered functions of mixing processes , with an application to the empirical distribution of the remainders in a continued fraction expansion . empirical processes play a very important role in large sample statistical inference . many statistical estimators and test statistics can be expressed as functionals of the empirical distribution . as a result , their asymptotic distribution can often be derived from empirical process limit theorems , combined with the continuous mapping theorem or a functional delta method . a well - known example is the kolmogorov - smirnov goodness - of - fit test , which uses the test statistic @xmath11)- \mu_0((-\infty , t])|$ ] in order to test the null hypothesis that @xmath12 is the marginal distribution of @xmath13 . under the null hypothesis , the limit distribution of @xmath14 is given by the supremum of the gaussian limit of the empirical process . another example are von - mises - statistics , also known as v - statistics . these are defined as @xmath15 , where @xmath16 is a symmetric kernel function . specific examples include the sample variance and gini s mean difference , where the kernel functions are given by @xmath17 and @xmath18 , respectively . @xmath19-statistics can be expressed as integrals with respect to the empirical distribution function , namely @xmath20 . the asymptotic distribution of @xmath21 can then be derived via a functional delta method from an empirical process central limit theorem ; see e.g. @xcite for some recent results . empirical process clts for @xmath22-valued i.i.d . data @xmath0 have first been studied by @xcite , @xcite , @xcite and @xcite . these authors consider the classical @xmath23-dimensional empirical process @xmath24 ) -\mu((-\infty , t]))$ ] , where @xmath25=\{x\in { { \mathbb r}}^d : x_1\leq t_1,\ldots , x_d\leq t_d\}$ ] , @xmath26 , denotes the semi - infinite rectangle in @xmath22 . @xcite , @xcite and @xcite studied weak convergence of the multivariate empirical process in the case of mixing data . @xcite initiated the study of empirical processes indexed by classes of sets , or more generally by classes of functions . this approach allows the study of empirical processes for very general data , not necessarily having values in euclidean space . clts for empirical processes indexed by classes of functions require entropy conditions on the size of the index set . for i.i.d . data , @xcite obtained the clt for empirical processes indexed by classes of sets satisfying an entropy condition with inclusion . @xcite used an entropy condition with bracketing to obtain results for empirical processes indexed by classes of functions . for the theory of empirical processes of i.i.d . data , indexed by classes of functions , see the book by @xcite . limit theorems for more general empirical processes indexed by classes of functions have also been studied under entropy conditions for general covering numbers , e.g. by @xcite who investigate empirical @xmath27-processes . in the case of strongly mixing data , @xcite were the first to obtain clts for empirical processes indexed by classes of functions . @xcite and @xcite study empirical processes for absolutely regular data . @xcite investigate the empirical process and the empirical @xmath27-process for data that can be represented as functionals of absolutely regular processes . for further results , see the survey article by @xcite , the book by @xcite , as well as the paper by @xcite . a lot of research has been devoted to the study of statistical properties of data arising from dynamical systems or from markov chains . a very powerful technique to prove clts and other limit theorems is the spectral gap method , using spectral properties of the perron - frobenius operator or the markov operator on an appropriate space of functions ; see @xcite . when the space of functions under consideration contains the class of indicator functions of intervals , standard tools can be used to establish the classical empirical process clt . finite - dimensional convergence of the empirical process follows from the clt for @xmath28}(x_i)$ ] , and tightness can be established using moment bounds for @xmath29}(x_i)$ ] . @xcite used this approach to establish the empirical process clt for expanding maps of the unit interval . the situation differs markedly when the clt and moment bounds are not directly available for the index class of the empirical process , but only for a different class of functions . recently , @xcite developed techniques to cover such situations . they were able to prove classical empirical process clts for @xmath4-valued data when the clt and moment bounds are only available for lipschitz functions . @xcite extended these techniques to @xmath22-valued data satisfying a multiple mixing condition for hlder continuous functions . under this condition , they proved the clt for the empirical process indexed by semi - infinite rectangles @xmath25 $ ] , @xmath30 . the multiple mixing condition is strictly weaker than the spectral gap condition . e.g. , ergodic torus automorphisms satisfy a multiple mixing condition , while generally they do not have a spectral gap . @xcite proved the empirical process clt for ergodic torus automorphisms . @xcite provide very general conditions under which the classical empirical process clt for @xmath22-valued data holds . the above mentioned papers study exclusively classical empirical processes , indexed by semi - infinite intervals or rectangles . it is the goal of the present paper to extend the techniques developed by @xcite to empirical processes indexed by classes of functions . let @xmath31 be a measurable space , let @xmath0 be a stationary process of @xmath32-valued random variables , and let @xmath1 be a uniformly bounded class of real - valued functions on @xmath32 . we consider the @xmath1-indexed empirical process @xmath33 . as in the above mentioned papers , we will assume that there exists some banach space @xmath34 of functions on @xmath32 such that the clt and a moment bound hold for partial sums @xmath35 , for all @xmath36 in some subset of @xmath34 ; see assumptions 1 and 2 . these conditions are satisfied , e.g. when the perron - frobenius operator or the markov operator acting on @xmath34 has a spectral gap . again , if the index class @xmath1 is a subset of @xmath34 , standard techniques for proving empirical process clts can be applied . in many examples , however , @xmath34 is some class of regular functions , while @xmath1 is a class of indicators of sets . it is the goal of the present paper to provide techniques suitable for this situation . empirical process invariance principles require a control on the size of the index class @xmath1 , as measured by covering or bracketing numbers ; see e.g. @xcite . in this paper , we will consider coverings of @xmath1 by @xmath34-brackets , i.e. brackets bounded by functions @xmath37 . because of the specific character of our moment bounds , we have to impose conditions on the @xmath34-norms of @xmath38 and @xmath39 . we will thus introduce a notion of bracketing numbers by counting how many @xmath34-brackets of a given @xmath40-size and with a given control on the @xmath34-norms of the upper and lower functions are needed to cover @xmath1 . the main theorem of the present paper establishes an empirical process clt under an integral condition on this bracketing number . this paper is organized as follows : section [ sec : main ] contains precise definitions as well as the statement of the main theorem . in section [ sec : examples ] , we will specifically consider the case when @xmath34 is the space of hlder continuous functions . we will give examples of classes of functions which satisfy the bracketing number assumption . in section [ sec : appli ] , we will give applications to ergodic torus automorphisms which extend the empirical process clt of @xcite to more general classes of sets . section [ sec : pf - ep - clt ] contains the proof of our main theorem , while proofs of technical aspects of the examples can be found in the appendix .

we study weak convergence of empirical processes of dependent data , indexed by classes of functions . our results are especially suitable for data arising from dynamical systems and markov chains , where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique . our results apply to the empirical process of data satisfying a multiple mixing condition . this includes dynamical systems and markov chains , if the perron - frobenius operator or the markov operator has a spectral gap , but also extends beyond this class , e.g. to ergodic torus automorphisms .

we study weak convergence of empirical processes of dependent data , indexed by classes of functions . our results are especially suitable for data arising from dynamical systems and markov chains , where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique . we are specifically interested in situations where the index class is different from the class of functions for which we have good properties of the observables . we introduce a new bracketing number to measure the size of the index class which fits this setting . our results apply to the empirical process of data satisfying a multiple mixing condition . this includes dynamical systems and markov chains , if the perron - frobenius operator or the markov operator has a spectral gap , but also extends beyond this class , e.g. to ergodic torus automorphisms .

1301.7131

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the possibility that irregular distributions of satellite galaxies may be a common feature of large galaxy halos was originally bolstered by several studies of the anisotropic distribution of our own galaxy s satellites . @xcite found that the magellanic stream along with sculptor and the draco - ursa minor stream and their associated dwarf spheroidal galaxies all appear to lie in the orbital plane of the magellanic clouds . in @xcite , all the then known dwarf spheroidal companions of the milky way are identified as lying in one of two streams . @xcite examined the likelihood of producing the observed disk - like distribution of milky way satellites from a spherical or oblate dark matter halo . from comparisons with theoretical isotropic satellite distributions produced from such a halo , they find that the chance of producing the observed distribution from the dark - matter sub - halos of cold - dark - matter ( cdm ) cosmology is less than 0.5 % . they examine various combinations of the inner most satellites and find a best - fit plane that is almost perpendicular to the plane of the milky way with a root - mean - square height ranging from only about @xmath3 to @xmath4 kpc . @xcite , whilst finding a similar plane to @xcite for the milky way satellites , disagree with their assumption that such a plane is unlikely to arise from a conventional cdm dark matter halo . they argue that that the most luminous satellites can not be taken for granted as forming randomly from the isotropic sub - halo distribution but instead , lie preferentially at smaller distances from the halo centre and co - planar with the major axis of the host halo . coupled with the finding that galaxies preferentially align themselves with their major - axis highly - inclined or even perpendicular to that of the surrounding matter ( e.g. @xcite ; @xcite ) , this would provide a good explanation for the observed orientation of the best fit plane . the arguments of @xcite are contended by @xcite however , and it should be noted that in contrast to @xcite , @xcite found that the distribution of the most massive subhaloes is not as flattened as the distribution of the milky way s satellites . more recently , @xcite , using the six halo models in the aquarius simulations @xcite , find that all six halos produce a significant population of sub - halos with quasi - planar orbits aligned with the main halo spin . this , they argue , is a natural explanation for the observed satellite distribution of the milky way . @xcite argue against this however . with the calculation of the angular momenta of 8 milky way satellites @xcite revealing a strong alignment between 6 of the orbital poles , @xcite examine the likelihood of randomly drawing 6 sub - halos from each of the 6 aquarius simulations ( among other halo simulations ) , and finding a similar degree of alignment . more precisely , they draw @xmath5 sets of 8 satellites from each of the 6 simulations , and select the 6 with the highest degree of alignment between their orbits , thus emulating the findings of @xcite . they then look at the degree of clumping of the orbital poles @xmath6 as well as the angular distance of the average of the orbital pole inclinations from the model equator @xmath7 and find that the actual degree of planarity observed for the six satellites identified by @xcite ( @xmath8 and @xmath9 ) are equalled or exceeded in the random draws in less than @xmath10 of cases when @xmath6 is considered and less than @xmath11 of cases for @xmath7 . @xcite also find that the degree of planarity observed for the milky way satellites is uncommon in all six of the aquarius halos ( see fig . 7 of that study ) . in addition to the revelation that the milky way s satellites appear to inhabit highly - inclined great planes , they also appear to corroborate the finding of @xcite , namely that the companions of spiral galaxies preferentially congregate at high galactic latitudes ( the holmberg effect ) , as observed in his study of 174 galaxy groups . it is not clear why this should be the case , or even if it truly is the case , although if the apparent adherence of satellite systems to polar great planes is typical of galaxies in general , then the holmberg effect seems to be an extension of this . @xcite proposed that dynamical friction may be responsible for the observed polar great planes , with those orbits spending the most time in close proximity to the galactic disk , experiencing the fastest decay , while those that take the most direct route through the disk environs , namely the polar orbits , experiencing the slowest orbital decay . it should be noted however that dynamical friction , whilst producing more polar orbits , would not produce planes of satellites . nor would it be effective on the young globular clusters which are shown to be co - planar with the `` vast plane of satellites '' identified around the milky way by @xcite . indeed , @xcite show that dynamical friction did not play a role in the formation of the milky way satellite orbits . besides the conjecture that satellite great planes trace the major - axis of the dark - matter halo in which the parent galaxy resides , there are other proposed mechanisms for their creation . one hypothesis is that these planes trace the orbits of ancient galaxies that have been cannibalized by the host galaxy . @xcite have investigated this hypothesis by looking for planes among groups of satellite galaxies and globular clusters in the milky way s outer halo and find various members to be co - planar with either the magellanic or sagittarius streams . the findings of @xcite are also consistent with such a hypothesis . indeed , it is this hypothesis which is most strongly supported by @xcite , wherein the @xmath6 and @xmath7 of satellites drawn from various tidal models equal or exceed @xmath12 and @xmath13 in over @xmath14 of draws in some cases . a similar hypothesis , which in some regards links the galaxy - cannibalization and dark - matter hypotheses , proposes that the observed planes result from the orientation of the large - scale filamentary structure of galaxy clusters ( e.g. @xcite ) , an orientation traced out by those minor galaxies which fall into the halo of a major galaxy . @xcite argue however that extra - galactic associations of dwarf galaxies are too extended to account for the high degree of planarity observed for the milky way satellites . this argument is supported by the findings of @xcite based on the aquarius simulations . the great obstacle to a conclusive resolution of these issues is the lack of systems for which reliable spatial ( and kinematic ) data exists . while some such data does exist for large galaxy clusters such as virgo and coma , accurate 3d distributions of galaxies within their halo have for a long time been known only for our own galaxy s halo , ascertainable due to our central position within it . it has only been in recent times that a second system has opened up to us - that of our counterpart in the local group , m31 . whilst various databases of photometry and other data have been available for m31 and some of its brighter companions for over a decade , it is the pan - andromeda archaeological survey ( pandas - @xcite ) - a deep photometric , 2-colour survey providing a uniform coverage of the m31 halo out to approximately 150 kpc - that has provided a new level of detail for this system . it is from this survey that we obtained our distances to m31 and 27 of its companions , following the method developed in @xcite ( henceforth cli11 ) and further adapted for this purpose in @xcite ( henceforth cil12 ) . the distances themselves and their associated uncertainty distributions are presented in cil12 and it is these distributions that are utilized for all analysis contained in this paper . with regard to previous studies of the anisotropy in the m31 satellite distribution , two investigations warrant consideration at this point . @xcite , making use of wide field camera ( wfc ) photometry from the isaac newton telescope ( int ) in what was essentially the forerunner to the pandas survey , focus on ghostly streams " of satellite galaxies following a similar approach as @xcite used for the milky way . in addition , they characterize the large degree of asymmetry in the satellite distribution , a feature also noted in cil12 , and examine the radial distribution of the satellites , noting a ( statistically insignificant ) larger average distance from m31 than that observed between the milky way and its satellites . they find a large number of candidate satellite streams , with some favoring the dwarf spheroidal members . utilize distance measurements from a variety of sources and focus particularly on planes of satellites and , whilst they do not find a particularly significant best fit plane when their whole satellite sample is considered , it is rather interesting that they find a 99.7 % statistical significance to their best fit plane when the then - known dwarf spheroidal galaxies dominate their sample . furthermore , this plane is near - polar - as has been observed for the milky way , although they find little support for the holmberg effect . @xcite utilize a particularly robust method in their search for high - significance planar fits to subsets of galaxies by considering every possible combination of a given number of satellites from their sample . in the current study we employ a similar approach , but with the great advantage of having a considerably extended sample of galaxies in our sample , with all distances derived by the same method and from the same data as described in cli11 and cil12 . ( distances are sampled from the distributions without the density prior applied - e.g. cil12 , table 1 , column 2 - whilst the best fit distances are drawn from cil12 , table 2 , column 4 . ) as a result , we are able to give full consideration to the effects of selection bias on the observed satellite distribution . this then presents an excellent opportunity to greatly improve our knowledge of the three - dimensional structure of the m31 satellite distribution , with important implications regarding the recent evolution of the system . a breakdown of the structure of the paper is as follows . in section [ s_method ] , we outline our method for plane fitting ( [ ss_method_pf ] ) and locating significant planes of satellites as well as the orientation , magnitude and significance of the asymmetry of the distribution . a method for generating random realizations of satellites subject to the same selection biases as the real data is also discussed in this section ( [ ss_method_rr ] ) as is the selection bias itself ( [ ss_method_sdb ] ) . [ s_results ] then presents the results of applying these methods , first to the sample as a whole , and then to subsets of galaxies . specifically , [ ss_bfp_all_sats ] presents a study of planarity within the satellite system when all satellites contribute to the determination of the best fit plane ; [ ss_asy_all_sats ] examines the asymmetry in a similar way ; [ sat_subsets ] examines the orientations of planes of smaller subsets of satellites within the distribution ; and [ great_plane ] concludes this section with a determination of the significance of a ` great plane ' of satellites emerging from the preceding sections . sections [ s_discussion ] and [ s_conclusions ] then follow with discussion and conclusions . note that this paper was written in conjunction with a shorter contribution ( @xcite ; hereafter ilc13 ) which announced some of the key discoveries resulting from the analysis we present here . in particular , the process of identifying the member satellites of the ` great plane ' discussed in ilc13 is described here in more detail . in this analysis however , we concern ourselves with the _ spatial _ structure of the satellite system only and so the reader should refer to ilc13 for the interesting insight provided by the addition of the velocity information .

we undertake an investigation into the spatial structure of the m31 satellite system utilizing the distance distributions presented in a previous publication . these distances make use of the unique combination of depth and spatial coverage of the pan - andromeda archaeological survey ( pandas ) to provide a large , homogeneous sample consisting of of m31 s satellites , as well as m31 itself . a similar analysis of the asymmetry of the m31 satellite distribution finds that it is also significantly larger than one would expect from a random distribution . in particular , it is remarkable that of the satellites most likely lie on the milky way side of the galaxy , with the asymmetry being most pronounced within the satellite subset forming the aforementioned disk .

we undertake an investigation into the spatial structure of the m31 satellite system utilizing the distance distributions presented in a previous publication . these distances make use of the unique combination of depth and spatial coverage of the pan - andromeda archaeological survey ( pandas ) to provide a large , homogeneous sample consisting of of m31 s satellites , as well as m31 itself . we find that the satellite distribution , when viewed as a whole , is no more planar than one would expect from a random distribution of equal size . a disk consisting of 15 of the satellites is however found to be highly significant , and strikingly thin , with a root - mean - square thickness of just kpc . this disk is oriented approximately edge on with respect to the milky way and almost perpendicular to the milky way disk . it is also roughly orthogonal to the disk like structure regularly reported for the milky way satellite system and in close alignment with m31 s giant stellar stream . a similar analysis of the asymmetry of the m31 satellite distribution finds that it is also significantly larger than one would expect from a random distribution . in particular , it is remarkable that of the satellites most likely lie on the milky way side of the galaxy , with the asymmetry being most pronounced within the satellite subset forming the aforementioned disk . this lopsidedness is all the more intriguing in light of the apparent orthogonality observed between the satellite disk structures of the milky way and m31 .

1301.7131

c

it is clear that whilst the satellites of m31 when taken as a whole are no more planar than one can expect from a random distribution , a subset consisting of roughly half the sample _ is _ remarkably planar . the presence of this thin disk of satellites has been conspicuous throughout the analysis contained in this paper . the degree of asymmetry determined from the satellite distribution is also found to be relatively high . of particular note , the orientation of the asymmetry is very significant , being aligned very strongly in the direction of the milky way . when this fact is combined with the apparent orthogonality observed between the milky way and m31 satellite distributions and the milky way disk , it appears that the two halos may in fact be coupled . regardless , the great plane of satellites identified in this study , and its clear degree of significance , is not directly expected from @xmath134 cosmology . this finding provides strong evidence that thin disks of satellites do indeed exist in galaxy halos , and whether or not the standard model can account for such structures remains to be seen . a. r. c. would like to thank macquarie university for their financial support through the macquarie university research excellence scholarship ( mqres ) and both the university of sydney and universit de strasbourg for the use of computational and other facilities . g. f. l. thanks the australian research council for support through his future fellowship ( ft100100268 ) and discovery project ( dp110100678 ) . r. a. i. and d. v. g. gratefully acknowledge support from the agence nationale de la recherche though the grant pommme ( anr 09-blan-0228 ) . based on observations obtained with megaprime / megacam , a joint project of cfht and cea / dapnia , at the canada - france - hawaii telescope ( cfht ) which is operated by the national research council ( nrc ) of canada , the institut national des science de lunivers of the centre national de la recherche scientifique ( cnrs ) of france , and the university of hawaii .

we find that the satellite distribution , when viewed as a whole , is no more planar than one would expect from a random distribution of equal size . a disk consisting of 15 of the satellites is however found to be highly significant , and strikingly thin , with a root - mean - square thickness of just kpc . this lopsidedness is all the more intriguing in light of the apparent orthogonality observed between the satellite disk structures of the milky way and m31 .

we undertake an investigation into the spatial structure of the m31 satellite system utilizing the distance distributions presented in a previous publication . these distances make use of the unique combination of depth and spatial coverage of the pan - andromeda archaeological survey ( pandas ) to provide a large , homogeneous sample consisting of of m31 s satellites , as well as m31 itself . we find that the satellite distribution , when viewed as a whole , is no more planar than one would expect from a random distribution of equal size . a disk consisting of 15 of the satellites is however found to be highly significant , and strikingly thin , with a root - mean - square thickness of just kpc . this disk is oriented approximately edge on with respect to the milky way and almost perpendicular to the milky way disk . it is also roughly orthogonal to the disk like structure regularly reported for the milky way satellite system and in close alignment with m31 s giant stellar stream . a similar analysis of the asymmetry of the m31 satellite distribution finds that it is also significantly larger than one would expect from a random distribution . in particular , it is remarkable that of the satellites most likely lie on the milky way side of the galaxy , with the asymmetry being most pronounced within the satellite subset forming the aforementioned disk . this lopsidedness is all the more intriguing in light of the apparent orthogonality observed between the satellite disk structures of the milky way and m31 .

0710.0964

i

using a suite of carefully controlled 3d hydrodynamic simulations , we have investigated the ram pressure stripping of hot gas in the halos of galaxies as they fall into groups and clusters . we have proposed a physically simple analytic model that describes the stripping seen in the simulations remarkably well . this model is analogous to the original formulation of gunn & gott ( 1972 ) , except that it is appropriate for the case of a spherical gas distribution ( as opposed to a face - on disk ) and takes into account that stripping is not instantaneous but occurs on approximately a sound crossing time . the only pieces of information that the model requires are the initial conditions of the orbiting galaxy ( its gas and dark matter profiles ) , the density profile of the icm and the orbit [ the latter two are needed to calculate @xmath101 ] . the model contains two tunable coefficients that are of order unity . fixing these coefficients to match the stripping in just one of our idealised uniform medium simulations ( see 3.2 ) leads to excellent agreement with all our other simulations . with the exception of cases where the mass of the galaxy is greater than about 10% of the mass of the group and its orbit is highly non - radial , the analytic model reproduces the mass loss in the simulations to @xmath1 accuracy at all times and for all the orbits , galaxy masses , and galaxy concentrations that we have explored . for cases where the mass of the galaxy exceeds 10% of the mass of the group , it will likely be necessary to factor in the effects of tidal stripping and gravitational shock heating , which are neglected by our model . we re - iterate that the numerical simulations with which our analytic model has been calibrated have been demonstrated to be robust to the adopted resolution and artificial viscosity strength ( see 2.2 ) . furthermore , as we have demonstrated that kh ( and rt ) instability stripping is expected to be unimportant , sph codes should be fully capable of tackling the problem of hot halo gas stripping in galaxies orbiting in groups and clusters . a direct comparison between the results using the gadget-2 and flash hydrodynamic codes for one of our runs ( see appendix a ) confirms this conclusion . the model we have derived has a number of potentially interesting applications , including modelling observed satellite galaxies and satellite galaxies in cosmological simulations . one application that we are currently pursuing is the incorporation of our ram pressure stripping model into a semi - analytic model of galaxy formation . as mentioned in 1 , recent observations ( weinmann et al . 2006 ; baldry et al . 2006 ) have revealed that current semi - analytic models predict satellite galaxies whose colours are too red compared to the observed systems . the implementation of ram pressure stripping in these models is unrealistically efficient since , by assumption , the hot halo of the satellite galaxy is instantly transferred to the more massive system as soon as the satellite galaxy enters the massive system s virial radius . in reality , the hot gaseous halo of the galaxy will remain intact for a while . for example , for the most common orbital parameters , we find that between 20%-40% of the initial hot halo of the galaxy can remain in place even after 10 gyr of orbiting inside the group or cluster ( see fig . 9 ; note , however , that the quoted numbers could be sensitive to the adopted hot gas distribution of the galaxy ) . we note that these predictions are in qualitative agreement with recent _ chandra _ x - ray observations of massive galaxies orbiting in hot clusters by sun et al . ( 2007b ) , who find that most of the galaxies have detectable hot gaseous halos . depending on the efficiency of feedback ( e.g. , from supernovae winds ) in the semi - analytic models , radiative cooling of the remaining hot halo gas will replenish the cold gaseous component at the centre of the galaxy , which in turn will allow star formation to continue for some time . this will have the effect of making the colour of model satellite galaxies bluer and could resolve the discrepancy between semi - analytic models and observations ( font et al . , in prep ) .

we use a large suite of carefully controlled full hydrodynamic simulations to study the ram pressure stripping of the hot gaseous halos of galaxies as they fall into massive groups and clusters . the sensitivity of the results to the orbit , total galaxy mass , and galaxy structural properties is explored . for typical structural and orbital parameters , we find that of the initial hot galactic halo gas can remain in place after 10 gyr . we propose a physically simple analytic model that describes the stripping seen in the simulations remarkably well . the model is analogous to the original formulation of gunn & gott ( 1972 ) , except that it is appropriate for the case of a spherical ( hot ) gas distribution ( as opposed to a face - on cold disk ) and takes into account that stripping is not instantaneous but occurs on a characteristic timescale . the model reproduces the results of the simulations to within at almost all times for all the orbits , mass ratios , and galaxy structural properties we have explored . the one exception involves unlikely systems where the orbit of the galaxy is highly non - radial and its mass exceeds about 10% of the group or cluster into which it is falling ( in which case the model under - predicts the stripping following pericentric passage ) .

we use a large suite of carefully controlled full hydrodynamic simulations to study the ram pressure stripping of the hot gaseous halos of galaxies as they fall into massive groups and clusters . the sensitivity of the results to the orbit , total galaxy mass , and galaxy structural properties is explored . for typical structural and orbital parameters , we find that of the initial hot galactic halo gas can remain in place after 10 gyr . we propose a physically simple analytic model that describes the stripping seen in the simulations remarkably well . the model is analogous to the original formulation of gunn & gott ( 1972 ) , except that it is appropriate for the case of a spherical ( hot ) gas distribution ( as opposed to a face - on cold disk ) and takes into account that stripping is not instantaneous but occurs on a characteristic timescale . the model reproduces the results of the simulations to within at almost all times for all the orbits , mass ratios , and galaxy structural properties we have explored . the one exception involves unlikely systems where the orbit of the galaxy is highly non - radial and its mass exceeds about 10% of the group or cluster into which it is falling ( in which case the model under - predicts the stripping following pericentric passage ) . the proposed model has several interesting applications , including modelling the ram pressure stripping of both observed and cosmologically - simulated galaxies and as a way to improve current semi - analytic models of galaxy formation . one immediate consequence is that the colours and morphologies of satellite galaxies in groups and clusters will differ significantly from those predicted with the standard assumption of complete stripping of the hot coronae . [ firstpage ] hydrodynamics methods : n - body simulations galaxies : clusters : general galaxies : evolution galaxies : structure cosmology : theory

cond-mat9607088

i

man - tailored semiconductor heterostructures @xcite offer , for the first time , the possibility to test quantum mechanics at a mesoscopic level @xcite . the scenario of systems which can be investigated is so rich that the art of their realization deserves the name of quantum design . in the simplest case , a quantum designer can grow sandwiches of different semiconductor alloys by choosing the number of atomic layers for each kind of alloy . in the resulting heterostructure , the conduction band profile along the growth direction forms steps whose height can be continuously varied by a proper choice of the alloy composition . typical widths and heights are of the order of tens of and tenths of ev , respectively . at low - temperature , the mean free path of carriers for scattering from crystal impurities is of the order of @xmath0 and for heterostructures smaller than this size the electric transport along the growth direction is a phase coherent quantum scattering from the conduction band discontinuities @xcite . due to the translational invariance in the plane orthogonal to the growth direction , the problem is one - dimensional . moreover , the carriers are described by an effective mass which accounts for the microscopic scattering with the periodic crystal sites and their wave function is an envelope wave function @xcite . in a homogeneous neutral conductor , the electron - electron interaction can be taken into account by a renormalization of the carrier effective masses @xcite and one deals with a transport problem like in a noninteracting case . in a heterostructure , even as simple as that described above , the breaking of translational invariance in the transport direction allows the electric neutrality to be locally violated . the corresponding interaction potential , obtained , at hartree level , by solving a proper poisson equation , can strongly modify the transport properties . the example of a double barrier heterostructure with the exterior regions doped with donors is illuminating @xcite . due to tunneling , electrons populate the resonance(s ) created by the double barrier and the region between the barriers becomes negatively charged . this generates an electric potential which decreases the tunneling probability of electrons in the double barrier region . as a consequence , current oscillations on the picosecond scale @xcite and chaotic behavior without classical counterpart @xcite have been predicted in a ballistic configuration in which electrons are injected at some chosen energy . experiments with ballistic electrons are difficult and measurements became available only recently @xcite . technologically simpler is the case of biased heterostructures where transport is due to the presence of reservoirs at thermal equilibrium with different chemical potentials . manifestations of the electron - electron interaction are known also in this configuration . for example , hysteresis in the current - voltage characteristics of double barrier heterostructures have been observed @xcite and recognized as a consequence of the accumulation of electrons in the resonance @xcite . in this case , however , one has the theoretical problem of attaching reservoirs at thermal equilibrium to a piece of conductor where quantum coherent transport takes place . in the recent paper @xcite we proposed an approach to this problem based on a mathematical method earlier applied in the framework of ballistic transport @xcite . we showed that for heterostructures with a single resonance our approach allows one _ i ) _ to obtain steady - state voltage - current characteristics having hysteresis or not in agreement with the experimental results @xcite and _ ii ) _ to predict time - dependent properties analogous to those studied in optically bistable systems @xcite . here , we develop the general mathematical scheme of this approach and discuss the case with several resonances where multistability phenomena can take place as in superlattices @xcite . for simplicity , consider the one - dimensional double barrier heterostructure discussed above . the idea is that due to the presence of resonances the corresponding schrdinger problem can be divided in two parts : a schrdinger equation for the barrier region and one for the exterior space , the two being weakly coupled by tunneling . this decomposition corresponds to the schematization of the transport process as a coherent process fed by reservoirs . in the exterior space ( reservoirs ) homogeneous and neutral , the electron - electron interaction is neglected and thermal equilibrium is taken into account by considering a continuous set of energy eigenstates distributed according to the fermi statistics . in the barrier region ( coherent conductor ) , the coulomb interaction is included in a self - consistent potential obtained by solving the poisson equation associated to the local charge density . under the assumption that the barriers are wide enough , the corresponding nonlinear schrdinger problem is discussed in two steps . in the first step we eliminate the potential well between the two barriers , by artificially increasing the potential there , and we solve the schrdinger equation asymptotically for the new potential by means of wkb - expansions . the resulting solution is then very small near the ( filled ) potential well , so we get only a small error in the schrdinger equation when we go back to the true potential . in the second step we correct for this small error by adding a wave function concentrated near the potential well . assuming a priori that the charge in the well changes slowly with time , the correcting wave function can be expected to be large only at energies close to the resonances , and be well approximated by some linear combination of the resonant states . in most of the paper we discuss the case in which only one resonance participates . the validity of this one - mode approximation has been tested numerically with excellent results in the ballistic configuation of @xcite . here , the coefficient of the one - mode approximation obeys an ordinary differential equation with respect to time in the infinite dimensional space of square integrable functions of energy . we study the stationary points of the corresponding vector field and their nature , whether they are attractive or not , and arrive at quite neat answers . for solutions of the dynamical system which have existed as bounded solutions for a long time and in a suitable asymptotic limit ( of wide barriers ) we derive a simplified scalar differential equation for the evolution of the sheet density of electrons trapped in well which gives good global understanding of the more complete dynamical system . using these results , we are able to discuss the phenomenon of hysteresis and we support and illustrate the discussion with several numerical results . the discussion includes the evolution of solutions away from fixed points which necessarily appears when there is hysteresis . we also discuss the case of several resonances , and get analogous results . from the mathematical point of view , the present paper could be a starting point for rigorous work on some fresh problems , involving semiclassical analysis , adiabatic theory , non - linear schrdinger equations and dynamical systems . a strong motivation for such an enterprise is the fact that the theory of electric transport in semiconductor devices offers many problems similar to that one we illustrate here @xcite . the plan of the paper is as follows . in section [ 2 ] we define the model . in section [ 3 ] we review the wkb expansion for slowly varying potentials . in sections [ 4 ] and [ 5 ] we determine the driving term and the ground resonant state , respectively , within the wkb approximation . the central equation of our paper is derived in section [ 6 ] and the general properties of the associated fixed points and linearizations are discussed in section [ 7 ] . in section [ 8 ] we introduce an approximation valid in the limit of small resonance width and discuss the corresponding fixed point solutions and linearizations . in section [ 9 ] we obtain a simplified differential equation describing the dynamics of the electron density in the well . a qualitative discussion of the hysteresis phenomenon in comparison with numerical results is given in section [ 10 ] . in section [ 11 ] we finally consider the case with several resonances .

we use an adiabatic approximation in terms of instantaneous resonances to study the steady - state and time - dependent transport properties of interacting electrons in biased resonant tunneling heterostructures . this approach leads , in a natural way , to a transport model of large applicability consisting of reservoirs coupled to regions where the system is described by a nonlinear schrdinger equation . from the mathematical point of view , this work is non - rigorous but may offer some fresh and interesting problems involving semiclassical approximation , adiabatic theory , non - linear schrdinger equations and dynamical systems .

we use an adiabatic approximation in terms of instantaneous resonances to study the steady - state and time - dependent transport properties of interacting electrons in biased resonant tunneling heterostructures . this approach leads , in a natural way , to a transport model of large applicability consisting of reservoirs coupled to regions where the system is described by a nonlinear schrdinger equation . from the mathematical point of view , this work is non - rigorous but may offer some fresh and interesting problems involving semiclassical approximation , adiabatic theory , non - linear schrdinger equations and dynamical systems .

1409.2529

c

we have proposed to use the natural concept of homotopy theory to extend the results from @xmath3-theory ( and the intermediate step of considering isomorphism classes of vector bundles ) to the non - stable regime , which includes _ all _ insulators independent of the number of occupied / empty bands . we showed that the definition of strong topological insulators has to be more restrictive in general in order to avoid the possibility of realising them by stacking lower - dimensional systems . furthermore , we demonstrated that there are topological insulators in @xmath0 dimensions that are `` truly @xmath0-dimensional '' despite being weak ( independently of how the latter is defined ) , meaning they can not be realized by stacking lower - dimensional systems . along the way , we derived some useful technical results : we showed how the factorization of topological invariants in the stable regime can be understood from the perspective of homotopy theory and proved that in general ( in the stable and non - stable regime ) , it is legitimate to replace a domain consisting of products of spheres by a single sphere of the same total dimension , even in the presence of @xmath24-equivariance .

we use homotopy theory to extend the notion of strong and weak topological insulators to the non - stable regime ( low numbers of occupied / empty energy bands ) . additionally , we prove some useful technical results , including the homotopy theoretic derivation of the factorization of invariants over the torus into invariants over spheres in the stable regime , as well as the rigorous justification of the parameter space replacements and used widely in the current literature .

we use homotopy theory to extend the notion of strong and weak topological insulators to the non - stable regime ( low numbers of occupied / empty energy bands ) . we show that for strong topological insulators in spatial dimensions to be `` truly-dimensional '' , i.e. not realizable by stacking lower - dimensional insulators , a more restrictive definition of `` strong '' is required . however , this does not exclude weak topological insulators from being `` truly-dimensional '' , which we demonstrate by an example . additionally , we prove some useful technical results , including the homotopy theoretic derivation of the factorization of invariants over the torus into invariants over spheres in the stable regime , as well as the rigorous justification of the parameter space replacements and used widely in the current literature .

1410.8329

i

fix the nonnegative integer @xmath2 and let @xmath3 be the grassmannian which parametrizes isotropic subspaces of dimension @xmath4 in the vector space @xmath5 , equipped with a symplectic form . in 2008 , buch , kresch , and tamvakis @xcite introduced the theta polynomials @xmath6 , a family of polynomials in independent variables @xmath7 , defined using young s raising operators @xcite . when the @xmath7 are mapped to the special schubert classes , which are the chern classes of the tautological quotient vector bundle over @xmath8 , then the @xmath6 represent the schubert classes in @xmath9 . these theta polynomials give a combinatorially explicit , intrinsic solution to the _ giambelli problem _ for the cohomology ring of @xmath8 . this paper is concerned with giambelli formulas for the _ equivariant schubert classes _ in the equivariant cohomology ring @xmath10 , where @xmath11 denotes a maximal torus of the complex symplectic group . as explained in @xcite and @xcite , such formulas are equivalent to corresponding ones within the framework of degeneracy loci of vector bundles @xcite . in 2009 , the theta polynomials @xmath6 were extended to obtain representing polynomials in the torus - equivariant cohomology ring of @xmath8 , and more generally , of any isotropic partial flag variety , by the first author @xcite . the _ equivariant giambelli problem _ was thus solved in a uniform manner for any classical @xmath12 space , in terms of positive combinatorial formulas which are _ native to @xmath12_. see @xcite for an exposition of this work . it has been known for some time ( cf . @xcite ) that the equivariant schubert classes of the usual ( type a ) grassmannian and of the lagrangian grassmannian @xmath13 may be represented by jacobi - trudi type determinants and schur pfaffians which generalize the classical results of @xcite . the polynomials which appear in these formulas are called double ( or factorial ) schur @xmath0-polynomials and @xmath1-polynomials , respectively ( cf . @xcite ) . our aim here is to use raising operators to define _ double theta polynomials _ @xmath14 , which advance the theory of the single theta polynomials @xmath6 from @xcite to the equivariant setting , and specialize to both of the aforementioned versions of double schur polynomials . the @xmath14 represent the equivariant schubert classes on @xmath8 , but differ from the equivariant giambelli polynomials given in @xcite . it follows that the two theories must agree up to the ideal @xmath15 of _ classical relations _ ( [ relations ] ) among the variables @xmath7 . we discuss this in detail in [ compare ] and corollary [ comp ] . for the rest of this paper , @xmath16 and @xmath17 will denote two families of commuting variables . we set @xmath18 and @xmath19 for any @xmath20 . for any integers @xmath21 and @xmath22 , define the elementary and complete symmetric polynomials @xmath23 and @xmath24 by the generating series @xmath25 respectively . let @xmath26 , @xmath27 , and @xmath28 , where @xmath29 denotes the kronecker delta . furthermore , if @xmath30 then define @xmath31 . we will work throughout with integer sequences @xmath32 which are assumed to have finite support , when they appear as subscripts . for any positive integers @xmath33 and integer sequence @xmath34 , define the operator @xmath35 by @xmath36 a _ raising operator _ @xmath37 is any monomial in the basic operators @xmath35 . the sequence @xmath38 is a _ composition _ if @xmath39 for all @xmath40 , and a _ partition _ if @xmath41 for all @xmath42 . as is customary , we will identify partitions with their young diagram of boxes . fix the nonnegative integer @xmath2 . for any integers @xmath43 and @xmath44 , define @xmath45 and for any integer sequences @xmath38 and @xmath46 , let @xmath47 given any raising operator @xmath37 , define @xmath48 . it is important that the variables @xmath49 in the monomials @xmath50 are regarded as _ noncommuting _ for the purposes of this action . we say that a partition @xmath51 is _ @xmath2-strict _ if all parts @xmath52 which are strictly greater than @xmath2 are distinct . to any such @xmath51 , we attach a finite set of pairs @xmath53 and a sequence @xmath54 defined by @xmath55 following @xcite , consider the raising operator expression @xmath56 given by @xmath57 [ thdef ] for any @xmath2-strict partition @xmath51 , the _ double theta polynomial _ @xmath58 is defined by @xmath59 the single theta polynomial @xmath6 of @xcite is given by @xmath60 we next relate the double theta polynomials @xmath14 to the torus - equivariant schubert classes on @xmath8 . let @xmath61 denote the standard symplectic basis of @xmath5 and let @xmath62 be the subspace spanned by the first @xmath40 vectors of this basis , so that @xmath63 for @xmath64 . let @xmath65 denote the stabilizer of the flag @xmath66 in the symplectic group @xmath67 , and let @xmath68 be the associated maximal torus in the borel subgroup @xmath65 . the @xmath68-equivariant cohomology ring @xmath69 is defined as the cohomology ring of the borel mixing space @xmath70 . the schubert varieties in @xmath8 are the closures of the @xmath65-orbits , and are indexed by the @xmath2-strict partitions @xmath51 whose young diagram fits in an @xmath71 rectangle . any such @xmath51 defines a schubert variety @xmath72 of codimension @xmath73 by @xmath74 since @xmath75 is stable under the action of @xmath68 , we obtain an _ equivariant schubert class _ @xmath76^{t_n}:=[et_n\times^{t_n}x_{\lambda}]$ ] in @xmath77 . following @xcite and @xcite , we consider the _ stable _ equivariant cohomology ring of @xmath78 , denoted by @xmath79 . the latter is defined by @xmath80 where the inverse limit of the system @xmath81 is taken in the category of graded algebras . the surjections ( [ embeddings ] ) are induced from the natural inclusions @xmath82 of the weyl groups of type c , as in @xcite , @xcite , and [ trans ] of the present work . moreover , the variables @xmath83 are identified with the characters of the maximal tori @xmath68 in a compatible fashion , as in loc . one then has that @xmath79 is a free @xmath84$]-algebra with a basis of stable equivariant schubert classes @xmath85^{t_n},\ ] ] one for every @xmath2-strict partition @xmath51 . we view @xmath86 as a @xmath84$]-module via the natural projection map @xmath84\to { { \mathbb z}}[t_1,\ldots , t_n]$ ] . consider the graded polynomial ring @xmath87:={{\mathbb z}}[c_1,c_2,\ldots]$ ] where the element @xmath7 has degree @xmath43 for each @xmath88 . let @xmath89 $ ] be the homogeneous ideal generated by the relations @xmath90 and define the quotient ring @xmath91/{i^{(k)}}$ ] . we call the graded polynomial ring @xmath92 $ ] the _ ring of double theta polynomials_. [ mainthm ] the polynomials @xmath14 , as @xmath51 runs over all @xmath2-strict partitions , form a free @xmath84$]-basis of @xmath92 $ ] . there is an isomorphism of graded @xmath84$]-algebras @xmath93\to { { \mathbb h}}_t(\operatorname{ig}_k)\ ] ] such that @xmath14 is mapped to @xmath94 , for every @xmath2-strict partition @xmath51 . for every @xmath95 , the morphism @xmath96 induces a surjective homomorphism of graded @xmath84$]-algebras @xmath97\to \operatorname{\mathrm{h}}_{t_n}^*(\operatorname{ig}(n - k,2n))\ ] ] which maps @xmath14 to @xmath76^{t_n}$ ] , if @xmath51 fits inside an @xmath71 rectangle , and to zero , otherwise . the equivariant cohomology ring @xmath86 is presented as a quotient of @xmath92 $ ] modulo the relations @xmath98 and @xmath99 where @xmath100 denotes the partition @xmath101 with @xmath43 parts . theorem [ mainthm ] has a direct analogue for the torus - equivariant cohomology ring of the odd orthogonal grassmannian @xmath102 ; this follows from known results ( cf . * and 5.1 ) ) . in this case , the polynomials @xmath103 represent the equivariant schubert classes , where @xmath104 denotes the number of parts @xmath52 of @xmath51 which are strictly bigger than @xmath2 . theorem [ mainthm ] therefore generalizes the results of pragacz @xcite and ikeda and naruse @xcite to the equivariant cohomology of all symplectic and odd orthogonal grassmannians . let @xmath105 denote the universal exact sequence of vector bundles over @xmath78 , with @xmath106 the trivial bundle of rank @xmath107 and @xmath108 the tautological subbundle of rank @xmath4 . the @xmath68-equivariant vector bundles @xmath108 , @xmath106 , and @xmath109 have equivariant chern classes , denoted by @xmath110 , @xmath111 , and @xmath112 , respectively . define @xmath113 by the equation of total chern classes @xmath114 . we now have the following chern class formula for the schubert classes @xmath76^t$ ] . [ maincor ] let @xmath51 be a @xmath2-strict partition that fits inside an @xmath71 rectangle . then we have @xmath115^t = { \theta}_{\lambda}(e - e'-f_{n+{\beta}({\lambda } ) } ) = r^{\lambda}\ , c^t_{\lambda}(e - e'-f_{n+{\beta}({\lambda})})\ ] ] in the equivariant cohomology ring @xmath77 . the equivariant chern polynomial in ( [ genkaz ] ) is interpreted as the image of @xmath116 under the @xmath117-linear map which sends the noncommutative monomial @xmath118 to @xmath119 , for every integer sequence @xmath38 . formula ( [ genkaz ] ) is a natural generalization of kazarian s multi - pfaffian formula ( * ? ? ? 1.1 ) to arbitrary equivariant schubert classes on symplectic grassmannians . the double theta polynomials @xmath14 were defined in wilson s 2010 university of maryland ph.d . thesis @xcite , whose aim was to apply the raising operator approach of @xcite to the theory of factorial schur polynomials and equivariant giambelli formulas . of @xmath14 in the ring of type c double schubert polynomials of @xcite . ] in her dissertation , wilson gave a direct proof that the @xmath14 satisfy the _ equivariant chevalley rule _ in @xmath79 , i.e. , the combinatorial formula for the expansion of the products @xmath120 in the basis of stable equivariant schubert classes . this fits into a program for proving theorem [ mainthm ] , and its extension to equivariant quantum cohomology , by applying mihalcea s characterization theorem ( * ? ? ? 8.2 ) ( see [ trans ] ) . the proof of the chevalley rule for the @xmath121 which we present in [ prelim ] and [ cfdtp ] uses the technical tools for working with raising operators constructed in @xcite , but the argument is much simpler , since it avoids the general substitution rule employed in op . it was conjectured by wilson in @xcite that the @xmath14 represent the stable equivariant schubert classes on @xmath8 . ikeda and matsumura ( 1.2 ) established this result , which is the main ingredient behind the new presentation of the equivariant cohomology ring @xmath77 displayed in theorem [ mainthm ] . a key technical achievement in their proof was to show that the @xmath122 are compatible with the action of the ( left ) divided difference operators on @xmath92 $ ] . in section [ gdds ] , we explain how this fact may be combined with a formula for the equivariant schubert class of a point on @xmath78 , to obtain a proof of wilson s conjecture ; the argument found in @xcite is different , and uses localization in equivariant cohomology . it is easy to see that the polynomial @xmath14 may be written formally as a sum of schur pfaffians ( cf . proposition [ kstricts ] ) . this identity was used in @xcite in the proof of the compatibility of the @xmath122 with divided differences , allowing them to avoid the language of raising operators entirely . in section [ dds ] , we eliminate this feature of their argument , and work directly with the raising operator expressions @xmath56 . this makes the proof more transparent , and leads naturally to a companion theory of _ double eta polynomials _ @xmath123 , which represent the equivariant schubert classes on even orthogonal grassmannians ; see @xcite for further details . the role of the first author of this article has been mostly expository , extending the point of view found in @xcite to the present setting , and comparing the results with various earlier formulas in the literature . we emphasize that the double theta polynomials @xmath14 defined here are new , and are not a special case of the type c double schubert polynomials of @xcite , which represent the equivariant schubert classes on complete symplectic flag varieties . recently , hudson , ikeda , matsumura , and naruse @xcite proved an analogue of our cohomological formula in connective @xmath124-theory , and anderson and fulton @xcite obtained related chern class formulas for more general degeneracy loci . the latter work features a powerful geometric approach to the theory , which is uniform for all four classical lie types . this paper is organized as follows . section [ prelim ] establishes some preliminary lemmas which are required in our proof that the @xmath14 obey the equivariant chevalley rule ( theorem [ cvthm ] ) ; the latter is completed in sections [ cfdtp ] and [ trans ] . although theorem [ cvthm ] is a consequence of theorem [ mainthm ] , it is a much earlier result , and the independent proof given here exhibits alternating properties of the @xmath14 which are useful in other contexts . section [ firstexs ] is concerned with how the @xmath14 behave in special cases , and relates them to some earlier formulas . in section [ dds ] , we observe that the action of the divided differences on @xmath92 $ ] lifts to @xmath125 $ ] , and prove the key result about them and the @xmath14 ( proposition [ uniq ] ) using our raising operator approach . the proof of theorem [ mainthm ] and its corollaries is contained in section [ gdds ] . we thank andrew kresch and leonardo mihalcea for helpful discussions related to this project , and takeshi ikeda and tomoo matsumura for informative correspondence regarding their joint work .

we use young s raising operators to introduce and study _ double theta polynomials _ , which specialize to both the theta polynomials of buch , kresch , and tamvakis , and to double ( or factorial ) schur-polynomials and-polynomials . these double theta polynomials give giambelli formulas which represent the equivariant schubert classes in the torus - equivariant cohomology ring of symplectic grassmannians , and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations .

we use young s raising operators to introduce and study _ double theta polynomials _ , which specialize to both the theta polynomials of buch , kresch , and tamvakis , and to double ( or factorial ) schur-polynomials and-polynomials . these double theta polynomials give giambelli formulas which represent the equivariant schubert classes in the torus - equivariant cohomology ring of symplectic grassmannians , and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations .

astro-ph9809056

i

in this paper we have presented new hst / wfpc2 images of 11 high redshift radio galaxies , all complemented with vla radio maps of comparable resolution . the images reveal a wide variety in the morphology of the host galaxies of these high redshift radio sources : in particular most objects have a clumpy , irregular appearance , consisting of a bright nucleus and a number of smaller components . the number of clumps seems to increase with increasing radio size . the uv continuum emission is generally elongated and aligned with the axis of the radio sources , however the characteristics of the `` alignment effect '' differ greatly from case to case . the new data confirm that none of the proposed models can satisfactorily explain the phenomenon and that most probably the aligned continuum emission is a mixture of star light , scattered light , and nebular continuum emission . our data show no significant evolution in the morphological properties over the redshift interval . finally , we compare the properties of our radio galaxies with those of the uv dropout galaxies and conclude that high redshift radio galaxies might be forming from aggregates of sub - clumps similar to the uv dropout galaxies and that they will probably evolve into present day brightest cluster galaxies . + in a future paper we will present complementary hst / nicmos data of an enlarged sample of high redshift radio galaxies . the new infrared observations will provide constrains to the age of the older stellar population of the host galaxies . with the high resolution we will be able to determine if also the older stellar population shows significant clumpy sub - structures and to what extent are the forming brightest cluster ellipticals already assembled and relaxed . this work is based on observations with the nasa / esa hubble space telescope , obtained at the space telescope science institute , which is operated by aura inc . under contract with nasa . we thank c. carilli for doing the reduction of the radio data for tx 1707 + 105 . hjar acknowledges support from an eu twinning project , a programme subsidy granted by the netherlands organization for scientific research ( nwo ) and a nato research grant . the work by wvb at igpp / llnl was performed under the auspices of the us department of energy under contract w-7405-eng-48 . lcccccccccc component & s@xmath88&s@xmath89&i@xmath88&i@xmath89&@xmath12&s@xmath90 & s@xmath91 & fp@xmath88 & fp@xmath89 & rm + & mjy & mjy & mjy / beam & mjy / beam & & mjy & mjy & % & % & rad m@xmath35 + + nw & 2.85 & 7.47 & 1.07 & 4.69 & 2.6 & & & & + se & 27.3 & 55.8 & 23.4 & 53.1 & 1.5 & 4.85&4.12 & 18&7.4 & 47 + lcccccccc component & s@xmath92&s@xmath89&s@xmath93&i@xmath92&i@xmath89&i@xmath93 & @xmath94 & @xmath95 + & mjy & mjy & mjy & mjy / beam & mjy / beam & mjy / beam & + + north lobe & 22.9 & 55.2 & 294 & 10.4 & 49.8 & 278 & 1.45&2.9 + core & 0.55 & & & 0.53 & & & + south lobe & 1.77 & 7.14 & 34.8 & 1.27 & 4.66 & 19.2 & 1.2 & 2.5 +

we present the images overlayed on vla radio maps of comparable resolution . we find that on the scale of the hst observations there is a wide variety of morphological structures of the hosting galaxies : most objects have a clumpy , irregular appearance , consisting of a bright nucleus and a number of smaller components , suggestive of merging systems . the uv continuum emission is generally elongated and aligned with the axis of the radio sources , however the characteristics of the `` alignment effect '' differ from case to case , suggesting that the phenomenon can not be explained by a single physical mechanism . we compare the properties of our radio galaxies with those of the uv dropout galaxies and conclude that ( i ) the most massive radio galaxies may well evolve from an aggregate of uv dropout galaxies and ( ii ) high redshift radio galaxies probably evolve into present day brightest cluster galaxies .

with the hubble space telescope we have obtained images of 9 of the most distant radio galaxies . the galaxies , which have redshifts between and , were observed with the wfpc2 camera in a broad band filter ( f606w or f707w , roughly equivalent to v or r - band ) , corresponding to the near ultraviolet emission in the rest frame of the radio galaxies . the total observing time was 2 orbits per object . in this paper we present the images overlayed on vla radio maps of comparable resolution . we also present previously unpublished images , taken from the hst archive , of two other high redshift radio galaxies , observed through similar broad band filters . we find that on the scale of the hst observations there is a wide variety of morphological structures of the hosting galaxies : most objects have a clumpy , irregular appearance , consisting of a bright nucleus and a number of smaller components , suggestive of merging systems . some observed structures could be due ( at least partly ) to the presence of dust distributed through the galaxies . the uv continuum emission is generally elongated and aligned with the axis of the radio sources , however the characteristics of the `` alignment effect '' differ from case to case , suggesting that the phenomenon can not be explained by a single physical mechanism . we compare the properties of our radio galaxies with those of the uv dropout galaxies and conclude that ( i ) the most massive radio galaxies may well evolve from an aggregate of uv dropout galaxies and ( ii ) high redshift radio galaxies probably evolve into present day brightest cluster galaxies .

hep-ph0107200

c

for the example of @xmath1-theory we have shown that self - consistent dyson resummations based on a @xmath0-derivable scheme can be renormalized with local counter terms defined on the self - consistently determined vacuum level . this result was obtained with help of weinberg s power counting theorem and using the bphz - renormalization scheme with the usual modifications for finite temperature diagram rules , which can be summarized in the simple rule that the `` contraction boxes '' defining the counter terms have to exclude sub - diagrams which contain any temperature line . the hidden subdivergence structure of the self - consistent scheme has been resolved . this leads to a bethe - salpeter equation for the vacuum four - point function compatible with the chosen @xmath0-approximation , which we have renormalized . the method is free of pinch singularities . closed equations could be formulated which resum the non - perturbative structure of both , the equations of motion , i.e. , the self - energies and also the non - perturbative counter - term structure . the complexity of these equations is comparable to standard dyson resummation schemes and therefore in principle does not imply new techniques . first numerical applications , which include the construction of the bs - kernel , the solution of the half - sided four - point function and thus the renormalized self - energies up to the self - consistent sunset self - energy , are presented in a second paper @xcite . the renormalization of the generating functional @xmath2 , c.f . [ sect - re - ga ] , shows that the derivation and thus the renormalized in - matter equations of motion equally apply to the general non - equilibrium case . this also proves that there is no arbitrariness in studying the in - medium modifications of model parameters like the mass and the coupling constants within this class of approximation schemes : it is sufficient to adjust them in the vacuum , for instance by fitting them to scattering data , in order to predict without ambiguity how they change in the dense and hot medium : _ the in - medium modifications are ruled completely by the model and its vacuum parameters alone , no further assumptions need to be made . _ although demonstrated for the @xmath1-theory , the method is in principle general , since the derivation only relies on the analytic and asymptotic form of the propagators . in particular the renormalization of hidden overlapping divergences in the logarithmically divergent bethe - salpeter equations is general . still , there is a number of restrictions of the self consistent dyson resummation within the @xmath0-derivable scheme , which concerns global and local symmetries and the corresponding conservation laws and ward - takahashi identities . the @xmath0-functional formalism only ensures the conservation laws for the expectation values of charges associated with the symmetry by noether s theorem . however , in general the ward - takahashi identities are violated for the self - energy and higher vertex functions . heuristically the problem can be traced back to the violation of crossing symmetry by the self - consistent scheme : our derivation shows that the self - consistent solution of the self - energy involves bethe - salpeter ladder resummations of the four - point function , but only in the @xmath79-chanel . the crossing symmetric @xmath80- and @xmath81- channel contributions to the four - point function are not included . the symmetry properties of the @xmath195$]-functional were already investigated by us with help of the here applied path - integral method @xcite . we show that it is always possible to define a _ non - perturbative _ approximation to the effective action @xmath196 $ ] which respects linearly realized symmetries of the classical action provided the symmetry is not anomalously broken . the self - energy and higher vertex functions defined from this improved approximation action formalism then fulfill the ward - takahashi identities of the underlying symmetry . as a result the effective action @xmath197 enforces that additional @xmath80- and @xmath81-channel bethe - salpeter resummations are needed to restore the crossing symmetry together with the ward - takahashi identities for the self - energy and the vertex functions . however , these vertex functions are not self - consistently calculated and thus some problems remain also within this approximation : for instance in the case of the linear sigma - model the o(n)-symmetry is restored for the vertex - functions and the goldstone - modes become massless . yet , the phase transition from the nambu - goldstone phase at low temperatures to the wigner - weyl phase at high temperatures results to be of @xmath198 order rather than @xmath199 order @xcite . in the case of a local gauge symmetry the problems become even more intricate : self - consistent schemes beyond the classical field level for the gauge fields generally violate local gauge symmetries for the same reasons as for global symmetries . however , this immediately causes the excitation of spurious modes of the gauge fields which leads to violation of the unitarity of the s - matrix , the positive definiteness of the statistical operator and the causality structure of green s functions . nevertheless a gauge invariant effective action @xmath197 within a background - field approach can be formulated which provides gauge covariant polarization functions @xcite . from a practical point of view the problem remains to calculate the self - consistent propagators needed for the symmetry - restoring bethe - salpeter resummation , which presently can only be solved in simple cases ( rpa bubble resummation ) . in @xcite we have presented a workaround in terms of a suitably chosen projection method onto the physical ( transverse ) degrees of freedom of the gauge - field polarization tensor . this procedure , of course , does not lead to a full restoration of local gauge theory but to causal green s functions and current conservation within the self - energies of matter - fields . alternative methods are restricted to the approximate solution of the self - consistent equations of motion , e.g. , in the sense of a hard thermal loop approximation @xcite or to a systematic expansion in terms of the coupling constant or @xmath29 @xcite . the proof of the renormalizability of @xmath0-derivable approximations opens a broad range of perspectives for effective field theory model applications describing the non - perturbative in - medium properties of particles in dense or finite - temperature matter with model parameters fixed at the vacuum level . further applications point towards the appropriate renormalization of non - equilibrium transport equations @xcite , where in particular the drift terms , which determine the equation of state , involve the real part of the self - energies which generally need renormalization .

within finite temperature field theory , we show that truncated non - perturbative self - consistent dyson resummation schemes can be renormalized with local counter terms defined at the vacuum level . this guarantees the standard-derivable properties like thermodynamic consistency and exact conservation laws also for the renormalized approximation schemes to hold .

within finite temperature field theory , we show that truncated non - perturbative self - consistent dyson resummation schemes can be renormalized with local counter terms defined at the vacuum level . the requirements are that the underlying theory is renormalizable and that the self - consistent scheme follows baym s-derivable concept . the scheme generates both , the renormalized self - consistent equations of motion and the closed equations for the infinite set of counter terms . at the same time the corresponding 2pi - generating functional and the thermodynamical potential can be renormalized , in consistency with the equations of motion . this guarantees the standard-derivable properties like thermodynamic consistency and exact conservation laws also for the renormalized approximation schemes to hold . the proof uses the techniques of bphz - renormalization to cope with the explicit and the hidden overlapping vacuum divergences .

1510.02896

i

let @xmath1 be a 3-manifold with positive ricci curvature . in this paper we obtain quantitative results about sweepouts of @xmath1 by 1-cycles and surfaces . [ main_cycle ] every closed 3-manifold @xmath1 of positive ricci curvature admits a map @xmath5 with fibers of length at most @xmath6 for a universal constant @xmath7 . moreover , the family of 1-cycles @xmath8 is continuous in a strong sense . we define this precisely in section 3 . the geometric invariant that we bound from above in theorem [ main_cycle ] is called the k - waist of a riemannian manifold . for an n - dimensional manifold @xmath1 and @xmath9 the k - waist , @xmath10 , is defined as @xmath11 , where the infimum is taken over all proper functions @xmath12 . waists have been defined by gromov and extensively studied in @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . gromov proved deep theorems about waists of manifolds ( see @xcite and @xcite for exposition of some of his work ) , yet many significant questions about waists remain open . in particular , existence of upper bounds for @xmath10 when @xmath13 is a widely open question . in @xcite guth asked if for every riemannian metric @xmath14 on a 3-torus @xmath15 the 1-waist @xmath16 is bounded from above by @xmath17 . more generally , one may ask analogous questions for any riemannian 3-manifold , that is , does there exist a constant @xmath18 , such that for any metric @xmath14 there exists a map @xmath5 with fibers of length at most @xmath19 ? if this is true , it would be a strong generalization of the systolic inequality . in theorem [ main_cycle ] we affirmatively answer this question for 3-manifolds @xmath1 under an additional assumption of @xmath20 . it is to our knowledge the first occasion where such a generalization is proved . theorem [ main_cycle ] generalizes , in the setting of positive ricci curvature , some fundamental theorems about the length of closed geodesics and stationary one - cycles in riemannian manifolds due to gromov and nabutovsky - rotman . as consequence of theorem [ main_cycle ] we obtain [ systolic ] let @xmath1 be a closed 3-manifold of positive ricci curvature . if @xmath1 is homeomorphic to @xmath21 then it contains a non - contractible closed geodesic of length at most @xmath22 . if @xmath23 then it contains a geodesic net of length at most @xmath22 . the first statement of corollary [ systolic ] is a special case of gromov s systolic inequality @xcite . the systole of a riemannian manifold is defined as the length of the shortest non - contractible geodesic loop . in the same paper gromov conjectured that every manifold contains a non - trivial closed geodesic of length at most @xmath24 . nabutovsky and rotman @xcite proved that every riemannian manifold contains a stationary 1-cycle of length at most @xmath25 ( a stationary 1-cycle need not be a closed geodesic ; it may look , for example , like a bouquet of geodesic loops all intersecting at a point with tangent vectors at that point summing up to @xmath26 , see @xcite for more examples ) . if @xmath1 is topologically a sphere , then a min - max argument yields an upper bound for the length of a stationary 1-cycle , giving an alternative proof of a special case of the result of nabutovsky and rotman . here we present a short ( and incomplete ) overview of previously known estimates for sweepouts of manifolds . in @xcite guth proved that every open subset of euclidean space @xmath27 admits a sweepout by relative @xmath28-cycles of volume at most @xmath29 . in general such inequalities do not hold for riemannian manifolds ( see appendix 5 in @xcite and @xcite ) . however , we may control volumes of @xmath30-cycles if we impose an additional requirement on the metric . in @xcite , among other results , it was shown that if @xmath1 is conformally equivalent to a manifold with non - negative ricci curvature then it admits a sweepout by @xmath30-cycles of volume at most @xmath31 ( in @xcite sabourau independently constructed a sweepout of @xmath1 with @xmath32 by @xmath30-cycles of controlled volume ) . + when @xmath20 and @xmath33 we show that we can simultaneously control the area and the genus of surfaces in the sweepout , which will be essential in the proof of theorem [ main_cycle ] . [ main theorem3 ] given a three manifold with an arbitrary metic @xmath0 of positive ricci curvature , i.e. @xmath34 , there exists a minimal surface @xmath35 , such that @xmath36 , for a universal constant @xmath7 . also we have * if @xmath37 is orientable , then the genus @xmath38 of @xmath39 satisfies @xmath40 , and there exists a smooth sweepout @xmath41}$ ] of @xmath0 , such that * * @xmath42 forms a heegaard splitting of @xmath43 , i.e. @xmath44 is an embedded surface of genus @xmath38 , for @xmath45 , and @xmath46 and @xmath47 are graphs ; * * @xmath48 for @xmath49 . * if @xmath37 is non - orientable , then the genus @xmath50 of its double cover @xmath51 satisfies @xmath52 . moreover , by removing @xmath39 from @xmath1 , we get a manifold with boundary @xmath53 with @xmath54 , and there exists a smooth sweepout @xmath55}$ ] of @xmath53 , such that * * @xmath42 forms a heegaard splitting of @xmath53 , i.e. @xmath44 is an embedded surface of genus @xmath38 lying in the interior of @xmath53 , for @xmath56 , and @xmath57 ; * * @xmath58 for @xmath49 . after this paper was submitted for publication ketover , marques and neves @xcite proved that the min - max minimal surface @xmath37 in 3-manifolds with positive ricci curvature must be orientable . this result follows from a clever application of the catenoid estimate they derive . therefore , the second case of theorem 1.3 does not occur . note that this simplifies our construction of a continuous sweepout of @xmath1 by 1-cycles as it rules out case 2 in the proof of theorem [ main_cycles1 ] ( see section 5 ) . we would like to compare our result with that of f. marques and a. neves @xcite . in @xcite , assuming @xmath34 and the scalar curvature lower bound @xmath59 , marques - neves produced a smooth sweepout @xmath55}$ ] , where the genus of @xmath44 is the heegaard genus . ] , and @xmath60 . the advantage of @xcite is that they have better estimates for the genus . however , from the point of view of area estimates ( e.g. for the application to prove theorem [ main_cycle ] ) , our result can be much better than that in @xcite while we still have a relatively good genus estimate . an example illustrating this fact is a long and thin 3-dimensional ellipsoid ; when we normalize the scalar curvature lower bound to be @xmath61 , the width can be very small ( compared to @xmath62 ) . the difference between our method with @xcite is that we use the almgren - pitts min - max theory @xcite for general sweepouts constructed in @xcite , while marques - neves used the colding - de lellis @xcite ( or simon - smith @xcite ) min - max method for smooth sweepouts given by heegaard splittings . we refer to [ open questions ] for more discussion . + the sweepout @xmath42 in theorem [ main theorem3 ] is used to construct a sweepout by 1-cycles of controlled length in theorem [ main_cycle ] . an important open question is whether one can construct a sweepout by _ closed curves _ of controlled length rather than 1-cycle ( see more discussion in [ open questions ] ) . one approach in this direction is to first construct a sweepout of @xmath1 by spheres or tori of controlled area and _ diameter_. for this purpose , we derive the following partial result . in particular , if we further assume a scalar curvature lower bound , we can get a uniform diameter upper bound for the min - max minimal surface . [ main theorem3.1 ] let @xmath0 be as in theorem [ main theorem3 ] ; if the scalar curvature of @xmath0 is bounded from below , i.e. @xmath63 , for some @xmath64 , then the diameter of @xmath35 ( when it is orientable ) or the diameter of its double cover ( when it is non - orientable ) is bounded from above by @xmath65 . the main idea of proving theorem [ main_cycle ] is a dimension reduction type argument . we first construct a nice sweepout by 2-surfaces with controlled area and genus by theorem [ main theorem3 ] . then we continuously sweep out these 2-surfaces by 1-cycles . a large portion of the argument ( in particular , proof of lemma [ main_cycles1 ] ) is devoted to making this family continuous in a strong sense ( cf . section [ intro to 1-cycles ] ) . this continuity is needed to obtain a geodesic net of controlled length as a limit of a min - max sequence of 1-cycles . theorem [ main theorem3 ] is proved by combining several ingredients . we apply the almgren - pitts min - max theory to the sweepout constructed in @xcite and get a min - max minimal surface of controlled area . by using one of the authors morse index bound @xcite , we can get the desired genus bound via schoen - yau genus estimates @xcite . the existence of good heegaard splitting follows from meeks - simon - yau @xcite . the diameter estimates ( theorem [ main theorem3.1 ] ) for the min - max surface comes from schoen - yau diameter estimates @xcite and the morse index estimate . our paper is organized as follows . in [ area and diameter estimates ] , we prove theorem [ main theorem3 ] and theorem [ main theorem3.1 ] . in [ intro to 1-cycles ] we give a precise definition of the sweepout by 1-cycles . in [ parametric sweepouts of surfaces ] , we show how to sweep out a family of surfaces simultaneously by continuous 1-cycles with lengths controlled by the genus and area . in [ proof of main_cycle ] we prove theorem [ main_cycle ] by combining results in [ area and diameter estimates ] and [ parametric sweepouts of surfaces ] . finally , we summarize several interesting open questions in [ open questions ] . : both authors would like to thank larry guth for getting them together and useful comments . y.l . would like to thank alexander nabutovsky and regina rotman for helpful discussions and alexander nabutovksy for pointing out a mistake in the earlier draft .

* abstract : * we prove that given a three manifold with an arbitrary metric of positive ricci curvature , there exists a sweepout of by surfaces of genus and areas bounded by . we use this result to construct a sweepout of by 1-cycles of length at most , giving a partial answer to a question of l. guth . the sweepout of surfaces is generated from a min - max minimal surface . if further assuming a positive scalar curvature lower bound , we can get a diameter upper bound for the min - max surface . [ section ] [ theorem]proposition [ theorem]corollary [ theorem]remark [ theorem]definition [ theorem]lemma * * 12pt [ runin ] * * .

* abstract : * we prove that given a three manifold with an arbitrary metric of positive ricci curvature , there exists a sweepout of by surfaces of genus and areas bounded by . we use this result to construct a sweepout of by 1-cycles of length at most , giving a partial answer to a question of l. guth . the sweepout of surfaces is generated from a min - max minimal surface . if further assuming a positive scalar curvature lower bound , we can get a diameter upper bound for the min - max surface . [ section ] [ theorem]proposition [ theorem]corollary [ theorem]remark [ theorem]definition [ theorem]lemma * * 12pt [ runin ] * * .

1408.4407

i

the classical and pioneering works in the field of stellar physics include dynamics of supernova explosion by baade @xmath5 zwicky @xcite , energy production in stars and stellar evolution by bethe @xcite , and , synthesis of elements in stars by burbidge et al . since then the microphysics of supernova explosion has come a long way . whereas the strong interactions ( fusion reactions ) are responsible for providing the fuel to the stars empowering them throughout their life cycles , it is precisely the weak interactions that play a decisive role in determining both the presupernova stellar structure and the nucleosynthesis . the weak interaction reactions lead to the initiation of the gravitational collapse of the core of a massive star triggering a supernova explosion , control the lepton - to - baryon fraction of the core throughout the course of stellar evolution , play a key role in neutronisation of the core material , and , affect the formation of heavy elements above iron via the r - process at the final stage of the supernova explosion . much to the advantage of the astrophysicists , the temperature during the late phases of stellar evolution is high enough for the matter composition to be given by nuclear statistical equilibrium . this means that one can get away with the need of reaction networks for the strong and electromagnetic interactions and the composition of the matter is given by the saha equation . weak interactions in presupernova stars are known to be dominated by allowed fermi ( vector - type ) and gamow - teller ( axial - vector type ) transitions . the calculation of weak - interaction rates is very sensitive to the distribution of the gt@xmath6 strength function . in the gt@xmath7 strength a proton is changed into a neutron whereas the gt@xmath8 strength is responsible for transforming a neutron into a proton . it was fuller , fowler , and newman ( ffn ) @xcite who first performed an extensive calculation of stellar weak rates including the capture rates , decay rates , neutrino energy loss rates and gamma heating rates for a wide density and temperature domain . they performed their detailed calculation for 226 nuclei in the mass range @xmath9 . the authors recognized the key role played by the gt giant resonance and noted that measured decay rates exploited only a small fraction of the total available strength . the centroids of the gt@xmath6 distribution functions determine the effective energy of the capture and decay reactions . ffn estimated the gt contributions to the rates by a parametrization based on the independent particle model . aufderheide et al . @xcite later extended the ffn work for heavier nuclei with a @xmath10 60 and took into consideration the quenching of the gt strength neglected by ffn . authors in ref . @xcite also stressed on the importance of @xmath0-decay rates in the iron core prior to the collapse . they tabulated the 71 top @xmath0-decay nuclei averaged throughout the stellar trajectory for @xmath11 ( see table 26 of ref . the measured data from various @xmath12 and @xmath13 experiments later revealed the misplacement of the gt centroids adopted in the parameterizations of ffn and subsequently used in the calculation of weak rates by ref . since then theoretical efforts were concentrated on the microscopic calculations of weak - interaction mediated rates of iron - regime nuclide . two such widely used models are the large - scale shell model ( lssm)(e.g . @xcite ) and the proton - neutron quasiparticle random phase approximation ( pn - qrpa ) theory ( e.g. @xcite ) . the pn - qrpa theory is an efficient way to generate gt strength distributions which constitute a primary and nontrivial contribution to the weak - interaction mediated rates among iron - regime nuclide . the shell model calculations perform a detailed study of the nuclear spectroscopy , however , the pn - qrpa model has two important advantages . it can handle any arbitrarily heavy system of nucleons since the calculation is performed in a luxurious model space of up to 7 major oscillator shells . the second advantage is even more important for the calculation of weak - interaction rates in stellar matter . because of the high temperatures prevailing during the presupernova and supernova phases of a massive star , there is a reasonable probability of occupation of parent excited states and the total weak interaction rates have a finite contribution form these excited states . thus , in calculating a stellar rate , one must know the gt strength distributions of the excited states of the parent nucleus . as experimental information about excited state strength functions seems inaccessible , aufderheide @xcite stressed much earlier the need to probe these strength functions theoretically . the pn - qrpa model calculates the gt strength distribution strengths of _ all _ excited states of parent nucleus in a microscopic fashion and this feature of the pn - qrpa model greatly enhances the reliability of the calculated rates in stellar matter . these excited states are like resonances having finite band width and contributions from many discrete states calculated microscopically within the pn - qrpa framework . the construction of these excited states and calculation of the relevant nuclear matrix elements will be shown in the next section . in this sense the pn - qrpa model allows a fully microscopic `` state - by - state '' calculation of stellar weak rates . the calculation of stellar weak rates on iron - regime nuclei are sensitive to both the weak low - energy and strong high - energy ground and parent excited state gt strength distributions . the low energy part is more important during the earlier phase and the high energy part becomes important during the late phases of stellar evolution @xcite . previous pn - qrpa calculations have shown that for certain nuclei the excited state rates can command the total weak rate ( e.g. @xcite ) . this clearly endorses the contribution of excited states in the calculation of total weak rates . other calculations revert to approximations like the so - called brink s hypothesis ( in the electron capture direction ) and back - resonances ( in the @xmath0-decay direction ) . brink s hypothesis states that gt strength distribution on excited states is _ identical _ to that from ground state , shifted _ only _ by the excitation energy of the state . gt back resonances are the states reached by the strong gt transitions in the inverse process ( electron capture ) built on ground and excited states . nabi and klapdor - kleingrothaus @xcite first reported the calculation of weak interaction rates for 709 nuclei with a = 18 to 100 in stellar environment using the pn - qrpa theory . these included capture rates , decay rates , gamma heating rates , neutrino energy loss rates , probabilities of beta - delayed particle emissions and energy rate of these particle emissions . the authors then presented a detailed calculation of stellar weak interaction rates over a wide range of temperature and density scale for sd- @xcite and fp / fpg - shell nuclei @xcite . these also included the weak interaction rates for nuclei with a = 40 to 44 ( not yet calculated by shell model ) . since then these calculations were further refined with use of more efficient algorithms , computing power , incorporation of latest data from mass compilations and experimental values , and fine - tuning of model parameters both in the sd- shell @xcite and fp - shell @xcite region . all theoretical calculations of stellar weak interactions have inherent uncertainties . the uncertainties associated with the pn - qrpa model were discussed in detail in ref . the reliability of the pn - qrpa calculation was discussed in length by nabi and klapdor - kleingrothaus @xcite . there the authors compared the measured data ( half lives and b(gt@xmath6 ) strength ) of thousands of nuclide with the pn - qrpa calculation and got fairly good comparison . earlier half - lives of @xmath14 decays were calculated systematically for about 6000 neutron - rich nuclei between the beta stability line and the neutron drip line using the pn - qrpa model @xcite . similarly half - lives for @xmath15/ec ( electron capture ) decays for neutron - deficient nuclei with atomic numbers z = 10 - 108 were calculated up to the proton drip line for more than 2000 nuclei using the same model @xcite . these microscopic calculations gave a remarkably good agreement with the then existing experimental data ( within a factor of two for more than 90@xmath16 ( 73@xmath16 ) of nuclei with experimental half - lives shorter than 1 s for @xmath14 ( @xmath15/ec ) decays ) . most nuclei of interest of astrophysical importance are the ones far from stability and one has to rely on theoretical models to estimate their beta decay properties . the accuracy of the pn - qrpa model increases with increasing distance from the @xmath0-stability line ( shorter half - lives ) @xcite . this is a promising feature with respect to the prediction of experimentally unknown half - lives ( specially those present in the stellar interior ) , implying that the predictions are made on the basis of a realistic physical model . the isotopes of iron , @xmath3fe , are mainly responsible for decreasing the electron - to - baryon ratio during the oxygen and silicon burning phases of massive stars through electron capture and positron decay processes . the electron captures on these iron isotopes are the dominant process . nevertheless , the @xmath0-decay rates for these isotopes of iron are also argued to be relevant during the presupernova evolution of massive stars in literature . because of their astrophysical importance @xmath4fe were included in the list of key @xmath0-decay nuclei that have a significant impact on the presupernova evolution of massive stars after core silicon burning for @xmath17 compiled by aufderheide and collaborators ( see tables 19 , 20 and 26 of ref . @xcite ) . later heger and collaborators @xcite studied the presupernova evolution of massive stars ( of masses @xmath18 and @xmath19 ) and found @xmath20fe in the list of top five nuclei that increase @xmath1 via @xmath0-decay and positron capture during the silicon burning phases of these massive stars . the authors employed the lssm @xmath0-decay rates in their simulation code . the calculations of gt@xmath6 strength distributions and stellar weak rates for these isotopes of iron were introduced earlier using the pn - qrpa theory with improved model parameters @xcite . there the author was able to reproduce fairly well the experimental centroids and the total strength distributions in both directions for the even - even iron isotopes , @xmath21fe where measurements were available . in this paper i discuss in detail the calculation of @xmath2 decay rates of @xmath3fe in stellar environment and its astrophysical implications . the main finding of this work is that @xmath0-decay rates on @xmath3fe are around 3 5 orders of magnitude smaller than previously assumed and are irrelevant for the determination of the evolution of @xmath1 during the presupernova phases of massive stars . in the following section i briefly describe the theoretical formalism used to calculate the stellar electron and positron decay rates . the stellar @xmath2-decay rates of @xmath3fe are presented in section 3 . here i also compare the pn - qrpa decay rates with earlier calculations . summary and conclusions are finally presented in section 4 .

-decay and positron decay are believed to play a consequential role during the late phases of stellar evolution of a massive star culminating in a supernova explosion . the-decay contributes in maintaining a respectable lepton - to - baryon ratio , , of the core prior to collapse which results in a larger shock energy to produce the explosion . the structure of the presupernova star is altered both by the changes in and the entropy of the core material . recently the microscopic calculation of weak - interaction mediated rates on key isotopes of iron was introduced using the proton - neutron quasiparticle random phase approximation ( pn - qrpa ) theory with improved model parameters . here i discuss in detail the improved calculation of decay rates for iron isotopes (fe ) in stellar environment . the pn - qrpa theory allows a microscopic `` state - by - state '' calculation of stellar rates as explained later in text . the main finding of this work includes that the stellar-decay rates offe are around 3 5 orders of magnitude smaller than previously assumed and hence irrelevant for the determination of the evolution of during the presupernova phase of massive stars . beta decay , gt strength distributions , pn - qrpa , core - collapse supernovas , massive stars 21.60.jz , 23.40.-s , 26.50.+x , 97.10.cv = 0.5 cm

-decay and positron decay are believed to play a consequential role during the late phases of stellar evolution of a massive star culminating in a supernova explosion . the-decay contributes in maintaining a respectable lepton - to - baryon ratio , , of the core prior to collapse which results in a larger shock energy to produce the explosion . the positron decay acts in the opposite direction and tends to decrease the ratio . the structure of the presupernova star is altered both by the changes in and the entropy of the core material . recently the microscopic calculation of weak - interaction mediated rates on key isotopes of iron was introduced using the proton - neutron quasiparticle random phase approximation ( pn - qrpa ) theory with improved model parameters . here i discuss in detail the improved calculation of decay rates for iron isotopes (fe ) in stellar environment . the pn - qrpa theory allows a microscopic `` state - by - state '' calculation of stellar rates as explained later in text . excited state gamow - teller distributions are much different from ground state and a microscopic calculation of decay rates from these excited states greatly increases the reliability of the total decay rate calculation specially during the late stages of stellar evolution . the reported decay rates are also compared with earlier calculations . the positron decay rates are in reasonable agreement with the large - scale shell model calculation . the main finding of this work includes that the stellar-decay rates offe are around 3 5 orders of magnitude smaller than previously assumed and hence irrelevant for the determination of the evolution of during the presupernova phase of massive stars . the current work discourages the inclusion offe in the list of key stellar-decay nuclei as suggested by former simulation results . beta decay , gt strength distributions , pn - qrpa , core - collapse supernovas , massive stars 21.60.jz , 23.40.-s , 26.50.+x , 97.10.cv = 0.5 cm

1408.4407

r

the improved pn - qrpa model , as described in the previous section , was used to calculate the gt@xmath6 strength distributions of iron isotopes in astrophysical conditions . the model incorporated experimental excitation energies , log@xmath33 values and deformations as discussed earlier . the gt strength distributions ( in both directions ) were calculated for ground and 245 excited states in @xmath31fe , 296 excited states in @xmath20fe and 265 excited states in @xmath32fe . the model was tested for the case of @xmath21fe where measured distributions were available . it was shown in ref . @xcite that the comparison of gt@xmath6 strength distributions for @xmath21fe with measured data was very good ( e.g. the total b(gt@xmath8 ) strength for @xmath31fe calculated within the framework of ref . @xcite was 9.33 which was narrowed down to 7.56 using the improved model of pn - qrpa @xcite in comparison with the measured value of 7.5 @xmath88 0.7 @xcite ) . the gt@xmath6 strength distributions for ground and all excited states of iron isotopes are available as ascii files and can be requested from the author . in order to highlight the important role of parent excited states in the calculation of total decay rates , i present in table 1 the values of the total beta decay ( @xmath89 ) and positron decay ( @xmath90 ) rates with contributions from all excited states in units of @xmath91 . these rates are calculated at selected values of astrophysical temperature and density shown in first column ( the first number within the parenthesis gives the value of density in units of @xmath92 and the second number denotes the stellar temperature in units of @xmath93 k ) . it is to be noted that the corresponding ground - state beta and positron decay rates are energetically forbidden . the excited states play a key role in the calculation of total decay rates which are very sensitive to the available phase space ( @xmath94 ) . the construction of excited states in the pn - qrpa model was discussed in the previous section . the phase space integrals given in eq . ( [ phase space ] ) increase considerably at higher stellar temperatures and cause orders of magnitude enhancement in the calculated decay rates . fine grid calculation of positron and @xmath0-decay rates for iron isotopes as a function of stellar temperature , density and fermi energy , suitable for core - collapse simulations and interpolation purposes , is available as ascii files and can be requested from the author . figures [ fig1 ] , [ fig2 ] and [ fig3 ] show the calculated @xmath0-decay rates for @xmath3fe , respectively . each figure shows four panels depicting the calculated @xmath0-decay rates at selected temperature and density domain . the upper left panel shows the decay rates in low - density region ( @xmath95 = 10^{0.5 } , 10^{1.5}$ ] and @xmath96 ) , the upper right in medium - low density region ( @xmath95 = 10^{3.5 } , 10^{4.5}$ ] and @xmath97 ) , the lower left in medium - high density region ( @xmath98 = 10^{6.5 } , 10^{7.5}$ ] and @xmath99 ) and finally the lower right panel depicts the calculated rates in high density region ( @xmath95 = 10^{9.5 } , 10^{10.5}$ ] and @xmath100 ) of stellar core of massive stars . the decay rates are given in logarithmic scales ( to base 10 ) in units of @xmath91 . t@xmath101 gives the stellar temperature in units of @xmath93 k. one should note the orders of magnitude increment in @xmath0-decay rates as the stellar temperature increases . the rates are almost superimposed on one another as a function of stellar density in the first three panels . this means that there is no appreciable change in the @xmath0-decay rates when increasing the density by an order of magnitude . however as the stellar matter moves from the medium high density region to high density region these rates start to peel off from one another . in high density regions the rates start to decrease appreciably due to a decrease in the phase space . there is a sharp exponential increase in the decay rates as the stellar temperature increases up to @xmath102 t@xmath101 = 5 . beyond this temperature the slope of the rates reduces drastically with increasing density . for a given density the @xmath0-decay rates increase monotonically with increasing temperatures . the calculated stellar @xmath0-decay rates are smallest for @xmath31fe and biggest for @xmath20fe . the calculated @xmath0-decay rates are smaller in magnitude compared to the corresponding positron capture rates ( in the same direction ) on these iron isotopes . figures [ fig4 ] , [ fig5 ] and [ fig6 ] again show four panels depicting the calculated positron decay rates at selected temperature and density domain for @xmath3fe , respectively . the upper left panel shows the positron decay rates in low - density region ( @xmath95 = 10^{0.5 } , 10^{1.5}$ ] and @xmath96 ) , the upper right in medium - low density region ( @xmath95 = 10^{3.5 } , 10^{4.5}$ ] and @xmath97 ) , the lower left in medium - high density region ( @xmath95 = 10^{6.5 } , 10^{7.5}$ ] and @xmath99 ) and finally the lower right panel depicts the calculated positron decay rates in high density region ( @xmath95 = 10^{9.5 } , 10^{10.5}$ ] and @xmath100 ) of the stellar core . the rates are given in logarithmic scales ( to base 10 ) in units of @xmath91 . the calculated positron decay rates are greater than the corresponding @xmath0-decay rates by orders of magnitude . due to an increase in phase space with increasing temperature , the positron decay rates increase by orders of magnitude as t@xmath101 increases . the increase is exponential up to @xmath102 t@xmath101 = 5 . as temperature rises ( the degeneracy parameter is negative for positrons ) , more and more high - energy positrons are created leading in turn to higher decay rates . it can be seen from these figures that the positron decay rates are almost the same as functions of the core density . the positron decay rates are biggest for @xmath20fe and smallest for @xmath32fe . it is worth mentioning that the beta and positron decay rates are very small numbers and can change by orders of magnitude by a mere change of 0.5 mev , or less , in parent or daughter excitation energies ( as the rates are very sensitive to the available q@xmath103 window ) and are more reflective of the uncertainties in the calculation of energies . it would be interesting to know how the reported decay rates of iron isotopes compare with earlier calculations for temperature and density domains of astrophysical interest . for the sake of comparison i took into consideration the pioneer calculations of ffn @xcite and those performed using the large - scale shell model ( lssm ) @xcite . the ffn rates were used in many simulation codes ( e.g. kepler stellar evolution code @xcite ) while lssm rates were employed in recent simulation of presupernova evolution of massive stars in the mass range 11 - 40 @xmath104 @xcite . positron decay rates of iron isotopes may decrease the @xmath1 value of massive stars . as mentioned before , the positron decay rate calculations do not differ appreciably by changing densities . figure [ fig7 ] depicts the comparison of positron decay rates for @xmath31fe with earlier calculations . the upper panel displays the ratio of calculated rates to the lssm rates , @xmath105 , while the lower panel shows a similar comparison with the ffn calculation , @xmath106 . the comparison is made for the selected temperature and density domain . here one sees that the pn - qrpa and lssm calculations are in overall reasonable agreement . the comparison is good at low temperatures while at higher temperatures , @xmath107 5 , the lssm rates are bigger around an order of magnitude . the ffn rates are much bigger than the reported positron decay rates by 1 2 orders of magnitude . it is reminded that ffn neglected the quenching of the gt strength in their rate calculation . further ffn did not take into effect the process of particle emission from excited states and their parent excitation energies extended well beyond the particle decay channel . these high lying excited states began to show their cumulative effect at high temperatures and densities . the comparison of reported positron decay rates of @xmath20fe with lssm calculation is somewhat different . for the first time one sees that the pn - qrpa calculated positron decay rate is bigger than lssm ( figure [ fig8 ] ) . this however is true only for @xmath108 = 1 . during the early phases of presupernova evolution the reported positron decay rates are bigger than lssm rates by an order of magnitude . for successive stages , 1.5 @xmath109 5 , the two calculations are in good agreement . at still higher temperatures the lssm rates exceed by more than an order of magnitude . excited state gt distributions in @xmath20fe play a key role in determining the total decay rate at various stages of presupernova evolution . it is reminded that the pn - qrpa makes a microscopic assessment of gt strength distribution for all such parent excited states . comparison with the ffn rates is fairly constant . it is to be noted that the ratios are shown on a linear scale ( bottom panel of figure [ fig8 ] ) . at @xmath108 = 10 , the ffn rates surpass the reported rates for reasons already mentioned . the comparison of reported positron decay rates of @xmath32fe with previous calculations is shown in figure [ fig9 ] . the lssm decay rates are bigger by as much as a factor of 20 ( the comparison is better at low temperatures ) . the lower panel of figure [ fig9 ] shows a whopping enhancement in the calculated positron decay rates of @xmath32fe of around 13 orders of magnitude compared to ffn rates at low temperatures . however there are reasons for this unusual suppression of ffn positron decay rates . unmeasured matrix elements for allowed transitions were assigned an average value of @xmath1105 in ffn calculations . later , @xmath13 experiments were performed for @xmath32fe @xcite which revealed a too small @xmath111 value assignment for allowed transitions employed by ffn . consequently the ffn rates are much too smaller than the lssm and pn - qrpa calculations at low temperatures ( lssm rates are also bigger than ffn rates by roughly 13 orders of magnitude at @xmath108 = 1.5 ) . the lssm positron decay rates for iron isotopes are bigger up to a factor of 20 as compared to the pn - qrpa rates . the overall comparison of positron decay rates of @xmath3fe shows that the calculated rates are in reasonable agreement with the lssm calculated rates . major differences are seen in the case of the @xmath0-decay rate calculations and are discussed in detail below . authors in ref . @xcite reported that , for even - even and odd - a nuclei , ffn systematically placed the back resonance at much lower energies and concluded that contribution of the back resonances to the @xmath0-decay rates for these nuclei decreases . they estimated that lssm @xmath0-decay rates as a result were smaller , on the average , by a factor of 20 ( 40 ) as compared to the ffn @xmath0-decay rates for even - even ( odd - a ) nuclei . figure [ fig10 ] depicts the comparison of @xmath0-decay rates for @xmath31fe with earlier calculations . as before the upper panel displays the ratio of calculated rates to the lssm rates , @xmath112 , while the lower panel shows a similar comparison with the ffn calculation , @xmath113 . the density scale is shown in the inset . a mutual comparison of lssm and ffn @xmath0-decay rates for the case of @xmath31fe reveals that for low temperatures ( t@xmath114 2 ) and densities @xmath95 \sim 10^{6}-10^{7}$ ] the lssm @xmath0-decay rates are bigger than the ffn rates by as much as four orders of magnitude . only at higher temperatures and densities are the lssm rates smaller than the ffn rates ( by around an order of magnitude ) . the lssm @xmath0-decay rates for @xmath31fe are 3 4 orders of magnitude bigger than the pn - qrpa rates for the physical conditions depicted in figure [ fig10 ] . the enhancement ratio increases with increasing densities . the reasons for bigger lssm @xmath0-decay rates are not very clear the q - values and phase space formulation for the two calculations appear the same . perhaps the contribution from back resonances needs a further cut back in the lssm calculation for the @xmath0-decay of even - even and odd - a nuclei ( i.e. the back resonances for these nuclei should be put at even higher energies than calculated by lssm ) . as mentioned before the reported rates do not employ these approximations and calculate all the excited state gt strength distributions in a microscopic fashion . the comparison with the ffn rates reveals that the reported decay rates are smaller by as much as four orders of magnitude ( at higher temperatures ) . at temperature t@xmath101 = 1 and density @xmath95 = 10^{6}$ ] , where the lssm @xmath0-decay rates are bigger than the ffn rates by roughly four orders of magnitude , the reported rates are bigger than the ffn rates by an order of magnitude . for all other points the ffn rates are much bigger than pn - qrpa rates for reasons mentioned before . the @xmath0-decay rates of @xmath20fe are believed to be important during the silicon burning stages of massive stars as per the simulation results of ref . figure [ fig11 ] shows a comparison of the three calculations for @xmath0-decay rates of @xmath20fe during the relevant temperature and density domain of stellar core . only at low temperatures and densities are the lssm decay rates smaller than the corresponding ffn numbers . at higher temperatures ( t@xmath115 5 ) and densities ( @xmath95 > 10^{7}$ ] ) the lssm rates surpass the ffn decay rates by more than two orders of magnitude . the comparison of pn - qrpa rates with lssm suggests that the lssm rates are bigger by 2 3 orders of magnitude ( at lower densities the comparison is relatively better ) . in fact the lssm @xmath0-decay rates are even bigger than their calculated positron capture rates ( only at higher temperatures their positron capture rates surpass the @xmath0-decay rates ) . in contrast the pn - qrpa calculated @xmath0-decay rates are suppressed as compared to the corresponding positron capture rates for all temperature and density scales . at higher densities the lssm @xmath0-decay rates are bigger by 3 5 orders of magnitude . the ffn rates are up to four orders of magnitude bigger than the pn - qrpa rates for reasons mentioned before . a study of lssm and the ffn rates reveals that lssm @xmath0-decay rates of @xmath32fe are much bigger than the ffn rates at high temperatures and densities ( by more than an order of magnitude ) . figure [ fig12 ] shows how the reported @xmath0-decay rates of @xmath32fe compare with earlier calculations for relevant physical conditions . it can be seen from the figure that the reported rates are much smaller ( up to four orders of magnitude ) than previous calculations . comparison with lssm calculation shows that at higher densities the reported @xmath0-decay rates are suppressed by up to three orders of magnitude . at higher temperatures ( t@xmath101 = 30 ) the comparison ratio improves . however the lssm rates are still bigger by around an order of magnitude . on the other hand the ffn rates are bigger than the reported @xmath0-decay rates by around four orders of magnitude at lower densities and temperatures . the comparison ratio improves with increasing temperatures and densities . the pn - qrpa rates are bigger than the ffn rates by around a factor of 8 at t@xmath101 = 30 . the comparison study of stellar @xmath0-decay rates of iron isotopes suggests that the lssm @xmath0-decay rates are bigger than the corresponding pn - qrpa rates by 3 5 orders of magnitude . earlier using the same model the electron capture rates due to @xmath3fe were calculated and were found to be in overall good comparison with the lssm electron capture rates @xcite . in fact the pn - qrpa electron capture rates on @xmath31fe were around a factor three bigger than lssm rates in the relevant astrophysical conditions ( see table 3 of ref . @xcite ) . using the same model , however , the calculated @xmath0-decay rates are smaller by 3 5 orders of magnitude . this suppression in the pn - qrpa @xmath0-decay rates comes from the excited state gt distributions ( smaller nuclear matrix elements ) which are much different from the ground state distribution . in order to gain a detailed insight of these distributions i took a representative temperature and density point of t@xmath101 = 10 and @xmath95 = 10^{7}$ ] ( weak interaction rates due to iron isotopes are argued to contribute effectively around such physical conditions during the presupernova evolution of massive stars ) . table 2 shows the data of the ground and first four excited states that have a finite partial decay rate contribution to the total @xmath0-decay rate of @xmath3fe within the framework of the model of magnitude greater than @xmath116 . the first column shows the calculated parent excited energy state in units of mev , the second column gives the product of occupation probability and partial @xmath0-decay rate from this state in units of @xmath91 . the third and fourth column give the centroid and total b(gt ) strength , respectively , from this parent state . the cut - off energy in daughter nuclei is 12 mev . it can be seen clearly from table 2 that for the even - even isotopes the centroids of the excited state gt strength distributions are shifted to much higher energies in the daughter . the @xmath117 strengths are also considerably smaller from the corresponding ground state strengths . these are mainly responsible for the smaller total @xmath0-decay rates . the excited state gt strength distributions are also much different from the ground state distribution for the case of @xmath20fe . the excited state gt distributions for @xmath3fe are shown graphically in figures [ fig13 ] , [ fig14 ] and [ fig15 ] , respectively . the ground state gt strength distributions were presented earlier in ref . @xcite . here i have shown only the first three excited state distributions that have a finite partial decay rate contribution of magnitude greater than @xmath116 . note the different scales of the b(gt ) strength values in the vertical panels of these figures . from these figures it is clear that the brink s hypothesis ( and back resonances for the calculation of @xmath0-decay ) is not a good approximation for calculation of stellar weak interaction rates of iron isotopes . these and similar finite partial decay rates lead to an overall suppression in the total @xmath0-decay rate which happens to be 5.44e-09 @xmath91 , 4.32e-07 @xmath91 and 2.62e-05 @xmath91 for @xmath31fe , @xmath20fe and @xmath32fe , respectively , under the physical conditions stated above .

excited state gamow - teller distributions are much different from ground state and a microscopic calculation of decay rates from these excited states greatly increases the reliability of the total decay rate calculation specially during the late stages of stellar evolution . the reported decay rates are also compared with earlier calculations . the positron decay rates are in reasonable agreement with the large - scale shell model calculation .

-decay and positron decay are believed to play a consequential role during the late phases of stellar evolution of a massive star culminating in a supernova explosion . the-decay contributes in maintaining a respectable lepton - to - baryon ratio , , of the core prior to collapse which results in a larger shock energy to produce the explosion . the positron decay acts in the opposite direction and tends to decrease the ratio . the structure of the presupernova star is altered both by the changes in and the entropy of the core material . recently the microscopic calculation of weak - interaction mediated rates on key isotopes of iron was introduced using the proton - neutron quasiparticle random phase approximation ( pn - qrpa ) theory with improved model parameters . here i discuss in detail the improved calculation of decay rates for iron isotopes (fe ) in stellar environment . the pn - qrpa theory allows a microscopic `` state - by - state '' calculation of stellar rates as explained later in text . excited state gamow - teller distributions are much different from ground state and a microscopic calculation of decay rates from these excited states greatly increases the reliability of the total decay rate calculation specially during the late stages of stellar evolution . the reported decay rates are also compared with earlier calculations . the positron decay rates are in reasonable agreement with the large - scale shell model calculation . the main finding of this work includes that the stellar-decay rates offe are around 3 5 orders of magnitude smaller than previously assumed and hence irrelevant for the determination of the evolution of during the presupernova phase of massive stars . the current work discourages the inclusion offe in the list of key stellar-decay nuclei as suggested by former simulation results . beta decay , gt strength distributions , pn - qrpa , core - collapse supernovas , massive stars 21.60.jz , 23.40.-s , 26.50.+x , 97.10.cv = 0.5 cm

1610.00814

i

let @xmath2 be a continuous endomorphism , and @xmath3 be a non - degenerate interval on the real line . let @xmath4 be an @xmath0th iteration of @xmath5 . a point @xmath6 is called a periodic point of @xmath5 with period @xmath7 if @xmath8 , @xmath9 for @xmath10 . the set of @xmath7 distinct points @xmath11 is called the orbit of @xmath12 , or briefly @xmath7-orbit or periodic @xmath7-cycle . in his celebrated paper @xcite , sharkovski discovered a law on the coexistence of periodic orbits of continuous endomorphisms on the real line . [ thm : sharkovskii ] @xcite let the positive integers be totally ordered in the following way : @xmath13 if a continuous endomorphism , @xmath14 , has a cycle of period @xmath0 and @xmath15 , then @xmath5 also has a periodic orbit of period @xmath7 . this result played a fundamental role in the development of the theory of discrete dynamical systems . a conceptually novel proof was given in @xcite . following the standard approach , we characterize each periodic orbit with cyclic permutations and directed graphs of transitions or _ digraphs_. consider @xmath7-orbit : @xmath16 if @xmath17 for @xmath18 , with @xmath19 , then @xmath20 is associated with cyclic permutation @xmath21 let @xmath22 be the order reversing permutation @xmath23 then , given a cyclic permutation @xmath24 , it s inverse is obtained as @xmath25 . in the sequel @xmath26 means either @xmath27 $ ] or @xmath28 $ ] . let @xmath29 $ ] . the digraph of @xmath7-orbit is a directed graph of transitions with vertices @xmath30 and oriented edges @xmath31 if @xmath32 . the inverse digraph of @xmath7-orbit is a digraph associated with inverse cyclic permutation @xmath33 . equivalently , inverse digraph is obtained from the digraph of @xmath7-orbit by replacing each @xmath34 with @xmath35 . proof of the sharkovskii s theorem significantly uses the concept of _ minimal orbit_. @xmath0-orbit of @xmath5 is called minimal if @xmath0 is the minimal period of @xmath5 in sharkovski s ordering . digraph of the @xmath7-orbit contains the red edge @xmath36 if @xmath37 . the structure of the minimal orbits is well understood @xcite . minimal odd orbits are called stefan orbits , due to the following characterization : [ thm : stefan ] @xcite the digraph of a @xmath38 minimal odd orbit has the unique structure given in figure [ fig : minodddigraph ] up to an inverse . similar characterization of @xmath39-orbits ( @xmath40 ) is given in @xcite . [ thm : abdulla13 ] @xcite the digraph of a minimal @xmath39-orbit ( @xmath40 ) has one of four types up to their inverses ( type i is shown in figure [ fig : min2timesodd ] ) . the main idea of the constructive proof of @xcite is based on the fact that each half of the minimal @xmath39-orbit is minimal @xmath41 orbit of @xmath42 . therefore , the digraph of the minimal @xmath39-orbit is designed as one of the possible four `` unions '' of two stefan digraphs of @xmath42 . the result of theorem [ thm : abdulla13 ] can be generalized as follows : [ thm : odd2n ] the digraph of any minimal @xmath43-orbit , @xmath40 , has one of @xmath44 types up to their inverses . furthermore , each digraph is strongly simple and can be constructed from the digraphs of two minimal and strongly simple @xmath45-orbits in @xmath42 . the main goal of this paper is the characterization of second minimal odd orbits . [ thm : secondmin ] an @xmath0-orbit , @xmath46 , of @xmath5 is called second minimal if @xmath0 is the successor of the minimal orbit of @xmath5 in the sharkovskii ordering . for example , if map has a second minimal @xmath1-orbit , then it has a minimal @xmath47-orbit , but no @xmath48-orbit . our main result reads : [ thm : secondmin7 ] the second minimal @xmath1-orbit has one of @xmath49 possible types up to their inverses . the associated cyclic permutations are listed in table [ tab : allsecmin7 ] ; digraphs and piecewise linear representatives are demonstrated in appendix 1 . .all second minimal @xmath1 cycles [ cols="^,^,^ " , ]

[ sec : abstract ] this paper introduces the notion of second minimal-periodic orbit of the continuous map on the interval according as if is a successor of the minimal period of the map in sharkovski ordering . we pursue classification of second minimal-orbits in terms of cyclic permutations and digraphs . it is proved that there are 9 types of second minimal orbits with accuracy up to inverses . it is revealed that by fixing the maximum number of appearances of the periodic windows there is a universal pattern of distribution . in particular , the first appearance of all the orbits is always a minimal orbit , while the second appearance is a second minimal orbit . it is observed that the second appearance of 7-orbit is a second minimal 7-orbit with type 1 digraph . abdulla/ ) . ]

[ sec : abstract ] this paper introduces the notion of second minimal-periodic orbit of the continuous map on the interval according as if is a successor of the minimal period of the map in sharkovski ordering . we pursue classification of second minimal-orbits in terms of cyclic permutations and digraphs . it is proved that there are 9 types of second minimal orbits with accuracy up to inverses . the result is applied to the problem on the distribution of periodic windows within the chaotic regime of the bifurcation diagram of the one - parameter family of unimodal maps . it is revealed that by fixing the maximum number of appearances of the periodic windows there is a universal pattern of distribution . in particular , the first appearance of all the orbits is always a minimal orbit , while the second appearance is a second minimal orbit . it is observed that the second appearance of 7-orbit is a second minimal 7-orbit with type 1 digraph . the reason for the relevance of the type 1 second minimal orbit is the fact that the topological structure of the unimodal map with single maximum is equivalent to the structure of the type 1 piecewise monotonic endomorphism associated with the second minimal 7-orbit . yet another important report of this paper is the revelation of the universal pattern dynamics with respect to increased number of appearances . abdulla/ ) . ]

cond-mat0110467

c

to conclude , we have implemented a field theoretic scheme to explore the structure of the dos close to the edge of the support of two linear non - hermitian operators describing a quantum particle subject to a random imaginary scalar and an imaginary vector potential . in doing so , we have provided a general scheme for the symmetry classification of non - hermitian operators . the field theoretic approach is easily generalized to the consideration of higher point spectral correlations of the fields . in the quasi - classical limit , where the real part of the energy is in excess of any other energy scale , the tails are dominated by ` optimal configurations ' of the real random scalar potential . in contrast to band tail states in semi - conductors , these states are quasi - classical in nature being localized on the length scale @xmath312 . as such the profile of the dos and their dimension is universal depending on just a few material parameters and independent of the nature of the impurity distribution . in the particular case of the constant imaginary vector potential , we have argued that tail states of the system are prohibited by the delocalization mechanism of hatano and nelson . m. b. hastings , preprint cond - mat/9909234 . r. a. janik , w. noerenberg , m. a. nowak , g. papp and i. zahed _ phys . e _ * 60 * , 2699 ( 1999 ) . a. v. izyumov and b. d. simons , _ europhys . lett . _ * 45 * , 290 ( 1999 ) . b. d. simons and a. altland , `` mesoscopic physics '' , to be published in the proceedings of the crm summer school _ theoretical physics at the end of the xxth century _ ( banff , canada , 1999 ) , crm series in mathematical physics , springer , berlin ( 2000 ) .

_ a statistical field theory is developed to explore the density of states and spatial profile of ` tail states ' at the edge of the spectral support of a general class of disordered non - hermitian operators . these states , which are identified with symmetry broken , instanton field configurations of the theory , are closely related to localized sub - gap states recently identified in disordered superconductors . by focusing separately on the problems of a quantum particle propagating in a random imaginary scalar potential , and a random imaginary vector potential

_ a statistical field theory is developed to explore the density of states and spatial profile of ` tail states ' at the edge of the spectral support of a general class of disordered non - hermitian operators . these states , which are identified with symmetry broken , instanton field configurations of the theory , are closely related to localized sub - gap states recently identified in disordered superconductors . by focusing separately on the problems of a quantum particle propagating in a random imaginary scalar potential , and a random imaginary vector potential , we discuss the methodology of our approach and the universality of the results . finally , we address potential physical applications of our findings . _

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[ se29:intro ] a rich chemical and physical interplay exists between gas and grains in which molecules are formed on grains , creating ice mantles that are preserved in environments ranging from quiescent dense molecular clouds to envelopes and disks around protostars . various processes , among which are bombardment by cosmic rays , ultraviolet irradiation , heating , and shocks , can physically or chemically alter the icy mantles , or return molecules into the gas phase ( see @xcite , and references therein ) . a study of the chemical evolution of dense clouds to planet forming disks would ideally involve observations of molecular gas and ices in a range of environments , from quiescent clouds to disk dominated protostars . the most pristine , initial conditions are presumably well sampled by field stars behind clouds tracing quiescent molecular cloud material ( e.g. @xcite ) . lines of sight to protostars are more difficult to characterize , however , since they may trace quiescent foreground material , in addition to the gas and ices in their envelopes and disks ( e.g. @xcite ) . it is thus crucial to characterize the line of sight conditions in order to locate the ices and derive physical conditions and eventually the evolution of the molecular gas and solid state in the interstellar medium . in this paper , we study the line of sight of the class i protostar elias 29 in the @xmath0 oph molecular cloud , using ( sub)millimeter single dish and interferometer gas phase observations . this object is one of the most luminous protostars ( @xmath8 ; @xcite ) in the nearby @xmath0 oph complex ( @xmath9 pc ; @xcite ) , yet little is known about its nature and line of sight conditions . abundant ice has been detected in its direction @xcite . a detailed analysis of ice band profiles indicates that the ices are not strongly thermally processed ( i.e. the ices are not crystallized or segregated ) , despite the presence of abundant warm molecular gas toward the object @xcite . this contrasts with high mass , luminous ( @xmath10 ) protostars , where significant thermal processing of the ices accompanies the presence of abundant warm molecular gas @xcite . the result obtained for elias 29 can only be understood once the location of the ices and the physical conditions of the various gas components are known . therefore , in this paper we try to identify any foreground material , the presence of a circumstellar envelope as well as the presence and orientation of a circumstellar disk , and the column density of each component . we will then address the question where the ices are located , and what their relation is to the young star . this study will , as a consequence , reveal important information on the nature and evolutionary stage of elias 29 , which has many interesting and unique properties @xcite . details of the single dish and interferometer observations are presented in 2 , and the maps and spectra are decomposed and interpreted in 3 . the physical conditions are determined for the different components along the line of sight , among which are two foreground clouds ( 4.1 ) , a remnant envelope and face - on disk ( 4.2 ) , as well as a dense ridge from which elias 29 probably formed ( 4.3 ) . the gas phase conditions and abundances are linked to the ice observations in 4.4 . the depletion of gas phase species is compared to young class 0 objects and quiescent clouds in 4.5 . the results are summarized in 5 .

a ( sub-)millimeter line and continuum study of the class i protostar elias 29 in the ophiuchi molecular cloud is presented , whose goals are to understand the nature of this source , and to locate the ices that are abundantly present along this line of sight . within 1560 beams , several different components contribute to the line emission . two different foreground clouds are detected , an envelope / disk system and a dense ridge ofrich material . the latter two components are spatially separated in millimeter interferometer maps . a small fraction of the ices is present in a ( remnant ) envelope of mass 0.120.33 , but most of the ices (% ) are present in cool ( k ) quiescent foreground clouds .

a ( sub-)millimeter line and continuum study of the class i protostar elias 29 in the ophiuchi molecular cloud is presented , whose goals are to understand the nature of this source , and to locate the ices that are abundantly present along this line of sight . within 1560 beams , several different components contribute to the line emission . two different foreground clouds are detected , an envelope / disk system and a dense ridge ofrich material . the latter two components are spatially separated in millimeter interferometer maps . we analyze the envelope / disk system by using inside - out collapse and flared disk models . the disk is in a relatively face - on orientation ( ) , which explains many of the remarkable observational features of elias 29 , such as its flat sed , its brightness in the near infrared , the extended components found in speckle interferometry observations , and its high velocity molecular outflow . it can not account for the ices seen along the line of sight , however . a small fraction of the ices is present in a ( remnant ) envelope of mass 0.120.33 , but most of the ices (% ) are present in cool ( k ) quiescent foreground clouds . this explains the observed absence of thermally processed ices ( crystallized ho ) toward elias 29 . nevertheless , the temperatures could be sufficiently high to account for the low abundance of apolar ( co , n , o ) ices . this work shows that it is crucial to obtain spectrally and spatially resolved information from single - dish and interferometric molecular gas observations in order to determine the nature of protostars and to interpret infrared iso satellite observations of ices and silicates along a pencil beam .

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the single dish maps show that elias 29 is not a particularly prominent center of molecular line emission . within a radius of @xmath37 , the @xmath2 3 - 2 and @xmath31 2 - 1 emission is highly structured , and quite differently distributed ( figs . [ f : hcop ] and [ f : csochan ] ) . [ f : hcop ] [ f : csochan ] the @xmath2 3 - 2 emission , tracing high densities ( @xmath38 @xmath39 ) , is concentrated in a remarkable ridge - like structure , oriented in the southeast - northwest direction ( fig . [ f : hcop ] ) , at a velocity of @[email protected] @xmath21 ( fig . [ f : csochan ] ) . it is likely no coincidence that the protostars elias 29 , wl 20 , lfam 26 , gy 210 as well as the 1.3 mm continuum protostellar condensations e - mm3 and e - mm5 are all located along this dense ridge ( fig . [ f : hcop ] ) . the ridge is also particularly prominent in the 800 @xmath20 continuum @xcite . star formation along dense filamentary structures is common in the @xmath0 oph cloud , and has been explained by the presence of magnetic field tubes or , more likely , by externally induced shocks ( see @xcite for a short discussion ) . the @xmath31 2 - 1 emission , a column rather than volume density tracer , shows that at least three clouds are present along the line of sight of elias 29 ( fig . [ f : csochan ] ) . the channel maps show a cloud at @xmath40@xmath41 @xmath21 that peaks to the northeast of elias 29 , and a cloud at @xmath40@xmath42 @xmath21 spread rather evenly over the map . the brightest cloud at @xmath55 @xmath21 peaks prominently near the south - southwest of the map , and is probably that in which the dense @xmath2 ridge resides , given the similar velocities . all these clouds are likely present in the foreground , since absorption in the @xmath43 emission lines is seen at these velocities ( fig . [ f : sdish ] ) . [ f : sdish ] llrcccccll & transition & frequency & @xmath44 & @xmath45 & fwhm & @xmath40 & beam @xmath46 & telescope & date + & & mhz & k & k.@xmath21 & @xmath21 & @xmath21 & arcsec & & + co & 2 - 1 & 230538.0 & 15.9 & 19.4 & 6 & 2/6.5 & 21 & jcmt & 03/1997 + & 3 - 2 & 345796.0 & 26 & 98 & 7 & 2/6.5 & 14 & jcmt & 02/1996 + & & & 9 & 46 & 6 & 2/6.5 & 21 & cso & 07/2000 + & 6 - 5 & 691473.0 & 17 & 124 & 12 & 1.8/6.5 & 7 & jcmt & 04/1995 + @xmath47co & 6 - 5 & 661067.4 & 10 & 39 & 3.6 & 4.78 & 7 & jcmt & 04/1995 + c@xmath48o & 1 - 0 triplet & 112358.7 & 0.40 & 0.99 & 2.3 & & 56 & nrao & 05/1995 + & & 112359.0 & 0.94 & 1.00 & 1.0 & 3.58 & & nrao & 05/1995 + & & 112360.0 & 0.68 & 1.05 & 1.5 & & & nrao & 05/1995 + & 2 - 1 multiplet & 224714.3 & 1.76 & 4.95 & 2.63 & 4.08 & 22 & jcmt & 03/1995 + c@xmath49o & 1 - 0 & 109782.2 & 6.34 & 11.0 & 2.0 & 3.6 & 57 & nrao & 05/1995 + & 2 - 1 & 219560.4 & 5.81 & 16.2 & 2.62 & 4.19 & 22 & jcmt & 03/1995 + & & & 9.6 & 21.3 & 2.41 & 4.05 & 35 & cso & 01/2001 + & 3 - 2 & 329330.6 & 4.3 & 10.4 & 2.28 & 4.24 & 15 & jcmt & 02/1996 + cs & 2 - 1 & 97981.0 & 1.28 & 3.55 & 2.61 & 3.96 & 64 & nrao & 05/1995 + & 5 - 4 & 244935.6 & 0.56 & 0.94 & 1.56 & 4.92 & 20 & jcmt & 03/1995 + & 7 - 6 & 342883.0 & @xmath500.09 & @xmath51 & & & 14 & jcmt & 02/1996 + c@xmath52s & 2 - 1 & 96412.9 & 0.14 & 0.23 & 1.62 & 3.29 & 65 & nrao & 05/1995 + h@xmath7co para & @xmath53 & 72838.0 & 0.62 & 1.44 & 2.19 & 3.80/5.39 & 86 & nrao & 05/1995 + & @xmath54 & 145603.0 & 0.56 & 0.89 & 2.0 & 3.67/4.10 & 43 & nrao & 05/1995 + & @xmath55 & 218222.2 & 0.39 & 0.66 & 1.89 & 4.60 & 22 & jcmt & 03/1995 + & @xmath56 & 218475.6 & @xmath500.06 & & & & 22 & jcmt & 03/1995 + h@xmath7co ortho & @xmath57 & 140839.5 & 0.80 & 1.85 & 2.17 & 4.20 & 45 & nrao & 05/1995 + & @xmath58 & 150498.4 & 0.80 & 1.85 & 2.17 & 4.20 & 42 & nrao & 05/1995 + & @xmath59 & 225697.8 & 0.41 & 0.76 & 1.76 & 4.93 & 21 & jcmt & 03/1995 + & & & 0.26 & 0.48 & 1.70 & 4.51 & 32 & cso & 07/2000 + ch@xmath60oh & 2 - 1 triplet & 96739.4 & 0.15 & 0.19 & 1.24 & & 65 & nrao & 05/1995 + & & 96741.4 & 0.18 & 0.33 & 1.68 & 3.59 & & nrao & 05/1995 + & & 96744.6 & @xmath500.03 & & & & & nrao & 05/1995 + & 5 - 4 multiplet & 241802.0 & @xmath500.05 & & & & 20 & jcmt & 03/1997 + & & & @xmath500.02 & & & & 31 & cso & 06/2000 + @xmath2 & 1 - 0 & 89188.5 & 3.57 & 5.69 & 3.3/1.0 & 4.3/4.7 & 70 & nrao & 05/1995 + & 3 - 2 & 267557.6 & 1.06 & 2.7 & 2.28 & 4.77 & 18 & jcmt & 05/1996 + & & & 4.34 & 4.9 & 1.05 & 4.63 & 28 & cso & 06/2000 + & 4 - 3 & 356734.3 & 0.30 & 0.80 & 2.5 & 4.5 & 21 & cso & 03/1999 + h@xmath47co@xmath61 & 1 - 0 & 86754.3 & 0.38 & 0.36 & 0.90 & 4.66 & 72 & nrao & 05/1995 + & 3 - 2 & 260255.5 & 0.04 ( 0.02 ) & 0.09 ( 0.02 ) & 2.15 & 4.89 & 29 & cso & 03/1999 + in various other single dish lines these clouds show up as discrete emission components ( fig . [ f : sdish ] ) . the 2.7 and 3.8 @xmath21components are primarily seen in the low - lying transitions of the cs , @xmath2 , @xmath18 , and @xmath19 molecules , indicating low densities and temperatures of these extended clouds . the emission at 5.0 @xmath21 , identified with the elias 29 core and ridge , shows up in the high excitation lines . given the fact that these emission components have different spatial distributions , their relative intensity also depends on beam size . the @xmath2 3 - 2 line is a factor of 4 stronger and a factor of 2 narrower in the cso data compared to jcmt : the larger cso beam ( 28@xmath1 versus 18@xmath1 ; table [ t : sdish ] ) picks up more emission from the ridge , which is bright in the west ( 3.2 ) . the wide beam nrao-12 m @xmath2 1 - 0 and h@xmath47co@xmath61 1 - 0 spectra have remarkably narrow 5.0 @xmath21 components , originating from the extended ridge . in contrast , the @xmath18 @xmath59 line may peak on the elias 29 core , rather than the ridge . it has the same width in the cso and jcmt beams , but the cso spectrum is weaker ( table [ t : sdish ] ) . we have performed a gaussian decomposition of the emission lines to separate the foreground clouds from the dense material around elias 29 . the derived relative strength of these blended lines depends sensitively on the assumed line width . the most reasonable solution , based on the @xmath18 and cs lines and the @xmath31 map , is to simultaneously fit the peak intensity with three gaussians , centered on @xmath40= 2.7 , 3.8 , and 5.0 @xmath21 , with fwhm=0.5 , 0.8 , and 1.5 @xmath21 . when needed , we allow a 30% variation in the fwhm , and 0.20 @xmath21 in @xmath40 . some of these small variations may be real , but given the complexity of the line profiles and spatial distribution of the various components , we will not seek a physical interpretation for this . the integrated intensities are summarized in table [ t : decomp ] , and the physical conditions are derived from these in 4 . we conclude that when studying individual protostars in the @xmath0 oph cloud , it must be realized that many different physical components are present within single dish beams of @xmath62 or larger . the spatial information of interferometers is essential here . [ f : ovromap ] [ f : ovrospec ] [ f : ovrochan ] in the ovro interferometer maps , bright @xmath22 and @xmath2 1 - 0 emission is detected in the direct neighbourhood of elias 29 ( fig . [ f : ovromap ] ) . the @xmath22 emission is most strongly peaked on the infrared position , and with a spatial fwhm@xmath63 ( 900 au ) it is resolved along the ew direction , where the ovro beam is smallest ( 3@xmath1 ) . this is the elias 29 core , which likely consists of a disk / envelope system ( 4.2 ) . an extension of @xmath64 toward the southwest is visible in both lines , but most prominently in @xmath21 - 0 , which must be attributed to the ridge discussed above . indeed , on this small scale the @xmath2 emission is also parallel to the iram-30 m 1.3 mm continuum emission ( fig . [ f : ovromap ] ) . the ovro @xmath22 and @xmath2 1 - 0 spectra peak between @xmath40=57 @xmath21 , and show no evidence for the strong , extended 2.7 and 3.8 @xmath21foreground emission seen in the single dish maps ( fig . [ f : ovrospec ] ) . the @xmath31 1 - 0 emission is _ spectacularly _ absent in the interferometer spectrum ; it is at least a factor of 10 weaker compared to the single dish data . after dilution to the nrao-12 m beam ( taking the emitting area from the ovro @xmath22 image ) , the ovro @xmath31 signal is even a factor of 60 weaker than the bright detection with the nrao-12 m telescope . this shows again that the bulk of the molecular material in the elias 29 line of sight is not associated with elias 29 , but instead is present in extended foreground clouds , which are resolved out by the ovro interferometer . it is interesting to note that the ovro line profiles are double peaked ( fig . [ f : ovrospec ] ) , with a 5.0 @xmath21 component emitting in the dense ridge ( fig . [ f : ovrochan ] ) . emission at the central infrared position is strongest at 6 @xmath21 . whether this is a true dynamical difference between the elias 29 core and the ridge , or artificially created by self - absorption at @xmath65 @xmath21 , is difficult to answer at present . in this paper we will keep referring to this as the 5.0 @xmath21 component . a disk / envelope / ridge decomposition is attempted in 4.2 .

the disk is in a relatively face - on orientation ( ) , which explains many of the remarkable observational features of elias 29 , such as its flat sed , its brightness in the near infrared , the extended components found in speckle interferometry observations , and its high velocity molecular outflow .

a ( sub-)millimeter line and continuum study of the class i protostar elias 29 in the ophiuchi molecular cloud is presented , whose goals are to understand the nature of this source , and to locate the ices that are abundantly present along this line of sight . within 1560 beams , several different components contribute to the line emission . two different foreground clouds are detected , an envelope / disk system and a dense ridge ofrich material . the latter two components are spatially separated in millimeter interferometer maps . we analyze the envelope / disk system by using inside - out collapse and flared disk models . the disk is in a relatively face - on orientation ( ) , which explains many of the remarkable observational features of elias 29 , such as its flat sed , its brightness in the near infrared , the extended components found in speckle interferometry observations , and its high velocity molecular outflow . it can not account for the ices seen along the line of sight , however . a small fraction of the ices is present in a ( remnant ) envelope of mass 0.120.33 , but most of the ices (% ) are present in cool ( k ) quiescent foreground clouds . this explains the observed absence of thermally processed ices ( crystallized ho ) toward elias 29 . nevertheless , the temperatures could be sufficiently high to account for the low abundance of apolar ( co , n , o ) ices . this work shows that it is crucial to obtain spectrally and spatially resolved information from single - dish and interferometric molecular gas observations in order to determine the nature of protostars and to interpret infrared iso satellite observations of ices and silicates along a pencil beam .

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we have analyzed infrared and millimeter wave line and continuum observations to construct a model of the class i protostar elias 29 and its environment . this model has to contain a number of different components ( summarized in fig . [ f : schem ] ) : a disk to account for the @xmath5250 @xmath20 sed , an envelope contributing to the emission at 1.3 mm ( in particular its size ) , a dense ridge from which elias 29 may have condensed , and foreground material which provides most of the extinction . elias 29 can then be well described by a 500 au radius face - on flared disk with a mass of 0.012 @xmath93 , embedded in a 6000 au radius , 0.12 @xmath93 envelope . this large disk provides the simplest explanation for the observed flat sed , weakly detected 3 mm continuum emission , and 400 au radius 5 @xmath20 thermal continuum emission . the present data does however not fully exclude models with smaller disks . the minimum possible disk has a 30 au radius and a mass of 0.002 @xmath93 surrounded by an envelope of 0.33 @xmath93 . in this case the combination of disk and envelope emission produces a flat sed . the entire system is embedded in a long , dense , cold , and @xmath2rich ridge . elias 29 is slightly offset from the crest of this ridge . in front of the disk , envelope , and ridge system are two foreground clouds at a few @xmath21 lower radial velocities that cover the entire field of view . the large column of the foreground clouds , corresponding to @xmath12111 , may be responsible for the ` class i ' appearance of elias 29 , which would otherwise appear as a t tauri or herbig ae / be star ( i.e. , optically visible ) . these same foreground clouds are also the most likely repository of most of the ices seen along the line of sight ( @xmath122% ) . the low temperature of the foreground clouds explains the observed absence of crystallized ices , and the presence of large abundances of polar , h@xmath7o rich ices , i.e. thermal processing did not play a major role for the ices toward elias 29 . the foreground cloud temperature ( 25@xmath3615 k ) could , however , be high enough to explain the low abundance of apolar , volatile co rich ices , presumably due to the proximity of a number of luminous b type stars . the important question as to whether the ices in the disk or envelope have experienced thermal processing , as is seen in the envelopes of massive objects @xcite , can not be addressed given the large column of foreground material and the face - on nature of the system . this work shows the value of spectrally and spatially resolved information offered by single - dish and interferometric molecular gas observations in interpreting infrared iso satellite observations of ices along a pencil beam . it shows that , at least for the @xmath0 oph cloud , it is crucial to disentangle the different physical components along the line of sight , since otherwise incorrect conclusions may be derived , for example , on the origin and evolution of interstellar and circumstellar ices . follow up observations with sensitive submillimeter interferometers ( e.g. alma ) or high frequency ( @xmath123 ghz ) single dish telescopes with small beams will clearly provide essential information on the structure , abundances , and depletion factor of species in the disk and outflow of elias 29 , and its relationship to the envelope and ridge . in turn , such studies would provide invaluable information on the initial conditions of planet formation . we thank remo tilanus , goeran sandell , and fred baas for carrying out part of the jcmt observations in service mode . the research of fm and acab at the caltech submillimeter observatory is funded by the nsf through contract ast-9980846 . the research of mrh is supported by the miller institute for basic research in science . andr , ph . , & montmerle , th . 1994 , , 420 , 837 blake , g.a . , sandell , g. , van dishoeck , e.f . , groesbeck , t.d . , mundy , l.g . , & aspin , c. 1995 , , 441 , 689 bohlin , r.c . , savage , b.d . , & drake , j.f . 1978 , , 224 , 132 boogert , a.c.a . , ehrenfreund , p. , gerakines , p.a . , tielens , a.g.g.m . , whittet , d.c.b . , et al . 2000a , , 353 , 349 boogert , a.c.a . , tielens , a.g.g.m . , ceccarelli , c. , boonman , a.m.s , van dishoeck , e.f . , et al . 2000b , , 360 , 683 ceccarelli , c. , castets , a. , caux , e. , hollenbach , d. , loinard , l. , molinari , s. , & tielens , a.g.g.m . 2000 , , 355 , 1129 ceccarelli , c. , loinard , l. , castets , a. , tielens , a.g.g.m . , caux , e. , lefloch , b. , & vastel , c. 2000 , , 372 , 998 chandler , c.j . , & richer , j.s . 2000 , , 530 , 851 chen , h. , myers , p.c . , ladd , e.f . , & wood , d.o.s . 1995 , , 445 , 377 chiang , e.i . , & goldreich , p. 1997 , , 490 , 368 chiang , e.i . , & goldreich , p. 1999 , , 519 , 279 chiar , j.e . , adamson , a.j . , kerr , t.h . , & whittet , d.c.b . 1995 , , 455 , 234 dickens , j.e . , & irvine , w.m . 1999 , , 518 , 733 dullemond , c.p . , dominik , c. , & natta , a. 2001 , , in press dutrey , a. , guilloteau , s. , & guelin , m. 1997 , , 317 , l55 elias , j.h . 1978 , , 224 , 857 greene , th.p . , & young , e.t . 1989 , , 339 , 258 hogerheijde , m.r . 1998 , ph . d. thesis , rijksuniversiteit leiden hogerheijde , m.r . , & sandell , g. 2000 , , 534 , 880 hogerheijde , m.r . , & van der tak , f.f.s . 2000 , , 362 , 697 imanishi , k. , koyama , k. , & tsuboi , y. 2001 , preprint ( astro - 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verlag , berlin , ed . g. serra , 81 zinnecker , h. , perrier , c. , & chelli , a. 1988 , in : eso conference and workshop proceedings no . 29 , noao - eso conference on high - resolution imaging by interferometry , ed . f. merkle , 505

this explains the observed absence of thermally processed ices ( crystallized ho ) toward elias 29 . nevertheless , the temperatures could be sufficiently high to account for the low abundance of apolar ( co , n , o ) ices . this work shows that it is crucial to obtain spectrally and spatially resolved information from single - dish and interferometric molecular gas observations in order to determine the nature of protostars and to interpret infrared iso satellite observations of ices and silicates along a pencil beam .

a ( sub-)millimeter line and continuum study of the class i protostar elias 29 in the ophiuchi molecular cloud is presented , whose goals are to understand the nature of this source , and to locate the ices that are abundantly present along this line of sight . within 1560 beams , several different components contribute to the line emission . two different foreground clouds are detected , an envelope / disk system and a dense ridge ofrich material . the latter two components are spatially separated in millimeter interferometer maps . we analyze the envelope / disk system by using inside - out collapse and flared disk models . the disk is in a relatively face - on orientation ( ) , which explains many of the remarkable observational features of elias 29 , such as its flat sed , its brightness in the near infrared , the extended components found in speckle interferometry observations , and its high velocity molecular outflow . it can not account for the ices seen along the line of sight , however . a small fraction of the ices is present in a ( remnant ) envelope of mass 0.120.33 , but most of the ices (% ) are present in cool ( k ) quiescent foreground clouds . this explains the observed absence of thermally processed ices ( crystallized ho ) toward elias 29 . nevertheless , the temperatures could be sufficiently high to account for the low abundance of apolar ( co , n , o ) ices . this work shows that it is crucial to obtain spectrally and spatially resolved information from single - dish and interferometric molecular gas observations in order to determine the nature of protostars and to interpret infrared iso satellite observations of ices and silicates along a pencil beam .

1612.08057

i

let @xmath0 be a simple and undirected graph . a subset @xmath15 is an _ independent set _ , respectively , a _ clique _ if no two , respectively , every two vertices of @xmath16 are adjacent . the complete graph with @xmath17 vertices is denoted by @xmath18 . the path and cycle with @xmath17 vertices of length @xmath19 , respectively , of length @xmath17 , is denoted by @xmath20 , respectively , @xmath21 . for a vertex @xmath22 we write @xmath23 for the set of its neighbors in @xmath1 . a _ universal _ vertex @xmath24 is one such that @xmath25 . for a subset @xmath15 we write @xmath26 $ ] for the subgraph of @xmath1 induced by @xmath16 and @xmath27 for the graph @xmath28 $ ] ; for a vertex @xmath24 we write @xmath29 rather than @xmath30 . given a graph class @xmath31 , a graph @xmath0 is called a _ probe @xmath31 graph _ ( or _ @xmath31 probe graphs _ ) if there exists an independent set @xmath32 ( _ i.e. _ , a set of _ nonprobes _ ) and a set of new edges @xmath33 between certain nonprobe vertices such that the graph @xmath34 is in the class @xmath31 , where @xmath35 stands for the set of all 2-element subsets of @xmath36 . a graph @xmath0 with a _ given _ independent set @xmath32 is said to be a _ partitioned probe @xmath31 graph _ if there exists a set @xmath33 such that the graph @xmath34 is in the class @xmath31 . in both cases , @xmath5 is called a @xmath31 _ embedding _ of @xmath1 . thus , a graph is a ( partitioned ) probe @xmath31 graph if and only if it admits a @xmath31 embedding . the most popular case is the class @xmath31 of interval graphs . the study of probe interval graphs was motivated from certain problems in physical mapping of dna in the computational biology ; see , _ e.g. _ , @xcite . recently , the concept of probe graphs has been generalized as a width parameter of graph class in @xcite . let @xmath31 be a class of graphs . the _ @xmath31-width _ of a graph @xmath1 is the minimum number @xmath2 of independent sets @xmath37 in @xmath1 such that there exists an embedding @xmath38 of @xmath1 such that for every edge @xmath39 in @xmath5 which is not an edge of @xmath1 there exists an @xmath40 with @xmath41 . a collection of such @xmath2 independent sets @xmath42 , is called a _ @xmath31 witness _ for @xmath1 . in the case @xmath43 , @xmath1 is a _ probe @xmath31-graph_. the _ @xmath31-width _ problem asks for a given graph @xmath1 and an integer @xmath2 if the @xmath31-width of @xmath1 is at most @xmath2 . graphs of @xmath31-width @xmath2 are also called @xmath2-probe @xmath31-graph . note that graphs in @xmath31 are , by convenience , @xmath44-probe @xmath31-graphs . in @xcite , the complete width and block - graph width have been investigated . the authors proved that , for fixed @xmath2 , graphs of complete width @xmath2 can be characterized by finitely many forbidden induced graphs . their proof is however not constructive . they also showed , implicitly , that complete width @xmath2 graphs and block - graph width @xmath2 graphs can be recognized in cubic time . the case @xmath43 , _ e.g. _ , probe complete graphs and probe block graphs , has been discussed in depth in @xcite . the case @xmath45 is discussed in @xcite . graphs that do not contain an induced subgraph isomorphic to a graph @xmath46 are called _ @xmath46-free_. more generally , a graph is _ @xmath47-free _ if it does not contain an induced subgraph isomorphic to one of the graphs @xmath48 . for two graphs @xmath1 and @xmath46 , we write @xmath49 for the disjoint union of @xmath1 and @xmath46 , and for an integer @xmath50 , @xmath51 stands for the disjoint union of @xmath52 copies of @xmath1 . the complete @xmath2-partite with @xmath53 vertices in color class @xmath40 is denoted by @xmath54 . for graph classes not defined here see , for example , @xcite . in this paper we study the complete width problem ( given @xmath1 and @xmath2 , is the complete width of @xmath1 at most @xmath2 ? ) . we show that * complete width is np - complete , even on @xmath7-free bipartite graphs , and * computing the complete width of a @xmath8-free bipartite graph ( chain graph ) , and more generally , of a @xmath55-free graph can be done in polynomial time , * computing the complete width of a @xmath8-free chordal graph ( split graph ) , and more generally , of a @xmath9-free graph can be done in polynomial time , * complete width admits a kernel with at most @xmath13 vertices . that is , any instance @xmath56 of complete width can be reduced in polynomial time to an equivalent instance @xmath57 of complete width with @xmath58 and @xmath5 has at most @xmath13 vertices . in particular , complete width is fixed - parameter tractable with respect to parameter @xmath2 . moreover , we give structural characterizations for graphs of complete width at most @xmath59 . in the next section we point out a relation between complete width and the more popular notion of edge clique cover of graphs . then we prove our results in the last four sections . as we will see , it follows from our results on complete width that edge clique cover is np - complete on @xmath10-free co - bipartite graphs and is polynomially solvable on @xmath11-free co - bipartite graphs .

the complete width of is the minimum such that there exist independent sets , , such that the graph obtained from by adding some new edges between certain vertices inside the sets , , is a complete graph . the complete width problem is to decide whether the complete width of a given graph is at most or not . in this paper we study the complete width problem . we show that the complete width problem is np - complete on-free bipartite graphs and polynomially solvable on-free bipartite graphs and on-free graphs . as a by - product , we obtain the following new results : the edge clique cover problem is np - complete on-free co - bipartite graphs and polynomially solvable on-free co - bipartite graphs and on-free graphs .

a complete graph is the graph in which every two vertices are adjacent . for a graph , the complete width of is the minimum such that there exist independent sets , , such that the graph obtained from by adding some new edges between certain vertices inside the sets , , is a complete graph . the complete width problem is to decide whether the complete width of a given graph is at most or not . in this paper we study the complete width problem . we show that the complete width problem is np - complete on-free bipartite graphs and polynomially solvable on-free bipartite graphs and on-free graphs . as a by - product , we obtain the following new results : the edge clique cover problem is np - complete on-free co - bipartite graphs and polynomially solvable on-free co - bipartite graphs and on-free graphs . we also give a characterization for-probe complete graphs which implies that the complete width problem admits a kernel of at most vertices . this provides another proof for the known fact that the edge clique cover problem admits a kernel of at most vertices . finally we determine all graphs of small complete width .

cs0405077

i

dynamic systems with many interacting components are encountered in areas ranging from econometric modeling to communications to material science . individual components and their local interactions may or may not be complex , but their mere multiplicity often creates a complexity which needs to be addressed . as an example , consider the flexchan feature , introduced in new releases of the tdma wireless system . according to the flexchan method , each base station that serves a cell in the service area , periodically checks interference level on different channels and dynamically rearranges the channels in the order of the increase of the expected interference . a new service request ( a new call or a handoff request from a mobile with the call in progress ) is allocated according to an accepted strategy taking into account availability of free channels and the segregation order . for instance , a `` greedy '' strategy places the call on the unoccupied channel with the least expected interference . all the channel capacity of the system is potentially available to each cell in flexchan in contrast with fixed allocation schemes , that are largely in use today . the latter prespecifies a partition of capacity among the cells in anticipation of the traffic . under the flexchan , cells themselves negotiate the capacity , dynamically forming an allocation pattern in response to the actual traffic . thus , the flexchan eliminates the manual planning , reduces the operating cost and presumably increases capacity and qos . does it ? will the channel segregation adapt to the traffic , or maybe it will instead oscillate somehow , with base stations `` fighting '' each other for the capacity ? if it does adapt , how long would the adaptation process take depending on algorithm parameters such as the frequency of checking the interference by base stations ? short of actual system deployment , only simulations can answer these questions . to be convincing , the simulation should be dynamic with base stations asynchronously working out their flexchan routine while users are moving in the service area . many base stations should be represented in order to demonstrate the algorithm viability ; it is possible to imagine how the algorithm would work for just a few base stations , but break down for , say , a hundred base stations . it becomes obvious that a crucial element of this simulation is the computing efficiency of the simulation algorithm . this paper reviews algorithmic and programming techniques which were used to answer the posed questions by simulation@xcite . the program was written so that , say , for about 100 base stations and 1000 active mobiles , simulating one hour of operation requires only several hours of processing on a single pc or workstation . similar algorithmic concepts and techniques for improving efficiency of _ discrete event _ simulations recur in diverse applications such as : - an econometric model of telephone companies , like at&t and mci , fighting among themselves for the residential telephone market quotas . to attract the customers , the players introduce various payment plans , like `` friends and family '' or `` volume discount . '' a question may be : which policy / discount works better given the same cost for the player . the customers behave randomly , their response is staggered , but they tend to behave in accordance with their individual economic interests . it was possible to simulate such systems with more than 100,000 customers with individualized connections and `` habits '' in calling each other during several simulated years with a single run taking a few minutes on a pc . - a dynamic routing in a wired network , like the long distance at&t network , where the use of an `` anticipating '' data structure allows one to cut processing time of simulating one operating hour from a hundred hours to under a ten hours and where further speed improvement using parallel processing shrinks the processing time to a few minutes . - multiparticle studies in computational physics , and material science , such as a model of magnetization , a model of particle deposition , a study of impurity - perturbed crystals . here `` lazy '' evaluation , poisson dispensing , and parallel processing lead to several order of magnitude speed improvements . some simulations previously thought impossible , as they would take years to complete under old techniques , with the new programming techniques move to the category of possible ones , those that can be completed in several hours . this paper is organized as follows . sections [ s : edr ] , [ s : lazy ] , and [ s : poisson ] present techniques and concepts for simulating multicomponent systems on a uniprocessor . the advantages and difficulties of event - driven processing are discussed in section [ s : edr ] , the balance between `` lazy '' and anticipatory methods of evaluation is explained in section [ s : lazy ] , and a general method of event - driven simulation which avoids , and which is faster than the event list method is introduced in section [ s : poisson ] . sections [ s : sequent ] and [ s : relax ] present technique for simulating efficiently large multicomponent systems on a multiprocessor . a `` cautious '' technique which avoids speculative computations is discussed in section [ s : sequent ] . if the latter technique is not feasible , one may resort to the synchronous relaxation method , presented in section [ s : relax ] . the lessons learned in the course of the practical application of the discussed methods are discussed in conclusion .

a computer simulation has to be fast to be helpful , if it is employed to study the behavior of a multicomponent dynamic system . this paper discusses modeling concepts and algorithmic techniques useful for creating such fast simulations . concrete examples of simulations that range from econometric modeling to communications to material science are used to illustrate these techniques and concepts . the algorithmic and modeling methods discussed include event - driven processing , `` anticipating '' data structures , and `` lazy '' evaluation , poisson dispenser , parallel processing by cautious advancements and by synchronous relaxations .

a computer simulation has to be fast to be helpful , if it is employed to study the behavior of a multicomponent dynamic system . this paper discusses modeling concepts and algorithmic techniques useful for creating such fast simulations . concrete examples of simulations that range from econometric modeling to communications to material science are used to illustrate these techniques and concepts . the algorithmic and modeling methods discussed include event - driven processing , `` anticipating '' data structures , and `` lazy '' evaluation , poisson dispenser , parallel processing by cautious advancements and by synchronous relaxations . the paper gives examples of how these techniques and models are employed in assessing efficiency of capacity management methods in wireless and wired networks , in studies of magnetization , crystalline structure , and sediment formation in material science , in studies of competition in economics .

0909.1276

i

the understanding of the structure of nuclei far from stability with extreme isospin is one the most exciting challenges of present nuclear physics . new experimental facilities with radioactive nuclear beams make it possible to investigate the nuclear chart to the very limits of nuclear binding . a wealth of structure phenomena in exotic nuclei have been reported and the next generation of radioactive - beam facilities will present new exciting opportunities to study not only the ground states but also excitations and spectra of these strongly interacting many - body systems . this situation has stimulated considerable new efforts on the theoretical side to understand the dynamics of the nuclear many - body problem by microscopic methods . exact solutions of the non - relativistic schrdinger equation based on the bare nucleon - nucleon interaction are used to study very light nuclei with @xmath0 by an ab initio " approach , modern shell - model calculations based on large scale diagonalization techniques and truncation schemes show considerable success in situations where configuration mixing calculations are possible , i.e. in light nuclei or in nuclei with single magic or doubly magic configurations . for the large majority of nuclei , however , a quantitative microscopic description is , so far , only possible by density functional theory ( dft ) and its extensions . although dft can , in principle , provide an exact description of many - body problems@xcite , if the exact density functional is known , in nuclear physics one is far from a microscopic derivation of this functional . in addition , nuclei are self - bound systems . as a consequence of translational invariance the density in the laboratory frame is constant in space . density functional theory in nuclei is therefore based on the intrinsic density , a concept that requires additional approximations @xcite . the most successful schemes of dft in nuclei use a phenomenological ansatz incorporating as many symmetries of the system as possible and adjusting the parameters of the functional to ground state properties of a few characteristic nuclei on the nuclear chart . considerable progress has been reported recently in constructing such functionals . for a recent review see @xcite . one of the underlying symmetries of qcd is lorentz invariance and therefore covariant density functionals @xcite are of particular interest in nuclear physics . this symmetry not only allows to describe the spin - orbit coupling , which has an essential influence on the underlying shell structure , in a consistent way , but it also puts stringent restrictions on the number of parameters in the corresponding functionals without reducing the quality of the agreement with experimental data . most of the nuclei are superfluid systems and therefore the inclusion of pairing correlations is essential for a correct description of structure phenomena in open - shell nuclei @xcite . hartree - bogoliubov theory provides a unified description of @xmath1- and @xmath2-correlations on a mean - field level by using two type of densities , the normal density matrix @xmath3 and antisymmetric pairing tensor @xmath4 @xcite . according to valatin these two densities can be combined to the generalized density matrix @xmath5 of double dimension @xcite . cdft theory for superfluid systems is therefore based on a generalized relativistic hartree - bogoliubov ( rhb ) energy density @xmath6 $ ] . the same is true for the landau - migdal theory and for all the methods discussed in this paper . for simplicity , however , we restrict all our considerations in this article to the case without pairing correlations . pairing correlations can be included on all steps by using super - matrices . details are given in ref . @xcite . only in the applications we present also calculations in isotopic chains of open shell nuclei , that include pairing correlations . a very successful example of a covariant density functional theory is the relativistic hartree - bogoliubov model @xcite . it combines a density dependence through a non - linear coupling between the meson fields @xcite with pairing correlations based on an effective interaction of finite range . a large variety of nuclear phenomena have been described over the years within this model : the equation of state in symmetric and asymmetric nuclear matter , ground state properties of finite spherical and deformed nuclei all over the periodic table @xcite from light nuclei @xcite to super - heavy elements @xcite , from the neutron drip line , where halo phenomena are observed @xcite to the proton drip line @xcite with nuclei unstable against the emission of protons @xcite . in principle density functional theory can be used for the description of all properties depending on the single - particle density . it is therefore not only limited to the description of the ground state properties . the same density functionals have also been applied for a very successful description of excited states , such as rotational bands in normal and super - deformed nuclei @xcite and collective vibrations @xcite . rotations are treated in the cranking approximation providing a quasi - static description of the nuclear dynamics in a rotating frame and for the description of vibrations a time - dependent mean field approximation is used by assuming independent particle motion in time - dependent average fields @xcite . in the small amplitude limit one obtains the relativistic random phase approximation ( rrpa ) @xcite and in superfluid nuclei the relativistic quasiparticle random phase approximation ( rqrpa ) @xcite . this method provides a natural framework to investigate collective and non - collective excitations of @xmath1- ( or @xmath7 ) character . it is successful in particular for the understanding of the position of giant resonances and spin- or / and isospin - excitations as the gamov teller resonance ( gtr ) or the isobaric analog resonance ( iar ) . recently it has been also used for a theoretical interpretation of low lying e1-strengths observed in neutron rich isotopes ( pygmy modes ) @xcite and for low - lying collective quadrupole excitations @xcite . density functional theory in nuclei is based on intrinsic densities and on the mean field approach . therefore it can not provide an exact treatment of the full nuclear dynamics . it breaks down in transitional nuclei , where the intrinsic density is not well defined and where one has to include correlations going beyond the mean field approximation by treating quantum fluctuations through a superposition of several mean field solutions , as for instance in the generator coordinate method ( gcm ) @xcite . but it also provides a poor approximation for the single - particle spectra particularly in ideal shell model nuclei such as @xmath8pb with closed protons and neutron shells . one finds in self - consistent mean field calculations usually a considerably enhanced hartree - fock gap in the single - particle spectrum and a reduced level density at the fermi surface as compared with the experiment . it is well known that this fact is connected with the relatively small effective mass in such models . mahaux and collaborators @xcite have shown that the effective mass in nuclear matter is roughly @xmath9 @xmath10 . in finite nuclei it should be modified by the coupling of the single - particle motion to low - lying collective surface vibrations . this leads , in the vicinity of the fermi surface , to an enhancement of @xmath9 @xmath11 . non - self - consistent models with the bare mass ( @xmath9 @xmath11 ) show indeed a single - particle spectrum with a level density close to the experiment . with a few exceptions , where the quadrupole motion has been studied within the relativistic generator coordinate method ( gcm ) @xcite , applications of covariant density functional theory to the description of excited states are limited to relativistic rpa , i.e. to configurations of @xmath12-nature . none of these methods , however , can be applied to the investigation of the coupling to more complicated configurations , as it occurs for instance in the damping phenomena causing the width of giant resonances . already before density functional theory has been introduced in the sixties for the description of quantum mechanical many - body problems by kohn and sham @xcite landau has developed in the fifties his fermi liquid theory ( flt ) @xcite for infinite systems . it has been extended to the theory of finite fermi systems ( tffs ) @xcite by migdal . this theory provides another very successful method for the description of low - lying nuclear excitations @xcite . it has several general properties in common with density functional theory . first , both theories are know to be exact , at least in principle , but in practice , in nuclear physics , the parameters entering these theories have to be determined in a phenomenological way by adjustment to experimental data . second , both theories are based on a single - particle concept . dft uses the mean field concept with slater determinants in an effective single - particle potential as a vehicle to introduce shell - effects in the exact density functional introduced by hohenberg and kohn @xcite . fermi liquid theory is based on the concept of quasi - particles obeying a dyson equation , which are defined as the basic excitations of the neighboring system with odd particle number . third , in practical applications both theories describe in the simplest versions the nuclear excitations in the rpa approximation , i.e. by a linear combination of @xmath1-configurations in an average nuclear potential . however , there are also essential differences between these two concepts . first , in contrast to dft , tffs does not attempt to calculate the ground state properties of the many - body system , but it describes the nuclear excitations in terms of landau quasi - particles and their interaction . therefore the experimental data used to fix the phenomenological parameters of the theories are bulk properties of the ground state in the case of density functional theory , and properties of single - particle excitations and of the collective excitations in the case of finite fermi systems theory . second , in dft the mean field is determined in a self - consistent way and therefore the rpa spectrum contains goldstone modes at zero energy . this is usually not the case in tffs calculations , which are based on a non - relativistic shell - model potential , whose parameters are adjusted to the experimental single - particle spectra . therefore , apart from a few approximate attempts to treat the goldstone modes by adjusting additional parameters in the effective quasiparticle interaction , there is no self - consistency in the rpa calculations of tffs and the goldstone modes do not separate from the other modes . they are distributed among the low - lying excitations . third , modern versions of tffs go much beyond the mean field approximation . the coupling between the particles and the phonons is investigated with green s function techniques . based on the phonons calculated in the framework of the rpa one has included particle - phonon coupling vertices and an energy dependence of the self - energy in the dyson equation @xcite . this leads also to an induced interaction in the bethe - salpeter equation caused by the exchange of phonons which also depends on the energy . the coupling of particles and phonons has also been derived from nuclear field theory ( nft ) and its extensions @xcite . many aspects of the coupling between the quasi - particles and the collective vibrations have been investigated with these techniques @xcite as well as with other kinds of approaches beyond rpa @xcite over the years . we give here an overview over recent attempts @xcite to find a combination of the basic ideas of covariant density functional theory and landau - migdal theory and show as examples corresponding applications . the concept is similar to earlier work in refs . @xcite , where specific non - relativistic energy functionals have been used to construct a self - consistent theory of finite fermi systems . the starting point is a covariant density functional @xmath13 $ ] widely used in the literature . it is adjusted to ground state properties of characteristic nuclei and , without any additional parameter , it provides the necessary input of finite fermi systems theory , such as the mean field and the single - particle spectrum as well as an effective interaction between the @xmath1-configurations in terms of the second derivative of the same energy @xmath13 $ ] with respect to the density . thus the phenomenological input of landau - migdal theory is replaced by the results of density functional theory . the same interaction is used to calculate the vertices for particle - vibration coupling @xcite . in a second step techniques of landau - migdal theory and its modern extensions are used to describe the coupling of one- and two - quasiparticle configurations . the main assumption of the quasiparticle - phonon coupling model is that two types of elementary excitations two - quasiparticle and vibrational modes are coupled in such a way that configurations of @xmath14 type with low - lying phonons strongly compete with simple @xmath12 configurations close in energy or , in other words , that quasiparticles can emit and absorb phonons with rather high probabilities . in this way a fully consistent description of the many - body dynamics is obtained . as a result an induced additional interaction between single - particle and vibrational excitations provides a strong fragmentation of the pure rqrpa states causing the spreading width of giant resonances and the redistribution of the pygmy strength . two essential approximations are used in this context : first , the _ time - blocking approximation _ ( tba ) @xcite , that has been extended to systems with pairing correlations ( qtba ) in ref . @xcite , blocks in a special time - projection technique the @xmath12-propagation through states which have a more complex structure than @xmath14 . the nuclear response can then be explicitly calculated on the @xmath15 level by summation of an infinite series of feynman s diagrams . second , a special _ subtraction technique _ guarantees , that there is no double counting between the additional correlations introduced by particle - vibration coupling and the ground state correlations already taken into account in the phenomenological density functional . these two tools are essential for the success of this method . tba introduces a consistent truncation scheme into the bethe - salpeter equation and without it it would be hard to solve the equations explicitly . the subtraction method is the essential tool to connect density functional theory so far used only on the level of mean field theory , i.e. on the rpa level , with the extended landau - migdal theory , where complex configurations are included through particle - vibration coupling . the structure of the paper is the following : in section ii we discuss shortly the general formalism of covariant density functional theory , we introduce in section iii the concept of the energy - dependent self - energy @xmath16 and the vertices of particle - vibration coupling in the relativistic framework , and we discuss in section iv the response formalism , the time blocking approximation and the subtraction mechanism for the response function . in section v we present recent numerical applications for the calculation of level densities at the fermi surface and the spreading width of the several nuclei . section vi contains a brief summary and an outlook for future applications .

both methods are in principle exact , but landau - migdal theory can not describe ground state properties and density functional theory does not take into account the energy dependence of the self - energy and therefore fails to yield proper single - particle spectra as well as the coupling to complex configurations in the width of giant resonances . starting from an energy functional , phonons and their vertices are calculated without any further parameters . they form the basis of particle - vibrational coupling leading to an energy dependence of the self - energy and an induced energy - dependent interaction in the response equation . a subtraction procedure avoids double counting .

a consistent combination of covariant density functional theory ( cdft ) and landau - migdal theory of finite fermi systems ( tffs ) is presented . both methods are in principle exact , but landau - migdal theory can not describe ground state properties and density functional theory does not take into account the energy dependence of the self - energy and therefore fails to yield proper single - particle spectra as well as the coupling to complex configurations in the width of giant resonances . starting from an energy functional , phonons and their vertices are calculated without any further parameters . they form the basis of particle - vibrational coupling leading to an energy dependence of the self - energy and an induced energy - dependent interaction in the response equation . a subtraction procedure avoids double counting . applications in doubly magic nuclei and in a chain of superfluid nuclei show excellent agreement with experimental data .

math0510204

i

there has been new and encouraging evidence that the language and methods of the theory of groupoids , after being successfully applied in other major mathematical fields , offer new insights and perspectives for applications in combinatorics and discrete and computational geometry . the groupoids ( groups of projectivities ) have recently appeared in geometric combinatorics in the work of m. joswig @xcite , see also a related paper with izmestiev @xcite and the references to these papers , where they have been applied to toric manifolds , branched coverings over @xmath1 , colorings of simple polytopes etc . the purpose of this paper is to show that these developments should not be seen as isolated examples . quite the opposite , they serve as a motivation for further extensions and generalizations . as a first application we show how a cubical extension of joswig s groupoid provides new insight about cubical complexes non - embeddable into cubical lattices ( a question related to a problem of s.p . novikov which arose in connection with the @xmath2-dimensional ising model ) @xcite @xcite . the second application leads to a generalization , from graphs to simplicial complexes , of a recent resolution of the lov ' asz conjecture by babson and kozlov @xcite @xcite . combinatorial groupoids also appear ( implicitly ) in other contemporary combinatorial constructions and applications , see babson et al . @xcite , recent work of barcelo et al . on atkin - homotopy @xcite @xcite one of our central objectives is to advocate their systematic use , and to propose their recognition as a valuable tool for geometric and algebraic combinatorics . our point of departure is an observation that different problems , including the question of existence of an embedding @xmath3 of riemann manifolds , the existence and classification of embeddings ( immersions ) of cubical surfaces into hypercubes or cubical lattices , the existence of a coloring of a graph @xmath4 with a prescribed number of colors etc . , can all be approached from a similar point of view . the unifying theme and a single point of view is provided by the concept of a groupoid . in each of the listed cases there is a groupoid naturally associated to an object of the given category . for each of these groupoids there is an associated parallel transport , holonomy groups and other related invariants which serve as obstructions for the existence of morphisms indicated in figure [ fig : sphere7 ] . if @xmath5 is a riemannian manifold , the associated groupoid @xmath6 has @xmath7 as the set of objects while the morphism set @xmath8 consists of all linear isomorphisms @xmath9 arising from a parallel transport along piece - wise smooth curves from @xmath10 to @xmath11 . one of the manifestations of gauss `` theorema egregium '' is that the associated holonomy group is a metric invariant , consequently an isometry depicted in figure [ fig : sphere7 ] ( a ) is not possible . in a similar fashion a cubical `` theorema egregium '' ( theorem [ thm : cub - thm - egr ] ) provides an obstruction for an embedding of a cubical complex into a hypercube or a cubical lattice . as a consequence the cubulation ( quadrangulation ) of an annulus depicted in figure [ fig : sphere7 ] ( b ) does not admit a cubical embedding into a hypercube for the same formal reason ( incompatible holonomy ) a spherical cap can not be isometrically represented in the plane . perhaps it comes as a surprise that the graph coloring problem , figure [ fig : sphere7 ] ( c ) , can be also approached from a similar point of view . an analysis of holonomies ( `` parallel transport '' ) of diagrams of @xmath0-complexes of graphs ( simplicial complexes ) over the associated joswig groupoid eventually leads to a general result ( theorem [ thm : main ] ) which includes the `` odd '' case of babson - kozlov - lovsz coloring theorem as a special case . our objective is to lay a foundation for a program of associating groupoids to posets , graphs , complexes , arrangements , configurations and other combinatorial structures . this `` geometrization of combinatorics '' should provide useful guiding principles and transparent geometric language introducing concepts such as combinatorial holonomy , discrete parallel transport , combinatorial curvature , combinatorial bundles etc . the first steps in this direction are made in section [ sec : groupoids ] . section [ sec : cubical ] , devoted to cubical complexes , is an elaboration of the theme depicted in figure [ fig : sphere7 ] ( b ) . a holonomy @xmath12-valued invariant @xmath13 of a cubical complex is introduced . this leads to a `` combinatorial curvature '' @xmath14 and associated `` cubical theorema egregium '' ( theorem [ thm : cub - thm - egr ] ) which provides necessary conditions for the existence of an embedding / immersion @xmath15 of cubical complexes . section [ sec : gen - lov - conj ] , formally linked to figure [ fig : sphere7 ] ( c ) , introduces groupoids to the problem of coloring graphs and complexes . the focus is on the lovsz @xmath0-conjecture and its ramifications . parallel transport of graph and more general @xmath0-complexes over graphs / complexes is introduced and fundamental invariance of homotopy types of maps is established in propositions [ prop : transport ] and [ prop : prenos ] . this ultimately leads to general results about coloring of graphs and simplicial complexes ( theorems [ thm : main ] and [ thm : paradigm ] , corollary [ cor : lbk - gen ] ) .

a foundation is laid for a theory of _ combinatorial groupoids _ , allowing us to use concepts like `` holonomy '' , `` parallel transport '' , `` bundles '' , `` combinatorial curvature '' etc . in the context of simplicial ( polyhedral ) complexes , posets , graphs , polytopes and other combinatorial objects . a new , holonomy - type invariant for cubical complexes is introduced , leading to a combinatorial `` theorema egregium '' for cubical complexes non - embeddable into cubical lattices .

a foundation is laid for a theory of _ combinatorial groupoids _ , allowing us to use concepts like `` holonomy '' , `` parallel transport '' , `` bundles '' , `` combinatorial curvature '' etc . in the context of simplicial ( polyhedral ) complexes , posets , graphs , polytopes and other combinatorial objects . a new , holonomy - type invariant for cubical complexes is introduced , leading to a combinatorial `` theorema egregium '' for cubical complexes non - embeddable into cubical lattices . parallel transport of-complexes and maps is used as a tool for extending babson - kozlov - lovsz graph coloring results to more general statements about non - degenerate maps ( colorings ) of simplicial complexes and graphs .

0710.3572

i

semiconductor based concepts for quantum information processing promise a high degree of scalability . however , the implementation of even a single qubit proves to be more difficult in semiconductors than in alternative approaches . @xcite the reasons lie in the short decoherence times and strong interactions between elementary excitations in solids . @xcite while a 7-qubit nmr quantum computer was realized already in 2001 , @xcite the first two - qubit quantum operation in semiconductors has been demonstrated with spin qubits only recently . @xcite at liquid helium temperatures and below , electrons in a gaas / algaas two - dimensional electron gas ( 2deg ) show mean free paths as well as phase - relaxation lengths of the order of @xmath0 which are remarkably long distances . @xcite indeed , bertoni _ _ et al.__@xcite and ionicioiu _ _ et al.__@xcite have proposed a scheme for quantum computation in semiconductors that exploits these long coherence lengths for electrons propagating through quantum wires ( qwrs ) . in this scheme , a single electron propagates through two parallel qwrs , that represent the qubit states @xmath1 and @xmath2 , respectively . a single - qubit rotation gate can be realized by an electronic directional coupler@xcite that may consist of a small window in the barrier between the two qwrs and is able to transfer the wave packet from one channel to the other . single - qubit structures have been studied theoretically by several authors@xcite who showed how the dimensions of the coupling window can be utilized to tailor the transfer process . this approach allows one , at least in principle , to apply multiple quantum gate operations to a qubit without exceeding the relevant coherence lengths . universal quantum computation requires not only one - qubit but also two - qubit gates @xcite that are responsible for the creation of entanglement between the qubits . to this end , two pairs of qwrs may be brought close to one another so that the electrons in the qwrs can interact with one another in a controlled way via their coulomb repulsion . this concept has been analyzed both for wave packets@xcite as well as for stationary states@xcite . the latter investigations used simple models that provide important proofs - of - principle , but do not provide quantitative predictions of realizable device structures that exhibit quantum gate operations . while two - qubit devices have only been addressed theoretically in the literature , one - qubit systems have also stimulated a number of experiments in both lateral@xcite as well as vertical@xcite nanostructures . however , quantum entanglement can clearly not be exposed in single - qubit systems as entanglement expresses the non - separability of a multi - qubit state . therefore , the following step towards the realization of a prototype quantum gate based on coupled qwrs consists of a two - qubit device . also , we should keep in mind that conceivable two - qubit experiments need realistic estimations of an entanglement witness that is experimentally accessible and a clear signature of the correct coherent functioning of the directional coupler . the aim of the present work is to theoretically study realistic single- and two - qubit quantum gates , by proposing conceptually simple and experimentally realizable novel semiconductor devices for quantum information processing . the system proposed is based on ballistic gaas / algaas qwrs and allows for the controlled generation and detection of entanglement between an all - electric mach - zehnder interferometer and an electrostatically defined single - electron double quantum dot . the mach - zehnder interferometer is realized by two electrostatically defined qwrs that are connected by two coupling windows . we model the electrons that are propagating through the interferometer by stationary scattering states , and our qubit state will be defined accordingly . in fact , the original proposal of bertoni _ et al . _ involves the injection of electron wave packets into the pair of qwrs . however , we chose here a time - independent approach since , ( i ) it is equivalent to the time - dependent one when the spatial dimension of the wave packet tends to be larger than the device ( indeed , this is the case in our devices due to the small source / drain biases leading to a well defined kinetic energy of the electron ) , ( ii ) the structures that implement the quantum gates operate in the same way , ( iii ) it allows us to compute realistic estimations of the charge - self - consistent ballistic @xmath3-@xmath4 characteristics of the device . in our calculations , we include the detailed charge - self consistent three - dimensional device geometry , material composition , doping profile and bias voltage . importantly , we demonstrate that the all - electric mach - zehnder interferometer can function as a fully controllable single - qubit gate for experimentally attainable parameters . in addition , we have developed a green s function method for the quantitative analysis of the entangled mach - zehnder and double quantum dot system that includes the coulomb interaction between the two qubits non - perturbatively . in order to gain better qualitative insight into the numerical results , we interpret them in terms of an analytical model that reproduces the computational results qualitatively . taking both results together , we are able to show that the degree of entanglement can be related to the dc @xmath3-@xmath4 characteristics of the interferometer and that the mach - zehnder double quantum dot device can be employed as an externally controllable two - qubit gate . this paper is organized as follows . in sec . [ method ] , we introduce a novel method for the quantum mechanical calculation of the ballistic current through an interacting two - particle system . this includes the calculation of the electronic structure and the determination of the ballistic transmission characteristics of the realistic three - dimensional nanostructure . in sec . [ numericaldetails ] , we discuss numerical details of the method . in sec . [ analyticalmodel ] we introduce a simplified analytical scheme able to gather the essential features of the two - qubit device . the results obtained with this approach will be compared , in the following section , with the calculations of the realistic device . [ results ] focuses on the results and the discussion of the ballistic current through the proposed gaas / algaas single- and two - qubit qwr devices . in the same section , the degree of entanglement between the two qubits is evaluated . the paper concludes with final remarks and a summary in sec . [ conclusions ] .

we have developed a two - particle green s function method of open systems and calculate the properties of three - dimensional interacting entangled systems non - perturbatively . we present concrete device designs and detailed , charge self - consistent predictions . one of the qubits is an all - electric mach - zehnder interferometer that consists of two electrostatically defined quantum wires with coupling windows , whereas the second qubit is an electrostatically defined double quantum dot located in a second two - dimensional electron gas beneath the quantum wires .

a gaas / algaas based two - qubit quantum device that allows the controlled generation and straightforward detection of entanglement by measuring a stationary current - voltage characteristic is proposed . we have developed a two - particle green s function method of open systems and calculate the properties of three - dimensional interacting entangled systems non - perturbatively . we present concrete device designs and detailed , charge self - consistent predictions . one of the qubits is an all - electric mach - zehnder interferometer that consists of two electrostatically defined quantum wires with coupling windows , whereas the second qubit is an electrostatically defined double quantum dot located in a second two - dimensional electron gas beneath the quantum wires . we find that the entanglement of the device can be controlled externally by tuning the tunneling coupling between the two quantum dots .

0710.3572

c

we have theoretically analyzed semiconductor single- and two - qubit quantum gates based on electrostatically defined qwrs and quantum dots . we predict the detailed three - dimensional geometry , material composition , doping profile , and bias voltage of an all - electric mach - zehnder interferometer . our calculations of the electronic structure and ballistic transport properties of this device show that the proposed device is a fully controllable single - qubit gate for electrons that propagate in qwrs . the fabrication of the mach - zehnder interferometer is within the reach of present - day technology but full control of the electronic beam - splitters requires a very high fabrication precision in the range of a few @xmath313 . based on this all - electric mach - zehnder interferometer , we predict a two - qubit quantum transport device where the electrons in the interferometer couple to a single - electron double quantum dot by coulomb interaction . we have calculated the ballistic transport properties of the three - dimensional two - qubit device with an interacting two - particle quantum transport method and designed the device geometry for optimal entanglement . by means of an analytical model of the device , we have illustrated the qualitative physics of the two - qubit device and showed that the visibility is a faithful measure of the entanglement . in particular , we have found that the visibility can be controlled externally by tuning the tunneling coupling between the two quantum dots of the second qubit . the device realizes a non - trivial two - qubit gate that allows the controlled generation and straightforward detection of entanglement from dc current - voltage characteristics .

a gaas / algaas based two - qubit quantum device that allows the controlled generation and straightforward detection of entanglement by measuring a stationary current - voltage characteristic is proposed . we find that the entanglement of the device can be controlled externally by tuning the tunneling coupling between the two quantum dots .

a gaas / algaas based two - qubit quantum device that allows the controlled generation and straightforward detection of entanglement by measuring a stationary current - voltage characteristic is proposed . we have developed a two - particle green s function method of open systems and calculate the properties of three - dimensional interacting entangled systems non - perturbatively . we present concrete device designs and detailed , charge self - consistent predictions . one of the qubits is an all - electric mach - zehnder interferometer that consists of two electrostatically defined quantum wires with coupling windows , whereas the second qubit is an electrostatically defined double quantum dot located in a second two - dimensional electron gas beneath the quantum wires . we find that the entanglement of the device can be controlled externally by tuning the tunneling coupling between the two quantum dots .

1110.1452

i

a model was developed for the excitation of the uibs by h atom impacts in the interstellar medium . it builds upon the fact that , in the presence of far uv radiation and hydrocarbon grains , the hydrogen gas will be partially dissociated and the grain surface will be partially hydrogenated and partially covered with free carbon bonds . under such a statistical equilibrium , h atoms from the gas will recombine with c atoms at the grain surface at some rate . at each recombination , the h atom deposits an energy of about 5 ev in the grain . half of this is directly converted into vibrational excitation , always distributed in the same way among the most tightly coupled vibration modes of the grain . absent frequent grain - grain collisions , the only outlet for this energy is ir reemission , part of it in the uibs , provided the chemical structure of the grains is adequate , and the other part in the continuum . the partition only depends upon the grain size , all grains being assumed to have the same constitution . only a fraction , @xmath216 , of the grains ( the smallest ones ) will contribute significantly to the uibs . it is shown quantitatively that h impacts are generally more efficient excitation agents than uv absorption because of the overwhelming abundance of hydrogen relative to uv photons . only very near young bright stars is this no longer true because photon flux then largely exceeds h atom flux . thus h impacts and fuv absorption are both necessary to understand the variety of observed uib * spectral intensities . * the model translates into a small number of equations enabling a quantitative comparison of its predictions with available astronomical observations , which have become exquisitely rich and accurate in the last two decades . section 7 collects some of the most crucial tests of the model : - it predicts uib emission even far from star bursts and in poor - fuv environments . - since the excitation agent and the deposited energy are always the same , uib reemission is independent of the temperature of the neighbouring illuminating stars . - in pdrs , it becomes possible to understand the observed very near spatial coincidence of hi and h@xmath123 emission with the peak of the uibs . - the model predicts the measured average power in the uibs ( 2 - 15 @xmath1 m ) emitted per ambient h atom ( in statistical equilibrium in the solar neighbourhood ) , 1.5 @xmath99 w , provided that 1/4 of the cosmic carbon be locked in uib - emitting grains , and the average h nuclei density be about 100 @xmath14 in the average emitting cloud . - agreement between computed and measured uib intensities is obtained for a given fuv illumination , provided the unknown h column densities are given adequate values , which are found to be quite plausible . this is true even for m 31 , which is presumed to be poor in fuv flux . - the assumption of a common carrier for the uibs and the underlying continuum makes it possible to quantitatively account for the observed large variations of the band - to - continuum contrast ratio : they are simply due to the steep variation of wien s law , at a given wavelength , for small variations of illumination . - the ir excess " paradox , posed long ago by the comparison of the overall galactic ir emission with the uv flux available for grain excitation , is resolved quantitatively by h atom excitation , provided the r.m.s . density of h nuclei throughout the galaxy be of order 200 @xmath217 . * in order to allow quantitative comparisons to be made with observations , a number of assumptions were necessary . the most restrictive one is the neglect of uv extinction and self - shielding of hydrogen ; the calculations are progressively invalidated as column densities along the sight lines exceed a few times 10@xmath218 @xmath18 . fortunately , this is not the case for most of the mesured uib spectra . * for lack of relevant experimental results or explicit values from theoretical computations , i relied heavily on numerical chemical simulation to obtain the values of cross - sections of grain - gas interactions . as the heating is proportional to the c - h recombination c - s ( eq . 8) , the latter needs measurement or at least confirmation by other means . it was also assumed that the importance of the dissociating uv flux relative to total uv does not depend on the total uv flux ; this only impacts the degree of dissociation of hydrogen , and most of our calculations are weakly sensitive to it . none of these assumptions notably affects the general nature and performance of this model . the notion of a fraction @xmath71 should not be taken to correspond to definite lower and upper size limits ; for the transition between emitting and non - emitting grains is progressive and likely to depend on grain structure and composition . some of the physical processes involved here are evidenced by molecular dynamic simulations . even for a 100-atom molecule , an important broadening mechanism is already at work and directly observable in chemical simulations : this is the interaction between modes and constant redistribution , between them , of the initially deposited energy . this widens lines into bands and confers wings to each line , so plateaus " appear below bands " in crowded spectral regions , and even an underlying continuum is visible ( see papoular 2000 ) . this effect is observed to increase quickly with the size , much more so than the number of modes . as the grain size increases , the number of skeletal vibration modes ( involving the whole structure ) increases more rapidly than the number of modes of functional groups ( involving only a small number of atoms of adequate chemical nature , and with adequate bonds to other atoms ) . the uibs are mainly due to the latter , while the continuum ( mainly in the farther ir ) is due to the former . thus , the fraction of energy deposited in the grain that goes into uib emission progressively decreases as the size increases . this is observed up to the largest structures attainable by the usual simulations ( of order of 10@xmath219 atoms ) . tailoring the composition and structure of the grains so that they carry the uibs is outside the scope of this paper ( it was treated at length by papoular 2011 ) . the validity of our conclusions do not depend on these properties , provided the grains bear enough carbon dangling bonds at their periphery , in steady state , so they can capture ambient h atoms .

a model was developed for the excitation of the uibs by h atom impacts in the interstellar medium . it builds upon the fact that , in the presence of far uv radiation and hydrocarbon grains , the hydrogen gas will be partially dissociated and the grain surface will be partially hydrogenated and partially covered with free carbon bonds . under such a statistical equilibrium , h atoms from the gas will recombine with c atoms at the grain surface at some rate . at each recombination , the h atom deposits an energy of about 5 ev in the grain . half of this is directly converted into vibrational excitation , always distributed in the same way among the most tightly coupled vibration modes of the grain . absent frequent grain - grain collisions , the only outlet for this energy is ir reemission , part of it in the uibs , provided the chemical structure of the grains is adequate , and the other part in the continuum . the partition only depends upon the grain size , all grains being assumed to have the same constitution . only a fraction , about 0.25 , of the grains ( among the smallest ones ) will contribute significantly to the uibs . it is shown quantitatively that h impacts are generally more efficient excitation agents than uv absorption because of the overwhelming abundance of hydrogen relative to uv photons . only very close to young bright stars is this no longer true because photon flux then largely exceeds h atom flux . thus h impacts and fuv absorption are both necessary to understand the variety of observed uib spectra . the model translates into a small number of equations enabling a quantitative comparison of its predictions with available astronomical observations , which have become exquisitely rich and accurate in the last two decades .

a model was developed for the excitation of the uibs by h atom impacts in the interstellar medium . it builds upon the fact that , in the presence of far uv radiation and hydrocarbon grains , the hydrogen gas will be partially dissociated and the grain surface will be partially hydrogenated and partially covered with free carbon bonds . under such a statistical equilibrium , h atoms from the gas will recombine with c atoms at the grain surface at some rate . at each recombination , the h atom deposits an energy of about 5 ev in the grain . half of this is directly converted into vibrational excitation , always distributed in the same way among the most tightly coupled vibration modes of the grain . absent frequent grain - grain collisions , the only outlet for this energy is ir reemission , part of it in the uibs , provided the chemical structure of the grains is adequate , and the other part in the continuum . the partition only depends upon the grain size , all grains being assumed to have the same constitution . only a fraction , about 0.25 , of the grains ( among the smallest ones ) will contribute significantly to the uibs . it is shown quantitatively that h impacts are generally more efficient excitation agents than uv absorption because of the overwhelming abundance of hydrogen relative to uv photons . only very close to young bright stars is this no longer true because photon flux then largely exceeds h atom flux . thus h impacts and fuv absorption are both necessary to understand the variety of observed uib spectra . the model translates into a small number of equations enabling a quantitative comparison of its predictions with available astronomical observations , which have become exquisitely rich and accurate in the last two decades . [ firstpage ]

astro-ph9612152

i

uit fuv images and derived stellar photometry for most of the bar of the smc are presented . the uv morphology of the smc s bar shows four concentrations of uv - bright stars spread from northeast to southwest at nearly equal ( @xmath130 arcmin ) spacings . one of the concentrations , near dem 55 , appears as a well - defined 8-arcmin diameter ring surrounded by a larger h@xmath2 ring and strongly suggests sequential star formation . fuv psf photometry , resulting in m(162 ) magnitudes , is obtained for 11,306 stars . we present a fuv luminosity function for the smc bar , complete to m(162 ) @xmath114.5 , and compare our photometry with the compiled ground - based data of av . detected objects are well correlated with other smc population i material ; 520 of 711 h@xmath2 emission - line stars and small nebulae within the uit fields of view are identified with fuv sources . for early type stars , the bluest observed ( m(162)@xmath3v ) colors for each spectral type agree well with values computed from unreddened galactic spectral atlas stars for types earlier than about a0 ; for later types , observed smc stars range significantly bluer , as predicted by low - metallicity models . we attribute redder colors for some stars of all spectral types to strong fuv extinction due to even small amounts of smc dust . internal smc reddenings are determined for all catalog stars . all stars with e(b@xmath3v)@xmath40.15 are within regions of visible h@xmath2 emission . fuv photometry is obtained for the resolved stars and for the total emission from dem hii regions in the smc bar . the flux - weighted ratio of total to stellar flux for dem hiiregions using the apertures of kh is 2.15 ; since only 22% more flux is contributed by stars fainter than m(162)=14.5 , most of the excess total flux is due to scattered fuv from dust . stellar and total emission from dem hii regions are well correlated with h@xmath2 fluxes measured by kh . we compute ratios of fuv to h@xmath2 flux for 42 smc hii regions and compare them with model results , finding that the observed ratios for all h@xmath2-selected hii regions are consistent with models of smc metallicity , ages from 1 - 5 myr , and moderate ( e(b - v)=0.0 - 0.1 mag ) extinction . we gratefully acknowledge the contributions made by the many people involved in the _ astro-2 _ mission . we thank robert s. hill , wayne landsman , and michael fanelli for useful discussions , greg bothun for the use of his h@xmath2 image of the smc , joel offenberg for help in preparing publication - quality images , and joan hollis and emily zamkoff for data entry and programming assistance . we also thank lister staveley - smith for kindly supplying a digital version of a part of his hi maps of the complete smc . funding for the uit project has been through the spacelab office at nasa under project number 440 - 51 . rwo gratefully acknowledges nasa support of portions of this research through grants nag5 - 700 and nagw-2596 to the university of virginia . observational data for stars from azzopardi & vigneau 1982 ( av ) which were observed by uit . av stars which are within uit fields but later in spectral type than a9 are omitted , since they are too faint in the fuv for detection by uit . m(162)values are derived from psf photometry as described in the text . e(b@xmath3v ) values are derived by comparing the unreddened ( m(162)@xmath3v ) color for the stellar spectral type with the observed color , using the reddening law of @xcite . fluxes for hii regions from the list of davies , elliott , & meaburn 1976 ( dem ) as measured by kennicutt & hodge 1986 ( kh ) which are completely contained within the uit field of view . `` fuv stellar '' flux is the total flux of stars within the aperture defined by kh ; `` fuv aperture '' flux is the total pixel flux contained within the aperture defined by kh ; and `` sky flux '' is an estimate for the non - smc sky contribution within the aperture , as described in the text . 1 & 00 47 33.7 -73 06 26.8 & 23.9,117.0 & 12/05/90 & 06:16 & 1713 & astro-1 2 & 00 50 58.3 -72 44 56.5 & 25.9,247.0 & 03/16/95 & 23:50 & 2110 & astro-2 3 & 00 56 41.2 -72 28 28.8 & 18.7,459.0 & 03/06/95 & 06:14 & 2517 & astro-2 4 & 00 59 24.2 -72 11 08.4 & 36.2,179.0,898 . & 03/09/95 & 04:04 & 4966 & astro-2 1 & 2b & 1686 & 00 43 51.2 -73 08 54 & b0 & 13.49 & 10.51 & 0.10 & 0.05 & 2 & 3 & 70 & 00 44 29.6 -72 58 00 & b0 & 14.12 & 12.65 & 0.07 & 0.17 & 3 & 6 & 1699 & 00 45 19.8 -73 15 18 & b0 & 13.46 & 10.25 & 0.08 & 0.03 & 4 & 7 & 1417 & 00 45 33.7 -73 04 31 & b0 & 14.57 & 10.85 & 0.06 & -0.01 & 5 & 8 & 1505 & 00 45 36.2 -72 59 12 & b8 & 13.53 & 12.07 & 0.05 & -0.02 & 6 & 9 & 1694 & 00 45 37.1 -73 14 07 & b2 & 13.05 & 11.25 & 0.07 & 0.08 & 7 & 11 & 1622 & 00 45 54.3 -73 16 13 & b2 & 13.56 & 11.48 & 0.05 & 0.05 & 8 & 12 & 1429 & 00 46 2.8 -73 06 19 & o9 & 13.21 & 10.11 & 0.07 & 0.06 & 9 & 14 & 321 & 00 46 33.2 -73 05 55 & o5 & 13.77 & 12.73 & 0.07 & 0.28 & 10 & 16 & 1558 & 00 46 55.8 -73 08 25 & b1 & 13.19 & 12.03 & 0.10 & 0.16 & 11 & 17 & 1522 & 00 47 3.6 -73 05 55 & b1 & 13.76 & 11.49 & 0.12 & 0.08 & 12 & 17a & 1679 & 00 47 4.8 -73 06 19 & b2 & 13.37 & 11.42 & 0.10 & 0.07 & 13 & 18 & 1433 & 00 47 13.1 -73 06 25 & b1 & 12.48 & 10.98 & 0.04 & 0.14 & 14 & 19 & 2766 & 00 47 15.8 -72 49 55 & b6 & 13.00 & 13.38 & 0.08 & 0.16 & 15 & 20 & 158 & 00 47 29.9 -73 01 25 & b8 : & 12.14 & 14.76 & 0.08 & 0.31 & 16 & 21 & 1690 & 00 47 32.6 -73 10 56 & b1 & 14.14 & 11.05 & 0.05 & 0.01 & 17 & 22 & 1444 & 00 47 39.8 -73 07 38 & b2 & 12.25 & 10.98 & 0.06 & 0.12 & 18 & 23 & 1670 & 00 47 40.1 -73 22 43 & b3 & 12.25 & 12.03 & 0.08 & 0.17 & 19 & 24 & 1675 & 00 47 42.9 -73 02 25 & o9 & 13.78 & 10.96 & 0.07 & 0.08 & 20 & 25 & 1597 & 00 47 47.5 -73 12 19 & b5 & 13.19 & 11.35 & 0.10 & -0.01 & 21 & 26 & 514 & 00 47 50.5 -73 08 14 & b0 & 12.55 & 12.56 & 0.07 & 0.28 & 22 & 27 & 947 & 00 47 51.6 -73 14 14 & a2 & 12.17 & 14.12 & 0.05 & 0.10 & 23 & 28 & 1662 & 00 47 55.0 -73 21 14 & b0 & 13.42 & 11.34 & 0.08 & 0.12 & 24 & 31 & 361 & 00 48 7.2 -73 06 32 & b8 : & 12.52 & 13.20 & 0.06 & 0.15 & 25 & 32 & 1939 & 00 48 10.2 -72 43 44 & b1 & 14.20 & 10.95 & 0.03 & 0.00 & 26 & 34 & -1 & 00 48 18.7 -73 23 38 & b2 : & 13.81 & 0.00 & 0.00 & -9.99 & 27 & 35 & -1 & 00 48 21.3 -72 51 02 & b0 : & 14.13 & 0.00 & 0.00 & -9.99 & 28 & 36 & 1934 & 00 48 23.9 -72 43 38 & b1 & 14.02 & 10.82 & 0.04 & 0.00 & 29 & 37 & 1676 & 00 48 27.0 -73 03 20 & b5 & 12.88 & 11.43 & 0.07 & 0.03 & 30 & 38 & 142 & 00 48 28.2 -73 00 32 & a0 & 12.85 & 13.89 & 0.05 & 0.11 & 31 & 39a & 1476 & 00 48 31.4 -73 15 32 & w6 & 99.99 & 10.94 & 0.06 & -9.99 & 32 & 41 & 1767 & 00 48 36.1 -72 52 50 & b2 & 14.56 & 11.05 & 0.06 & -0.06 & 33 & 42 & 2265 & 00 48 40.2 -72 58 14 & b5 & 13.49 & 12.16 & 0.09 & 0.03 & 34 & 43 & 1875 & 00 48 48.8 -72 46 14 & b1 & 14.08 & 10.34 & 0.04 & -0.04 & 35 & 45 & -1 & 00 48 50.3 -73 22 02 & b8 & 14.15 & 0.00 & 0.00 & -9.99 & 36 & 48 & 1709 & 00 49 3.4 -73 21 32 & b3 & 11.03 & 9.45 & 0.12 & 0.06 & 37 & 50 & 1765 & 00 49 16.7 -72 52 32 & b1 & 13.11 & 10.15 & 0.05 & 0.02 & 38 & 51 & 1777 & 00 49 32.8 -72 51 08 & b0 & 14.05 & 10.81 & 0.03 & 0.03 & 39 & 53 & 2495 & 00 49 45.8 -72 52 38 & a0 & 12.96 & 14.41 & 0.05 & 0.14 & 40 & 55 & 1197 & 00 49 48.2 -73 17 45 & b5 & 13.40 & 13.66 & 0.06 & 0.16 & 41 & 56 & 1751 & 00 49 51.7 -72 55 39 & b5 & 11.17 & 9.14 & 0.04 & -0.02 & 42 & 58 & 1769 & 00 49 58.3 -72 51 45 & b2 & 14.38 & 10.83 & 0.04 & -0.06 & 43 & 59 & 788 & 00 49 58.4 -73 11 33 & a0 & 13.40 & 13.54 & 0.06 & 0.04 & 44 & 62 & 2361 & 00 50 1.2 -72 55 03 & b3 & 14.34 & 11.64 & 0.03 & -0.03 & 45 & 63 & 672 & 00 50 0.9 -73 10 08 & a0 & 13.48 & 13.12 & 0.06 & 0.00 & 46 & 65 & 1448 & 00 50 6.9 -73 07 39 & b6 & 11.13 & 11.18 & 0.08 & 0.14 & 47 & 66 & 1642 & 00 50 6.7 -73 16 21 & b0 & 13.48 & 11.27 & 0.09 & 0.11 & 48 & 67 & 2039 & 00 50 11.9 -72 32 27 & b0 & 13.66 & 10.22 & 0.07 & 0.01 & 49 & 69 & 2442 & 00 50 18.0 -72 53 21 & b0 & 13.35 & 10.01 & 0.21 & 0.02 & 50 & 70 & 2007 & 00 50 18.9 -72 38 03 & b0 & 12.38 & 8.86 & 0.04 & 0.00 & 51 & 73 & 1677 & 00 50 28.4 -73 03 09 & b0 & 14.08 & 10.95 & 0.07 & 0.04 & 52 & 75 & 1757 & 00 50 32.9 -72 52 27 & o9 & 12.79 & 9.30 & 0.04 & 0.03 & 53 & 77 & 1827 & 00 50 34.2 -72 47 39 & b0 & 13.91 & 10.41 & 0.03 & 0.01 & 54 & 80 & 1824 & 00 50 44.3 -72 47 33 & b1 & 13.33 & 9.83 & 0.04 & -0.02 & 55 & 82 & 1925 & 00 50 49.8 -72 42 33 & b2 & 14.13 & 10.09 & 0.06 & -0.10 & 56 & 83 & 1930 & 00 50 52.8 -72 42 03 & b1 & 13.38 & 9.80 & 0.04 & -0.03 & 57 & 85 & 2440 & 00 51 0.8 -72 52 57 & b0 & 13.75 & 11.33 & 0.05 & 0.09 & 58 & 87 & 3400 & 00 51 8.3 -72 40 03 & b0 & 13.90 & 12.76 & 0.11 & 0.19 & 59 & 89 & 1121 & 00 51 10.4 -73 15 45 & b0 & 14.47 & 14.03 & 0.09 & 0.25 & 60 & 90 & -1 & 00 51 12.1 -72 28 09 & a5 & 12.58 & 0.00 & 0.00 & -9.99 & 61 & 91 & 2325 & 00 51 11.5 -72 55 09 & b8 & 12.61 & 12.36 & 0.04 & 0.08 & 62 & 93 & 1940 & 00 51 20.8 -72 41 21 & b0 : & 14.13 & 11.07 & 0.06 & 0.04 & 63 & 94 & 1783 & 00 51 20.6 -72 49 33 & b1 & 13.99 & 10.15 & 0.04 & -0.05 & 64 & 95 & 1885 & 00 51 21.9 -72 44 03 & o9 & 13.91 & 9.90 & 0.04 & -0.01 & 65 & 99 & 1809 & 00 51 25.4 -72 47 45 & b2 & 13.01 & 10.59 & 0.04 & 0.03 & 66 & 100 & 1743 & 00 51 26.9 -72 57 45 & b3 & 14.29 & 11.44 & 0.05 & -0.04 & 67 & 102 & 2036 & 00 51 36.5 -72 32 09 & b2 & 14.29 & 11.11 & 0.07 & -0.03 & 68 & 103 & 1847 & 00 51 36.7 -72 45 57 & b1 & 13.36 & 10.27 & 0.03 & 0.01 & 69 & 104 & 1800 & 00 51 39.0 -72 47 57 & b1 & 13.13 & 9.93 & 0.04 & 0.00 & 70 & 105 & 3885 & 00 51 41.9 -72 27 57 & a0 & 12.24 & 12.61 & 0.07 & 0.05 & 71 & 106 & 2005 & 00 51 44.1 -72 37 15 & b1 & 14.32 & 11.28 & 0.05 & 0.01 & 72 & 109 & 1976 & 00 51 50.6 -72 39 15 & b1 & 13.73 & 10.43 & 0.03 & -0.01 & 73 & 110 & 1877 & 00 51 52.8 -72 44 03 & a0 & 12.17 & 12.06 & 0.06 & 0.02 & 74 & 111 & 2035 & 00 51 56.7 -72 31 57 & b1 & 13.84 & 10.52 & 0.06 & -0.01 & 75 & 112 & 2031 & 00 51 58.5 -72 33 15 & b0 & 14.15 & 10.27 & 0.05 & -0.02 & 76 & 114 & 1971 & 00 52 3.1 -72 39 22 & b0 & 14.93 & 11.10 & 0.05 & -0.02 & 77 & 122 & 1932 & 00 52 25.7 -72 40 52 & b6 & 12.79 & 11.44 & 0.05 & 0.02 & 78 & 123 & 1758 & 00 52 27.8 -72 50 58 & b8 & 13.22 & 11.59 & 0.06 & -0.03 & 79 & 126 & 1961 & 00 52 31.7 -72 39 22 & b0 & 13.47 & 10.26 & 0.04 & 0.03 & 80 & 131 & 1841 & 00 52 41.0 -72 45 10 & b8 & 12.61 & 11.94 & 0.07 & 0.05 & 81 & 133 & 2003 & 00 52 44.2 -72 36 46 & b0 & 13.91 & 9.87 & 0.04 & -0.04 & 82 & 137 & 1865 & 00 52 53.0 -72 44 04 & b4 & 12.33 & 10.46 & 0.04 & 0.01 & 83 & 138 & 1784 & 00 52 52.9 -72 48 22 & b0 & 14.28 & 10.94 & 0.05 & 0.02 & 84 & 141 & 3723 & 00 53 10.5 -72 33 40 & b3 : & 14.50 & 12.79 & 0.04 & 0.05 & 85 & 143 & 1970 & 00 53 27.1 -72 38 22 & b0 & 14.12 & 10.32 & 0.04 & -0.02 & 86 & 144 & 1742 & 00 53 33.7 -72 56 16 & b0 & 14.06 & 10.13 & 0.08 & -0.03 & 87 & 145 & 1983 & 00 53 36.0 -72 37 40 & b1 & 13.35 & 10.10 & 0.05 & 0.00 & 88 & 148 & 1882 & 00 53 42.5 -72 42 28 & b0 & 14.28 & 10.40 & 0.04 & -0.02 & 89 & 149 & 1774 & 00 53 53.6 -72 48 22 & b2 & 13.96 & 10.79 & 0.07 & -0.03 & 90 & 150 & 4034 & 00 53 58.3 -72 28 35 & b6 & 12.72 & 11.44 & 0.05 & 0.03 & 91 & 151 & 1807 & 00 53 59.6 -72 45 53 & b5 & 12.30 & 10.27 & 0.06 & -0.02 & 92 & 152 & 5913 & 00 54 4.2 -72 31 40 & a3 & 11.87 & 12.23 & 0.03 & -0.06 & 93 & 153 & 3451 & 00 54 4.0 -72 37 17 & b8 & 13.58 & 12.51 & 0.06 & 0.01 & 94 & 154 & 3095 & 00 54 9.9 -72 41 34 & b0 : & 13.55 & 13.30 & 0.05 & 0.26 & 95 & 155 & 2975 & 00 54 15.2 -72 42 59 & b0 : & 14.34 & 14.54 & 0.10 & 0.30 & 96 & 156 & 4384 & 00 54 19.4 -72 17 47 & a0 & 14.17 & 13.99 & 0.06 & 0.01 & 97 & 157 & 3938 & 00 54 23.0 -72 17 05 & b0 & 14.33 & 11.07 & 0.05 & 0.03 & 98 & 158 & 3946 & 00 54 23.5 -72 18 59 & b1 & 14.06 & 10.22 & 0.04 & -0.05 & 99 & 160 & 2340 & 00 54 43.5 -72 51 53 & b0 & 13.99 & 11.34 & 0.07 & 0.07 & 100 & 164 & 4070 & 00 55 16.0 -72 41 17 & b0 & 14.16 & 10.71 & 0.04 & 0.01 & 101 & 165 & 3994 & 00 55 19.9 -72 25 59 & b7 & 12.79 & 11.65 & 0.06 & 0.03 & 102 & 169 & 4077 & 00 55 40.3 -72 44 54 & b1 & 13.90 & 11.07 & 0.05 & 0.03 & 103 & 171 & 6141 & 00 55 48.2 -72 39 23 & b6 & 13.34 & 12.77 & 0.05 & 0.09 & 104 & 172 & 10309 & 00 55 54.9 -72 08 54 & b8 & 13.37 & 13.37 & 0.06 & 0.10 & 105 & 174 & 8802 & 00 56 38.6 -72 01 36 & a7 & 12.43 & 14.52 & 0.07 & -0.06 & 106 & 175 & 4036 & 00 56 38.9 -72 36 30 & b1 & 13.65 & 11.12 & 0.05 & 0.05 & 107 & 176 & 3947 & 00 56 39.7 -72 25 06 & b1 & 13.93 & 11.10 & 0.04 & 0.03 & 108 & 177 & 6454 & 00 56 45.1 -72 03 30 & o5 & 14.60 & 10.50 & 0.03 & 0.03 & 109 & 178 & 3970 & 00 56 48.6 -72 28 42 & b2 & 14.38 & 11.29 & 0.06 & -0.03 & 110 & 180 & 7161 & 00 56 55.8 -72 24 12 & b5 & 13.19 & 12.27 & 0.09 & 0.07 & 111 & 181 & 11001 & 00 56 57.8 -72 17 36 & a0 & 13.83 & 13.15 & 0.04 & -0.03 & 112 & 182 & 6480 & 00 57 1.6 -72 08 06 & b0 & 14.33 & 10.81 & 0.03 & 0.00 & 113 & 184 & 7143 & 00 57 14.4 -72 22 24 & b8 & 14.18 & 12.57 & 0.07 & -0.03 & 114 & 185 & 6777 & 00 57 24.4 -72 01 30 & b8 : & 13.28 & 11.97 & 0.06 & 0.00 & 115 & 186 & 4000 & 00 57 27.8 -72 33 06 & o9 & 14.10 & 10.63 & 0.05 & 0.03 & 116 & 189 & 3958 & 00 57 33.3 -72 28 48 & b1 & 14.51 & 11.21 & 0.05 & -0.01 & 117 & 190 & 11396 & 00 57 35.1 -72 26 48 & b6 & 13.52 & 13.19 & 0.08 & 0.11 & 118 & 191 & 6482 & 00 57 37.8 -72 13 00 & b1 & 13.63 & 10.42 & 0.06 & 0.00 & 119 & 192 & 6490 & 00 57 38.2 -72 21 49 & o9 & 14.58 & 11.06 & 0.05 & 0.03 & 120 & 193 & 7152 & 00 57 38.1 -72 24 31 & b1 & 15.40 & 12.54 & 0.09 & 0.03 & 121 & 194 & 4008 & 00 57 45.6 -72 35 30 & b0 & 13.95 & 10.73 & 0.04 & 0.03 & 122 & 195 & 5818 & 00 57 45.5 -72 40 12 & b5 & 13.53 & 14.47 & 0.04 & 0.21 & 123 & 196 & 7163 & 00 57 54.7 -72 27 37 & b1 & 13.93 & 11.57 & 0.09 & 0.07 & 124 & 199 & 5274 & 00 58 2.3 -72 35 37 & a3 & 13.41 & 13.45 & 0.03 & -0.09 & 125 & 200 & 4025 & 00 58 8.1 -72 38 25 & a0 & 12.10 & 11.37 & 0.05 & -0.03 & 126 & 201 & 6479 & 00 58 10.7 -72 10 55 & b0 & 14.04 & 10.76 & 0.06 & 0.02 & 127 & 202 & 6855 & 00 58 15.5 -72 07 25 & b0 & 14.33 & 11.20 & 0.05 & 0.04 & 128 & 204 & 3998 & 00 58 22.5 -72 35 13 & b1 : & 14.38 & 11.65 & 0.05 & 0.04 & 129 & 205 & 11006 & 00 58 23.5 -72 21 31 & a2 & 12.30 & 13.41 & 0.04 & 0.03 & 130 & 207 & 6429 & 00 58 33.7 -71 55 43 & o7 & 14.37 & 10.70 & 0.06 & 0.05 & 131 & 208 & 5549 & 00 58 33.7 -72 39 25 & o9 & 14.10 & 11.49 & 0.10 & 0.10 & 132 & 209 & 7135 & 00 58 35.9 -72 24 55 & b0 & 14.46 & 11.45 & 0.04 & 0.05 & 133 & 210 & 6484 & 00 58 36.7 -72 16 19 & b3 & 12.67 & 10.73 & 0.05 & 0.03 & 134 & 211 & 7145 & 00 58 41.8 -72 26 13 & b8 & 11.50 & 11.67 & 0.07 & 0.11 & 135 & 213 & 6607 & 00 58 55.1 -71 56 31 & a2 & 12.11 & 12.14 & 0.08 & -0.05 & 136 & 214 & 6964 & 00 58 55.2 -72 13 13 & b3 & 13.34 & 12.15 & 0.03 & 0.09 & 137 & 215 & 3955 & 00 58 55.9 -72 32 01 & b1 & 12.76 & 10.15 & 0.06 & 0.05 & 138 & 217 & 7050 & 00 59 2.3 -72 18 49 & b2 : & 14.59 & 11.48 & 0.03 & -0.03 & 139 & 218 & 7064 & 00 59 5.2 -72 19 37 & b1 & 13.80 & 11.29 & 0.05 & 0.06 & 140 & 219 & 7010 & 00 59 7.7 -72 16 31 & b2 : & 14.50 & 11.23 & 0.06 & -0.04 & 141 & 220 & 6446 & 00 59 10.9 -72 05 43 & o9 & 14.50 & 10.52 & 0.06 & -0.01 & 142 & 221 & 3945 & 00 59 10.2 -72 31 31 & b0 & 13.46 & 11.34 & 0.05 & 0.12 & 143 & 222 & 7080 & 00 59 13.5 -72 20 55 & b3 & 13.20 & 11.42 & 0.06 & 0.04 & 144 & 223 & 4007 & 00 59 13.6 -72 38 55 & b0 & 13.67 & 10.64 & 0.02 & 0.04 & 145 & 224 & 6732 & 00 59 16.9 -72 04 37 & b0 & 14.22 & 11.23 & 0.05 & 0.05 & 146 & 225 & 4313 & 00 59 16.8 -72 29 43 & b8 & 14.05 & 13.07 & 0.03 & 0.02 & 147 & 226 & 6481 & 00 59 21.4 -72 17 07 & b0 & 14.42 & 10.72 & 0.05 & -0.01 & 148 & 229 & 8932 & 00 59 27.5 -72 09 49 & o9 & 11.86 & 10.98 & 0.16 & 0.24 & 149 & 230 & 6435 & 00 59 30.1 -72 01 02 & b2 & 12.75 & 10.64 & 0.05 & 0.05 & 150 & 231 & 10562 & 00 59 29.6 -72 20 14 & a0 & 14.20 & 13.54 & 0.04 & -0.03 & 151 & 232 & -1 & 00 59 32.9 -72 10 43 & o9 & 12.36 & 0.00 & 0.00 & -9.99 & 152 & 234 & 6440 & 00 59 44.3 -72 04 13 & b2 & 12.98 & 10.75 & 0.04 & 0.04 & 153 & 237 & 7019 & 00 59 53.5 -72 19 02 & b8 & 12.59 & 12.26 & 0.03 & 0.07 & 154 & 238 & 6475 & 00 59 56.0 -72 13 32 & o9 & 13.77 & 10.28 & 0.06 & 0.03 & 155 & 240 & 3921 & 01 00 1.1 -72 23 50 & b3 & 14.03 & 11.42 & 0.05 & -0.02 & 156 & 242 & 9303 & 01 00 7.9 -72 13 56 & b1 & 12.11 & 12.61 & 0.07 & 0.29 & 157 & 245 & 6543 & 01 00 16.7 -71 57 02 & b5 & 13.34 & 11.74 & 0.06 & 0.01 & 158 & 250 & 6589 & 01 00 23.8 -71 59 26 & b9 & 12.82 & 12.71 & 0.08 & 0.06 & 159 & 251 & 3932 & 01 00 22.9 -72 30 44 & o7 & 14.75 & 11.09 & 0.06 & 0.05 & 160 & 252 & 6450 & 01 00 30.6 -72 10 56 & b2 : & 13.07 & 10.63 & 0.05 & 0.03 & 161 & 257 & 6486 & 01 00 45.1 -72 23 50 & b3 & 12.79 & 10.87 & 0.05 & 0.03 & 162 & 260 & 6453 & 01 00 52.0 -72 13 32 & b1 & 13.28 & 10.67 & 0.05 & 0.05 & 163 & 263 & 6813 & 01 01 6.9 -72 12 57 & b6 & 12.85 & 12.07 & 0.04 & 0.07 & 164 & 264 & 6422 & 01 01 8.4 -71 59 56 & b0 & 12.36 & 9.50 & 0.12 & 0.06 & 165 & 266 & 6489 & 01 01 10.1 -72 27 26 & b1 & 12.63 & 9.90 & 0.06 & 0.04 & 166 & 267 & 6439 & 01 01 16.0 -72 06 32 & o8 & 14.92 & 10.72 & 0.03 & -0.01 & 167 & 268 & 6448 & 01 01 16.4 -72 12 39 & b0 : & 13.14 & 10.42 & 0.06 & 0.07 & 168 & 270 & 6477 & 01 01 17.5 -72 17 27 & b9 & 11.42 & 10.60 & 0.05 & 0.00 & 169 & 271 & 6476 & 01 01 20.5 -72 17 15 & b0 & 13.46 & 10.59 & 0.06 & 0.06 & 170 & 272 & 9881 & 01 01 23.4 -72 20 09 & b2 & 14.52 & 12.41 & 0.05 & 0.05 & 171 & 273 & 7855 & 01 01 27.9 -72 07 03 & a1 & 12.18 & 13.32 & 0.04 & 0.07 & 172 & 274 & 6483 & 01 01 29.8 -72 23 15 & b0 & 14.02 & 10.60 & 0.06 & 0.01 & 173 & 277 & 6824 & 01 01 33.0 -72 14 39 & b2 & 14.07 & 11.20 & 0.05 & -0.01 & 174 & 279 & 6428 & 01 01 35.2 -72 03 03 & o9 & 14.20 & 10.46 & 0.04 & 0.01 & 175 & 280 & 6580 & 01 01 39.9 -72 02 27 & b0 & 14.66 & 11.74 & 0.05 & 0.05 & 176 & 282 & 6447 & 01 01 50.4 -72 13 09 & o7 & 14.83 & 10.81 & 0.04 & 0.02 & 177 & 286 & 6605 & 01 01 57.8 -72 04 15 & a5 & 12.35 & 13.64 & 0.07 & -0.06 & 178 & 287 & 6444 & 01 01 57.5 -72 12 39 & b0 & 12.86 & 9.65 & 0.05 & 0.03 & 179 & 290 & 6561 & 01 02 0.8 -72 02 21 & b1 & 13.93 & 11.49 & 0.07 & 0.06 & 180 & 291 & 6816 & 01 02 4.6 -72 15 27 & b1 & 14.78 & 12.61 & 0.04 & 0.08 & 181 & 292 & 6923 & 01 02 4.5 -72 19 03 & b3 & 13.10 & 11.52 & 0.05 & 0.06 & 182 & 296 & 6445 & 01 02 8.8 -72 13 15 & o8 & 14.38 & 10.77 & 0.03 & 0.04 & 183 & 297 & 6419 & 01 02 10.4 -72 00 21 & b7 & 12.18 & 10.63 & 0.06 & 0.00 & 184 & 298 & 6560 & 01 02 12.7 -72 02 51 & a0 & 12.52 & 12.58 & 0.09 & 0.03 & 185 & 299 & 6980 & 01 02 13.9 -72 22 09 & o9 & 14.61 & 12.34 & 0.07 & 0.13 & 186 & 300 & 6442 & 01 02 14.8 -72 11 15 & b0 & 14.46 & 10.69 & 0.05 & -0.01 & 187 & 301 & 6512 & 01 02 15.7 -71 59 45 & b0 & 14.24 & 11.24 & 0.08 & 0.05 & 188 & 302 & 9841 & 01 02 19.3 -72 22 03 & b0 & 14.35 & 14.54 & 0.11 & 0.30 & 189 & 303 & 6418 & 01 02 22.3 -72 00 15 & b1 & 12.81 & 10.16 & 0.07 & 0.05 & 190 & 312 & 6441 & 01 02 45.8 -72 12 03 & b0 & 13.62 & 9.93 & 0.02 & -0.01 & 191 & 314 & 8760 & 01 02 48.6 -72 16 39 & b2 & 12.90 & 10.57 & 0.12 & 0.04 & 192 & 315 & 6438 & 01 02 50.0 -72 10 09 & b8 & 10.92 & 10.82 & 0.07 & 0.09 & 193 & 318 & 6436 & 01 02 54.8 -72 09 51 & o9 & 13.59 & 10.53 & 0.05 & 0.06 & 194 & 321 & 6433 & 01 02 57.8 -72 08 03 & o9 & 13.88 & 10.42 & 0.06 & 0.03 & 1 & 14 & 5.00 & 1.00 & 1.047 & 0.081 & 2.397 & 0.074 & 0.022 2 & 15 & 0.70 & 0.30 & 1.067 & 0.101 & 1.665 & 0.061 & 0.015 3 & 17 & 1.70 & 0.20 & 0.190 & 0.016 & 0.559 & 0.036 & 0.006 4 & 18 & 4.70 & 0.70 & 1.158 & 0.105 & 1.781 & 0.061 & 0.015 5 & 20 & 2.90 & 0.50 & 0.755 & 0.048 & 1.987 & 0.063 & 0.015 6 & 22 & 2.00 & 0.20 & 0.498 & 0.032 & 1.059 & 0.036 & 0.006 7 & 23 & 8.40 & 1.00 & 0.253 & 0.015 & 1.017 & 0.041 & 0.006 8 & 24 & 2.80 & 0.30 & 0.146 & 0.010 & 0.747 & 0.036 & 0.006 9 & 30 & 2.90 & 0.30 & 0.405 & 0.033 & 1.217 & 0.037 & 0.006 10 & 31 & 2.50 & 0.30 & 0.201 & 0.013 & 0.378 & 0.025 & 0.002 11 & 32 & 40.00 & 4.00 & 6.202 & 0.463 & 11.814 & 0.107 & 0.040 12 & 35 & 1.30 & 0.20 & 1.362 & 0.085 & 4.246 & 0.084 & 0.030 13 & 36 & 0.90 & 0.20 & 1.793 & 0.108 & 4.655 & 0.085 & 0.030 14 & 37 & 10.00 & 1.50 & 1.978 & 0.116 & 4.825 & 0.085 & 0.030 15 & 40 & 4.20 & 0.50 & 0.779 & 0.050 & 1.446 & 0.037 & 0.006 16 & 42 & 3.20 & 0.40 & 0.364 & 0.024 & 0.943 & 0.036 & 0.006 17 & 43 & 6.50 & 0.70 & 0.545 & 0.032 & 1.177 & 0.037 & 0.006 18 & 45 & 11.50 & 1.20 & 0.589 & 0.049 & 1.203 & 0.050 & 0.010 19 & 46 & 2.30 & 0.30 & 0.720 & 0.033 & 2.779 & 0.034 & 0.022 20 & 47 & 1.70 & 0.20 & 0.556 & 0.026 & 1.069 & 0.018 & 0.006 21 & 49 & 5.00 & 1.30 & 1.669 & 0.114 & 3.286 & 0.061 & 0.015 22 & 54 & 28.00 & 2.80 & 3.557 & 0.212 & 7.581 & 0.052 & 0.040 23 & 55 & 25.00 & 2.50 & 4.585 & 0.217 & 8.356 & 0.047 & 0.040 24 & 56 & 4.00 & 0.40 & 3.464 & 0.151 & 5.338 & 0.051 & 0.030 25 & 57 & 0.70 & 0.20 & 0.312 & 0.020 & 0.745 & 0.018 & 0.006 26 & 63 & 40.00 & 4.00 & 22.478 & 1.077 & 33.105 & 0.097 & 0.139 27 & 69 & 19.50 & 4.00 & 4.353 & 0.217 & 13.957 & 0.059 & 0.075 28 & 77 & 1.30 & 0.20 & 2.616 & 0.139 & 2.750 & 0.024 & 0.010 29 & 80 & 13.50 & 1.50 & 5.744 & 0.326 & 8.266 & 0.044 & 0.089 30 & 83 & 0.40 & 0.20 & 0.670 & 0.039 & 0.550 & 0.009 & 0.006 31 & 84 & 0.20 & 0.20 & 0.379 & 0.024 & 0.276 & 0.006 & 0.002 32 & 85 & 1.30 & 0.30 & 2.562 & 0.156 & 2.369 & 0.022 & 0.030 33 & 86 & 1.30 & 0.30 & 2.145 & 0.126 & 2.465 & 0.022 & 0.030 34 & 90 & 1.70 & 0.50 & 2.534 & 0.129 & 2.916 & 0.015 & 0.040 35 & 93 & 0.80 & 0.20 & 1.575 & 0.091 & 1.552 & 0.016 & 0.015 36 & 94 & 1.30 & 0.20 & 1.603 & 0.080 & 1.653 & 0.016 & 0.015 37 & 98 & 2.00 & 0.20 & 0.387 & 0.014 & 0.383 & 0.004 & 0.002 38 & 100 & 0.70 & 0.20 & 0.299 & 0.017 & 0.353 & 0.005 & 0.006 39 & 101 & 0.60 & 0.20 & 0.363 & 0.023 & 0.387 & 0.005 & 0.006 40 & 102 & 3.40 & 0.40 & 1.410 & 0.070 & 1.251 & 0.009 & 0.010 41 & 103 & 155.00 & 16.00 & 29.339 & 1.747 & 93.790 & 32.747 & 0.158 42 & 111 & 1.20 & 0.20 & 0.521 & 0.037 & 0.491 & 0.006 & 0.006

a mosaic of four uit far - uv ( fuv ) ( = 1620 ) images , with derived stellar and hii region photometry , is presented for most of the bar of the smc . the uv morphology of the smc s bar shows that recent star formation there has left striking features including : a ) four concentrations of uv - bright stars spread from northeast to southwest at nearly equal ( arcmin=0.5 kpc ) spacings ; b ) one of the concentrations , near dem 55 , comprises a well - defined 8-arcmin diameter ring surrounded by a larger h ring , suggestive of sequential star formation . fuv psf photometry is obtained for 11,306 stars in the fuv images , resulting in magnitudes m(162 ) . we present a fuv luminosity function for the smc bar , complete to m(162).5 . detected objects are well correlated with other smc population i material ; of 711 h emission - line stars and small nebulae within the uit fields of view , 520 are identified with fuv sources . the bluest observed colors for each type agree well with colors computed from unreddened galactic spectral atlas stars for types earlier than about a0 ; for later spectral types the observed smc stars range significantly bluer , as predicted by comparison of low - metallicity and galactic - composition models . redder colors for some stars of all spectral types are attributed to the strong fuv extinction arising from even small amounts of smc dust . internal smc reddenings are determined for all catalog stars . all stars with e(bv).15 are within regions of visible h emission . fuv photometry for 42 h-selected hii regions in the smc bar is obtained for stars and for total emission ( as measured in hii - region - sized apertures ) . the flux - weighted average ratio of total to stellar fuv flux is 2.15 ; consideration of the stellar fuv luminosity function indicates that most of the excess total flux is due to scattered fuv radiation , rather than stars fainter than m(162)=14.5 . both stellar and total emission are well correlated with h fluxes measured by kennicutt and hodge ( 1986 ; hereafter kh ) , yielding fuv / h flux ratios that are consistent with models of smc metallicity , ages from 1 - 5 myr , and moderate ( e(bv)=0.00.1 mag ) internal smc extinction .

a mosaic of four uit far - uv ( fuv ) ( = 1620 ) images , with derived stellar and hii region photometry , is presented for most of the bar of the smc . the uv morphology of the smc s bar shows that recent star formation there has left striking features including : a ) four concentrations of uv - bright stars spread from northeast to southwest at nearly equal ( arcmin=0.5 kpc ) spacings ; b ) one of the concentrations , near dem 55 , comprises a well - defined 8-arcmin diameter ring surrounded by a larger h ring , suggestive of sequential star formation . fuv psf photometry is obtained for 11,306 stars in the fuv images , resulting in magnitudes m(162 ) . we present a fuv luminosity function for the smc bar , complete to m(162).5 . detected objects are well correlated with other smc population i material ; of 711 h emission - line stars and small nebulae within the uit fields of view , 520 are identified with fuv sources . the fuv photometry is compared with available ground - based catalogs of supergiants , yielding 191 detections of 195 supergiants with spectral type earlier than f0 in the uit fields . the ( m(162)v ) color for supergiants is a sensitive measure of spectral type . the bluest observed colors for each type agree well with colors computed from unreddened galactic spectral atlas stars for types earlier than about a0 ; for later spectral types the observed smc stars range significantly bluer , as predicted by comparison of low - metallicity and galactic - composition models . redder colors for some stars of all spectral types are attributed to the strong fuv extinction arising from even small amounts of smc dust . internal smc reddenings are determined for all catalog stars . all stars with e(bv).15 are within regions of visible h emission . fuv photometry for 42 h-selected hii regions in the smc bar is obtained for stars and for total emission ( as measured in hii - region - sized apertures ) . the flux - weighted average ratio of total to stellar fuv flux is 2.15 ; consideration of the stellar fuv luminosity function indicates that most of the excess total flux is due to scattered fuv radiation , rather than stars fainter than m(162)=14.5 . both stellar and total emission are well correlated with h fluxes measured by kennicutt and hodge ( 1986 ; hereafter kh ) , yielding fuv / h flux ratios that are consistent with models of smc metallicity , ages from 1 - 5 myr , and moderate ( e(bv)=0.00.1 mag ) internal smc extinction .

1212.3446

c

instrumental distortion of the total intensity profile introduces significant systematic timing errors that are correlated between pulsars observed with the same instrument . therefore , high - fidelity polarimetry is inextricably linked with the long - term objectives of high - precision timing and the primary goals of pulsar timing array experiments . this paper presents a novel method of calibrating the instrumental response by exploiting the long - term stability of the polarized emission from a millisecond pulsar . full - polarization timing observations of psrj0437@xmath04715 are used to model variations in the 21-cm multibeam and h - oh receivers and down - conversion system at parkes . over a period of @xmath149 years , temporal and spectral variations of the receptor cross - coupling introduces systematic error of the order of 1@xmath3s in arrival time estimates derived from observations of psrj1022@xmath11001 . high - fidelity calibration of instrumental polarization and arrival time estimation using matrix template matching is demonstrated to correct systematic error and double the sensitivity of the experiment , yielding significant detections of the shapiro delay , the precession of periastron , and the secular variation of the projected semi - major axis of the orbit . the improvements in both timing accuracy and precision are consistent with the hypothesis that the average polarized emission from these millisecond pulsars is stable over the timescales of relevance to this experiment . with a modest instrumental bandwidth of 64 mhz and median integration length of 64 minutes , psrj1022@xmath11001 yields arrival time residuals with an uncertainty - weighted standard deviation of only @xmath90 = 880ns , roughly five orders of magnitude smaller than the pulsar spin period @xmath150 s. the remarkable timing precision of these observations supports the conclusions of @xcite , who argue that the psrj1022@xmath11001 profile variations observed by @xcite and @xcite are not intrinsic to the pulsar and are most likely due to instrumental calibration errors . the precision of the arrival time data presented in this paper places psrj1022@xmath11001 among the top ten sources regularly observed as part of the parkes pulsar timing array ( ppta ) project . other ppta sources may also benefit from the application of metm ; e.g. six of the sources listed in table [ tab : theory ] have instrumental distortion susceptibility factors ranging from 200 ns to 340 ns . referring to the measured values of the non - orthogonality parameters plotted in figure [ fig : coupling ] , these factors roughly correspond to systematic timing error variations of the order of 1 @xmath3s . therefore , the timing precision of these pulsars would likely be improved by application of the methods developed for this study . for psr j1744@xmath01134 and psr j2129@xmath05721 , matrix template matching is predicted to yield arrival times with greater uncertainty than those derived from the total intensity profile . in these two cases , the best results would be obtained by calibrating using metm and deriving arrival times using sti . in a wider analysis of all of the pulsars in the ppta , @xcite calibrated approximately 5.9 years of psrj1022@xmath11001 timing observations using mem - derived solutions . using typical integration lengths of 64 min and an instrumental bandwidth of 256 mhz , this study yielded arrival time residuals with an uncertainty - weighted standard deviation of 1.7 @xmath3s and reduced @xmath151 . the residual noise is approximately 2.4 times greater than expected based on simple extrapolation of the metm sti results presented in table [ tab : residual ] . this discrepancy may be explained by the fact that the pdfb instruments have a non - linear response that introduces over - polarization , which can not be calibrated using mem . the accuracy of mem - based calibration is also limited by the fact that there is no unique solution to the measurement equation when the only constraining transformation is the geometric rotation of the receptors with respect to the sky @xcite . to constrain the otherwise degenerate boost along the stokes @xmath80 axis , it is necessary to include observations of a source that is assumed to have negligible circular polarization ( e.g. , for the mem fits performed in [ sec : template ] , observations of hydra a were used to constrain @xmath46 ) . however , the use of an unpulsed source of radiation as a constraint is intrinsically susceptible to variability in other contributions to the system temperature , including receiver noise and ground spillover . that is , it is safer to assume that the polarized emission from psrj0437@xmath04715is constant than it is to rely on a source of unpulsed radiation during calibration . the results presented by @xcite may also be limited by the instability of the receptor cross - coupling , as shown in figure [ fig : coupling ] . the temporal variations of the non - orthogonality parameters are sufficiently resolved only via application of the metm method presented in this paper , which yields a seven - fold increase in the number of available calibrator solutions . as noted in previous studies @xcite , other phenomena may also currently limit the timing precision of psrj1022@xmath11001 . for example , owing to its low ecliptic latitude , the radio signal from this pulsar is subject to significant dispersive delays in the solar wind @xcite . no corrections for dispersion variations were applied to the data presented in this work ; however , observations made when the line of sight to psrj1022@xmath11001 passes near the sun ( around late august of each year ) were excluded from the data set . in addition to fluctuations in dispersion , the observed flux of the pulsar varies as a function of time and radio frequency due to scintillation in the interstellar medium . the average profile of psrj1022@xmath11001 varies significantly as a function of radio frequency ( e.g. see figure 1 of * ? ? ? * ) and , when modulated by interstellar scintillation , the frequency - integrated mean profile may fluctuate with time . the potentially significant arrival time estimation errors induced by this effect could be mitigated through the use of a frequency - dependent template profile . it is reasonable to expect that psrj1022@xmath11001 timing will be improved by the current generation of instrumentation at the parkes observatory . the data presented in this paper were observed using a system with low dynamic range that performed analog - to - digital conversion of the radio signal with only two bits per sample . two - bit quantization is an intrinsically non - linear process and the techniques applied to restore linearity ( dynamic output level setting ) and mitigate quantization noise ( scattered power correction ) are based on a linear approximation to the response of the digitizer @xcite . as discussed in appendix [ app : spc ] , the scattered power correction algorithm applied in this analysis is accurate only to first order and it is possible that overestimation of the unpolarized scattered power contributes to the over - polarization noted in figure [ fig:0437 ] . to overcome the limitations of two - bit sampling , a new baseband recording and processing system with greater dynamic range was commissioned at the parkes observatory in 2010 april . designed in collaboration with the center for astronomy signal processing and electronics research ( casper ) at berkeley , the casper - parkes - swinburne recorder ( caspsr ) digitizes a dual - polarization 400 mhz band with eight bits per sample and performs real - time radio frequency interference excision based on the spectral kurtosis estimator @xcite . this system currently operates in parallel with the third and fourth generations of the parkes digital filter bank ( pdfb ) instruments as part of the ppta program . the calibration techniques applied in this paper could potentially be improved by reducing the number of degrees of freedom in the estimates of the receptor cross - coupling parameters . for example , this could be achieved by fitting an analytic model to the temporal and spectral variations of the calibrator solutions or by employing a lossy compression algorithm that can be applied to irregularly sampled data . the smoothing effected by such a transformation would reduce the instantaneous noise in the applied calibrator solutions and might also provide a robust means of interpolating between solutions . however , further refinements of the calibration technique may yield only marginal improvements to arrival time accuracy and precision . table [ tab : residual ] demonstrates that , even when the data are calibrated using the inaccurate ideal - feed assumption , matrix template matching ( mtm ) yields arrival time estimates that are nearly as good as those derived from data calibrated using the new metm technique described in this paper . this seems to suggest that , as long as the signal is not depolarized by integrating over time and/or frequency , the fidelity of the calibration technique has negligible impact on arrival times derived using mtm . this conclusion may have a significant impact on the design of the next generation of instrumentation for high - precision timing . for example , it has been asserted that in order to achieve timing precision of the order of 100 ns , the square kilometre array ( ska ) must achieve net polarization purity corresponding to @xmath152 @xcite . however , even when non - orthogonality as large as @xmath153 is ignored , as is the case when the 21-cm multibeam receiver at parkes is assumed to be ideal , mtm yields accurate arrival time estimates . that is , by relaxing the requirement for polarization purity , the application of mtm has the potential to reduce the cost of ska development for high - precision timing . all of the software required to perform measurement equation modeling , polarimetric calibration , and matrix template matching is freely available as part of psrchive @xcite ; the use of this software is demonstrated by @xcite and more fully documented online . i am grateful to matthew bailes , paul demorest , mike keith , michael kramer , stefan osowski , and john reynolds for helpful discussions during this research project and for insightful comments that greatly improved this report . joris verbiest implemented and assisted with tempo2 support of ecliptic coordinates and the orthometric parameterization of the shapiro delay . the parkes observatory is part of the australia telescope which is funded by the commonwealth of australia for operation as a national facility managed by csiro .

a new method of polarimetric calibration is presented in which the instrumental response is derived from regular observations of psrj0437 based on the assumption that the mean polarized emission from this millisecond pulsar remains constant over time . application of the new technique followed by arrival time estimation using matrix template matching yields post - fit residuals with an uncertainty - weighted standard deviation of 880 ns , two times smaller than that of arrival time residuals obtained via conventional methods of calibration and arrival time estimation . the precision achieved by this experiment yields the first significant measurements of the secular variation of the projected semi - major axis , the precession of periastron , and the shapiro delay ; it also places psrj1022 among the ten best pulsars regularly observed as part of the parkes pulsar timing array ( ppta ) project . these limitations may be overcome by adopting the techniques presented in this work , which relax the demand for instrumental polarization purity and thereby have the potential to reduce the development cost of next - generation telescopes such as the square kilometre array .

a new method of polarimetric calibration is presented in which the instrumental response is derived from regular observations of psrj0437 based on the assumption that the mean polarized emission from this millisecond pulsar remains constant over time . the technique is applicable to any experiment in which high - fidelity polarimetry is required over long time scales ; it is demonstrated by calibrating 7.2 years of high - precision timing observations of psrj1022 made at the parkes observatory . application of the new technique followed by arrival time estimation using matrix template matching yields post - fit residuals with an uncertainty - weighted standard deviation of 880 ns , two times smaller than that of arrival time residuals obtained via conventional methods of calibration and arrival time estimation . the precision achieved by this experiment yields the first significant measurements of the secular variation of the projected semi - major axis , the precession of periastron , and the shapiro delay ; it also places psrj1022 among the ten best pulsars regularly observed as part of the parkes pulsar timing array ( ppta ) project . it is shown that the timing accuracy of a large fraction of the pulsars in the ppta is currently limited by the systematic timing error due to instrumental polarization artifacts . more importantly , long - term variations of systematic error are correlated between different pulsars , which adversely affects the primary objectives of any pulsar timing array experiment . these limitations may be overcome by adopting the techniques presented in this work , which relax the demand for instrumental polarization purity and thereby have the potential to reduce the development cost of next - generation telescopes such as the square kilometre array .

hep-th9903259

i

it has been proposed @xcite that supergravity ( and string theory ) on @xmath6 times a compact manifold is equivalent to a conformal field theory living on the boundary of @xmath6 . supergravity ( and string theory ) on asymptotically @xmath6 space - time is also expected @xcite to be realized by a boundary field theory on a suitable vacuum . investigations of the bulk quantum gravity , in particular , to clarify @xmath7 black holes as the quantum gravitational states , will lead a deeper insight on our understanding of the @xmath8 correspondence . in three dimensions , a description of gravity in terms of gauge theory @xcite is known , by which quantum gravity in three dimensions seems to be tractable and thus consideration about such quantum gravitational states will become possible . in this article we present a quantization of @xmath0-dimensional gravity with negative cosmological constant @xmath9 and thereby investigate quantum aspects of the @xmath0-dimensional black holes from the perspective of the @xmath10 correspondence . the quantization consists of the following two procedures . the first is related with quantization of the asymptotic virasoro symmetry found by brown and henneaux @xcite . to describe it we introduce in section 2 a notion of the virasoro deformation of three - geometry . it is originally defined as a deformation of a suitable non - compact simply - connected region in @xmath5 and is given by the coadjoint action of the virasoro groups @xmath11 . induced metric from @xmath5 provides a metric of the deformed region . given a black hole , we can take , as such a non - compact simply - connected region , a covering space of the exterior of the outer horizon . its virasoro deformation commutes with the projection . therefore we can obtain a family of the deformed quotients . this defines the virasoro deformation of the exterior of the outer horizon of the black hole . the induced metric of the deformed region becomes that of its quotient , which can be understood as a deformed metric of the black hole obtained in @xcite . the asymptotic virasoro symmetry turns out to be the infinitesimal form of this deformation . the family of the deformed quotients will be identified with a product of the coadjoint orbits of the virasoro groups @xmath2 labeled by mass and angular momentum of the black hole together with the cosmological constant . in section 3 we provide quantization of the asymptotic virasoro symmetry or the virasoro deformation of the exterior of the outer horizon . it is prescribed by quantization of the corresponding coadjoint orbits . quantization of each orbit turns out to give a unitary irreducible representation of the virasoro algebra with central charge @xmath12 . state of the black hole becomes the primary state of the representation . all the excited states of the representation correspond to the local excitations by the virasoro deformation , that is , as noted in @xcite , gravitons . @xmath13 is also examined . quantization of its virasoro deformation turns out to provide a unitary irreducible representation , in which state of @xmath13 is the @xmath14-invariant primary state . although the quantization of the virasoro deformations leads the unitary irreducible representations of the virasoro algebra , it is not sufficient as a quantization of three - dimensional gravity . if one takes the perspective of @xmath14 chern - simons gravity @xcite , the virasoro deformation corresponds to local degrees of freedom of the theory and , in order to make the quantization complete , we must take into account of the global degrees of freedom , _ i.e. _ , holonomies . we start section 4 by discussing their quantization . having quantized holonomies , it is very reasonable to expect from the perspective of the @xmath15 correspondence that , by an identification of these topological operators with the zero modes of a suitable two - dimensional quantum field , the unitary irreducible representations obtained by the quantization of the virasoro deformations could be reproduced as the hilbert space of the two - dimensional conformal field theory . this expectation turns out to be true at least when @xmath4 . the holonomy variables will be identified with the zero modes of a real scalar field @xmath16 , which is interpreted as the liouville field . the state of @xmath13 is identified with the @xmath17-invariant vacuum of this conformal field theory . the states of the black holes are the primary states obtained from the vacuum by an operation of the corresponding vertex operators . excitations of the liouville field are equivalent with the generators of the virasoro deformation . this field theory admits to have continuous spectrums as expected . having obtained the quantization of three - dimensional gravity by the liouville field theory , the following question may arise : how can one understand the three - dimensional black holes in two dimensions where the liouville field lives ? the liouville field theory is living on a two - sphere which is obtained by a compactification of the boundary cylinder at infinity of the black holes . to answer this question it becomes necessary to understand how the virasoro deformation recognizes the outer horizon of the black hole . it will be shown at the end of section 4 that it recognizes the outer horizon not as a two - dimensional object but as a one - dimensional object in @xmath5 . therefore the boundary liouville field @xmath16 can see the outer horizon only as a one - dimensional object . compactification of the boundary cylinder to a sphere simultaneously makes the solid cylinder to a three - ball , by which this one - dimensional outer horizon intersects with the boundary sphere at two points . the boundary liouville field theory recognizes its intersections as insertions of the corresponding vertex operator . a microscopic description of the three - dimensional black holes has been proposed by carlip @xcite and strominger @xcite using a conformal field theory on the horizon . since the virasoro deformation of the exterior of the outer horizon can not recognize the horizon as a two - dimensional object , the boundary liouville field theory can not be equivalent to the horizon conformal field theory . in section 5 we propose two possible descriptions of the horizon conformal field theory . one is a string theory in the background of a macroscopic string living in @xmath5 . worldvolume of the macroscopic string should be identified with the two - dimensional outer horizon , by which we can obtain the virasoro algebra on the horizon as its fluctuation by microscopic string . the other is a conformal field theory which could be obtained by the virasoro deformation of the region between the inner and outer horizons of the black hole . the second possibility seems to imply an interpretation of the conjecture made by maldacena @xcite , in terms of a gauge transformation .

a quantization of-dimensional gravity with negative cosmological constant is presented and quantum aspects of the-dimensional black holes are studied thereby . the quantization consists of two procedures . one is related with quantization of the asymptotic virasoro symmetry . a notion of the virasoro deformation of-geometry is introduced . for a given black hole , its quantization provides unitary irreducible representations of the virasoro algebra , in which state of the black hole becomes primary . to make the quantization complete , , the global degrees of freedom , are taken into account . by an identification of these topological operators with zero modes of the liouville field , the aforementioned unitary representations reveal , as far as , as the hilbert space of this two - dimensional conformal field theory . this conformal field theory , living on the cylinder at infinity of the black hole and having continuous spectrums , can recognize the outer horizon only as a _ one - dimensional object _ in and realize it as insertions of the corresponding vertex operator . therefore it can not be a conformal field theory on the horizon . two possible descriptions of the horizon conformal field theory are proposed . -5 mm -2 mm addtoresetequationsection ou - het 317 + hep - th/9903259 + march 1999 * three - dimensional black holes * + * and * + * liouville field theory * t. nakatsu , h. umetsu and n. yokoi +

a quantization of-dimensional gravity with negative cosmological constant is presented and quantum aspects of the-dimensional black holes are studied thereby . the quantization consists of two procedures . one is related with quantization of the asymptotic virasoro symmetry . a notion of the virasoro deformation of-geometry is introduced . for a given black hole , the deformation of the exterior of the outer horizon is identified with a product of appropriate coadjoint orbits of the virasoro groups . its quantization provides unitary irreducible representations of the virasoro algebra , in which state of the black hole becomes primary . to make the quantization complete , , the global degrees of freedom , are taken into account . by an identification of these topological operators with zero modes of the liouville field , the aforementioned unitary representations reveal , as far as , as the hilbert space of this two - dimensional conformal field theory . this conformal field theory , living on the cylinder at infinity of the black hole and having continuous spectrums , can recognize the outer horizon only as a _ one - dimensional object _ in and realize it as insertions of the corresponding vertex operator . therefore it can not be a conformal field theory on the horizon . two possible descriptions of the horizon conformal field theory are proposed . -5 mm -2 mm addtoresetequationsection ou - het 317 + hep - th/9903259 + march 1999 * three - dimensional black holes * + * and * + * liouville field theory * t. nakatsu , h. umetsu and n. yokoi +

1401.7939

i

the ability to store a quantum state over long times is a desirable feature in many quantum information protocols . in the optical domain , quantum memories ( qm ) are necessary to implement the quantum repeaters needed for future long - distance quantum networks , and are the object of active research @xcite . quantum memories at microwave frequencies have also become of great interest in recent years because of the development of superconducting qubits which have their resonance frequency in the ghz range , in the perspective of implementing holographic quantum computing @xcite . for such schemes , the memory should act as an ideal multi - qubit register , able to store over long times the state of large numbers of qubits and to retrieve them on - demand . spin ensembles have emerged as promising candidates for a multi - mode microwave quantum memory because of their long coherence time @xcite and of the multiple collective modes that a spin ensemble withstands . existing proposals @xcite ( inspired by optical quantum memory protocols @xcite ) proceed in two distinct steps . first , the microwave field prepared in a well - defined quantum state @xmath1 ( for instance by a superconducting qubit ) is absorbed by the spin ensemble . this generates a transverse magnetisation which decays rapidly in a time @xmath2 due to the spread of resonance frequencies in the ensemble . given the weakness of the coupling constant of a single spin to the microwave field , efficient absorption requires embedding the ensemble in a high - quality factor microwave resonator in order to reach the so - called high - cooperativity regime @xcite . the second step of the memory operation consists in retrieving the initial state , by a series of operations after which the spins emit a microwave pulse in a quantum state as close as possible to @xmath1 . in @xcite , this is achieved by a hahn - echo - like sequence consisting of two consecutive @xmath3 pulses on the spins , combined with dynamical tuning of the resonator frequency and quality factor . the maximum storage time of the memory is approximately the hahn - echo decay time @xmath4 , so that the maximal number of stored quantum states is of order @xmath5 , a figure which can be very large in many spin systems . the first step of this protocol ( quantum state transfer ) has been demonstrated at the single - photon level in recent experiments @xcite ; the remaining obstacle to a microwave quantum memory is therefore the implementation of hahn - echo refocusing sequences at the quantum level in a hybrid quantum circuit . the object of this work is precisely to identify the challenges posed by this task and to demonstrate experimentally that they can be solved . for simplicity , we consider from now on a protocol simpler than the full qm @xcite but which constitutes an essential building block : the two - pulse echo ( 2pe ) . as depicted in fig . [ fig1]a , the 2pe consists in storing weak pulses @xmath6 into the spin ensemble at times @xmath7 , and applying a single refocusing pulse at time @xmath8 which triggers the emission of echo pulses @xmath9 at times @xmath10 ( therefore in reverse order ) in the detection waveguide @xcite . performing the 2pe at the quantum level imposes a number of requirements which represent experimental challenges . for quantum states to be well defined , thermal excitations should be absent from the system . this implies both that the spin ensemble has a high degree of polarisation and that the microwave field is in its ground state with high probability , which can only be achieved if the experiments are performed at millikelvin temperatures . at these temperatures however , spins tend to relax very slowly towards their ground state , and an active spin reset is therefore needed in order to repeat the experimental sequence at a reasonable rate ( @xmath11hz ) as requested by experiments at the single photon level . then , applying refocusing pulses to the spins requires large microwave powers potentially incompatible with the detection of quantum fields . finally , the echo emitted by the spins should faithfully restore the initial field , which implies that the echo recovery efficiency @xmath12 , that we define as the ratio of the energy radiated during the echo to the energy of the incoming pulse , should be close to @xmath13 . to summarise , reaching the quantum regime requires a mean excitation per mode ( both microwave and spin ) @xmath14 , input microwave fields with intra - cavity photon number @xmath15 , and an echo efficiency @xmath12 close to @xmath13 . are stored in the spin ensemble . a refocusing pulse @xmath16 acts as time - reversal for the spins and triggers the retrieval of the stored pulses as echoes @xmath9 in reverse order . top and bottom time lines show the applied , and the reflected and echo signals , respectively . ( b ) setup placed in a dilution refrigerator : the cavity is a lumped - element parallel lc resonator in niobium coupled to a coplanar waveguide by a capacitor @xmath17 . it consists of an interdigitated capacitor @xmath18 and a meander wire inductor @xmath19 creating the ac magnetic field shown in inset , for a @xmath20 incident microwave power at resonance . the spin ensemble consists of nv centres in a diamond monocrystal pressed on top of the inductor . laser pulses can be shone on it through an optical fibre glued to its top face . a tunable dc magnetic field @xmath21 is applied parallel to the @xmath22 $ ] direction of the crystal . ( c ) negatively - charged nv centres in diamond consist of a nitrogen atom next to a vacancy of the diamond lattice , having trapped an electron . their electronic spin @xmath23 is coupled by hyperfine interaction to the nitrogen nuclear spin @xmath24 ( for the @xmath25 isotope ) . half of the electronic spins ( sub - ensemble denoted n - orth in blue ) make an angle @xmath26 with @xmath21 , whereas the other half ( sub - ensemble orth in red ) is orthogonal to the field . ( d ) nv simplified energy diagram ( top ) showing the ground @xmath27 and the excited @xmath28 electronic states as well as the zeeman and hyperfine structure of @xmath27 , with @xmath29ghz the zero - field splitting . ( bottom ) magnetic field dependence of the allowed transitions for both n - orth ( blue ) and orth ( red ) sub - ensembles , showing respectively a linear and quadratic zeeman effect . nvs can be optically repumped in their @xmath30 ground state by application of green ( @xmath31 nm ) laser pulses exciting the @xmath27 - @xmath28 transition.,width=377 ] these stringent requirements have never been met in an experiment , by far . the multi - mode character of the 2pe has been recently benchmarked in the classical regime @xcite with an ensemble of phosphorus donors in silicon at @xmath32k in the three - dimensional microwave cavity of an electron paramagnetic resonance spectrometer . that experiment reached @xmath33 , @xmath34 , and an echo recovery efficiency @xmath35 . here we use negatively - charged nitrogen - vacancy ( nv ) centres in diamond , which are colour centres consisting of a substitutional nitrogen atom sitting next to a vacancy of the lattice ( see fig . [ fig1]c ) with properties suitable for a quantum memory : their spin triplet ( @xmath23 ) electronic ground state has a long coherence time @xcite and can be optically repumped in the spin ground state @xmath36 ( see figs . [ fig1]c and d ) . we re - visit the 2pe protocol with an ensemble of nv centres at @xmath37mk coupled to a planar superconducting resonator , in a setup compatible with hybrid quantum circuits , with active reset of the spin at the beginning of each experimental sequence , and we demonstrate the storage of multiple pulses at the picowatt level for @xmath38 , three orders of magnitude longer than in earlier experiments @xcite . our experiment reaches @xmath39 , @xmath40 , @xmath41 , and @xmath42 , and comes therefore closer to the quantum regime than previous work by several orders of magnitude . we quantitatively identify the present limitations and show that they can be solved in future experiments , opening the way to the implementation of quantum memory protocols .

a quantum memory at microwave frequencies , able to store the state of multiple superconducting qubits for long times , is a key element for quantum information processing . electronic and nuclear spins are natural candidates for the storage medium as their coherence time can be well above one second . benefiting from these long coherence times requires to apply the refocusing techniques used in magnetic resonance , a major challenge in the context of hybrid quantum circuits . here we report the first implementation of such a scheme , using ensembles of nitrogen - vacancy ( nv ) centres in diamond coupled to a superconducting resonator , in a setup compatible with superconducting qubit technology . this enables the storage of multiple microwave pulses at the picowatt level and their retrieval after up tos , a three orders of magnitude improvement compared to previous experiments .

a quantum memory at microwave frequencies , able to store the state of multiple superconducting qubits for long times , is a key element for quantum information processing . electronic and nuclear spins are natural candidates for the storage medium as their coherence time can be well above one second . benefiting from these long coherence times requires to apply the refocusing techniques used in magnetic resonance , a major challenge in the context of hybrid quantum circuits . here we report the first implementation of such a scheme , using ensembles of nitrogen - vacancy ( nv ) centres in diamond coupled to a superconducting resonator , in a setup compatible with superconducting qubit technology . we implement the active reset of the nv spins into their ground state by optical pumping and their refocusing by hahn echo sequences . this enables the storage of multiple microwave pulses at the picowatt level and their retrieval after up tos , a three orders of magnitude improvement compared to previous experiments .

1502.06562

i

glass transition on surface / interface / thin film has drawn intensive interests in the last two decades for two reasons . firstly , the importance of surface and thin film in materials science and engineering requires a better understanding of its dynamic and thermodynamic properties . secondly , the confined geometry is a good approach to understand the mysterious glass transition itself , especially to explore the dynamic properties within the thin film geometry . numerous works of both experimental and simulation / calculation approaches have been done [ 1 - 23 ] on glass transition on surface / thin film . keddie and co - workers firstly investigated the supported thin film of ps by ellipsometric measurements @xcite . they prepared several ps films supported by silicon wafers with thickness from @xmath0 to @xmath1 . the measurements indicated the deduction of the @xmath2 with thickness under 400@xmath3 . an empirical relationship between thickness and @xmath2 was given as : @xmath4 , \label{tg_reduction}\ ] ] where @xmath5 is the glass transition temperature of the bulk ps . the @xmath6 and @xmath7 are the parameters fitted to be @xmath8 and @xmath9 respectively , @xmath10 is the thickness of the film . following keddie s work , many researches have been done on different supported thin films of various polymers [ 2 - 3 ] by different characterization methods , such as x - ray reflectivity , positron annihilation , and dielectric @xcite . most results have demonstrated the same phenomenon that for liner polymer the @xmath2 decreases with the thickness of films . however the supported film has a considerable film - substrate interaction , which makes the conclusion controversial . strong attractive interaction between the substrate and thin film may increase the @xmath2 of thin film above the bulk @xmath2 [ 6 ] . van zanten et al . measured @xmath2 of poly-2-vinylpyridine on oxide - coated si substrates , and found it increase by @xmath11k than the bulk @xmath2 , for a @xmath12 film [ 7 ] . forrest and co - workers have done pioneer works in measuring the @xmath2 of free - standing thin films [ 8 - 10 ] . they measured the @xmath2 of free - standing ps films with thickness from @xmath13 to @xmath14 and different molecular weights by brillouin light scattering and transmission ellipsometry . their results showed that the @xmath2 decreases with the thickness of ps thin film with a much larger magnitude : for example , the @xmath2 of @xmath15 film with _ mw _ within @xmath16k to @xmath17k reduces by 70k below the @xmath5 , while this magnitude is around @xmath18k for supported films . the empirical equation ( eq.[tg_reduction ] ) derived for supported film still holds for the low _ mw _ free - standing films . with @xmath19 , the parameter @xmath20 was found to be @xmath21 which is twice of it found for supported films . other than the experimental work , computer simulation and calculation have also been developed to investigate the glass transition on surface / thin film , and most of them are for polymer systems [ 11 - 23 ] . molecular dynamics ( md ) and monte carlo ( mc ) method were usually employed with various modelings , and confirmed the @xmath2 decrease with the thickness reduction for both supported and free standing film , or @xmath2 increase in some particular substrate - film cases . the eq.[tg_reduction ] derived from experiments can also be validated by simulations , nevertheless the explanation for the mechanism of @xmath2 reduction is still a matter of debate . most md simulations verified the experimental observation that for supported film , a strong substrate - film interaction will increase the @xmath2 above the value in the bulk , while the weak substrate - film interaction will lead @xmath2 reduction , and free - standing film shows a much larger @xmath2 reduction than supported film with weak substrate - film interaction [ 12 ] . mattice and co - workers firstly reported the mc simulation in this field [ 13 , 14 ] . de pablo et.al . reported a mc simulation on free - standing films of both linear and cyclic polymeric chains [ 23 ] . basically the @xmath2 reduction with smaller thickness was confirmed in the above investigations . in de pablo and co - workers work , the relation between @xmath2 and @xmath5 can also be fitted to be eq . [ tg_reduction ] with slight difference on the parameters fitting . although the glass transition is not a unique property of polymers , most simulation works were modeled for polymer systems , while the works on small molecule systems are very rare . meanwhile , ising spins glass has also been widely utilized to study the glass transition [ 24 - 37 ] . by different lattices adopted and interactions setup the ising model is capable to describe various systems , such as gas , liquid , crystal and glass , and consequently the phase transitions , like melting and glass transition [ 24 ] . however , very few of the efforts were on surface / thin film glass transition of ising spin glass . this field had been explored by gujrati et al . by applying ising model on a modified bethe lattice to describe the thermodynamics of polymer systems near surface [ 38 - 42 ] . in this paper we follow the similar method to study the glass transition of ising spin glass on the surface / thin film on a specially constructed recursive lattice .

a quasi 2-dimensional recursive lattice formed by planar elements have been designed to investigate the surface thermodynamics of ising spin glass system with the aim to study the metastability of supercooled liquids and the ideal glass transition . the recursive properties of the lattices were adopted to achieve exact calculations . the model has an anti - ferromagnetic interaction to give rise to an ordered phase identified as crystal , and a metastable solution representing the amorphous / metastable phase . free energy and entropy of the ideal crystal and supercooled liquid state of the model on the surface are calculated by the partial partition function . by analyzing the free energies and entropies of the crystal and supercooled liquid state , we are able to identify the melting transition and the second order ideal glass transition on the surface .

a quasi 2-dimensional recursive lattice formed by planar elements have been designed to investigate the surface thermodynamics of ising spin glass system with the aim to study the metastability of supercooled liquids and the ideal glass transition . the lattice is constructed as a hybrid of partial husimi lattice representing the bulk and 1d single bonds representing the surface . the recursive properties of the lattices were adopted to achieve exact calculations . the model has an anti - ferromagnetic interaction to give rise to an ordered phase identified as crystal , and a metastable solution representing the amorphous / metastable phase . interactions between particles farther away than the nearest neighbor distance are taken into consideration . free energy and entropy of the ideal crystal and supercooled liquid state of the model on the surface are calculated by the partial partition function . by analyzing the free energies and entropies of the crystal and supercooled liquid state , we are able to identify the melting transition and the second order ideal glass transition on the surface . the results show that due to the coordination number change , the transition temperature on the surface decreases significantly compared to the bulk system . our calculation agrees with experimental and simulation results on the thermodynamics of surfaces and thin films conducted by others .

1311.3818

i

the deformation properties of bulk metallic glasses are characterised by two temperature regimes @xcite . at high temperatures , close to the glass transition temperature , bmgs deform homogeneously with the strain rate properties being reasonably well understood by thermally activated plastic flow . at lower temperatures , plasticity becomes heterogeneous and highly localized to shear - bands , resulting in limited ductility in tension and a few percent plastic strain prior to failure for some bmgs in compression . in this regime of temperatures both athermal @xcite and thermal @xcite theories of plasticity have been proposed . in the early work of turnbull and cohen @xcite these different regimes emerged as a result of two competing processes , one diffusional and one stress induced . later , spaepen @xcite attributed the homogeneous regime as resulting from a steady state of free volume creation and diffusional related free volume annihilation . heterogeneous flow on the other hand was dominated by free volume creation since annihilation became diffusion limited at low temperatures . that two distinct temperature scales exist was also recognised by argon @xcite , who from the perspective of thermal activation developed two microscopic theories where at high temperature , plasticity was mediated by small atomic reconfigurations that resulted in a localized shear strain event , and at lower temperature , a less local structural excitation that was akin to the nucleation of a dislocation loop occurs . such mechanisms could give insight into the observed high temperature homogeneous plasticity and the low temperature heterogeneous plasticity . some time after this , high - strain rate atomistic simulation revealed the generality of such local structural excitations , where the transition from homogeneous to heterogeneous plasticity ( as a function of decreasing temperature or increasing strain rate ) emerged from the degree of correlation between such localized activity @xcite . due to their high strain rates and stresses , both dynamic and static atomistic simulations primarily probe the athermal region of plasticity . out of the earliest of this work by falk and langer @xcite arose the concept of the shear transformation zone ( stz ) which was assumed to exist in regions of the material not so well relaxed the so - called liquid like regions @xcite . the corresponding structural transformations were assumed to be activated _ athermally _ once a critical stress had been reached . such ideas developed into what is now called the effective temperature theories in which the quenched disorder well below the glass transition temperature is characterised by an effective temperature that describes the structural fluctuations within the material , see for example @xcite . this approach has been quite successful in developing a quantitative understanding of experimental low temperature deformation data . it was recognised by bouchbinder , langer and falk @xcite that if one were to assume a thermal activation picture within the stz framework @xcite , the density of such local structural excitations would have to be `` improbably '' large if the corresponding activation energies are in the realistic range of hundreds of kilojoule per mole ( or electron volts per excitation ) . in going to an athermal picture , this difficulty was avoided and indeed stzs could now occur at quite dilute densities @xcite , a picture that is compatible with the distinct liquid like regions assumed to exist within the glass . understanding the nature of such stzs , or more generally localized structural excitations , has also been influenced by the physics of the under - cooled liquid regime , in which two distinct relaxation time - scales are evidenced : that of the slow @xmath0-relaxation and the fast @xmath1-relaxation processes first introduced by goldstein @xcite . at temperatures near the glass transition and below , the @xmath0-relaxation processes rapidly increase their time - scale with the traditional viewpoint being that they freeze out of the structural dynamics in the amorphous solid regime @xcite . despite such an assumed freezing , the terminology of @xmath0- and @xmath1-relaxation has also been applied to the amorphous solid regime where the microscopic @xmath1 processes mediate escape from the mega - basins of the elastic energy landscape of the @xmath0 processes @xcite . the present work also exploits this terminology and uses an additional fundamental feature of the under - cooled liquid regime , that the number of structural transformations available will scale exponentially with respect to a fundamental volume measure @xcite . such ideas have been used in the early thermodynamic theories of adam and gibbs @xcite and kirkpatrick _ et al _ @xcite see also the review article by heuer @xcite and provides a sufficient number of structural transformations to justify a thermal activation picture for temperatures that are well below the glass transition . in particular the present work develops further the ideas of refs . @xcite . in these works , a thermal activation picture was developed in which the characteristic plastic rate with respect to an internal heterogeneous volume scale was determined from a distribution of @xmath0-relaxation barriers whose number scales exponentially with the size of the heterogeneous volume and whose statistics is described by a distribution with extensive first and second cumulants . by doing this , the average time scale associated with plasticity at a particular temperature becomes intimately connected to the @xmath0-relaxation potential energy landscape ( mediated by @xmath1 relaxation processes ) which is thermally accessible at that temperature . at zero loading or any given shear stress , two critical temperature scales emerge : 1 ) the plastic transition temperature which indicates a significant drop in the amount of plastic activity as the temperature is reduced , and 2 ) the ( lower ) kinetic freezing temperature where ( on average ) a single structural transformation becomes available per heterogeneous volume . the transition between the higher temperature regime of 1 ) and the lower temperature region of 2 ) corresponds to a change of statistics , from that of the most probable to that of the extreme . importantly , before the critical freezing temperature is reached , there will exist a low temperature regime ( below the glass transition ) where a sufficient number of thermally activated events exist to mediate a non - negligible macroscopic plasticity . thus , in ref . @xcite , the low temperature heterogeneous and high temperature homogeneous deformation regimes are distinguished by a transition in the nature of the statistics of the @xmath0-relaxation energy landscape . put in other words , the freezing out of @xmath0-relaxation processes is only complete at temperatures far below that of the glass transition temperature , and until this extreme regime is reached thermal activation remains the underlying phenomenon controlling macroscopic plasticity a fact that has been established experimentally for a variety of deformation geometries @xcite including most recently that of shear band nucleation , propagation and arrest @xcite . the present work extends on the work done in ref . @xcite ( which will be referred to as paper i ) by using a log - normal distribution to describe the statistics of the @xmath0-relaxation barrier energy landscape to develop a mean - field description of thermally activated plasticity for the high temperature / low strain rate regime of homogeneous deformation . [ secsummary ] gives a brief overview of the current theory , secs . [ seclnd]-[sectheory ] develops the theory for the positive valued log - normal distribution of barrier energies , sec . [ secmfp ] develops a mean field description of plasticity suitable for the high temperature homogeneous regime of deformation , and sec . [ secexp ] applies the theory to well known experimental data for the well studied bmg vitroley-1 . in the discussion and conclusion , sec . [ secconc ] , two results are summarized : 1 ) when comparing with the usual arrhenius description of high temperature homogeneous plasticity , the corresponding activation volume parameter is found to be equivalent to the variance of the internal slipped volume distribution , and 2 ) the fragility of the material is a measure of the number of structural transformations available at the glass transition temperature .

2013 , doi : 10.1080/14786435.2013.826396 ) which uses a distribution of barrier energies and some aspects of under - cooled liquid physics is developed further . in particular , a log - normal distribution is now employed to describe the statistics of barrier energies . a high temperature mean - field description of homogeneous macro - plasticity is then developed and is shown to be similar to a thermal activation picture employing a single characteristic activation energy and activation volume . in making this comparison , the activation volume is interpreted as being proportional to the average mean - square - value of the plastic shear strain magnitude within the material . also , the kinetic fragility at the glass transition temperature is shown to represent the effective number of irreversible structural transformations available at that temperature .

a thermal activation model to describe the plasticity of bulk metallic glasses ( derlet and maa , phil . mag . 2013 , doi : 10.1080/14786435.2013.826396 ) which uses a distribution of barrier energies and some aspects of under - cooled liquid physics is developed further . in particular , a log - normal distribution is now employed to describe the statistics of barrier energies . a high temperature mean - field description of homogeneous macro - plasticity is then developed and is shown to be similar to a thermal activation picture employing a single characteristic activation energy and activation volume . in making this comparison , the activation volume is interpreted as being proportional to the average mean - square - value of the plastic shear strain magnitude within the material . also , the kinetic fragility at the glass transition temperature is shown to represent the effective number of irreversible structural transformations available at that temperature .

astro-ph0407270

i

[ sec : intro ] among the suite of molecules observed in interstellar and circum - protostellar media , methane is relatively poorly studied . ch@xmath1 has no permanent dipole moment , and therefore can not be observed by pure rotational transitions at radio wavelengths . its strongest fundamental ( @xmath8 ) ro - vibrational transitions , i.e. the @xmath9 c - h stretching and the @xmath10 c - h bending modes , occur at 3.32 and 7.67 @xmath5 , respectively . strong telluric absorption at these wavelengths hinders interstellar ch@xmath1 studies . gas and solid phase ch@xmath1 have been positively identified at 7.67 @xmath5 in three sight - lines : the massive protostars ngc 7388 : irs9 , w 33a , and gl 7009 s @xcite . ch@xmath1 was detected only in the solid state in low resolution _ infrared space observatory ( iso ) _ observations at 7.67 @xmath5 toward @xmath11 low mass protostars @xcite , the galactic center @xcite , and an external galactic nucleus @xcite . a recent low resolution _ spitzer space telescope _ spectrum shows absorption by solid ch@xmath1 toward the young , low mass protostar iras 08242 + 5100 @xcite . observations of ch@xmath1 provide insight into the basic principles of astrochemical networks . in the presence of atomic c and h , ch@xmath1 is rapidly formed on cold grains , just as h@xmath12o is formed from atomic o and h ( e.g. @xcite ) . perhaps surprisingly , the observed gas and solid state ch@xmath1 abundances are low , not more than a few percent with respect to co and h@xmath12o @xcite . this points to relatively low atomic c abundances at the time of ch@xmath1 formation , when most c is already locked up in co. in this same scenario the high ch@xmath13oh abundances in several lines of sight may be explained by hydrogenation of abundantly present co @xcite , and large co@xmath12 abundances by oxidation of co. the dominance of a co- and o - based , rather than c - based , chemistry is stressed by the profiles of solid co@xmath12 absorption bands which show an intimate mixture of co@xmath12 , ch@xmath13oh , and h@xmath12o in the ices @xcite . nevertheless , even at relatively low abundances methane is proposed to be the starting point of a rich chemistry leading to the complex organic molecules observed in dense clouds like the tmc-1 ridge @xcite . to further test the role of ch@xmath1 in interstellar chemistry , basic diagnostics , such as gas phase abundances , gas / solid phase abundance ratios , gas phase temperatures and abundance gradients need to be measured in more lines of sight . in this work we report the discovery of absorption and emission features arising from the 3.32 @xmath5 c - h stretching mode of gas and solid phase ch@xmath1 . a previous attempt to detect ch@xmath1 at this wavelength was unsuccessful and indicated an abundance of @xmath14 with respect to h in the orion bn object @xcite . we study the line of sight toward the massive protostar ngc 7538 : irs9 , because its high radial velocity of @xmath15 @xmath4(e.g . @xcite ) facilitates the separation of telluric and interstellar lines . ngc 7538 : irs9 is a well studied massive protostar , known for its large ice and gas columns @xcite and a prominent molecular outflow ( mitchell , maillard , & hasegawa 1991 ) . the r(0 ) and r(2 ) lines of the bending mode of gas phase ch@xmath1 and solid state ch@xmath1 were detected in this source by @xcite . unambiguous confirmations of these detections were obtained with space based observations @xcite . here we independently confirm the presence of ch@xmath1 by detecting many lines from the 3.32 @xmath5 stretching mode as well as the ice band . the high spectral resolution of these observations permits a line profile analysis in which we determine gas phase conditions in the various environments along the line sight , such as the envelope , the hot core , and the outflow . this , in turn , elucidates the formation and destruction pathways of ch@xmath1 in these different environments . the keck / nirspec and iso / sws observations are discussed in [ sec : obs ] , and presented in [ sec : feat ] . a profile analysis of the gas phase absorption and emission ch@xmath1 lines , in conjunction with previously reported high resolution co observations , is presented in [ sec : ch4 ] and [ sec : co ] . the ice band profile is discussed and compared with the c - h bending mode in [ sec : ice ] . the implications of these results on the formation and evolution history of the ch@xmath1 molecule in protostellar environments are discussed in [ sec : disc1 ] . the origin of the low velocity , warm gas , responsible for the p cygni - like line profiles , is further constrained in [ sec : disc2 ] . we conclude with future prospects in [ sec : concl ] . [ f : obs ]

it is concluded that the gas phase ch abundance varies by an order of magnitude in this line of sight : it is low in the envelope and the outflow ( x[ch] ) , and at least a factor of 10 larger in the central core . the discovery of solid ch in independent ground and space based data sets shows that methane is nearly entirely frozen onto grains in the envelope .

absorption and emission lines originating from the c - h stretching manifold of gas phase ch were discovered in the high resolution (=25,000 ) infrared l band spectrum along the line of sight toward ngc 7538 : irs9 . these observations provide a diagnostic of the complex dynamics and chemistry in a massive star forming region . the line shapes resemble p cygni profiles with the absorption and emission components shifted by with respect to the systemic velocity . similar velocity components were observed in co at 4.7 , but in contrast to ch , the co shows deep absorption due to a high velocity outflow as well as absorption at the systemic velocity due to the cold outer envelope . it is concluded that the gas phase ch abundance varies by an order of magnitude in this line of sight : it is low in the envelope and the outflow ( x[ch] ) , and at least a factor of 10 larger in the central core . the discovery of solid ch in independent ground and space based data sets shows that methane is nearly entirely frozen onto grains in the envelope . it thus appears that ch is formed by grain surface reactions , evaporates into the gas phase in the warm inner regions of protostellar cores and is efficiently destroyed in shocks related to outflows .

astro-ph0407270

c

[ sec : disc ] [ t : col ] lllccccl species & location@xmath65 & phase@xmath66 & @xmath104&@xmath105 & @xmath106 & @xmath107@xmath108 & notes + & & & @xmath4 & k & @xmath109 @xmath48 & 10@xmath110 & + ch@xmath1 & outflow & gas / abs . & @xmath2216 & 270 & @xmath111 & @xmath112 & assumed @xmath80 + ch@xmath1 & core & gas / abs . & @xmath226 & 55@xmath8315 & [email protected] & @xmath113 & @xmath78=0.53 @xmath4 + ch@xmath1 & envelope & gas / abs . & 0 & 20 & @xmath111 & @xmath114 & assumed @xmath80 + ch@xmath1 & core & gas / em . & + 8 & 600@xmath83100 & 0.58@xmath115 & 1.9@xmath116 & @xmath117 ster@xmath118 + ch@xmath119 & envelope & ice / abs . & & & 1.3@xmath120 & 2.1@xmath121 & + co & outflow & gas / abs . & @xmath2216 & 270@xmath83100 & 35@xmath8310 & 100 & opt . thin @xmath90 + co & core & gas / abs . & @xmath226 & 55 & @xmath122 & 100 & assumed @xmath80 + co & envelope & gas / abs . & 0 & 20@xmath837 & 35@xmath8310 & 100 & opt . thin @xmath90 + co & core & gas / em . & + 8 & 500@xmath83150 & 30@xmath8310@xmath123 & 100 & @xmath117 ster@xmath118 + co@xmath124 & envelope & ice / abs . & & & 18@xmath832 & 36@xmath125 & + [ sec : disc1 ] the detections of both the 3.32 @xmath5 stretch and the 7.67 @xmath5 bend modes firmly establish the presence of both gaseous and solid ch@xmath1 in the interstellar medium . the gas+solid state abundance averaged along the line of sight of the massive protostar ngc 7538 : irs9 is x[ch@xmath1]=@xmath126 , and the absorption gas / solid state abundance ratio is 0.23 . a combined high resolution co / ch@xmath1 line profile analysis shows that the abundance as well as the gas / solid state abundance ratio of ch@xmath1 vary _ locally _ by an order of magnitude , however ( table [ t : col ] ) . an ice+gas abundance of a few @xmath127 is found in the outer envelope ( systemic velocity ) , the inner envelope ( 6 @xmath4 ) and the compact emission component ( + 8 @xmath4 ) . in the outer envelope , at 20 k the coldest component along the line of sight , at least 90% of the ch@xmath1 is frozen on grains . within the envelope a segregation is present as well ; ch@xmath1 is intimately mixed with h@xmath12o and ch@xmath13oh rather than with co. in the warm , low velocity inner regions of ngc 7538 : irs9 , ch@xmath1 is mostly present in the gas phase . these enhanced abundances strongly contrast to the absence of ch@xmath1 in the massive , high velocity 16 @xmath4 , warm ( @xmath128 k ) outflow seen in co ( x[ch@xmath1]@xmath7 ) . these new results provide clues to the formation and evolutionary history of ch@xmath1 , refining previous scenarios @xcite . in time dependent pure gas phase models , such as the ` standard ' dense cloud model of @xcite and the umist model of @xcite , gas phase ch@xmath1 abundances of a few 10@xmath110 are achieved after 10@xmath129 years . although this is comparable to the observed abundances , there are strong arguments against a gas phase origin : the low gas / solid state ratio and the intimate mixture between ch@xmath1 and h@xmath12o in the ices in the envelope . for comparison , co , a typical gas phase product , has an order of magnitude larger gas / solid phase ratio compared to ch@xmath1 , even in the cold gas phase . h@xmath12o on the other hand is formed by hydrogenation of oxygen on the grains at low temperatures ( e.g. @xcite ) , and a low gas / solid state ratio is observed @xcite . it is thus likely that ch@xmath1 was formed on grain surfaces by hydrogenation of atomic c , at the same time as h@xmath12o was formed from atomic o. the low observed ch@xmath1 abundance points to a formation time when most of the c was already locked up in co @xcite . the gas phase ch@xmath1 abundance in the 6 and + 8 @xmath4 components is comparable to the solid state abundance in the envelope . the ch@xmath1 may thus have been released from the grains in the inner regions near the protostar . because ch@xmath1 is diluted in an h@xmath12o rich ice ( fig . [ f : ice ] ) , its sublimation temperature is close to that of h@xmath12o , 90 k @xcite . the temperature of the + 8 @xmath4emission component is indeed well above 90 k. the gas at 6 @xmath4 , however , has a temperature of 55@xmath8315 k only . while in one massive protostar , w 33a , ch@xmath1 gas at @xmath130 k was detected @xcite , a lower temperature of 50 k was also measured in another line of sight , gl 7009s @xcite . we speculate that the icy mantles did not thermally evaporate as a result of protostellar radiative heating , but by a mild shock that has briefly heated the grains above 90 k. subsequently the gas has cooled radiatively or adiabatically as the inner ngc 7538 : irs9 core expands outwards with a ( current ) velocity of 6 @xmath4 . shocks may have had a profound impact on the higher velocity gas . ch@xmath1 is absent at velocities of 16 @xmath4 and above , where warm co ( @xmath131 k ) is very abundant . after release from the grains , ch@xmath1 must have efficiently chemically reacted . the observed co velocities up to 35 @xmath4 point to the presence of non - destructive c - type shocks . these shocks have been well studied in the context of molecular chemistry . at initial densities of 10@xmath129 @xmath132 , shocks with velocities of 1535 @xmath4 have peak temperatures of 500 - 2500 k ( bergin , neufeld , & melnick 1998 ) . at such high temperatures , and at the low expected shock pressures bar assuming a factor of 2 density enhancement at a maximum temperature of 2500 k for a 35 @xmath4 shock @xcite ] , ch@xmath1 is rapidly driven into co @xcite . the occurrence of a c shock therefore explains the absence of ch@xmath1 in the outflow of ngc 7538 : irs9 . the importance of shock chemistry in high mass cores was deduced from low gas / solid co@xmath12 abundance ratios as well @xcite , and by the abundance ratios of s bearing species @xcite . one distinct characteristic of shock chemistry is the formation of h@xmath12o at temperatures above 250 k ( e.g. @xcite ) . h@xmath12o columns comparable to co are expected , i.e. 2.8@xmath133 @xmath48 for the ngc 7538 : irs9 outflow ( table [ t : col ] ) . this is a factor of 3.5 larger than the upper limit to the gas phase h@xmath12o abundance derived from iso / sws 6 @xmath5 spectra @xcite . high resolution 6 @xmath5spectra are required to further investigate the presence of h@xmath12o in the line of sight of ngc 7538 : irs9 , the effect of blended emission and absorption components and the relation to shocks and outflows . [ sec : disc2 ] the elevated temperature of the low velocity , 6 - 8 @xmath4 , absorption and emission components seen both in co and ch@xmath1 locates this gas in the inner regions of the protostellar core . the present observations further constrain the origin of this material . the area of the radiating physical body can be calculated by matching the @xmath93 column density derived from the isotope ratio ( [ sec : co ] ) to the emission line fluxes . the measured @xmath93line fluxes need to be corrected for continuum absorption by foreground material . we derive a 4.7 @xmath5 continuum extinction magnitude of @xmath134 using the interstellar extinction law of @xcite and an optical extinction @xmath46 of 60@xmath8320 . the latter is derived from the @xmath46/@xmath135 relation of @xcite and the average line of sight hydrogen column density from this work ( table [ t : col ] ) and that from the silicate absorption band @xcite . the extinction corrected line fluxes are thus matched at a solid angle of 4.3@xmath49 sterad . at the distance to ngc 7538 : irs9 of 2.8 kpc @xcite this corresponds to a radius of 70 au . the similarity of co and ch@xmath1 velocities and emission line widths ( fig . [ f : velo]c ) indicates that both molecules are present in this same volume . the co and ch@xmath1 gas detected on these scales may well be related to the dense material that was inferred on scales of @xmath136 au from unresolved 3 millimeter continuum interferometer images @xcite . this intense emission is unrelated to the envelope , and instead may originate from a dense shell or circumstellar disk . the p cygni - like line profiles in both co and ch@xmath1 may suggest that an expanding envelope is the preferred geometry . in these line profiles the absorption component is red - shifted by the shell expansion velocity , and the emission component is centered at the systemic velocity and is broadened by twice the expansion velocity . @xcite argue that the co emission component is blue - shifted due to the effect of deep absorption by foreground material at the systemic velocity . this hypothesis can be tested by the profiles of the highest @xmath32 transitions , which should not have foreground contributions . we find that even the lines from the highest rotational levels of co are blue shifted by + 8 @xmath4(fig . [ f : velo]c ) . furthermore , the ch@xmath1 emission lines peak at this same velocity and are also unresolved . perhaps the ngc 7538 : irs9 core moves within its envelope by 8 @xmath4 and the observed peak of the emission line is the center of the p cygni profile . in this scenario the shell expansion velocity is 14 @xmath4 and an emission line width of 28 @xmath4 is expected . such broad lines are inconsistent with the observations as well . we conclude that the line profiles are not true p cygni type profiles and thus are inconsistent with a spherically symmetric expanding envelope . it is unclear what the relation is between the 55 k absorbing gas seen at @xmath226 @xmath4 and the 500 k emitting gas at + 8 @xmath4 . more detailed dynamical models of circumstellar disks , accretion flows and outflows are required . the possibility of photon excitation needs to be considered as well . as noted in @xcite the similarity of co absorption and emission line profiles between a number of massive protostars suggests that these lines trace a fundamental property of their inner cores . this is strengthened by the similarity of the ch@xmath1 line profiles .

absorption and emission lines originating from the c - h stretching manifold of gas phase ch were discovered in the high resolution (=25,000 ) infrared l band spectrum along the line of sight toward ngc 7538 : irs9 . the line shapes resemble p cygni profiles with the absorption and emission components shifted by with respect to the systemic velocity .

absorption and emission lines originating from the c - h stretching manifold of gas phase ch were discovered in the high resolution (=25,000 ) infrared l band spectrum along the line of sight toward ngc 7538 : irs9 . these observations provide a diagnostic of the complex dynamics and chemistry in a massive star forming region . the line shapes resemble p cygni profiles with the absorption and emission components shifted by with respect to the systemic velocity . similar velocity components were observed in co at 4.7 , but in contrast to ch , the co shows deep absorption due to a high velocity outflow as well as absorption at the systemic velocity due to the cold outer envelope . it is concluded that the gas phase ch abundance varies by an order of magnitude in this line of sight : it is low in the envelope and the outflow ( x[ch] ) , and at least a factor of 10 larger in the central core . the discovery of solid ch in independent ground and space based data sets shows that methane is nearly entirely frozen onto grains in the envelope . it thus appears that ch is formed by grain surface reactions , evaporates into the gas phase in the warm inner regions of protostellar cores and is efficiently destroyed in shocks related to outflows .

astro-ph0407270

c

[ sec : concl ] using high resolution l and m band spectroscopy we have shown that the gas phase ch@xmath1 abundance varies considerably in the line of sight of the massive protostar ngc 7538 : irs9 . this is a result of specific molecule formation and destruction processes : ch@xmath1 is formed on grain surfaces and subsequently evaporated from the grains in the inner regions of the protostar . both evaporation by photon heating and by shocks likely have played roles . ch@xmath1 efficiently ` burns ' to co in the highest velocity shocked regions . the gas phase ch@xmath1/co abundance ratio is thus a sensitive tracer of shock chemistry . high spectral resolution observations of other species , in particular h@xmath12o at 6 @xmath5 , are required to further test the importance and nature of shock chemistry in high mass star forming regions . the _ stratospheric observatory for infrared astronomy ( sofia ) _ will offer this opportunity in the near future @xcite . observations of the 3.32 @xmath5 ch@xmath1 stretch mode in more lines of sight require high radial velocities ( @xmath137 @xmath4 ) or temperatures that populate the highest @xmath32 levels to avoid saturated telluric lines . in this respect , the 7.67 @xmath5 bending mode may be more suited . future observations of ch@xmath1 toward low mass protostars are particularly exciting , as the ch@xmath1/co ratio is a sensitive tracer of protoplanets . in the relatively low pressure solar nebula low ch@xmath1/co abundances occur . the higher pressure in protoplanetary sub - nebulae , however , favors higher ch@xmath1/co ratios @xcite . variations of the ch@xmath1 abundance between solar system comets possibly trace such physical condition gradients in the early protoplanetary disk @xcite . the research of a.c.a.b . and g.a.b . is supported by the spitzer legacy science program and by the owens valley radio observatory through nsf grant ast-0228955 . the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community . we are most fortunate to have the opportunity to conduct observations from this mountain . we thank jan cami for providing a copy of iso / sws ia , fred lahuis for an independent reduction , tim brooke for suggesting the possible ethane identification , and yuk yung for help with the wavelength calibration . suggestions by an anonymous referee have led to an improved presentation of the data . alexander , r. d. , casali , m. m. , andr ' e , p. , persi , p. , & eiroa , c. 2003 , , 401 , 613 allamandola , l.j , sandford , s.a . , tielens , a.g.g.m . , & herbst , t.m . 1992 , , 399 , 134 bergin , e. a. , neufeld , d. a. , & melnick , g. j. 1998 , , 499 , 777 bohlin , r. c. , savage , b. d. , & drake , j. f. 1978 , , 224 , 132 boogert , a. c. a. , et al . 1996 , , 315 , l377 boogert , a. c. a. , schutte , w. a. , helmich , f. p. , tielens , a. g. g. m. , & wooden , d. h. 1997 , , 317 , 929 boogert , a. c. a. , helmich , f. p. , van dishoeck , e. f. , schutte , w. a. , tielens , a. g. g. m. , & whittet , d. c. b. 1998 , , 336 , 352 boogert , a.c.a . , ehrenfreund p. , gerakines , p.a . , tielens , a.g.g.m . , whittet , d.c.b . , et al . 2000 , , 353 , 349 boogert , a. c. a. , blake , g. a. , & tielens , a. g. g. m. 2002 , , 577 , 271 boogert , a. c. a. , et al . 2004 , , 154 , in press boonman , a. m. s. & van dishoeck , e. f. 2003 , , 403 , 1003 boudin , n. , schutte , w. a. , & greenberg , j. m. 1998 , , 331 , 749 brooke , t. y. , sellgren , k. , & smith , r. g. 1996 , , 459 , 209 brooke , t. y. , sellgren , k. , & geballe , t. r. 1999 , , 517 , 883 brown , p. d. , charnley , s. b. , & millar , t. j. 1988 , , 231 , 409 brown , l. r. , et al . 2003 , journal of quantitative spectroscopy and radiative transfer , 82 , 219 ceccarelli , c. , loinard , l. , castets , a. , tielens , a. g. g. m. , caux , e. , lefloch , b. , & vastel , c. 2001 , , 372 , 998 charnley , s. b. & kaufman , m. j. 2000 , , 529 , l111 chiar , j. e. , tielens , a. g. g. m. , whittet , d. c. b. , schutte , w. a. , boogert , a. c. a. , lutz , d. , van dishoeck , e. f. , & bernstein , m. p. 2000 , , 537 , 749 crampton , d. , georgelin , y. m. , & georgelin , y. p. 1978 , , 66 , 1 dartois , e. , dhendecourt , l. , boulanger , f. , jourdain de muizon , m. , breitfellner , m. , puget , j .- l . , & habing , h. j. 1998 , , 331 , 651 dartois , e. , schutte , w. , geballe , t. r. , demyk , k. , ehrenfreund , p. , & dhendecourt , l. 1999 , , 342 , l32 dartois , e. , dhendecourt , l. , thi , w. , pontoppidan , k. m. , & van dishoeck , e. f. 2002 , , 394 , 1057 de graauw , t. , et al . 1996 , , 315 , l49 draine , b. t. 2003 , , 41 , 241 fischer , j. , gamache , r. r. , goldman , a. , rothman , l. s. , & perrin , a. 2003 , journal of quantitative spectroscopy and radiative transfer , 82 , 401 genzel , r. 1992 , in the galactic interstellar medium , ed . d. pfenniger & p. bartholdi ( berlin : springer ) , 275 gerakines , p.a . , whittet , d.c.b . , ehrenfreund , p. , boogert , a.c.a . , tielens , a.g.g.m . , et al . 1999 , , 522 , 357 gibb , e. l. , mumma , m. j. , dello russo , n. , disanti , m. a. , & magee - sauer , k. 2003 , icarus , 165 , 391 g " urtler , j. , klaas , u. , henning , t. , ' abrah ' am , p. , lemke , d. , schreyer , k. , & lehmann , k. 2002 , , 390 , 1075 hatchell , j. & viti , s. 2002 , , 381 , l33 hiraoka , k. , miyagoshi , t. , takayama , t. , yamamoto , k. , & kihara , y. 1998 , , 498 , 710 jones , a. p. & williams , d. a. 1984 , , 209 , 955 kaufman , m. j. & neufeld , d. a. 1996 , , 456 , 250 kessler , m. f. , et al . 1996 , , 315 , l27 knacke , r. f. , noll , k. s. , geballe , t. r. , & tokunaga , a. t. 1985 , , 298 , l67 lacy , j. h. , carr , j. s. , evans , n. j. , baas , f. , achtermann , j. m. , & arens , j. f. 1991 , , 376 , 556 lacy , j. h. , knacke , r. , geballe , t. r. , & tokunaga , a. t. 1994 , , 428 , l69 lacy , j. h. , richter , m. j. , greathouse , t. k. , jaffe , d. t. , & zhu , q. 2002 , , 114 , 153 lee , h .- h . , bettens , r. p. a. , & herbst , e. 1996 , , 119 , 111 lide , d. r. , 1994 , crc handbook of chemistry and physics ( 75th ed . ; boca raton : crc press ) markwick , a. j. , millar , t. j. , & charnley , s. b. 2000 , , 535 , 256 mclean , i.s . , becklin , e.e . , bendiksen , o. , brims , g. , & canfield , j. 1998 , proc . spie , 3354 , 566 millar , t. j. , farquhar , p. r. a. , & willacy , k. 1997 , , 121 , 139 mitchell , g.f . , maillard , j .- , allen , m. , beer , r. , & belcourt , k. 1990 , , 363 , 554 mitchell , g. f. , maillard , j .- p . , & hasegawa , t. i. 1991 , , 371 , 342 noriega - crespo , a. , et al . 2004 , , 154 , in press pendleton , y. j. , sandford , s. a. , allamandola , l. j. , tielens , a. g. g. m. , & sellgren , k. 1994 , , 437 , 683 pontoppidan , k. m. , dartois , e. , van dishoeck , e. f. , thi , w .- f . , & dhendecourt , l. 2003 , , 404 , l17 prinn , r. g. , & fegley , b. 1989 , in : `` origin and evolution of planetary and satellite atmospheres '' , eds . s. k. atreya , j. b. pollack , m. s. matthews , the university of arizona press , p. 78 rothman , l. s. , et al . 2003 , journal of quantitative spectroscopy and radiative transfer , 82 , 5 sandford , s.a . , & allamandola , l.j . 1988 , icarus , 76 , 201 sandford , s. a. , allamandola , l. j. , tielens , a. g. g. m. , sellgren , k. , tapia , m. , & pendleton , y. 1991 , , 371 , 607 schutte , w. a. , gerakines , p. a. , geballe , t. r. , van dishoeck , e. f. , & greenberg , j. m. 1996 , , 309 , 633 spitzer , l. 1978 , physical processes in the interstellar medium ( new york : wiley ) spoon , h. w. w. , keane , j. v. , tielens , a. g. g. m. , lutz , d. , & moorwood , a. f. m. 2001 , , 365 , l353 tielens , a.g.g.m . , tokunaga , a.t . , geballe , t.r . , & baas , f. 1991 , , 381 , 181 tielens , a. g. g. m. & whittet , d. c. b. 1996 , iau symp . 178 : molecules in astrophysics : probes & processes , 178 , 45 van der tak , f.f.s . , van dishoeck , e.f . , evans , n.j . , & blake , g.a . 2000 , , 537 , 283 white , g. j. , et al . 2000 , , 364 , 741 whittet , d.c.b . , schutte , w.a . , tielens , a.g.g.m . , boogert , a.c.a . , de graauw , t. , et al . 1996 , , 315 , 357 willner , s. p. , et al . 1982 , , 253 , 174 wilson , t.l . , & rood , r.t . 1994 , , 32 , 191

these observations provide a diagnostic of the complex dynamics and chemistry in a massive star forming region . it thus appears that ch is formed by grain surface reactions , evaporates into the gas phase in the warm inner regions of protostellar cores and is efficiently destroyed in shocks related to outflows .

absorption and emission lines originating from the c - h stretching manifold of gas phase ch were discovered in the high resolution (=25,000 ) infrared l band spectrum along the line of sight toward ngc 7538 : irs9 . these observations provide a diagnostic of the complex dynamics and chemistry in a massive star forming region . the line shapes resemble p cygni profiles with the absorption and emission components shifted by with respect to the systemic velocity . similar velocity components were observed in co at 4.7 , but in contrast to ch , the co shows deep absorption due to a high velocity outflow as well as absorption at the systemic velocity due to the cold outer envelope . it is concluded that the gas phase ch abundance varies by an order of magnitude in this line of sight : it is low in the envelope and the outflow ( x[ch] ) , and at least a factor of 10 larger in the central core . the discovery of solid ch in independent ground and space based data sets shows that methane is nearly entirely frozen onto grains in the envelope . it thus appears that ch is formed by grain surface reactions , evaporates into the gas phase in the warm inner regions of protostellar cores and is efficiently destroyed in shocks related to outflows .

1107.0678

i

according to recent models , the accretion disk and the bh in agns are surrounded by a clumpy torus , with its inner radius governed by the dust sublimation process . regarding the inner radius of the torus , there was a systematic deviation between the observational results and the theory . in paper i , we showed that the anisotropy of the disk emission resolves this conflict for a typical type-1 agn . we found that the anisotropy makes the torus inner region closer to the central bh and concave . furthermore , the innermost edge of the torus may connect with the outermost edge of the accretion disk continuously . in this study , we have calculated the nir flux variation of the torus in response to a uv flash for various geometries of the disk , the torus and the observer . anisotropic illumination by the disk and the effect of the torus self - occultation contrast our study with earlier works . we have found that both the waning effect of each clump and the torus self - occultation selectively reduce the emission from the region with a short delay . thus , the resultant nir time response shows a @xmath19-dependent delay and an asymmetric profile with a negative skewness , opposing to the results for optically thin tori ( barvainis 1992 ) . by contrast with the fiducial viewing angle of 25 , a small viewing angle results in a short time delay with a narrow and peaky response . on the other hand , a more inclined viewing angle leads to a longer delay with a broader profile and to a redder nir - to - optical color . we propose that the red nir - optical color of type-1.8/1.9 objects is caused by not only the dust extinction but also intrinsically red color . the computed range of @xmath24 coincides with the observed one . compared with the modest torus thickness of @xmath108 , both a thick and a thin tori display the weaker nir emission , consistent with the work by nenkova et al . in other words , a modest thickness of the torus leads to the strongest nir emission . a selection bias is thus expected such that nir - selected agns tend to possess moderately thick tori . for thin tori , we have found that the nir fluence is in proportion to @xmath100 . as the torus becomes thicker , the nir response shows a slightly longer delay with a narrower and more heavily skewed profile due to the torus self - occultation . this trend opposes to the viewing angle dependency , where the delay and the width are positively correlated each other . on the contrary , as the torus gets thinner , the nir response becomes more rapid , narrower , closer to time - symmetric and low . in contrast to a thin disk with a sub - eddington accretion rate , a super - eddington accretion rate leads to a much weaker ( @xmath156 ) nir emission due to the disk self - occultation and the disk truncation by the self - gravity , and to a low time response . among the three dependencies examined for aligned tori , only the variation of the viewing angles reproduces the observed range of the delay . therefore , we propose that the viewing angle is the key parameter responsible for the observed scatter about the regression line in the @xmath24 v.s . @xmath6 conversely , the measurements of @xmath24 are potentially useful to estimate the inclination angles . we have also investigated the consequences of the misalignment between the torus and the disk axes . a variety of @xmath24 is achieved , which is wide enough to cover the observed range . a short delay is obtained for a small misalignment and is associated with a small nir fluence , while a a long delay is obtained for a largely misaligned torus with an usual fluence . this trend contrasts with the viewing angle dependency for aligned tori , where a short delay is associated with a normal nir fluence while a long delay means a large nir fluence . for highly misaligned cases , contributions from @xmath62 increase the time delay ( up to @xmath146 ) , the nir fluence and the skewness ( @xmath145 ) . in case the torus is optically thin ( with an inefficient self - occultation ) , the time delay of the nir emission from an aligned torus becomes longer . moreover , misaligned optically thin tori show a yet longer delay ( @xmath155 ) with a huge nir flux . from an observational point of view , these numerical results are summarised as follows . if the observed time delay of the nir emission is short , it will mean either a small viewing angle , a geometrically thin torus , or a slightly misaligned torus . if the delay is long , on the other hand , it indicates either an inclined viewing angle , an aligned optically thin torus , or a largely misaligned torus . an extremely long delay ( @xmath155 ) would mean a misaligned optically thin torus . as to the nir flux , a blue nir - to - optical color ( i.e. , a weak nir emission such as in hot - dust - poor agns in the weakest cases ) indicates either a geometrically thin torus , a geometrically thick torus , a super - eddington accretion , or a slight misalignment between the torus and the disk . on the other hand , a red nir - optical color ( a large nir emission ) means a large viewing angle with a modest geometrical thickness of the torus or a largely misaligned optically thin torus . we are grateful to t. minezaki , m. umemura , m. shirahata , t. nakagawa , m. gaskell , j. fukue , h. takahashi , y. miki , m. kishimoto and s. koshida for useful discussions , and the anonymous referee for helpful comments . this work was partly supported by the grants - in - aid of the ministry of education , science , culture , and sport ( 19740105 , 21244013 ) .

according to recent models , the accretion disk and black hole in active galactic nuclei are surrounded by a clumpy torus . we investigate the nir flux variation of the torus in response to a uv flash for various geometries . both the waning effect of each clump and the torus self - occultation selectively reduce the emission from the region with a short delay . therefore , the nir delay depends on the viewing angle ( where a more inclined angle leads to a longer delay ) and the time response shows an asymmetric profile with a negative skewness , opposing to the results for optically thin tori . we also propose that the red nir - to - optical color of type-1.8/1.9 objects is caused by not only the dust extinction but also the intrinsically red color . a selection bias is thus expected such that nir - selected agns tend to possess moderately thick tori . a super - eddington accretion rate leads to a much weaker nir emission due to the disk self - occultation and the disk truncation by the self - gravity . a long delay is expected from an optically thin and/or a largely misaligned torus . a very weak nir emission , such as in hot - dust - poor active nuclei , can arise from a geometrically thin torus , a super - eddington accretion rate or a slightly misaligned torus .

according to recent models , the accretion disk and black hole in active galactic nuclei are surrounded by a clumpy torus . we investigate the nir flux variation of the torus in response to a uv flash for various geometries . anisotropic illumination by the disk and the torus self - occultation contrast our study with earlier works . both the waning effect of each clump and the torus self - occultation selectively reduce the emission from the region with a short delay . therefore , the nir delay depends on the viewing angle ( where a more inclined angle leads to a longer delay ) and the time response shows an asymmetric profile with a negative skewness , opposing to the results for optically thin tori . the range of the computed delay coincides with the observed one , suggesting that the viewing angle is primarily responsible for the scatter of the observed delay . we also propose that the red nir - to - optical color of type-1.8/1.9 objects is caused by not only the dust extinction but also the intrinsically red color . compared with the modest torus thickness , both a thick and a thin tori display the weaker nir emission . a selection bias is thus expected such that nir - selected agns tend to possess moderately thick tori . a thicker torus shows a narrower and more heavily skewed time profile , while a thin torus produces a rapid response . a super - eddington accretion rate leads to a much weaker nir emission due to the disk self - occultation and the disk truncation by the self - gravity . a long delay is expected from an optically thin and/or a largely misaligned torus . a very weak nir emission , such as in hot - dust - poor active nuclei , can arise from a geometrically thin torus , a super - eddington accretion rate or a slightly misaligned torus .

0801.2068

c

the mass values and the corresponding nuclear binding energies represent the underlying forces acting in a nucleus . in analogy to the ionization potential of atoms , plots of separation energies for the removal of two protons , @xmath272 or two neutrons , @xmath273 reveal the details of nuclear shell structure . in this representation , odd - even effects caused by the usually much larger binding energy for even proton or neutron numbers are removed . + figures [ bbb ] and [ ddd ] show the behavior of separation energies for protons and neutrons versus the respective nucleon number . the results from ame1995 and ame2003 are compared . a global trend , a continuous decrease in the separation energies , is observed . discontinuities from this general trend appear at shell closures or in regions of strong deformations . the @xmath274 shell closure is clearly visible in fig . the much less pronounced strength of the proton shell closure @xmath275 is larger at the magic neutron number @xmath276 than at the half - filled shell @xmath24 ( fig . [ ddd ] ) . the main differences between the appearance of the mass surface in 1995 and 2003 are substantially reduced uncertainties , due to improved as well as numerous new experimental input data . these are isoltrap measurements in the isotopic chains of mercury , platinum , lead , polonium , radon , and radium @xcite as well as a large number of new results obtained at the esr by the schottky technique @xcite . in particular the small uncertainty ( @xmath277 ) of data from isoltrap along the extended isotopic chain from @xmath278 to @xmath279 enables the observation of finer nuclear structure effects and to study , for example , nuclear shape coexistence and the influence of low - excitation intruder states @xcite . note the pronounced irregularities in the thallium isotopic chain at neutron numbers @xmath280 and 114 . + the two - proton separation energies are much more regular and smooth in this region of the nuclear chart than those for two neutrons . however , a peculiar irregularity in the two - proton separation energies appears for the isotopes of thallium ( @xmath281 ) with even neutron numbers @xmath282 from 98 to 108 : some separation energies are nearly equal or even higher than those of the isotopes with one additional neutron . such an effect is very uncommon in the chart of nuclei . in order to investigate this anomalous behavior further , the two - proton separation energies are plotted in fig . [ stagg ] against the neutron number and the isotopes of an element are connected by lines . this presentation shows very clearly in the region of the lead isotopes at @xmath43 the steadily increasing shell gap from mid - shell @xmath24 towards the magic neutron number @xmath274 . also the above mentioned irregularities in the thallium and ( less pronounced ) in the mercury isotopes are revealed distinctly as an odd - even effect . the reason for this effect is unclear . three possible explanations are discussed in the following : + 1 . the masses of @xmath101 and @xmath105 , determined with isoltrap cyclotron resonances and included in ame2003 , were wrongly assigned to the ground states of these isotopes . however , an assignment to the isomeric states ( excitation energy @xmath283 or @xmath284 ) would increase the staggering even further . also the mass of @xmath4 is determined to more than 60 % by isoltrap , but in this case indirectly via the @xmath264 value of the @xmath33 decay and the directly measured mass of @xmath123 . also here , a possible problem in the assignment can be excluded due to the sufficiently high resolving power for separating ground and isomeric states and because of the short half - life of the isomer . some mass measurements might be wrong . this explanation is contradictory to the fact that the mass values in question stem from different sources and mass spectrometric techniques ( isoltrap , esr , @xmath264 values , etc . ) . furthermore , the odd - even staggering is not an isolated phenomenon but appears also ( less pronounced ) in the neighboring isotopic chain of mercury . this is not easily explicable since the mercury and the thallium isotopes are not interconnected by @xmath266 values . the odd - even staggering of the two - proton separation energies in the thallium isotopic chain is factual and represents a nuclear structure effect . this would imply a stronger pairing effect for neutron numbers around mid - shell ( @xmath24 ) than for nearly double - magic thallium nuclei with @xmath23 and @xmath285 . similar to the drastic odd - even staggering observed in the charge radii of the mercury isotopes ( caused by shape transitions ) @xcite these phenomena depend very critically on the neutron and proton configurations and very elaborated calculations are required to reproduce these effects . nuclear masses , as determined in this work , and nuclear charge radii , as obtained by optical spectroscopy from isotope shift measurements , illustrate global and collective properties of nuclear matter . the mass is a measure of the binding energy and the charge radii reflect static and dynamic deformations . the nuclear system , as a function of protons and neutrons , maximizes the binding energy according to the nuclear shape . the pairing energy , which is about @xmath38 , plays an important role in this context and becomes visible as odd - even effect in the masses as well as in the charge radii . + due to the complex interplay of single - particle and collective degrees of freedom , one observes in the region of the nuclear chart investigated in this work both a spherical and deformed shape in the same nucleus at low excitation energies . this phenomenon , first observed at isolde / cern in the seventies by optical spectroscopy @xcite is known as `` shape coexistence or shape isomerism '' and nowadays an extensive bulk of data has been collected throughout the chart of nuclei @xcite . shape coexistence is visible in the vicinities of the shell closures @xcite . most clearly it is observed in the slope of charge radii in the mercury isotopic chain : it manifests itself in a huge odd - even shape staggering near the @xmath286 mid - shell region @xcite and a large difference in charge radii and therefore nuclear deformation between the ground and isomeric state in @xmath287 @xcite ( fig . [ figrad3 ] ) . since the nuclear states of very different configuration are ( almost ) degenerate in energy , one can not expect to observe any major trend in the slope of the masses or binding energies as a function of neutron number . in fact , only now the high mass resolving power @xmath288 of up to @xmath37 of the penning trap mass spectrometer isoltrap enables one to resolve ground and isomeric states in this region of the nuclear chart where shape coexistence occurs . the implementation of many new accurate mass data from isoltrap ( @xcite and this work ) to the data base of the atomic mass evaluation and a large data set of charge radii @xcite stimulated an attempt to reveil a correlation between masses and charge radii @xcite . + in order to uncover finer details in the binding energies regarding odd - even staggering and general trends @xcite , the absolute values of the deduced pairing gap parameters for neutrons are plotted instead of the experimental masses . + the three - point indicator for the shell gap parameter @xmath289 is given by @xmath290,\ ] ] which is often interpreted as a measure of the empirical pairing gap @xcite . another commonly used relation is the four - point indicator , which averages the @xmath291 values and is given by @xmath292 . \vspace*{-0.045cm}\ ] ] figures [ figrad1 ] - [ figrad5 ] compare the results from mass measurements with those from isotope shift determinations for the isotopic chains of platinum , gold , mercury , thallium , and lead . for these elements , extensive data are available . in order to allow an easy comparison of pairing gap parameters with charge radii for the different elements , the scales of the x- and y - axis are made identical in figs . [ figrad1 ] - [ figrad5 ] . the pairing gap parameters of the isotopic chains of the even-@xmath293 elements lead , mercury and platinum are nearly identical in the regions of @xmath294 and @xmath295 , respectively . the pairing gap parameters for the isotopes of the odd-@xmath293 elements thallium and gold show a similar behavior but the pairing gap is smaller in magnitude . around mid - shell neutron number @xmath24 a minimum develops which is barely visible in the case of lead . it becomes sharper and is shifted to @xmath296 for thallium , distinct and narrow for mercury but very much pronounced and broader for gold and platinum . + the slopes of the isotopic chains of lead , thallium and mercury are almost identical in the region where data are available for the same neutron number . only the odd - even staggering is much more pronounced for the thallium isotopes as compared to those of lead and mercury . + down to neutron number @xmath297 , the slope of the charge radii for the isotopes of gold and platinum is a little bit smaller than for the other isotopic chains which indicates increasing deformation . for @xmath296 in the case of mercury and @xmath298 for gold a sudden change of the charge radii is observed which is due to a sharp change in deformation . + comparing the slopes of pairing parameters with those of the charge radii , one observes that the extremely sharp discontinuity at the shell closure at @xmath274 is visible in the charge radii only as a small kink in their slope , that the pronounced increase of deformation at mid - shell @xmath24 is reflected as a local minimum in the pairing parameters , and that the size of the odd - even staggering in the pairing parameter @xmath291 is not reflected in the size of the one of the charge radii . there is an interesting observation : for thallium an unusual odd - even staggering between the isomers below mass number @xmath299 is observed . + the new data obtained in this region of the nuclear chart near @xmath300 will hopefully stimulate theoretical work to develop nuclear models further which are able to describe simultaneously experimental mean square charge radii as well as experimental masses of spherical and deformed nuclei in a satisfactory way . up to now , only little attention has been paid to such investigations which concern model - independent gross properties of nuclear matter . for example , the charge radii are only poorly described by macroscopic - microscopic mass models which are , however , quite successful in describing nuclear masses @xcite . even more demanding is a description of the odd - even staggering of masses and of the charge radii as attempted in ref . @xcite for spherical nuclei .

the masses of , , , , , , , , , , , and were determined . the resulting separation energies are discussed and the mass spectrometric and laser spectroscopic data are examined for possible correlations . , , , , , , , , , and atomic mass , binding energy , penning trap , radionuclide , isomer , cesium , thallium , lead , bismuth , radium , francium + 07.75.+h mass spectrometers , 21.10.dr binding energies and masses , 27.70.+q , 27.80.+w , 27.90.+b

accurate atomic mass measurements of neutron - deficient and neutron - rich nuclides around the doubly - magic and of neutron - rich cesium isotopes were performed with the penning trap mass spectrometer isoltrap at isolde / cern . the masses of , , , , , , , , , , , and were determined . the obtained relative mass uncertainty in the range of to is not only required for safe identification of isomeric states but also allows mapping the detailed structure of the mass surface . a mass adjustment procedure was carried out and the results included into the atomic mass evaluation . the resulting separation energies are discussed and the mass spectrometric and laser spectroscopic data are examined for possible correlations . , , , , , , , , , and atomic mass , binding energy , penning trap , radionuclide , isomer , cesium , thallium , lead , bismuth , radium , francium + 07.75.+h mass spectrometers , 21.10.dr binding energies and masses , 27.70.+q , 27.80.+w , 27.90.+b

1302.5234

i

despite of the simplicity of the molecular ion h@xmath0 and its isotopomers d@xmath0 and t@xmath0 , the three body schrdinger equation can not be solved exactly . however , high precision results , for energies and wave functions , have been obtained using different approaches . for the particular case of the deuterium molecular ion , the ground state as well as some vibrational states are well known . in this sense , non - adiabatic dissociation energies were calculated for several vibrational - rotational levels for the ground electronic state @xcite and also in the first excited electronic state . in particular , the value of the ground state energy has been improved in several papers @xcite . energies for the zero vibrational band for different values of the total angular momentum @xmath3 have been also considered @xcite . complete bound vibrational states below the dissociation limit @xmath4 were presented in @xcite for the two lowest rotational quantum numbers @xmath5 . extension to this work until @xmath6 is presented in @xcite but with a different value of the deuteron mass . some expectation values have been also provided like the mean radii @xcite or quadrupole moments @xcite . however , nowadays quadrupole transitions has not been considered for this system . in this paper , the e2 transitions are obtained from the three - body wave functions calculated with the lagrange - mesh method in perimetric coordinates @xcite , with which the calculation is particularly simple and very precise . the lagrange - mesh method is an approximate variational calculation using a basis of lagrange functions and the associated gauss quadrature . it has the high accuracy of a variational approximation and the simplicity of a calculation on a mesh . in section [ sec : lmtp ] , some expressions are presented but detailed explanations are given in @xcite . in section [ sec : res ] , energies are given for the lowest four vibrational states over the full rotational bands and e2 transition probabilities are tabulated . conclusions are presented in section [ sec : conc ] . throughout atomic units

after the study of the three body molecular system h ( _ j . phys . opt . phys . _ * 45 * 065101 ) , its isotopomer , the deuterium molecular ion d is studied . the three - body schrdinger equation is solved using the lagrange - mesh method in perimetric coordinates .

after the study of the three body molecular system h ( _ j . phys . b : at . mol . opt . phys . _ * 45 * 065101 ) , its isotopomer , the deuterium molecular ion d is studied . the three - body schrdinger equation is solved using the lagrange - mesh method in perimetric coordinates . energies and wave functions for four vibrational states and bound or quasibound states for total orbital momenta from 0 to 56 are calculated . the 1986 fundamental constant is used . the obtained energies have an accuracy from about 13 digits for the lowest vibrational state to at least 9 digits for the third vibrational excited state . quadrupole transition probabilities per time unit between those states over the whole rotational bands were calculated . extensive results are presented with six significant figures .

gr-qc0311018

i

the description of the motion of binary black holes within the einsteinian theory of gravity is quite a challenging problem and is likely to be finally solvable by numerical means only . however , it seems that the full success will occur in a far future . it has turned out in the past that analytical developments did always influence numerical investigations , hence further progress on the analytical side is certainly desirable . in this paper we shall present an analytical solution of truncated einstein equations for systems of non - spinning point - like objects , which shows many aspects of past numerical models . most of the numerical computations of quasi - stationary initial data for black - hole binaries are based on the assumption that the space metric is conformally flat @xcite . they often assume as well the existence of an approximate helical killing vector which permits to elaborate some manageable formulations ( also see @xcite ) . recently , the time - symmetric initial data of brill - lindquist @xcite were generalized by adding a non - conformally flat contribution that incorporates pieces of information provided by perturbative post - newtonian calculations @xcite . the present paper does not follow this line , but rather keeps the spirit of former numerical simulations . we essentially perform two simplifications : on the one hand , we adopt the often - used assumption of conformally flat space metric ; on the other hand , we attribute the non - linear gravitational field energy to point - like sources , which are modeled as dirac delta distributions . we solve the resulting truncated einstein equations analytically and get what we call , in reference to the point - like character of the source support , the _ skeleton _ solution . the calculations are carried out for @xmath0 bodies in a mathematically sound way by working from the beginning in @xmath1-dimensional space - time and taking the limit @xmath2 in the final expressions , so that the dirac delta distributions modeling the sources are consistently handled @xcite . our space metric and extrinsic curvature may be regarded as some generalization of the multiple black hole brill - lindquist initial value solution @xcite to arbitrary black - hole momenta preserving its original asymptotic structure . moreover , the energy function can be used as generator of the dynamical evolution of the system . we shall refer to it as the skeleton hamiltonian associated to the full arnowitt - deser - misner ( adm ) hamiltonian . it agrees with the exact hamiltonian _ ( i ) _ in the test body limit , _ ( ii ) _ at the first post - newtonian ( 1pn ) approximation , i.e. , at the first order in powers of @xmath3 ( @xmath4 denoting the speed of light ) beyond the newtonian dynamics , and _ ( iii ) _ in the limit of special relativity in absence of gravity . the skeleton gravitational field is given explicitly , as well as the equations of motion . as a first application , we shall investigate the location of the last stable ( circular ) orbit ( lso ) for binary systems . in order to reach a proper accuracy , the last - stable - orbit parameters are worked out up to the tenth post - newtonian ( 10pn ) approximation within the skeleton model . the comparison of second and third post - newtonian results derived from the skeleton model with the corresponding quantities computed in the full einstein theory delivers a quantitative measure for the missing terms in the skeleton approach .

an approximate analytical and non - linear solution of the einstein field equations is derived for a system of multiple non - rotating black holes . the associated space - time has the same asymptotic structure as the brill - lindquist initial data solution for multiple black holes . _ approximate solution of the einstein field equations . explicit calculations for the determination of the last stable circular orbit of the binary system are performed up to the tenth post - newtonian order within the skeleton model .

an approximate analytical and non - linear solution of the einstein field equations is derived for a system of multiple non - rotating black holes . the associated space - time has the same asymptotic structure as the brill - lindquist initial data solution for multiple black holes . the system admits an arnowitt - deser - misner ( adm ) hamiltonian that can particularly evolve the brill - lindquist solution over finite time intervals . the gravitational field of this model may properly be referred to as a _ skeleton _ approximate solution of the einstein field equations . the approximation is based on a conformally flat truncation , which excludes gravitational radiation , as well as a removal of some additional gravitational field energy . after these two simplifications , only source terms proportional to dirac delta distributions remain in the constraint equations . the skeleton hamiltonian is exact in the test - body limit , it leads to the einsteinian dynamics up to the first post - newtonian approximation , and in the time - symmetric limit it gives the energy of the brill - lindquist solution exactly . the skeleton model for binary systems may be regarded as a kind of analytical counterpart to the numerical treatment of orbiting misner - lindquist binary black holes proposed by gourgoulhon , grandclment , and bonazzola , even if they actually treat the corotating case . along circular orbits , the two - black - hole skeleton solution is quasi - stationary and it fulfills the important property of equality of komar and adm masses . explicit calculations for the determination of the last stable circular orbit of the binary system are performed up to the tenth post - newtonian order within the skeleton model .

1505.00708

i

in some solids thermal energy can be transmitted by the mechanism of _ wave - like _ propagation of heat , unlike the usual mechanism of conduction by diffusion . this phenomenon of heat conduction as waves , known as _ second sound _ , has been observed experimentally ( see , for example , @xcite for an extensive survey of experimental works involving propagation of heat as a thermal wave ) . the classical theory of heat conduction based on _ fourier s law _ , fails to model the second sound phenomenon . moreover , the classical theory permits _ infinite speed _ of propagation of parts of a localized initial heat pulse , which is paradoxical from a physical point of view . as a result , efforts have been made in an attempt to find a consistent model of heat propagation that is capable of capturing the second sound phenomenon with finite speed ( see , for example , @xcite for a review of models of heat conduction as waves ) . one of the alternative theory for formulating the propagation of heat in a general way that aims at capturing the second sound phenomenon was proposed by green and naghdi @xcite . the theory of green and naghdi is based on three types of constitutive relations for the heat flux : type i is equivalent to the classical theory based on fourier s law . type ii permits the propagation of a localized heat signal as thermal wave without dissipation ( see @xcite for a remark on the appropriateness of this classification ) . type iii is the most general theory , which includes both type i and ii as special cases , in which second sound phenomenon is supported while dissipation is incorporated in the process . the thermomechanical coupling of non - classical heat conduction with classical elasticity is the subject of non - classical thermoelasticity . extensive overviews of the non - classical thermoelasticity of green and naghdi can be found in @xcite . theoretical results concerning the non - classical theory have been addressed in several research works . in @xcite exact solutions are obtained for thermal wave propagation in one dimension . results on the existence and uniqueness of solutions of non - classical problem of thermoelasticity can be found , for example , in @xcite and the references therein . designing a robust and efficient numerical solution strategy for strongly coupled problems of hyperbolic - type is challenging . this is particularly the case for the non - classical theory of thermoelasticity where hyperbolic ( or nearly hyperbolic ) heat conduction equation is coupled with the classical hyperbolic elasticity problem . a standard approach for solving such time - dependent problems is the method of lines ( mol ) in which the governing partial differential equation is first discretised in space using the finite element method ( fem ) leading to a system of ordinary differential equations , which can then be solved using the finite difference method . despite its popularity , mol struggles to accurately solve problems involving propagation of sharp gradients or discontinuities @xcite . recently , a great deal of attention has been invested in designing a spatial discontinuous galerkin ( dg ) approach for convection - dominated problems ; see for example @xcite . however , these methods , like mol , are based on decoupling space and time in the sense that space and time are treated differently . hulbert and hughes @xcite introduced a powerful scheme based on a space - time dg finite element methodology for linear elastodynamics problems . in their approach , space and time are treated simultaneously and the unknown fields are allowed to be discontinuous in time while continuous in space . recently , the space - time dg method has been used in @xcite for classical thermoelasticity , using a monolithic approach where all the unknown fields are solved for simultaneously . recently , in @xcite a numerical solution approach based on mol was proposed for non - classical thermoelasticity in which time integration was done in two ways : continuous galerkin fem for type ii and iii , while mixed - discountinuous galerkin fem for the classical problem based on the fourier s law of heat conduction . in their approach a streamline - upwind numerical stabilization was added to localize numerical oscillations due to the propagation of sharp thermal wave . in the current work , we extend the existence and uniqueness results otained in @xcite for type ii theory to the more general problem of type iii thermoelasticity . we also present a novel numerical algorithm for the non - classical thermoelasticity based on an operator - splitting technique motivated by armero and simo @xcite for classical thermoelasticity , coupled with a space - time dg methodology that extends the work of hulbert and hughes @xcite which was formulated for linear elastodynamics . the major contributions of this work are twofold : ( i ) the adaptation of the operator - splitting strategy for classical thermoelasticity first proposed by @xcite to the non - standard theory , in which the operator defining non - standard thermoelasticity is split in a way that the resulting sub - operators retain the same contractive behaviour as the global operator ; and ( ii ) the time - dg formulation in which continuity of the unknown fields is enforced weakly by using an @xmath0-inner product in contrast to the energy - norm used in @xcite . the rest of this paper is organized as follows . in section [ sec : gnt ] , the governing equations of the non - classical theory are summarized in a general framework of type iii thermoelasticity . well - posedness and physically meaningful boundary and initial conditions are also discussed in this section . an operator - splitting strategy for the problem of type iii thermoelasticity is proposed and the resulting sub - operators are analysed in section [ sec : alg - os ] . in section [ sec : t - dg ] , time - dg formulation is proposed for the sub - problems and stability of the individual algorithms and the global one is analysed in detail . a number of numerical examples both in 1d and 2d are presented in section [ sec : numresults ] to demonstrate the excellent performance and capability of the proposed numerical scheme . finally , concluding remarks and some open problems are discussed in section [ sec : conclusion ] .

the non - classical theory incorporates three models ; the classical theory based on fourier s law of heat conduction resulting in a hyperbolic - parabolic coupled system , a non - classical theory of a fully hyperbolic extension , and a combination of the two . a number of numerical examples are presented to demonstrate the performance and capability of the method .

an efficient time - stepping algorithm is proposed based on operator - splitting and the space - time discontinuous galerkin finite element method for problems in the non - classical theory of thermoelasticity . the non - classical theory incorporates three models ; the classical theory based on fourier s law of heat conduction resulting in a hyperbolic - parabolic coupled system , a non - classical theory of a fully hyperbolic extension , and a combination of the two . the general problem is split into two contractive sub - problems , namely the mechanical phase and the thermal phase . each sub - problem is discretised using space - time discontinuous galerkin finite element method resulting each to be stable which then leads to unconditional stability of the global product algorithm . a number of numerical examples are presented to demonstrate the performance and capability of the method . operator - splitting , space - time discontinuous galerkin finite element , non - classical theory of thermoelasticity , fourier s law , second sound , contractive .

1505.00708

c

an operator - splitting strategy coupled with a space - time discontinuous galerkin finite element method for the solution of transient and fully coupled initial - boundary problem of generalized thermoelasticity was presented . well - posdeness of the problem in the general setting ( type iii ) is proven using the theory of semigroups . the defining operator is split additively so that the first sub - operator represents an isentropic ( adiabatic ) elasticity in which the entropy density is held fixed , and the other is a non - standard heat conduction at fixed configuration . both of the sub - problems are also shown to inherit the same contractivity property as the full problem . each sub - problem is then discretised separately using a time - discontinuous galerkin finite element method where the unknown fields are allowed to be discontinuous along the interfaces of each space - time slab . weak continuity of the unknown fields is enforced using an @xmath0-inner product which differs from the original time - discontinuous formulation using an energy - inner product @xcite which was formulated for linear elastodynamics problem . the unconditionally stability behaviour of each of the algorithms is proven without the need to add extra ` artificial viscosity ' . the algorithm for the global problem is finally obtained by way of lie - trotter - kato product formula , leading to an unconditional stability . the results presented in this paper are demonstrated by a number of numerical examples in both one and two dimensional cases . the efficiency of the current numerical scheme were examined by comparing the rate of convergence of with the corresponding monolithic approach . the result shows that the splitting scheme not only it retains the accuracy of the monolithic scheme but also it improves the efficiency as two smaller problems are solved sequentially at each time - step . the capability of the splitting algorithm is tested using problems involving propagation of heat waves driven by a pulsing laser heat source and an initial temperature disturbance in one and two dimensions respectively . furthermore , the capability of the non - standard thermoelasticity and the proposed numerical method to model the phenomenon of second sound in some solids is demonstrated by considering the quasi - static expansion of an infinitely long thick walled cylinder in plane stress . the dg formulation proposed in this work may be extended to the non - linear regime without the need to eliminate the displacement - velocity relation in the formulation . hence , a full recovery of the numerical dissipation in the non - linear case is possible . this will be the subject of a forthcoming work in which issues such as non - linear stability and the existence of lyapunov function are discussed . * acknowledgments . * the work reported in this paper has been supported by the national research foundation of south africa through the south african research chair in computational mechanics . this support is acknowledged with thanks . the authors also thank professor s. bargmann for discussions which led to various improvements in the work .

an efficient time - stepping algorithm is proposed based on operator - splitting and the space - time discontinuous galerkin finite element method for problems in the non - classical theory of thermoelasticity . each sub - problem is discretised using space - time discontinuous galerkin finite element method resulting each to be stable which then leads to unconditional stability of the global product algorithm . operator - splitting , space - time discontinuous galerkin finite element , non - classical theory of thermoelasticity , fourier s law , second sound , contractive .

an efficient time - stepping algorithm is proposed based on operator - splitting and the space - time discontinuous galerkin finite element method for problems in the non - classical theory of thermoelasticity . the non - classical theory incorporates three models ; the classical theory based on fourier s law of heat conduction resulting in a hyperbolic - parabolic coupled system , a non - classical theory of a fully hyperbolic extension , and a combination of the two . the general problem is split into two contractive sub - problems , namely the mechanical phase and the thermal phase . each sub - problem is discretised using space - time discontinuous galerkin finite element method resulting each to be stable which then leads to unconditional stability of the global product algorithm . a number of numerical examples are presented to demonstrate the performance and capability of the method . operator - splitting , space - time discontinuous galerkin finite element , non - classical theory of thermoelasticity , fourier s law , second sound , contractive .

1403.1130

i

let @xmath0 be a coxeter group . an element @xmath1 is said to be fully commutative ( fc ) if any reduced word for @xmath2 can be obtained from any other one by only short braid relations . these elements were introduced and studied independently by fan in @xcite , graham in @xcite and stembridge in @xcite , where among other things , he classified the coxeter groups with a finite number of fully commutative elements and enumerated them in each case . fully commutative elements appear naturally in the context of ( generalized ) temperley - lieb algebras , as they index a linear basis of the latter . recently , in @xcite , biagioli , jouhet and nadeau refined stembridge s enumeration by counting fully commutative elements according to their coxeter length in any finite or affine coxeter group . + in this paper , we focus on a certain subset of fully commutative elements , the cyclically fully commutative elements ( which we denote cfc from now on ) . these are elements for which every cyclic shift of any reduced expression is a reduced expression of a fc element . they were introduced by boothby et al . in @xcite , where they classified the coxeter groups with a finite number of cfc elements ( they showed that the latter are the groups with a finite number of fc elements ) and enumerated them . the main goal of @xcite is to establish necessary and sufficient conditions for a cfc element @xmath1 to be logarithmic , i.e to satisfy @xmath3 for any positive integer @xmath4 . this is the first step towards studying a cyclic version of matsumoto s theorem for cfc elements . here we will focus on the combinatorics of cfc elements . we introduce a new operation on heaps , which will allow us to give a new characterization of cfc elements in all coxeter groups ( see theorem [ propcfc ] ) . for finite or affine coxeter groups , this characterization can be reformulated in terms of words , by using the work on fc elements from @xcite . it allows us to enumerate the cfc elements by taking into account their coxeter length . we will also prove that the number of cfc involutions is finite in all coxeter groups . this paper is organized as follows . we recall in section 1 some definitions and properties on coxeter groups , we introduce a new operation on heaps , the _ cylindric transformation _ , in two different ways , and we deduce a new characterization of cfc elements in terms of pattern - avoidance for cylindric transformed heaps ( see theorem [ propcfc ] ) . in section 2 , we use this characterization in order to obtain a complete classification ( in terms of words ) of cfc elements in the affine group @xmath5 . we also deduce a classification of cfc elements in the group @xmath6 , and use this to enumerate cfc elements in both groups , according to their coxeter length . the same work is done for the groups @xmath7 , and @xmath8 in section 3 . in section 4 , we will focus on cfc involutions . the main result is that there is a finite number of cfc involutions in all coxeter groups . we also give a characterization of cfc involutions for all coxeter groups and enumerate them in finite and affine types . [ sec : intro ]

these elements were studied by boothby _ et al_. in particular the authors enumerated cyclically fully commutative elements in all coxeter groups having a finite number of them . in this work we characterize and enumerate cyclically fully commutative elements according to their coxeter length in all finite or affine coxeter groups by using a new operation on heaps , the cylindric transformation . in finite types , this refines the work of boothby _ et al . _ , by adding a new parameter . in affine type , we study afterwards the cyclically fully commutative involutions and prove that their number is finite in all coxeter groups .

an element of a coxeter group is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators . an element of a coxeter group is cyclically fully commutative if any of its cyclic shifts remains fully commutative . these elements were studied by boothby _ et al_. in particular the authors enumerated cyclically fully commutative elements in all coxeter groups having a finite number of them . in this work we characterize and enumerate cyclically fully commutative elements according to their coxeter length in all finite or affine coxeter groups by using a new operation on heaps , the cylindric transformation . in finite types , this refines the work of boothby _ et al . _ , by adding a new parameter . in affine type , all the results are new . in particular , we prove that there is a finite number of cyclically fully commutative logarithmic elements in all affine coxeter groups . we study afterwards the cyclically fully commutative involutions and prove that their number is finite in all coxeter groups . [ multiblock footnote omitted ]

1106.3424

i

the zero - temperature random - field ising model ( zt - rfim ) is a prototype model for systems that exhibit avalanche dynamics when slowly driven accross athermal first - order phase transitions @xcite . examples of such transitions include the condensation of fluids in porous media @xcite , the martensitic transformation @xcite , or magnetisation reversal of ferromagnets ( barkhausen effect ) @xcite . for experimentally reasonable time scales , thermal fluctuations do not play an important role and the dynamics in this kind of system proceed along a non - equilibrium path consisting of metastable states @xcite . avalanches are a manifestation of such behaviour and correspond to the driven - induced passage between two metastable states . the zt - rfim predicts that the properties of avalanches ( e.g. their size or duration ) are drastically affected by the degree of quenched disorder in the system . more explicitly , the model predicts three different regimes for avalanche behaviour depending on the degree of disorder . the main feature of the small - disorder regime is the existence of an avalanche that is infinite in extent . in contrast , high degrees of disorder lead to a regime where all avalanches are small . these two regimes are separated by an intermediate situation where the model exhibits critical behaviour ( i.e. a continuous phase transition ) @xcite . for any degree of disorder , avalanche - like dynamics lead to spacial correlations in observable properties averaged over quenched disorder ( e.g. magnetisation , stress or fluid density ) , which extend over a typical length scale , called the correlation length . this quantity is expected to diverge in the critical regime , and this has been shown to be the case in the systems described by mean field @xcite and hypercubic lattices @xcite . concerning other topologies , including a bethe lattice , the situation is less clear . previous studies of the zt - rfim on the bethe lattice have obtained exact results for the magnetisation @xcite , the avalanche size distribution @xcite , the different contributions to the energy @xcite , and the number of metastable states @xcite . in particular , it has been demonstrated that for coordination numbers @xmath0 the system exhibits a discontinuity in its magnetisation hysteresis loop for small amounts of disorder associated with the infinite avalanche . the universality class of the critical point on such a lattice has been suggested to be the same as that of the mean - field system ( complete graph ) @xcite , with the critical exponents for the order parameter , @xmath1 and @xmath2 , being identical to the mean - field values @xcite . the spatial correlations of the model have also been investigated in the past but mostly using numerical simulations @xcite or approximate analytical methods based on mean - field descriptions and/or renormalization group ( rg ) techniques @xcite . in this paper , we derive an exact analytical expression for the spin - spin correlation function corresponding to the zt - rfim with spins placed on a bethe lattice . our results are exact for any coordination number and confirm the validity of the functional form for the correlation function derived independently in @xcite , where it is assumed that the extension from the one - dimensional case ( i.e. with @xmath3 ) is valid . it is well established that in a hypercubic lattice of dimensionality @xmath4 around criticality , the correlation function , @xmath5 , decays exponentially with distance @xmath6 for the zt - rfim , @xmath7 where @xmath8 obeys a power law for large @xmath6 , and @xmath9 is the correlation length . at the critical point , @xmath10 , and so @xmath8 represents the critical behaviour of the correlation function , found to be @xmath11 with anomalous dimension @xmath12 @xcite . in a prototype loopless topology , the cayley tree or bethe lattice , the correlation function exhibits additional exponential behaviour due to the associated hyperbolic topology , @xmath13 where @xmath14 is the coordination number and @xmath15 is a power - law function for large values of chemical distance @xmath6 . the exponential prefactor @xmath16 in gives the number of lattice sites within @xmath6 shells of the bethe lattice , and plays the role of the factor @xmath17 in the equivalent expression for hypercubic lattices . the function @xmath18 with some exponent @xmath19 ( found below to be @xmath20 ) , accounts for all power - law behaviour of the correlation function . there is some inconsistency in the definition of the correlation length for two - state models ( the equilibrium ising model , the zt - rfim and percolation ) on the bethe lattice . in the equilibrium ising model @xcite and the zt - rfim @xcite on the bethe lattice , the prefactor @xmath21 in has been absorbed into the definition of the correlation length @xmath22 with @xmath23 chosen to be the correlation length . in an alternative definition , for both percolation @xcite and the ising model @xcite , the correlation length has instead been identified with @xmath9 in , thus separating the length scale of dynamic correlations from the intrinsic topological length scale of the bethe lattice , @xmath24 . below , we demonstrate that ( i ) the value of @xmath9 diverges at the critical point according to a power law , in contrast to @xmath25 which remains finite at criticality and ( ii ) the critical exponents governing the divergence of @xmath9 around the critical point are consistent with the mean - field values for the divergence of correlation length in hypercubic lattices above the upper critical dimension , @xmath26 . the exponents , describing the divergence of @xmath9 in terms of the external field @xmath27 and degree of disorder @xmath28 , @xmath29 and @xmath30 , are found to be @xmath31 and @xmath32 . as argued below , these values can be related to those in the mean - field model , @xmath33 and @xmath34 , where @xmath35 originates from fractal dimension of a random walk @xcite . therefore these findings motivate us to choose @xmath9 as the correlation length describing the critical behaviour of the zt - rfim on a bethe lattice . such a definition , while in contrast with that used in @xcite , separates the topological contribution of the bethe lattice to the correlation length , and allows its comparison with the correlation length for hypercubic lattices .

separated from the intrinsic topological length scale of the bethe lattice is shown to diverge as a power law at the critical point . the critical exponents governing the behaviour of the correlation length are consistent with the mean - field values found for a hypercubic lattice with dimension greater than the upper critical dimension .

an exact expression for the spin - spin correlation function is derived for the zero - temperature random - field ising model defined on a bethe lattice of arbitrary coordination number . the correlation length describing dynamic spin - spin correlations and separated from the intrinsic topological length scale of the bethe lattice is shown to diverge as a power law at the critical point . the critical exponents governing the behaviour of the correlation length are consistent with the mean - field values found for a hypercubic lattice with dimension greater than the upper critical dimension .

hep-th0308052

i

in a recent paper @xcite simple theorems pertaining to the correspondence between quantum and classical dynamics are proved for the general multi - particle quantum mechanical systems with discrete eigenvalues . the theorems relate quantum mechanical eigenvalues and eigenfunctions to the properties of the classical mechanical system at equilibrium . corresponding to each quantum eigenfunction , a ` _ classical eigenfunction _ ' is defined whose eigenvalue is given by the ` main part ' , that is the order @xmath3 part , of the quantum eigenvalue . for the ` _ elementary excitations _ ' these classical and quantum eigenvalues are nothing but the eigenfrequencies of the normal modes of the small oscillations at equilibrium . we apply these theorems to the calogero and sutherland @xcite systems , typical integrable multi - particle dynamics with long range interactions based on root systems @xcite . the theorems provide an analytic proof for the interesting observations made by corrigan - sasaki @xcite concerning the classical and quantum integrability in calogero and sutherland systems . explicit forms of the classical and quantum eigenfunctions for the elementary excitations are presented for the calogero and sutherland systems based on any root systems . these exemplify another aspect of the close relationship between the classical and quantum integrability in calogero and sutherland systems . this paper is organised as follows . in section two , basic formulation of multi - particle quantum mechanics in terms of the _ prepotential _ @xcite is briefly reviewed . after the reformulation of the quantum mechanical wavefunctions at equilibrium , the main theorem of the loris - sasaki paper @xcite is recapitulated . in section three the basic concepts of the calogero and sutherland systems are summarised . section four and five are the main part of this paper , presenting the classical and quantum eigenfunctions of the calogero systems ( section four ) and sutherland systems ( section five ) . the final section is for summary and comments .

an interesting observation was reported by corrigan - sasaki that all the frequencies of small oscillations around equilibrium are _ quantised _ " for calogero and sutherland ( c - s ) systems , typical integrable multi - particle dynamics . we present an analytic proof by applying recent results of loris - sasaki . explicit forms of ` classical ' and quantum eigenfunctions are presented for c - s systems based on any root systems .

an interesting observation was reported by corrigan - sasaki that all the frequencies of small oscillations around equilibrium are _ quantised _ " for calogero and sutherland ( c - s ) systems , typical integrable multi - particle dynamics . we present an analytic proof by applying recent results of loris - sasaki . explicit forms of ` classical ' and quantum eigenfunctions are presented for c - s systems based on any root systems . yukawa institute kyoto + yitp-03 - 51 + hep - th/0308052 + august 2003 * quantum & classical eigenfunctions in + calogero & sutherland systems + + * i. loris and r. sasaki yukawa institute for theoretical physics , + kyoto university , kyoto 606 - 8502 , japan + dienst theoretische natuurkunde , vrije universiteit brussel , + pleinlaan 2 , b-1050 brussels , belgium

1612.01201

i

nanoscale electronic devices in contact with superconducting contacts ( s ) exhibit a large variety of fundamental physical phenomena and play , for example , a central role in schemes for quantum computation.@xcite if s is in contact with the many quantum channels of a normal metal ( n ) , phase - coherent andreev reflections lead to electron pairing and an induced superconducting gap in n. this proximity effect has recently been demonstrated also for one - dimensional semiconducting nanowires,@xcite where strong spin - orbit interactions can give rise to majorana bound states.@xcite if s is strongly coupled to a single channel quantum dot ( qd ) , new subgap eigenstates form , which are known as andreev bound states ( abs).@xcite abss carry the supercurrent in josephson junctions,@xcite and thus constitute a model system to investigate the superconducting proximity effect in qds.@xcite abss might also be exploited as andreev quantum bits @xcite and have recently attracted considerable attention in both , theoretical @xcite and experimental work.@xcite previous experiments on abss were focused on qds with two contacts,@xcite and only few were possible in multi - terminal devices.@xcite three - terminal abs devices allow for new transport mechanisms , so that nonlocal processes like cooper pair splitting@xcite compete with local mechanisms like andreev tunneling.@xcite such mechanisms are expected to result in new effects like the triplet blockade @xcite , characteristic patterns in the electrical conductance,@xcite and other nonlocal effects.@xcite in addition , even the most basic abs characteristics are expected to be determined by the barrier strengths , such as the abs resonance broadening@xcite and whether a quantum phase transition in the s - qd many - body ground state occurs as a function of the gate and bias voltages . however , the coupling strengths can only be accessed unambiguously in three terminal geometries.@xcite here we report experiments on three - terminal qd devices fabricated on carbon nanotubes ( cnts ) with one central superconducting contact inducing abss on the qd . a scanning electron microscopy ( sem ) image is shown in fig . we investigate three devices with different coupling strengths @xmath0 to s to investigate several of these open questions and demonstrate a series of new effects and measurement configurations . we use pb as superconductor , which results in a large energy gap @xmath1 and thus in a high relative spectroscopic resolution,@xcite crucial for our experiments . the paper is structured as follows : in sec . [ sec : abs_basics ] we summarize our intuitive picture of abss and abs - mediated electron transport in two - terminal devices . [ sec : fab ] describes the sample fabrication and measurement setup . then we characterize each of the investigated devices by transport spectroscopy in sec . [ sec : abs - devicecharacteriation ] , from which we extract the tunnel coupling strengths to the individual contacts and demonstrate that @xmath0 determines the system ground state and abs dispersion , while the coupling strengths to the normal metal contacts , @xmath2determine the spectroscopic width of the andreev resonances ( ars).@xcite one device shows replicas of the lowest - energy abs resonance at higher energies , which we ascribe to transitions to excited odd - parity qd states in sec . [ sec : abs - addbias - excitedar ] . in sec . [ sec : abs - serialtransport ] , we analyze the transport between the two normal metal contacts mediated by abss , in which competing local and nonlocal transport mechanisms give rise to a characteristic pattern of positive and negative differential conductance . these findings are well captured by a simple rate equation model , which allows us to identify cooper pair splitting@xcite and a new three - terminal subgap process we call _ resonant abs tunneling_. in sec . [ sec : abs - floatings - proximity ] we show that depending on the boundary conditions imposed in the experiments , the measured conductance either exhibits abs or coulomb blockade characteristics , or both at the same time , which we tentatively attribute to finite frequency coherent oscillations between the single particle basis states when an abs is excited by single electron tunneling .

andreev bound states ( abss ) are well - defined many - body quantum states that emerge from the hybridization of individual quantum dot ( qd ) states with a superconductor and exhibit very rich and fundamental phenomena . we demonstrate several new electron transport phenomena mediated by abss that form on three - terminal carbon nanotube ( cnt ) qds , with one superconducting ( s ) contact in the center and two adjacent normal metal ( n ) contacts . in addition , we ascribe replicas of the lowest - energy abs resonance to transitions between the abs and odd - parity excited qd states , a process we call excited state abs resonances . in the conductance between the two n contacts we find a characteristic pattern of positive and negative differential subgap conductance , which we explain by considering two nonlocal processes , the creation of cooper pairs in s by electrons from both n terminals , and a novel transport mechanism called resonant abs tunneling , possible only in multi - terminal qd devices . in the latter process , electrons are transferred via the abs without effectively creating cooper pairs in s. the three - terminal geometry also allows spectroscopy experiments with different boundary conditions , for example by leaving s floating . these experiments demonstrate that three - terminal experiments on a single complex quantum object can also be useful to investigate charge dynamics otherwise not accessible due to the very high frequencies .

andreev bound states ( abss ) are well - defined many - body quantum states that emerge from the hybridization of individual quantum dot ( qd ) states with a superconductor and exhibit very rich and fundamental phenomena . we demonstrate several new electron transport phenomena mediated by abss that form on three - terminal carbon nanotube ( cnt ) qds , with one superconducting ( s ) contact in the center and two adjacent normal metal ( n ) contacts . three - terminal spectroscopy allows us to identify the coupling to the n contacts as the origin of the andreev resonance ( ar ) linewidths and to determine the critical coupling strengths to s , for which a ground state ( or quantum phase ) transition in such s - qd systems can occur . in addition , we ascribe replicas of the lowest - energy abs resonance to transitions between the abs and odd - parity excited qd states , a process we call excited state abs resonances . in the conductance between the two n contacts we find a characteristic pattern of positive and negative differential subgap conductance , which we explain by considering two nonlocal processes , the creation of cooper pairs in s by electrons from both n terminals , and a novel transport mechanism called resonant abs tunneling , possible only in multi - terminal qd devices . in the latter process , electrons are transferred via the abs without effectively creating cooper pairs in s. the three - terminal geometry also allows spectroscopy experiments with different boundary conditions , for example by leaving s floating . surprisingly , we find that , depending on the boundary conditions and the device parameters , the experiments either show single - particle coulomb blockade resonances , abs characteristics , or both in the same measurements , seemingly contradicting the notion of abss replacing the single particle states as eigenstates of the qd . we qualitatively explain these results as originating from the finite time scale required for the coherent oscillations between the superposition states after a single electron tunneling event . these experiments demonstrate that three - terminal experiments on a single complex quantum object can also be useful to investigate charge dynamics otherwise not accessible due to the very high frequencies .

1612.01201

c

in summary , we investigate electronic transport mediated by andreev bound states in a three - terminal s - qd device and identify the coupling to the normal metal leads as the main source for the spectroscopic broadening of the andreev resonances , and establish how the coupling to the superconductor determines the ground state of the system . in addition , we present ` excited state andreev resonances ' at higher energies , with transitions between abss and odd parity excited qd states . we also report pronounced sign changes in the abs - mediated transport between the two normal metal contacts , which we explain in an intuitive rate equation model as resulting from the competition between the nonlocal creation of a cooper pair in s and the new process of ` resonant abs tunneling ' , i.e. , the subgap transport of single electrons through the s - qd system , only allowed in multi - terminal devices . surprisingly , we find that depending on the imposed boundary conditions in the experiments it is possible to observe either abss or coulomb blockade resonances , or both in the same experiment , which we tentatively attribute to the competition between coherent oscillations in the abs ( exchange of cooper pairs with s ) and the relaxation of the system by single electron tunneling into the second normal terminal . we believe that experiments on complex quantum systems using multiple terminals provide a novel and clear experimental probe for many old and new phenomena , here for example the onset of the superconducting proximity effect on a qd and the formation of many - body quantum states . such experiments also give direct access to the strength of the coupling between s and the qd , and a way to probe time dependent phenomena like the finite - frequency coherent oscillation between superposition states , without resorting to high - frequency and time domain experiments . in particular , we envisage experiments in similar structures with gate - tunable tunnel barriers@xcite to investigate in more depths the presented physical mechanisms . we thank a. levy yeyati , j. schindele and p. makk for fruitful discussions . this work was financially supported by the swiss national science foundation ( snf ) , the swiss nanoscience institute ( sni ) , the swiss nccr qsit , the erc project quest and the eu fp7 project se@xmath262nd .

three - terminal spectroscopy allows us to identify the coupling to the n contacts as the origin of the andreev resonance ( ar ) linewidths and to determine the critical coupling strengths to s , for which a ground state ( or quantum phase ) transition in such s - qd systems can occur . surprisingly , we find that , depending on the boundary conditions and the device parameters , the experiments either show single - particle coulomb blockade resonances , abs characteristics , or both in the same measurements , seemingly contradicting the notion of abss replacing the single particle states as eigenstates of the qd . we qualitatively explain these results as originating from the finite time scale required for the coherent oscillations between the superposition states after a single electron tunneling event .

andreev bound states ( abss ) are well - defined many - body quantum states that emerge from the hybridization of individual quantum dot ( qd ) states with a superconductor and exhibit very rich and fundamental phenomena . we demonstrate several new electron transport phenomena mediated by abss that form on three - terminal carbon nanotube ( cnt ) qds , with one superconducting ( s ) contact in the center and two adjacent normal metal ( n ) contacts . three - terminal spectroscopy allows us to identify the coupling to the n contacts as the origin of the andreev resonance ( ar ) linewidths and to determine the critical coupling strengths to s , for which a ground state ( or quantum phase ) transition in such s - qd systems can occur . in addition , we ascribe replicas of the lowest - energy abs resonance to transitions between the abs and odd - parity excited qd states , a process we call excited state abs resonances . in the conductance between the two n contacts we find a characteristic pattern of positive and negative differential subgap conductance , which we explain by considering two nonlocal processes , the creation of cooper pairs in s by electrons from both n terminals , and a novel transport mechanism called resonant abs tunneling , possible only in multi - terminal qd devices . in the latter process , electrons are transferred via the abs without effectively creating cooper pairs in s. the three - terminal geometry also allows spectroscopy experiments with different boundary conditions , for example by leaving s floating . surprisingly , we find that , depending on the boundary conditions and the device parameters , the experiments either show single - particle coulomb blockade resonances , abs characteristics , or both in the same measurements , seemingly contradicting the notion of abss replacing the single particle states as eigenstates of the qd . we qualitatively explain these results as originating from the finite time scale required for the coherent oscillations between the superposition states after a single electron tunneling event . these experiments demonstrate that three - terminal experiments on a single complex quantum object can also be useful to investigate charge dynamics otherwise not accessible due to the very high frequencies .

0801.1111

i

characterizing old stellar populations provide important constraints on the early star formation histories of galaxies . only the satellite galaxies of the milky way ( mw ) are sufficiently close to resolve individual stars well below the oldest main - sequence turn - offs , which is a pre - condition for accurate photometric age dating of old stellar populations . all local group galaxies , for which adequate data exist , appear to contain stars older than 10 gyr @xcite . this result is based on main - sequence turn - off photometry of globular clusters and field populations in galactic satellites and a few more distant local group galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ) , as well as the detection of horizontal branch stars ( including rr lyrae variables ) in the local group and beyond . globular clusters are preferred as the basis for old stellar population age tracers since they are usually single - age , single - metallicity objects facilitating comparative studies . moreover , while globular cluster systems exhibit a range of ages ( e.g. , * ? ? ? * ) , the oldest ones may belong to the most ancient surviving stellar systems to have completed their formation in the youthful universe ( e.g. , * ? ? ? in those nearby galaxies where relative age dating based on main - sequence photometry was carried out in comparison to the oldest globular clusters in the milky way , no age difference within the measurement accuracy was found ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . the relative age dating of the oldest identifiable population ii objects thus indicates a common epoch of substantial early star formation in the milky way and its companions , although information about a putative , even older population iii remains to be uncovered in these objects . a galaxy that may _ not _ share this common epoch of early star formation at least not with respect to its globular clusters ( e.g. , * ? ? ? * ) is the small magellanic cloud ( smc ) . the smc is one of the closest and therefore best studied dwarf galaxies orbiting our galaxy . while the smc hosts a large number of intermediate - age and young star clusters , it only contains one old globular cluster , ngc121 , which is also the most massive star cluster . ngc121 is located @xmath7 2.4 ( @xmath73 kpc ) west of the smc bar at ( @xmath8 , @xmath9 ) = ( @xmath10 , @xmath11 ) . ngc121 is the only cluster in the smc that is sufficiently old to have developed an extended red horizontal branch @xcite and to contain rr lyrae stars . indeed , whether or not to call a star cluster a globular cluster is a matter of definition . in this case we refer to @xcite who consider lindsay1 as having a stumpy red clump and not a red horizontal branch . three rr lyrae stars were discovered in ngc121 by @xcite . @xcite found a fourth rr lyrae variable in the cluster and an additional 75 in a 1 @xmath12 1.3 square degree field centered on ngc121 . studies of various clusters in the lmc and in the mw showed that the presence of rr lyrae variables indicates that the parent population is as old as or older than @xmath7 10 gyr . an important question is whether ngc121 is as old as the typical old globular clusters in the large magellanic cloud ( lmc ) and in the mw . previous studies found ages ranging from 8 to 14 gyr for ngc121 @xcite using a variety of different techniques . studies based on the deepest available color - magnitude diagrams from hubble space telescope ( hst ) observations with the wide field and planetary camera 2 ( wfpc2 ) indicate an age of 10 to 10.6 gyr for ngc121 , suggesting that this globular cluster is several gyr younger than the oldest globulars in other nearby galaxies and in the mw @xcite . the capabilities of the advanced camera for surveys ( acs ) provide an improvement in both sensitivity ( depth ) as well as angular resolution , which is essential for a reliable photometric age determination in this dense star cluster . here we present deep photometry of ngc121 obtained with acs aboard the hst . we determine the age of ngc121 utilizing both absolute and relative methods ( e.g. , * ? ? ? the current study is the first in a series of papers based on hst studies of rich intermediate - age and old star clusters in the smc . in addition to ngc121 , six intermediate - age smc star clusters have been observed as part of our program : lindsay1 , kron3 , ngc339 , ngc416 , lindsay38 and ngc419 . we will derive fiducial ridgelines and fit isochrones to obtain accurate ages for each cluster using the same reduction techniques and isochrone models as described here ( see @xmath13 [ sec : obs]-[sec : age ] ) , and will present our results in future papers . in table [ tab : journalobs ] we list the cluster identification , date of observation , passband , exposure times and location of all clusters in our hst program ( go-10396 ; principal investigator : j. s. gallagher ) . in the next section we describe the data reduction procedure . in @xmath13 [ sec : cmd ] we present the color - magnitude diagram ( cmd ) of ngc121 and discuss its main features . in @xmath13 [ sec : age ] we describe our age derivation methods and present our results .

we derive the age of ngc121 using both absolute and relative age - dating methods . therefore we confirm the earlier finding that the oldest globular cluster in the small magellanic cloud , ngc121 , is a few gyr younger than its oldest counterparts in the milky way and in other nearby dwarf galaxies such as the large magellanic cloud , fornax , and sagittarius .

as first paper of a series devoted to study the old stellar population in clusters and fields in the small magellanic cloud , we present deep observations of ngc121 in the f555w and f814w filters , obtained with the advanced camera for surveys on the _ hubble space telescope_. the resulting color - magnitude diagram reaches mag below the main - sequence turn - off ; deeper than any previous data . we derive the age of ngc121 using both absolute and relative age - dating methods . fitting isochrones in the acs photometric system to the observed ridge line of ngc121 , gives ages of gyr ( teramo ) , gyr ( padova ) and gyr ( dartmouth ) . the cluster ridge line is best approximated by the-enhanced dartmouth isochrones . placing our relative ages on an absolute age scale , we find ages of gyr ( from the magnitude difference between the main - sequence turn - off and the horizontal branch ) and gyr ( from the absolute magnitude of the horizontal branch ) , respectively . these five different age determinations are all lower by 2 3 gyr than the ages of the oldest galactic globular clusters of comparable metallicity . therefore we confirm the earlier finding that the oldest globular cluster in the small magellanic cloud , ngc121 , is a few gyr younger than its oldest counterparts in the milky way and in other nearby dwarf galaxies such as the large magellanic cloud , fornax , and sagittarius . if it were accreted into the galactic halo , ngc121 would resemble the `` young halo globulars '' , although it is not as young as the youngest globular clusters associated with the sagittarius dwarf . the young age of ngc121 could result from delayed cluster formation in the small magellanic cloud or result from the random survival of only one example of an initially small number star clusters .

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we derived ages for the old smc globular cluster ngc121 based on our high dynamic range hst / acs photometry that extends at least three magnitudes below its msto . in order to obtain absolute ages , we applied three different isochrone models . these isochrone models yielded ages of @xmath110 gyr ( padova ) , @xmath114 gyr ( teramo ) , and @xmath3 gyr ( dartmouth ) . we find the @xmath4-enhanced dartmouth isochrones provide the closest approximation to the ms , sgb , and rgb , whereas the other models can not reproduce the slope of the upper rgb . high - resolution spectroscopy indicates that ngc121 is indeed @xmath4-enhanced @xcite , a property that it shares with many of the old outer galactic halo globulars . given the proximity of ngc121 to the smc on the sky and its distance , its physical association with the smc seems well - established . our determinations of relative ages for ngc121 are consistent with the results of our absolute age determination . relative age estimates , when converted to an absolute age scale , are @xmath5 gyr ( @xmath111 ) , @xmath93 ( @xmath90 ) and @xmath6 gyr ( @xmath115 ) . these numbers agree well with the absolute age derivations . our results confirm that ngc121 is 23 gyr younger than the oldest mw and lmc clusters ( as also found in earlier wfpc2 studies ) . ngc121 is similar in age to the youngest globular cluster in the fornax dsph @xcite , and to several of the young galactic halo clusters . on the other hand , ngc121 is not as young as some of the sgr dwarf galaxy s globular clusters or the youngest galactic globular clusters . it is intriguing that the smc in contrast to other galactic companion dwarf galaxies with globulars does not contain any old classical globular clusters . but given the existence of only one cluster and the question of star cluster survival , this could be a result of the one survivor from the smc s epoch of globular cluster formation randomly sampling an initial distribution of star cluster ages . on the other hand , in low - mass galaxies without bulges , spiral density waves , and shear it is much more difficult to destroy globular clusters through external effects . that this cluster is both younger than the galactic mean and enhanced in @xmath4-elements may have interesting implications for the early development of the smc . it also is intriguing that the only globular cluster in the smc is not very metal - poor . the smc must have experienced substantial enrichment prior to the formation of ngc121 . in the lmc , where two main epochs of the formation of populous compact star clusters have been found ( e.g. , * ? ? ? * ) , a few globular clusters are found that are old enough to exhibit blue hbs . interestingly , these globular clusters , which are similarly old as the oldest galactic globulars @xcite , have a similar metallicity to ngc 121 @xcite ( e.g. , ngc1898 , ngc2019 ) , indicating very early chemical enrichment . the mw also contains old classical globular clusters ( with blue hbs ) that have similarly high metallicities as the somewhat younger ngc 121 . evidently , the conditions for and the efficiency of star formation varied in these three galaxies at early epochs . after ngc 121 formed there was a hiatus in surviving stars clusters and thus possibly in cluster formation activity in the smc : the second oldest smc cluster is lindsay 1 with an age of @xmath7 8 gyr ( glatt et al . 2007 , in preparation ) . since then compact populous star clusters formed fairly continuously until the present day in the smc ( e.g. , da costa 2002 ) in contrast to both the lmc and the mw . in forthcoming papers on our acs photometry of smc clusters and field populations we will explore the evolutionary history of the smc in more detail . clearly , clues about the early star formation history of the smc will have to come from its old field populations . we thank the anonymous referee for extremely useful suggestions to improve our paper . we gratefully acknowledge support by the swiss national science foundation through grant number 200020 - 105260 and 200020 - 113697 . support for program go-10396 was provided by nasa through a grant from the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas 5 - 26555 . we warmly thank paolo montegriffo to provide his software and leo girardi for the padova isochrones in the standard acs color system . gisella clementini and monica tosi have been partially supported by prin - miur-2004 and prin - inaf-2005 , and jay gallagher also obtained helpful additional support from the university of wisconsin graduate school . , d. r. , & sarajedini , a. 1999 , , 511 , 225 , m. , fusi pecci , f. , ferraro , f. r. , galleti , s. , catelan , m. , & landsman , w. b. 2001 , , 122 , 2569 , g. , mateo , m. , chiosi , c. , & bressan , a. 1992 , , 388 , 400 , e. , & arnouts , s. 1996 , , 117 , 393 , t. m. , et al . 2007 , , 658 , 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1993 , the globular cluster - galaxy connection , asp conf . smith , g. h. & brodie , j. p. ( san francisco : asp ) , 302

as first paper of a series devoted to study the old stellar population in clusters and fields in the small magellanic cloud , we present deep observations of ngc121 in the f555w and f814w filters , obtained with the advanced camera for surveys on the _ hubble space telescope_. the resulting color - magnitude diagram reaches mag below the main - sequence turn - off ; deeper than any previous data . the cluster ridge line is best approximated by the-enhanced dartmouth isochrones . placing our relative ages on an absolute age scale , we find ages of gyr ( from the magnitude difference between the main - sequence turn - off and the horizontal branch ) and gyr ( from the absolute magnitude of the horizontal branch ) , respectively . if it were accreted into the galactic halo , ngc121 would resemble the `` young halo globulars '' , although it is not as young as the youngest globular clusters associated with the sagittarius dwarf . the young age of ngc121 could result from delayed cluster formation in the small magellanic cloud or result from the random survival of only one example of an initially small number star clusters .

as first paper of a series devoted to study the old stellar population in clusters and fields in the small magellanic cloud , we present deep observations of ngc121 in the f555w and f814w filters , obtained with the advanced camera for surveys on the _ hubble space telescope_. the resulting color - magnitude diagram reaches mag below the main - sequence turn - off ; deeper than any previous data . we derive the age of ngc121 using both absolute and relative age - dating methods . fitting isochrones in the acs photometric system to the observed ridge line of ngc121 , gives ages of gyr ( teramo ) , gyr ( padova ) and gyr ( dartmouth ) . the cluster ridge line is best approximated by the-enhanced dartmouth isochrones . placing our relative ages on an absolute age scale , we find ages of gyr ( from the magnitude difference between the main - sequence turn - off and the horizontal branch ) and gyr ( from the absolute magnitude of the horizontal branch ) , respectively . these five different age determinations are all lower by 2 3 gyr than the ages of the oldest galactic globular clusters of comparable metallicity . therefore we confirm the earlier finding that the oldest globular cluster in the small magellanic cloud , ngc121 , is a few gyr younger than its oldest counterparts in the milky way and in other nearby dwarf galaxies such as the large magellanic cloud , fornax , and sagittarius . if it were accreted into the galactic halo , ngc121 would resemble the `` young halo globulars '' , although it is not as young as the youngest globular clusters associated with the sagittarius dwarf . the young age of ngc121 could result from delayed cluster formation in the small magellanic cloud or result from the random survival of only one example of an initially small number star clusters .

hep-ph0402263

i

at high temperature and/or baryon density , the system governed by qcd will show a transition from an ordinary hadronic phase to a chirally symmetric , deconfined plasma phase@xcite . the main objective of the heavy ion programs at rhic and at future lhc is to create this long - sought plasma state and to study collective properties of this many body assembly@xcite . these two phases would have to be separated by a boundary with singularity if chiral symmetry and/or confinement of qcd were exact symmetry . in reality , dynamical quarks with finite masses @xmath0 make both symmetries only approximate , and their order parameters , the quark condensate and the polyakov loop , have non vanishing values everywhere in the phase diagram . thus the plasma state may be smoothly connected with the ordinary hadronic state , even though they would possess qualitatively different properties from each other . recently a strong possibility of a critical point in the real qcd phase diagram was suggested@xcite , based on model calculations@xcite as well as lattice qcd results@xcite . it is the endpoint of the first order line , inferred from the crossover behavior along the temperature ( @xmath1 ) axis and the first order transition along the axis of the baryon number chemical potential ( @xmath2 ) , and is a genuine singular point with the same criticality as the z@xmath3 ising model . its location , which is sensitive to the strange quark mass @xmath4 , is expected to be within the reach of current experimental facilities . observable implications of this z@xmath3 critical point ( z@xmath3cp)cp instead of cep , to indicate the symmetry of the point . ] in heavy ion experiments have been discussed in the literature@xcite such as large fluctuations of the low momentum particle distributions , and the limitations on them due to the finite space time geometry of collision events . this z@xmath3cp will become a critical cornerstone in the qcd phase diagram once its location is confirmed in experiments . based on the approximate chiral symmetry , the scalar density is usually taken as the order parameter of the ginzburg landau ( gl ) effective potential to describe the critical behavior at the z@xmath3cp . in this description all the singularities associated with the z@xmath3cp seemingly originate from softening of the scalar density fluctuations as the effective potential becomes flat there . especially , it might be concluded that the sigma meson becomes massless as an immediate consequence of this critical point . as a basic fact , however , we should strictly distinguish between the chiral critical point with @xmath5 and the z@xmath3cp with @xmath6 even within the chiral effective models . when the chiral symmetry is exact , the @xmath1@xmath2 plane is divided into two domains of the symmetric and broken phases with a boundary _ line_. but the symmetry argument is unable to fix the order of the singularity of this line , especially the possible existence of the tri critical point ( tcp ) on this line . since the z@xmath3cp at finite quark mass is the remnant of this tcp , the relation of the z@xmath3cp to the chiral symmetry is very obscure . in fact , z@xmath3 is not the symmetry of the underlying interactions , but of the thermodynamic potential at this particular isolated _ point _ in the @xmath1@xmath2 phase diagram@xcite . from this point of view the z@xmath3cp is different from the chiral critical point . the flat effective potential indicates the large fluctuation of the scalar density . at the chiral critical point , this is related to softening of the sigma meson mode , so as to form a symmetric excitation spectrum together with the pions . on the other hand , there is no such symmetry constraint at the z@xmath3cp . actually the scalar fluctuation _ linearly _ mixes with fluctuations of baryon number density and energy density , and therefore not only the sigma mode but also the hydrodynamic mode are to be taken into account there to study the associated soft mode . consequently the fate of the sigma meson mode at the z@xmath3cp is non trivial . recent calculations of the dynamic mode in the scalar channel using the chiral models@xcite indeed showed that the sigma meson is massive at the z@xmath3cp . furthermore , another scalar mode with space like momentum dispersion is identified as the soft mode associated with the z@xmath3cp in the nambu jona - lasinio ( njl ) model@xcite . in this paper we shall confirm the result of ref . @xcite on the more general ground using the time dependent ginzburg landau ( tdgl ) approach , and extend the study to discuss the changeover of the soft modes along the critical line in the @xmath1@xmath2@xmath7 space within the tdgl approach as well as the njl model . our investigation is based on two fundamental observations about the z@xmath3cp . the first point is that the proper ordering density at the z@xmath3cp is a linear combination of the scalar , baryon number and internal energy densities@xcite , as mentioned above . because of this mixing all the susceptibilities of these densities diverge with the same critical exponent at the z@xmath3cp . in contrast , in the chiral critical transition , the susceptibility of the scalar density diverges with exponent @xmath8 of the o(4 ) model in the two flavor case , while the other susceptibilities of the baryon number and the energy have the smaller exponent @xmath9 . the second is a consequence on the dynamics following from the conservation of the baryon number and the energy . the fluctuations of these conserved densities are intrinsically soft and constitute the hydrodynamic modes , whose excitation energies vanish as the wavevector @xmath10 goes to zero . susceptibilities of these conserved densities in turn have the spectral contributions solely from these hydrodynamic modes when expressed as a sum of mode spectra@xcite . hence the divergence of the susceptibility of a conserved density must be accompanied by critical slowing of a hydrodynamic mode . the spectral contribution from this hydrodynamic mode may well be involved in the scalar susceptibility through the mixing at the z@xmath3cp . at an o(4 ) critical point ( o(4)cp ) the importance of the hydrodynamic mode depends on which phase we start from . the hydrodynamic mode plays no critical role in the symmetric phase whereas the scalar condensate makes the mixing possible in the broken phase . the situation becomes more subtle at the tcp , where the o(4 ) critical line shifts to the first order line . only the scalar susceptibility diverges due to the softening of the sigma meson at the tcp if it is approached from the symmetric phase . otherwise , the hydrodynamic soft mode causes the divergence in the susceptibilities of the baryon number and energy as well as the scalar one . this paper is organized as follows . in the next section we briefly review generic properties of the phase diagram of qcd with two flavors near the tcp using the gl effective potential . it is stressed that at the tcp there are two relevant order parameters , the scalar condensate and a conserved density which is a linear combination of the baryon number and entropy densities . then we include the dynamics using the tdgl model . writing the susceptibilities as a spectral sum , we discuss the relative weight of the spectral contributions from the sigma and hydrodynamic modes . it is pointed out that the hydrodynamic contribution generates the discontinuity of the baryon number and entropy susceptibilities at the o(4)cp , and that this hydrodynamic mode gives the divergence at the tcp approached from the broken phase and also at the z@xmath3cp . in iii we perform the same analysis using the njl model as an illustration . the gl effective potential with two ordering densities are numerically constructed there . the flat directions at the critical points are shown and discussed in relation to the divergences of the susceptibilities . the spectral origins of these divergences are studied with the relative weight of the mode spectra , and in detail based on the poles and residues of the scalar response function . sections iv and v are devoted to discussions and summary . in appendix a we prove the relation between the susceptibility and the response function , and in appendix b we present the explicit formulas of the response functions in the njl model . the results with the chiral quark model is briefly reported in appendix c.

assuming a tricritical point of the two flavor qcd in the space of temperature , baryon number chemical potential and quark mass , we study the change of the associated soft mode along the critical line within the ginzburg landau approach and the nambu jona - lasinio model . the ordering density along the chiral critical line is the scalar density whereas a linear combination of the scalar , baryon number and energy densities becomes the proper ordering density along the critical line with finite quark masses . it is shown that the critical eigenmode shifts from the sigma like fluctuation of the scalar density to a hydrodynamic mode at the tricritical point , where we have two ordering densities , the scalar density and a linear combination of the baryon number and energy densities .

assuming a tricritical point of the two flavor qcd in the space of temperature , baryon number chemical potential and quark mass , we study the change of the associated soft mode along the critical line within the ginzburg landau approach and the nambu jona - lasinio model . the ordering density along the chiral critical line is the scalar density whereas a linear combination of the scalar , baryon number and energy densities becomes the proper ordering density along the critical line with finite quark masses . it is shown that the critical eigenmode shifts from the sigma like fluctuation of the scalar density to a hydrodynamic mode at the tricritical point , where we have two ordering densities , the scalar density and a linear combination of the baryon number and energy densities . we argue that appearance of the critical eigenmode with hydrodynamic character is a logical consequence of divergent susceptibilities of the conserved densities .

1408.0168

i

in this paper , we investigate numerically the behavior of granular material at the surface of an asteroid during close approach to the earth . we focus on the specific case of asteroid ( 99942 ) apophis , which will come as close as @xmath3 earth radii on april 13th , 2029 . we provide predictions about possible reshaping and spin - alteration of and surface effects on apophis during this passage , as a function of plausible properties of the constituent granular material . studies of possible future space missions to apophis are underway , including one by the french space agency cnes calling for international partners ( e.g. , michel et al . 2012 ) , with the aim of observing this asteroid during the 2029 close encounter and characterizing whether reshaping , spin - alteration , and/or surface motion occur . the numerical investigations presented here allow for estimation of the surface properties that could lead to any observed motion ( or absence of motion ) during the actual encounter . apophis made a passage to the earth at @xmath4 earth radii in early 2013 . at that time , the herschel space telescope was used to refine the determination of the asteroid s albedo and size ( mller et al . according to these observations , the albedo is estimated to be about 0.23 and the longest dimension about @xmath5 m , which is somewhat larger than previous estimates ( @xmath6 m , according to delbo et al . concurrent radar observations improved the astrometry of the asteroid , ruling out the possibility of a collision with the earth in 2036 to better than 1 part in @xmath7 . however , wlodarczyk et al . ( 2013 ) presented a possible path of risk for 2068 . this finding has put off any crisis by @xmath8 years and makes exploring apophis in 2029 to be more for scientific interest . to date , nothing is known about the asteroid s surface mechanical properties , and this is why its close passage in 2029 offers a great opportunity to visit it with a spacecraft , determine its surface properties , and , for the first time , observe potential modifications of the surface due to tidal effects . and as apophis approaches , it is likely that international interest in a possible mission will increase , since such close approaches of a large object are relatively rare . the case for tidally induced resurfacing was made by binzel et al . ( 2010 ; also see demeo et al . 2013 ) and discussed by nesvorn et al . ( 2010 ) to explain the spectral properties of near - earth asteroids ( neas ) belonging to the q taxonomic type , which appear to have fresh ( unweathered ) surface colors . dynamical studies of these objects found that those bodies had a greater tendency to come close to the earth , within the earth - moon distance , than bodies of other classes in the past @xmath9 kyr . the authors speculated that tidal effects during these passages could be at the origin of surface material disturbance leading to the renewed exposure of unweathered material . we leave a more general and detailed investigation of this issue for future work , but if this result is true for those asteroids , it may also be true for apophis , which will approach earth on friday , april 13 , 2029 no closer than about 29,500 km from the surface ( i.e. , @xmath10 earth radii , or @xmath3 earth radii from the center of the planet ; giorgini et al . it is predicted to go over the mid - atlantic , appearing to the naked eye as a moderately bright point of light moving rapidly across the sky . our aim is to determine whether , depending on assumed mechanical properties , it could experience surface particulate motions , reshaping , or spin - state alteration due to tidal forces caused by earth s gravity field . the classical roche limit for a cohesionless fluid body of bulk density @xmath11 @xmath12 to not be disrupted by tidal forces is @xmath13 earth radii , so we do not expect any violent events to occur during the rocky asteroid s 2029 encounter at @xmath3 earth radii . the presence of granular material ( or regolith ) and boulders at the surface of small bodies has been demonstrated by space missions that visited or flew by asteroids in the last few decades ( e.g. , veverka et al . 2000 ; fujiwara et al . it appears that all encountered asteroids to date , from the largest one , the main belt asteroid ( 4 ) vesta by the dawn mission , to the smallest one , the nea ( 25143 ) itokawa , sampled by the hayabusa mission , are covered with some sort of regolith . in fact , thermal infrared observations support the idea that most asteroids are covered with regolith , given their preferentially low thermal inertia ( delbo et al . there even seems to be a trend as a function of the asteroid s size based on thermal inertia measurements : larger objects are expected to have a surface covered by a layer of fine regolith , while smaller ones are expected to have a surface covered by a layer of coarse regolith ( clark et al . this trend is consistent with observations by the near - shoemaker spacecraft of the larger ( @xmath14 km mean diameter ) eros , whose surface is covered by a deep layer of very fine grains , and by the hayabusa spacecraft of the much smaller ( @xmath15 m mean diameter ) itokawa , whose surface is covered by a thin layer of coarse grains . however , interpretation of thermal inertia measurements must be made with caution , as we do not yet have enough comparisons with actual asteroid surfaces to verify that the suggested trend is systematically correct . thus , we are left with a large parameter space to investigate possible surface motion during an earth close approach of an asteroid with unknown surface mechanical properties . our approach is to consider a range of simple and well - controlled cases that certainly do not cover all possibilities regarding apophis surface mechanical properties , but rather aim at demonstrating whether , even in a simple and possibly favorable case for surface motion , some resurfacing event can be expected to occur during the passage . for instance , instead of considering a flat granular surface , we consider a sandpile consisting of a size distribution of spherical grains ( section [ s : initialcondition ] ) and vary the grain properties in order to include more or less favorable cases for motion ( from a fluid - like case to a case involving rough particles ) . slight disturbances may manifest as very - small - scale avalanches in which grain connections readjust slightly , for example . the forces acting on the sandpile are obtained by measuring all `` external '' perturbations during the encounter , including body spin magnitude and orientation changes , for cases in which the global shape remains nearly fixed , and again assuming simple and favorable configurations of the asteroid . indirectly , the encounter may also lead to internal reconfigurations of the asteroid , which in turn produce seismic vibrations that could propagate to the surface and affect the regolith material . these secondary modifications are not modeled here , although it may be possible in future work to account for this by shaking the surface in a prescribed manner . in any case , for this particular encounter , we demonstrate ( section [ s : reshpeff ] ) that any global reconfiguration will likely be small to negligible in magnitude . in the following , we first present , in section [ s : method ] , the numerical method used to perform our investigation , including the initial conditions of the sandpile adopted to investigate surface motion , the representation of the encounter , and the mechanical environment . results are described in section [ s : results ] , including potential reshaping of the asteroid , tidal disturbances for apophis encounter in 2029 , which is a function of the sandpile properties , spin orientation changes , and the dependency of the location of the sandpile on the asteroid to the outcome of the encounter . we also show the responses of the sandpiles for artificially close approaches ( @xmath16 and @xmath17 earth radii ) to demonstrate that our method does predict significant alteration of the sandpiles when this is certainly expected to happen . the investigation is discussed in section [ s : discuss ] and conclusions are presented in section [ s : concl ] .

asteroid ( 99942 ) apophis close approach in 2029 will be one of the most significant small - body encounter events in the near future and offers a good opportunity for exploration to determine the asteroid s surface properties and measure any tidal effects that might alter its regolith configuration . resurfacing mechanics has become a new focus for asteroid researchers due to its important implications for interpreting surface observations , including space weathering effects .

asteroid ( 99942 ) apophis close approach in 2029 will be one of the most significant small - body encounter events in the near future and offers a good opportunity for exploration to determine the asteroid s surface properties and measure any tidal effects that might alter its regolith configuration . resurfacing mechanics has become a new focus for asteroid researchers due to its important implications for interpreting surface observations , including space weathering effects . this paper provides a prediction for the tidal effects during the 2029 encounter , with an emphasis on whether surface refreshing due to regolith movement will occur . the potential shape modification of the object due to the tidal encounter is first confirmed to be negligibly small with systematic simulations , thus only the external perturbations are taken into account for this work ( despite this , seismic shaking induced by shifting blocks might still play a weak role and we will look into this mechanism in future work ) . a two - stage approach is developed to model the responses of asteroid surface particles ( the regolith ) based on the soft - sphere implementation of the parallel-body gravity tree code ` pkdgrav ` . a full - body model of apophis is sent past the earth on the predicted trajectory to generate the data of all forces acting at a target point on the surface . a sandpile constructed in the local frame is then used to approximate the regolith materials ; all the forces the sandpile feels during the encounter are imposed as external perturbations to mimic the regolith s behavior in the full scenario . the local mechanical environment on the asteroid surface is represented in detail , leading to an estimation of the change in global surface environment due to the encounter . typical patterns of perturbation are presented that depend on the asteroid orientation and sense of rotation at perigee . we find that catastrophic avalanches of regolith materials may not occur during the 2029 encounter due to the small level of tidal perturbation , although slight landslides might still be triggered in positions where a sandpile s structure is weak . simulations are performed at different locations on apophis surface and with different body- and spin - axis orientations ; the results show that the small - scale avalanches are widely distributed and manifest independently of the asteroid orientation and the sandpile location . we also include simulation results of much closer encounters of the apophis with earth than what is predicted to occur in 2029 , showing that much more drastic resurfacing takes place in these cases . * keywords * : asteroids , dynamics ; asteroids , surfaces ; near - earth objects ; regoliths ; tides , solid body

1408.0168

r

richardson et al . ( 1998 ) showed the complicated behavior of a rotating rubble pile due to a tidal encounter . generally , a variation in rotational state is usually induced because of the coupling effect between librational motion and orbital motion . especially for elongated bodies like apophis , the terrestrial torques during its flyby may force a strong alteration of the rotational state in a short period ( scheeres et al . the modifications in structure of the rubble pile are also notable . the tidal encounter outcomes show a strong dependence on the progenitor elongation , the perigee distance , and the time spent within the roche sphere . the outcomes also depend on the spin orientation of the progenitor to the extent that retrograde rotators always suffer less catastrophic consequences than prograde ones ( richardson et al . 1998 ) . however , the parameter space considered in that study ( richardson et al . 1998 ) did not include the apophis scenario in 2029 , since the focus was on the conditions for full - scale distortion or disruption . for this study , we first redid the flyby simulations ( section [ s : representme ] ) using a true ( non - rigid ) rubble pile model with the apophis encounter parameters of @xmath3 @xmath52 and @xmath53 km / s . we quantified the tidal reshaping effect by measuring the maximum net change of a normalized shape factor defined by eq . ( [ e : shpfact ] ) ( hu et al . 2004 ) , in which @xmath58 , @xmath59 , @xmath60 are the principal inertia moments , with @xmath61 : @xmath62 the shape factor @xmath63 $ ] roughly describes the mass distribution of the body ( 0=oblate , 1=prolate ) . the initial rubble - pile model ( equilibrated , prior to the encounter ) had @xmath64 , and its relative change , defined by @xmath65 , was measured for several simulations parameterized by bulk density and constituent material type ( table [ t : fullscalechange ] ) . the corresponding roche limits are provided in the table , with the values estimated for the case of a circular orbit . the reshaping effects show consistent results for all three types of materials , varying between a noticeable change in shape ( magnitude @xmath66 ) and a negligible change in shape ( magnitude @xmath67 ) , with the sharpest transition occurring for a critical bulk density @xmath68 @xmath12 with corresponding roche limit of @xmath3 @xmath52 . this agrees with the analysis of apophis disruption limit using a continuum theory for a body made up of solid constituents ( holsapple et al . note we find that even a bulk density as low as @xmath69 @xmath12 did not dislodge any particles enough for them to end up in a new geometrical arrangement , due to the very short duration of the tidal encounter . figure [ f : newpile ] . table [ t : fullscalechange ] . since idealized equal - size spheres can arrange into crystal - like structures that may artificially enhance the shear strength of the body ( tanga et al . 2009 ; walsh et al . 2012 ) , we carried out a second suite of simulations with rubble piles made up of a bimodal distribution of spheres ( fig . [ f : newpile ] ) . the model consisted of @xmath70 big particles of radius @xmath71 m and @xmath72 small particles of radius @xmath73 m , bounded by the same tri - axial ellipsoid as shown in fig . [ f : rubblepile ] , and arranged randomly . as before , the bulk density and material type were varied , with notable differences in outcome compared to the equal - size - sphere cases ( table [ t : fullscalechange ] ) . particles using the smooth " parameter set are not able to hold the overall shape of apophis ( the 1.4:1.0:0.8 tri - axial ellipsoid ) ; instead the rubble pile collapses and approaches a near - spherical shape . for the other two parameter sets , however , the overall shape of the rubble pile is maintained . figure [ f : comp ] shows the relative change of shape factor @xmath74 as a function of bulk density for both types of simulations . as shown , the results from the unimodal and bimodal rubble piles present similar trends : before reaching the roche limit density , the reshaping remains small ( the magnitudes are @xmath75@xmath76 with a certain amount of stochasticity ) ; after that , the reshaping effect sharply increases to around @xmath77 . the bimodal rubble piles show less resistance to the tidal disturbance , since they have less - well - organized crystalline structures compared to the unimodal cases ; they therefore exhibit larger shape modifications ( walsh et al . 2012 ) . figure [ f : comp ] . regardless , catastrophic events were not detected for either type of rubble pile until the bulk density was as small as @xmath69 @xmath12 , for all material parameter sets . therefore , the reshaping effects on apophis in 2029 should be negligibly small for bulk densities in the likely range ( @xmath78@xmath79 @xmath12 ) . even so , minor internal reconfigurations resulting from the tidal encounter may produce seismic waves that could propagate and affect the configuration of surface regolith . since we are only looking at localized areas on the surface , isolated from the rest of the body , vibrations emanating from other regions , in or on the asteroid , are not evaluated in the current work , although again we expect these to be small or non - existent for the specific case of the apophis 2029 encounter . only the external " forces acting directly on the surface particles in the considered localized region are taken into account . we will look into the effects of seismic activity , which may stem from other regions of the asteroid , on an actual high - resolution rubble pile in future work . in this study , we focus on external forces , outside of the rubble pile itself ; the rigid rubble pile model is employed as a reasonable simplification for the purpose of measuring the external forces on a surface sandpile . the right - hand side of eq . ( [ e : locdynequ ] ) provides a description of the constituents of the mechanical environment experienced by a sandpile particle , including local gravity , tidal force , centrifugal force , ltif , and the coriolis effect . note the coriolis force does not play a role before particle motion begins , so it can be ignored when examining the causes of avalanches / collapses . the centrifugal force and ltif show weak dependence on local position @xmath36 , so these terms can be simplified by substituting @xmath80 , since the dimension of the sandpile is much smaller than that of the asteroid . this leads to an approximation of the resultant environmental force ( eq . ( [ e : envirforce ] ) ) , which provides a uniform expression of the field force felt throughout the sandpile : @xmath81 the variation of @xmath82 shows a common pattern for different encounter trajectories and different locations on the asteroid : it stays nearly invariable for hours as the planet is approaching / departing and shows a rapid single perturbation around the rendezvous . this feature enables us to evaluate the mechanical environment by measuring the short - term change of @xmath82 , and moreover , to make a connection between these external stimuli with the responses of sandpiles in the simulations . spherical coordinates @xmath83 are used to represent the resultant force ( fig . [ f : accs ] ) , in which @xmath84 is the magnitude and @xmath85 and @xmath86 denote the effective gravity slope angle and deflection angle , respectively . figure [ f : accs ] . the effects of the environment force must be considered in the context of the sandpile modeling . a rough approximation of the catastrophic slope angle ( eq . ( [ e : critcang ] ) ) can be derived from the conventional theory of the angle of repose for conical piles ( brown et al . 1966 ) , namely that avalanches will occur when the effective gravity slope angle @xmath85 exceeds @xmath87 : @xmath88 here @xmath89 denotes the resting angle of the ( conical ) sandpile , assumed to be less than the repose angle @xmath90 , and @xmath91 denotes the critical slope angle , which , if exceeded by the effective gravity slope angle , results in structural failure of the sandpile , i.e. , an avalanche . essentially , the avalanches of the sandpile depend primarily on the instantaneous change of the slope angle @xmath85 of the resultant force and should be only weakly related to the magnitude @xmath84 and deflection angle @xmath86 . the global distribution of changes of @xmath85 were examined for the duration of the encounter ( see below ) . it would be difficult to make an exhaustive search over all asteroid encounter orientations due to the coupling effects between the asteroid s rotation and the orbital motion . instead , we chose @xmath92 representative trajectories along the symmetry axes of the apophis model ( at perigee ) for study , which serves as a framework for understanding the influence of encounter orientation and the strength of tidal disturbance at different locations . ( the technique used to match the orientation of the asteroid at perigee is to first run a simulation in reverse starting at perigee with the required orientation , thereby providing the correct asteroid state for a starting position far away . ) figure [ f : traject ] shows the @xmath92 trajectories along three mutually perpendicular planes , including both the prograde and retrograde cases . this choice is based on the symmetry of the dynamical system , and all @xmath92 trajectories have a speed of @xmath53 km / s at a perigee distance of @xmath3 @xmath52 . figure [ f : traject ] . figure [ f : traject ] presents these representative trajectories in order , each corresponding to a special possible orientation of the encounter . the effective gravity slope angle changes throughout the surface of the tri - axial ellipsoid model were recorded for each trajectory , with particular attention to the location and time of the maximum change . we found that these maximum changes concentrate at several minutes around perigee . figure [ f : maxcsa ] shows the maximum values of slope angle change along these trajectories , indicating that the largest perturbation on slope angle is less than @xmath93 . we verified that the achievable range of the effective gravity slope angle during the encounter ( section [ s : representme ] ) is within the safe limit predicted by eq . ( [ e : critcang ] ) for all three sandpiles ( section [ s : initialcondition ] ) ; that is , a slope change below @xmath93 is not enough to trigger any massive avalanches throughout the sandpiles . figure [ f : maxcsa ] . figure [ f : maxcsa ] also shows a general dependence of the tidal perturbation on the orientation of the encounter trajectory , namely that the three step levels seen in the figure correspond to three directions of the relative planet motion at perigee . trajectories @xmath94@xmath95 correspond to the direction of the body long axis , which leads to relatively strong tidal effects ; trajectories @xmath96@xmath97 correspond to the direction of the intermediate axis , which leads to moderate effects ; while trajectories @xmath98@xmath92 correspond to the direction of the short axis , which leads to relatively weak tidal effects . figure [ f : patterns ] illustrates the distribution of effective gravity slope angle change at perigee for the three trajectory sets , with a denoting trajectories @xmath94@xmath95 , b denoting trajectories @xmath96@xmath97 , and c denoting trajectories @xmath98@xmath92 . we confirm that the four trajectories in each set present visually the same distribution around perigee , therefore we chose the patterns due to trajectories 1 , 5 , and 9 for demonstration of sets a , b , and c , and generalize these three patterns as representative . several points can be inferred from fig . [ f : patterns ] . first , the direction to the planet at perigee largely determines the global distribution of tidal perturbation , that is , the strongest effects tend to occur along the long axis and the weakest effects along the short axis . second , at perigee , the largest slope change occurs near areas surrounding the pole for the most favorable orientation ( set a ) , while the largest slope change at the pole occurs several minutes before or after perigee ( see fig . [ f : pxpypz ] ) . these maximum slope changes are about equal in magnitude ( the change is only slightly smaller at the pole compared to the area immediately surrounding it , but not enough to make a difference to the avalanches ) , so for simplicity we just use the poles themselves as our testing points . third , the duration of strong tidal effects depends on the eccentricity of the encounter trajectory . we checked the trajectories shown in fig . [ f : traject ] and found the duration of strong perturbation , defined for illustration as the period when the force magnitude stays above @xmath99 of the peak value , is tens of minutes long ; correspondingly , the responses of the surface material are also transitory and weak . figure [ f : patterns ] . the three poles p@xmath100 , p@xmath101 and p@xmath102 ( see figure [ f : traject ] ) were chosen as the test locations for the sample sandpiles since the effective gravity slope angle is near zero at these locations , which enables the sandpiles to hold their initial shapes . figure [ f : pxpypz ] illustrates the time variation of slope angle at the poles during the encounter . the results derived from all @xmath92 trajectories in fig . [ f : traject ] are shown and to the same scale . the most perturbed pole is p@xmath101 on the medium axis , which gains the largest slope angle change for most cases , except in trajectories \{3,4,9,10}. it is notable that trajectories in the same plane always share similar variation at the three poles , such as \{1,2,5,6 } , \{3,4,9,10 } , and \{7,8,11,12}. the curves at p@xmath100 and p@xmath101 show similar doublet shapes since those poles are both located in the rotational plane . the variation at p@xmath102 is more complicated due to the fact that the spin pole is not perpendicular to the orbit plane . figure [ f : pxpypz ] . figure [ f : pxpypz ] can be used to estimate the changes in the mechanical environment at these pole locations . since the slope angle change is the primary mechanism to drive avalanches on the sandpiles , the results derived from different trajectories serve as a framework to determine the magnitude of tidal perturbation at the candidate locations and locations in between , which turns out to be quite small for the apophis encounter ( less than about @xmath103 in slope for these cases ) . as analyzed above , catastrophic avalanches solely due to external forces acting directly on the surface particles may never occur on apophis during the 2029 encounter since the tidal perturbation will be very weak , however it might still have the potential to trigger some local tiny landslides of the surface materials . generally , the regolith experiences more activity than the constituents deep in the asteroid due to the dynamical environment on the surface , including the microgravity , maximum tidal and centrifugal acceleration , and smaller damping forces from the surroundings due to the smaller confining pressure , therefore we conclude that resurfacing due to external perturbations should occur before wholesale reshaping during the encounter ( discussed in section [ s : reshpeff ] ) . in the following sections we detail our numerical examination of the local sandpiles for the predicted apophis encounter scenario to estimate the limit and magnitude of the material responses in 2029 . one problem that must be confronted in the numerical simulations is that the soft - sphere sandpiles exhibit slow outward spreading due to the accumulation of velocity noise , and this slow spreading may eventually lead to a collapse of the whole sandpile . unfortunately , our integration time required is long ( hours ) for a granular system , thus the numerical noise has to be well limited in our method . two techniques are used in this study . first , as described in section [ s : initialcondition ] , a rough pallet made up of closely packed spheres in a disk fixed to the ground is used to reduce spreading of the bottom particles . second , we introduce a critical speed in the code , below which all motions are considered to be noise and are forced to zero . we found a speed threshold @xmath104 m / s by launching hours - long simulations of a static sandpile for different material properties and finding the corresponding minimum value of the critical speed that limits the numerical spreading . simulations of sandpiles without a flyby were carried out first , serving as a reference for subsequent flyby simulations . sandpiles in this situation were confirmed to stay equilibrated for a long period , which suggests the avalanches in our numerical experiments ( if any ) would be attributed entirely to the effects of the tidal encounter . we performed local simulations with the sandpiles generated in section [ s : initialcondition ] to consider the response of different materials to the apophis encounter . the local frames at the three poles p@xmath100 , p@xmath101 and p@xmath102 ( fig . [ f : traject ] ) were used to position the sample sandpiles , and the flyby simulation data derived from the @xmath92 fiducial trajectories were employed as the source of external perturbations for the local simulations . there are @xmath105 possible combinations in total for different materials , locations , and trajectory orientations , covering a wide range of these undetermined parameters . in all the simulations , the start distance of apophis from the center of the planet was set to @xmath106 @xmath52 to ensure the sandpiles were equilibrated fully before it approached the perigee . for our study , we concentrated on the connections between the sandpile s responses and significant variables , e.g. , spin orientation and sandpile locations , by which we will get a better sense of the surface processes due to a tidal encounter . the sandpiles located at pole p@xmath101 for trajectory 2 ( fig . [ f : traject ] ) were examined in detail , which is the case suffering the strongest tidal perturbation ( fig . [ f : pxpypz ] ) . figure [ f : avalanches ] illustrates the disturbances on sandpiles of three different materials for this configuration . each panel includes a snapshot of the sandpile after the tidal encounter with the displaced particles highlighted and a diagram showing the time variations of the total kinetic energy with planetary distance . intensive events can be identified by noting the peaks of the kinetic energy curves , and the particles involved in these events are marked with different colors to show the correspondence between the disturbed site and occurrence time . figure [ f : avalanches ] . the scale of the disturbances proves to be tiny even for the strongest " case in fig . [ f : avalanches ] . as illustrated , the displaced particles are few in number and mostly distributed on the surface of the sandpile . we found that the maximum displacement of a given particle is less than @xmath107 times its radius ; that is , these disturbances only resulted from the collapse of some local weak structures and the small chain reaction among the surrounding particles . moreover , the gravel " sandpile proves to be unaffected by the tidal encounter , because the static friction is quite large ( table [ t : ssdemparams ] ) and all particles are locked in a stable configuration , in which case the total kinetic energy of the sandpile remains very small ( @xmath108 j ) . the glass beads " sandpile suffered small but concentrated disturbances that involved very few particles . accordingly , the displacements of these particles are relatively large . smooth " sandpile presents near - fuild properties with many surface particles experiencing small - amplitude sloshing . the motion eventually damps out and the displacements of the involved particles are tiny ( smaller than that of glass beads " ) . based on the detailed analysis of this representative scenario , we performed simulations at other poles and at other orientations , constructing a database to reveal any connections between disturbances and these parameters . however , the results show little association between these two : for the gravel " sandpile , no disturbances were detected for any location and any orientation due to the large @xmath109 ; for the glass beads " sandpile , some small - scale avalanches can always be triggered , while the occurrence ( site and time ) of these avalanches seems to be widely distributed and independent of location and orientation ; and for the smooth " sandpile , surface particles can feel the tidal perturbation and show slight sloshing , but no visible avalanches eventually resulted , because the initial low slope angle in this case imposes a relatively stable structure that is always able to recover from the external perturbations . to illustrate the effect of stronger perturbations on our model sandpiles , we carried out a few simulations of closer approaches , specifically at @xmath16 and @xmath17 @xmath52 , for the same encounter speed of @xmath53 km / s . these scenarios , though unlikely to occur in 2029 based on current understanding of apophis orbit , are presented here to illustrate the magnitudes of tidal resurfacing effects over a wider range of perturbation strengths . flyby simulations of rubble piles were first carried out to quantify the reshaping effects , using the monodisperse and bidisperse particle rubble - pile models ( see section [ s : reshpeff ] ) . the bulk densities for both models were set to @xmath110 g / cc . table [ t : reshp24 ] presents the relative net changes of shape factor @xmath65 for these rubble piles of different materials at different perigee distances . table [ t : reshp24 ] . table [ t : reshp24 ] shows results consistent with section [ s : reshpeff ] , namely that the bimodal rubble pile shows greater fluidity and larger shape changes during the encounter . to be more specific , the magnitude of reshaping effects at perigee distance @xmath16 @xmath52 ( still larger than the roche limit of @xmath111 @xmath52 for a fluid body ) remains small ( @xmath112 ) , and that at perigee distance @xmath17 @xmath52 ( smaller than the roche limit ) achieves a significant level of @xmath113@xmath77 . it is notable that the rubble piles did not experience any irreversible distortion even for an approach distance as close as @xmath17 @xmath52 , because the duration of strong tidal effects is relatively short ( see section [ s : reshpeff ] ) . anyway , in this section we still adopt the assumption of rigidity to measure the quantities required by the local simulations , which is acceptable since these scenarios are fictitious and only designed to exhibit some massive resurfacing effects . for the same reason , in this section we do not present systematic local simulations for different sandpile locations and orientations as done for the real encounter scenario ; instead , we simply place the sample sandpiles ( see fig . [ f : sandpile ] ) at the pole of the body long axis p@xmath100 , and choose trajectory @xmath94 ( see fig . [ f : traject ] ) as the encounter trajectory to give rise to the maximum tidal effects . figures [ f : refrsh4f ] and [ f : refrsh2f ] illustrate the responses of the three sandpiles for perigee distance @xmath16 @xmath52 and @xmath17 @xmath52 , respectively . each panel includes snapshots of the sandpile before and after the tidal encounter , and the overall shape of the sandpile is traced with a white border for emphasis . diagrams showing the time variation of the total kinetic energy with planetary distance are also included , in which the intensive avalanches can be identified by noting the peaks of the kinetic energy curves ( note the different vertical scales ) . figure [ f : refrsh4f ] . [ f : refrsh2f ] . as illustrated in fig . [ f : refrsh4f ] , the shape changes of the three sandpiles are still small ( but visible ) for the encounter at @xmath16 @xmath52 perigee , and involve many more particles than the encounter at @xmath3 @xmath52 ( fig . [ f : avalanches ] ) . and accordingly , the magnitudes of the total kinetic energy increase consistently for the three sandpiles of different materials , except for the gravel case at @xmath16 @xmath52 , where it appears the frictional `` lock '' established when the pile was first created is disturbed enough to cause a stronger distortion than might otherwise be expected , causing a sharp spike in the kinetic energy plot . essentially , the `` gravel '' sandpile experienced a more significant collapse than the `` glass bead '' sandpile . as stated in section [ s : initialcondition ] , the three sandpiles were constructed in a manner that allowed for some inherent randomness , thus their energy state and strength may differ to some degree , and appearently the `` gravel '' sandpile was closer to its failure limit than the other two . this adds an extra element of stochasticity to the results , an aspect to be explored in future work . figure [ f : refrsh2f ] shows the results from the encounter at @xmath17 @xmath52 perigee , for which the encounter trajectory partly entered the roche sphere ( @xmath111 @xmath52 ) . in this case , the shapes of the three sandpiles are highly distorted during the encounter , with the involved particles slumped towards the direction where the planet is receding . the corresponding changes in total kinetic energy become extremely large when the massive avalanches occur , especially for the `` smooth '' particles . the results of this section suggest that a @xmath16 @xmath52 encounter may alter the regolith on apophis surface slightly , and a @xmath17 @xmath52 encounter may produce a strong resurfacing effect ( of course , we would expect considerable global distortion as well , if the asteroid is a rubble pile ) . the shape changes of the sandpiles depend on the orientation of the encounter trajectory ; i.e. , particles can be dragged away along the direction in which the planet recedes .

the potential shape modification of the object due to the tidal encounter is first confirmed to be negligibly small with systematic simulations , thus only the external perturbations are taken into account for this work ( despite this , seismic shaking induced by shifting blocks might still play a weak role and we will look into this mechanism in future work ) . a sandpile constructed in the local frame is then used to approximate the regolith materials ; all the forces the sandpile feels during the encounter are imposed as external perturbations to mimic the regolith s behavior in the full scenario . we find that catastrophic avalanches of regolith materials may not occur during the 2029 encounter due to the small level of tidal perturbation , although slight landslides might still be triggered in positions where a sandpile s structure is weak . simulations are performed at different locations on apophis surface and with different body- and spin - axis orientations ; the results show that the small - scale avalanches are widely distributed and manifest independently of the asteroid orientation and the sandpile location .

asteroid ( 99942 ) apophis close approach in 2029 will be one of the most significant small - body encounter events in the near future and offers a good opportunity for exploration to determine the asteroid s surface properties and measure any tidal effects that might alter its regolith configuration . resurfacing mechanics has become a new focus for asteroid researchers due to its important implications for interpreting surface observations , including space weathering effects . this paper provides a prediction for the tidal effects during the 2029 encounter , with an emphasis on whether surface refreshing due to regolith movement will occur . the potential shape modification of the object due to the tidal encounter is first confirmed to be negligibly small with systematic simulations , thus only the external perturbations are taken into account for this work ( despite this , seismic shaking induced by shifting blocks might still play a weak role and we will look into this mechanism in future work ) . a two - stage approach is developed to model the responses of asteroid surface particles ( the regolith ) based on the soft - sphere implementation of the parallel-body gravity tree code ` pkdgrav ` . a full - body model of apophis is sent past the earth on the predicted trajectory to generate the data of all forces acting at a target point on the surface . a sandpile constructed in the local frame is then used to approximate the regolith materials ; all the forces the sandpile feels during the encounter are imposed as external perturbations to mimic the regolith s behavior in the full scenario . the local mechanical environment on the asteroid surface is represented in detail , leading to an estimation of the change in global surface environment due to the encounter . typical patterns of perturbation are presented that depend on the asteroid orientation and sense of rotation at perigee . we find that catastrophic avalanches of regolith materials may not occur during the 2029 encounter due to the small level of tidal perturbation , although slight landslides might still be triggered in positions where a sandpile s structure is weak . simulations are performed at different locations on apophis surface and with different body- and spin - axis orientations ; the results show that the small - scale avalanches are widely distributed and manifest independently of the asteroid orientation and the sandpile location . we also include simulation results of much closer encounters of the apophis with earth than what is predicted to occur in 2029 , showing that much more drastic resurfacing takes place in these cases . * keywords * : asteroids , dynamics ; asteroids , surfaces ; near - earth objects ; regoliths ; tides , solid body

1408.0168

c

this paper provides a numerically derived prediction for the surface effects on ( 99942 ) apophis during its 2029 approach , which is likely to be one of the most significant asteroid encounter events in the near future . a two - stage scheme was developed based on the soft - sphere code implementation in ` pkdgrav ` to mimic both a rubble pile s ( rigid and flexible ) responses to a planetary flyby and a sandpile s responses to all forms of perturbations induced by the encounter . the flyby simulations with the rubble pile indicate that reshaping effects due to the tidal force on apophis in 2029 will be negligibly small for bulk densities in the expected range ( @xmath78@xmath79 @xmath12 ) . the resultant environmental force felt by the sandpile on the asteroid surface was approximated with a uniform analytical expression , which led to an estimate of the changes in the global mechanical environment . three typical patterns of perturbation were presented based on the asteroid body and spin orientation at perigee , showing a general dependence of the magnitude of tidal perturbation on the orientation of the trajectory . twelve fiducial trajectories were used to calculate the magnitude of the tidal perturbation at three poles of the tri - axial ellipsoid model , indicating that the strongest tidal perturbation appears where the local slope is originally steep and that the duration of the strong perturbation is short compared with the whole process . the tidal perturbation on surface materials is confirmed to be quite weak for the 2029 encounter , therefore large - scale avalanches may never occur . however , we showed this weak perturbation does trigger some local tiny landslides on the sample sandpiles , though the involved particles are few in number and are distributed on the surface of the sandpile . these small - scale avalanches result from the breaking of weak substructures by slight external perturbations , therefore the occurrence of these local avalanches is widely distributed and presents little dependence on the encounter parameters . the simulations of closer approaches show that an encounter at @xmath17 @xmath52 is capable of triggering some massive avalanches of the sandpiles , i.e. , to alter the regolith on apophis surface significantly ( although the entire body would also undergo significant shape change in this case ) . further research will be performed to generalize our work over a wide range of possible asteroid - planet encounter conditions . and as stated above , we will also investigate whether even slight internal perturbations in the asteroid during tidal encounters may contribute to noticeable surface motions .

this paper provides a prediction for the tidal effects during the 2029 encounter , with an emphasis on whether surface refreshing due to regolith movement will occur . the local mechanical environment on the asteroid surface is represented in detail , leading to an estimation of the change in global surface environment due to the encounter . typical patterns of perturbation are presented that depend on the asteroid orientation and sense of rotation at perigee .

asteroid ( 99942 ) apophis close approach in 2029 will be one of the most significant small - body encounter events in the near future and offers a good opportunity for exploration to determine the asteroid s surface properties and measure any tidal effects that might alter its regolith configuration . resurfacing mechanics has become a new focus for asteroid researchers due to its important implications for interpreting surface observations , including space weathering effects . this paper provides a prediction for the tidal effects during the 2029 encounter , with an emphasis on whether surface refreshing due to regolith movement will occur . the potential shape modification of the object due to the tidal encounter is first confirmed to be negligibly small with systematic simulations , thus only the external perturbations are taken into account for this work ( despite this , seismic shaking induced by shifting blocks might still play a weak role and we will look into this mechanism in future work ) . a two - stage approach is developed to model the responses of asteroid surface particles ( the regolith ) based on the soft - sphere implementation of the parallel-body gravity tree code ` pkdgrav ` . a full - body model of apophis is sent past the earth on the predicted trajectory to generate the data of all forces acting at a target point on the surface . a sandpile constructed in the local frame is then used to approximate the regolith materials ; all the forces the sandpile feels during the encounter are imposed as external perturbations to mimic the regolith s behavior in the full scenario . the local mechanical environment on the asteroid surface is represented in detail , leading to an estimation of the change in global surface environment due to the encounter . typical patterns of perturbation are presented that depend on the asteroid orientation and sense of rotation at perigee . we find that catastrophic avalanches of regolith materials may not occur during the 2029 encounter due to the small level of tidal perturbation , although slight landslides might still be triggered in positions where a sandpile s structure is weak . simulations are performed at different locations on apophis surface and with different body- and spin - axis orientations ; the results show that the small - scale avalanches are widely distributed and manifest independently of the asteroid orientation and the sandpile location . we also include simulation results of much closer encounters of the apophis with earth than what is predicted to occur in 2029 , showing that much more drastic resurfacing takes place in these cases . * keywords * : asteroids , dynamics ; asteroids , surfaces ; near - earth objects ; regoliths ; tides , solid body

1610.05810

i

topology has played an increasingly important role in condensed matter physics . an early application involved using topological chern - simons theory in quantum hall systems @xcite . more recently , topological ideas applied to brillouin zone ( bz ) have played a significant role in classifying topological phases of matter ( for example , see @xcite and references therein ) . whereas the former application involved a dynamical gauge field in physical spacetime , the latter cases were formulated in the momentum space and involved topological aspects of static gauge fields such as the berry connection . it is thus natural to ask : can dynamical gauge fields naturally occur in a condensed matter system ? the aim of this paper is to argue that this can be done at least in the context of three - dimensional ( 3d ) time reversal invariant ( tri ) superconductors , where ( a slightly modified version of ) berry s gauge field can be viewed as dynamical in the brillouin zone and where the action of the theory is the chern - simons action . in this class of theories , the fluctuations of the berry connection are induced from quantum fluctuations of the superconductor s pairing amplitude . more interestingly , 3d tri superconductors are known to be capable of having gapless nodal lines in the bz @xcite , and we will show that they play exactly the role of wilson loop observables in the chern - simons theory . in nature , a large class of materials belongs to 3d tri superconductors , which includes most of the conventional s - wave superconductors described by the bardeen - cooper - schrieffer theory , and a large number of unconventional superconductors such as many heavy fermion superconductors , cuprates , iron - based superconductors and the 3d topological superconductors . in particular , unconventional superconductors often exhibit a highly anisotropic pairing amplitude such as p - wave , d - wave or their hybridizations with s - wave , and have much richer phenomena in experiments @xcite . previous studies have shown from different perspectives the significance of topology in these tri unconventional superconductors . gapped 3d tri superconductors are shown to have a @xmath0 topological classification @xcite , which fit in the generic k - theory classification framework of gapped topological phases @xcite . topological superconductors are defined as such superconductors with a nonzero topological number ( e.g. the @xmath1he - b phase ) , and are shown to support gapless topological majorana fermions on the surfaces @xcite . on the other hand , the gapless 3d tri superconductors contain gapless nodal lines in the bz that are allowed by the time - reversal symmetry , with known examples such as the heavy fermion superconductor cept@xmath2si and the cuprates @xcite . several recent studies show that topological numbers can also be defined for these nodal - line superconductors in terms of k - theory @xcite , which give rise to various types of topological majorana surface states @xcite . in addition , in analogy to 3d weyl semimetals which can be viewed as intermediate phases between 3d topological insulators and trivial insulators @xcite , nodal - line superconductors can also arise as gapless intermediate phases between conventional tri and topological tri superconductor phases @xcite . these different while related facts strongly indicate the existence of a unified topological field theory that describes both gapped and gapless 3d tri topological superconductors . this also motivates us to consider the dynamical chern - simons theory in the 3d bz for these superconductors . there is already a hint that a topological field theory in bz can be physically relevant . in particular , it has been shown @xcite that for an insulator in odd spatial dimensions , the value of the chern - simons ( cs ) action for berry connection of the filled bands in the bz computes the effective theta angle of the corresponding @xmath3 term ( or a gravitational analog proportional to @xmath4 in the 3d case @xcite ) , where @xmath5 is the electromagnetic field in the physical spacetime . in other words , there is a coupling of the form @xmath6\times \left[\int_{\mathbb r^{2n } } f^n\right]\ .\]]we will be specializing to the case of @xmath7 , i.e. , 3d space in this paper . ( a similar term can be used to compute the gravitational response involving @xmath8 ) . note that the cs action , which is an angular quantity and has a shift ambiguity , has the correct structure to be the coefficient in front of @xmath9 , which needs to be defined only up to shift symmetry . in this context , it is very natural to ask whether the berry connection @xmath10 can fluctuate . in particular , we can imagine having in physical spacetime a pulse where @xmath11 , which leads to an effective level of @xmath12 for the cs theory in the bz . can the berry connection behave as if it were dynamical ? for the answer to be yes , the classical background for @xmath10 must be flat , as is demanded by the cs equations of motion . this is certainly not the case in general . however , as we shall see , it is the case for tri superconductors . with this encouraging result , one may then ask whether there are natural objects in the bz corresponding to wilson loops of the cs theory . indeed , for tri superconductors the symmetries allow gapless nodal lines in the bz , which we will see end up playing the role of wilson lines for the cs theory . for superconductors , the relevant term to compute is the gravitational @xmath13 term . but with the gapless modes , as would be the case if we have nodal lines , the theta angle is ambiguous . it turns out the choices of resolving this ambiguity by introducing infinitesimal time reversal breaking perturbations to get rid of gapless modes are in 1 - 1 correspondence with allowed basic charges for the cs theory ! we thus find that dressing up the nodal lines with this data gives an unambiguous theta angle for the coefficient of @xmath4 term , which is identified with the free energy of cs theory in the presence of wilson loops and leads to physically measurable effects . in particular , as we shall see , when the nodal lines change from linked to unlinked , the theta angle changes in units of @xmath14 . our main discussion is in the context of the abelian chern - simons theory , but we also indicate briefly how the theory gets extended to non - abelian case , and how in particular the @xmath15 case can be potentially realized in experiments . we will also connect aspects of our discussions with topological strings @xcite which is formulated in a 6-dimensional symplectic manifold ( which is typically taken to be a calabi - yau 3-fold ) . in our context , the phase space which is @xmath16 turns out to play the role of this 6d symplectic space . in the topological string setup , cs theory lives on 3d lagrangian defects ( ` d - branes ' ) . if they are oriented along the bz @xmath17 , they give rise to the cs theory we find in the bz . in this context , wilson loop observables arise from a pair of such 3d defects intersecting along a loop @xcite . we will show that line defects in physical space lead to such 3d lagrangian branes and can also give rise to wilson loop observables ( nodal lines ) in the bz . the organization of this paper is as follows . in section ii , we review the hamiltonian of tri superconductors and define the modified berry connection . in section iii , we relate the topological @xmath18 angle to a chern - simons term . section iv incorporates the nodal lines as wilson loops , first in minimal model and then in multi - band system . in section v , we consider non - abelian nodal lines , first in a @xmath15 example and then generalize to the @xmath19 example . in section vi we discuss connections with topological strings and show how line defects in physical space also lead to effective wilson loops in the bz . we present our conclusions in section vii , and in the appendix are some computational details .

berry connection is conventionally defined as a static gauge field in the brillouin zone . here we show that for three - dimensional ( 3d ) time - reversal invariant superconductors , a generalized berry gauge field behaves as a dynamical fluctuating field of a chern - simons gauge theory . the gapless nodal lines in the momentum space play the role of wilson loop observables , while their linking and knot invariants modify the gravitational theta angle .

berry connection is conventionally defined as a static gauge field in the brillouin zone . here we show that for three - dimensional ( 3d ) time - reversal invariant superconductors , a generalized berry gauge field behaves as a dynamical fluctuating field of a chern - simons gauge theory . the gapless nodal lines in the momentum space play the role of wilson loop observables , while their linking and knot invariants modify the gravitational theta angle . this angle induces a topological gravitomagnetoelectric effect where a temperature gradient induces a rotational energy flow . we also show how topological strings may be realized in the 6 dimensional phase space , where the physical space defects play the role of topological d - branes .

math0506242

i

reduced decompositions of permutations are classical objects in combinatorics that appear throughout the literature . following the work of rodica simion and frank schmidt in @xcite , the study of permutation patterns , particularly pattern avoidance , has become a frequently studied field as well . in @xcite , sara billey , william jockusch , and richard stanley relate these two concepts , possibly for the first time . there they show that @xmath0-avoiding permutations are exactly those permutations where the subsequence @xmath1 never occurs in a reduced decomposition . relatedly , victor reiner shows in @xcite that the number of @xmath2 occurrences in reduced decompositions of the longest element in the symmetric group , which has the maximal number of occurrences of @xmath0 , is equal to the number of such reduced decompositions . stanley had previously shown that this is the number of standard young tableaux of a staircase shape in @xcite . inspired by these results , and more generally by the relationship they suggest between the two aspects of permutations , this paper studies elements of the symmetric group from the combined perspectives of their reduced decompositions and their patterns . while these aspects of a permutation appear extensively in combinatorial literature , they are not often treated together . this paper strives to remedy that fact , addressing several questions where reduced decompositions and permutation patterns together lead to interesting results . after introducing basic terminology and notation in section [ section defs ] , section [ section vexillary ] generalizes the result of billey , jockusch , and stanley , via a new characterization of vexillary permutations in theorem [ vexthm ] . this characterization is based on the reduced decompositions of the permutations _ containing _ the permutation in question , and is strikingly different from all previous equivalent characterizations . in addition to requiring that each of the permutations containing the vexillary permutation has a certain kind of reduced decomposition , the proof of theorem [ vexthm ] explicitly constructs such a reduced decomposition . there are three algorithms which appear in this paper , the first of which occurs in the proof of theorem [ vexthm ] . it should be noted that these are not deterministic , and include a certain amount of choice . for instance , example [ vex ex ] describes only one possible route that the algorithm may take on a particular input . there is an equivalence relation , sometimes known as the commutation relation , on the set of reduced decompositions of a particular permutation . this and an associated graph are discussed in section [ section eq ] . theorem [ 1 lbm ] and corollary [ 1 ulbm ] characterize permutations with graphs and commutation classes having certain properties . these results are strengthened in theorem [ chainthm ] . the results in sections [ section polygon ] and [ section poset ] discuss permutation patterns with respect to a polygon defined by serge elnitsky in @xcite . the rhombic tilings of this polygon are in bijection with the commutation classes of a permutation . new results include that the number of commutation classes of a permutation is monotonically increasing with respect to pattern containment ( theorem [ monotone ] ) , and several results pertaining to a poset associated with tilings of the polygon . finally , section [ section fb ] completely describes this poset in the case of a freely braided permutation , as defined by richard green and jozsef losonczy in @xcite and @xcite .

this paper generalizes that result with a detailed study of permutations via their reduced decompositions and the notion of pattern containment . additionally , the combined frameworks yield several new results about the commutation classes of a permutation . in particular , these describe structural aspects of the corresponding graph of the classes and the zonotopal tilings of a polygon defined by elnitsky that is associated with the permutation .

billey , jockusch , and stanley characterized-avoiding permutations by a property of their reduced decompositions . this paper generalizes that result with a detailed study of permutations via their reduced decompositions and the notion of pattern containment . these techniques are used to prove a new characterization of vexillary permutations in terms of their principal dual order ideals in a particular poset . additionally , the combined frameworks yield several new results about the commutation classes of a permutation . in particular , these describe structural aspects of the corresponding graph of the classes and the zonotopal tilings of a polygon defined by elnitsky that is associated with the permutation .

math-ph0401054

i

in recent years there has been an increasing interest in the geometric treatment of non - holonomic mechanical systems , see , e.g. , @xcite . in particular , it has been recognised that the hamiltonian formulation of such systems can be stated in terms of an _ almost - poisson _ bracket , that is , a biderivation of functions of phase space , antisymmetric in its arguments but which does not necessarily fulfil the jacobi identity ( see , e.g. , @xcite ) . therefore , for researchers in this field , it seems to be usual the conceptual association of the hamiltonian formulation of non - holonomic mechanical systems with almost - poisson structures . on the other hand , there exist non - holonomic systems which , after certain reductions are performed , admit a hamiltonian formulation after a rescaling of time is carried out , by means of rescaling factors ( sometimes called _ invariant measures _ ) of the reduced vector field of the system . this is the case for the so - called lr systems , which are systems formulated on compact lie groups endowed with a left - invariant metric and right - invariant non - holonomic constraints . after a rescaling of time , their corresponding reduced systems become integrable hamiltonian systems describing geodesic flows on unit spheres @xcite . in @xcite , a necessary and sufficient condition for the existence of an invariant measure for the reduced dynamics of generalized chaplygin systems of mechanical type another recent work on this line is @xcite . for a classic treatment of the theory of chaplygin s reducing multiplier , see section iii-12 of @xcite . thus , it could be conceptually associated as well the existence of specific rescaling factors for these reduced systems with the possibility of formulating them in a hamiltonian way . in addition , borisov , mamaev and kilin @xcite have recently found a poisson structure for each studied case of reduced non - holonomic systems , such that the reduced system becomes hamiltonian , with respect to such a structure , after a rescaling , the hamiltonian function being the reduced energy . the examples treated by them are classical in the literature , consisting mainly of rolling bodies without slipping , namely a rigid body of revolution rolling on a plane , in particular the _ routh s sphere _ ( see section [ sect_routh_sphere ] ) , the rolling disk ( to be treated in section [ sect_rolling_disk ] ) , the motion of a homogeneous ball on a surface of revolution ( called sometimes _ routh s problem _ , see , e.g. , @xcite ) , and other cases . there is a strong emphasis in these references in the sense that the poisson structure for each case can be found _ after a rescaling of time _ of the reduced vector field . our primary motivation for this work was to understand the origin of the two integrals of motion appearing in the mentioned problem of a ball rolling without slipping inside a surface of revolution , which are not given , in general , in an explicit form but being related to the solutions of a system of first order non - autonomous differential equations @xcite . this also happens in the other mentioned cases . the results of @xcite suggest that such systems can be interpreted as the equations providing a set of functionally independent casimir functions of the poisson structure they find for each specific case . therefore , it seemed to be worth investigating further such poisson structures , in particular to clarify their domain of definition and basic properties . let us note that another recent approach , devoted to the study of poisson structures which can be associated to never vanishing vector fields on manifolds of arbitrary dimension @xmath0 , with _ fibrating periodic flows _ , is given in @xcite . it follows that the previously mentioned poisson structures have a rather peculiar form . in particular , the associated characteristic distributions have rank two in the open sets of the reduced spaces considered in @xcite . this property implies that such poisson structures , when multiplied by a never vanishing function , are again poisson structures of the same type . the immediate consequence is that the above mentioned reduced non - holonomic systems are already hamiltonian with respect to one of these poisson structures without any need of rescaling . other interesting result is that , in the cases studied , the poisson structures obtained can be extended from their original domains of definition , namely ( open sets of ) semialgebraic subvarieties of @xmath1 , to an open set of the ambient space . such extended poisson structures become zero only at the so - called _ singular equilibria _ of the reduced systems . moreover , the existence of these ( extended ) poison structures , from an algebraic point of view , is only caused by the existence of integrals of motion of the reduced vector field related to the solutions of the mentioned systems of first order differential equations . this paper is organized as follows . in section [ pois_str_rank2 ] we briefly review some notions of poisson geometry and in particular , of poisson structures of rank two . in section [ pois_str_rank2 ] we show the explicit expressions of certain bivectors in @xmath2 and @xmath1 , determined up to a non - vanishing factor function , by choosing the 1-forms in their kernels to have a specific form , and we prove that they are in fact poisson bivectors of rank two . section [ sect_of_examples ] is devoted to show the application of the previous results in specific examples , namely , the rolling disk , the routh s sphere , and the ball rolling on a surface of revolution . we will use the formulation of @xcite , @xcite and @xcite , respectively , of these problems , rather than that of @xcite . however , we point out the equivalence of both treatments in the last case . we also treat the special case of a ball rolling inside a cylinder . finally , we end with some conclusions and an outlook for further research .

borisov , mamaev and kilin have recently found certain poisson structures with respect to which the reduced and rescaled systems of certain non - holonomic problems , involving rolling bodies without slipping , become hamiltonian , the hamiltonian function being the reduced energy . we show that they are determined , up to a non - vanishing factor function , by the existence of a system of first - order differential equations providing two integrals of motion .

borisov , mamaev and kilin have recently found certain poisson structures with respect to which the reduced and rescaled systems of certain non - holonomic problems , involving rolling bodies without slipping , become hamiltonian , the hamiltonian function being the reduced energy . we study further the algebraic origin of these poisson structures , showing that they are of rank two and therefore the mentioned rescaling is not necessary . we show that they are determined , up to a non - vanishing factor function , by the existence of a system of first - order differential equations providing two integrals of motion . we generalize the form of that poisson structures and extend their domain of definition . we apply the theory to the rolling disk , the routh s sphere , the ball rolling on a surface of revolution , and its special case of a ball rolling inside a cylinder . # 1#2 ( # 1 # 2 ) # 1#2

astro-ph0312315

c

that significant and correlated star - to - star differences in c and n , as well as o , na , al , and mg have been found among the sgb , msto , and ms stars of several clusters ( see references above ) , implies the operation of some process external to the present stars , presumably having taken place early in the cluster history . the discussion of @xcite includes a comprehensive look at various possibilities . it is worth while however , to revisit a few of the more critical constraints on any theory of the origin of the abundance variations . first , whatever mechanism has altered the light - element compositions of the cluster stars has left the heavy elements essentially untouched , at least to the limits of our ability to determine them the analysis of m5 by @xcite is an excellent example . this alone would seem to exclude the possibility of the light - element variations arising from the merger of two distinct proto - cluster clouds ( as has been pointed out by numerous authors ) . second , these abundance variations appear to be almost ubiquitous among the population of galactic globular clusters . to highlight this , we have plotted in figure [ fig_6 ] the [ c / fe ] and [ n / fe ] values for the present sample of m13 msto stars , the 47 tuc ms stars of @xcite , the m5 sgb stars of @xcite , and the msto stars of m71 from bc01 . note that bc01 did not directly extract c and n abundances from their observed indices we have converted them here following the procedure outlined in @xcite and using the indices and models presented in bc01 ; the values are given in appendix a , table [ tbl - a3 ] . third , the elements which are observed to vary are associated with proton capture nucleosynthesis under conditions of cn and on - cycling . the source / site must process these cno - group elements and return this material to the cluster to be incorporated into the present population of low mass stars either before , during , or after their formation . a popular model which fits these constraints is the incorporation of ejecta from intermediate mass ( 3 - 6@xmath21 ) agb stars undergoing hot bottom burning and third dredge - up ( see * ? ? ? * ) , although difficulties such as the establishment of an o - na anti - correlation remain ( see for example * ? ? ? however , as is discussed in @xcite and bcs02 , the quantities of material required to produce the observed star - to - star differences among the low luminosity stars ( most notably extreme c depletions ) , which are clearly not diluted as the convective envelopes deepen during rgb ascent , rules out any sort of simple accretion model . indeed , for the present m13 stars , roughly 70% of a c - poor ms star s total mass must be captured ejecta if the accreted matter is completely free of c ( see bcs02 ) . it is of course unclear how such an enormous amount of material can be returned to the cluster without appealing to a shallow initial mass function ( see * ? ? ? * ) , nor how the present stars can sweep up the necessary mass of ejecta . we note in figure [ fig_6 ] that the depletions in c do appear smaller in the more metal - rich clusters m71 and 47 tuc in accord with the prediction the of agb ejecta models of @xcite . yet at the same time , if one presumes the highest [ c / fe ] sgb / msto stars in m13 and m5 to represent the original ( accretion free ) c abundance of the cluster stars , they are still some 0.4 dex more c - poor than their 47 tuc / m71 counterparts , implying either truly primordial ( i.e. , pre - accretion ) differences in at least c or that nearly all the present stars in m13 and m5 have undergone at least some accretion of c - poor material . however , the spread in [ n / fe ] is essentially identical among all four clusters . clearly , knowledge of the patterns of [ o / fe ] and [ na / fe ] among the present stars would help constrain the agb ejecta theories . an interesting counterpoint to this model is the scenario suggested by @xcite and @xcite to explain similar results among m92 and m15 sgb stars that the stars of these clusters were inhomogeneously `` polluted '' by an injection of raw c from intermediate mass agb stars which is subsequently converted into n in the present stars before sgb evolution thereby explaining both the c deficiencies and large n enhancements as well as star - to - star differences in ( c+n ) . this has the additional advantage of requiring considerably more modest composition modifications ( a factor of 4 or so in c from star to star ) , which in turn lowers the mass of captured ejecta required . however , to explain the large c depletions already in place by the msto , significant processing of the envelope through a region of cn - cycling must have taken place while the stars occupied the ms . one then returns to the difficulty of mixing in such stars discussed above . another site of the proton - capture reactions has recently been suggested by @xcite , who note that the high mass ( 250 - 300 @xmath21 ) zero metallicity models of @xcite tend to mix he and he - burning products into their h - burning shells during the later stages of he - burning . this fresh c , n , o is partially processed into n while at the same time , the stars expand into red supergiants . if mass loss also occurs at this point , the cluster could be seeded with freshly produced c / o - poor , n - rich material . such a scenario is presented within the context of the cluster formation history of that the gcs formed from primordial material ( zero - metal ) that was subsequently enriched by the supernovae of massive stars before low mass stars could form . however , the problem remains that the production / seeding and mixing of the heavy - elements must be decoupled from that of the light - elements in order to explain the remarkable homogeneity of fe , ti , ca , etc . within the gcs . in the context of gc formation in a well mixed supershell ( e.g. * ? ? ? * ) this is difficult to explain if the cno - modified material is ejected prior to the driving supernovae and subsequent supershell expansion / mixing . the entire keck / lris user community owes a huge debt to jerry nelson , gerry smith , bev oke , and many other people who have worked to make the keck telescope and lris a reality . we are grateful to the w. m. keck foundation , and particularly its late president , howard keck , for the vision to fund the construction of the w. m. keck observatory . we also wish to express our thanks to roger bell whose ssg code was instrumental in this project and the anonymous referee for their suggestions . partial support was provided by the national science foundation under grant ast-0098489 to mmb and grant ast-0205951 to jgc and by the f. john barlow professorship and uw oshkosh faculty development program ( mmb ) .

we presume these differences to be the result of some process operating early in the cluster history . a comparison of c abundances derived using the present methods and those from the literature yield a mean difference of 0.03.14 dex for four stars in common with and 0.14.07 dex for stars also observed by ( if one extreme case is removed ) . we conclude that the lower surface c abundances of these luminous giants as compared to the subgiants and main - sequence stars are likely the result of mixing rather than a difference in our abundance scales . finally , the most recent c and n abundances for main - sequence , main - sequence turn - off , and subgiant stars in 47 tuc , m71 , m5 , and the present m13 data are compared .

building upon earlier observations which demonstrate substantial star - to - star differences in the carbon abundances of m13 subgiants , we present new keck lris spectra reaching more that 1.5 mag below the m13 main - sequence turn - off ( to v 20 ) . our analysis reveals a distribution of c abundances similar to that found among the subgiants , implying little change in the compositions of the m13 stars at least through the main - sequence turn - off . we presume these differences to be the result of some process operating early in the cluster history . additional spectra of previously studied bright m13 giants have been obtained with the hale 5-m . a comparison of c abundances derived using the present methods and those from the literature yield a mean difference of 0.03.14 dex for four stars in common with and 0.14.07 dex for stars also observed by ( if one extreme case is removed ) . we conclude that the lower surface c abundances of these luminous giants as compared to the subgiants and main - sequence stars are likely the result of mixing rather than a difference in our abundance scales . nh band strengths have also been measured for a handful of the most luminous m13 turn - off stars . while molecular band formation in such stars is weak , significant star - to - star nh band strength differences are present . moreover , for the stars with both c and n measurements , differences between stars in these two elements appear to be anticorrelated . finally , the most recent c and n abundances for main - sequence , main - sequence turn - off , and subgiant stars in 47 tuc , m71 , m5 , and the present m13 data are compared .

0905.3933

c

we present robust evidence for the prevailing existence of hot gas clumps , whose characteristic scales are @xmath0 @xmath1 kpc , in the central regions of a sample of nine intermediate - redshift galaxy clusters . these substructures are not caused by uncertainties in background subtraction , response calibration , and spectral model fittings , nor are they related to background flares , the effects of which have been largely corrected and/or fixed . the possibility of the temperature substructures arising from the non - thermal emission of low - mass x - ray binaries ( lmxbs ) can also be eliminated , since the icm is only sparsely filled with galaxies . our results confirm the previous findings in the coma cluster ( schuecker et al . 2004 ) and the virgo cluster ( shibata et al . 2001 ) . we estimate the excess thermal energy @xmath137 in each high temperature substructure ( table 4 ) , where the gas temperature is typically higher than environment by @xmath138 kev ( fig . 4 ) , by defining the thermal energy as the product of gas pressure @xmath111 and volume @xmath139 , and assuming a spherical or elliptical geometry for the volume of the hot gas clump . we obtain @xmath140 @xmath141 erg for each clump , where @xmath142 is the electron density determined in the deprojected spectral analysis ( 3.2.2 ) . to account for azimuthal variations in the electron density , we include a 10% systematic error ( sanders et al . 2004 ) in the calculations . as the projection effect is not corrected for @xmath143 , the obtained excess energy may have been underestimated by a factor of about two . the origin of this energy excess is discussed as follows . the detected temperature substructures may have been formed due to the spatially inhomogeneous cooling in the icm . since for a unit volume , the radiative cooling rate is @xmath144 , where @xmath145 is the cooling function calculated by including both continuum and line emissions , the radiative cooling time of the substructures can be estimated as @xmath146 typically , the calculated cooling time is @xmath147 yrs , and is thus nearly an order of magnitude longer than the characteristic time for thermal conduction alone to smear out the temperature substructures ( table 4 ) , which is given by @xmath148 where @xmath149 is the enthalpy excess per unit volume in the hot substructure , @xmath150 is the conductive flux , @xmath151 is the characteristic scale length of the substructure , @xmath119 = 0.2 is the factor to assess the suppression caused by magnetic fields ( e.g. , narayan & medvedev 2001 ) , and @xmath152 is the thermal conductivity of hydrogen plasma , @xmath153\ ] ] for coulomb gas ( spitzer 1962 ) . this indicates that , unless the conduction process is extremely hindered ( i.e. , when @xmath154 ; maron et al . 2004 ; markevitch et al . 2003 ) , the effect of the radiative cooling is less important in the formation and evolution of temperature substructures . hence the possibility of the substructures being formed by inhomogeneous cooling can be excluded . the destruction of a temperature substructure via conduction can be accelerated by turbulence ( dennis & chandran 2005 ) , whose effect can be evaluated by introducing a modified conductivity ( cho et al . 2003 ; voigt & fabian 2004 ) @xmath155 where @xmath156 is the ratio of the turbulence velocity @xmath157 to the local adiabatic sound speed @xmath158 ( @xmath159 ) , and @xmath160 is the turbulence length scale . to calculate @xmath161 , the turbulence velocity @xmath157 can be estimated as follows . as there is evidence that heating continuously compensates a large fraction of radiative cooling since z@xmath162 ( bauer et al . 2005 ; mcnamara & nulsen 2007 ) , we may assume an approximate energy balance between heating and cooling for the cluster ( e.g. , kim & narayan 2003 ; zakamska & narayan 2003 ; dennis & chandran 2005 ) as @xmath163 where @xmath164 is the radiative loss , and @xmath165 , @xmath166 and @xmath167 are the heating rates of thermal conduction , viscous dissipation and turbulent diffusion , respectively . since @xmath168 @xmath169 and @xmath170 where @xmath171 is the gas mass density , @xmath172 is the specific entropy , @xmath173 is a dimensionless constant ( dennis & chandran 2005 and references therein ) , and @xmath174 @xmath175 @xmath6 is the diffusion coefficient ( rebusco et al . 2006 ) . by subtracting eqs.@xmath176 into eq.(9 ) , we have @xmath177\frac{l}{c_{\rm diss } \rho}\}^{\frac{1}{3}},\ ] ] if we define @xmath178 and @xmath179 ( dennis & chandran 2005 ) . using the observed deprojected gas density and temperature profiles ( fig . 3 ) , we calculate @xmath157 and show the results as a function of turbulence scale @xmath160 in figure 7 . if we follow schuecker et al . ( 2004 ) to adopt the lengths indicated by the turnover points on the wavelet spectra ( fig . 5 ) as the turbulence scale , we find that @xmath157 commonly ranges from 200 to 400 @xmath180 @xmath181 , or @xmath182 , for our sample . by using eqs.(6 ) and ( 8) , we find that the inclusion of turbulence reduces the characteristic time @xmath183 to the modified characteristic time @xmath184 by a factor of about 3 ( table 4 ) , which sets an upper limit ( @xmath185 yrs ) on the duty cycles of the dominating heating sources . the calculated @xmath137 ( table 4 ) suggests that the hot clumps may be dying or remnants of buoyant bubbles , into which the central agn has injected energy via shocks and turbulences ( e.g. , mcnamara & nulsen 2007 ) . such agn - induced hot gas clumps are predicted in numerical simulations ( e.g. , quilis et al . 2001 ; dalla vecchia et al . 2004 ; vernaleo & reynolds 2006 ) and theoretical calculations ( e.g. , pizzolato & soker 2006 ) , whose scales are expected to be about 100 kpc by the end of the buoyancy process . by examining the temperature maps ( fig . 4 ) , we note that some hot clumps appear in pairs on both sides of the central galaxies ( @xmath1861 and @xmath1862 in a1201 ; @xmath1861 and @xmath1862 in a2204 ; @xmath1861 and @xmath1863 in a3112 ) , which suggests possible agn activity taking place at the cluster cores . however , can an agn - heated temperature substructure survive conduction and turbulence destructions ? to answer this question , we estimate the characteristic rising time of a bubble as @xmath187 , where @xmath188 is the travel distance measured from the cluster center to the location of a hot clump , and @xmath189 is the travel velocity . in terms of the keplerian velocity @xmath190 , a large bubble is expected to travel through the ambient gas at @xmath191 where @xmath192 is the drag coefficient and @xmath193 is the bubble radius in kpc ( churazov et al . this yields @xmath194 smaller or very close to @xmath184 for @xmath195% of the substructures ( table 4 ) , indicating that such substructures can survive in most cases . even for the clumps with @xmath196 , the agn heating scenario may still stand , if the distance traveled by the bubble is shorter ( e.g. , gardini 2007 ) . it should be noted that no hot gas clump is found associated with any known radio source in all sample clusters . this , however , does not contradict with the agn heating scenario , because the lack of radio sources can be explained by the fact that the relativistic electrons may have lost most of their energy when the bubbles arose , as inferred by @xmath197 ( @xmath198 yrs ; 2 ) . numerical and theoretical works show that shocks caused by supersonic galaxy motion in the icm can generate observable temperature substructures via gas compression and friction ( e.g. , el - zant et al . 2004 ) , and even the low - speed galaxy infalling may also be a heating source candidate due to the efficient energy conversion via the magnetohydrodynamic turbulence and magnetic reconnection ( makishima et al . 2001 ; reviewed by markevitch & vikhlinin 2007 ) . the energy dissipation rate in the central region of a cluster is as large as @xmath199 erg @xmath6 ( fujita et al . 2004 ) , which assembles @xmath200 erg during the merger process and is thus sufficient to account for the excess energy in the observed temperature substructures . nevertheless , no convincing evidence is given in numerical studies that the non - filamentary morphologies of the observed temperature substructures can appear commonly in merger process ( e.g. , poole et al . 2006 ) . we also have estimated the contribution of supernova feedback to the icm heating over the substructure lifetime ( @xmath184 ) using the observed rate of type ia supernovae ( sne ia ; sharon et al . 2007 ) , which implies that about @xmath201 sn ia explosions have occurred in the central 400 @xmath1 kpc region of each sample cluster . since the averaged dynamic energy injection into the environment is @xmath202 erg per sn ia event ( spitzer 1978 ) , and @xmath79 % of this energy is assumed to have been used to heat the gas ( e.g. , thornton et al . 1998 ) , the total sn ia heating is found to be @xmath203 erg per cluster per @xmath204 yrs . the contribution of type ii supernovae ( sne ii ) will not significantly increase the total supernova heating , since even in the ir - luminous cluster a1068 , where the ratio of sn ii rate to sn ia rate is @xmath205 ( wise et al . 2004 ) , the sn ii heating is still lower than @xmath206 erg . therefore , we conclude that the supernova heating is insufficient to create the observed high temperature substructures . high temperature substructures may also occur as a result of the inverse - comptonization ( ic ) of the cosmic microwave background ( cmb ) photons that are scattered by the relativistic electrons in the icm , since in a acis spectrum it is difficult to distinguish between such an ic component and a high temperature thermal component ( petrosian et al . if in the icm @xmath207% of the electrons are relativistic ( @xmath208 ; eilek 2003 ) , we will have an ic energy conversion rate of @xmath209 erg @xmath6 , where @xmath210 is the thomson cross section and @xmath211 is the cmb energy density at the location of the cluster ( sarazin 1999 ) , which implies that the flux of the ic component is sufficiently high to bias the temperature measurements by about @xmath212 kev . however , a tight correlation between the high temperature substructures and radio synchrotron sources is required in this scenario , which is actually not observed , possibly due to the lack of relativistic electrons . therefore , the ic mechanism for the formation of the temperature substructures can be eliminated .

since if there were no significant non - gravitational heating sources , these substructures would have perished in yrs due to thermal conduction and turbulent flows , whose velocity is found to range from about 200 to 400 km , we conclude that the substructures can not be created and sustained by inhomogeneous radiative cooling .

by analyzing the gas temperature maps created from the archive data , we reveal the prevailing existence of temperature substructures on kpc scales in the central regions of nine intermediate - redshift ( ) galaxy clusters , which resemble those found in the virgo and coma clusters . each substructure contains a clump of hot plasma whose temperature is about kev higher than the environment , corresponding to an excess thermal energy of erg per clump . since if there were no significant non - gravitational heating sources , these substructures would have perished in yrs due to thermal conduction and turbulent flows , whose velocity is found to range from about 200 to 400 km , we conclude that the substructures can not be created and sustained by inhomogeneous radiative cooling . we also eliminate the possibilities that the temperature substructures are caused by supernova explosions , or by the non - thermal x - ray emission due to the inverse - comptonization of the cmb photons . by calculating the rising time of agn - induced buoyant bubbles , we speculate that the intermittent agn outbursts ( erg per burst ) may have played a crucial role in the forming of the high temperature substructures . our results are supported by recent study of mcnamara & nulsen ( 2007 ) , posing a tight observational constraint on future theoretical and numerical studies .

0905.3933

i

we reveal the prevailing existence of temperature substructures on @xmath0 @xmath1 kpc scales in the central regions of nine intermediate - redshift galaxy clusters . by comparing the characteristic destruction times of the temperature substructures with the gas cooling times and the rising times of agn - induced bubbles , we conclude that the agn outbursts may have played a crucial role in the forming of the observed temperature substructures . our results agree with those found earlier in the virgo and coma clusters ( shibata et al . 2001 ; schuecker et al . 2004 ) .

by analyzing the gas temperature maps created from the archive data , we reveal the prevailing existence of temperature substructures on kpc scales in the central regions of nine intermediate - redshift ( ) galaxy clusters , which resemble those found in the virgo and coma clusters . each substructure contains a clump of hot plasma whose temperature is about kev higher than the environment , corresponding to an excess thermal energy of erg per clump . we also eliminate the possibilities that the temperature substructures are caused by supernova explosions , or by the non - thermal x - ray emission due to the inverse - comptonization of the cmb photons . by calculating the rising time of agn - induced buoyant bubbles , we speculate that the intermittent agn outbursts ( erg per burst ) may have played a crucial role in the forming of the high temperature substructures .

by analyzing the gas temperature maps created from the archive data , we reveal the prevailing existence of temperature substructures on kpc scales in the central regions of nine intermediate - redshift ( ) galaxy clusters , which resemble those found in the virgo and coma clusters . each substructure contains a clump of hot plasma whose temperature is about kev higher than the environment , corresponding to an excess thermal energy of erg per clump . since if there were no significant non - gravitational heating sources , these substructures would have perished in yrs due to thermal conduction and turbulent flows , whose velocity is found to range from about 200 to 400 km , we conclude that the substructures can not be created and sustained by inhomogeneous radiative cooling . we also eliminate the possibilities that the temperature substructures are caused by supernova explosions , or by the non - thermal x - ray emission due to the inverse - comptonization of the cmb photons . by calculating the rising time of agn - induced buoyant bubbles , we speculate that the intermittent agn outbursts ( erg per burst ) may have played a crucial role in the forming of the high temperature substructures . our results are supported by recent study of mcnamara & nulsen ( 2007 ) , posing a tight observational constraint on future theoretical and numerical studies .

astro-ph0012365

i

ultraluminous infrared galaxies ( ulirgs ) with @xmath16 @xmath3 @xmath4 @xmath5 discovered by the @xmath17 @xmath18 @xmath19 ( iras ) are considered to be ideal laboratories for studying the formation of dusty starburst galaxies , elliptical galaxies , and qsos . detailed morphological studies of these ulirgs have revealed that virtually all show evidence of galaxy merging or strong tidal interaction ( e.g. , tidal tails and double or multiple nucleus ) and thus that most of ulirgs are formed by galaxy interaction / merging ( e.g. , sanders et al . 1988 ; clements et al . 1996 ; murphy et al . 1996 ; scoville et al . 1999 ; surace et al . 1999 ) . these ulirgs are observed to show mixtures of two types of activity ; starburst and active galactic nuclei ( agn ) ( sanders et al 1988 ; sanders & mirabel 1996 ; veilleux , kim , & sanders 1999 ; zheng et al . it is still controversial whether dust - enshrouded agn or a nuclear dusty starburst is the primary energy source for the huge luminosity of ulirgs ( sanders & mirabel 1996 for a review ; genzel et al . 1998 ; lutz et al . 1998 ; smith , lonsdale , & lonsdale 1998 ) . all of these of ulirgs are rich in molecular gas with the total mass of @xmath20 ranging from @xmath21 @xmath22 to 3 @xmath23 @xmath24 @xmath22 ( sanders et al . 1988 ) , and a large fraction of this gas is in very dense regions near nuclei ( solomon et al . ulirgs appear to be about 2 times more numerous than optically selected qsos and the luminosity function of ulirgs is nearly similar to that of qsos ( soifer et al . these results give rise to the argument that quasar - like activity is also being initiated in major mergers and ulirgs can finally evolve into qsos ( sanders et al . 1988 ) . recent high - resolution optical / infrared imaging of ulirgs have revealed a number of very bright star - forming knots probably formed by merging processes and furthermore clarified diverse morphology reflecting their varying stages of galactic dynamical evolution ( scoville et al . 1999 ; surace et al . 1999 ) . these peculiar phenomena seen in ulirgs ( i.e. , ongoing major mergers ) have raised theoretical interests in the interstellar gas and star formation process in gas - rich major mergers with strong dusty starbursts . numerical simulations of galaxy mergers including gas dynamics indicate that a large amount of gas , as much as @xmath21 @xmath22 , can be funneled to the central kpc region of the remnant ( barnes & hernquist 1991 , 1992 ; olson & kwan 1990 ; noguchi 1991 ) . mihos et al . ( 1992 ) suggested that the gas accumulated in the central kpc region of galaxy mergers can be converted into massive starburst , though the magnitude of the starburst is not strong enough to explain that observed in ulirgs . by using a more sophisticated model of hydrodynamical evolution of galaxy mergers , mihos & hernquist ( 1996 ) demonstrated that the strength and the epoch of massive starburst depend on initial internal structure of merger progenitor disks rather than initial orbital configurations of the disks . bekki & noguchi ( 1994 ) and bekki ( 1995 ) demonstrated that the dynamical heating by two sinking cores and subsequent dissipative cloud - cloud collisions in a merger can drive a large fraction of disk interstellar gas into the central tens pc where nuclear starburst and agn activity are observed for ulirgs . gerritsen & icke ( 1999 ) investigated how thermal energy from supernovae ( i.e. , feedback of supernovae ) affects the dynamical evolution of interstellar gas and consequently controls the star formation rate in a major merger and found that the supernovae energy from the nuclear high density gas region ( @xmath3 1 kpc ) can prevent the star formation in the outer region of the merger . although these theoretical studies have been quite successful in clarifying the detailed processes of gaseous inflow to the central 10 @xmath3 1000 pc and in describing star formation history for gas - rich major galaxy mergers , they have not yet investigated at all how photometric and spectroscopic properties of merger evolve with time . therefore it remains highly uncertain how the photometric properties of ulirgs is determined by the detailed dynamical processes of major galaxy merging . the purpose of the present paper is to investigate numerically morphological , structural , and photometric evolution of gas - rich major mergers with dusty starburst in an explicitly self - consistent manner . by using a new numerical code for calculating the spectral energy distribution ( sed ) of a galaxy with dusty interstellar gas ( bekki & shioya 2000a ) , we particularly investigate the following five points : ( 1 ) the fundamental properties of seds of dusty starburst mergers with the infrared luminosity larger than @xmath7 @xmath5 , ( 2 ) time evolution of the relative spatial distribution of dusty gas and old and new stars , ( 3 ) the dependence of @xmath13 on stellar ages and radius from the center of mass of a merger , ( 4 ) time evolution of colors , absolute magnitude , surface brightness , dust temperature , and the degree of dust extinction ( @xmath13 ) , and ( 5 ) physical correlations between @xmath0 , central gas density , luminosity ratio ( e.g. , @xmath25 , where @xmath26 is the @xmath8 band luminosity ) , star formation efficiency , and morphology during starburst . the above points have not been investigated extensively by previous theoretical studies . accordingly , the present numerical results can provide new clues to the nature and the origin of ulirgs . the layout of this paper is as follows . in 2 , we summarize numerical models used in the present study and describe briefly the methods for deriving the seds corrected by internal dust extinction . in 3 , we present numerical results on the time evolution of morphology , sed , and photometric properties in a gas - rich major merger . in 4 , we discuss the nature of ulirgs , in particular , the importance of the relative distribution of gas and stars in determining photometric and spectroscopic properties of ulirgs . the conclusions of the preset study are given in 5 .

main results are the following five . \(1 ) global colors and absolute magnitudes the during dusty starburst of a major merger do not change with time significantly , because interstellar dust heavily obscures young starburst populations that could cause rapid evolution of photometric properties of the merger . our numerical results clearly demonstrate that dynamical processes of major galaxy merging , which can control the time evolution of the relative spatial distribution of dusty interstellar gas and young stars , play a vital role in determining photometric properties of ulirgs .

by performing n - body simulations of chemodynamical evolution of galaxies with dusty starbursts , we investigate photometric evolution of gas - rich major mergers in order to explore the nature of ultraluminous infrared galaxies ( ulirgs ) with the total infrared luminosity ( for m ) of . main results are the following five . \(1 ) global colors and absolute magnitudes the during dusty starburst of a major merger do not change with time significantly , because interstellar dust heavily obscures young starburst populations that could cause rapid evolution of photometric properties of the merger . \(2 ) dust extinction of stellar populations in a galaxy merger with large infrared luminosity ( ) is selective in the sense that younger stellar populations are preferentially obscured by dust than old ones . this is because younger populations are located in the central region where a larger amount of dusty interstellar gas can be transferred from the outer gas - rich regions of the merger . \(3 ) both and the ratio of to band luminosity ) increases as the star formation rate increase during the starburst of the present merger model , resulting in the positive correlation between and . the dust temperature and the flux ratio of also increase with the increase of the star formation rate , because a larger number of young stars formed by massive starburst can heat the dusty interstellar gas as the star formation becomes larger . \(4 ) the star formation efficiency , total gas mass , the degree of dust extinction ( ) , , , , and depend strongly on the separation of two cores of the merger , which clearly reflects the fact that dynamical processes of galaxy merging play an important role in determining the photometric evolution of dusty starbursts in the merger . \(5 ) the two - dimensional distribution of global colors ( e.g. , ) shows a negative color gradient during starburst , mainly because central young starburst populations are preferentially and very heavily obscured by dust . furthermore , the peak of the two dimensional distribution of coincides with that of . these results clearly suggest that the spatial distribution of gas and stars is one important determinant of the two - dimensional photometric properties of major mergers . our numerical results clearly demonstrate that dynamical processes of major galaxy merging , which can control the time evolution of the relative spatial distribution of dusty interstellar gas and young stars , play a vital role in determining photometric properties of ulirgs .

astro-ph0012365

c

we have investigated in detail morphological , structural , and photometric evolution of gas - rich major mergers with strong dusty starburst in an explicitly self - consistent manner in order to explore the nature of ulirgs . main conclusions are summarized as follows . \(1 ) a massive starburst triggered by dynamical processes of major galaxy merging is found to be very heavily obscured by high density , dusty gas in a gas - rich merger . as a result of this , the merger can become a very luminous infrared galaxy with @xmath0 of @xmath3 5.9 @xmath23 @xmath7 @xmath5 at the maximum starburst epoch which corresponds to the epoch of the formation of central high density core of an elliptical galaxy . \(2 ) dust extinction of stellar populations in a galaxy merger with large infrared luminosity ( @xmath0 @xmath6 @xmath7 @xmath5 ) is selective in the sense that younger stellar populations are preferentially obscured by dust than old ones . this is because younger populations are located in the central region where a larger amount of dusty interstellar gas can be transferred from the outer gas - rich regions in the merger . this selective extinction provides physical basis for the origin of spectroscopic properties of ulirgs , such as e(a ) spectra recently revealed by poggianti & wu ( 1999 ) . \(3 ) both @xmath0 and @xmath10 increase as the star formation rate increases during starburst of the present merger model , resulting in the positive correlation between @xmath0 and @xmath10 in the merger model . the time evolution of the relative distribution of dusty gas and young starburst components is crucially important for the time evolution of these properties . \(4 ) the dust temperature @xmath110 and the flux ratio of @xmath12 increase with the increase of the star formation rate . this is because a larger number of young stars formed by massive starburst can heat the dusty interstellar gas as the star formation rate becomes higher . \(5 ) global colors ( e.g. , @xmath14 ) are redder in the central region than in the outer one at the epoch of strong starburst in a dusty major merger , which implies that most ulirgs show negative color gradients . furthermore the central surface brightness ( e.g. , in @xmath84 band ) becomes higher as the star formation rate becomes larger during starburst primarily because both old and young stellar populations are more strongly concentrated in the nuclear region of the merger . two - dimensional color distributions seem to be more strongly disturbed than surface brightness maps . the peak of two dimensional distribution of @xmath15 flux coincides with that of @xmath13 . these results clearly suggest that spatial distribution of gas and stars is one of important determinants for the two - dimensional photometric properties of major mergers . \(6 ) the star formation efficiency in a major merger becomes higher as the separation of the two cores becomes smaller , which clearly reflects the fact that dynamical processes of galaxy merging play an important role in determining the time evolution of dusty starburst in the merger ( e.g. , the time evolution of chemical components and local gaseous density in the starburst regions ) . total gas mass , the degree of dust extinction ( @xmath13 ) , @xmath11 , @xmath0 , @xmath10 , and @xmath12 also depend strongly on the separation of two cores of the merger . \(7 ) the star formation efficiency during starburst in a merger is found to correlate positively with the flux ratio of @xmath12 , which is consistent with recent observational results by young ( 1999 ) . furthermore @xmath10 and @xmath12 are found to be proportional to @xmath0 and to the central gaseous density of the merger , which is also consistent with fundamental properties observed in ulirgs ( e.g. , sanders & mirabel 1996 ) . we lastly stress that better understanding of dissipative and dissipationless dynamics of galaxy merging furthermore leads us to clarify more thoroughly the photometric properties of ulirgs . we are grateful to the referee guy worthey for valuable comments , which contribute to improve the present paper thanks the japan society for promotion of science ( jsps ) research fellowships for young scientists .

by performing n - body simulations of chemodynamical evolution of galaxies with dusty starbursts , we investigate photometric evolution of gas - rich major mergers in order to explore the nature of ultraluminous infrared galaxies ( ulirgs ) with the total infrared luminosity ( for m ) of . \(2 ) dust extinction of stellar populations in a galaxy merger with large infrared luminosity ( ) is selective in the sense that younger stellar populations are preferentially obscured by dust than old ones . this is because younger populations are located in the central region where a larger amount of dusty interstellar gas can be transferred from the outer gas - rich regions of the merger . \(3 ) both and the ratio of to band luminosity ) increases as the star formation rate increase during the starburst of the present merger model , resulting in the positive correlation between and . the dust temperature and the flux ratio of also increase with the increase of the star formation rate , because a larger number of young stars formed by massive starburst can heat the dusty interstellar gas as the star formation becomes larger . \(4 ) the star formation efficiency , total gas mass , the degree of dust extinction ( ) , , , , and depend strongly on the separation of two cores of the merger , which clearly reflects the fact that dynamical processes of galaxy merging play an important role in determining the photometric evolution of dusty starbursts in the merger . furthermore , the peak of the two dimensional distribution of coincides with that of . these results clearly suggest that the spatial distribution of gas and stars is one important determinant of the two - dimensional photometric properties of major mergers .

by performing n - body simulations of chemodynamical evolution of galaxies with dusty starbursts , we investigate photometric evolution of gas - rich major mergers in order to explore the nature of ultraluminous infrared galaxies ( ulirgs ) with the total infrared luminosity ( for m ) of . main results are the following five . \(1 ) global colors and absolute magnitudes the during dusty starburst of a major merger do not change with time significantly , because interstellar dust heavily obscures young starburst populations that could cause rapid evolution of photometric properties of the merger . \(2 ) dust extinction of stellar populations in a galaxy merger with large infrared luminosity ( ) is selective in the sense that younger stellar populations are preferentially obscured by dust than old ones . this is because younger populations are located in the central region where a larger amount of dusty interstellar gas can be transferred from the outer gas - rich regions of the merger . \(3 ) both and the ratio of to band luminosity ) increases as the star formation rate increase during the starburst of the present merger model , resulting in the positive correlation between and . the dust temperature and the flux ratio of also increase with the increase of the star formation rate , because a larger number of young stars formed by massive starburst can heat the dusty interstellar gas as the star formation becomes larger . \(4 ) the star formation efficiency , total gas mass , the degree of dust extinction ( ) , , , , and depend strongly on the separation of two cores of the merger , which clearly reflects the fact that dynamical processes of galaxy merging play an important role in determining the photometric evolution of dusty starbursts in the merger . \(5 ) the two - dimensional distribution of global colors ( e.g. , ) shows a negative color gradient during starburst , mainly because central young starburst populations are preferentially and very heavily obscured by dust . furthermore , the peak of the two dimensional distribution of coincides with that of . these results clearly suggest that the spatial distribution of gas and stars is one important determinant of the two - dimensional photometric properties of major mergers . our numerical results clearly demonstrate that dynamical processes of major galaxy merging , which can control the time evolution of the relative spatial distribution of dusty interstellar gas and young stars , play a vital role in determining photometric properties of ulirgs .

1204.3222

m

many of the standard methods of combinatorial game theory have proven ineffective for analyzing the effects of a pass in a game . in the case of nim , for instance , bouton s @xcite elegant solution method ( along with the more general sprague - grundy theory which generalizes it @xcite ) breaks down when a pass is introduced and offers very little additional insight . here , the failure can be attributed to the fact that a pass move renders the game non - decomposable . the situation is still more difficult in the case of chomp , a more complex game for which standard solution methods have failed even in the absence of a pass . given this situation , we will utilize a recently introduced dynamical - systems - based methodology for analyzing games that focuses on identifying the underlying geometry of a game and its recursive structure @xcite . a fundamental virtue of this approach is that it does not rely on a game being solvable or decomposable , and hence can address complex situations , whether it be a simple game which becomes complex by virtue of the introduction of a pass ( as in nim ) or a game which is intrinsically complex even in the absence of a pass ( as in chomp ) . ( in a dynamical systems context , this recursive - based approach is somewhat akin to exploiting the fact that a chaotic system is often describable by a simple , underlying iterative mapping . ) in this section we present a basic conceptual overview of the recursive methodology to be used in this paper , restricting first for simplicity to the case of games without the passes , and then describing the complete methodology which incorporates the modifications necessary to capture the effects of passes . ( we refer the reader to @xcite for a more detailed introduction to the general dynamical - systems - based approach to combinatorial games . ) we begin with some general considerations . to start , we note that in the two games considered here 3-pile nim and 3-row chomp a game position can be represented by a triplet of integers @xmath0 $ ] . in nim , these integers specify the number of tokens in each pile ; in chomp they are related to the number of tokens in each row . ( the specific rules for each game will be described later . ) hence both games have a three - dimensional `` position space '' @xcite . if we imagine marking the locations in position space of all p - positions of the game , they form a type of three - dimensional geometric object , which we dub the `` p - set '' . the p - set of the game is the main object of interest ; knowledge of a game s p - set allows one to define a winning strategy for the game . interestingly , for many combinatorial games , the p - set turns out to be neither a collection of randomly dispersed points in position space , nor a completely regular geometric object @xcite . rather , it lies intermediate between the two : it displays an overall geometric structure , but with local scatter ( disorder ) . ( as will be discussed later , the p - set of 3-pile nim without a pass proves to be exceptional in this regard ; its p - set geometry is completely regular . ) a recursion - based analysis of a game s p - set is carried out by foliating the game s three - dimensional position space @xmath1 by sets of horizontal planes ( `` sheets '' ) , with @xmath2 serving as the index for the sheets and @xmath3 as the coordinates within a sheet ( fig . there are two basic classes of foliating sheets that are of interest : `` loser sheets '' @xmath4 and `` instant - winner sheets '' @xmath5 . the loser sheets mark the locations of the game s p - positions . more formally , the loser sheet at level @xmath2 , denoted @xmath6 , is defined to be a two - dimensional , semi - infinite ( boolean ) matrix consisting of zeros and ones , with ones marking the location of the game s p - positions at the specified @xmath2 value : @xmath7 if position @xmath0 $ ] is a p - position and @xmath8 otherwise . we will be interested in the geometric patterns formed by the p - positions on these loser sheets . taken collectively , the loser sheets describe the full geometric structure of the p - set in the three - dimensional position space of the game . the second class of foliating sheets , the instant - winner sheets , also play a key role in the analysis . these are defined as follows : first , define a game position @xmath0 $ ] to be an _ instant winner _ if there exists a legal move from that position to some p - position @xmath9 $ ] on a lower sheet ( i.e. , with @xmath10 ) . the instant - winner _ sheets _ are planes in the game s three - dimensional position space on which are marked the locations of these instant winners . more formally , we define the instant - winner sheet at level @xmath2 , denoted @xmath11 , to be a two - dimensional , semi - infinite ( boolean ) matrix consisting of zeros and ones , with ones marking the location of the instant winners having the specified @xmath2 value : @xmath12 if position @xmath0 $ ] is an instant - winning position and @xmath8 otherwise . just as for the loser sheets , we will be interested in geometric patterns that may arise within these instant - winner sheets . the basis of the recursive analysis is the finding that there exist relationships between these various foliating sheets . ( zeilberger @xcite was among the first to recognize the importance of such recursive structures in games . ) in particular , one can construct a recursion operator , denoted here @xmath13 , that relates an instant - winner sheet at one level to the instant - winner sheet at the next higher level : @xmath14 . thus , starting from the lowest - level instant - winner sheet , one can recursively generate all higher instant - winner sheets by repeatedly applying the @xmath13 operator . moreover , one can construct another operator , denoted here @xmath15 ( dubbed the `` supermex operator '' ) , that relates an instant - winner sheet at level @xmath2 to the associated loser sheet at that same level : @xmath16 . hence , once the instant - winner sheets have been recursively generated , the loser sheets can be readily found ( and thus the game s p - set determined ) . this situation is depicted in fig . [ f3 ] . for this reason , we can think of the instant - winner sheets as effectively encoding all information about a game s p - set , and thus much of our subsequent analysis will focus on the recursive structure of the instant - winner sheets . we note that the operators @xmath13 and @xmath15 for the games of 3-pile nim and 3-row chomp can be explicitly computed from their respective game rules , as will be illustrated shortly . adapting the recursive strategy outlined above to games with passes requires some modifications to the operators and foliating sheets , but the overall recursive framework remains intact . before discussing this process explicitly , we first mention two basic considerations concerning the introduction of passes into games : 1 . by definition , the pass move can not be invoked if no tokens remain in the game ( i.e. , more generally , the pass can not be used from any position that would be deemed a terminal position in the corresponding game without a pass ) . without this restriction , a pass would be uninteresting since a game - with - pass " could then be viewed as nothing more than a game - without - pass " played in disjunctive sum with a nim pile containing a single token a situation which is readily handled by standard game - theoretic techniques . once the pass move has been invoked by either player ( thereby rendering it unavailable from that point forward ) , the analysis of the game - with - pass reduces to that of the corresponding game - without - pass . notationally , a hat @xmath17 will henceforth indicate that the pass has not yet been used ; the absence of a hat signifies that it has been used . for example , if @xmath18 denotes an arbitrary position in the game when the pass move is still available , @xmath19 will denote that same position when the pass is no longer available . if @xmath19 is a ( non - terminal ) p - position , then @xmath18 is necessarily an n - position , since by invoking the pass a player at @xmath18 could move his / her opponent to losing position @xmath19 . thus , under optimal play , the only time a player at an arbitrary position @xmath18 will invoke the pass move is if position @xmath19 is a p - position . the introduction of a pass move into a game also necessitates some modification to the recursive formulation described earlier . observe first that it is possible to continue to view the position spaces of 3-pile nim - with - a - pass and 3-row chomp - with - a - pass as three - dimensional provided we also explicitly specify whether or not the pass remains available . we do so by using the hat / no - hat notation described above , i.e. , if @xmath18 denotes a position with the pass available , @xmath19 denotes that same position when the pass is unavailable . in cases where we wish to specify explicitly the coordinates of a position , we do so by appending an auxiliary boolean coordinate as follows : @xmath20 $ ] and @xmath21 $ ] , where the 1 indicates the pass is available and 0 that it has been used . the introduction of a pass move into a game also requires that we now distinguish between two distinct classes of loser sheets , @xmath4 and @xmath22 ; the former describe the p - positions of the game at the specified @xmath2-level when the pass is not available , the latter when the pass is available . a similar distinction must be made for instant - winner sheets : @xmath23 and @xmath24 . more precisely , when the pass is not available , we define a position @xmath21 $ ] to be an _ instant winner _ if from that position one can move to a p - position @xmath25 $ ] at a lower @xmath2-level ( i.e. , with @xmath26 ) . sheets @xmath23 mark the locations of such instant winners . likewise , when the pass is available , a position @xmath20 $ ] is defined to be an instant winner if one can move to a p - position @xmath27 $ ] at a lower @xmath2-level . sheets @xmath24 mark the locations of these instant winners in this case .

we consider two well known combinatorial games , 3-pile nim and 3-row chomp . in the case of nim = 1.5 combinatorial games like chess , checkers , go , nim , and chomp have been the focus of considerable attention in the fields of computer science , mathematics , artificial intelligence , and most recently , chaos and dynamical systems theory . in traditional combinatorial games ( under `` normal play '' ) , two players alternate moves until one player reaches a terminal position from which no legal move is available , thereupon losing the game . an intriguing but surprisingly difficult question in combinatorial game theory centers on what happens when standard game rules are modified so as to allow for a one - time pass i.e. , a pass move which may be used at most once in a game , and not from a terminal position . once the pass has been used by either player , it is no longer available . indeed , the effect of a pass on even the simplest , most canonical of combinatorial games nim remains an important open question in combinatorial game theory that has defied traditional approaches , and the late mathematician david gale even offered a monetary prize to the first person to develop a solution for 3-pile nim with a pass . in this paper we take up this question of the effects of a pass via two well studied combinatorial games , 3-pile nim and 3-row chomp . the first of these games , 3-pile nim , is a simple combinatorial game which has been fully solved ( without the pass ) ; a complete solution was presented by bouton over a century ago . we find that the introduction of a pass has dramatically different effects on these two games . in the former [ f1 ] , which will be explained later ) . the essential feature of our approach is the recognition that combinatorial games like those considered here can be recast as dynamical systems described by a form of recursive mapping , allowing us to invoke tools and notions from dynamical systems theory such as attractors and sensitivity to initial conditions . the paper is organized as follows : in sect . 1 we describe a recently developed recursive technique for analyzing combinatorial games that will also prove useful for analyzing games with passes . this methodology is based on the construction of recursion operators that describe a game s underlying geometric structure . in sect . 2 we discuss some general considerations regarding games with passes , and describe how the recursive approach can be adapted to handle passes . in sect .

by treating combinatorial games as dynamical systems , we are able to address a longstanding open question in combinatorial game theory , namely , how the introduction of a `` pass '' move into a game affects its behavior . we consider two well known combinatorial games , 3-pile nim and 3-row chomp . in the case of nim , we observe that the introduction of the pass dramatically alters the game s underlying structure , rendering it considerably more complex , while for chomp , the pass move is found to have relatively minimal impact . we show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations . from these recursion relations we are able to identify underlying structural connections between these `` games with passes '' and a recently introduced class of `` generic ( perturbed ) games . '' this connection , together with a ( non - rigorous ) numerical stability analysis , allows one to understand and predict the effect of a pass on a game . = 1.5 combinatorial games like chess , checkers , go , nim , and chomp have been the focus of considerable attention in the fields of computer science , mathematics , artificial intelligence , and most recently , chaos and dynamical systems theory . in traditional combinatorial games ( under `` normal play '' ) , two players alternate moves until one player reaches a terminal position from which no legal move is available , thereupon losing the game . an intriguing but surprisingly difficult question in combinatorial game theory centers on what happens when standard game rules are modified so as to allow for a one - time pass i.e. , a pass move which may be used at most once in a game , and not from a terminal position . once the pass has been used by either player , it is no longer available . although this question has been raised in various contexts ( see , e.g. , ) , it touches upon some deep issues relating to the underlying structure and computational complexity of a game , and to date it remains largely unanswered . indeed , the effect of a pass on even the simplest , most canonical of combinatorial games nim remains an important open question in combinatorial game theory that has defied traditional approaches , and the late mathematician david gale even offered a monetary prize to the first person to develop a solution for 3-pile nim with a pass . in this paper we show how tools from dynamical systems theory ( wherein we treat `` games with passes '' as a type of dynamical system ) can be used to address such issues . we take up this question of the effects of a pass via two well studied combinatorial games , 3-pile nim and 3-row chomp . the first of these games , 3-pile nim , is a simple combinatorial game which has been fully solved ( without the pass ) ; a complete solution was presented by bouton over a century ago . the second , 3-row chomp ( without the pass ) , is an unsolved , intrinsically more complex combinatorial game . we find that the introduction of a pass has dramatically different effects on these two games . in the former , the pass leads to a radical change in the game s underlying structure and complexity , while in the latter we find no such dramatic changes ( see , e.g. , fig . [ f1 ] , which will be explained later ) . in this paper we attempt to analyze and explain this phenomena not by offering an explicit `` solution '' to nim - with - a - pass or chomp - with - a - pass indeed our analysis indicates why this may not be possible but rather by showing that these combinatorial games with passes are related to a class of perturbed " ( also called generic " ) combinatorial games . this transcription of the problem from one of passes to one of perturbations is based on our discovery of an underlying equivalence in the recursive algorithms used to analyze passes and perturbations in games . the main focus of this paper is to describe this underlying connection . this finding will in turn provide critical insight into why a simple game like nim is radically transformed by the introduction of a pass , while a game like chomp is not . the essential feature of our approach is the recognition that combinatorial games like those considered here can be recast as dynamical systems described by a form of recursive mapping , allowing us to invoke tools and notions from dynamical systems theory such as attractors and sensitivity to initial conditions . the paper is organized as follows : in sect . 1 we describe a recently developed recursive technique for analyzing combinatorial games that will also prove useful for analyzing games with passes . this methodology is based on the construction of recursion operators that describe a game s underlying geometric structure . in sect . 2 we discuss some general considerations regarding games with passes , and describe how the recursive approach can be adapted to handle passes . in sect . 3 we analyze the game of 3-pile nim - with - a - pass . in sect . 4 we focus on 3-row chomp - with - a - pass . in sect . 5 the underlying connections between these two games with passes and so - called `` generic games '' is revealed , and we discuss the implications of these relationships .

1403.0032

i

cassiopeia a ( cas a ; 3c 461 , g111.7 - 2.1 ) is the 2nd - youngest - known supernova remnant ( snr ) and , at a distance of 3.4 kpc @xcite , it lies just beyond the perseus arm of the galaxy . with the discovery of light echoes from the explosion , we now know that cas a resulted from a type iib explosion @xcite . cas a is one of the strongest synchrotron radio - emitting objects in the sky and has been observed extensively with the very large array ( vla ) since its commissioning in 1980 . the morphology of cas a is quite complex with structure distributed over a variety of spatial scales . the terminology used to describe some of these structures was coined in some of the earliest papers describing the optical and resolved radio images @xcite . the most prominent feature is the almost circular `` bright ring '' at a radius of @xmath5 which is generally regarded as marking the location of ejecta that have interacted with the reverse shock ( see e.g. * ? ? ? a fainter `` plateau '' of radio emission is seen out to a radius of @xmath6 . to the northeast , where the shell becomes broken , is the `` jet '' and extending in the opposite direction to the southwest is the counter - jet . the jet and counter - jet do not represent outflow in the classical sense . instead they describe locations where the fastest - moving ejecta are observed well beyond the plateau and farthest from the explosion center ( see e.g. * ? ? ? to the southeast , iron - rich ejecta extend beyond the bright ring and into the plateau . the jets and extended iron - rich structure are likely the result of an asymmetric explosion of the progenitor ( see e.g. * ? ? ? * ; * ? ? ? the light echo data also indicate an asymmetric explosion @xcite . in addition to the shocked ejecta described above , there is a class of ejecta still interior to the reverse shock in cas a. these `` unshocked ejecta '' were discovered via absorption of low frequency ( @xmath0100 mhz ) radio emission @xcite and are also seen to radiate in the infrared in the emission lines of [ ] , [ ] , [ ] , and [ ] @xcite . the term `` unshocked '' is somewhat of a misnomer because all of the ejecta were originally shocked by the passage of the blast wave through the star . however , the ejecta cooled during the subsequent expansion of the snr . what we consider to be shocked ejecta today are those ejecta that have crossed through the reverse shock with the term `` unshocked ejecta '' referring to those ejecta that are still interior to the reverse shock . thus the simple cartoon of cas a s structure is that of cold ejecta in the interior of a roughly spherical shell composed of shocked gas that radiates strongly in multiple bands . at low frequencies , the radio emission from the far side of the shocked shell is absorbed by the cold , unshocked ejecta in the interior . in [ sec : geometry ] we provide a more thorough description of the geometry assumed for our analysis . the infrared emission from the unshocked ejecta in cas a occurs because they have been photoionized . an analysis of the unshocked ejecta observed in the supernova remnant sn1006 , based on calculations of photoionization cross - sections and bethe parameters , showed that the unshocked ejecta are photoionized by two primary sources @xcite . ambient ultraviolet starlight is all that is necessary to photoionize and since their ionization potentials are below the lyman limit . for ions with ionization potentials above the lyman limit , the ultraviolet and soft x - ray radiation field of the shocked ejecta in sn1006 is such that species up to ( 54.9 ev ) , ( 45.1 ev ) , and ( 54.8 ev ) can be photoionized . a similar analysis was performed for cas a assuming photoionization equilibrium , abundances appropriate for a core - collapse snr , a thermal bremsstrahlung spectrum for the shocked ejecta , and utilizing a simple one - dimensional hydrodynamic model to follow cas a s evolution @xcite . this simplified model predicts that [ ] ( 34.8 ) and [ ] ( 25.9 ) should be the dominant infrared lines in the unshocked ejecta , directly in line with observations @xcite . in addition , @xcite predicts strong [ ] ( 88.4 ) , which we will show in [ sec : composition ] is present in spectra from the _ infrared space observatory _ ( _ iso _ , * ? ? ? * ; * ? ? ? * ) . cas a was spectrally mapped with the _ spitzer _ space telescope and doppler shifts were measured which allowed a 3d mapping of the ejecta distribution , including the unshocked component @xcite . we now know based on the _ spitzer _ data that the ejecta are organized into a `` thick disk '' structure , tilted at @xmath7 from the line - of - sight , providing further evidence that the explosion , or subsequent evolution of the snr prior to the reverse shock encounter , must have been asymmetric . an upper limit of 100 @xmath8 was determined for the electron density of the unshocked ejecta based on infrared [ ] line ratios , but the actual density is likely much lower @xcite . the absorption seen in the low frequency radio observations provides a means to probe the density and mass of the unshocked ejecta because the free - free optical depth ( @xmath9 ) is related to emission measure and thus density . @xcite attempted to determine the total mass of the unshocked ejecta , but due to using a @xmath9 appropriate for a hydrogenic gas and a temperature that was too high , arrived at 19m@xmath3 , which is unreasonably large considering that the total ejecta mass is likely only 2 - 4 m@xmath3 @xcite . given the role that the ejecta play in the evolution of snrs , it is important to provide an accurate census of the total mass present , thus prompting a new look at the low frequency absorption analysis of @xcite . the upgraded karl g. jansky very large array ( vla ) primarily accesses frequencies above 1 ghz through its broadband cassegrain focus systems . its predecessor , hereafter the `` legacy '' vla , also accessed two relatively narrow bands below 1 ghz through its primary focus systems @xcite . these included the `` p band '' and `` 4 band '' systems operating at 330 mhz ( 1990 - 2009 ) and 74 mhz ( 1998 - 2009 ) , respectively . both systems provided sub - arcminute resolution imaging and were widely used over their lifetimes . these systems were removed during the vla upgrade and have been only recently replaced with a new `` low band '' receiving system @xcite . the first call for proposals using the 330 mhz band of this new system was issued by nrao in february , 2013 . the legacy 74 mhz and 330 mhz vla systems achieved their maximum angular resolution of @xmath10 and @xmath11 in the a configuration ( maximum baseline @xmath436 km ) , respectively . an 8-antenna prototype of the 74 mhz system was used to observe cas a @xcite early on , adding to the body of vla work extending from 330 mhz to higher frequencies . as the resolution was still relatively poor compared to shorter wavelengths , nrl and nrao added a 74 mhz feed system to the pie town antenna , utilizing an optical fiber link connecting the innermost very large baseline array ( vlba ) antenna to the legacy vla . with a maximum baseline of 73 km , this improved the angular resolution at 74 mhz and 330 mhz by a factor of two to approximately 9@xmath1 and 3@xmath1 respectively . cas a was thereafter reobserved with the full ( 27-antenna ) 74 mhz and the 330 mhz legacy systems using the pie town link capability in august 2003 . in this paper , we report on legacy vla observations in 1997 - 1998 at frequencies of 5 ghz , 1.4 ghz , 330 mhz , and 74 mhz , and on the pie town link , observations at frequencies of 330 mhz and 74 mhz in 2003 . we use our derived images to determine the degree of free - free absorption present , applying the appropriate formula for when the composition of the gas is not dominated by hydrogen . finally , we calculate the density and mass of the unshocked ejecta and discuss the implications of our results .

low frequency ( mhz ) radio observations of sufficient angular resolution offer a unique probe of unshocked ejecta revealed via free - free absorption against the synchrotron emitting shell . this is a significant improvement over the 100 upper limit offered by infrared [ ] line ratios from the _ spitzer space telescope_. our estimates are sensitive to a number of factors including temperature and geometry . however using reasonable values for each

characterizing the ejecta in young supernova remnants is a requisite step towards a better understanding of stellar evolution . in cassiopeia a the density and total mass remaining in the unshocked ejecta are important parameters for modeling its explosion and subsequent evolution . low frequency ( mhz ) radio observations of sufficient angular resolution offer a unique probe of unshocked ejecta revealed via free - free absorption against the synchrotron emitting shell . we have used the very large array plus pie town link extension to probe this cool , ionized absorber at 9 and 185 resolution at 74 mhz . together with higher frequency data we estimate an electron density of 4.2 and a total mass of 0.39 m with uncertainties of a factor of . this is a significant improvement over the 100 upper limit offered by infrared [ ] line ratios from the _ spitzer space telescope_. our estimates are sensitive to a number of factors including temperature and geometry . however using reasonable values for each , our unshocked mass estimate agrees with predictions from dynamical models . we also consider the presence , or absence , of cold iron- and carbon - rich ejecta and how these affect our calculations . finally we reconcile the intrinsic absorption from unshocked ejecta with the turnover in cas a s integrated spectrum documented decades ago at much lower frequencies . these and other recent observations below 100 mhz confirm that spatially resolved thermal absorption , when extended to lower frequencies and higher resolution , will offer a powerful new tool for low frequency astrophysics .

1403.0032

c

we have imaged cas a from 5 ghz to 74 mhz in all four configurations of the legacy vla with follow - up observations at 74 and 330 mhz with the legacy vla+pt link . our spatially resolved spectral index maps confirm the interior spectral flattening measured earlier , but at higher signal - to - noise and resolution . comparison with _ spitzer _ infrared spectra confirms the earlier hypothesis that the spectral flattening is due to thermal absorption by cool , unshocked ejecta photoionized by x - ray radiation from cas a s reverse shock . we use the spectral flattening to measure the free - free optical depth . next , using priors of electron temperature , atomic number , and electron to ion ratios , we derive an emission measure from the measured optical depth . with an assumed geometry , informed from three - dimensional modeling based on higher frequency studies , we use the emission measure to place constraints on both the density and total mass of the unshocked ejecta . we consider modest , physically plausible variations in both our priors and the assumed geometry , and find that the effect on the total mass is relatively modest , varying by a factor of about two . furthermore , our derived total mass is consistent with recent model predictions @xcite . after accounting for the relative ages of cas a and sn1006 , our derived mass density is much higher than found in sn1006 , not unexpected since cas a ( type iib ) and sn1006 ( type ia ) emerged from two fundamentally different supernova explosion types . however , if there is a systematic difference in unshocked ejecta density for core collapse vs. type ia snrs , low frequency radio data can be used to test this hypothesis . finally , we consider the contribution of the intrinsic thermal absorption to the known turnover of cas a s integrated spectrum at much lower frequencies . we find that the intrinsic thermal absorption from the unshocked ejecta , combined with extrinsic absorption from a known , patchy distribution of low density ism gas , are completely consistent with the low frequency turnover . the promise of the emerging instruments is expanding the population of snrs , young and old , that can be probed for intrinsic and extrinsic thermal absorption and shock acceleration variations beyond pathologically bright sources like cas a. more generally , the seemingly ubiquitous detection of resolved thermal absorption by the 74 mhz legacy vla against the galactic background @xcite and towards , discrete non thermal sources ( e.g. see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) confirms the phenomena will continue to emerge as a powerful tool for low frequency astrophysics . the vla is operated by the national radio astronomy observatory , which is a facility of the national science foundation , operated under cooperative agreement by associated universities , inc . all sub - ghz systems on the vla have been developed cooperatively between the national radio astronomy observatory and the naval research laboratory . basic research in radio astronomy at the naval research laboratory is supported by 6.1 base funding . partial funding for this research at west virginia wesleyan college was provided by chandra grant go0 - 11089x and by the nasa - west virginia space grant consortium . _ facility : _

finally we reconcile the intrinsic absorption from unshocked ejecta with the turnover in cas a s integrated spectrum documented decades ago at much lower frequencies . these and other recent observations below 100 mhz confirm that spatially resolved thermal absorption , when extended to lower frequencies and higher resolution , will offer a powerful new tool for low frequency astrophysics .

characterizing the ejecta in young supernova remnants is a requisite step towards a better understanding of stellar evolution . in cassiopeia a the density and total mass remaining in the unshocked ejecta are important parameters for modeling its explosion and subsequent evolution . low frequency ( mhz ) radio observations of sufficient angular resolution offer a unique probe of unshocked ejecta revealed via free - free absorption against the synchrotron emitting shell . we have used the very large array plus pie town link extension to probe this cool , ionized absorber at 9 and 185 resolution at 74 mhz . together with higher frequency data we estimate an electron density of 4.2 and a total mass of 0.39 m with uncertainties of a factor of . this is a significant improvement over the 100 upper limit offered by infrared [ ] line ratios from the _ spitzer space telescope_. our estimates are sensitive to a number of factors including temperature and geometry . however using reasonable values for each , our unshocked mass estimate agrees with predictions from dynamical models . we also consider the presence , or absence , of cold iron- and carbon - rich ejecta and how these affect our calculations . finally we reconcile the intrinsic absorption from unshocked ejecta with the turnover in cas a s integrated spectrum documented decades ago at much lower frequencies . these and other recent observations below 100 mhz confirm that spatially resolved thermal absorption , when extended to lower frequencies and higher resolution , will offer a powerful new tool for low frequency astrophysics .

0906.0803

i

we identify 39,668 clusters of galaxies in the redshift range @xmath107 using photometric redshifts of galaxies from the sdss dr6 . a cluster is recognized if more than eight member galaxies of @xmath2 are found within a radius of 0.5 mpc and a photometric redshift gap between @xmath79 . this is the largest cluster catalog to date . our sample is much deeper in redshift than the previous cluster catalogs from the sdss . cluster redshifts are estimated with an uncertainty less than 0.022 . using the sdss spectroscopic data , we also estimate the contamination rate and completeness of member galaxy candidates to be about 20% and 90% , respectively . monte carlo simulations show that the cluster detection rate depends on richness , but is approximately constant to redshift @xmath249 . the detection rate is @xmath360% for clusters with a mass @xmath243 , which corresponds to a mean output richness @xmath250 . the detection rate increases to be 90% for clusters with a mass @xmath4 , which corresponds to a mean @xmath251 . the false detection rate of clusters is @xmath35% for our algorithm . we compare our catalog with the published abell , ce , maxbcg , and _ rosat _ x - ray cluster catalogs . we find that our catalog includes 77% abell clusters and 77% _ rosat _ x - ray selected clusters at @xmath180 . rich clusters are more likely detected by our method . with luminous member galaxies discriminated , we get the richness , @xmath92 , the summed luminosity , @xmath218 , and the gross galaxy number @xmath106 within a cluster radius ( @xmath105 ) for clusters in our catalog up to @xmath174 . we find that they are tightly related to the x - ray luminosity and temperature , and can trace the cluster mass with the relations , @xmath6 and @xmath7 . by cross - identification with the _ rosat _ x - ray source list , we obtain _ 685 new candidates _ of x - ray clusters , of which 60% are likely true . we thank the anonymous referee , professor y. y. zhou and shude mao for valuable comments that helped to improve the paper . the authors are supported by the national natural science foundation ( nnsf ) of china ( 10773016 , 10821061 and 1083303 ) and the national key basic research science foundation of china ( 2007cb815403 ) and the doctoral foundation of synu of china ( 054 - 55440105020 ) . funding for the sdss and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the u.s . department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society , and the higher education funding council for england . the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium for the participating institutions . the participating institutions are the american museum of natural history , astrophysical institute potsdam , university of basel , cambridge university , case western reserve university , university of chicago , drexel university , fermilab , the institute for advanced study , the japan participation group , johns hopkins university , the joint institute for nuclear astrophysics , the kavli institute for particle astrophysics and cosmology , the korean scientist group , the chinese academy of sciences ( lamost ) , los alamos national laboratory , the max planck institute for astronomy ( mpia ) , the max planck institute for astrophysics ( mpa ) , new mexico state university , ohio state university , university of pittsburgh , university of portsmouth , princeton university , the united states naval observatory , and the university of washington . rrrrrccrrrrrl whl j000006.0@xmath120152547 & 0.02482 & 15.42990 & 0.1735&@xmath252 & 16.58 & 15 & 11.30 & 9.07 & 0.50 & 62.69 & 6.97 & maxbcg + whl j000007.1@xmath127092909 & 0.02957&@xmath253 & 0.3963&@xmath252 & 19.11 & 19 & 15.88 & 14.44 & 0.71 & 81.33 & 8.56 & + whl j000007.6@xmath120155003 & 0.03177 & 15.83423 & 0.1489 & 0.1528 & 16.00 & 17 & 13.40 & 13.20 & 0.71 & 54.61 & 6.55 & abell , maxbcg + whl j000020.1@xmath120160859 & 0.08358 & 16.14976 & 0.4591&@xmath252 & 19.88 & 20 & 18.56 & 29.40 & 1.58 & 107.01 & 6.02 & + whl j000021.7@xmath120150611 & 0.09053 & 15.10328 & 0.2883&@xmath252 & 17.67 & 20 & 18.17 & 22.88 & 1.50 & 94.66 & 9.43 & maxbcg + whl j000025.1@xmath127093452 & 0.10453&@xmath254 & 0.3648&@xmath252 & 18.44 & 16 & 9.29 & 9.65 & 0.71 & 74.32 & 4.90 & + whl j000027.6@xmath127010140 & 0.11617&@xmath255 & 0.4491 & 0.4387 & 18.62 & 25 & 20.07 & 20.07 & 1.00 & 124.10 & 8.81 & + whl j000048.3@xmath127011204 & 0.18509&@xmath256 & 0.4373 & 0.4392 & 18.76 & 18 & 14.44 & 13.33 & 0.87 & 82.12 & 5.01 & + whl j000050.5@xmath120004705 & 0.21051 & 0.78477 & 0.2458&@xmath252 & 17.64 & 22 & 20.10 & 26.16 & 1.22 & 105.53 & 5.94 & nscs , ce , maxbcg + whl j000050.7@xmath120004704 & 0.21134 & 0.78470 & 0.4889&@xmath252 & 19.73 & 10 & 6.40 & 7.10 & 0.50 & 51.92 & 5.69 & + whl j000052.9@xmath120160520 & 0.22045 & 16.08902 & 0.1986&@xmath252 & 16.88 & 12 & 10.22 & 11.33 & 1.22 & 46.42 & 5.44 & + whl j000059.1@xmath120004841 & 0.24642 & 0.81162 & 0.3551&@xmath252 & 19.18 & 18 & 14.80 & 13.60 & 0.87 & 70.64 & 4.93 & nscs + whl j000111.3@xmath120151839 & 0.29608 & 15.30418 & 0.4053&@xmath252 & 19.10 & 21 & 19.10 & 30.26 & 1.58 & 125.22 & 6.50 & + whl j000116.2@xmath127093137 & 0.31767&@xmath257 & 0.3383 & 0.3693 & 18.29 & 24 & 19.83 & 31.65 & 1.41 & 112.56 & 7.03 & + whl j000117.5@xmath120142848 & 0.32297 & 14.48012 & 0.3815&@xmath252 & 19.68 & 17 & 12.84 & 11.92 & 0.71 & 45.04 & 4.93 & + ccrrcccl rxs j000522.7@xmath120161306 & whl j000524.0@xmath120161309 & 1.34987 & 16.21922 & 0.1115 & 0.04 & 0.076 & rxc + rxs j001739.4@xmath127005150 & whl j001740.0@xmath127005314 & 4.40670 & @xmath258 & 0.2340 & 0.21 & 0.037 & + rxs j002302.3@xmath120144645 & whl j002300.7@xmath120144656 & 5.75279 & 14.78240 & 0.3826 & 0.13 & 0.033 & + rxs j002815.2@xmath120135601 & whl j002819.8@xmath120135459 & 7.08254 & 13.91657 & 0.1516 & 0.23 & 0.026 & + rxs j003209.2@xmath127003932 & whl j003212.1@xmath127003950 & 8.04672 & @xmath259 & 0.2175 & 0.14 & 0.013 & + rxs j003417.8@xmath120005145 & whl j003419.1@xmath120004948 & 8.59684 & 0.85723 & 0.2035 & 0.27 & 0.024 & + rxs j004149.7@xmath127091817 & whl j004148.2@xmath127091703 & 10.46029 & @xmath260 & 0.0560 & 0.01 & 4.079 & rxc + rxs j010101.1@xmath127095726 & whl j010101.5@xmath127095717 & 15.25645 & @xmath261 & 0.1457 & 0.03 & 0.035 & + rxs j010243.0@xmath120010805 & whl j010243.1@xmath120010810 & 15.67950 & 1.13633 & 0.1345 & 0.01 & 0.052 & rx + rxs j010649.5@xmath120010317 & whl j010650.5@xmath120010410 & 16.71051 & 1.06970 & 0.2527 & 0.21 & 0.187 & rxc + rxs j010717.9@xmath120141635 & whl j010721.9@xmath120141623 & 16.84109 & 14.27322 & 0.0963 & 0.10 & 0.020 & + rxs j010921.7@xmath120005457 & whl j010923.1@xmath120005429 & 17.34616 & 0.90818 & 0.2723 & 0.14 & 0.029 & + rxs j011006.0@xmath120135849 & whl j011001.3@xmath120135555 & 17.51321 & 13.97815 & 0.0712 & 0.06 & 0.061 & rxc + rxs j011202.7@xmath127004355 & whl j011204.1@xmath127004351 & 18.01689 & @xmath262 & 0.2119 & 0.07 & 0.053 & + rxs j011940.0@xmath120145303 & whl j011938.3@xmath120145352 & 19.90952 & 14.89799 & 0.1289 & 0.12 & 0.096 & rxc + , r. , gunn , j. e. , ivezi , z. , knapp , g. r. , & kent , s. 2001 , in astronomical society of the pacific conference series , vol . 238 , astronomical data analysis software and systems x , ed . f. r. harnden , jr . , f. a. primini , & h. e. payne , 269

clusters of galaxies in most previous catalogs have redshifts . using the photometric redshifts of galaxies from the sloan digital sky survey data release 6 ( sdss dr6 ) , we identify 39,668 clusters in the redshift range with more than eight luminous ( ) member galaxies . cluster redshifts are estimated accurately with an uncertainty less than 0.022 . monte carlo simulations show that the cluster detection rate is more than 90% for massive ( ) clusters of . we obtain the richness , the summed luminosity , and the gross galaxy number within the determined radius for identified clusters . they are tightly related to the x - ray luminosity and temperature of clusters .

clusters of galaxies in most previous catalogs have redshifts . using the photometric redshifts of galaxies from the sloan digital sky survey data release 6 ( sdss dr6 ) , we identify 39,668 clusters in the redshift range with more than eight luminous ( ) member galaxies . cluster redshifts are estimated accurately with an uncertainty less than 0.022 . the contamination rate of member galaxies is found to be roughly 20% , and the completeness of member galaxy detection reaches to% . monte carlo simulations show that the cluster detection rate is more than 90% for massive ( ) clusters of . the false detection rate is% . we obtain the richness , the summed luminosity , and the gross galaxy number within the determined radius for identified clusters . they are tightly related to the x - ray luminosity and temperature of clusters . cluster mass is related to the richness and summed luminosity with and , respectively . in addition , 685 new candidates of x - ray clusters are found by cross - identification of our clusters with the source list of the _ rosat _ x - ray survey .

1010.5116

i

we consider here the cauchy problem for the general scalar balance law @xmath4 this kind of equation has already been intensively studied : a fundamental result is the one of s. n. krukov ( * ? ? ? * theorem 1 & 5 ) , stating the existence and uniqueness of a weak entropy solution for an initial data @xmath5 . furthermore , krukov describes the dependence of the solutions with respect to the initial condition : if @xmath6 and @xmath7 are two initial data , then the associated entropy solutions @xmath8 and @xmath9 satisfy @xmath10 furthermore , a huge literature on this subject is available in the special case the flow @xmath1 depend only on @xmath8 and not on the variables @xmath11 and @xmath12 and there is no source @xmath13 ( see for example @xcite ) . we are interested here in the dependence of the solution with respect to flow @xmath1 and source @xmath2 in the case these functions depend on the three variables @xmath11 , @xmath12 and @xmath8 . this dependence with respect to flow and source has already been investigated : this question was first addressed from the point of view of numerical analysis by b. lucier @xcite who studied the case of an homogeneous flow ( @xmath14 ) , without source term ( @xmath13 ) . more recently f. bouchut & b. perthame @xcite improved this result , always in the case of an homogeneous flow and without source . chen & k. karlsen @xcite also studied this dependence , for a flow depending also on @xmath12 , but the estimate they obtained was depending on an a priori ( unknown ) bound on @xmath15 . the purpose of the present paper is to improve the recent result of r. colombo , m. mercier & m. rosini @xcite , which provided an estimate of the total variation in the general case ( with flow and source depending on the three variables @xmath11 , @xmath12 and @xmath8 ) and of the @xmath16 distance between solutions . in particular , this estimate can be compared to the one of krukov ( [ eq : kru ] ) that give a bound on the @xmath16 distance between solutions with different initial data ( but with same flow and source ) . the estimates ( [ eq : kru ] ) and ( * ? ? ? * theorem 2.6 ) look similar but in @xcite , the coefficient @xmath17 given by krukov in ( [ eq : kru ] ) is replaced by @xmath18 . consequently , we do not recover ( [ eq : kru ] ) from @xcite in the case @xmath13 ( because @xmath19 whereas @xmath20 a priori ) . in the same setting as in @xcite , we provide here an estimate on the total variation of the solution to ( [ eq : probv ] ) , and on the dependence of the solutions to ( [ eq : probv ] ) on the flow @xmath1 , on the source @xmath2 , with better hypotheses and coefficients than in @xcite . the advances are twofold . firstly , we relax hypotheses , and thus widely extend the usability of our results . more precisely , we require here less regularity in time than in @xcite , which is very useful for applications ( see @xcite ) . furthermore , we recover the same estimate as krukov when we consider the dependence toward initial conditions only . this note is organized as follows . in section [ sec : thm ] we state the main results and compare them to those in @xcite . in section [ sec : prelim ] , we give some tools on functions with bounded variations ; in sections [ sec : prooftv ] and [ sec : proofcomparison ] we prove theorems [ teo : tv ] and [ teo : estimates ] ; finally section [ sec : tech ] contains some technical lemmas used in the preceding sections .

consider the general scalar balance law in several space dimensions . the aim of this note is to improve the results of colombo , mercier , rosini who gave an estimate of the dependence of the solutions from the flow and from the source . the improvements are twofold : first the expression of the coefficients in these estimates are more precise ; second , we eliminate some regularity hypotheses thus extending significantly the applicability of our estimates . _ 2000 mathematics subject classification : _ 35l65 . _ keywords : _ multi - dimensional scalar conservation laws , krukov entropy solutions , estimate .

consider the general scalar balance law in several space dimensions . the aim of this note is to improve the results of colombo , mercier , rosini who gave an estimate of the dependence of the solutions from the flow and from the source . the improvements are twofold : first the expression of the coefficients in these estimates are more precise ; second , we eliminate some regularity hypotheses thus extending significantly the applicability of our estimates . _ 2000 mathematics subject classification : _ 35l65 . _ keywords : _ multi - dimensional scalar conservation laws , krukov entropy solutions , estimate .

1505.07607

i

a fundamental statistical problem is shrinkage estimation of a multivariate normal mean . see , for example , the february 2012 issue of _ statistical science _ for a broad range of theory , methods , and applications . let @xmath0 be multivariate normal with _ unknown _ mean vector @xmath1 and _ known _ variance matrix @xmath2 . consider the problem of estimating @xmath3 by an estimator @xmath4 under the loss @xmath5 , where @xmath6 is a _ known _ positive definite , symmetric matrix . the risk of @xmath7 is @xmath8 . the general problem can be transformed into a canonical form such that @xmath2 is diagonal and @xmath9 , the identity matrix ( e.g. , lehmann and casella @xcite , problem 5.5.11 ) . for simplicity , assume except in section [ sec3.2 ] that @xmath2 is @xmath10 and @xmath11 , where @xmath12 for a column vector @xmath13 . the letter @xmath14 is substituted for @xmath2 to emphasize that it is diagonal . for this problem , we aim to develop shrinkage estimators that are both minimax and capable of effective risk reduction over the usual estimator @xmath15 even in the heteroscedastic case ( i.e. , @xmath16 are not equal ) . an estimator of @xmath3 is minimax if and only if , _ regardless of _ @xmath17 , its risk is always no greater than @xmath18 , the risk of @xmath19 . for @xmath20 , minimax estimators different from and hence dominating @xmath21 are first discovered in the homoscedastic case where @xmath22 ( i.e. , @xmath23 ) . james and stein @xcite showed that @xmath24 is minimax provided @xmath25 . stein @xcite suggested the positive - part estimator @xmath26 , which dominates @xmath27 . throughout , @xmath28 . shrinkage estimation has since been developed into a general methodology with various approaches , including empirical bayes ( efron and morris @xcite ; morris @xcite ) and hierarchical bayes ( strawderman @xcite ; berger and robert @xcite ) . while these approaches are prescriptive for constructing shrinkage estimators , minimaxity is not automatically achieved but needs to be checked separately . for the heteroscedastic case , there remain challenging issues on how much observations with different variances should be shrunk relatively to each other ( e.g. , casella @xcite , morris @xcite ) . for the empirical bayes approach ( efron and morris @xcite ) , the coordinates of @xmath29 are shrunk directly in proportion to their variances . but the existing estimators are , in general , non - minimax ( i.e. , may have a greater risk than the usual estimator @xmath21 ) . on the other hand , berger @xcite proposed minimax estimators , including admissible minimax estimators , such that the coordinates of @xmath29 are shrunk inversely in proportion to their variances . but the risk reduction achieved over @xmath21 is insubstantial unless all the observations have similar variances . to address the foregoing issues , we develop novel minimax estimators for multivariate normal means under heteroscedasticity . there are two central ideas in our approach . the first is to develop a class of minimax estimators by generalizing a geometric argument essentially in stein @xcite ( see also brandwein and strawderman @xcite ) . for the homoscedastic case , the argument shows that @xmath27 can be derived as an approximation to the best linear estimator of the form @xmath30 , where @xmath31 is a scalar . in fact , the optimal choice of @xmath31 in minimizing the risk is @xmath32 . replacing @xmath33 by @xmath34 leads to @xmath35 with @xmath36 . this derivation is highly informative , even though it does not yield the optimal value @xmath37 . our class of minimax estimators are of the linear form @xmath38 , where @xmath39 is a nonnegative definite , diagonal matrix indicating the direction of shrinkage and @xmath31 is a scalar indicating the magnitude of shrinkage . the matrix @xmath39 is open to specification , depending on the variance matrix @xmath14 but _ not _ on the data @xmath29 . for a fixed @xmath39 , the scalar @xmath31 is determined to achieve minimaxity , depending on both @xmath14 and @xmath29 . @xcite minimax estimator corresponds to the special choice @xmath40 , thereby leading to the unusual pattern of shrinkage discussed above . the second idea of our approach is to choose @xmath39 by approximately minimizing the bayes risk with a normal prior in our class of minimax estimators . the bayes risk is used to measure average risk reduction for @xmath3 in an elliptical region as in berger @xcite . it turns out that the solution of @xmath39 obtained by our approximation strategy has an interesting simple form . in fact , the coordinates of @xmath29 are automatically segmented into two groups , based on their bayes `` importance '' ( berger @xcite ) , which is of the same order as the coordinate variances when the specified prior is homoscedastic . the coordinates of high bayes `` importance '' are shrunk inversely in proportion to their variances , whereas the remaining coordinates are shrunk in the direction of the bayes rule . this shrinkage pattern may appear paradoxical : it may be expected that the coordinates of high bayes `` importance '' are to be shrunk in the direction of the bayes rule . but that scheme is inherently aimed at reducing the bayes risk under the specified prior and , in general , fails to achieve minimaxity ( i.e. , it may lead to even a greater risk than the usual estimator @xmath21 ) . in addition to simplicity and minimaxity , we further show that the proposed estimator is scale adaptive in reducing the bayes risk : it achieves close to the minimum bayes risk , with the difference no greater than the sum of the 4 highest bayes `` importance '' of the coordinates of @xmath29 , simultaneously over a scale class of normal priors ( including the specified prior ) . to our knowledge , the proposed estimator seems to be the first one with such a property in the general heteroscedastic case . previously , in the homoscedastic case , @xmath41 is known to achieve the minimum bayes risk up to the sum of 2 ( equal - valued ) bayes `` importance '' of the coordinates over the scale class of homoscedastic normal priors ( efron and morris @xcite ) . the rest of this article is organized as follows . section [ sec2 ] gives a review of existing estimators . section [ sec3 ] develops the new approach and studies risk properties of the proposed estimator . section [ sec4 ] presents a simulation study . section [ sec5 ] provides concluding remarks . all proofs are collected in the .

consider the problem of estimating a multivariate normal mean with a known variance matrix , which is not necessarily proportional to the identity matrix . the coordinates are shrunk directly in proportion to their variances in efron and morris ( _ j . amer . statist . assoc . _ statist . _ * 4 * ( 1976 ) 223226 ) minimax estimators . we propose a new minimax estimator , by approximately minimizing the bayes risk with a normal prior among a class of minimax estimators where the shrinkage direction is open to specification and the shrinkage magnitude is determined to achieve minimaxity . the proposed estimator has an interesting simple form such that one group of coordinates are shrunk in the direction of berger s estimator and the remaining coordinates are shrunk in the direction of the bayes rule . moreover , the proposed estimator is scale adaptive : it can achieve close to the minimum bayes risk simultaneously over a scale class of normal priors ( including the specified prior ) and achieve close to the minimax linear risk over a corresponding scale class of hyper - rectangles .

consider the problem of estimating a multivariate normal mean with a known variance matrix , which is not necessarily proportional to the identity matrix . the coordinates are shrunk directly in proportion to their variances in efron and morris ( _ j . amer . statist . assoc . _ * 68 * ( 1973 ) 117130 ) empirical bayes approach , whereas inversely in proportion to their variances in berger s ( _ ann . statist . _ * 4 * ( 1976 ) 223226 ) minimax estimators . we propose a new minimax estimator , by approximately minimizing the bayes risk with a normal prior among a class of minimax estimators where the shrinkage direction is open to specification and the shrinkage magnitude is determined to achieve minimaxity . the proposed estimator has an interesting simple form such that one group of coordinates are shrunk in the direction of berger s estimator and the remaining coordinates are shrunk in the direction of the bayes rule . moreover , the proposed estimator is scale adaptive : it can achieve close to the minimum bayes risk simultaneously over a scale class of normal priors ( including the specified prior ) and achieve close to the minimax linear risk over a corresponding scale class of hyper - rectangles . for various scenarios in our numerical study , the proposed estimators with extreme priors yield more substantial risk reduction than existing minimax estimators .

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gammma - ray bursts ( grbs ) were discovered serendipitously as flashes in the sky " by the nuclear test - ban monitoring satellites vela ( us ) and konus ( ussr ) . these were soon recognized to be a natural phenomenon . the first data were publicly released by @xcite and @xcite . the mysterious origin of this new astrophysical transient , lasting up to tens of seconds in non - thermal gamma - ray radiation , gradually triggered a huge interest from the high - energy astrophysics community . the burst and transient source experiment ( batse ) on nasa s compton gamma - ray observatory , launched in 1991 , identified an isotropic distribution in the sky and revealed the existence of short and long gamma - ray bursts , whose durations are broadly distributed around 0.3s and 30s , respectively @xcite . moreover , the sensitivity of batse showed a deficit in faint bursts in the number - versus - intensity distribution distinct from a -3/2 powerlaw . @xcite hereby established that gamma - ray bursts are of cosmological origin . more recently , the italian - dutch satellite bepposax discovered x - ray flashes @xcite , which may be closely related to long gamma - ray bursts . their non - thermal emissions are herein best accounted for by internal shocks in ultrarelativistic outflows in the fireball model proposed by @xcite with unknown inner engine producing these baryon - poor outflows . if gravitationally powered , their compact size , based on short - time scale variabilities in the gamma - ray lightcurves , may involve the formation of black holes surrounded by a high - mass torus @xcite . at cosmological distances , the isotropic equivalent luminosities are on the order of @xmath0 erg s@xmath1 as bright as @xmath2 solar luminosities . these events , widely acclaimed as the biggest bang since the big bang " , are now known to light up the universe about once per minute . what , then , is their origin ? the most powerful observational method to encyst the enigmatic energy source is through _ calorimetry on all radiation channels_. this involves measuring the energy output in emissions in the electromagnetic spectrum , as well as in yet unseen " emissions in neutrinos and gravitational waves . perhaps thus shall we be in a position to determine their constitution , and decide whether their inner engine is _ baryonic _ , such as a neutron star , or _ non - baryonic _ : a black hole energized by spin . the exact solution of roy p. @xcite see table i for a parametrization shows an energy per unit mass which is anomalously large at high spin rates , exceeding that of a rapidly spinning neutron by an order of magnitude . by this potentially large and baryon - free energy reservoir , a kerr black hole is a leading candidate for the inner engine of grbs , at least those of long - durations in view of the following observational results . the italian - dutch bepposax satellite , launched in 1996 , dramatically changed the landscape of long grbs with the discovery by e. costa @xcite of x - ray afterglows in grb970228 ( confirmed in observations by the asca and rosat satellites ) . these lower - energy emissions permit accurate localizations , enabling follow - up by optical telescopes . thus , j. van paradijs @xcite pointed the isaac newton telescope and the william herschel telescope to further discover optical emissions associated with the same event . these lower - energy x - ray and optical emissions agree remarkably well with the expected decay of shocks in ultrarelativistic , baryon - poor outflows in the previously developed fireball model . even lower - energy , radio - afterglows have been observed in some cases , including grb970228 @xcite . the same optical observations have now led to tens of grbs with individually measured redshifts , up to @xmath3 in grb050904 detected by the recently launched swift satellite @xcite . optical follow - up identified the association of long grbs with supernovae . in particular , the unambiguous association of grb030329 with sn2003dh ( @xmath4mpc ) @xcite identified type ib / c supernovae as the parent population of long grbs ( fig . 1 ) . this observation confirms their association with core - collapse supernovae of massive stars , proposed by s. woosley @xcite . massive stars have characteristically short lifetimes , whereby grb - supernovae track the cosmological event rate . this conclusion is quantitatively confirmed by consistent estimates of the true - to - observed grb - sne event rate of 450 - 500 deduced from two independent analysis , based on geometrical beaming factors @xcite and locking to the star - formation rate @xcite . these event rates correspond to a true cosmological event rate of about one per minute . core - collapse supernovae from massive stars are believed to produce neutron stars or black holes . the former are perhaps best known from the rapidly spinning neutron star ( a pulsar ) in the crab nebula , a remnant of what was probably a type ii supernova in 1054 . the second is more difficult to ascertain with certainty , such as in the more recent type ii event sn1987 in the large megellanic cloud . at close proximity , its burst in mev - neutrinos was detected by kamiokanda and imb ( see @xcite ) which provided direct evidence for the formation of matter at nuclear densities . the absence of a pulsar or a point - source of x - ray radiation in this case suggests continued collapse probably into a stellar mass black hole . type ii and type ib / c supernovae are both believed to represent the endpoint of massive stars @xcite , and possibly so in binaries such as the type ii / ib event sn1993j @xcite . this binary association suggests a hierarchy , wherein hydrogen - rich , envelope retaining snii are associated with wide binaries , while hydrogen - poor , envelope stripped snib and snic are associated with increasingly compact binaries @xcite . by tidal coupling , the primary star in the latter will rotate at approximately the orbital period at the moment of core collapse . with an evolved core @xcite , these type ib / c events in particular are believed to produce a spinning black hole @xcite . however , the branching ratio of type ib / c into grb - supernovae is small , @xmath5 as calculated from the true grb - supernova event rate relative to the observed event rates of supernovae of type ii and type ib / c . this ratio is remarkably small , suggesting a higher - order down - selection process . the identification of long grbs with core - collapse supernovae brings into scope their potential gravitational - wave emissions . this may be due to a variety of mechanisms associated with rapidly rotating fluids . in particular , emissions will be produced by bar - mode instabilities producing fragmentation of orbiting matter prior to the formation of a compact object @xcite or in the formation of multiple compact objects @xcite , non - axisymmetries in accretion disks @xcite , as well as after the formation into a rapidly spinning black hole . the first may produce an initial splash " of gravitational radiation @xcite , the second a bi - model spectrum containing high - frequency emissions produced by non - axisymmetric perturbations of the black hole and low - frequency emissions from the disk , whereas the latter creates a long - duration burst of low - frequency gravitational radiation . calculations this time - evolving spectrum of gravitational - radiation are only beginning to be addressed by computational hydro- and magnetohydrodynamics , during initial collapse @xcite and in the formation of a non - axisymmetric torus @xcite . emissions from matter surrounding a stellar - mass black hole are in the frequency range of sensitivity of the gravitational - wave experiments ligo @xcite and virgo @xcite . understanding their wave - forms can serve strategic search - and - detection algorithms , triggered by gamma - ray observations @xcite or by their associated supernova . the latter is more common by the aforementioned true - to - observed event ratio of grb - supernovae . in this review , we shall discuss a theory of grbs from rotating black holes . current phenomenology on grb - supernovae poses a challenge to model * the durations of long grbs of tens of seconds * the formation of an associated aspherical supernova * the launch of ultrarelativistic jets with beamed gamma - ray emissions * a small branching ratio of type ib / c supernovae into grbs of less than @xmath6 this phenomenology will be directly linked to the spin energy of the black hole . quite generally , we are left with the task of modeling all possible radiation channels produced by the putative kerr black hole , including gravitational radiation , mev - neutrino emissions , and magnetic winds . just as lower - energy x - ray and optical afterglow emissions linked long grbs to supernovae , we expect that the detection of a contemporaneous burst in gravitational - radiation will lift the veil on their enigmatic inner engine . a precise determination of this link will enable the identification of a kerr black hole . the same might apply to type ii and type ib / c supernovae . understanding the energy source of long grbs might tell us also about the constitution of short grbs . the first _ faint _ x - ray afterglows have been detected of the short event grb050509b by swift @xcite and grb050709 by hete - ii @xcite , further providing the low redshifts @xmath7 and @xmath8 , respectively . the nature of host galaxies , preferably ( older ) elliptical galaxies for short bursts versus ( young ) star - forming galaxies for long bursts , supports the origin of short events in binary coalescence of black holes and neutron stars @xcite . after completion of this review , @xcite outlined a very interesting prospect for long - lasting supernova - like signatures to short bursts from the debris of a neutron star binary coalescence with a similar partner or black hole . earlier , we explained the dichotomy of short and long events in terms of hyper- and suspended accreting onto slowly and rapidly rotating black holes , respectively @xcite . thus , we predicted similar x - ray afterglows with the property that those to short events are relatively faint : _ the short burst is identical to the final moment of a long burst of gamma - rays_. hyperaccreting , slowly rotating black holes can be produced through various channels . black - hole neutron star binaries are remnants of core - collapse events in binaries ( @xcite ; unless formed by capture in dense stellar clusters ) . any rapidly spinning black hole will be spun - down in the core - collapse process , possibly representing a prior grb - supernova event ( this review ) . the merger of a neutron star onto a black hole @xcite could produce a state of hyperaccretion and produce a short gamma - ray burst . this scenario is consistent with the predominance of long over short events . less clear is the potential for gamma - ray emissions from the merger of two neutron stars ( e.g. @xcite ) which , however , should also produce a slowly rotating black hole . finally , slowly rotating black holes can also be produced in core - collapse of massive star in isolation or a wide binary . here as well , may detection of their gravitational - wave emissions provide a probe to differentiate amongst these different types of short bursts . in 2 , we give a practical outlook on kerr black holes and gravitational radiation . in 3 , we review the formation and evolution ( kick - velocities , growth and spin down ) of kerr black holes in core collapse supernovae . this includes the process of converting spin energy into radiation and estimates on the lifetime of rapid spin . in 4 , we calculate multipole mass - moments in a torus due to a hydrodynamical instability . by asymptotic analysis , the energies in the various radiation channels are given in 5 . an mechanism for launching ultrarelativistic jets from rotating black holes by a small fraction of black hole spin is given in 6 . specific predictions for these flashes " as sources for the observatories ligo and virgo are given in 7 .

current phenomenology suggests the presence of a compact baryon - poor energy source to cosmological gamma - ray bursts reacting to high - density matter . the association of short and long gamma - ray bursts with hyper- and suspended accretion onto slowly and rapidly spinning black holes predicts weak x - ray afterglow emissions from short bursts , as in grb050509b and grb050709 . this predicts a long - duration burst of gravitational radiation powered by the spin - energy of the black hole . detection of the expected chirps and long - duration bursts of gravitational waves promises identification of kerr black holes as the most luminous objects in the universe .

current phenomenology suggests the presence of a compact baryon - poor energy source to cosmological gamma - ray bursts reacting to high - density matter . the association of short and long gamma - ray bursts with hyper- and suspended accretion onto slowly and rapidly spinning black holes predicts weak x - ray afterglow emissions from short bursts , as in grb050509b and grb050709 . long gamma - ray bursts are probably the birth place of rapidly spinning high - mass black holes in core - collapse of massive stars , as in grb030329 with supernova sn2003dh . this predicts a long - duration burst of gravitational radiation powered by the spin - energy of the black hole . in contrast to mev - neutrino emissions , as in sn1987a , this can be tested by the advanced detectors ligo and virgo about once per year up to distances of about 100mpc . detection of the expected chirps and long - duration bursts of gravitational waves promises identification of kerr black holes as the most luminous objects in the universe .

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the sixties saw two independent discoveries : the first grb670702 by the vela satellite and the exact solution of rotating black holes by roy p. kerr . through observational campaigns with batse , bepposax , the interplanetary network ( ipn ) , hete - ii and now swift , we have come to understand the phenomenology of grbs . long bursts are association with supernovae , representing a rare and extraordinary powerful cosmological transient , taking place about once a minute and reaching the earliest epochs in the universe . independently , through the works of @xcite and others , we have come to understand the potential significance of kerr black holes as compact , baryon - free energy sources with certain universal properties . the applications to high - energy astrophysics should be enormous ( e.g. @xcite ) , of supermassive black holes in active galactic nuclei including our own galaxy @xcite , and of stellar mass black holes in microquasars @xcite and , possibly , gamma - ray bursts . while the formation - process of supermassive black holes remains inconclusive , the birthplace of stellar mass black holes is most probably core - collapse supernovae of massive stars . evidence for kerr black holes as the energy source to high - energy astrophysical processes remains elusive , however . recent measurements on frame - dragging by x - ray spectroscopy , typically during inactive states of the black hole , are encouraging in this respect ( a.c . fabian , these proceedings ) . specically , we propose that grb - supernovae are powered by rapidly rotating black holes , wherein ( 1 ) the durations of long grbs of tens of seconds are identified with the lifetime of rapid spin of the black hole in a state of suspended accretion , ( 2 ) an accompanying supernova is radiation - driven by magnetic winds from a torus in suspended accretion , ( 3 ) ultrarelativistic outflows are launched by gravitational spin - orbit coupling with charged particles along open magnetic field - lines , and ( 4 ) a small branching ratio of type ib / c supernovae into grbs is attributed to the small probability of producing black holes with small kick velocities . modeling short grbs from slowly rotating black holes , we predicted x - ray afterglows very similar but weaker ( in total energies ) than those of long bursts . the recent discovery of faint x - ray afterglows to grb050509b and grb050709 fit well within this scheme . gamma - ray bursts present a potentially powerful link between rotating black holes and gravitational radiation . strategic searches for their chirps in binary coalescence of neutron stars and black holes , or long - duration bursts in gravitational radiation during radiative spin - down of a high - mass black hole , can be pursued by the advanced detectors ligo and virgo . strategic searches are preferrably pursued in combination with upcoming optical - radio supernova surveys , e.g. , pan - starrs in hawaii @xcite in combination with the low frequency array @xcite , see also @xcite . in light of the proposed supernova - like signatures from the debris of a neutron star @xcite , these strategies might apply to both short and long bursts . * acknowledgment . * the author thanks a. levinson , r.p . kerr , r. preece , and david wiltshire for constructive comments . this research is supported by the ligo observatories , constructed by caltech and mit with funding from nsf under cooperative agreement phy 9210038 . the ligo laboratory operates under cooperative agreement phy-0107417 . this paper has been assigned ligo document number ligo - 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( reprinted from @xcitethe american astronomical society . ) * fig . 2 * ( @xmath201 ) black holes with small kick velocities remain centered in core - collapse of a uniformly rotating massive star . shown is the accumulated specific angular momentum of the central object ( arbitrary units ) versus dimensionless orbital period @xmath202 . arrows indicate the evolution as a function of time . kerr black holes exist _ inside _ the outer curve ( diamonds ) . a black hole forms in a first - order transition following the formation and collapse of a torus . this produces a short burst in gravitational radiation . when centered , the black hole surges to a high - mass object by direct infall of matter with relatively low specific angular momentum , up to the inner continuous curve ( isco ) . at this point , the black hole either spins up by continuing accretion or spins down radiatively against gravitational radiation emitted by a surrounding non - axisymmetric torus . in this state , the black hole creates a baryon poor jet as input to grb - afterglow emissions . this continues until the angular velocity of the black hole equals that of the torus ( dot - dashed line ) . this scenario fails for black holes with typical kick velocities with inevitable escape from the high - density core , prohibiting the formation of a high mass black hole surrounded by a high - mass torus . the probability of small kick velocities defines the branching ratio of type ib / c supernovae into long grbs . ( @xmath203 ) the black hole mass @xmath19 and rotational energy @xmath104 of formed after surge in case of small kick velocities , expressed relative to the mass @xmath204 of the progenitor he - star . the results are shown in cylindrical geometry ( continuous ) and spherical geometric ( dashed ) . note the broad distribution of high - mass black holes with large rotational energies of @xmath205 ( spherical to cylindrical ) of @xmath206 . ( reprinted from @xcitethe american astronomical society . ) * fig . 3 * a uniformly magnetized torus represented by two counter - oriented current rings around a black hole ( c ) forms out of both core - collapse ( a1-b1 ) in massive stars and tidal break - up ( a2-b2 ) in black hole - neutron star coalescence , followed by a single reconnection event ( b2-c ) . ( reprinted from @xcitethe american astronomical society . ) * fig . 4 * the stability diagram showing the neutral stability curves for non - axisymmetric buckling modes in a torus of an inviscid incompressible fluid of arbitrary minor - to - major radius @xmath93 . curves of critical rotation index @xmath207 are labeled with azimuthal quantum number @xmath208 , where instability sets in above and stability sets in below . ( reprinted from @xcitethe american astrophysical society . ) * fig . 5 * ( @xmath201 ) grb - supernovae from rotating black holes predict a contemporaneous long - duration burst in gravitational radiation within the range of sensitivity of upcoming gravitational - wave detectors ligo and virgo within a distance of about 100mpc . the corresponding event rate is about one per year . blue bar " denotes the distribution of dimensionless strain amplitudes for a distribution of sources corresponding to a range of black hole masses and efficiency factors , assuming matched filtering . in practice , the sensitivity will depend less on using time frequency trajectory methods with correlations in the spectral domain . @xmath193 the cosmological distribution of grb - supernovae is locked to the star - formation rate @xmath209 . this enables the calculation of the expected contribution to the stochastic background in gravitational radiation , here shown in terms of the spectral energy - density @xmath210 , the strain amplitude @xmath211 and the spectral closure density @xmath187 . the results are calculated for uniform mass - distributions of the black hole , @xmath212 ( top curves ) and @xmath213 ( lower curves ) with @xmath214 ( solid curves ) and @xmath215 ( dashed curves ) . the extremal value of @xmath187 is in the neighborhood of maximal sensitivity of ligo and virgo . ( reprinted from @xcite2004 american physical society . )

long gamma - ray bursts are probably the birth place of rapidly spinning high - mass black holes in core - collapse of massive stars , as in grb030329 with supernova sn2003dh . in contrast to mev - neutrino emissions , as in sn1987a , this can be tested by the advanced detectors ligo and virgo about once per year up to distances of about 100mpc .

current phenomenology suggests the presence of a compact baryon - poor energy source to cosmological gamma - ray bursts reacting to high - density matter . the association of short and long gamma - ray bursts with hyper- and suspended accretion onto slowly and rapidly spinning black holes predicts weak x - ray afterglow emissions from short bursts , as in grb050509b and grb050709 . long gamma - ray bursts are probably the birth place of rapidly spinning high - mass black holes in core - collapse of massive stars , as in grb030329 with supernova sn2003dh . this predicts a long - duration burst of gravitational radiation powered by the spin - energy of the black hole . in contrast to mev - neutrino emissions , as in sn1987a , this can be tested by the advanced detectors ligo and virgo about once per year up to distances of about 100mpc . detection of the expected chirps and long - duration bursts of gravitational waves promises identification of kerr black holes as the most luminous objects in the universe .

1503.04066

r

we consider data sets describing contacts between individuals , collected by the sociopatterns collaboration ( http://www.sociopatterns.org ) in three different settings : a workplace ( office building , invs ) @xcite , a high school ( thiers13 ) @xcite and a scientific conference ( sfhh ) @xcite . these data correspond to the close face - to - face proximity of individuals equipped with wearable sensors , at a temporal resolution of @xmath0 seconds @xcite . table [ tab : data ] summarises the characteristics of each data set . the contact data are represented by temporal networks , in which nodes represent the participating individuals and a link between two nodes @xmath1 and @xmath2 at time @xmath3 indicates that the two corresponding persons were in contact at that time . these three data sets were chosen as representative of different types of day - to - day contexts and of different contact network structures : the sfhh data correspond to a rather homogeneous contact network ; the invs and thiers13 populations were instead structured in departments and classes , respectively . moreover , high school classes ( thiers13 ) are of similar sizes while the invs department sizes are unequal . finally , the high school contact patterns ( thiers13 ) are constrained by strict and repetitive school schedules , while contacts in offices are less regular across days . to quantify how the incompleteness of data , assumed to stem from a uniformly random participation of individuals to the data collection , affects the outcome of simulations of dynamical processes , we consider as ground truth the available data and perform population resampling experiments by removing a fraction @xmath4 of the nodes uniformly at random . ( note that the full data sets are also samples of all the contacts that occurred in the populations , as the participation rate was lower than @xmath5 in each case . in the thiers13 case however , the participation rate was quite high . ) we then simulate on the resampled data the paradigmatic susceptible - infectious - recovered ( sir ) and the susceptible - infectious - susceptible ( sis ) models of epidemic propagation . in these models , a susceptible ( s ) node becomes infectious ( i ) at rate @xmath6 when in contact with an infectious node . infectious nodes recover spontaneously at rate @xmath7 . in the sir model , nodes then enter an immune recovered ( r ) state , while in the sis model , nodes become susceptible again and can be reinfected . the quantities of interest are for the sir model the distribution of epidemic sizes , defined as the final fraction of recovered nodes , and for the sis model the average fraction of infectious nodes @xmath8 in the stationary state . we also calculate for the sir model the fraction of epidemics that infect more than 20% of the population and the average size of these epidemics . for the sis model , we determine the epidemic threshold @xmath9 for different values of @xmath7 : it corresponds to the value of @xmath6 that separates an epidemic - free state ( @xmath10 ) for @xmath11 from an endemic state ( @xmath12 ) for @xmath13 , and is thus an important indicator of the epidemic risk . we refer to the methods section for further details on the simulations . we then present several methods for constructing surrogate data using only information contained in the resampled data . we compare for each data set the outcomes of simulations performed on the whole data set , on resampled data sets with a varying fraction of nodes removed , @xmath4 , and on the reconstructed data sets built using these various methods . missing data are known to affect the various properties of contact networks in different ways . in particular , the number of neighbours ( degree ) of a node decreases as the fraction @xmath4 of removed nodes increases , since removing nodes also removes links to these nodes . under the hypothesis of uniform sampling , the average degree @xmath14 becomes @xmath15 for the resampled network @xcite . as a result , the density of the resampled aggregated contact network , defined as the number of links divided by the total number of possible links between the nodes , does not depend on @xmath4 . the same reasoning applies to the density @xmath16 of links between groups of nodes @xmath17 and @xmath18 , defined as the number of links @xmath19 between nodes of group @xmath17 and nodes of group @xmath18 , normalised by the maximum possible number of such links , @xmath20 , where @xmath21 is the number of nodes of group @xmath17 ( for @xmath22 , the maximum possible number of links is @xmath23 ) : both the expected number of neighbours of group @xmath18 for nodes of group @xmath17 ( given by @xmath24 ) and the number @xmath25 of nodes in group @xmath18 are indeed reduced by a factor @xmath26 , so that @xmath16 remains constant . this means that the link density contact matrix , which gathers these densities and gives a measure of the interaction between groups ( here classes or departments ) , is stable under uniform resampling . we illustrate these results on our empirical data sets in supplementary figures 1 , 2 , 4 and 5 . table [ tab : sim_cm ] and supplementary figure 2 show in particular that the similarities between the original and resampled matrices are high for all data sets ( see supplementary figures 45 for the contact matrices themselves ) . finally , the temporal statistics of the contact network are not affected by population sampling , as already noted in @xcite for other data sets : the distributions of contact and inter - contact durations ( the inter - contact durations are the times between consecutive contacts on a link ) , of number of contacts per link and of cumulated contact durations ( i.e. , of the link weights in the aggregated network ) do not change when the network is sampled uniformly ( supplementary figure 1 ) . in the case of structured population , an interesting property is moreover illustrated in supplementary figures 67 : although the distributions of contact durations occurring between members of the same group or between individuals belonging to different groups are indistinguishable , this is not the case for the distributions of the numbers of contacts per link nor , as a consequence , for the distributions of cumulated contact durations . in fact , both cumulated contact durations and numbers of contacts per link are more broadly distributed for links joining members of the same group . the figures show that this property is stable under uniform resampling . despite the robustness of these properties , the outcome of simulations of epidemic spread is strongly affected by the resampling . as fig . [ sampling ] illustrates for instance , the probability of large outbreaks in the sir model decreases strongly as @xmath4 increases and even vanishes at large @xmath4 . as mentioned above , such a result is expected , since the removed nodes act as if they were immunised : sampling hinders the propagation in simulations by removing transmission routes between the remaining nodes . as a consequence , the prevalence and the final size of the outbreaks are systematically underestimated by simulations of the sir model on the resampled network with respect to simulations on the whole data set ( for the sis model , the epidemic threshold is overestimated ) : resampling leads overall to a systematic underestimation of the epidemic risk , and fig . [ sampling ] illustrates the extent of this underestimation for the data at hand . we now present a series of methods to improve the estimation of the epidemic risk in simulations of epidemic spread on temporal network data sets in which nodes ( individuals ) are missing uniformly at random . note that we do not address here the problem of link prediction @xcite as our aim is not to infer the missing contacts . the hierarchy of methods we put forward uses increasing amounts of information corresponding to increasing amounts of detail on the group and temporal structure of the contact patterns , as measured in the resampled network . we moreover assume that the timelines of scheduled activity are known ( i.e. , nights and weekends , during which no contact occurs ) . for each data set , considered as ground truth , we create resampled data sets by removing at random a fraction @xmath4 of the @xmath27 nodes . we then measure on each resampled data set a series of statistics of the resulting contact network and construct stochastic , surrogate versions of the missing part of the network by creating for each missing node a surrogate instance of its links and a synthetic timeline of contacts on each surrogate link , in the different ways described below ( see supplementary information and methods section for more details on their practical implementation ) . method 0 . as discussed above , the first effect of missing data is to decrease the average degree of the aggregate contact network , while keeping its density constant . hence , the simplest approach is to merely compensate this decrease . we therefore measure the density of the resampled contact network @xmath28 , as well as the average aggregate duration of the contacts , @xmath29 . we then add back the missing nodes and create surrogate links between these nodes and between these nodes and the nodes of the resampled data set at random , with the only constraint to keep the overall link density fixed to @xmath28 . we then attribute to each surrogate link the same weight @xmath29 and create for each link a timeline of randomly chosen contact events of equal length @xmath30 ( the temporal resolution of the data set ) whose total duration gives back @xmath29 . method w. the heterogeneity of aggregated contact durations is known to play a role in the spreading patterns of model diseases @xcite . we therefore refine method 0 by collecting in the resampled data the list @xmath31 of aggregate contact durations , or weights ( w ) . we build the surrogate links and surrogate timelines of contacts on each link as in method 0 , except that each surrogate link carries a weight extracted at random from @xmath31 , instead of the average @xmath29 . method ws . the fact that the population is divided into groups of individuals such as classes or departments can have a strong impact on the structure of the contact network @xcite and on spreading processes @xcite . we thus measure here the link density contact matrix of the resampled data , and construct surrogate links in a way to keep this matrix fixed ( equal to the value measured in the resampled data ) , in the spirit of stochastic block models with fixed numbers of edges between blocks @xcite . we collect in the resampled data two separate lists of aggregate contact durations : @xmath32 gathers the weights of links between individuals belonging to the same group , and @xmath33 is built with the weights of links joining individuals of different groups . for each surrogate link , its weight is extracted at random either from @xmath32 if it joins individuals of the same group or from @xmath33 if it associates individuals of different groups . timelines are then attributed to links as in w. this method assumes that the number of missing nodes in each group is known , and preserves the group structure ( s ) of the population . method wt . several works have investigated how the temporal characteristics of networks ( such as burstiness ) can slow down or accelerate spreading @xcite . in order to take these characteristics into account , we measure in the resampled data the distributions of number of contacts per link and of contact and inter - contact durations , in addition to the global network density . we build surrogate links as in method w , and construct on each link a synthetic timeline in a way to respect the measured temporal statistics ( t ) of contacts . more precisely , we attribute at random a number of contacts ( taken from the measured distribution ) to each surrogate link , and then alternate contact and inter - contact durations taken at random from the respective empirical distributions . method wst . this method conserves the distribution of link weights ( w ) , the group structure ( s ) , and the temporal characteristics of contacts ( t ) : surrogate links are built and weights assigned as in method ws , and contact timelines on each link as in method wt . each of these methods uses a different amount of information gathered from the resampled data . methods 0 , w and wt include an increasing amount of detail on the temporal structure of contacts : method 0 assumes homogeneity of aggregated contact durations , while w takes into account their heterogeneity , and wt reproduces heterogeneities of contact and inter - contact durations . on the other hand , neither of these three methods assume any knowledge of the population group structure . this can be due either to an effective complete lack of knowledge about the population structure , as in the sfhh data , or also to the lack of data on the repartition of the missing nodes in the groups . methods ws and wst on the other hand reproduce the group structure as in a stochastic block model with fixed number of links within and between groups , and take into account the difference between the distributions of numbers of contacts and aggregate durations between individuals of the same or of different groups . indeed , links within groups correspond on average to larger weights , as found empirically in @xcite and discussed above ( supplementary figures 56 ) . overall , method wst is the one that uses most information measured in the resampled data . ( additional properties such as the transitivity -which is also stable under resampling procedure , see supplementary figure 3- can also be measured in the resampled data and imposed in the construction of surrogate links , as detailed in the supplementary information . this comes however at a strong computational cost and we have verified that it does not impact significantly our results , as shown in the supplementary figure 20 . ) we check in table [ tab : sim_cm ] and supplementary figures 813 that the statistical properties of the resulting reconstructed ( surrogate ) networks , obtained by the union of the resampled data and of the surrogate links , are similar to the ones of the original data for the wst method . we emphasise again that our aim is not to infer the true missing contacts , so that we do not compare the detailed structures of the surrogate and original contact networks . figures [ fig : inferinvs ] , [ fig : inferthiers ] , [ fig : infersfhh ] and supplementary figures 1619 display the outcome of sir spreading simulations performed on surrogate networks obtained using the various reconstruction methods , compared with the outcome of simulations on the resampled data sets , for various values of @xmath4 . method 0 leads to a clear overestimation of the outcome and does not capture well the shape of the distribution of outbreak sizes . method w gives only slightly better results . the overall shape of the distribution is better captured for the three reconstruction methods using more information : ws , wt and wst ( note that for the sfhh case the population is not structured , so that w and ws are equivalent , as are wt and wst ) . the wst method matches best the shape of the distributions and yields distributions much more similar to those obtained by simulating on the whole data set than the simulations performed on the resampled networks . we also show in fig . [ fig : dist ] the fraction of outbreaks that reach at least @xmath34 of the population and the average epidemic size for these outbreaks . in the case of simulations performed on resampled data , we rapidly lose information about the size and even the existence of large outbreaks as @xmath4 increases . simulations using data reconstructed with methods 0 and w , on the contrary , largely overestimate these quantities , which is expected as infections spread easier on random graphs than on structured graphs @xcite , especially if the heterogeneity of the aggregated contact durations is not considered @xcite . taking into account the population structure or using contact sequences that respect the temporal heterogeneities ( broad distributions of contact and inter - contact durations ) yield better results ( ws and wt cases , respectively ) . overall , the wst method , for which the surrogate networks respect all these constraints , yields the best results . we show in the supplementary information that similar results are obtained for different values of the spreading parameters . moreover , as shown in fig . [ fig : sis_invs ] and supplementary figures 1415 , the phase diagram obtained for the sis model when using reconstructed networks is much closer to the original than for resampled networks . overall , simulations on networks reconstructed using the wst method yield a much better estimation of the epidemic risk than simulations using resampled network data , for both sis and sir models . even when simulations are performed on reconstructed contact patterns built with the wst method , the maximal outbreak sizes are systematically overestimated ( figs . [ fig : inferinvs ] - [ fig : infersfhh ] ) , as well as , in most cases , the probability and average size of large outbreaks , especially for the sfhh case ( figs . [ fig : infersfhh ] - [ fig : dist ] ) . these discrepancies might stem from structural and/or temporal correlations present in empirical contact data that are not taken into account in our reconstruction methods . in order to test this hypothesis , we construct several reshuffled data sets and use them as initial data in our resampling and reconstruction procedure . we use both structural and temporal reshuffling as described in the methods section , in order to remove either structural correlations , temporal correlations , or both , from the original data sets . we then proceed to a resampling and reconstruction procedure ( using the wst method ) as for the original data , and perform numerical simulations of sir processes . as for the original data , simulations on resampled data lead to a strong underestimation of the process outcome , and simulations using the reconstructed data gives much better results . we show in the supplementary figures 21 - 22 that we still obtain discrepancies , and in particular overestimations of the largest epidemic sizes , when we use temporally reshuffled data in which the link structure of the contact network is maintained . if on the other hand we use data in which the network structure has been reshuffled in a way to cancel structural correlations within each group , the reconstruction procedure gives a very good agreement between the distributions of epidemic sizes of original and reconstructed data , as shown in fig . [ fig : infercmshuffled ] . more precisely we consider here `` cm - shuffled '' data , i.e. , contact networks in which the links have been reshuffled randomly but separately for each pair of groups , i.e. , a link between an individual of group @xmath17 and an individual of group @xmath18 is still between groups @xmath17 and @xmath18 in the reshuffled network . the difference with the case of non - reshuffled empirical data is particularly clear for the sfhh case . this indicates that the overestimation observed in figs . [ fig : inferinvs ] - [ fig : infersfhh ] is mostly due to the fact that the reconstructed data does not reproduce small scale structures of the contact networks : such structures might be due to e.g. groups of colleagues or friends , whose composition is neither available as metadata nor detectable in the resampled data sets . when the fraction @xmath4 of nodes excluded by the resampling procedure becomes large , the properties of the resampled data may start to differ substantially from those of the whole data set ( figs . s1 & s2 ) . as a result , the distributions of epidemic sizes of sir simulations show stronger deviations from those obtained on the whole data set ( fig . [ fig : infer_high ] ) , even if the epidemic risk evaluation is still better than for simulations on the resampled networks ( fig . [ fig : dist ] ) . most importantly however , the information remaining in the resampled data at large @xmath4 can be insufficient to construct surrogate contacts . this happens in particular if an entire class or department is absent from the resampled data or if all the resampled nodes of a class / department are disconnected ( see methods for details ) . we show in the bottom plots of fig . [ fig : dist ] the failure rate , i.e. , the fraction of cases in which we are not able to construct surrogate networks from the resampled data . the failure rate increases gradually with @xmath4 for the invs data since the groups ( departments ) are of different sizes . for the thiers13 data , all classes are of similar sizes so that the failure rate reaches abruptly a large value at a given value of @xmath4 . for the sfhh data , we can always construct surrogate networks as the population is not structured . another limitation of the reconstruction method lies in the need to know the number of individuals missing in each department or class . if these numbers are completely unknown , giving an estimation of outbreak sizes is impossible as adding arbitrary numbers of nodes and links to the resampled data can lead to arbitrarily large epidemics . the methods are however still usable if only partial information is available . for instance , if only the overall missing number of individuals is available , it is possible to use the wt method , which still gives sensible results . moreover , if @xmath4 is only approximately known , e.g. , @xmath4 is known to be within an interval of possible values @xmath35 $ ] , it is possible to perform two reconstructions using the respective hypothesis @xmath36 and @xmath37 and to give an interval of estimates . we provide an example of such procedure in supplementary figure 23 .

data describing human interactions often suffer from incomplete sampling of the underlying population . as a consequence , the study of contagion processes using data - driven models can lead to a severe underestimation of the epidemic risk . here we present a systematic method to alleviate this issue and obtain a better estimation of the risk in the context of epidemic models informed by high - resolution time - resolved contact data . we consider several such data sets collected in various contexts and perform controlled resampling experiments . we show how the statistical information contained in the resampled data can be used to build a series of surrogate versions of the unknown contacts . we discuss limitations and potential improvements of our method . human interactions play an important role in determining the potential transmission routes of infectious diseases and other contagion phenomena . their measure and characterisation thus represent an invaluable contribution to the study of transmissible diseases . the resulting measuring infrastructures register contacts specifically within the closed population formed by the participants wearing sensors , with typically high spatial and temporal resolutions . in the recent years , several data gathering efforts have used such methods to obtain , analyse and publish data sets describing the contact patterns between individuals in various contexts in the form of temporal networks : nodes represent individuals and , at each time step , a link is drawn between pairs of individuals who are in contact . despite the relevance and interest of such detailed data sets , as illustrated by these recent investigations , they suffer from the intrinsic limitation of the data gathering method : contacts are registered only between participants wearing sensors . hence , the collected data only contains information on contacts occurring among a fraction of the population under study . population sampling is well - known to affect the properties of static networks : various statistical properties and mixing patterns of the contact network of a fraction of the population of interest may differ from those of the whole population , even if the sampling is uniform , and several works have focused on inferring network statistics from the knowledge of incomplete network data . both structural and temporal properties of time - varying networks might as well be affected by missing data effects . for instance , in simulations of epidemic spreading , excluded nodes are by definition unreachable and thus equivalent to immunised nodes . due to herd vaccination effects , the outcome of simulations of epidemic models on sampled networks is thus expected to be underestimated with respect to simulations on the whole network . we proceed through resampling experiments on empirical data sets in which we exclude uniformly at random a fraction of the individuals ( nodes of the contact network ) . we measure how relevant network statistics vary under such uniform resampling and confirm that , although some crucial properties are stable , numerical simulations of spreading processes performed using incomplete data lead to strong underestimations of the epidemic risk . , we do not try to infer the true sequence of missing contacts . investigate how their outcomes vary depending on the amount of information included in the reconstruction method . we show that it is possible to obtain outcomes close to the results obtained on the complete data set , while , as mentioned above , using only the incomplete data severely underestimates the epidemic risk . we show the efficiency of our procedure using three data sets collected in widely different contexts and representative of very different population structures found in day - to - day life : a scientific conference , a high school and a workplace . we finally discuss the limitations of our method in terms of sampling range , model parameters and population sizes .

data describing human interactions often suffer from incomplete sampling of the underlying population . as a consequence , the study of contagion processes using data - driven models can lead to a severe underestimation of the epidemic risk . here we present a systematic method to alleviate this issue and obtain a better estimation of the risk in the context of epidemic models informed by high - resolution time - resolved contact data . we consider several such data sets collected in various contexts and perform controlled resampling experiments . we show how the statistical information contained in the resampled data can be used to build a series of surrogate versions of the unknown contacts . we simulate epidemic processes on the resulting reconstructed data sets and show that it is possible to obtain good estimates of the outcome of simulations performed using the complete data set . we discuss limitations and potential improvements of our method . human interactions play an important role in determining the potential transmission routes of infectious diseases and other contagion phenomena . their measure and characterisation thus represent an invaluable contribution to the study of transmissible diseases . while surveys and diaries in which volunteer participants record their encounters have provided crucial insights ( see however for recent investigations of the memory biases inherent in self - reporting procedures ) , new approaches have recently emerged to measure contact patterns between individuals with high resolution , using wearable sensors that can detect the proximity of other similar devices . the resulting measuring infrastructures register contacts specifically within the closed population formed by the participants wearing sensors , with typically high spatial and temporal resolutions . in the recent years , several data gathering efforts have used such methods to obtain , analyse and publish data sets describing the contact patterns between individuals in various contexts in the form of temporal networks : nodes represent individuals and , at each time step , a link is drawn between pairs of individuals who are in contact . such data has been used to inform models of epidemic spreading phenomena used to evaluate epidemic risks and mitigation strategies in specific , size - limited contexts such as schools or hospitals , finding in particular outcomes consistent with observed outbreak data or providing evidence of links between specific contacts and transmission events . despite the relevance and interest of such detailed data sets , as illustrated by these recent investigations , they suffer from the intrinsic limitation of the data gathering method : contacts are registered only between participants wearing sensors . contacts with and between individuals who do not wear sensors are thus missed . in other words , as most often not all individuals accept to participate by wearing sensors , many data sets obtained by such techniques suffer from population sampling , despite efforts to maximise participation through e.g. scientific engagement of participants . hence , the collected data only contains information on contacts occurring among a fraction of the population under study . population sampling is well - known to affect the properties of static networks : various statistical properties and mixing patterns of the contact network of a fraction of the population of interest may differ from those of the whole population , even if the sampling is uniform , and several works have focused on inferring network statistics from the knowledge of incomplete network data . both structural and temporal properties of time - varying networks might as well be affected by missing data effects . in addition , a crucial though little studied consequence of such missing data is that simulations of dynamical processes in data - driven models can be affected if incomplete data are used . for instance , in simulations of epidemic spreading , excluded nodes are by definition unreachable and thus equivalent to immunised nodes . due to herd vaccination effects , the outcome of simulations of epidemic models on sampled networks is thus expected to be underestimated with respect to simulations on the whole network . ( we note however , that in the different context of transportation networks , it was found in that the inclusion of the most important transportation nodes can be sufficient to describe the global worldwide spread of influenza - like illnesses , at least in terms of times of arrival of the spread in various cities . ) how to estimate the outcome of dynamical processes on contact networks using incomplete data remains an open question . here we make progresses on this issue for incompletely sampled data describing networks of human face - to - face interactions , collected by infrastructures based on sensors , under the assumption that the population participating to the data collection is a uniform random sample of the whole population of interest . ( we do not therefore address here the issue of non - uniform sampling of contacts that may result from other measurement methods such as diaries or surveys . ) we proceed through resampling experiments on empirical data sets in which we exclude uniformly at random a fraction of the individuals ( nodes of the contact network ) . we measure how relevant network statistics vary under such uniform resampling and confirm that , although some crucial properties are stable , numerical simulations of spreading processes performed using incomplete data lead to strong underestimations of the epidemic risk . our goal and main contribution consists then in putting forward and comparing a hierarchy of systematic methods to provide better estimates of the outcome of models of epidemic spread in the whole population under study . to this aim , we do not try to infer the true sequence of missing contacts . instead , the methods we present consist in the construction of surrogate contact sequences for the excluded nodes , using only structural and temporal information available in the resampled contact data . we perform simulations of spreading processes on the reconstructed data sets , obtained by the union of the resampled and surrogate contacts , and investigate how their outcomes vary depending on the amount of information included in the reconstruction method . we show that it is possible to obtain outcomes close to the results obtained on the complete data set , while , as mentioned above , using only the incomplete data severely underestimates the epidemic risk . we show the efficiency of our procedure using three data sets collected in widely different contexts and representative of very different population structures found in day - to - day life : a scientific conference , a high school and a workplace . we finally discuss the limitations of our method in terms of sampling range , model parameters and population sizes .

1503.04066

c

the understanding of epidemic spreading phenomena has been vastly improved thanks to the use of data - driven models at different scales . high resolution contact data in particular have been used to evaluate epidemic risk or containment policies in specific populations or to perform contact tracing @xcite . in such studies , missing data due to population sampling might represent however a serious issue : individuals absent from a data set are indeed equivalent to immunised individuals when epidemic processes are simulated . feeding sampled data into data - driven models can therefore lead to severe underestimations of the epidemic risk and might even a priori affect the evaluation of mitigation strategies if for instance some at - risk groups are particularly undersampled . here we have put forward a set of methods to obtain a better evaluation of the outcome of spreading simulations for data - driven models using contact data from a uniformly sampled population . to this aim , we have shown how it is possible , starting from a data set describing the contacts of only a fraction of the population of interest ( uniformly sampled from the whole population ) , to construct surrogate data sets using various amounts of accessible information , i.e. , quantities measured in the sampled data . we have shown that the simplest method , which consists in simply compensating for the decrease in the average number of neighbours due to sampling , yields a strong overestimation of the epidemic risk . when additional information describing the group structure and the temporal properties of the data is added in the construction of surrogate data sets , simulations of epidemic spreading on such surrogate data yield results similar to those obtained on the complete data set . ( we note that the issue of how much information should be included when constructing the surrogate data is linked to the general issue of how much information is needed to get an accurate picture of spreading processes on temporal networks @xcite . ) some discrepancies in the epidemic risk estimation are however still observed , due in particular to small scale structural correlations of the contact network that are difficult or even impossible to measure in the resampled data : these discrepancies are indeed largely suppressed if we use as original data a reshuffled contact network in which such correlations are absent . the methods presented here yield much better results than simulations using resampled data , even when a substantial part of the population is excluded , in particular in estimating the probability of large outbreaks . it suffers however from limitations , especially when the fraction @xmath4 of excluded individuals is too large . first , the construction of the surrogate contacts relies on the stability of a set of quantities with respect to resampling , but the measured quantities start to deviate from the original ones at large @xmath4 . the shape of the distribution of epidemic sizes may then differ substantially from the original one . second , large values of @xmath4 might even render the construction of the surrogate data impossible due to the loss of information on whole categories of nodes . finally , at least an estimate of the number of missing individuals in the population is needed in order to create a surrogate data set . an interesting avenue for future work concerns possible improvements of the reconstruction methods , in particular by integrating into the surrogate data additional information and complex correlation patterns measured in the sampled data . for instance , the number of contacts varies significantly with the time of day in most contexts : the corresponding activity timeline might be measured in the sampled data ( overall or even for each group of individuals ) , assumed to be robust to sampling and used in the reconstruction of contact timelines . more systematically , it might also be possible to use the temporal network decomposition technique put forward in @xcite on the sampled data , in order to extract mesostructures such as temporally - localized mixing patterns . the surrogate contacts could then be built in a way to preserve such patterns . indeed , correlations between structure and activity in the temporal contact network are known to influence spreading processes @xcite but are notoriously difficult to measure . if the group structure of the population is unknown , recent approaches based on stochastic block models @xcite might be used to extract groups from the resampled data ; this extracted group structure could then be used to build the corresponding contact matrix and surrogate data sets . we finally recall that we have assumed an uniform sampling of nodes , corresponding to an independent random choice of each individual of the population to take part or not to the data collection . other types of sampling or data losses can however also be present in data collected by wearable sensors , such as partial coverage of the premises of interest by the measuring infrastructure , non - uniform sampling depending on individual activity ( too busy persons or , on the contrary , asocial individuals , might not want to wear sensors ) , on group membership , or due to clusters of non - participating individuals ( e.g. , groups of friends ) . in addition , other types of data sets such as the ones obtained from surveys or diaries correspond to different types of sampling , as each respondent provides then information in the form of an ego - network @xcite . such data sets involve potentially additional types of biases such as underreporting of the number of contacts and overestimation of contact durations @xcite : how to adapt the methods presented here is an important issue that we will examine in future work . finally , the population under study is ( usually ) not isolated from the external world , and it would be important to devise ways to include contacts with outsiders in the data and simulations , for instance by using other data sources such as surveys .

we simulate epidemic processes on the resulting reconstructed data sets and show that it is possible to obtain good estimates of the outcome of simulations performed using the complete data set . such data has been used to inform models of epidemic spreading phenomena used to evaluate epidemic risks and mitigation strategies in specific , size - limited contexts such as schools or hospitals , finding in particular outcomes consistent with observed outbreak data or providing evidence of links between specific contacts and transmission events . contacts with and between individuals who do not wear sensors are thus missed . in other words , as most often not all individuals accept to participate by wearing sensors , many data sets obtained by such techniques suffer from population sampling , despite efforts to maximise participation through e.g. scientific engagement of participants . in addition , a crucial though little studied consequence of such missing data is that simulations of dynamical processes in data - driven models can be affected if incomplete data are used . ( we note however , that in the different context of transportation networks , it was found in that the inclusion of the most important transportation nodes can be sufficient to describe the global worldwide spread of influenza - like illnesses , at least in terms of times of arrival of the spread in various cities . ) how to estimate the outcome of dynamical processes on contact networks using incomplete data remains an open question . here ( we do not therefore address here the issue of non - uniform sampling of contacts that may result from other measurement methods such as diaries or surveys . ) our goal and main contribution consists then in putting forward and comparing a hierarchy of systematic methods to provide better estimates of the outcome of models of epidemic spread in the whole population under study . to this aim instead , the methods we present consist in the construction of surrogate contact sequences for the excluded nodes , using only structural and temporal information available in the resampled contact data . we perform simulations of spreading processes on the reconstructed data sets , obtained by the union of the resampled and surrogate contacts , and

data describing human interactions often suffer from incomplete sampling of the underlying population . as a consequence , the study of contagion processes using data - driven models can lead to a severe underestimation of the epidemic risk . here we present a systematic method to alleviate this issue and obtain a better estimation of the risk in the context of epidemic models informed by high - resolution time - resolved contact data . we consider several such data sets collected in various contexts and perform controlled resampling experiments . we show how the statistical information contained in the resampled data can be used to build a series of surrogate versions of the unknown contacts . we simulate epidemic processes on the resulting reconstructed data sets and show that it is possible to obtain good estimates of the outcome of simulations performed using the complete data set . we discuss limitations and potential improvements of our method . human interactions play an important role in determining the potential transmission routes of infectious diseases and other contagion phenomena . their measure and characterisation thus represent an invaluable contribution to the study of transmissible diseases . while surveys and diaries in which volunteer participants record their encounters have provided crucial insights ( see however for recent investigations of the memory biases inherent in self - reporting procedures ) , new approaches have recently emerged to measure contact patterns between individuals with high resolution , using wearable sensors that can detect the proximity of other similar devices . the resulting measuring infrastructures register contacts specifically within the closed population formed by the participants wearing sensors , with typically high spatial and temporal resolutions . in the recent years , several data gathering efforts have used such methods to obtain , analyse and publish data sets describing the contact patterns between individuals in various contexts in the form of temporal networks : nodes represent individuals and , at each time step , a link is drawn between pairs of individuals who are in contact . such data has been used to inform models of epidemic spreading phenomena used to evaluate epidemic risks and mitigation strategies in specific , size - limited contexts such as schools or hospitals , finding in particular outcomes consistent with observed outbreak data or providing evidence of links between specific contacts and transmission events . despite the relevance and interest of such detailed data sets , as illustrated by these recent investigations , they suffer from the intrinsic limitation of the data gathering method : contacts are registered only between participants wearing sensors . contacts with and between individuals who do not wear sensors are thus missed . in other words , as most often not all individuals accept to participate by wearing sensors , many data sets obtained by such techniques suffer from population sampling , despite efforts to maximise participation through e.g. scientific engagement of participants . hence , the collected data only contains information on contacts occurring among a fraction of the population under study . population sampling is well - known to affect the properties of static networks : various statistical properties and mixing patterns of the contact network of a fraction of the population of interest may differ from those of the whole population , even if the sampling is uniform , and several works have focused on inferring network statistics from the knowledge of incomplete network data . both structural and temporal properties of time - varying networks might as well be affected by missing data effects . in addition , a crucial though little studied consequence of such missing data is that simulations of dynamical processes in data - driven models can be affected if incomplete data are used . for instance , in simulations of epidemic spreading , excluded nodes are by definition unreachable and thus equivalent to immunised nodes . due to herd vaccination effects , the outcome of simulations of epidemic models on sampled networks is thus expected to be underestimated with respect to simulations on the whole network . ( we note however , that in the different context of transportation networks , it was found in that the inclusion of the most important transportation nodes can be sufficient to describe the global worldwide spread of influenza - like illnesses , at least in terms of times of arrival of the spread in various cities . ) how to estimate the outcome of dynamical processes on contact networks using incomplete data remains an open question . here we make progresses on this issue for incompletely sampled data describing networks of human face - to - face interactions , collected by infrastructures based on sensors , under the assumption that the population participating to the data collection is a uniform random sample of the whole population of interest . ( we do not therefore address here the issue of non - uniform sampling of contacts that may result from other measurement methods such as diaries or surveys . ) we proceed through resampling experiments on empirical data sets in which we exclude uniformly at random a fraction of the individuals ( nodes of the contact network ) . we measure how relevant network statistics vary under such uniform resampling and confirm that , although some crucial properties are stable , numerical simulations of spreading processes performed using incomplete data lead to strong underestimations of the epidemic risk . our goal and main contribution consists then in putting forward and comparing a hierarchy of systematic methods to provide better estimates of the outcome of models of epidemic spread in the whole population under study . to this aim , we do not try to infer the true sequence of missing contacts . instead , the methods we present consist in the construction of surrogate contact sequences for the excluded nodes , using only structural and temporal information available in the resampled contact data . we perform simulations of spreading processes on the reconstructed data sets , obtained by the union of the resampled and surrogate contacts , and investigate how their outcomes vary depending on the amount of information included in the reconstruction method . we show that it is possible to obtain outcomes close to the results obtained on the complete data set , while , as mentioned above , using only the incomplete data severely underestimates the epidemic risk . we show the efficiency of our procedure using three data sets collected in widely different contexts and representative of very different population structures found in day - to - day life : a scientific conference , a high school and a workplace . we finally discuss the limitations of our method in terms of sampling range , model parameters and population sizes .

1503.04066

m

we consider data sets collected using the sociopatterns proximity sensing platform ( http://www.sociopatterns.org ) based on wearable sensors that detect close face - to - face proximity of individuals wearing them . informed consent was obtained from all participants and the french national bodies responsible for ethics and privacy , the commission nationale de linformatique et des liberts ( cnil , http://www.cnil.fr ) , approved the data collections . the high school ( thiers13 ) data set @xcite is structured in 9 classes , forming three subgroups of three classes corresponding to their specialisation in mathematics - physics ( mp , mp@xmath381 , mp@xmath382 with respectively @xmath39 , @xmath40 and @xmath41 students ) , physics ( pc , pc@xmath38 , psi with respectively @xmath42 , @xmath43 and @xmath44 students ) , or biology ( 2bio1 , 2bio2 , 2bio3 with respectively @xmath45 , @xmath46 and @xmath43 students ) . the workplace ( invs ) data set @xcite is structured in @xmath47 departments : disq ( scientific direction , @xmath48 persons ) , dmct ( department of chronic diseases and traumatisms , @xmath49 persons ) , dse ( department of health and environment , @xmath44 persons ) , srh ( human resources , @xmath50 persons ) and sfle ( logistics , @xmath51 persons ) . for the conference data ( sfhh ) , we do not have metadata on the participants , and the aggregated network structure was found to be homogeneous @xcite . simulations of sir and sis processes on the temporal networks of contacts ( original , resampled or reconstructed ) are performed using the temporal gillespie algorithm described in @xcite . for each run of the simulations , all nodes are initially susceptible ; a node is chosen at random as the seed of the epidemic and put in the infectious state at a point in time chosen at random over the duration of the contact data . a susceptible node in contact with an infectious node becomes infectious at rate @xmath6 . infectious nodes recover at rate @xmath7 : in the sir model they then enter the recovered state and can not become infectious again , while in the sis model they enter the susceptible state again . if needed , the sequence of contacts is repeated in the simulation @xcite . for sir processes , we run each simulation , with the seed node chosen at random , until no infectious individual remains ( nodes are thus either still susceptible or have been infected and then recovered ) . we consider values of @xmath6 and @xmath7 yielding a non - negligible epidemic risk , i.e. , such that a rather large fraction of simulations lead to a final size larger than @xmath52 of the population ( see figs . [ sampling][fig : infersfhh ] ) : @xmath53 , @xmath54 ( invs ) or @xmath55 ( sfhh and thiers13 ) . other parameter values are explored in the supplementary information . for each set of parameters , the distribution of epidemic sizes is obtained by performing @xmath56 simulations . for sis processes , simulations are performed using the quasi - stationary approach of @xcite . they are run until the system enters a stationary state as witnessed by the mean number of infected nodes being constant over time . simulations are then continued for 50,000 time - steps while recording the number of infected nodes . for each set of parameters , the simulations are performed once with each node of the network chosen as the seed node . we consider a population @xmath57 of @xmath27 individuals , potentially organised in groups . we assume that all the contacts occurring among a subpopulation @xmath58 of these individuals , of size @xmath59 , are known . this constitutes our resampled data from which we need to construct a surrogate set of contacts concerning the remaining @xmath60 individuals for which no contact information is available : these contacts can occur among these individuals and between them and the members of @xmath58 . we assume that we know the group of each member of @xmath61 , and the overall activity timeline , i.e. , the intervals during which contacts take place , separated by nights and weekends . to construct the surrogate data ( wst method ) , we first compute from the activity timeline the total duration @xmath62 of the periods during which contacts can occur . then , we measure in the sampled data : * the density @xmath63 of links in the aggregated contact network ; * a row - normalised contact matrix @xmath64 , in which the element @xmath65 gives the probability for a node of group @xmath17 to have a link to a node of group @xmath18 ; * the list @xmath66 of contact durations ; * the lists @xmath67 and @xmath68 of inter - contact durations for internal and external links , _ i.e. _ , for links between nodes of the same group and links between nodes that belong to two different groups , respectively ; * the lists @xmath69 and @xmath70 of numbers of contacts per link , respectively for internal ( within groups ) and external ( between groups ) links ; * the list @xmath71 of initial times between the start of the data set and the first contact between two nodes . given @xmath63 , we compute the number @xmath72 of additional links needed to keep the network density constant when we add the @xmath73 excluded nodes . we then construct each link according to the following procedure : * a node @xmath1 is randomly chosen from the set @xmath61 of excluded nodes ; * knowing the group @xmath17 that @xmath1 belongs to , we extract at random a target group @xmath18 with probability given by @xmath65 ; * we draw a target node @xmath2 at random from @xmath18 ( if @xmath74 , we take care that @xmath75 ) such that @xmath1 and @xmath2 are not linked ; * depending on whether @xmath1 and @xmath2 belong to the same group or not , we draw from @xmath69 or @xmath70 the number of contact events @xmath76 taking place over the link @xmath77 ; * from @xmath71 , we draw the initial waiting time before the first contact ; * from @xmath66 , we draw @xmath76 contact durations @xmath78 , @xmath79 ; * from @xmath67 or @xmath68 , we draw @xmath80 inter - contact durations @xmath81 , @xmath82 ; * if @xmath83 , we repeat steps ( d ) to ( g ) until we obtain a set of values such that @xmath84 ; * from @xmath85 and the @xmath78 and @xmath81 , we build the contact timeline of the link @xmath77 ; * finally , we insert in the contact timeline the breaks defined by the global activity timeline . the construction of the surrogate version of the missing links uses as an input the group structure of the subgraph that remains after sampling , as given by the contact matrix of the link densities between the different groups of nodes that are present in the subpopulation @xmath58 . depending on the characteristics of @xmath58 and of the corresponding contacts , the construction method can fail in several cases : ( i ) if an entire group ( class / department ) of nodes in the population is absent from @xmath58 ; ( ii ) if the remaining nodes of a specific group ( class / department ) are all isolated in @xmath58 s contact network ; ( iii ) if , during the algorithm , a node of @xmath61 is selected in a certain group @xmath17 but can not create any more links because it already has links to all nodes in the groups @xmath18 such that @xmath86 ; ( iv ) if there are either no internal ( within groups ) or external ( between groups ) links in the contact network of @xmath58 : in this case one of the lists of link temporal characteristics is empty and the corresponding structures can not be reconstructed . cases ( i ) and ( ii ) correspond to a complete loss of information about the connectivity of a group ( class / department ) of the population , due to sampling . it is then impossible to reconstruct a sensible connectivity pattern for these nodes . case ( iii ) is more subtle and occurs in situations of very low connectivity between groups . for instance , within the contact network of @xmath57 , a group @xmath17 has links only with another specific group @xmath18 , and both @xmath17 and @xmath18 are small ; it is then possible that the nodes of @xmath87 exhaust the set of possible links to nodes of @xmath18 during the reconstruction algorithm . if a node of @xmath88 is again chosen to create a link , such a creation is not possible and the construction fails . case ( iv ) usually corresponds to situations in which the links between individuals of different groups which remain in the resampled data set correspond to pairs of individuals who have had only one contact event : in such cases , @xmath68 is empty and external links with more than one contact can not be built . in order to test the effect of correlations in the temporal network , we use four shuffling methods , based on the ones defined in @xcite . link shuffling . the contact timelines associated with each link are randomly redistributed among the links . correlations between timelines of links adjacent to a given node are destroyed , as well as correlations between weights and topology . the structure of the network is kept , as well as the global activity timeline . time shuffling . from the contact data we build the lists @xmath66 , @xmath89 and @xmath90 of , respectively , contact durations , inter - contact durations and number of contacts per link . we also measure the list @xmath71 of initial times between the start of the data set and the first contact between two nodes . for each link , we draw randomly a starting time @xmath85 , a number @xmath76 of contacts from @xmath90 , @xmath76 contact durations from @xmath66 and @xmath80 inter - contact durations from @xmath89 , so that the total duration of the timeline does not exceed the total available time @xmath62 . we then construct the contact timelines , thus destroying the temporal correlations among contacts . the structure of the network is instead kept fixed . cm shuffling . we perform a link rewiring separately on each compartment of the contact matrix , _ i.e. _ , we randomly redistribute links with their contact timelines within each group , and within each pair of groups . we thus destroy the structural correlations inside each compartment of the contact matrix , while preserving the group structure of the network as given by the link density contact matrix and the contact matrix of total contact times between groups . cm - time shuffling . we perform both a cm shuffling and a time shuffling . 10 url # 1`#1`urlprefix[2]#2 [ 2][]#2 , & _ _ ( , ) . , , , & . _ _ * * , ( ) . , & . _ _ * * , ( ) . , & . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . , , & . _ _ * * , ( ) . , , , & . _ _ * * , ( ) . , , & . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . _ et al . _ . in _ _ ( , , ) . _ et al . _ . in _ _ , vol . , ( ) . , & . _ _ * * , ( ) . , , & . in _ _ , mswim 10 , ( , , ) . _ et al . _ . _ _ * * , ( ) . , & . in _ _ , ( ) . _ et al . _ . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . et al . _ . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . _ et al . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . & . _ _ * * , ( ) . & . _ _ * * , ( ) . , , & . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . & . _ _ * * , ( ) . & . _ _ * * ( ) . , & . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . . _ _ * * , ( ) . . _ _ * * , ( ) . , , & . in _ _ , stoc 05 , ( , , ) . . _ _ * * , ( ) . , & . _ _ * * , ( ) . & . _ _ * * , ( ) . & . _ _ * * , ( ) . _ _ * * , ( ) . , & . _ _ * * , ( ) . , & . _ _ * * , ( ) . , , & ( ) . , & _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . _ _ * * , ( ) . & . _ _ * * , ( ) . , & _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . & _ _ * * , ( ) . _ _ * * , ( ) . , & . _ _ * * , ( ) . , , _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . , , & ( ) . ( ) . , & . _ _ * * , ( ) . , , & ( ) . & ( ) . , & _ _ * * , ( ) .

while surveys and diaries in which volunteer participants record their encounters have provided crucial insights ( see however for recent investigations of the memory biases inherent in self - reporting procedures ) , new approaches have recently emerged to measure contact patterns between individuals with high resolution , using wearable sensors that can detect the proximity of other similar devices . we make progresses on this issue for incompletely sampled data describing networks of human face - to - face interactions , collected by infrastructures based on sensors , under the assumption that the population participating to the data collection is a uniform random sample of the whole population of interest .

data describing human interactions often suffer from incomplete sampling of the underlying population . as a consequence , the study of contagion processes using data - driven models can lead to a severe underestimation of the epidemic risk . here we present a systematic method to alleviate this issue and obtain a better estimation of the risk in the context of epidemic models informed by high - resolution time - resolved contact data . we consider several such data sets collected in various contexts and perform controlled resampling experiments . we show how the statistical information contained in the resampled data can be used to build a series of surrogate versions of the unknown contacts . we simulate epidemic processes on the resulting reconstructed data sets and show that it is possible to obtain good estimates of the outcome of simulations performed using the complete data set . we discuss limitations and potential improvements of our method . human interactions play an important role in determining the potential transmission routes of infectious diseases and other contagion phenomena . their measure and characterisation thus represent an invaluable contribution to the study of transmissible diseases . while surveys and diaries in which volunteer participants record their encounters have provided crucial insights ( see however for recent investigations of the memory biases inherent in self - reporting procedures ) , new approaches have recently emerged to measure contact patterns between individuals with high resolution , using wearable sensors that can detect the proximity of other similar devices . the resulting measuring infrastructures register contacts specifically within the closed population formed by the participants wearing sensors , with typically high spatial and temporal resolutions . in the recent years , several data gathering efforts have used such methods to obtain , analyse and publish data sets describing the contact patterns between individuals in various contexts in the form of temporal networks : nodes represent individuals and , at each time step , a link is drawn between pairs of individuals who are in contact . such data has been used to inform models of epidemic spreading phenomena used to evaluate epidemic risks and mitigation strategies in specific , size - limited contexts such as schools or hospitals , finding in particular outcomes consistent with observed outbreak data or providing evidence of links between specific contacts and transmission events . despite the relevance and interest of such detailed data sets , as illustrated by these recent investigations , they suffer from the intrinsic limitation of the data gathering method : contacts are registered only between participants wearing sensors . contacts with and between individuals who do not wear sensors are thus missed . in other words , as most often not all individuals accept to participate by wearing sensors , many data sets obtained by such techniques suffer from population sampling , despite efforts to maximise participation through e.g. scientific engagement of participants . hence , the collected data only contains information on contacts occurring among a fraction of the population under study . population sampling is well - known to affect the properties of static networks : various statistical properties and mixing patterns of the contact network of a fraction of the population of interest may differ from those of the whole population , even if the sampling is uniform , and several works have focused on inferring network statistics from the knowledge of incomplete network data . both structural and temporal properties of time - varying networks might as well be affected by missing data effects . in addition , a crucial though little studied consequence of such missing data is that simulations of dynamical processes in data - driven models can be affected if incomplete data are used . for instance , in simulations of epidemic spreading , excluded nodes are by definition unreachable and thus equivalent to immunised nodes . due to herd vaccination effects , the outcome of simulations of epidemic models on sampled networks is thus expected to be underestimated with respect to simulations on the whole network . ( we note however , that in the different context of transportation networks , it was found in that the inclusion of the most important transportation nodes can be sufficient to describe the global worldwide spread of influenza - like illnesses , at least in terms of times of arrival of the spread in various cities . ) how to estimate the outcome of dynamical processes on contact networks using incomplete data remains an open question . here we make progresses on this issue for incompletely sampled data describing networks of human face - to - face interactions , collected by infrastructures based on sensors , under the assumption that the population participating to the data collection is a uniform random sample of the whole population of interest . ( we do not therefore address here the issue of non - uniform sampling of contacts that may result from other measurement methods such as diaries or surveys . ) we proceed through resampling experiments on empirical data sets in which we exclude uniformly at random a fraction of the individuals ( nodes of the contact network ) . we measure how relevant network statistics vary under such uniform resampling and confirm that , although some crucial properties are stable , numerical simulations of spreading processes performed using incomplete data lead to strong underestimations of the epidemic risk . our goal and main contribution consists then in putting forward and comparing a hierarchy of systematic methods to provide better estimates of the outcome of models of epidemic spread in the whole population under study . to this aim , we do not try to infer the true sequence of missing contacts . instead , the methods we present consist in the construction of surrogate contact sequences for the excluded nodes , using only structural and temporal information available in the resampled contact data . we perform simulations of spreading processes on the reconstructed data sets , obtained by the union of the resampled and surrogate contacts , and investigate how their outcomes vary depending on the amount of information included in the reconstruction method . we show that it is possible to obtain outcomes close to the results obtained on the complete data set , while , as mentioned above , using only the incomplete data severely underestimates the epidemic risk . we show the efficiency of our procedure using three data sets collected in widely different contexts and representative of very different population structures found in day - to - day life : a scientific conference , a high school and a workplace . we finally discuss the limitations of our method in terms of sampling range , model parameters and population sizes .

gr-qc0508118

c

we have demonstrated that , from currently available information , there is no contradiction between models and the full theory of loop quantum gravity . instead , a consistent picture emerges when all cases , isotropic as well as anisotropic or inhomogeneous models and the full theory , are considered . there is a technical difference between inverse volume in the full theory and similar operators in models , in that the expectation values in the full theory are unbounded on states of vanishing volume eigenvalue , while they are zero on such states in models considered so far . however , in the full theory the distinction of singular states , in particular their identification with certain degenerate states , is blurred due to non - abelian effects . there is then no crucial difference between a model where curvature is unbounded _ close to singular states _ ( i.e. for quantum labels close to those of a singular state ) and the full theory where inverse volume is unbounded _ on degenerate states _ which can be argued to be close to , but not necessarily identical with singular states . any claim of a contradiction between models and the full theory , based on these properties , is thus unsubstantiated . nevertheless , there are certainly differences by design since models capture behavior only in a particular , geometrically selected sector of physical interest . what is clear from the considerations is that so far inhomogeneities can not be made responsible for any explicit discrepancy between models and the full theory . to judge what implications particular properties of inverse volume operators have , their geometrical and dynamical roles must be clear before rushing to conclusions . the latter can be derived , e.g. , through difference equations representing the hamiltonian constraint or observables . but also from the classical side one needs to provide knowledge on the general singularity structure , which becomes exceedingly complicated when symmetry assumptions are dropped ( see , e.g. , @xcite ) . the main issue to be checked in further investigations is implications of non - abelian effects not studied so far in the different sectors of loop quantum gravity ( including not only cosmological situations but also black hole horizons and other models or semiclassical states , all of which often exploit possible eliminations of non - abelian terms in explicit calculations ) . in models , understanding the meaning of degenerate configurations is achieved by the selection of an appropriate sector of the theory displaying the configurations of interest explicitly . we emphasize again that at this point crucial information from the full theory enters through distributional states . the symmetry is specified for the physical context , and then the relevant quantum degrees of freedom result through derivation . ( similarly , imposing black hole horizons selects the appropriate degrees of freedom , again in an effectively abelian manner , through physical conditions for an isolated horizon @xcite relating horizon degrees of freedom to full flux operators . ) this suggests that non - abelian degrees of freedom are not always crucial physically , which can also be seen from the fact that diagonal homogeneous models , which in contrast to non - diagonal ones can effectively be abelianized , show the complete behavior of cosmological evolution @xcite . still , additional correction terms in a non - abelian situation can provide characteristic effects . better understanding non - abelian behavior is thus not just important for the relation between models and the full theory , i.e. as a test of approximations , but can also provide new physical insights . the non - commutative behavior of quantum geometry @xcite so far has not been made use of in cosmological investigations in loop quantum gravity , while non - commutative geometry itself has given rise to several cosmological applications ( see , e.g. , @xcite ) . combining loop cosmological phenomenology with non - commutative behavior thus has the potential of providing further scenarios for the very early universe .

significant features of this unboundedness are not a consequence of inhomogeneities but of non - abelian effects which can also be included in homogeneous models . aei2005133 + gr qc/0508118 + degenerate configurations , singularities and + the non - abelian nature of loop quantum gravity + martin bojowald + max - planck - institut fr gravitationsphysik , albert - einstein - institut , + am mhlenberg 1 , d-14476 potsdam , germany

degenerate geometrical configurations in quantum gravity are important to understand if the fate of classical singularities is to be revealed . however , not all degenerate configurations arise on an equal footing , and one must take into account dynamical aspects when interpreting results : while there are many degenerate spatial metrics , not all of them are approached along the dynamical evolution of general relativity or a candidate theory for quantum gravity . for loop quantum gravity , relevant properties and steps in an analysis are summarized and evaluated critically with the currently available information , also elucidating the role of degrees of freedom captured in the sector provided by loop quantum cosmology . this allows an outlook on how singularity removal might be analyzed in a general setting and also in the full theory . the general mechanism of loop quantum cosmology will be shown to be insensitive to recently observed unbounded behavior of inverse volume in the full theory . moreover , significant features of this unboundedness are not a consequence of inhomogeneities but of non - abelian effects which can also be included in homogeneous models . aei2005133 + gr qc/0508118 + degenerate configurations , singularities and + the non - abelian nature of loop quantum gravity + martin bojowald + max - planck - institut fr gravitationsphysik , albert - einstein - institut , + am mhlenberg 1 , d-14476 potsdam , germany

1606.03392

c

at ambient temperature and strong tension , bubbles in metastable water are essentially voids in the liquid whose shape can deviate significantly from the assumption of a spherical nucleus made in cnt , depending on their size . nonetheless , provided the dependence of the surface tension on the average curvature is included , the free energetics of bubble formation can be quantitatively described in the framework of cnt . we find that the curvature contribution favors the cavity over the droplet , i.e. , @xmath74 , in agreement with experimental results @xcite . in light of conflicting results on the sign of @xmath36 in water , further study is required to elucidate the influence of the chosen water model and biasing towards certain cavity shapes on the obtained value of @xmath36 . by including the effect of thermal fluctuations in the rayleigh plesset equation , we obtain an estimate for the bubble diffusivity that accurately reproduces the pressure dependence found in simulation and scales inversely with the viscosity of the liquid . combining the kinetic pre - factor determined for this diffusivity with the equilibrium bubble density yields a cnt expression for the cavitation rate that reproduces the nucleation rates very well for negative pressures . however , the microscopic mechanism for cavitation is expected to change for higher pressures and temperatures , where the saturated vapor density is significantly higher than at the temperature studied here . at those conditions , similarly to droplet nucleation @xcite , the transport of molecules across the interface via evaporation and condensation will have a stronger influence on the kinetics of bubble growth , thereby diminishing the influence of viscous damping on the dynamics of the bubble . the estimate for the cavitation pressure obtained from our rate calculations agrees well with the data from inclusion experiments , thus calling the conflicting results harvested by other techniques into question . since the latter methods greatly underestimate the stability of water under tension , heterogeneous cavitation due to impurities is a likely explanation for this discrepancy .

comparison of the observed bubble dynamics to the predictions of the macroscopic rayleigh plesset ( rp ) equation , augmented with thermal fluctuations , demonstrates that the growth of nanoscale bubbles is governed by viscous forces . combining the dynamical prefactor determined from the rp equation with cnt based on kramers formalism yields an analytical expression for the cavitation rate that reproduces the simulation results very well over a wide range of pressures . computer simulations are a natural choice to investigate cavitation in water with molecular resolution on the time - scales governing the emergence of microscopic bubbles in the liquid .

despite its relevance in biology and engineering , the molecular mechanism driving cavitation in water remains unknown . using computer simulations , we investigate the structure and dynamics of vapor bubbles emerging from metastable water at negative pressures . we find that in the early stages of cavitation , bubbles are irregularly shaped and become more spherical as they grow . nevertheless , the free energy of bubble formation can be perfectly reproduced in the framework of classical nucleation theory ( cnt ) if the curvature dependence of the surface tension is taken into account . comparison of the observed bubble dynamics to the predictions of the macroscopic rayleigh plesset ( rp ) equation , augmented with thermal fluctuations , demonstrates that the growth of nanoscale bubbles is governed by viscous forces . combining the dynamical prefactor determined from the rp equation with cnt based on kramers formalism yields an analytical expression for the cavitation rate that reproduces the simulation results very well over a wide range of pressures . furthermore , our theoretical predictions are in excellent agreement with cavitation rates obtained from inclusion experiments . this suggests that homogeneous nucleation is observed in inclusions , whereas only heterogeneous nucleation on impurities or defects occurs in other experiments . due to its pronounced cohesion , water remains stable under tension for long times . experimentally , strongly negative pressures exceeding can be sustained before the system decays into the vapor phase via cavitation , i.e. , bubble nucleation . recently , cavitation in water under tension has drawn research interest due to its importance in biological processes , like water transport in natural and synthetic trees , spore propagation of ferns , and poration of cell membranes . furthermore , cavitation in water appears to be the driving force behind the sonocrystallization of ice and preventing its occurrence remains a challenge in turbine and propeller design . studying the onset of cavitation has also proven to be a valuable tool to locate the line of density maxima in metastable water , which contributes to the ongoing effort of explaining the origin of water s anomalies . interest in the topic is magnified by the startling discrepancy arising when cavitation in water is investigated using different experimental methods . while agreement between different methods is excellent in the high - temperature regime , where the liquid is unable to sustain large tension , a significantly higher degree of metastability is reached when studying cavitation in inclusions along an isochoric path compared to other techniques at low temperatures . due to the short time - scale on which the transition takes place and the small volume of the critical bubble at experimentally feasible conditions , direct observation of cavitation at the microscopic level remains elusive . however , cavitation rates are directly accessible in experiment and some microscopic insight into the cavitation transition can be obtained from these data by means of the nucleation theorem , which relates the variation in the height of the free energy barrier separating the metastable liquid from the vapor phase upon change of external parameters to properties of the critical bubble . the microscopic information that can be inferred is limited and , since not all quantities entering the nucleation theorem are known , _ ad hoc _ assumptions have to be introduced . for state - points where cavitation is a rare event , classical nucleation theory ( cnt ) can be invoked to provide a qualitative understanding of the transition . however , while cnt provides a physically meaningful and appealingly simple picture of nucleation processes , the estimates for the nucleation rates obtained from cnt are known to differ substantially ( up to many orders of magnitude ) from those measured in experiments . computer simulations are a natural choice to investigate cavitation in water with molecular resolution on the time - scales governing the emergence of microscopic bubbles in the liquid . while cavitation in simple liquids has been studied extensively using computer simulations , simulation studies of cavitation in water were focused on methodological aspects or performed at state points in vicinity of the vapor liquid spinodal . in this work , we apply a combination of several complementary computer simulation methods to identify the molecular mechanism of cavitation . a statistical committor analysis carried out on reactive trajectories reveals that the volume of the largest bubble in the system constitutes a good reaction coordinate for bubble nucleation . we compute the dynamics of nanoscale bubbles along this reaction coordinate and demonstrate that the pressure dependence of the bubble diffusivity can be reproduced by rayleigh plesset ( rp ) theory generalized to include thermal fluctuations , thereby elucidating the crucial influence of viscous damping on bubble growth . based on kramers formalism and the rp equation we obtain an analytical expression for the nucleation rate that yields excellent agreement with numerical results obtained for a wide range of pressures with a method akin to the bennett chandler approach for the computation of reaction rate constants . the obtained rates are validated for selected points by comparison to estimates from transition interface sampling and support estimates obtained from inclusion experiments . to augment the microscopic picture of cavitation we characterize the morphology of bubbles in water under tension and analyze the bubble surface in terms of its hydrogen bonding structure .

1606.03392

m

we simulate @xmath75 water molecules in the isothermal isobaric ensemble at a temperature of @xmath24 using the rigid , non - polarisable tip4p/2005 model @xcite , where the long - range interactions are treated with ewald summation . the rate computations are carried out using molecular dynamics by integrating the equations of motion with a time step of @xmath76 using a time - reversible quaternion based integrator that maintains the rigid geometry of water molecules @xcite . constant pressure is ensured by a barostat based on the andersen approach @xcite coupled to a nos hoover thermostat chain @xcite . equilibrium free energies are computed by use of umbrella sampling ( us ) in conjunction with the hybrid monte carlo ( hmc ) @xcite scheme . here , we employ a modified version of the miller integrator @xcite with a liouville operator decomposition according to omelyan @xcite , which reduces fluctuations in the total energy significantly , thereby allowing the use of a time step of @xmath77 . each hmc step consists of three md integration steps , constant pressure was implemented by isotropic volume fluctuations according to the metropolis criterion and sampling was enhanced by replica exchange moves @xcite between neighboring windows . for the direct computation of cavitation rates we employ transition interface sampling ( tis ) @xcite , where we implemented time reversal and replica exchange moves in addition to shooting moves ( described in detail in refs . the probability histograms for the individual windows in us and tis were spliced together using a self consistent histogram method @xcite . we study homogeneous bubble nucleation from over - stretched metastable water using the volume of the largest bubble as a local order parameter . estimates for the volume @xmath8 of each bubble present in the system are obtained by use of the v - method , which was developed to give thermodynamically consistent estimates for the bubble volume @xcite./@xmath78 in this work corresponds to @xmath79 in ref . the v - method is a grid - based clustering approach to bubble detection @xcite , calibrated such that its estimate @xmath8 for the volume of a bubble corresponds to the average change in system volume due to the presence of such a bubble : @xmath80 here , @xmath78 is the preliminary bubble volume estimate from the grid - based method , i.e. , the total volume of all vapor - like grid cubes belonging to the bubble , and @xmath81 is the average volume of the system when @xmath82 bubbles of size @xmath78 are present . as such , @xmath83 corresponds to the average change in system volume when a single bubble of size @xmath78 is added to or removed from the system . for large bubbles , i.e. , for bubble volumes where @xmath84 is either zero or one and there are no larger bubbles present in the system , eq . ( [ eq : vmethod1 ] ) becomes @xmath85 where @xmath86 is the average volume of the system when the largest bubble is of size @xmath78 and @xmath87 is the average volume of the unconstrained metastable liquid at the thermodynamic state point . on average , since the vapor density in the interior of bubbles is negligible , volume estimates obtained by eq . ( [ eq : vmethod ] ) are equal to those obtained by computing the equimolar dividing surface between liquid and the largest cavity for each configuration . as a result , the obtained estimates for the bubble volume fulfill the nucleation theorem @xcite , i.e. , @xmath88 , and @xmath89 corresponds to the mechanical work gained with respect to the metastable liquid by expanding the system volume at negative pressures . details on the calibration of the v - method for the state - points investigated in this work are given in the appendix . to compute the equilibrium bubble density @xmath23 , we first carry out a straightforward molecular dynamics simulation and compute @xmath90 , the average number of bubbles with a volume in a narrow interval @xmath91 $ ] . to compute @xmath92 for larger bubbles which do not form spontaneously on the timescale of the simulation , we carry out umbrella sampling simulations with a bias on the volume of the largest bubble . the resulting curves are joined , thus yielding @xmath93 over a wide range of bubble volumes . we identify molecules as belonging to the bubble surface when they are within @xmath94@xmath95 of the bubble . this cutoff radius is identical to the radius of the exclusion spheres used to determine occupied grid points during the evaluation of the order parameter ( for an in - depth description see ref . @xcite ) and thus all water molecules forming the boundary layer in our bubble detection procedure are part of the interface . when analyzing whether two water molecules form a hydrogen bond with each other , we employ the criterion used in ref . @xcite in a study of the flat vapor liquid interface in order to facilitate easy comparison between the obtained results . for molecule @xmath96 to be considered as donating a hydrogen bond to molecule b , two criteria have to be fulfilled simultaneously : the distance between the oxygens @xmath97@xmath95 and the maximum angle @xmath98 . we employ a method based on the bennett chandler approach @xcite to obtain rates estimates without any assumptions about the dynamics of the bubble in the liquid . in addition to the states @xmath96 ( metastable liquid ) and @xmath99 ( far enough to the right of the free energy barrier such that the system is committed to transitioning to the vapor phase ) , we introduce a state @xmath100 around the dividing surface , akin to the approach taken in the divided - saddle method @xcite . an ensemble of trajectories , each @xmath101 steps long , is generated by propagating checkpoints selected from the region @xmath100 forward and backward in time . from these trajectories one then computes the time correlation function @xmath102 , which is the conditional probability to find the system in @xmath99 at time @xmath103 provided it is in @xmath96 at time zero , @xmath104}\right\rangle_{g } \frac{\langle h_{s } \rangle}{\langle h_{a } \rangle } \ , .\ ] ] here , @xmath105 is @xmath106 when the system is in state @xmath107 and zero else , @xmath108 $ ] is the number of configurations of a trajectory @xmath109 in the saddle domain and @xmath110 denotes an average over the trajectories generated from points in @xmath100 . the ratio @xmath111 is the equilibrium probability of finding the system in @xmath100 relative to the equilibrium probability of state @xmath96 and it can be determined from the free energy @xmath26 . the transition rate constant @xmath112 is then obtained by computing the numerical derivative @xmath113 in the time range where @xmath102 is linear . nucleation rates calculated at @xmath114 and @xmath115 using transition interface sampling @xcite ( tis , red circles in fig . [ fig : rates ] ) agree with the estimates of the bc - based approach up to statistical error . as an additional point of comparison , we used the bc - based approach to compute rates at @xmath116 and @xmath117 , where nucleation is spontaneous on the time - scale of an unconstrained molecular dynamics simulation starting in the metastable liquid . the estimate @xmath118 obtained from straight - forward md simulations in ref . @xcite agrees well with the bc - based estimate of @xmath119 . since the volume of the largest bubble is a good reaction coordinate for the transition , its diffusivity can be computed via mean first passage times @xcite . assuming that the diffusion coefficient does not change significantly in the barrier region , i.e. , @xmath120 , to second order it can be expressed as @xmath121 where @xmath122 is the distance of the absorbing boundary from the top of the free energy barrier , approximated by an inverted parabola with curvature @xmath46 , and @xmath123 is the mean first passage time for a given value of @xmath122 . as a starting point at the top of the barrier we used equilibrium configurations created by umbrella sampling where the system contained a cluster of critical size and drew the particle velocities as well as the thermostat and barostat velocities at random from the appropriate maxwell - boltzmann distributions . as a point of comparison , we obtain an estimate for the cavitation rates from cnt with a constant surface tension @xmath124 for tip4p/2005 water @xcite . the cnt estimate for the rate is given by @xmath125 the equation above was obtained from equs . ( [ eq : rates2 ] ) and ( [ eq : drp ] ) , where @xmath126 and the probability density @xmath127 . here , @xmath128 and the normalization constant was chosen as @xmath129 , where @xmath63 and @xmath64 are the number density of the metastable liquid and the number density of the vapor at coexistence @xcite , respectively . note that the prefactor @xmath28 is not uniquely defined in the framework of cnt and various choices have been employed in the literature @xcite . these choices lead to estimates ranging from @xmath130 to @xmath131 at @xmath132 ( we obtain @xmath133 from the simulation data shown in fig . [ fig : fs ] ) . the resulting predictions for the cavitation rates underestimate the values determined from simulation by @xmath134 orders of magnitude . we thank s. garde , p. geissler , v. molinero , a. patel , a. trster , e. vanden - eijnden , and s. venkatari for insightful comments . the work of g.m . , p.g . , and c.d . was supported by the austrian science foundation ( fwf ) under grant p24681-n20 and within the sfb vicom ( grant no . p.g . also acknowledges financial support from fwf grant p22087-n16 and f.c . from erc under the european fp7 grant agreement 240113 . c.v . acknowledges financial support from a marie curie integration grant 322326-cosaac - fp7-people - cig-2012 and a ramon y cajal tenure track . the team at madrid acknowledges funding from the mcinnc grant fis2013 - 43209-p . calculations were carried out on the vienna scientific cluster ( vsc ) . 10 green jl , durben dj , wolf gh , angell ca ( 1990 ) water and solutions at negative pressure : raman spectroscopic study to -80 megapascals . 249(4969):649652 . zheng q , durben dj , wolf gh , angell ca ( 1991 ) liquids at large negative pressures - water at the homogeneous nucleation limit . 254(5033):829832 . alvarenga ad , grimsditch m , bodnar rj ( 1993 ) elastic properties of water under negative pressures . 98(11):83928396 . azouzi mem , ramboz c , lenain jf , caupin f ( 2013 ) a coherent picture of water at extreme negative pressure . 9(1):3841 . mercury l , shmulovich k ( 2014 ) experimental superheating and cavitation of water and solutions at spinodal - like negative pressures in _ transport and reactivity of solutions in confined hydrosystems _ , nato science for peace and security series c : environmental security , eds . mercury l , tas n , zilberbrand m. 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( 2002 ) symplectic quaternion scheme for biophysical molecular dynamics . 116:8649 . omelyan ip , mryglod i m , folk r ( 2002 ) optimized verlet - like algorithms for molecular dynamics simulations . 65:056706 . geyer cj , thompson ea ( 1995 ) annealing markov chain monte carlo with appplications to ancestral inference . 90:909920 . van erp ts ( 2007 ) reaction rate calculation by parallel path swapping . 98:268301 . bolhuis pg ( 2008 ) rare events via multiple reaction channels sampled by path replica exchange . 129(11):114108 . ferrenberg am , swendsen rh ( 1989 ) optimized monte carlo data analysis . 63:11951198 . berezhkovskii am , szabo a ( 2013 ) diffusion along the splitting / commitment probability reaction coordinate . 117(42):1311513119 . lu j , vanden - eijnden e ( 2014 ) exact dynamical coarse - graining without time - scale separation . 141(4):044109 . vega c , de miguel e ( 2007 ) surface tension of the most popular models of water by using the test - area simulation method . 126(15):154707 . tolman rc ( 1949 ) the effect of droplet size on surface tension . 17(3):333337 . rowlinson js , widom b ( 1989 ) _ molecular theory of capillarity_. ( dover publications , new york ) . troster a , oettel m , block b , virnau p , binder k ( 2012 ) numerical approaches to determine the interface tension of curved interfaces from free energy calculations . 136(6):064709 . joswiak mn , duff n , doherty mf , peters b ( 2013 ) size - dependent surface free energy and tolman - corrected droplet nucleation of tip4p/2005 water . 4(24):42674272 . bruot n , caupin f ( 2016 ) curvature dependence of the liquid - vapor surface tension beyond the tolman approximation . 116:056102 . sedlmeier f , netz rr ( 2012 ) the spontaneous curvature of the water - hydrophobe interface . 137(13):135102 . vaikuntanathan s , geissler pl ( 2014 ) putting water on a lattice : the importance of long wavelength density fluctuations in theories of hydrophobic and interfacial phenomena . 112:020603 . factorovich mh , molinero v , scherlis da ( 2014 ) vapor pressure of water nanodroplets . 136(12):45084514 . wilhelmsen o , bedeaux d , reguera d ( 2015 ) communication : tolman length and rigidity constants of water and their role in nucleation . 142(17):171103 . lau gv , hunt pa , mller ea , jackson g , ford ij ( 2015 ) water droplet excess free energy determined by cluster mitosis using guided molecular dynamics . 143(24 ) . nevins d , spera fj ( 2007 ) accurate computation of shear viscosity from equilibrium molecular dynamics simulations . 33(15):12611266 . gonzalez ma , abascal jlf ( 2010 ) the shear viscosity of rigid water models . 132(9):096101 . caupin f ( 2015 ) escaping the no man s land : recent experiments on metastable liquid water . 407:441 448 . 7th idmrcs : relaxation in complex systems . pallares g , gonzalez ma , abascal jlf , valeriani c , caupin f ( 2016 ) equation of state for water and its line of density maxima down to -120 mpa . . stan ca et al . ( 2016 ) negative pressures and spallation in water drops subjected to nanosecond shock waves . 7(11):20552062 .

, we apply a combination of several complementary computer simulation methods to identify the molecular mechanism of cavitation . chandler approach for the computation of reaction rate constants . the obtained rates are validated for selected points by comparison to estimates from transition interface sampling and support estimates obtained from inclusion experiments . to augment the microscopic picture of cavitation

despite its relevance in biology and engineering , the molecular mechanism driving cavitation in water remains unknown . using computer simulations , we investigate the structure and dynamics of vapor bubbles emerging from metastable water at negative pressures . we find that in the early stages of cavitation , bubbles are irregularly shaped and become more spherical as they grow . nevertheless , the free energy of bubble formation can be perfectly reproduced in the framework of classical nucleation theory ( cnt ) if the curvature dependence of the surface tension is taken into account . comparison of the observed bubble dynamics to the predictions of the macroscopic rayleigh plesset ( rp ) equation , augmented with thermal fluctuations , demonstrates that the growth of nanoscale bubbles is governed by viscous forces . combining the dynamical prefactor determined from the rp equation with cnt based on kramers formalism yields an analytical expression for the cavitation rate that reproduces the simulation results very well over a wide range of pressures . furthermore , our theoretical predictions are in excellent agreement with cavitation rates obtained from inclusion experiments . this suggests that homogeneous nucleation is observed in inclusions , whereas only heterogeneous nucleation on impurities or defects occurs in other experiments . due to its pronounced cohesion , water remains stable under tension for long times . experimentally , strongly negative pressures exceeding can be sustained before the system decays into the vapor phase via cavitation , i.e. , bubble nucleation . recently , cavitation in water under tension has drawn research interest due to its importance in biological processes , like water transport in natural and synthetic trees , spore propagation of ferns , and poration of cell membranes . furthermore , cavitation in water appears to be the driving force behind the sonocrystallization of ice and preventing its occurrence remains a challenge in turbine and propeller design . studying the onset of cavitation has also proven to be a valuable tool to locate the line of density maxima in metastable water , which contributes to the ongoing effort of explaining the origin of water s anomalies . interest in the topic is magnified by the startling discrepancy arising when cavitation in water is investigated using different experimental methods . while agreement between different methods is excellent in the high - temperature regime , where the liquid is unable to sustain large tension , a significantly higher degree of metastability is reached when studying cavitation in inclusions along an isochoric path compared to other techniques at low temperatures . due to the short time - scale on which the transition takes place and the small volume of the critical bubble at experimentally feasible conditions , direct observation of cavitation at the microscopic level remains elusive . however , cavitation rates are directly accessible in experiment and some microscopic insight into the cavitation transition can be obtained from these data by means of the nucleation theorem , which relates the variation in the height of the free energy barrier separating the metastable liquid from the vapor phase upon change of external parameters to properties of the critical bubble . the microscopic information that can be inferred is limited and , since not all quantities entering the nucleation theorem are known , _ ad hoc _ assumptions have to be introduced . for state - points where cavitation is a rare event , classical nucleation theory ( cnt ) can be invoked to provide a qualitative understanding of the transition . however , while cnt provides a physically meaningful and appealingly simple picture of nucleation processes , the estimates for the nucleation rates obtained from cnt are known to differ substantially ( up to many orders of magnitude ) from those measured in experiments . computer simulations are a natural choice to investigate cavitation in water with molecular resolution on the time - scales governing the emergence of microscopic bubbles in the liquid . while cavitation in simple liquids has been studied extensively using computer simulations , simulation studies of cavitation in water were focused on methodological aspects or performed at state points in vicinity of the vapor liquid spinodal . in this work , we apply a combination of several complementary computer simulation methods to identify the molecular mechanism of cavitation . a statistical committor analysis carried out on reactive trajectories reveals that the volume of the largest bubble in the system constitutes a good reaction coordinate for bubble nucleation . we compute the dynamics of nanoscale bubbles along this reaction coordinate and demonstrate that the pressure dependence of the bubble diffusivity can be reproduced by rayleigh plesset ( rp ) theory generalized to include thermal fluctuations , thereby elucidating the crucial influence of viscous damping on bubble growth . based on kramers formalism and the rp equation we obtain an analytical expression for the nucleation rate that yields excellent agreement with numerical results obtained for a wide range of pressures with a method akin to the bennett chandler approach for the computation of reaction rate constants . the obtained rates are validated for selected points by comparison to estimates from transition interface sampling and support estimates obtained from inclusion experiments . to augment the microscopic picture of cavitation we characterize the morphology of bubbles in water under tension and analyze the bubble surface in terms of its hydrogen bonding structure .

1606.03392

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to put the cavitation pressure presented in fig . [ fig : rates ] into context , we discuss its relation to experimental results obtained from different setups . our estimate for the cavitation pressure , obtained from the cavitation rates calculated for typical experimental conditions , @xmath236 , can help disentangle the experimental situation . as discussed before @xcite , experiments fall in two major groups . on the one hand , a set of very different techniques reach similar @xmath65 around @xmath237 . on the other hand , only one technique ( water inclusions in quartz ) seems to reach beyond @xmath238 @xcite . a first possible explanation for this discrepancy in measured cavitation pressures is that the pressure reported for the inclusion experiments is not correct , because an extrapolated equation of state is used to infer @xmath65 from the fluid density in the inclusion and the temperature @xmath239 at which cavitation occurs . this explanation was excluded based on direct measurement of the cavitation density @xcite and , more recently , by a direct experimental determination of the pressure reached in inclusions @xcite . the latter work provides an equation of state down to @xmath240 at around @xmath241 ; a short extrapolation then confirms that pressures close to @xmath242 have been reached the experiments discussed in ref . @xcite . two different scenarios can explain the discrepancy between experiments @xcite : ( i ) either homogeneous cavitation occurs in water close to @xmath237 , and water in inclusions is stabilized by some unknown mechanism , or ( ii ) homogeneous cavitation occurs close to @xmath242 , and , apart from the inclusion work , nucleation occurs heterogeneously in experiments at lower tensions because of ubiquitous impurities . the cavitation rates obtained in the present work based on molecular simulation of pristine water , which result in @xmath236 , support the second scenario . this result is in good agreement with density functional theory calculations @xcite , while cnt with the prefactor employed in the rate prediction shown in fig . [ fig : rates ] yields a stronger tension of @xmath243 . based on theoretical predictions , the second scenario is therefore more likely , although the nature of the impurities inducing cavitation at @xmath237 is still unclear . a recent shock pulse study @xcite proposes that , for extremely fast pressure ramps , homogeneous cavitation beyond @xmath238 could occur concurrently with heterogeneous cavitation because the bubbles from heterogeneous nucleation forming at around @xmath237 do not have enough time to grow sufficiently to release the tension in the system . finally , we note that heterogeneous nucleation also occurs in some inclusions . indeed , in a given quartz sample containing many inclusions with similar liquid density , a wide range of @xmath65 has been observed @xcite . analyzing the details of the nucleation statistics in a given inclusion @xcite clearly shows that the scatter of nucleation temperatures for a given inclusion is fully consistent with nucleation theory : cavitation is a stochastic event depending on the thermodynamic conditions , the sample volume , and its cooling rate . however , the distribution of nucleation temperatures in a given inclusion is quite narrow , around @xmath244 , one order of magnitude less than the broad range observed between different inclusions with the same density . it must be concluded that heterogeneous nucleation ( on dissolved impurities or surface defects ) is responsible for the scatter of @xmath65 in inclusions . however , in the inclusions with the largest @xmath65 , it is assumed that nucleation occurs homogeneously . it is of course possible that further experiments would find inclusions exhibiting an even more negative @xmath65 . however the numerous experiments already performed and the consistent trend observed for the most negative @xmath65 vs. density suggests that the homogeneous nucleation limit has been reached . the value obtained for @xmath65 in the present work supports that this is indeed the case .

this suggests that homogeneous nucleation is observed in inclusions , whereas only heterogeneous nucleation on impurities or defects occurs in other experiments . recently , cavitation in water under tension has drawn research interest due to its importance in biological processes , like water transport in natural and synthetic trees , spore propagation of ferns , and poration of cell membranes .

despite its relevance in biology and engineering , the molecular mechanism driving cavitation in water remains unknown . using computer simulations , we investigate the structure and dynamics of vapor bubbles emerging from metastable water at negative pressures . we find that in the early stages of cavitation , bubbles are irregularly shaped and become more spherical as they grow . nevertheless , the free energy of bubble formation can be perfectly reproduced in the framework of classical nucleation theory ( cnt ) if the curvature dependence of the surface tension is taken into account . comparison of the observed bubble dynamics to the predictions of the macroscopic rayleigh plesset ( rp ) equation , augmented with thermal fluctuations , demonstrates that the growth of nanoscale bubbles is governed by viscous forces . combining the dynamical prefactor determined from the rp equation with cnt based on kramers formalism yields an analytical expression for the cavitation rate that reproduces the simulation results very well over a wide range of pressures . furthermore , our theoretical predictions are in excellent agreement with cavitation rates obtained from inclusion experiments . this suggests that homogeneous nucleation is observed in inclusions , whereas only heterogeneous nucleation on impurities or defects occurs in other experiments . due to its pronounced cohesion , water remains stable under tension for long times . experimentally , strongly negative pressures exceeding can be sustained before the system decays into the vapor phase via cavitation , i.e. , bubble nucleation . recently , cavitation in water under tension has drawn research interest due to its importance in biological processes , like water transport in natural and synthetic trees , spore propagation of ferns , and poration of cell membranes . furthermore , cavitation in water appears to be the driving force behind the sonocrystallization of ice and preventing its occurrence remains a challenge in turbine and propeller design . studying the onset of cavitation has also proven to be a valuable tool to locate the line of density maxima in metastable water , which contributes to the ongoing effort of explaining the origin of water s anomalies . interest in the topic is magnified by the startling discrepancy arising when cavitation in water is investigated using different experimental methods . while agreement between different methods is excellent in the high - temperature regime , where the liquid is unable to sustain large tension , a significantly higher degree of metastability is reached when studying cavitation in inclusions along an isochoric path compared to other techniques at low temperatures . due to the short time - scale on which the transition takes place and the small volume of the critical bubble at experimentally feasible conditions , direct observation of cavitation at the microscopic level remains elusive . however , cavitation rates are directly accessible in experiment and some microscopic insight into the cavitation transition can be obtained from these data by means of the nucleation theorem , which relates the variation in the height of the free energy barrier separating the metastable liquid from the vapor phase upon change of external parameters to properties of the critical bubble . the microscopic information that can be inferred is limited and , since not all quantities entering the nucleation theorem are known , _ ad hoc _ assumptions have to be introduced . for state - points where cavitation is a rare event , classical nucleation theory ( cnt ) can be invoked to provide a qualitative understanding of the transition . however , while cnt provides a physically meaningful and appealingly simple picture of nucleation processes , the estimates for the nucleation rates obtained from cnt are known to differ substantially ( up to many orders of magnitude ) from those measured in experiments . computer simulations are a natural choice to investigate cavitation in water with molecular resolution on the time - scales governing the emergence of microscopic bubbles in the liquid . while cavitation in simple liquids has been studied extensively using computer simulations , simulation studies of cavitation in water were focused on methodological aspects or performed at state points in vicinity of the vapor liquid spinodal . in this work , we apply a combination of several complementary computer simulation methods to identify the molecular mechanism of cavitation . a statistical committor analysis carried out on reactive trajectories reveals that the volume of the largest bubble in the system constitutes a good reaction coordinate for bubble nucleation . we compute the dynamics of nanoscale bubbles along this reaction coordinate and demonstrate that the pressure dependence of the bubble diffusivity can be reproduced by rayleigh plesset ( rp ) theory generalized to include thermal fluctuations , thereby elucidating the crucial influence of viscous damping on bubble growth . based on kramers formalism and the rp equation we obtain an analytical expression for the nucleation rate that yields excellent agreement with numerical results obtained for a wide range of pressures with a method akin to the bennett chandler approach for the computation of reaction rate constants . the obtained rates are validated for selected points by comparison to estimates from transition interface sampling and support estimates obtained from inclusion experiments . to augment the microscopic picture of cavitation we characterize the morphology of bubbles in water under tension and analyze the bubble surface in terms of its hydrogen bonding structure .

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compact x - ray sources exhibit a wide range of temporal variabilities ( from milliseconds to years ) . perhaps none of these is as exotic and diverse as the x - ray temporal variability observed from the black hole microquasar grs 1915 + 105 ( castro - tirado , brandt , & lund 1992 ; greiner , morgan , & remillard 1996 ; morgan , remillard & greiner 1997 ; muno , morgan & remillard 1999 ) . this object is one of two known galactic x - ray sources that exhibit superluminal radio jets ( mirabel & rodrigues 1994 ) . the combination of relativistic constraints and radio measurements at hi indicate that the source lies behind the sagittarius arm at a distance of 12.5 @xmath3 kpc ( mirabel & rodrigues 1994 ) . interstellar extinction limits optical / ir studies to weak detections at wavelengths less than 1 micron ( mirabel et al . the source is suspected to be a black hole binary because of its spectral and temporal similarities with the other galactic x - ray source with superluminal radio jets , gro j1655 - 40 ( zhang et al 1994 ) , which has a binary mass function indicative of a black hole system ( bailyn et al . 1995 ) . estimates for the mass of the compact object in grs 1915 + 105 range from 7 to 33 @xmath4 . even with the uncertainty in distance , its peak x - ray luminosity is unusually high , i.e. , @xmath5 ergs / sec , which is around the eddington luminosity for a @xmath6 @xmath4 object . in spite of several attempts it has proven especially illusive to interpret the x - ray light curves of grs1915 . it is not yet clear that even the basic time scales exhibited by the variability have been successfully explained . belloni et al . ( 1997a , b ) accounted for the observations with an empirical model in which the inner disk region `` disappears '' in the low count rate state , and is then replenished on a viscous time scale . the parameters of their model are : the inner disk radius , @xmath7 ; the corresponding effective temperature of the disk @xmath8 , and an ad - hoc non - thermal power law ( which is possibly produced in the disk corona ) . although no detailed physical model for the instability was given , very interesting patterns of behavior for @xmath7 and @xmath8 , as well as several other observables , were deduced from the data , and the shakura - sunyaev viscosity parameter was found to be unexpectedly low ( which may mean that the standard viscosity prescription is invalid for this source ) . the rather small values of @xmath7 found by these authors can be used to discriminate between different models of the accretion flow in grs 1915 ( see appendix and [ sect : previous ] ) . abramowicz , chen & taam ( 1995 ) suggested a model for the low frequency quasi - periodic oscillations ( qpo ) observed in selected x - ray binaries , in which a corona above the _ standard accretion _ disk leads to a mild oscillatory behavior . with some modifications , this model could reasonably be expected to account for at least some of the temporal variability in grs 1915 + 105 as well ( taam , chen & swank 1997 ) . however , it appears to us that the analysis of abramowicz et al . ( 1995 ) and taam et al . ( 1997 ) contains an error in the heating / cooling equation for the disk which , when corrected , constrains their model to have the same stability characteristics as a standard shakura - sunyaev disk , and is therefore unlikely to explain the grs 1915 + 105 observations ( see appendix a ) . we show more generally in appendix [ sect : geometry ] that neither a hot central region , nor an advection - dominated flow , nor a `` slim '' accretion disk are compatible with the observations of grs 1915 + 105 . ( `` slim '' accretion disk theory was developed in the most detail by abramowicz et al . 1988 ; it is similar to a standard thin shakura - sunyaev disk , except for the energy equation , which incorporates the radial advection of energy into the black hole . ) in this paper , we attempt to undertake a more systematic study of the variability patterns in grs 1915 + 105 within the context of the `` cold disk@xmath9hot corona '' picture . in [ sect : framework ] we present a general discussion that will guide us in our selection of a novel ( though somewhat ad - hoc ) prescription for the viscosity in cases where the radiation pressure is substantial ( [ sect : prescription ] ) . the details of our numerical algorithm to solve the time - dependent disk equations with the use of this new viscosity prescription are given in [ sect : code ] . in [ sect : results ] , we present the results of our time - dependent disk calculations . the calculated light curves are found to agree qualitatively with many observational features of grs 1915 + 105 . in particular , the characteristic cycle times and duty cycles are in reasonable agreement with the observations . moreover , the trend in the cycle time with the average accretion rate , @xmath10 has the correct sense . however , there are important disagreements as well . we therefore introduce a more elaborate model in [ sect : jet ] , where , in accordance with the observations ( e.g. , mirabel & rodrigues 1994 ) , the inner disk is allowed to expel some of its energy in the form of a non - steady jet . we assume that the ejected energy is not observed in x - rays , but rather that it ultimately produces radio emission . we show that this more elaborate model agrees with the grs 1915 + 105 observations much better , perhaps indicating that we are finally developing a zeroth order understanding of the geometry and the most important processes in this enigmatic source . in [ sect : previous ] we discuss our results in the light of the earlier work on grs 1915 , and in [ sect : discussion ] we summarize our conclusions .

during the past two years , the galactic black hole microquasar grs 1915 + 105 has exhibited a bewildering diversity of large amplitude , chaotic variability in x - rays . although it is generally accepted that the variability in this source results from an accretion disk instability , the exact nature of the instability remains unknown .

during the past two years , the galactic black hole microquasar grs 1915 + 105 has exhibited a bewildering diversity of large amplitude , chaotic variability in x - rays . although it is generally accepted that the variability in this source results from an accretion disk instability , the exact nature of the instability remains unknown . here we investigate different accretion disk models and viscosity prescriptions in order to provide a basic explanation for the exotic temporal behavior in grs 1915 + 105 . we discuss a range of possible accretion flow geometries . based on the fact that the overall cycle times are very much longer than the rise / fall time scales in grs 1915 , we rule out the geometry of advection dominated accretion flow ( adaf ) or a hot quasi - spherical region plus a cold outer disk for this source . a cold disk extending down to the last inner stable orbit plus a hot corona above it , on the other hand , is allowed . we thus concentrate on geometrically thin ( though not necessarily standard ) shakura - sunyaev type disks ( shakura & sunyaev 1973 ; hereafter ss73 ) . we argue that x - ray observations clearly require a quasi - stable accretion disk solution at high accretion rates where radiation pressure begins to dominate , which excludes the standard-viscosity prescription . to remedy this deficiency , we have therefore devised a modified viscosity law that has a quasi - stable upper branch , and we have developed a code to solve the time - dependent equations to study such an accretion disk . via numerical simulations , we show that the model does account for several gross observational features of grs 1915 + 105 , including its overall cyclic behavior on time scales of 100 - 1000 s. on the other hand , the rise / fall time scales are not short enough , no rapid oscillations on time scales 10 s emerge naturally from the model , and the computed cycle - time dependence on the average luminosity is stronger than is found in grs 1915 + 105 . we then consider , and numerically test , several effects as a possible explanation for the residual disagreement between the model and the observations . a hot corona with the energy input rate being a function of the local cold disk state and a radius - dependent-parameter do _ not _ appear to be promising in this regard . however , a more elaborate model that includes the cold disk , a corona , and plasma ejections from the inner disk region allows us to reproduce several additional observed features of grs 1915 + 105 . we conclude that the most likely structure of the accretion flow in this source is that of a cold disk with a modified viscosity prescription , plus a corona that accounts for much of the x - ray emission , and unsteady plasma ejections that occur when the luminosity of the source is high . the disk is geometrically thin due to the fact that most of the accretion power is drained by the corona and the jet .

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there are essentially 5 free parameters of the model that need to be specified and/or systematically varied to ascertain the sensitivity of the results to their value : ( _ i _ ) the black hole mass , @xmath15 , ( _ ii _ ) the average accretion rate , @xmath10 , ( _ iii _ ) the viscosity parameter , @xmath69 , for the gas - dominated region , ( _ iv _ ) @xmath87 , the critical ratio @xmath82 , and ( _ v _ ) @xmath67 , the fraction of the luminosity transported vertically by mechanisms other than radiative diffusion . we adopt a mass of @xmath89 , which is typical of the measured masses of galactic black hole candidates in binary systems ( see , e.g. , barret et al . 1996 ) . for purposes of illustration , we chose a value of @xmath69 equal to @xmath90 , since it leads to viscous time scales comparable to those observed . similarly , we selected a value of @xmath91 , in part because it produces a reasonable s - curve . however , we checked a range of other values as well ( from @xmath92 to @xmath93 ) , and found that @xmath94 reproduces the grs 1915 + 105 observations most closely . we present a computed light curve and variations of the disk parameters through one complete cycle in figures ( [ fig : feqo ] ) and ( [ fig : temp_tau_distr ] ) , respectively , for one set of illustrative parameters : @xmath95 and @xmath96 . in figure ( [ fig : temp_tau_distr ] ) , panels ( a ) , ( b ) , ( c ) and ( d ) show the evolution of the disk effective temperature @xmath97 , the ratio of radiation to gas pressure ( @xmath85 ) , the disk thomson optical depth @xmath24 and the radial flow velocity @xmath98 . note that since we are currently concerned only with the luminosity integrated over all photon energies , we define @xmath97 to include the emissivity from _ both _ the corona and the disk , i.e. , @xmath99 , where @xmath100 is given by equation ( [ fvert ] ) . in each panel , the series of curves are for temporal increments of @xmath101 sec . the curve with the smallest luminosity @xmath102 ( where @xmath103 ) in the upper left panel of figure ( [ fig : temp_tau_distr ] ) corresponds to the state of the disk right after the end of an outburst . an outburst starts in the innermost region of the disk , when a few zones at the smallest radii quickly make a transition to the upper stable branch of the solution ( high state ) . although this is not obvious from fig . ( [ fig : temp_tau_distr ] ) , the viscosity of the high state is larger than it is in the low state , because @xmath104 , and so even though @xmath0 decreases , the increase in @xmath76 and @xmath16 leads to a larger @xmath37 . the larger viscosity allows the gas to dispose of its angular momentum faster , thus allowing a faster inflow of the gas into the black hole . because of angular momentum conservation , the angular momentum of the gas that plummets into the black hole is transferred to larger @xmath17 , where it produces an excess of angular momentum , and thus some matter actually flows to larger @xmath17 , which is seen in panel ( d ) of figure ( [ fig : temp_tau_distr ] ) as positive spikes in the radial velocity distribution . a heating wave is initiated and propagates from the inner disk to larger @xmath17 ( fig . [ fig : temp_tau_distr]a ) . this wave is often referred to as either a ` density wave ' or a ` transition wave ' . as the wave propagates outward , the material on the inside of the wave becomes hot and shifts into the high viscosity state . it then rapidly loses its angular momentum and is transferred into the innermost disk , where the material continuously plummets into the hole . since the innermost stable region is a `` bottle - neck '' for the accreting gas , some material builds up there , which explains the bump in @xmath13 in that region as seen in figure ( [ fig : temp_tau_distr]c ) during the outburst . when the density wave reaches @xmath105 , the instability saturates , since the disk in that region is always gas - pressure dominated ( for the chosen @xmath10 and @xmath67 ) . as the density wave dies away , the outermost unstable regions are cleared of the excess mass and cool down . they are now in the low viscosity state , and thus the rate of mass inflow in the innermost region of the disk drops . an inward propagating cooling wave is initiated and the outburst dies out as the last excess mass in the innermost disk sinks into the hole . finally , a new cycle begins with the accumulation of new excess mass ( since the local accretion rate is smaller than the average rate ) . we now systematically explore the disk behavior with the new viscosity prescription , in order to eventually compare the results with the gross temporal behavior of grs 1915 + 105 . we adopted an energy transport fraction @xmath67 = 0.9 . as with the other parameters , we also tested a wide range of values of @xmath67 , and found that @xmath106 is the most appropriate due to the following considerations . the threshold for the onset of the instability is about @xmath107 0.26 for @xmath108 0.9 , and scales as : @xmath109^{-9/8 } \label{dmo}\ ] ] ( see svensson & zdziarski 1994 and note that their definition of @xmath10 differs from ours by a factor that accounts for the standard disk radiative efficiency , i.e. , @xmath110 ) . it seems clear observationally that @xmath111 should be larger by a factor of at least @xmath112 than the value of @xmath113 corresponding to the transition from the gas- to radiation pressure - dominated regime in the standard theory . indeed , if we assume that the maximum luminosity of @xmath114 erg sec@xmath115 observed in grs 1915 + 105 corresponds to the eddington luminosity for this source , then the instability seems to exist only for @xmath116 . in addition , galactic black hole candidates ( gbhcs ) with a lower persistent luminosity , or weaker transient behavior , have not shown such violent instabilities as grs 1915 + 105 does , and yet many of them are as bright as @xmath117 ( see , e.g. , barret , mcclintok & grindlay 1996 ) . we will discuss the parameter @xmath67 further in [ sect : gross ] . in figure ( [ fig : lcurves ] ) we show the evolution of the integrated disk luminosity as a function of time for the nominal values of the parameters listed above and for @xmath118 @xmath119 , 0.31 , 0.5 , and 1.5 . in agreement with equation ( [ dmo ] ) , we find that for values of @xmath10 below 0.26 the disk is stable , because there is no radiation - pressure dominated zone in the disk . we also find , in general , that as the value of @xmath10 increases , the duty cycle ( i.e. , the fraction of time that the source spends in the high state ) increases , from about 5% at @xmath118 0.26 , to approximately 50% when @xmath118 1.5 . similarly , the ratio of maximum to minimum luminosity through the cycle grows with increasing @xmath10 . the time for the disk to complete one of its cycles ( defined as the cycle time ) also increases with @xmath10 , except for very near the minimum value of @xmath118 0.26 where the instability first sets in ; here the cycle time actually decreases with increasing @xmath10 . the reason is that there seems to exist a `` critical excess mass '' in the inner disk region , such that the instability will appear only when the mass in the disk exceeds this value . the rate for building up the excess mass is @xmath120 , where @xmath111 is the maximum stable accretion rate given by equation ( [ dmo ] ) , and therefore a small increase in @xmath10 above @xmath111 can bring about a substantial decrease in the cycle time . for larger accretion rates , the cycle time starts to increase with increasing accretion rate , because a larger region of the disk is unstable , and it takes longer to clear the excess mass during the outburst . we will now compare our results shown in figure ( [ fig : lcurves ] ) with observations of grs 1915 + 105 . we should acknowledge at the start that it seems a daunting task to try to explain with a single model the diverse , unstable behavior exhibited by this source , but we expect to benefit from a comparison of even the gross properties of our model with the observations . panel ( a ) of figure ( [ fig : lcurves ] ) is similar to figure ( [ fig : data]a ) in that in both figures there exists a long phase of mass accumulation ( the low state ) . furthermore , an examination of the grs 1915 + 105 data reveals that the cyclic limiting behavior disappears when the average count rate is lower than @xmath121 counts / sec , so that there indeed exists a threshold accretion rate below which the instability does not operate . in simulations , the spiky nature of the outburst is explained by the fact that only a very narrow region in @xmath17 within the disk is unstable , and it takes very little time to get rid of the excess mass in that region . the differences with fig . ( 1a ) are mainly due to the fact that an outburst in the data does not seem to completely clear the excess mass , and that more oscillations follow with a gradually declining amplitude , whereas in the simulations the hot state persists until all the excess mass is swallowed by the black hole . panel ( b ) of fig . ( [ fig : data ] ) is also somewhat similar to panels ( b ) and ( c ) of the simulations ( fig . [ fig : lcurves ] ) . the progression of time scales and the average luminosity ( i.e. , roughly speaking , the count rate ) from pattern ( a ) to ( b ) in fig . ( [ fig : data ] ) is similar to that from panel ( a ) to ( b ) in fig . ( [ fig : lcurves ] ) as well . the physical reason for the cycle time getting shorter with increasing @xmath10 is that the excess mass can build up faster for larger accretion rates . next , panel ( c ) in figure ( [ fig : data ] ) is most interesting from the point of view that the high state is long lasting and is clearly stable at least for the first half . panel ( d ) of the simulations can account for some of the properties of this variability pattern . in particular , the duty cycle increased in both the simulations and the data as the accretion rate increased . the cycle time did not increase nearly enough in the simulations , however . it could be made longer by choosing a smaller @xmath69 , but then the time scales in all the other panels would increase as well , which does not appear acceptable . also , a general trend seen in the first three panels ( a - c ) in both the data and the simulations is that the minimum of each light curve shows a slow gradual rise in luminosity before the instability sets in . this is a clear indication of the disk accumulating mass until it reaches a global instability . we note that panel ( d ) of figure ( [ fig : data ] ) seems to be rather different from the other three variability examples . if the high state is again defined as the one with larger count rates , then a peculiarity of panel ( d ) is that the count rate first decreases and then increases by the end of a high state episode . this is the opposite of what is seen for the other states , such as that shown in panel ( c ) of fig . ( 1 ) . from a theoretical point of view , this is a highly significant observation . in all of our simulations , we observed that as the outburst progresses , the amount of mass in the inner disk region builds up because of the greater and greater inflow of mass from larger radii . once in a given ( high or low ) stable state , the local disk luminosity is proportional to the local column density @xmath13 , which can be seen from figure ( [ fig : scurve ] ) . therefore , in an outburst , the luminosity decreases only after the outer disk cools down and the influx of mass stops , leading to a decrease in @xmath13 in the inner disk . thus , the outburst profile in the simulations is such that the second time derivative @xmath122 is always negative , not positive as seen in panel ( d ) of figure 1 . we do not see a clear explanation for this disagreement in the context of the current model , but we will show that there might be a natural cause for this phenomenon if a jet is allowed ( [ sect : jet ] ) . finally , we note that in a number of the panels in figure ( [ fig : lcurves ] ) , the first ( and sometimes also the second ) peak in the light curve is of slightly different intensity or duration than the peaks that come later in the sequence . this is simply a consequence of the disk adjusting to its quasi - steady cyclic pattern after starting from the imposed initial conditions that are given by the shakura - sunyaev solution corresponding to the mean value of @xmath10 . let us now discus the rather extreme value for @xmath67 invoked here , which requires that as much as @xmath123 of the accretion power is carried out of the disk by processes other than the usual radiative diffusion . if this process is magnetic buoyancy or some other mhd process heating the corona , then there appears to be a contradiction with the observations of grs 1915 + 105 . the blackbody component in the spectrum of this source is generally small in the low state , but it can be dominant in the outburst state ( muno , morgan & remillard 1999 ) . one could argue here that as much as @xmath124 of the x - rays emitted by the corona towards the disk may be reprocessed into the soft disk radiation , ( e.g. , magdziarz & zdziarski 1995 ) if the reflection / reprocessing takes place in a _ neutral medium _ ( neutral in the sense of high - z elements like oxygen and iron , that are important for the reflection spectrum ) . thus , in principle , the blackbody flux could exceed that in the non - thermal component by the ratio @xmath125 ( see eqs . 15 in haardt & maraschi 1991 with their parameters @xmath126 and @xmath127 ) . the rather high inferred disk temperature in grs 1915 + 105 , e.g. , @xmath128 kev ( belloni et al . 1997a ) , rules out the possibility of neutral reflection , however . as found by nayakshin & dove ( 1998 ) and nayakshin ( 1998 ) , the integrated albedo of the reflected spectrum in the case of a strongly ionized disk can be much higher than the standard 0.1 - 0.2 of the neutral reflector ( i.e. , @xmath129 in eqs . 15 of haardt & maraschi 1991 ) , and thus it seems difficult to have @xmath67 as large as 0.9 given the large amount of soft power observed from the disk . a possible way out of this dilemma is that the high value of @xmath67 does not necessarily represent the energy flux from the disk into the corona . indeed , equation ( [ fvert ] ) states that the energy transport out of the disk may be @xmath56 times faster than that given by the standard diffusion of radiation . convection of energy in the vertical direction ( e.g. , bisnovatiy - kogan & blinnikov 1977 ; goldman & wandel 1995 ; and references therein ) is one physical mechanism that can speed up the transfer of energy out of the disk . mhd waves dissipating their energy before they reach the corona could be another . in addition , there is no proof that the vertical averaging procedure ( i.e. , a one zone approximation ) used in the standard accretion disk theory does not lead to a substantial underestimation of the vertical radiation flux out of the disk . for example , if @xmath130 , where @xmath131 few , then @xmath111 becomes @xmath132 ( see sz94 ) , and thus one may have @xmath111 as large as that observed in grs 1915 + 105 due to a faster or an additional disk cooling mechanism rather than due to a transfer of most of the disk power into the corona . it is notable that panel ( c ) of fig . ( [ fig : data ] ) shows rapid chaotic oscillations as fast as @xmath1 tens of seconds at the end of the high state , whereas our simulations do not show a similar behavior . furthermore , note that the rise / fall time scales are shorter in the data than they are in our model . it appears that the heating of the unstable region in grs1915 happens on a time scale that is very much shorter than the cycle time . by contrast , the outermost part of the unstable region of the simulated disk becomes unstable only after @xmath133 the duration of the hot phase , that is of the order of the disk viscous time for the high luminosity cases . one explanation here could be that these fast oscillations are failed attempts by the disk to make a state transition from the high to the low state . further , during these fast oscillations the disk is always brighter than it is in the low state , which could be interpreted as an indication that only a part of the disk ( most likely the innermost region ) in grs 1915 + 105 takes part in these rapid oscillations , whereas the rest of the unstable region is still on the upper stable branch of the s - curve . for this to be true , the inner disk must be able to decouple to some extent from the rest of the unstable disk , and be variable on a much shorter time scale than the outer disk . to test this idea , we have carried out several simulations in which the @xmath0-parameter is a decreasing function of radius . our hope is that in this case , since @xmath0 in the inner disk is larger than that in the outer disk , the time scale for oscillations in the inner disk may be much shorter than the overall viscous time , which has the dependence @xmath134 ( e.g. , frank et al . 1992 ) . in figure ( [ fig : variable_alpha ] ) we show one such simulation , in which the viscosity parameter was chosen to be @xmath135 $ ] . this functional form allows the @xmath0-parameter to be roughly @xmath136 in the inner disk region ( @xmath137 ) , and to be @xmath138 for @xmath139 . the presence of these two time scales is obvious just from a perusal of the resulting light curves . oscillations of the inner disk produce the precursor seen before each major outburst in fig . ( [ fig : variable_alpha ] ) . its relative magnitude is small because the amount of excess mass stored in the inner disk is small compared to that in the outer disk . furthermore , in the simulations the inner disk decouples from the outer one only at the beginning of the outburst , not at the end as is seen in the grs 1915 + 105 data . this is due to the fact that once the outer disk makes a transition to the high state , the large mass supply to the inner disk forces the latter to go into the high state as well and remain there . in summary , varying the @xmath0-parameter with radial distance does not appear to be a viable explanation for the rapid chaotic oscillations seen in the data which are superposed on the more regular long time scale disk evolution . an alternative way of explaining the fast oscillations could have been provided if the heating / cooling fronts stall and are reflected back as cooling / heating fronts . this behavior was observed in simulations of the classical thermal ionization disk instability by cannizzo ( 1993 , see text above his equation 5 ) , where it happened to be an unwanted result . however , we have not been able to see such stalled transition fronts ( except for the case presented in fig . [ fig : variable_alpha ] ) , which should not be too surprising , due to the fact that the underlying physics of the ionization instability and the one explored in this paper are vastly different , and one does not expect a direct correspondence between these two instabilities . although we will not present any light curves , we should mention that we have also attempted to allow the fraction @xmath67 to be a function of @xmath85 , since the transition from @xmath140 to @xmath141 means a substantial change in the physical conditions in the disk . our hope was that the freedom in choosing @xmath142 might help to reproduce the disk flickering . however , all our attempts in this regard ( with @xmath67 decreasing or increasing across the transition in @xmath85 ) have been unsuccessful .

here we investigate different accretion disk models and viscosity prescriptions in order to provide a basic explanation for the exotic temporal behavior in grs 1915 + 105 . based on the fact that the overall cycle times are very much longer than the rise / fall time scales in grs 1915 , we rule out the geometry of advection dominated accretion flow ( adaf ) or a hot quasi - spherical region plus a cold outer disk for this source . we thus concentrate on geometrically thin ( though not necessarily standard ) shakura - sunyaev type disks ( shakura & sunyaev 1973 ; hereafter ss73 ) . we then consider , and numerically test , several effects as a possible explanation for the residual disagreement between the model and the observations . a hot corona with the energy input rate being a function of the local cold disk state and a radius - dependent-parameter do _ not _ appear to be promising in this regard .

during the past two years , the galactic black hole microquasar grs 1915 + 105 has exhibited a bewildering diversity of large amplitude , chaotic variability in x - rays . although it is generally accepted that the variability in this source results from an accretion disk instability , the exact nature of the instability remains unknown . here we investigate different accretion disk models and viscosity prescriptions in order to provide a basic explanation for the exotic temporal behavior in grs 1915 + 105 . we discuss a range of possible accretion flow geometries . based on the fact that the overall cycle times are very much longer than the rise / fall time scales in grs 1915 , we rule out the geometry of advection dominated accretion flow ( adaf ) or a hot quasi - spherical region plus a cold outer disk for this source . a cold disk extending down to the last inner stable orbit plus a hot corona above it , on the other hand , is allowed . we thus concentrate on geometrically thin ( though not necessarily standard ) shakura - sunyaev type disks ( shakura & sunyaev 1973 ; hereafter ss73 ) . we argue that x - ray observations clearly require a quasi - stable accretion disk solution at high accretion rates where radiation pressure begins to dominate , which excludes the standard-viscosity prescription . to remedy this deficiency , we have therefore devised a modified viscosity law that has a quasi - stable upper branch , and we have developed a code to solve the time - dependent equations to study such an accretion disk . via numerical simulations , we show that the model does account for several gross observational features of grs 1915 + 105 , including its overall cyclic behavior on time scales of 100 - 1000 s. on the other hand , the rise / fall time scales are not short enough , no rapid oscillations on time scales 10 s emerge naturally from the model , and the computed cycle - time dependence on the average luminosity is stronger than is found in grs 1915 + 105 . we then consider , and numerically test , several effects as a possible explanation for the residual disagreement between the model and the observations . a hot corona with the energy input rate being a function of the local cold disk state and a radius - dependent-parameter do _ not _ appear to be promising in this regard . however , a more elaborate model that includes the cold disk , a corona , and plasma ejections from the inner disk region allows us to reproduce several additional observed features of grs 1915 + 105 . we conclude that the most likely structure of the accretion flow in this source is that of a cold disk with a modified viscosity prescription , plus a corona that accounts for much of the x - ray emission , and unsteady plasma ejections that occur when the luminosity of the source is high . the disk is geometrically thin due to the fact that most of the accretion power is drained by the corona and the jet .

1509.08688

i

[ smotivation ] for a smooth projective variety @xmath0 of dimension @xmath1 and a birational self - map @xmath2 one can define a tuple of real numbers @xmath3 where @xmath4 for @xmath5 , called the dynamical degrees of @xmath6 . see section [ sprelim ] for two equivalent definitions of these numbers . the dynamical degrees turn out to be invariant under birational conjugacy , so that the dynamical spectrum @xmath7 is a birational invariant of the variety @xmath0 . one thus might , for example , try to use it to distinguish very general ( conjecturally irrational ) cubic fourfolds @xmath0 from @xmath8 . here are some ideas how the spectra might differ : 1 . as point sets , that is , there might be a tuple @xmath9 in the spectrum of @xmath8 which is not in the spectrum of @xmath0 . this could for example be proven if one could show that on @xmath0 the dynamical degrees have to satisfy other additional inequalities , coming from the geometry of @xmath0 , which can be violated on @xmath8 . for this one has , in particular , to develop certain semi - continuity properties and computational tools for dynamical degrees , which we start doing in the subsequent sections . 2 . as metric ( or only topological ) spaces : for example , one can consider , for every @xmath10 , the smallest gap in the spectrum after @xmath11 , if there is any , that is , @xmath12 it is possible that these numbers differ for @xmath0 and @xmath8 , but the drawback is that it is not easy to see how to relate them to accessible geometric features of @xmath0 . in other words , it seems very hard to compute or estimate them . on the topological side , it may happen that the cantor - bendixson ranks of the dynamical spectra or some linear slices in them differ , but again these are very hard to access . arithmetically : it could happen that for both @xmath0 and @xmath8 , the dynamical degrees are algebraic integers , but for @xmath0 they may satisfy some additional arithmetic constraints . for example , the number fields generated by each tuple of dynamical degrees on @xmath0 might differ from the ones for @xmath8 . but also this seems very hard to detect . we thus concentrate on ( 1 ) for the moment . the paper is organised as follows . in section [ sprelim ] we recall some basic facts about cycles and dynamical degrees . in section [ ssemicont ] we prove theorem [ tsemicontinuity ] which says that the dynamical degrees are lower - semicontinuous functions for families over _ smooth _ bases , if one understands lower semi - continuity in a slightly generalized sense , namely that ( downward ) jumps may occur on countable unions of closed subsets . moreover , in proposition [ pcubicfourfolds ] we collect some results that point out the importance of the questions of existence of ( finite - dimensional , connected , non - trivial ) algebraic subgroups in @xmath13 and of the classification of birational automorphisms with all dynamical degrees equal to @xmath11 for the irrationality problem for cubic fourfolds @xmath0 . in section [ supperdeg ] we study the relationship between dynamical degrees and cremona degrees under some assumptions and prove a sub - multiplicativity result , theorem [ tsubmultiplicative ] . section [ sittrans ] is devoted to reflections on cubic fourfolds and in theorem [ tgeneralpointsequality ] we show , under some assumptions , the equality of the first and second dynamical degrees of a composition of reflections on a smooth cubic fourfold . this is a sample of a type of result which says that special dynamical degrees can only arise in the presence of special dynamics . such implications are very important for making progress on the irrationality problem for cubics , too . finally , in the appendix we describe generalized picard - manin spaces which might prove useful in the study of dynamical degrees on fourfolds in the future . * conventions . * we work over the field of complex numbers @xmath14 throughout the paper . a variety is a reduced and irreducible scheme of finite type over @xmath15 . a prime @xmath10-cycle on a variety is an ( irreducible ) subvariety of dimension @xmath10 . a @xmath10-cycle is a formal linear combination of prime @xmath10-cycles . if @xmath16 is a rational map between varieties , we denote by @xmath17 the largest open subset of @xmath0 on which @xmath6 is a morphism . the graph @xmath18 of @xmath6 is the closure of the locus of points @xmath19 with @xmath20 . * acknowledgments . * we would like to thank miles reid for useful discussions .

dynamical degrees and spectra can serve to distinguish birational automorphism groups of varieties in quantitative , as opposed to only qualitative , ways . we also specialize to cubic fourfolds and show that under certain genericity assumptions the first and second dynamical degrees of a composition of reflections in points on the cubic coincide . supported by heisenberg - stipendium bo 3699/1 - 2 of the dfg ( german research foundation ) during the initial stages of this work ] partially supported by the rtg 1670 of the dfg ( german research foundation ) ]

dynamical degrees and spectra can serve to distinguish birational automorphism groups of varieties in quantitative , as opposed to only qualitative , ways . we introduce and discuss some properties of those degrees and the cremona degrees , which facilitate computing or deriving inequalities for them in concrete cases : ( generalized ) lower semi - continuity , sub - multiplicativity , and an analogue of picard - manin / zariski - riemann spaces for higher codimension cycles . we also specialize to cubic fourfolds and show that under certain genericity assumptions the first and second dynamical degrees of a composition of reflections in points on the cubic coincide . supported by heisenberg - stipendium bo 3699/1 - 2 of the dfg ( german research foundation ) during the initial stages of this work ] partially supported by the rtg 1670 of the dfg ( german research foundation ) ]

physics0310031

r

the post - stimuli dynamics are constrained by the steady states of the system . for loops longer than a minimum length @xmath75 , sustained reentries are stable attractor of the system . these sustained reentries can be either periodic ( period-1 ) or quasiperiodic ( qp ) , and hold a single ( sw ) , two ( dw ) or more traveling activation fronts . table [ table_attract ] lists the stable solutions of both the i d and il models for @xmath76 . the number and nature of the sustained reentries change with @xmath77 and condition the outcomes of the stimulations . [ c]|l|ll|interval & reentry type & + @xmath77 ( cm ) & sw ( single wave ) & + @xmath78 & @xmath79 & + @xmath80 & qp , mode-0 & + @xmath81 & @xmath82{c}% \text{qp},\text{mode-0}\\ \text{qp},\text{mode-1}% \end{array } $ ] & + @xmath83 & period-1 & + @xmath84 & @xmath85 & + @xmath86 & period-1 & + @xmath87 & @xmath85 & + qp reentries are characterized by a spatial oscillation of @xmath27 , with a wavelength @xmath88 that is an irrational fraction of @xmath77 ( fig . [ quasi_sust]a and c ) . for both sw and dw reentries , there is an interval of @xmath77 in which two different qp solutions coexist . these solutions , labeled mode-0 and mode-1 , have a similar structure but @xmath89 . in sw reentries , the passage of each activation front is associated with a profile of @xmath90 and @xmath27 holding one ( mode-0 , panel a ) or multiple ( mode-1 , panel c ) maxima and minima over two turns . for dw qp reentries , @xmath77 is twice the value for sw qp reentries , such that mode-0 has one maximum and one minimum over one turn . [ quasi_sust]a ) . successive activations at each site alternate between long and short @xmath90 and @xmath27 values , except at a number of nodes corresponding to the boundaries from which the phase of the alternation is inverted . as illustrated by the time - course of @xmath16 and @xmath17(fig . [ quasi_sust ] , right panels ) , the quasi - periodic nature of the propagation makes the position of the extrema and of the nodes to drift slowly in the direction inverse to the propagation of the activation fronts . qp reentry is thus constituted by discordant alternans @xcite with boundaries between short and long apd moving around the loop . [ ptb ] fig_di_quasisust.eps period-1 sw reentry can be annihilated by an isolated stimulus applied in the narrow vulnerable window in which the stimulation produces only a retrograde front , corresponding to the well - known mechanism of unidirectional block @xcite . in a previous paper , we have also described other modes of annihilation as well as different transient dynamics that were induced by two successive stimuli @xcite . these new modes of annihilation were compatible with experimental observations , and relevant to antiarrhythmic pacing therapy @xcite . however , that study was restricted to a specific length of the loop ( @xmath91 cm @xmath92 ) . in the following , we present a systematic investigation of the outcomes of double stimuli applied on period-1 sw reentry as a function of the timing of the pulses and the length of the loop . complex dynamics can be induced by a second stimulus @xmath74 thanks to the asymmetric profile in @xmath17 left by the interaction of first stimulus @xmath72 with the reentry activation front @xmath93 . when @xmath71 , the time between the last passage of @xmath93 and the onset of @xmath72 , is beyond the vulnerable window , @xmath72 produces both a retrograde ( @xmath94 ) and an antegrade ( @xmath95 ) activation front . as illustrated in fig . [ fig_gradtrep]a , the key factor determining the dynamics that can be induced by @xmath74 is the region located between the site of stimulation ( @xmath96 ) and the site of the collision between @xmath93 and @xmath94 ( @xmath97 , identified by the arrow in [ fig_gradtrep]a ) . @xmath42 ( thin line in fig . [ fig_gradtrep]a ) is minimum near @xmath96 and reaches its maximum at @xmath97 . the il and i d models produce the same profile of @xmath17 ( thin lines and triangles , respectively , in [ fig_gradtrep]a ) , showing that the i d model that was initially developed to describe sustained reentry also provides an appropriate low - dimensional representation of the dynamics when stimulations are applied . the location of @xmath97 as well as the profile of @xmath17 around @xmath96 depend on both @xmath71 and @xmath77 . figure [ fig_gradtrep]b shows the profile of @xmath17 obtained from loops of different lengths stimulated at the same diastolic interval @xmath98 , where @xmath99 is the action potential duration of the stable reentry for each @xmath77 . the position of @xmath97 is shifted to the left ( arrows in [ fig_gradtrep]b ) because the collision is delayed on longer loops . however all the loops have the same invariant profile of @xmath42 in the time and space interval that they share before the collision . if @xmath72 is applied at larger @xmath100 value , the distance between @xmath96 and @xmath97 is shortened , @xmath101 and @xmath102 are increased , such that the extent and depth of the cusp in @xmath42 around @xmath96 are diminished . [ ptb ] fig_gtrep.eps the spatial profile of @xmath42 for short @xmath71 is asymmetrical , with a sharp gradient between @xmath96 and @xmath103 , and a more gradual increase at the right of @xmath96 . owing to this asymmetry , the outcome of @xmath74 depends on @xmath73 , the time interval between the onset of the two stimuli . figure [ blockr2 ] illustrates a case in which @xmath74 is applied after the collision of @xmath93 and @xmath94 , at an instant where @xmath95 , the antegrade front created by @xmath72 , still has not reached @xmath97 . @xmath74 creates both an antegrade ( @xmath104 ) and a retrograde ( @xmath105 ) front , but @xmath105 is blocked between @xmath96 and @xmath97 . thereupon , the system is left with two antegrade fronts ( @xmath95 and @xmath104 ) . this occurs as long as @xmath105 does not propagate beyond @xmath97 to collide with @xmath95 , in which case @xmath104 is left alone to perpetuate the reentry . this is an alternate scenario of unidirectional block that creates a propagating wave in the same direction as @xmath93 and @xmath95 . [ ptb ] fig_blr2.eps all the complex dynamics occur in the range of @xmath73 for which @xmath105 is blocked between @xmath96 and @xmath97 whereas @xmath104 propagates . this depends on the profile of @xmath106 left by @xmath72 , which was shown to be an invariant function of @xmath100 in fig . [ fig_gradtrep]b . figure [ fig_map_super ] shows the global characteristics of the dynamics in the [ @xmath107 ] parameters plane for two values of @xmath77 ( @xmath108 and @xmath109 cm @xmath110 ) . the parameter plane can be divided in three areas . in the region labeled @xmath111 at low @xmath73 values , @xmath112is applied during the refractory period and does not produce a response . in the upper region , labeled `` @xmath113 '' , @xmath105 propagates beyond @xmath97 , collides with @xmath95 , and @xmath104 is left alone to maintain the reentry . in the middle area , @xmath105 is blocked between @xmath97 and @xmath114and complex dynamics may occur . the phase plane area in which @xmath105 is blocked , for @xmath100 , between @xmath115 and @xmath116 ms ( dotted vertical line ) , is the same for the two @xmath77 . the specific subsets in which complex dynamics occurs ( represented by different shaded areas in fig . [ fig_map_super ] ) change with @xmath77 , and are discussed later . the article is focused on the area where @xmath105 is blocked , between @xmath117 and @xmath118 ms . the lower limit of the area coincides with @xmath119 and is close to the action potential duration restitution curve @xmath120 . the upper limit is nearly a constant , around @xmath121 ms . appendix [ appendix_a ] shows that this @xmath73 upper limit is set by the locus between @xmath114 and @xmath97 where @xmath122 is equal to the maximum conduction time . this locus , indicated by a circle for the specific @xmath100 illustrated in [ fig_gradtrep]b , does not depend on @xmath77 , which explains why the upper limit for the block of @xmath105 is identical in the two loops . @xmath122 remains everywhere below the maximum conduction time if @xmath100 is too long , explaining why the area of complex dynamics disappears beyond @xmath100 @xmath118 ms . it is also demonstrated in appendix [ appendix_a ] that the maximum slope of the @xmath123 function must be greater than @xmath124 to allow the block of @xmath105 . the same condition that controls the stability of the period-1 reentry @xcite thus determines if complex dynamics can be induced by @xmath74 . [ ptb ] fig_map_super.eps once @xmath104 has started to propagate and @xmath105 has been blocked , there are 4 possible outcomes : \1 ) @xmath95 is blocked and @xmath104 perpetuates the sw reentry , \2 ) @xmath104 is blocked , and @xmath95 maintains the sw reentry , \3 ) @xmath95 and @xmath104 are blocked , and reentry is annihilated \4 ) neither @xmath95 nor @xmath104 are blocked , and there is a transition to dw reentry . as seen in table [ table_attract ] , this last option can only occur if @xmath125 . the next section discusses the cases 3 ) and 4 ) in which the system does not return to the original sw period-1 reentry . the three new scenarios of reentry annihilation reported in @xcite are shown in fig . [ fig_bloc_aa ] . these scenarios of termination differ in the number of revolution made by the @xmath95 and @xmath126activation fronts before they are blocked . accordingly , we introduce the notation @xmath127 , meaning that the front @xmath128 is blocked ( @xmath129 ) after @xmath130 turns around the loop . let consider the simplest case ( [ @xmath131 , fig . [ fig_bloc_aa]a ) in which both @xmath95 and @xmath104 are blocked after one rotation . @xmath95 is blocked first near @xmath96 , when it reaches the refractory tail left by @xmath104 and @xmath105 . this occurs because @xmath95 has already completed a fraction of its rotation when @xmath74 is applied . as a consequence , @xmath95 comes back to reactivate the sites near @xmath96 after a time interval much shorter than the period of rotation @xmath132 of the stable reentry . the block of @xmath104 takes place between @xmath97 and the site where @xmath105 was blocked . when @xmath95 travels in this zone , the region has last been excited by @xmath94 , and a time interval longer than @xmath132 has elapsed since this last excitation . besides , both the action potential and @xmath42 associated to @xmath94 were short since @xmath72 was premature . as a consequence , @xmath95 produces action potentials that are longer than those of the stable reentry . since the time between the passage of @xmath95 and the return of @xmath104 is also shorter than @xmath132 , @xmath104 is blocked . the block of @xmath104 thus results from a process of amplified alternation in a region between @xmath96 and @xmath97 . the premature @xmath94 that creates short action potential is followed by the late @xmath95 generating long action potential . in the two other scenarios , @xmath95 nor @xmath104 are blocked after either 2 or 3 turns ( [ @xmath133 , [ @xmath134 in fig . [ fig_bloc_aa]b - c ) . each passage of @xmath95 leaves around @xmath96 a convex profile of @xmath42 , which is turned to a concave profile by the subsequent passage of @xmath104 . the panel b and c illustrates the process of alternans amplification . alternans amplification may end up by the blockade of one of the front , in which case propagation reverts to sw reentry . it may saturated , leading to dw reentry , or may lead to annihilation . the next section explores the conditions for annihilation . [ ptb ] fig_bloc_aa.eps [ [ the - scenarios - of - annihilation - by - alternans - amplification - occurs - over - limited - range - of - l ] ] the scenarios of annihilation by alternans amplification occurs over limited range of @xmath77 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the conditions leading to each type of alternans amplification annihilation vary with @xmath77 , and none exists for @xmath77 @xmath135@xmath136 cm . the four panels of fig . [ fig_map_single ] picture the extent of the different zones of annihilation for @xmath77 ranging from @xmath137 , the minimum @xmath77 with stable period-1 sw reentry , to the limiting @xmath138 cm value . at short @xmath77 ( @xmath139 @xmath140 , fig . [ fig_map_single]a ) , there is a large [ @xmath100 , @xmath73 ] area with [ @xmath131 block located at low @xmath100 values , and a small adjacent area of [ @xmath133 block . the zone of [ @xmath133 always remains minimal ; and it is the first to disappear at @xmath142 cm . the zone of [ @xmath143 appears at intermediate @xmath77 , expands , and is the last to disappear . [ h ] fig_map1.eps [ [ disappearance - of - thea_11dashva_21dashv - annihilation ] ] disappearance of the [ @xmath131 annihilation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + taken together , the maps of fig . [ fig_map_super ] and fig . [ fig_map_single ] a - c shows that the area of [ @xmath144 is embedded in the larger region , invariant with respect to @xmath77 , in which @xmath105 is blocked between @xmath97 and @xmath96 . the zone of [ @xmath131 is located at intermediate @xmath145 , just over the area , labeled [ @xmath146 ( fig . [ fig_map_super ] ) , in which @xmath95can propagated beyond @xmath96 , but @xmath104 is stopped by the refractory tail left by @xmath95 . as @xmath77 is increased , the lower boundary of [ @xmath131 is lifted , thus diminishing the [ @xmath131 area until it disappears completely . the lost of [ @xmath131 annihilation is thus caused by the inability of the system to block @xmath95 at its first return . appendix [ appendix_b ] provides the conditions needed for @xmath95 to block in the tail of @xmath104 and proves that there is a limiting @xmath77 beyond which this can not happen . to summarize : 1 ) increasing @xmath73 produces longer action potential for @xmath74 , delays @xmath102 , and thus augments the likelihood of @xmath95 to be blocked . 2 ) however , the increase of @xmath73 and @xmath102 are bounded by the condition of @xmath105 being blocked between @xmath96 and @xmath97 , and these limiting values are independent of @xmath77 . 3 ) since increasing @xmath77 delays the return of @xmath95 , there is a length @xmath77 from which the return of @xmath95 always occurs after the limiting @xmath102 value . [ [ opening - the - way - to - more - than - one - rotation - for - both - fronts ] ] opening the way to more than one rotation for both fronts + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + figure [ fig_map_single ] shows that , for each value of @xmath77 , the zone of [ @xmath131 and [ @xmath147 blocks are disjoint , being respectively located at low and high @xmath100 . as seen in fig [ fig_map_single]d , the zones of [ @xmath148 remain located at large @xmath100 even at values of @xmath77 for which [ @xmath131 does not exists anymore . however , enlarging @xmath77 also leads to the appearance and extension of areas in which @xmath95 and @xmath104 persist together for an increasing number of turns before one of the front is blocked ( up to 7 activations in [ fig_map_single]d ) . as @xmath77 is increased , the areas of the parameters plane associated to these other forms of transient complex dynamics extend toward low @xmath100 , to finally cover all the range from @xmath117 to @xmath149 ms when @xmath150 cm , a value close to @xmath151 from which dw reentry becomes possible . the areas of the parameters plane associated to these behaviors with coexistence of @xmath95 and @xmath104 for multiple turns form a sequence of contiguous parallel tongues . all the higher modes of block and complex dynamics require that both @xmath95 and @xmath104 travel beyond @xmath114 , at least at their first return . conditions for the block of @xmath104 are derived in appendix [ appendix_c ] based on an approximation using the @xmath120 equation and rules for @xmath95 and @xmath104 return cycles . it shows that the block of @xmath104 depends on the balance between the return cycle of @xmath95 and @xmath104 . in both the i d and il model , annihilation by alternans amplification is impossible from @xmath77 @xmath152 30.5 cm . in fact , for 31 @xmath153 30.5 cm , only transient complex dynamics with a final return to period-1 sw reentry are observed . however , the maximum number of turns with @xmath95 and @xmath104 co - traveling grows , just as the number of tongues in the [ @xmath154 ] plane associated to different numbers of turns during which the two fronts coexist . each new tongue appears at low @xmath100 and @xmath73 values , and expands as @xmath77 is further increased . finally , a first transition to qp mode-0 dw reentry is detected at @xmath155 cm ( fig . [ map_6200]a ) . transition to dw reentry thus appears as the asymptotic limit of the prolongation of the transient propagation with two fronts . however , transition to dw reentry begins much beyond @xmath156 @xmath157 cm , the value at which sustained mode-0 qp dw reentry starts to exist . in fact , at @xmath155 cm , the system is rather in the range of @xmath77 for which both dw mode-0 and mode-1 solutions coexists . [ tbh ] fig_map_6200.eps for longer @xmath77 , ( fig . [ map_6200]b ) the region with transition to mode-0 dw reentry expands , as it was the case for all the zones associated to two fronts transient propagation created at shorter @xmath77 . transition to mode-1 dw reentry also appears . however , the transition to mode-1 appears at low @xmath100 , but at two disjoint intermediate @xmath73 values , giving rise to two separated areas . for still longer @xmath77 , the areas of mode-0 and mode-1 transition enlarge until covering , from @xmath150 cm , all the area where @xmath105 is blocked . hence , from this value of @xmath158transition to dw reentry occurs in all instances where @xmath105 is blocked . finally , beyond @xmath159 cm , all transitions go to period-1 dw reentry , which is the only dw solution that remains . increasing the length of the pathway correlates with an increase in the complexity of the dynamics and to the transition to sustained dw reentry . the results presented in fig . [ map_6200 ] show that changing the timing of the stimuli when @xmath160 can induce the transition either to the mode-0 or mode-1 dw qp reentry . the il model was simulated with @xmath150 cm ( @xmath161 but near @xmath151 ) to circumscribe the basin of attraction in the parameter space associated to each dw qp reentry . the large area in which two antegrade fronts are created ( @xmath162 ms ) is separated between two regions , with lower @xmath73 converging to mode-0 , and higher @xmath73 to mode-1 . in this last section , we compare the transient dynamics leading to each of two modes of dw qp reentry . figure [ fig_di_quasi ] shows the spatial profile of @xmath27 associated to @xmath95 and @xmath104 for two cases converging respectively to mode-0 ( panel a ) and mode-1(panel b ) . in this representation,@xmath163the passage of @xmath104 ( @xmath164 , dotted line ) at each location is followed by that of @xmath95 ( @xmath165full line ) . at first , @xmath104 propagates with a short @xmath27 , producing brief action potentials everywhere along the loop . as a consequence , @xmath95 travelling afterwards meets long @xmath166 . @xmath167 has its first maximum ( p@xmath168 ) when it travels between @xmath97 and the point where @xmath105was blocked , such that @xmath169 has a minimum at the same location from which it increases slowly until its next return in the same region . from there , @xmath164 starts to alternate between long and short values with a spatial period close to 2 turns , and @xmath167 follows a complementary profile . this first phase of the propagation , lasting for approximately 8 turns , can be labeled as concordant alternans since @xmath169 and @xmath170 have alternating values and that each remains short or long for at least one complete rotation . this pattern does not correspond neither to mode-0 nor mode-1 , since both stabilized dw qp solutions have a wavelength less than @xmath77 . however , from the beginning ( @xmath171 ) , @xmath170 already shows a second spatial oscillation in @xmath27 that is superimposed to the concordant alternans . the structure of this oscillation , which has a wavelength close to one turn , makes the difference between the cases converging to mode-0 and mode-1 . in the former case , the oscillation embeds two peaks \{p@xmath168 , p@xmath172 } ( @xmath170 for mode-0 , continuous line in fig . [ fig_di_quasi]a ) , while in the later case , it has three peaks \{p@xmath168 , p@xmath172 , p@xmath173 } ( @xmath170 for mode-1 , continuous line in fig . [ fig_di_quasi]b ) . these superimposed spatial variations persist while the concordant alternans dissipate . during this process , the position of the peaks does not change much . as the amplitude of the concordant alternans decreases , the respective height of the discordant alternans increases up to a point where the boundary with large gradient in @xmath27 begins to move around the loop due to the quasiperiodic nature of the propagation . this travelling mechanism is akin to the propagation of paced discordant alternans on a cable of cardiac tissue@xcite . [ tbh ] fig_di_quasi.eps the main difference between the transition to mode-0 and mode-1 is the presence of the third peak p@xmath173 in @xmath170 . the p@xmath173 peak of @xmath170 is induced by the increase of @xmath73 . on one hand , a larger @xmath73 produces a longer @xmath42 such that @xmath95 meets a lower @xmath27 and the amplitude of p@xmath168 is reduced . but a larger @xmath73 also implies that @xmath104 travels faster , comes back sooner to the stimulation sites , and set the stage for a new maximum in the @xmath170 profile . of course the transition from two to three peaks is continuous process since a similar variation of @xmath27 with less amplitude already exists with the transition to mode-0 in fig . [ fig_di_quasi]a . it means that there must be a minimal spatial profile that corresponds to the boundary between the two basins of attraction ( the transition to either mode-0 or mode-1 ) .

it follows that reentry can be stopped over a limited interval of the pathway length and that increasing the length beyond the upper limit of this interval yields to a transition to sustained double - wave reentry . a similar dynamical mechanism , labeled alternans amplification , is found to be responsible for both behaviors .

electrical pacing is a common procedure that is used in both experimental and clinical settings for studying and/or annihilating anatomical reentry . in a recent study [ comtois and vinet , chaos * 12 * , 903 ( 2002 ) ] , new ways to terminate the one - dimensional reentry using a simple protocol consisting of only two stimulations were discovered . the probability of annihilating the reentrant activity is much more probable by these new scenarios than by the usual local unidirectional block . this paper is an extension of the previous study in which the sensitivity of the new scenarios of annihilation to the pathway length is studied . it follows that reentry can be stopped over a limited interval of the pathway length and that increasing the length beyond the upper limit of this interval yields to a transition to sustained double - wave reentry . a similar dynamical mechanism , labeled alternans amplification , is found to be responsible for both behaviors .

1312.4891

c

the well - known static mp systems in gr include gravitational and electromagnetic fields and electrically charged dust with equal densities @xmath323 . we have considered a generalization of mp systems , including a dilatonic scalar field ( the dilatonic mp , or dmp systems ) with an arbitrary coupling function @xmath1 in the lagrangian ( [ l ] ) , where the scalar field @xmath2 may be normal or phantom . as in mp systems , the metric has a conformally flat spatial part . a dmp system is characterized by the metric function @xmath70 , the electric potential @xmath324 , the dilaton field @xmath325 , and three densities , those of mass , @xmath326 , electric charge , @xmath327 , and scalar charge , @xmath328 . let us enumerate the main results obtained . 1 . it has been shown that static configurations are possible with arbitrary functions @xmath329 ( @xmath4 ) and @xmath85 , for any regular coupling function @xmath1 , without any assumption of spatial symmetry . 2 . for general static systems , the field equations imply that the functions @xmath330 , @xmath325 and @xmath69 are related , so that if , say , @xmath331 , then @xmath5 and @xmath84 . it is thus unnecessary to postulate the existence of such functional relations , as is often done . 3 . there are purely scalar analogs of mp systems , but only with phantom scalar fields . however , the corresponding point sources are different : an extreme bh for mp , a singularity for scalar mp . it has been shown that sufficiently general configurations with nontrivial scalar fields can not be obtained without a nonzero scalar charge density @xmath25 ; this , however , does not forbid the existence of special solutions with @xmath99 for particular @xmath1 . 5 . there is a universal balance condition , ( [ charges ] ) , between the schwarzschild mass and the electric and scalar charges , valid for any dmp systems , including those with horizons and/or singularities . it generalizes the results previously obtained for special cases ( see , e.g. , @xcite ) . 6 . in the case of spherical symmetry , the existence conditions have been formulated for bh and quasi - bh ( qbh ) configurations with smooth matter , electric charge and scalar charge density distributions . it turns out that horizons in dmp systems are second - order ( extremal ) , in agreement with the general properties of qbhs @xcite . 7 . for qbhs containing a small parameter @xmath9 whose nonzero value distinguishes them from bhs , different limiting transitions @xmath208 are analyzed . they lead to universal solutions independent of the particular choice of a qbh configuration . the limiting metrics coincide with those obtained previously for mp systems . examples of bh and qbh solutions have been obtained . among them are phantom - free ones , that is , the mass density and the energy densities of both scalar and electromagnetic fields are nonnegative . 9 . for cylindrically symmetric configurations , the conditions at a regular center , a possible horizon and at flat infinity have been formulated . it has been shown that neither bhs nor stringlike dmp configurations with a regular axis ( in particular , stringlike qbhs ) can be phantom - free . some of these results have been briefly presented in @xcite , viz . , items 1 , 3 , 5 , and 8 ( partly ) . in addition , in @xcite , polycentric configurations , possible in the dmp framework , were discussed , with any number of mass concentrations . for instance , one can consider the metric ( [ ds ] ) in cartesian coordinates @xmath332 ( so that @xmath333 ) and choose [ f_n ] ^-(x^i)f(x^i ) = _ a=1^n f_a ( x_a ) , where @xmath334 are functions of @xmath335 , @xmath336 being the ( fixed ) coordinates of the @xmath314-th center . as @xmath334 , one can take any functions providing solutions , e.g. , bhs or qbhs . a complete solution is obtained after choosing the function @xmath337 , or equivalently @xmath338 , which should be regular at all relevant values of @xmath213 and decay sufficiently rapidly at spatial infinity , as @xmath339 . in @xcite , an example is given of such a system with two mass concentrations , where each `` center '' can be a bh or a qbh . we would like to stress that it is in general rather difficult to find sources that admit qbh configurations since matter begins to collapse long before approaching a would - be horizon . it is the freedom in choosing the metric function @xmath107 that enables us to keep dmp configurations static even extremely closely to emergence of a horizon . bronnikov , j.c . fabris , r. silveira , and o.b . zaslavskii , phys . d * 89 * , 107501 ( 2014 ) ; arxiv : 1405.6116 . bronnikov and s.g . rubin , _ black holes , cosmology , and extra dimensions _ ( world scientific , 2012 ) . r. geroch , commun . * 13 * , 180 ( 1969 ) .

electrically charged dust is considered in the framework of einstein - maxwell - dilaton gravity with a lagrangian containing the interaction term , where is an arbitrary function of the dilaton scalar field , which can be normal or phantom . without assumption of spatial symmetry , we show that static configurations exist for arbitrary functions ( ) and . if , the classical majumdar - papapetrou ( mp ) system is restored . it turns out that cylindrical bhs and qbhs can not exist without negative energy density somewhere in space . however , in general , bhs and qbhs can be phantom - free , that is , can exist with everywhere nonnegative energy densities of matter , scalar and electromagnetic fields .

electrically charged dust is considered in the framework of einstein - maxwell - dilaton gravity with a lagrangian containing the interaction term , where is an arbitrary function of the dilaton scalar field , which can be normal or phantom . without assumption of spatial symmetry , we show that static configurations exist for arbitrary functions ( ) and . if , the classical majumdar - papapetrou ( mp ) system is restored . we discuss solutions that represent black holes ( bhs ) and quasi - black holes ( qbhs ) , deduce some general results and confirm them by examples . in particular , we analyze configurations with spherical and cylindrical symmetries . it turns out that cylindrical bhs and qbhs can not exist without negative energy density somewhere in space . however , in general , bhs and qbhs can be phantom - free , that is , can exist with everywhere nonnegative energy densities of matter , scalar and electromagnetic fields .