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[ "# Relationship among velocity speed and direction\n\n### Speed and Velocity", null, "This is one of the essential differences between speed and velocity. Speed is a scalar quantity and does not keep track of direction; velocity is a vector quantity. Explain the relationships between instantaneous velocity, average velocity, instantaneous speed, . One major difference is that speed has no direction. Get an answer for 'What's the difference between speed and velocity?' and find Velocity is the rate at which an object changes position in a certain direction.\n\nDisplacement, however, not only refers to the distance between two places but their relative locations as well. Say you walk 5m to the east, for example. At this point, your displacement and distance covered would both be 5m.\n\n## Speed and Velocity\n\nIf you were to turn around and walk 5m back west, you would have walked a distance of 10m. Your displacement, however, would be 0m because your position has not changed. If you were only to walk back 3m, your displacement would be 2m. See Image 1 Speed vs.\n\n### The Relationship Between Velocity and Acceleration. by Katie Crocker on Prezi\n\nSpeed is a measure of how quickly something is moving, but it does not consider the direction in which the movement is happening. Velocity, on the other hand, takes into consideration displacement and time taken for that displacement. The equation for velocity is very similar to the equation for speed: Even though the equations are very similar, they can produce very different results.\n\nLet's use the example from before. If it takes you 3 seconds s to walk 5m to the east, your speed and velocity are the same number: Your velocity would be stated as 1. Displacement is not just measured back and forth, however.\n\nIt can be measured in two or three dimensions. If a person in motion wishes to maximize their velocity, then that person must make every effort to maximize the amount that they are displaced from their original position.\n\nEvery step must go into moving that person further from where he or she started.", null, "For certain, the person should never change directions and begin to return to the starting position. Velocity is a vector quantity. As such, velocity is direction aware. When evaluating the velocity of an object, one must keep track of direction. One must include direction information in order to fully describe the velocity of the object. This is one of the essential differences between speed and velocity.\n\n## Speed & Velocity\n\nSpeed is a scalar quantity and does not keep track of direction; velocity is a vector quantity and is direction aware. Determining the Direction of the Velocity Vector The task of describing the direction of the velocity vector is easy.\n\n• Direction and speed: velocity\n\nThe direction of the velocity vector is simply the same as the direction that an object is moving. It would not matter whether the object is speeding up or slowing down. If an object is moving rightwards, then its velocity is described as being rightwards.\n\nIf an object is moving downwards, then its velocity is described as being downwards. Note that speed has no direction it is a scalar and the velocity at any instant is simply the speed value with a direction.\n\n### Displacement, velocity, acceleration\n\nCalculating Average Speed and Average Velocity As an object moves, it often undergoes changes in speed. For example, during an average trip to school, there are many changes in speed. Rather than the speed-o-meter maintaining a steady reading, the needle constantly moves up and down to reflect the stopping and starting and the accelerating and decelerating. The average speed during an entire motion can be thought of as the average of all speedometer readings.\n\nIf the speedometer readings could be collected at 1-second intervals or 0. Now that would be a lot of work. And fortunately, there is a shortcut. The average speed during the course of a motion is often computed using the following formula: In contrast, the average velocity is often computed using this formula Let's begin implementing our understanding of these formulas with the following problem: While on vacation, Lisa Carr traveled a total distance of miles.\n\nHer trip took 8 hours. What was her average speed? To compute her average speed, we simply divide the distance of travel by the time of travel. Lisa Carr averaged a speed of 55 miles per hour. Yet, she averaged a speed of 55 miles per hour. The above formula represents a shortcut method of determining the average speed of an object.", null, "Average Speed versus Instantaneous Speed Since a moving object often changes its speed during its motion, it is common to distinguish between the average speed and the instantaneous speed." ]

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[ "## Problem 4.08 – Sales and Growth\n\n0\n(0)\nWhat is the maximum increase in sales measured in dollars that can be sustained if no new equity is issued?\n\n### Experts Have Solved This Problem\n\n• Search Terms: income $(%) (%) net (do (lo] , , current assets fixed , ,$ - ..) [lo] a and answer are assets assets total $, ,$, long-term assuming balance be calculations can co. company constant costs current debt debt equity total $, ,$, assets debt-equity decimal dividend dollar e.g., equity financial fixed for growth here: here:page in income income $, ,$ income taxes increase intermediate is issued? long-term maintains maximum most net new no not payout percent places, problem proportional ratio ratio. recent round sales sales. sheet sheet sales costs taxable shown statement statement balance statements sustained taxable taxes that the to total tran what your" ]

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[ "### Angle of deviation of prism experiment readings\n\nAccurate measurement of the refractive index of materials can be obtained using a prism made out of the material of interest. Rays passing though the prism deviate because of refraction.\n\nSimple ray geometrical optics and the use of Snell's law show that the index is directly related to the minimum angular deviation of a ray that passes through the prism at different angles of incidence.", null, "You can measure this minimum deviation directly for any ray that comprises a single wavelength of light a monochromatic beam. A cadmium lamp provides a few discrete wavelengths and for each wavelength the minimum deviation angle will be different.\n\nThis is because the prism refractive index varies with wavelength the property known as dispersion and so you will also be able to measure this property as well. There is an ISE where you can practice this experiment before your session. It can be found here. The script for this experiment can be found in the lab script book or on DUO.\n\nDell xps 7590 overheating\n\nDepartment of Physics. The Prism Spectrometer Introduction Fig. Script The script for this experiment can be found in the lab script book or on DUO. Photo Gallery - Prism Spectrometer. Equpiment provided for the session.Aim To trace the path of the rays of light through a glass prism.\n\nMaterials Required A glass prism, some drawing pins, white paper, a drawing board, adhesive tape, a protractor, a sharp pencil and a measuring scale.\n\nIt has two triangular bases and three rectangular lateral surfaces. Note: When a ray of light passes through a prism, it bends towards the thicker part of the prism. Sources of Error 1. Pins may not be exactly perpendicular to the paper. The feet of the pins may not be in same straight line. In observing images of P 1 and P 2eye may be very close to the pins.\n\nPrism may not be fixed properly. Question 1.", null, "Define angle of deviation. Question 2. List the factors on which the angle of deviation through a prism depend. Question 3. Why does a ray of light bend towards the base when it passes through a glass prism? Question 4. Why does white light split into different colours when passes through a glass prism? Question 5. Why does white light not split into different colours when it passes through a glass slab?\n\nHence, the refracted light rays suffer equal amount of deviation. Question 6. How can you define an angle of prism? Answer: The angle between two lateral faces of prism is called an angle of prism. Question 7. What precaution must be taken for the refracting faces of glass prism while tracing the path of ray of light through it?\n\nQuestion 8.\n\nDirilis ertugrul season 1 episode 11 in urdu\n\nName the process by which when a white light passes through a prism splits into its constituent seven colours.\n\nAnswer: Dispersion of light. Question 9. Give the range of angle of incidence to complete this experiment accurately and successfully. Question Your email address will not be published. Comments Nyc.Login Now. A prism is a wedge-shaped body made from a refracting medium bounded by two plane faces inclined to each other at some angle. The two plane faces are called are the refracting faces and the angle included between these two faces is called the angle of prism or the refracting angle.\n\nIn the absence of the prism, the incident ray KL would have proceeded straight, but due to refraction through the prism, it changes its path along the direction PMN. The angle of deviation of a ray of light in passing through a prism not only depends upon its material but also upon the angle of incidence.\n\nThe above figure shows the nature of variation of the angle of deviation with the angle of incidence. The minimum value of the angle of deviation when a ray of light passes through a prism is called the angle of minimum deviation. The figure below shows the prism ABC, placed in the minimum deviation position. If a plane mirror M is placed normally in the path of the emergent ray MN the ray will retrace its original path in the opposite direction NMLK so as to suffer the same minimum deviation dm.\n\nHence, the ray which suffers minimum deviation possess symmetrically through the prism and is parallel to the base BC. I think not necessarily. If you are looking for answer to specific questions, you can search them here. We'll find the best answer for you. If you are looking for good study material, you can checkout our subjects. Hundreds of important topics are covered in them. Download our mobile app and study on-the-go.\n\nYou'll get subjects, question papers, their solution, syllabus - All in one app.\n\nMobil lubricants price list\n\nLogin You must be logged in to read the answer. Go ahead and login, it'll take only a minute. Derive the formula for angle of minimum deviation for the prism.\n\nFollow via messages Follow via email Do not follow. Please log in to add an answer. Next up Read More Questions If you are looking for answer to specific questions, you can search them here. Study Full Subject If you are looking for good study material, you can checkout our subjects.\n\nKnow More. Engineering in your pocket Download our mobile app and study on-the-go.It seems a mysterious and even a magical force. Magnetism's ability to serve mankind especially lies in its relationship to electricity. That means, magnetism and electricity are so closely related to each other.\n\n## CBSE Class 10 Science Practical Skills – Refraction Through Prism\n\nThe concept of this can be applied in many technologies for an effective productivity. The study of this lab revolves around the generation, propagation and reception of mechanical waves and vibrations.\n\nRefraction Through Prism & Finding Angle of Deviation Experiment\n\nThese concepts embody the study of tiny subatomic particles or lightening fast speeds. They find applications in technologies such as atomic energy or semiconductors. It explains the study of optical properties for different material by adopting laser devices and handling basic aspects of interferometry. It also gives the dynamics of special type of non-linear systems. Torque and angular acceleration of a fly wheel Torsional oscillations in different liquids Moment of Inertia of Flywheel Newton's Second Law of Motion Ballistic Pendulum Collision balls Projectile Motion Elastic and Inelastic Collision Electric Circuits Virtual Lab Pilot An electric circuit is composed of individual electrical components such as resistors, inductors, capacitors etc to trace the current that flows through it.\n\nThe combination of electrical components can perform various simple and compound electrical operations. Optics usually describes the behavior of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.\n\nResolving power of a prism Angle of the prism using Spectrometer Spectrometer i-i' curve Spectrometer: i-d curve Spectrometer- Determination of Cauchy's constants Spectrometer, Refractive Index of the material of a prism Spectrometer,Dispersive power of a prism Diffraction Grating Solid State Physics Virtual Lab Solid-state physics is a study of rigid matter or solids.\n\nThis part Includes theoretical description of crystal and electronic structure, lattice dynamics, and optical properties of different materials. Nodal Center student's feed back. Harmonic Motion and Waves Virtual Lab Harmonic Motion and Wave lab is the interdisciplinary science that deals with the study of sound, ultrasound and infrasound all mechanical waves in gases, liquids, and solids. Modern Physics Virtual Lab Modern physics refers to the post-Newtonian conception of physics developed in the first half of the 's.\n\nLaser Optics Virtual Lab This lab is thoroughly outfitted for experiments in introductory and advanced laser physics. Mechanics Virtual Lab Pilot It concerns with the dynamics of mechanical systems mainly rotational dynamics. Electric Circuits Virtual Lab Pilot An electric circuit is composed of individual electrical components such as resistors, inductors, capacitors etc to trace the current that flows through it.\n\nAdvanced Mechanics Virtual Lab The laboratory is concerned with the issues of advanced dynamics in mechanical systems dealing with describing motions, as well as the causes of motion.\n\nOptics Virtual Lab Optics is the study of the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it.Refraction through a Prism for Small Angle of incidence. A prism is a wedge-shaped body made from a refracting medium bounded by two plane faces inclined to each other at some angle.\n\nThe two plane faces are called are the refracting faces and the angle included between these two faces is called the angle of prism or the refracting angle. In the absence of the prism, the incident ray KL would have proceeded straight, but due to refraction through the prism, it changes its path along the direction PMN.\n\nAgain, in quadrilateral ALOM. The angle of deviation of a ray of light in passing through a prism not only depends upon its material but also upon the angle of incidence. The above figure 2 shows the nature of variation of the angle of deviation with the angle of incidence. The minimum value of the angle of deviation when a ray of light passes through a prism is called the angle of minimum deviation. The figure 3 shows the prism ABC, placed in the minimum deviation position. If a plane mirror M is placed normally in the path of the emergent ray MN the ray will retrace its original path in the opposite direction NMLK so as to suffer the same minimum deviation d m.\n\nHence, the ray which suffers minimum deviation possess symmetrically through the prism and is parallel to the base BC. Equation 7 gives the relation between the refractive index of the material of the prism and the minimum deviation. This is known as grazing incidence. This happens when the light ray strikes the second face of the prism at the critical angle for glass - air. This is known as grazing emergence. When a ray of light suffers minimum deviation through a prism.\n\nAngle of incidence is equal to the angle of emergence. Ray of light passing through the prism is parallel to the base of the prism. Angle of refraction inside the material of prism is equal to half the angle of prism.", null, "Different parts are coated with pure pigment colours in the order: violet, indigo, blue, green, yellow, orange and red. On rotating the disc at a high speed, all the colours merge into one another due to persistence of vision, giving a resultant white impression. This splitting up of light into its constituent colours is called dispersion.\n\nRefractive index of a transparent medium depends upon the nature of light i. Medium has greater refractive index for light and smaller wave-length. Since violet light has smaller wave-length than that for red light, i.\n\nIn case of a prism. Watch this Video for more reference. As a ray of light is incident on one of the refracting faces of a prism and proceeds through the prism, it undergoes following two changes:. This phenomenon is called dispersion. So, different wave-lengths contained in the incident ray suffer different deviations.\n\nAfter passing through the prism, it splits up into its constituent colours with violet and red as its extreme colours.Refraction through a Prism for Small Angle of incidence. A prism is a wedge-shaped body made from a refracting medium bounded by two plane faces inclined to each other at some angle.\n\nThe two plane faces are called are the refracting faces and the angle included between these two faces is called the angle of prism or the refracting angle. In the absence of the prism, the incident ray KL would have proceeded straight, but due to refraction through the prism, it changes its path along the direction PMN. Again, in quadrilateral ALOM. The angle of deviation of a ray of light in passing through a prism not only depends upon its material but also upon the angle of incidence.\n\nThe above figure 2 shows the nature of variation of the angle of deviation with the angle of incidence.\n\nThe minimum value of the angle of deviation when a ray of light passes through a prism is called the angle of minimum deviation.\n\nCar valuation oman\n\nThe figure 3 shows the prism ABC, placed in the minimum deviation position. If a plane mirror M is placed normally in the path of the emergent ray MN the ray will retrace its original path in the opposite direction NMLK so as to suffer the same minimum deviation d m.\n\nHence, the ray which suffers minimum deviation possess symmetrically through the prism and is parallel to the base BC. Equation 7 gives the relation between the refractive index of the material of the prism and the minimum deviation. This is known as grazing incidence. This happens when the light ray strikes the second face of the prism at the critical angle for glass - air.\n\nThis is known as grazing emergence.", null, "When a ray of light suffers minimum deviation through a prism. Angle of incidence is equal to the angle of emergence. Ray of light passing through the prism is parallel to the base of the prism. Angle of refraction inside the material of prism is equal to half the angle of prism. Different parts are coated with pure pigment colours in the order: violet, indigo, blue, green, yellow, orange and red. On rotating the disc at a high speed, all the colours merge into one another due to persistence of vision, giving a resultant white impression.\n\nThis splitting up of light into its constituent colours is called dispersion. Refractive index of a transparent medium depends upon the nature of light i.It has two triangular bases and three rectangular lateral surfaces which are inclined to each other as shown in the given figure. It has six vertices and nine edges.\n\nSince base of this prism is in triangular shape, it is called a triangular prism. AIM To trace the path of the rays of light through a glass prism. Question 1. What is a triangular prism? Answer: It is a piece of homogeneous, transparent refracting material enclosed by three rectangular refracting faces and two triangular bases. Question 2.\n\n### The Prism Spectrometer\n\nWhat is meant by the angle of prism? Answer: It is the angle of inclination between the two rectangular refracting faces of the prism. Question 3. What do you mean by the refracting edge of the prism? Answer: It is the edge where two rectangular refracting faces of the prism meet. Question 4. What is the angle of refraction? Question 5. What is the angle of emergence? Answer: The angle between the emergent ray and normal at the second refracting face of the prism is called the angle of emergence.\n\nQuestion 6. Question 7. How many faces are there in a prism? Answer: Three rectangular faces and two triangular bases. Question 8." ]

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[ "# Evidence of orbitals?\n\nHow do we know that there are different types of orbitals? For example, what evidence is there for the existence of $$\\mathrm{p}$$ orbitals instead of there being multiple $$\\mathrm{s}$$ orbitals (for example, why isn't the electronic configuration of sodium $$\\mathrm{1s^1, 2s^2, 2s^2, 2s^2, 2s^2, 3s^2}$$ instead of $$\\mathrm{1s^2 2s^2 2p^6 3p^1}$$)?\n\nLet me approach this another way than the others: orbitals are NOT physical objects! They do not exist in physical sense, they are theoretical constructs, chemical concepts that help understand / visualize / etc. mathematical solutions of Schrodinger / Dirac / Kohn–Sham / etc. equations.\n\nOrbitals are not unique: given linear combinations are equivalent with each other, and there is no \"correct orbitals\", one can choose whichever they like. Canonical orbitals, natural orbitals etc are all good to go.\n\nWhat is the evidence they exist? They do not exist, they are just mathematical solutions for given equations, and it is a purely mathematical question if they are good solutions for those equations or not. The theories themselves are consistent with experimental data, e.g. spectroscopic properties, geometries, reactivity.\n\n• The answer that I personally was looking for. – LordStryker Sep 12 '14 at 16:42\n• @Greg wote, \"They do not exist\" Electrons exist and they occupy regions with probabilities defined by equations. These regions may have different spatial functions. \"Orbital\" is a word that can be used to describe that spatial function\\distribution of electrons. In that sense of the word, orbitals exist, just as the spatial distribution of the electron exists. Perhaps this is more a semantics issue. – ron Sep 13 '14 at 1:39\n• It is not semantic issue. Which one exist? Canonical ones? NBOs? Or some of their rotations? – Greg Sep 13 '14 at 2:52\n• \"given linear combinations are equivalent with each other\" if we are talking about energy – user1420303 Sep 24 '16 at 12:46\n• 1. Not all linear combinations are equivalent, only unitary transformations. 2. This also only holds true if we look at the overall energy of the system. If we were to assume that orbital energies themselves are physically meaningful, then this does not hold anymore. 3. The question whether orbital energies are in any way observable harks back to the old question about how to interpret the photoelectron spectrum of methane and its distinct 3+1 intensity pattern. To this day I still haven't found a satisfying answer. – Antimon Nov 11 at 5:02\n\nThe answer lies with experimental chemistry, specifically successive ionisation energies (i.e how much energy is required to remove the first electron, the second electron, the third electron and so on).", null, "Each point on the graph corresponds to an element. The first one is hydrogen, the second is helium. The height of each point shows how much energy is required to remove the first electron.\n\nYou can note that generally, the energy increases over a period. This is because In each successive element there is one more proton, and this stronger nuclear charge 'hold the outer electrons' more tightly.\n\nNow to explain your question. Observe that within a period (e.g the 3rd dot to the 10th), it is not a constant increase. You can see that between the 4th-5th there is a slight drop, likewise between the 7th-8th.\n\nThe explanation for this is the sub orbitals. Some knowledge you need to know is that electron sub shells are only stable when empty, full, or half full (if you need an explanation for this comment on it later).\n\nLet us examine the fourth dot, which represents Beryllium. It has an electronic config of 1s2 2s2. All of its sub orbitals are full, meaning that it is quite stable. Compare it to the 5th dot, boron. Boron has a config of 1s2 2s2 3p1. Now the P orbital has room for 6 electrons, but this only has 1! It is not happy. It is not full or half full. Because of this it is trying to 'get rid' of the electron to be more stable. That's why it doesn't require so much energy to remove the outer electron.\n\nThe decrease between the 6th and 7th is explained by the fact that the p orbital is stable when empty, full or half full. The 7th dot (nitrogen) has 3 electrons in its p orbital (half full). Contrasted with oxygen, which has 4/6. This is not stable, so less energy is required to remove it.\n\nTL-DR: By analysing the ionisation energy graphs, we can see patterns that can be explained by sub orbitals.\n\nIf you need a more basic/complex explanation, comment.\n\n• I'd be a bit careful about the whole sub-shells being stable with empty, half-filled, and filled description. Slater's rules suggest these are general tendencies, but not the only story when you get into the transition metals and lanthanides/actinides. But that's a minor quibble. – Geoff Hutchison Sep 12 '14 at 13:47\n\nOh, I suspect someone could come up with a \"s-orbital only\" view of chemistry, much like people came up with complicated models for an Earth-centered view of the universe.\n\nLet's start with the simple fact that solving quantum mechanics for a hydrogen atom gives you solutions for s, p, d, f, g.. orbitals, and even the degeneracy (i.e., that there is one s-type orbital, three p-type, five d-type, etc.) based on the angular momentum.\n\nAs described above, solutions for the many-electron equations (albeit approximate) match up very nicely with experimental observations for ionization energies, electron affinities, etc.\n\nSo an elegant theory and experiment agree to a remarkable degree.\n\nBeyond that, we know that there must be non-spherical orbitals because we see molecular shapes with bond angles. I cannot come up with any way to describe a tetrahedral methane (let alone anything else) without some sort of non-spherical orbital.\n\nMoreover, when we look at reactivities of molecules, we see reactions occurring where we predict lone pairs or radical spin-density, etc.\n\nWhile orbitals are truly a mathematical construct, we find they are incredibly predictive of a wide range of chemistry. So my response would be \"how can we think there are not different kinds of orbitals?\"\n\n• Given the simplicity of s orbitals, people have tried using 's-orbital only' basis sets and just placing a crap ton of these orbitals all over a molecular space rather than using a smaller number of more 'accurate' atomic orbitals. Interesting concept. – LordStryker Sep 12 '14 at 14:01\n\nHow do we know that there are different types of orbitals?\n\nIn the early days of atomic spectroscopy scientists were able to explain the spectral properties of elements like hydrogen and sodium. Atoms of these elements had a single electron in the outermost shell and produced spectra that fit with (what we now view as) the relatively simplistic Bohr theory of the atom. When more complex atoms were examined spectroscopically, fine structure was observed that the Bohr theory could not explain. These spectral lines were often referred to as sharp, principal, diffuse and fundamental.\n\nAs quantum mechanics emerged, the concept of the 4 quantum numbers being needed to describe an electron surfaced. It was found that the spectral fine structure could now be explained when these various quantum states were taken into account. To honor the work done by the early spectroscopists (that led to the development of a better theory), scientists used their \"s, p, d, f\" notation to describe the various values for the angular momentum quantum number $\\ell$.\n\nSo although the observations made by the early spectroscopists couldn't be explained at the time, it turns out that they were actually observing spectral transitions involving other orbitals (p, d, f) in addition to those involving the s orbital. The observation of this spectral fine structure requires the presence of orbitals that are different from s orbitals.\n\nWhile I really like the other answers, the 21th century evidence here is that we have images of these orbits thanks to atomic force microscopy (AFM).", null, "As getafix pointed out, these are actually images of spatial electron density distributions by the examined molecules. The actual evidence here, that the model which uses the orbital term predicts the same distributions as we measured with AFM.\n\n• What you evidence of orbitals, can just as easily be called evidence of regions of electron density. ;) So is orbital a real thing or just a description of the spatial distribution of electrons. Electrons are real, regions with a high probability of finding them are real, but are orbitals? food for thought.. – getafix Sep 25 '16 at 1:30\n• @getafix Nice thoughts. :-) This depends on the definition of orbitals. So yes these images are evidence that there is a spatial electron density distribution and that distribution is very similar to the prediction of the limited mathematical model which uses this orbital term. Usually this is enough to prove a hypothesis. E.g. by general relativity such a prove was that by a solar eclipse they were able to check how the gravity of the Sun bends the space and so starlight... – inf3rno Sep 25 '16 at 2:59\n• @getafix But yes you are right, I could rephrase the text. So these are not images of orbitals, but images of electron density distributions which are the same as the model predicts. – inf3rno Sep 25 '16 at 3:01" ]

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[ "Cody\n\n# Problem 2784. Y=X\n\nSolution 1975720\n\nSubmitted on 14 Oct 2019 by Nguyen manh Duy\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1   Pass\nx = 1; y_correct = 1; assert(isequal(setEqual(x),y_correct))\n\n2   Pass\nx = 5; y_correct = 5; assert(isequal(setEqual(x),y_correct))" ]

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[ "# Range of Int in C\n\nIn this article, we are going to discuss the range of int in C with programs.\n\n## Range of Int\n\nData types, and their size and range, play an important role in C programming. The sizeof() operator gives the number of bytes required to store a value in memory of a specific form. However, to prevent overflow and underflow errors in programming, we must be aware of a type's range. C specifies the exact minimum storage size for each integer form. For example, short requires at least two bytes, and long requires at least four bytes. The compiler determines the size and range of a data type. As a result, we should not hardcode the size and range values in the program.\n\n### Find a range of data types manually without a C library\n\nThe given formula defines the minimum and maximum range of a signed type:\n\n• -(2N-1) to 2N-1 - 1 (Where N is sizeof(type) * 8 (it is the total number of bits used by integer type)).\n\nThe given formula defines the minimum and maximum range of an unsigned type:\n\n• 0 to (2N-1) + (2N-1 - 1)\n\nExamples:\n\nLet's take an example to find the range of integers in C programming.\n\nOutput: After executing this code, we will get the output as shown below:\n\n```Range of int = -2147483648 to 2147483647\nRange of unsigned int = 0 to 4294967295\nRange of char = -128 to 127\nRange of unsigned char = 0 to 255\nRange of long = -9223372036854775808 to 9223372036854775807\nRange of unsigned long = 0 to 18446744073709551615\nRange of short = -32768 to 32767\nRange of unsigned short = 0 to 65535\nRange of long long = -9223372036854775808 to 9223372036854775807\nRange of unsigned long long = 0 to 18446744073709551615\n```\n\n### Find the range of data types using a C library\n\nThe method described above for obtaining any form of the range is interesting, but it is not recommended for use. Using the power of a pre-defined C library is often recommended.\n\nLimits.h and float.h are two header files in C programming that define minimum and maximum constants. limits.h specifies constants for integer type and character types. Small and maximum size ranges, total bits, and many others. The float.h defines the floating-point numbers.\n\nExample:\n\nNow, let's take an example to understand how we can find the range of integers by using the C library.\n\nOutput: After executing this above code, we will get the output as shown below:\n\nThe output is:\n\n```Range of signed int -2147483648 to 2147483647\nRange of unsigned int 0 to 4294967295\nRange of signed char -128 to 127\nRange of unsigned char 0 to 255\nRange of signed long int -9223372036854775808 to 9223372036854775807\nRange of unsigned long int 0 to 18446744073709551615\nRange of signed short int -32768 to 32767\nRange of unsigned short int 0 to 65535\nRange of float 1.175494e-38 to 3.402823e+38\nRange of double 2.225074e-308 to 1.797693e+308\nRange of long double 3.362103e-4932 to 1.189731e+4932\n```\n\nExample:\n\nWrite a program that reads an integer type and checks it against the given range to see where it belongs.\n\nOutput: After executing this above code, we will get the output as shown below:\n\nThe Output is:\n\n```Input an integer: 28\nRange [26,50]\n```\n\nExample:\n\nThe sizeof() operator is used to determine the size of an integer form or some other type. The given below program shows how to use the sizeof() operator to determine the sizes of various integer types in the system.\n\nOutput: After executing this above code, we will get the output as shown below:\n\nThe output is:\n\n```sizeof(short) = 2 bytes\nsizeof(int) = 4 bytes\nsizeof(unsigned int) = 4 bytes\nsizeof(long) = 8 bytes\n```\n\n### Feedback", null, "", null, "", null, "" ]

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[ "# Banach algebra\n\n\nBanach algebra\n\nIn mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space norm are required to be related by the following inequality:", null, "$\\forall x, y \\in A : \\|x \\, y\\| \\ \\leq \\|x \\| \\, \\| y\\|$\n\n(i.e., the norm of the product is less than or equal to the product of the norms). This ensures that the multiplication operation is continuous. This property is found in the real and complex numbers; for instance, |-6×5| ≤ |-6|×|5|.\n\nIf in the above we relax Banach space to normed space the analogous structure is called a normed algebra.\n\nA Banach algebra is called \"unital\" if it has an identity element for the multiplication whose norm is 1, and \"commutative\" if its multiplication is commutative. Any Banach algebra A (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra Ae so as to form a closed ideal of Ae. Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering Ae and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity.\n\nThe theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.\n\nBanach algebras can also be defined over fields of p-adic numbers. This is part of p-adic analysis.\n\n## Examples\n\nThe prototypical example of a Banach algebra is C0(X), the space of (complex-valued) continuous functions on a locally compact (Hausdorff) space that vanish at infinity. C0(X) is unital if and only if X is compact. The complex conjugation being an involution, C0(X) is in fact a C*-algebra. More generally, every C*-algebra is a Banach algebra.\n\n• The set of real (or complex) numbers is a Banach algebra with norm given by the absolute value.\n• The set of all real or complex n-by-n matrices becomes a unital Banach algebra if we equip it with a sub-multiplicative matrix norm.\n• Take the Banach space Rn (or Cn) with norm ||x|| = max |xi| and define multiplication componentwise: (x1,...,xn)(y1,...,yn) = (x1y1,...,xnyn).\n• The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.\n• The algebra of all bounded real- or complex-valued functions defined on some set (with pointwise multiplication and the supremum norm) is a unital Banach algebra.\n• The algebra of all bounded continuous real- or complex-valued functions on some locally compact space (again with pointwise operations and supremum norm) is a Banach algebra.\n• The algebra of all continuous linear operators on a Banach space E (with functional composition as multiplication and the operator norm as norm) is a unital Banach algebra. The set of all compact operators on E is a closed ideal in this algebra.\n• If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L1(G) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy(g) = ∫ x(h) y(h−1g) dμ(h) for x, y in L1(G).\n• Uniform algebra: A Banach algebra that is a subalgebra of C(X) with the supremum norm and that contains the constants and separates the points of X (which must be a compact Hausdorff space).\n• Natural Banach function algebra: A uniform algebra whose all characters are evaluations at points of X.\n• C*-algebra: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some Hilbert space.\n• Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact group, where the product of two measures is given by convolution.\n\n## Properties\n\nSeveral elementary functions which are defined via power series may be defined in any unital Banach algebra; examples include the exponential function and the trigonometric functions, and more generally any entire function. (In particular, the exponential map can be used to define abstract index groups.) The formula for the geometric series remains valid in general unital Banach algebras. The binomial theorem also holds for two commuting elements of a Banach algebra.\n\nThe set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous, (and hence homeomorphism) so that it forms a topological group under multiplication.\n\nIf a Banach algebra has unit 1, then 1 cannot be a commutator; i.e.,", null, "$xy - yx \\ne \\mathbf{1}$  for any x, y ∈ A.\n\nThe various algebras of functions given in the examples above have very different properties from standard examples of algebras such as the reals. For example:\n\n• Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions. Hence, the only complex Banach algebra which is a division algebra is the complexes. (This is known as the Gelfand-Mazur theorem.)\n• Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.\n• Every commutative real unital Noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.\n• Permanently singular elements in Banach algebras are topological divisors of zero, i.e., considering extensions B of Banach algebras A some elements that are singular in the given algebra A have a multiplicative inverse element in a Banach algebra extension B. Topological divisors of zero in A are permanently singular in all Banach extension B of A.\n\n## Spectral theory\n\nUnital Banach algebras over the complex field provide a general setting to study spectral theory. The spectrum of an element x ∈ A, denoted by σ(x), consists of all those complex scalars λ such that x − λ1 is not invertible in A. The spectrum of any element x is a closed subset of the closed disc in C with radius ||x|| and center 0, and thus is compact. Moreover, the spectrum σ(x) of an element x is non-empty and satisfies the spectral radius formula:", null, "$\\sup \\{ |\\lambda| : \\lambda \\in \\sigma(x) \\} = \\lim_{n \\to \\infty} \\|x^n\\|^{1/n}.$\n\nGiven x ∈ A, the holomorphic functional calculus allows to define ƒ(x) ∈ A for any function ƒ holomorphic in a neighborhood of σ(x). Furthermore, the spectral mapping theorem holds:\n\nσ(f(x)) = f(σ(x)).\n\nWhen the Banach algebra A is the algebra L(X) of bounded linear operators on a complex Banach space X  (e.g., the algebra of square matrices), the notion of the spectrum in A coincides with the usual one in the operator theory. For ƒ ∈ C(X) (with a compact Hausdorff space X), one sees that:", null, "$\\sigma(f) = \\{ f(t) : t \\in X \\}.$\n\nThe norm of a normal element x of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.\n\nLet A  be a complex unital Banach algebra in which every non-zero element x is invertible (a division algebra). For every a ∈ A, there is λ ∈ C such that a − λ1 is not invertible (because the spectrum of a is not empty) hence a = λ1 : this algebra A is naturally isomorphic to C (the complex case of the Gelfand-Mazur theorem).\n\n## Ideals and characters\n\nLet A  be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A. Since a maximal ideal", null, "$\\mathfrak m$ in A is closed,", null, "$A / \\mathfrak m$ is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ(A) of all nonzero homomorphisms from A  to C. The set Δ(A) is called the \"structure space\" or \"character space\" of A, and its members \"characters.\"\n\nA character χ is a linear functional on A which is at the same time multiplicative, χ(ab) = χ(a) χ(b), and satisfies χ(1) = 1. Every character is automatically continuous from A  to C, since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (i.e., operator norm) of a character is one. Equipped with the topology of pointwise convergence on A (i.e., the topology induced by the weak-* topology of A), the character space, Δ(A), is a Hausdorff compact space.\n\nFor any xA,", null, "$\\sigma(x) = \\sigma(\\hat x)$\n\nwhere", null, "$\\hat x$ is the Gelfand representation of x defined as follows:", null, "$\\hat x$ is the continuous function from Δ(A) to C given by", null, "$\\hat x(\\chi) = \\chi(x).$  The spectrum of", null, "$\\hat x,$ in the formula above, is the spectrum as element of the algebra C(Δ(A)) of complex continuous functions on the compact space Δ(A). Explicitly,", null, "$\\sigma(\\hat x) = \\{ \\chi(x) : \\chi \\in \\Delta(A) \\}$.\n\nAs an algebra, a unital commutative Banach algebra is semisimple (i.e., its Jacobson radical is zero) if and only if its Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when A is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between A and C(Δ(A)) .\n\n## See also\n\nWikimedia Foundation. 2010.\n\n### Look at other dictionaries:\n\n• Banach-Algebra — berührt die Spezialgebiete Mathematik Topologie Abstrakte Algebra Lineare Algebra Funktionalanalysis ist Spezialfall von Abels …   Deutsch Wikipedia\n\n• banach algebra — ˈbäˌnäḵ , nək noun Usage: usually capitalized B Etymology: after Stefan Banach died 1945 Pol. mathematician : a linear algebra over the field of real or complex numbers that is also a Banach space for which the norm of the product of x and y is… …   Useful english dictionary\n\n• Banach algebra cohomology — In mathematics, Banach algebra cohomology of a Banach algebra with coefficients in a bimodule is defined in a similar way to Hochschild cohomology of an abstract algebra, except that one takes the topology into account so that all cohains and so… …   Wikipedia\n\n• Amenable Banach algebra — A Banach algebra, A , is amenable if all bounded derivations from A into dual Banach A bimodules are inner (that is of the form amapsto a.x x.a for some x in the dual module).An equivalent characterization is that A is amenable if and only if it… …   Wikipedia\n\n• Algebra (disambiguation) — Algebra is a branch of mathematics.Algebra may also mean: * elementary algebra * abstract algebra * linear algebra * universal algebra * computer algebraIn addition, many mathematical objects are known as algebras. * In logic: ** Boolean algebra… …   Wikipedia\n\n• Algebra (Begriffsklärung) — Algebra bezeichnet in der Mathematik: Algebra, ein Teilgebiet der Mathematik mit den weiteren Teilgebieten Elementare Algebra Abstrakte Algebra Lineare Algebra Kommutative Algebra Universelle Algebra Computeralgebra Außerdem bezeichnet man mit… …   Deutsch Wikipedia\n\n• Banach function algebra — In functional analysis a Banach function algebra on a compact Hausdorff space X is unital subalgebra, A of the commutative C* algebra C(X) of all continuous, complex valued functions from X , together with a norm on A which makes it a Banach… …   Wikipedia\n\n• Banach-Raum — Bạnach Raum   [nach S. Banach], Mathematik: fundamentaler Begriff der Funktionalanalysis, Verallgemeinerung des der analytischen Geometrie zugrunde liegenden Begriffs des n dimensionalen Vektorraumes. Ist auf einem reellen oder komplexen… …   Universal-Lexikon\n\n• Banach space — In mathematics, Banach spaces (pronounced [ˈbanax]) is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every… …   Wikipedia\n\n• Banach-Raum — Ein Banach Raum, benannt nach dem Mathematiker Stefan Banach, ist ein vollständiger normierter Vektorraum. Banach Räume gehören zu den zentralen Studienobjekten der Funktionalanalysis. Die interessantesten Banach Räume sind unendlichdimensionale… …   Deutsch Wikipedia\n\n### Share the article and excerpts\n\n##### Direct link\nDo a right-click on the link above\nand select “Copy Link”\n\nWe are using cookies for the best presentation of our site. 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[ "Cody\n\n# Problem 42. Find the alphabetic word product\n\nIf the input string s is a word like 'hello', then the output word product p is a number based on the correspondence a=1, b=2, ... z=26. Assume the input will be a single word, although it may mixed case. Note that A=a=1 and B=b=2.\n\nSo\n\n` s = 'hello'`\n\nmeans\n\n` p = 8 * 5 * 12 * 12 * 15 = 86400`\n\nBonus question: How close can you get to a word product of one million?\n\n### Solution Stats\n\n55.09% Correct | 44.91% Incorrect\nLast Solution submitted on Jun 04, 2020" ]

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[ "# Cohn's criterion\n\nLet", null, "$p$ be a prime number, and", null, "$b\\geq 2$ an integer. If", null, "$\\overline{p_np_{n-1}\\cdots p_1p_0}$ is the base-", null, "$b$ representation of", null, "$p$, and", null, "$0\\leq p_i, then", null, "$$f(x)=p_nx^n+p_{n-1}x^{n-1}+\\cdots+p_1x+p_0$$ is irreducible.\n\n## Proof\n\nThe following proof is due to M. Ram Murty.\n\nWe start off with a lemma. Let", null, "$g(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0\\in \\mathbb{Z}[x]$. Suppose", null, "$a_n\\geq 1$,", null, "$|a_i|\\leq H$. Then, any complex root of", null, "$g(x)$,", null, "$\\phi$, has a non positive real part or satisfies", null, "$|\\phi|<\\frac{1+\\sqrt{1+4H}}{2}$.\n\nProof: If", null, "$|z|>1$ and Re", null, "$z>0$, note that:", null, "$$|\\frac{g(z)}{z^n}|\\geq |a_n+\\frac{a_{n-1}}{z}|-H(\\frac{1}{|z|^2}+\\cdots+\\frac{1}{|z|^n})>Re(a_n+\\frac{a_{n-1}}{z})-\\frac{H}{|z|^2-|z|}\\geq 1-\\frac{H}{|z|^2-|z|}=\\frac{|z|^2-|z|-H}{|z|^2-|z|}$$ This means", null, "$g(z)>0$ if", null, "$|z|\\geq \\frac{1+\\sqrt{1+4H}}{2}$, so", null, "$|\\phi|<\\frac{1+\\sqrt{1+4H}}{2}$.\n\nIf", null, "$b\\geq 3$, this implies", null, "$|b-\\phi|>1$ if", null, "$b\\geq 3$ and", null, "$f(\\phi)=0$. Let", null, "$f(x)=g(x)h(x)$. Since", null, "$f(b)=p$, one of", null, "$|g(b)|$ and", null, "$h(b)$ is 1. WLOG, assume", null, "$g(b)=1$. Let", null, "$\\phi_1, \\phi_2,\\cdots,\\phi_r$ be the roots of", null, "$g(x)$. This means that", null, "$|g(b)|=|b-\\phi_1||b-\\phi_2|\\cdots|b-\\phi_r|>1$. Therefore,", null, "$f(x)$ is irreducible.\n\nIf", null, "$b=2$, we will need to prove another lemma:\n\nAll of the zeroes of", null, "$f(x)$ satisfy Re", null, "$z<\\frac{3}{2}$.\n\nProof: If", null, "$n=1$, then the two polynomials are", null, "$x$ and", null, "$x\\pm 1$, both of which satisfy our constraint. For", null, "$n=2$, we get the polynomials", null, "$x^2$,", null, "$x^2\\pm x$,", null, "$x^2\\pm 1$, and", null, "$x^2\\pm x\\pm 1$, all of which satisfy the constraint. If", null, "$n\\geq 3$,", null, "$$|\\frac{f(z)}{z^n}|\\geq |1+\\frac{a_{n-1}}{z}+\\frac{a_{n-2}}{z^2}|-(\\frac{1}{|z|^3}+\\cdots+\\frac{1}{|z|^n})>|1+\\frac{a_{n-1}}{z}+\\frac{a_{n-2}}{z^2}|-\\frac{1}{|z|^2(|z|-1)}$$\n\nIf Re", null, "$z\\geq 0$, we have Re", null, "$\\frac{1}{z^2}\\geq 0$, and then", null, "$$|\\frac{f(z)}{z^n}|>1-\\frac{1}{|z|^2(|z|-1)}$$ For", null, "$|z|\\geq \\frac{3}{2}$, then", null, "$|z|^2(|z|-1)\\geq(\\frac{3}{2})^2(\\frac{1}{2})=\\frac{9}{8}>1$. Therefore,", null, "$z$ is not a root of", null, "$f(x)$.\n\nTo finish the proof, let", null, "$f(x)=g(x)h(x)$. Since", null, "$f(2)=p$, one of", null, "$g(2)$ and", null, "$h(2)$ is 1. WLOG, assume", null, "$g(2)=1$. By our lemma,", null, "$|z-2|>|z-1|$. Thus, if", null, "$\\phi_1, \\phi_2,\\cdots,\\phi_r$ are the roots of", null, "$g(x)$, then", null, "$|g(2)|=|2-\\phi_1||2-\\phi_2|\\cdots|2-\\phi_n|>|1-\\phi_1||1-\\phi_2|\\cdots|1-\\phi_n|=|g(1)| < 1$. This is a contradiction, so", null, "$f(x)$ is irreducible." ]

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[ "# Circle Limit Exploration\n\nObjective: Explore Escher's Circle Limit prints to develop an intuition for hyperbolic geometry\n\n## Materials", null, "All four Circle Limit prints, dimmed: File:Four-dim-circle-limits.pdf\n\n## Circle Limit I\n\nRecall that in Spherical Geometry, the shortest path between two points is along a great circle. These shortest paths are called geodesics, and the geodesics play the same role as do straight lines in Euclidean geometry. Escher's Circle Limit prints are based on a new kind of geometry, Hyperbolic Geometry.\n\nThe red lines shown on Circle Limit I are geodesics in this new geometry. These curves will play the role of straight lines. Each red line follows the spines of a line of fish.\n\n1. There are two types of red line marked in the Circle Limit I figure. Describe them. Draw more geodesics by following the spines of other rows of fish. Describe the curves that result.\n\nIn these pictures of hyperbolic geometry, geodesics come in two forms, either straight lines through the center of the disk, or arcs of circles that meet the disk's edge at 90°. Segments of geodesics form the sides of polygons. Polygons in hypebolic geometry will look \"pinched\" to our Euclidean eyes.\n\n1. What type of polygons do you see in Circle Limit I?\n2. Compare the angle sum of one of these polygons to the corresponding angle sum for Euclidean geometry.\n\nCircle Limit I is a picture of a surface called \"hyperbolic space\", but it is a distorted picture. In actual hyperbolic space, these fish would all have the same size and shape.\n\n1. What is the highest order of rotation symmetry for this print?\n2. Describe the geometric tessellation underlying Circle Limit I.\n\n## Circle Limits II and IV\n\nFor Circle Limit II:\n\n1. What is the highest order of rotation?\n2. Draw geodesics in this figure. Describe the underlying geometric tessellation.\n\nFor Circle Limit IV:\n\n1. What is the highest order of rotation? What other orders of rotation are present?\n2. Draw geodesics in this figure. Describe the underlying geometric tessellation.\n3. Draw a geodesic NOT passing through the center point.\n4. How many geodesics can you draw through the center point, so that the new geodesic does not meet the geodesic you picked in the previous question? Another way to ask the same question: How many geodesics pass through the point so that the new geodesic is parallel to the first geodesic?\n\n## Circle Limit III\n\nThis Circle Limit is the most subtle. The white lines look like the geodesics in the other Circle Limit prints, but they are not the same. A closer look shows that these white lines are not geodesics at all.\n\n1. Pick a triangle and determine its corner angles by considering the number of polygons at a vertex. Assume all angles at each vertex are equal (they are, though the distortion makes this harder to believe).\n2. What is the sum of the angles in the triangle? Is this possible in hyperbolic geometry?\n3. Choose a white line and trace it to the point where it meets the boundary of the disk. Carefully measure the angle it makes with the edge of the disk (you may want to draw tangent lines to the disk and the white line). What angle did you get?\n\nHandin: Marked up Circle Limit prints and a sheet with answers to all questions." ]

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[ "Main Content\n\n# atmoshwm\n\nImplement horizontal wind model\n\n## Syntax\n\n``wind = atmoshwm(latitude,longitude,altitude)``\n``wind = atmoshwm(latitude,longitude,altitude,Name,Value)``\n\n## Description\n\nexample\n\n````wind = atmoshwm(latitude,longitude,altitude)` implements the U.S. Naval Research Laboratory Horizontal Wind Model (HWM™) routine to calculate the meridional and zonal components of the wind for one or more sets of geophysical data: `latitude`, `longitude`, and `altitude`. ```\n\nexample\n\n````wind = atmoshwm(latitude,longitude,altitude,Name,Value)` uses additional options specified by one or more `Name,Value` pair arguments.```\n\n## Examples\n\ncollapse all\n\nCalculate the total horizontal wind model for a latitude of 45 degrees south, longitude of 85 degrees west, and altitude of 25,000 m above mean sea level (msl). The date is the 150th day of the year, at 11 am UTC, using an Ap index of 80. The horizontal model version is 14.\n\n`w = atmoshwm(-45,-85,25000,'day',150,'seconds',39600,'apindex',80,'model','total', 'version', '14')`\n```w = 3.2874 25.8735```\n\nCalculate the quiet horizontal wind model for a latitude of 50 degrees north, two altitudes of 100,000 m and 150,000 m above msl, and a longitude of 20 degrees west. The date is midnight UTC of January 30. The default horizontal model version is 14.\n\n`w = atmoshwm([50;50],[-20;-20],[100000;150000],'day',[30;30])`\n```w = -42.9350 -40.3693 29.1106 0.6253```\n\nCalculate the disturbed horizontal wind model for an altitude of 150,000 m above msl at latitude 70 degrees north, longitude 65 degrees west. The date is midnight UTC of June 15. The default horizontal model version is 14.\n\n`dw = atmoshwm(70,-65,150000,'day',166,'model','disturbance')`\n```dw = 1.7954 -1.7130```\n\n## Input Arguments\n\ncollapse all\n\nGeodetic latitudes, in degrees, specified as a scalar or M-by-1 array, where M is one or more sets of geophysical data.\n\nExample: -45\n\nData Types: `double`\n\nGeodetic longitudes, in degrees, specified as a scalar or M-by-1 array, where M is one or more sets of geophysical data.\n\nExample: -85\n\nData Types: `double`\n\nGeopotential heights, in meters, within the range of 0 to 500 km, specified as a scalar or M-by-1 array. M is one or more sets of geophysical data.\n\nExample: 25000\n\nData Types: `double`\n\n### Name-Value Pair Arguments\n\nSpecify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.\n\nExample: `'apindex',80,'model','total'` specifies that the total horizontal wind model be calculated for an Ap index of 80.\n\nAp index for the Universal Coordinated Time (UTC) at which `atmoshwm` evaluates the model, specified as an M-by-1 array of zeroes, a scalar, or an M-by-1 array. M is one or more sets of geophysical data. Select the index from the NOAA National Geophysical Data Center, which contains three-hour interval geomagnetic disturbance index values. If the Ap index value is greater than zero, the model evaluation accounts for magnetic effects.\n\nSpecify the Ap index as a value from 0 through 400. Specify an Ap index value for only the disturbance or total wind model type.\n\nData Types: `double`\n\nDay of year in UTC. Specify the day as a value from 1 through 366 (for a leap year), specified as an M-by-1 array of zeroes, a scalar, or an M-by-1 array. M is one or more sets of geophysical data.\n\nData Types: `double`\n\nElapsed seconds since midnight for the selected day, in UTC, specified as specified as an M-by-1 array of zeroes, a scalar, or an M-by-1 array. M is one or more sets of geophysical data.\n\nSpecify the seconds as a value from 0 through 86,400.\n\nData Types: `double`\n\nHorizontal wind model type for which to calculate the wind components. This setting applies to all the sets of geophysical data in M.\n\n• `'quiet'`\n\nCalculates the horizontal wind model without the magnetic disturbances. Quiet model types do not account for Ap index values. For this model type, do not specify an Ap index value when using this model type.\n\n• `'disturbance'`\n\nCalculates the effect of only magnetic disturbances in the wind. For this model type, specify Ap index values greater than or equal to zero.\n\n• `'total'`\n\nCalculates the combined effect of the quiet and magnetic disturbances. for this model type, specify Ap index values greater than or equal to zero.\n\nData Types: `char` | `string`\n\nFunction behavior when inputs are out of range, specified as one of these values. This type applies to all the sets of geophysical data in M.\n\nValueDescription\n`'None'`No action.\n`'Warning'`Warning in the MATLAB® Command Window, model simulation continues.\n`'Error'`MATLAB returns an exception, model simulation stops.\n\nData Types: `char` | `string`\n\nImplements specified horizontal wind model type.\n\n• `'14'`\n\nHorizontal Wind Model 14.\n\n• `'07'`\n\nHorizontal Wind Model 07.\n\nData Types: `char` | `string`\n\n## Output Arguments\n\ncollapse all\n\nMeridional and zonal wind components of the horizontal wind model, returned as an M-by-2 array, in m/s.\n\n## See Also\n\nIntroduced in R2016b\n\n## Support", null, "Get trial now" ]

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[ "# FCFS Scheduling\n\nIn case of multiprogramming, CPU needs to be scheduled, so that multiple works can be performed simultaneously in less time or at a same time. By CPU scheduling it is decided which of the processes in the ready queue is to be allocated in the CPU. Thus, there are so many CPU-scheduling algorithms in order to schedule CPU. FCFS scheduling is one of the CPU-scheduling algorithms.\n\n## FCFS (FIRST-COME, FIRST-SERVED) Scheduling\n\nFCFS is considered as simplest CPU-scheduling algorithm. In FCFS algorithm, the process that requests the CPU first is allocated in the CPU first. The implementation of FCFS algorithm is managed with FIFO (First in first out) queue. FCFS scheduling is non-preemptive. Nonpreemptive means, once the CPU has been allocated to a process, that process keeps the CPU until it executes a work or job or task and releases the CPU, either by requesting I/O.\n\n## Real Life Example Of FCFS Scheduling\n\nAs a real life example of FCFS scheduling a billing counter system of shopping mall can be observed. The first person in the line gets the bill done first and then the next person gets the chance to get the bill and make payment and so on. If no priority is given to the VIP customers then the billing system will go on like this (means the first person (first task) in the line will get the bill first and after finishing (executing) the first customer’s payment the counter boy(CPU) will pay attention to other customers (separate tasks) as they are in the line). As FCFS is non-preemptive type so no priority will be given to the random important tasks.\n\n## FCFS Scheduling Mathematical Examples\n\nIn CPU-scheduling problems some terms are used while solving the problems, so for conceptual purpose the terms are discussed as follows −\n\n• Arrival time (AT) − Arrival time is the time at which the process arrives in ready queue.\n\n• Burst time (BT) or CPU time of the process − Burst time is the unit of time in which a particular process completes its execution.\n\n• Completion time (CT) − Completion time is the time at which the process has been terminated.\n\n• Turn-around time (TAT) − The total time from arrival time to completion time is known as turn-around time. TAT can be written as,\n\nTurn-around time (TAT) = Completion time (CT) – Arrival time (AT) or, TAT = Burst time (BT) + Waiting time (WT)\n\n• Waiting time (WT) − Waiting time is the time at which the process waits for its allocation while the previous process is in the CPU for execution. WT is written as,\n\nWaiting time (WT) = Turn-around time (TAT) – Burst time (BT)\n\n• Response time (RT) − Response time is the time at which CPU has been allocated to a particular process first time.\n\nIn case of non-preemptive scheduling, generally Waiting time and Response time is same.\n\n• Gantt chart − Gantt chart is a visualization which helps to scheduling and managing particular tasks in a project. It is used while solving scheduling problems, for a concept of how the processes are being allocated in different algorithms.\n\n### Problem 1\n\nConsider the given table below and find Completion time (CT), Turn-around time (TAT), Waiting time (WT), Response time (RT), Average Turn-around time and Average Waiting time.\n\nProcess ID\n\nArrival time\n\nBurst time\n\nP1\n\n2\n\n2\n\nP2\n\n5\n\n6\n\nP3\n\n0\n\n4\n\nP4\n\n0\n\n7\n\nP5\n\n7\n\n4\n\n### Solution\n\nGantt chart", null, "For this problem CT, TAT, WT, RT is shown in the given table −\n\nProcess ID\n\nArrival time\n\nBurst time\n\nCT\n\nTAT=CT-AT\n\nWT=TAT-BT\n\nRT\n\nP1\n\n2\n\n2\n\n13\n\n13-2= 11\n\n11-2= 9\n\n9\n\nP2\n\n5\n\n6\n\n19\n\n19-5= 14\n\n14-6= 8\n\n8\n\nP3\n\n0\n\n4\n\n4\n\n4-0= 4\n\n4-4= 0\n\n0\n\nP4\n\n0\n\n7\n\n11\n\n11-0= 11\n\n11-7= 4\n\n4\n\nP5\n\n7\n\n4\n\n23\n\n23-7= 16\n\n16-4= 12\n\n12\n\nAverage Waiting time = (9+8+0+4+12)/5 = 33/5 = 6.6 time unit (time unit can be considered as milliseconds)\n\nAverage Turn-around time = (11+14+4+11+16)/5 = 56/5 = 11.2 time unit (time unit can be considered as milliseconds)\n\n### Problem 2\n\nConsider the given table below and find Completion time (CT), Turn-around time (TAT), Waiting time (WT), Response time (RT), Average Turn-around time and Average Waiting time.\n\nProcess ID\n\nArrival time\n\nBurst time\n\nP1\n\n2\n\n2\n\nP2\n\n0\n\n1\n\nP3\n\n2\n\n3\n\nP4\n\n3\n\n5\n\nP5\n\n4\n\n5\n\n### Solution\n\nGantt chart −", null, "For this problem CT, TAT, WT, RT is shown in the given table −\n\nProcess ID\n\nArrival time\n\nBurst time\n\nCT\n\nTAT=CT-AT\n\nWT=TAT-BT\n\nRT\n\nP1\n\n2\n\n2\n\n4\n\n4-2= 2\n\n2-2= 0\n\n0\n\nP2\n\n0\n\n1\n\n1\n\n1-0= 1\n\n1-1= 0\n\n0\n\nP3\n\n2\n\n3\n\n7\n\n7-2= 5\n\n5-3= 2\n\n2\n\nP4\n\n3\n\n5\n\n12\n\n12-3= 9\n\n9-5= 4\n\n4\n\nP5\n\n4\n\n5\n\n17\n\n17-4= 13\n\n13-5= 8\n\n8\n\nAverage Waiting time = (0+0+2+4+8)/5 = 14/5 = 2.8 time unit (time unit can be considered as milliseconds)\n\nAverage Turn-around time = (2+1+5+9+13)/5 = 30/5 = 6 time unit (time unit can be considered as milliseconds)\n\n*In idle (not-active) CPU period, no process is scheduled to be terminated so in this time it remains void for a little time.\n\n• It is an easy algorithm to implement since it does not include any complex way.\n\n• Every task should be executed simultaneously as it follows FIFO queue.\n\n• FCFS does not give priority to any random important tasks first so it’s a fair scheduling.\n\n• FCFS results in convoy effect which means if a process with higher burst time comes first in the ready queue then the processes with lower burst time may get blocked and that processes with lower burst time may not be able to get the CPU if the higher burst time task takes time forever.\n\n• If a process with long burst time comes in the line first then the other short burst time process have to wait for a long time, so it is not much good as time-sharing systems.\n\n• Since it is non-preemptive, it does not release the CPU before it completes its task execution completely.\n\n*Convoy effect and starvation sounds similar but there is slight difference, so it is advised to not treat these both terms as same words.\n\n## Conclusion\n\nAlthough FCFS is simple to implement and understand, FCFS is not good for interactive system and not used in modern operating systems. The Convoy effect in the FCFS can be prevented using other CPU-scheduling preemptive algorithms such as ‘Round-robin scheduling’.\n\nUpdated on: 05-Apr-2023\n\n16K+ Views", null, "" ]

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[ "", null, "A directory of Objective Type Questions covering all the Computer Science subjects. Here you can access and discuss Multiple choice questions and answers for various compitative exams and interviews.\n\n## Discussion Forum\n\n Que. The 2^n vertices of a graph G corresponds to all subsets of a set of size n, for n >= 6 . Two vertices of G are adjacent if and only if the corresponding sets intersect in exactly two elements. The number of vertices of degree zero in G is: a. 1 b. n c. n+1 d. 2^n Answer:n+1" ]

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[ "# Weaker totality condition (T1)\n\nIn document Context-indexed counterfactuals and non-vacuous counterpossibles (Page 165-184)\n\n## Definition 5.7: Let (C) be a condition predicable of an ordering frame (e.g like the\n\n### 5.4.2 Weaker totality condition (T1)\n\nTotality of ≲𝑖 need not be abandoned altogether to avoid commitment to (5.10). It suffices to\n\ncondition (T2) with a weaker one (T1). Note that systems that invalidate (5.10), by virtue of satisfying the weaker comparability condition (T1) aren’t somehow particularly contrived. At least, they are not any more contrived than some aspects of the model theory that are already in place. That is, we’ve already distinguished the elements of 𝑁 and 𝑊\\𝑁 at the level of models, by stipulating distinct conditions for 𝜌 (and consequently ⊩). So, it doesn’t seem all that more contrived to distinguish the elements of 𝑁 and 𝑊\\𝑁 at level of ordering frames, by allowing distinct conditions for ≲.\n\nProposition 5.2.1: 𝑝 > 𝑞, 𝑞 > 𝑝 ⊭𝐂𝐒3∗ (𝑝 > 𝑟) ≡ (𝑞 > 𝑟)\n\nProof : Let 𝔄 = (𝑊, 𝑁, {≲𝑖: 𝑖 ∈ 𝑁}, 𝜌), be a 𝐂𝐒3∗ model such that 𝑊 = {𝑖, 𝑥, 𝑦, 𝑧}, 𝑁 = {𝑖}, the\n\nfollowing ordering assignment ≲𝑖 = {(𝑖, 𝑥), (𝑖, 𝑦), (𝑖, 𝑧), (𝑥, 𝑦), (𝑦, 𝑥), (𝑖, 𝑖), (𝑥, 𝑥), (𝑦, 𝑦), (𝑧, 𝑧)},\n\nand ⊩ = {(𝑖, ~𝑝), (𝑖, ~𝑞), (𝑖, ~𝑟), (𝑥, 𝑝), (𝑥, 𝑞), (𝑥, 𝑟), (𝑦, 𝑞), (𝑧, 𝑝), (𝑧, 𝑞)}. See diagram below (indication of reflexivity has been omitted for better readability).\n\nNow, 𝑧 ∈ [𝑝] ∩ [𝑞] = {𝑥, 𝑧}, and ↓.𝑧 ∩ [𝑝] = {𝑧} ⊆ [𝑞] = {𝑥, 𝑦, 𝑧}, and ↓.𝑧 ∩ [𝑞] = {𝑧} ⊆ [𝑝] =\n\n{𝑥, 𝑧}. Hence 𝑖 ⊩ {𝑝 > 𝑞, 𝑞 > 𝑝}. Also 𝑥 ∈ [𝑝] = {𝑥, 𝑧} and ↓.𝑥 ∩ [𝑝] = {𝑥} ⊆ [𝑟] = {𝑥}, therefore\n\n𝑖 ⊩ 𝑝 > 𝑟. But, ↓.𝑤 ∩ [𝑞] ⊈ [𝑟], for all 𝑤 ∈ [𝑞], i.e. ↓.𝑥 ∩ [𝑞] = ↓.𝑦 ∩ [𝑞] = {𝑥, 𝑦} ⊈ [𝑟] = {𝑥}, and\n\n↓.𝑧 ∩ [𝑞] = {𝑧} ⊈ [𝑟] = {𝑥}. Hence 𝑖 ⊮ 𝑞 > 𝑟, as required. □\n\n5.4.3 (T1) and Adjunction of Consequents\n\nNow we turn to another motivation for (T1). Lifting unrestricted comparability over impossible worlds is a way of invalidating inferences that have at least one pair of counterpossible premises with the same antecedent, and whose validity hinges on all counterpossibles with the same antecedents being evaluated on the same relevant set of worlds. So, for example, inferences like (5.11) – namely, Adjunction of Consequents, discussed extensively in §4 – will be valid in all conditional logics where the antecedent is the only parameter that determines the range of the accessibility relation.\n\n(5.11) 𝐴 > 𝐵, 𝐴 > 𝐶 ⊨ 𝐴 > (𝐵 ∧ 𝐶)\n\nThis is formally valid on any labelled transition system model (𝑊, 𝑁, {𝑅𝐴: 𝐴 ∈ 𝐹𝑜𝑟}, 𝜌) where\n\n𝜌 and ⊩ are defined exactly the same as in definition 5.3, where (𝑊, {𝑅𝐴: 𝐴 ∈ 𝐹𝑜𝑟}), 𝑅𝐴, and\n\n𝑓𝐴(𝑤) are as in §2.1.3, and the truth conditions for > are: 𝑤 ⊩ 𝐴 > 𝐵 iff 𝑓𝐴(𝑤) ⊆ [𝐵].196\n\nBut consider the following instance (which appears to be a clear counterexample) containing Goodman-inspired counteridenticals:197\n\n(1) If the number 2 was Sherlock Holmes, then 2 would be a detective.\n\n(2) If Sherlock Holmes was the number 2, then Sherlock Holmes wouldn’t be a detective. (3) Therefore, if 2 was S. Holmes, then 2 would be a detective and S. Holmes wouldn’t\n\nbe a detective.\n\nBoth premises are non-vacuously true counterpossibles, however the conclusion (3) is clearly not true.198 CS* systems with (T1) instead of (T2) give a correct analysis of this\n\ncounterexample and ones like it, i.e. there are countermodels to (5.11) in each system weaker than 𝐂𝐒4∗, and I explicitly give one in Proposition 5.3, below. One may object to admitting\n\nboth premises, on account of apparent radical context-shift required for that, but then one would have to decide how much of a context shift between premises is allowed. At least it’s not obviously clear that the freedom of context-shift employed in the counterexample to (5.10), discussed earlier, would not justify the context shifts in the invalidation of (5.11). And (5.11) is formally valid in all the alternative non-vacuist systems that Weiss (2017)\n\nendorses.199\n\nThe inference is still valid for systems with the stronger comparability condition (T2).\n\nProposition 5.3: 𝐴 > 𝐵, 𝐴 > 𝐶 ⊨𝐂𝐒4∗ 𝐴 > (𝐵 ∧ 𝐶)\n\nProof : Let 𝔄 = (𝑊, 𝑁, {≲𝑖}𝑖∈𝑁, 𝜌), be a 𝐂𝐒4∗ model and let 𝑖 ⊩ {𝐴 > 𝐵, 𝐴 > 𝐶} for arbitrary 𝑖 ∈\n\n𝑊. Then ∃𝑘 ∈ [𝐴] such that ↓.𝑘 ∩ [𝐴] ⊆ [𝐵], and ∃𝑘′ ∈ [𝐴] such that ↓.𝑘′ ∩ [𝐴] ⊆ [𝐶]. Now,\n\n196 This follows from Proposition 4.12, (and footnote 168) which shows that (5.11) is valid for CS and C. 197 Goodman (1983, p.6). Note that (1) and (2) are equivalent – the order has been inverted only for emphasis. 198 I’m assuming that being a detective is an essential property of Sherlock Holmes (i.e. Holmes has it in all possible worlds), and the number 2 is not a detective in any possible world, hence the identity is a counterpossible identity.\n\n199 Although Weiss builds the alternative non-vacuist proposal on conditional logics like C and C+ (i.e. labelled transition systems that take only the antecedent as a parameter in the accessibility relation), he does entertain similar systems, which also include the consequent as a parameter in characterizing the accessibility relation, drawing on Gabbay’s (1972) proposal – an approach that would allow for the invalidation of (5.11).\n\neither 𝑘 ≲𝑖 𝑘′ or 𝑘′ ≲𝑖𝑘 by (T2). If 𝑘 ≲𝑖 𝑘′, then clearly ↓.𝑘 ⊆ ↓.𝑘′, which in conjunction with\n\nthe hypothesis implies ↓.𝑘 ∩ [𝐴] ⊆ [𝐶]. So, we have both ↓.𝑘 ∩ [𝐴] ⊆ [𝐵] and ↓.𝑘 ∩ [𝐴] ⊆ [𝐶],\n\nwhich jointly imply ↓.𝑘 ∩ [𝐴] ⊆ [𝐵] ∩ [𝐶]. Hence 𝑖 ⊩ 𝐴 > (𝐵 ∧ 𝐶), as required. A very similar\n\nargument holds for the case when 𝑘′ ≲𝑖 𝑘. □\n\nHowever, it fails once we weaken the comparability condition to (T1).\n\nProposition 5.3.1: 𝑝 > 𝑞, 𝑝 > 𝑟 ⊭𝐂𝐒3∗ 𝑝 > (𝑞 ∧ 𝑟)\n\nProof : Let 𝔄 = (𝑊, 𝑁, {≲𝑖: 𝑖 ∈ 𝑁}, 𝜌), be a 𝐂𝐒3∗ model such that 𝑊 = {𝑖, 𝑗, 𝑘}, 𝑁 = {𝑖}, letting\n\n≲𝑖 = {(𝑖, 𝑗), (𝑖, 𝑘), (𝑖, 𝑖), (𝑗, 𝑗), (𝑘, 𝑘), }, and ⊩ = {(𝑖, ~𝑝), (𝑖, ~𝑞), (𝑖, ~𝑟), (𝑗, 𝑝), (𝑗, 𝑞), (𝑘, 𝑝), (𝑘, 𝑟)}.\n\nSee diagram below (indication of reflexivity has been omitted for better readability).\n\nNow, 𝑗 ∈ [𝑝] and ↓.𝑗 ∩ [𝑝] = {𝑗} ⊆ [𝑞] = {𝑗}, and also 𝑘 ∈ [𝑝] and ↓.𝑘 ∩ [𝑝] = {𝑘} ⊆ [𝑟] = {𝑘}.\n\nHence, 𝑖 ⊩ {𝑝 > 𝑞, 𝑝 > 𝑟}. But ↓.𝑤 ∩ [𝑝] ⊈ [𝑞 ∧ 𝑟] = ∅, for all 𝑤 ∈ [𝑝]. So, 𝑖 ⊮ 𝑝 > (𝑞 ∧ 𝑟), as\n\nrequired. □\n\n5.4.4 (T1) and mere counterfactuals\n\nThe benefits of (T1) are not confined to counterpossibles. That is, all CS* systems weaker than 𝐂𝐒4∗ invalidate (5.11) even when it is confined to counterfactuals. As exemplified in §4.5\n\nthere exist intuitive counterexamples to (5.11) confined to mere counterfactuals, which nevertheless go through in C and CS because the premises can never be jointly true at any possible world (see claim 4.5.2 and f.21 in §4.5). That is, and this is a crucial point (reiterated from §4.5), the counterexamples go through vacuously because the combined truth of both counterfactual premises at some world – particularly in cases of said counterexamples – implies inconsistent situations, once the imported information is accounted for, and so if the analysis is restricted to possible worlds, the premises can’t be jointly true (since inconsistent situations can’t be accommodated). But on CS* systems with the weaker comparability condition (T1), there is a way of preserving all the intuitive scope of (5.11)’s applicability (it\n\nonly fails in cases when the truth of the premises depends on radical context shifts), and at the same time allow it to fail by giving an accurate analysis of the counterexamples. This is achieved by:\n\n(i) Accommodating the implied inconsistent situations – necessitated by the truth of both premises – by allowing them to hold at impossible worlds.\n\n(ii) Lifting comparability from impossible worlds to block the truth of the conclusion. That is, the inference is analysed as invalid for counterfactuals, as it should be, for the same general reasons that motivate non-vacuism. Therefore, returning to the counterexample discussed at length in §4.5, there is a CS* model where (1) and (2) are true at the actual world, and (3) is false, as required – just take Proposition 5.3.1 as the corresponding countermodel.\n\n(1) If Everest was in New Zealand, Everest would be in the Southern Hemisphere. (2) If Everest was in New Zealand, New Zealand would be in the Northern Hemisphere. (3) Therefore, if Everest was in New Zealand, then Everest would be in the Southern\n\nHemisphere and New Zealand would be in the Northern Hemisphere.\n\nLifting totality does seem like a substantial change, even if restricted to impossible worlds, and some more reflection is required. Nevertheless (5.12), which is a (SI2) salvaged version of (5.10) is valid in 𝐂𝐒3∗. A similar solution will not work for (5.11), since the antecedent in\n\ncases where the inference fails need not express an impossibility, e.g. Mount Everest being in New Zealand is a perfectly possible scenario.\n\n(5.12) ◊𝐴,.◊𝐵, 𝐴 > 𝐵, 𝐵 > 𝐴 ⊨ (𝐴 > 𝐶) ≡ (𝐵 > 𝐶) Proposition 5.4: ◊𝐴,.◊𝐵, 𝐴 > 𝐵, 𝐵 > 𝐴 ⊨𝐂𝐒3∗ (𝐴 > 𝐶) ≡ (𝐵 > 𝐶)\n\nProof : It suffices to observe that the premises, if true, in conjunction with (SI2) will not require the evaluation of > formulae to access any impossible worlds. Only possible worlds will be accessed – and those are totally preordered by (T1). So, the inference goes through, since we need failure of comparability for a counterexample, by proposition 5.2.1, but with (T1) in place only impossible worlds can be incomparable. □\n\nConclusion\n\nIn the culminating chapters (4 and 5) of this thesis I have shown that there are accessible modifications of Lewis’ (1981) ordering semantics for analyses of counterfactuals that resolve some persistent issues regarding contextual ambiguity, and avoid the inadequacy of treating all counterpossibles as vacuously true or false. Moreover, in each case it has been indicated to what extent each modification preserves the logic that serves as its basis.\n\nThe account of context-indexed counterfactuals, proposed in chapter 4, not only addresses the context related concerns identified by Goodman (1954) and Quine (1966), but also offers a meaningful notion of semantic consequence based on the idea of contextual information preservation. Although the approach chosen may not be optimal, given that it also modifies the object language, nevertheless there is some evidence that it appears to be a natural move. But even if the offered account is vulnerable to the charge of not providing a direct solution to the pertinent contextual issues – as it departs from the analysis of a single conditional – at least it offers a framework of thinking about those issues, which can be viewed as going some way toward providing such a solution.\n\nThe account of counterfactuals, proposed in chapter 5, avoids a vacuist account of counterpossibles, whilst preserving much of Lewis’ analysis of mere counterfactuals. The application of an impossible world semantics in this case has been partly motivated and justified by the redeeming roles that such semantics have played in other areas of philosophical analysis and logic. However, although I have replied to Lewis’ defense of vacuism and his objection to impossible worlds, I admit that I have not given a comprehensive defense to all the existing objections. I have defended the feasibility of the comparative similarity of worlds interpretation of impossible world ordering semantics for counterfactuals against some recent objections, by showing that counterexamples arming such objections can be invalidated on systems where impossible worlds satisfy a weaker comparability condition, i.e. partial preorderhood. However, there do remain other questions regarding some key ordering conditions, underlying the extended domain that could be addressed in the future.\n\nA natural step would be to combine the results from chapters 4 and 5, and fashion a system that gives both an adequate analysis of counterpossibles, whilst accounting for contextual distinctions. That is, we can employ the benefits of the system CS2+ and modify its definition so it is based on CS* models instead of (vacuism-burdened) CS models, noting that 𝐂𝐒3∗ has\n\nbeen argued to be the optimal CS* system. This way we would have a system that inherits the advantages of both CS2+ and 𝐂𝐒3∗. This and a more comprehensive participation in the defense\n\nof impossible world semantics in general would constitute a well-motivated inclusion to future research.\n\nAppendix\n\nThe following appendix contains the proof of the equivalence of the class of CS models and the class of S models. That is, these classes validate the same sets of formulae and inferences. This proof is based on a proof sketch given by Lewis (1973, p.49). First, let us recall the relevant definitions, so the formulation of the theorem is clear.\n\nDefinition 2.15: Let ⊨𝐒 ⊆ ℘(𝐹𝑜𝑟) × 𝐹𝑜𝑟, and define Σ ⊨𝐒𝐴 iff for all models (𝑊, \\$, [.]), and\n\nall 𝑤 ∈ 𝑊, if 𝑤 ⊩ 𝐵 for all 𝐵 ∈ Σ, then 𝑤 ⊩ 𝐴. That is, valid inference is defined as truth preservation at all worlds in all systems of spheres models. A formula 𝐴 ∈ 𝐹𝑜𝑟 is said to be valid iff ∅ ⊨𝐒𝐴. Call this logic S.200\n\nDefinition 4.2.7: Let ⊨𝐂𝐒 ⊆ ℘(𝐹𝑜𝑟) × 𝐹𝑜𝑟, and define Σ ⊨𝐂𝐒 𝐴 iff for all models (𝑊, ≲, 𝜌), and\n\nall 𝑖 ∈ 𝑊, if 𝑖 ⊩ 𝐵 for all 𝐵 ∈ Σ, then 𝑖 ⊩ 𝐴. We say an inference from Σ to 𝐴 is valid iff Σ ⊨𝐂𝐒𝐴. That is, valid inference is defined as truth preservation at all worlds in all CS-\n\nmodels. A formula 𝐴 ∈ 𝐹𝑜𝑟 is said to be valid iff ∅ ⊨𝐂𝐒 𝐴. Call this logic CS.\n\nTheorem A.1.1: Σ ⊨𝐒𝐴 iff Σ ⊨𝐂𝐒 𝐴\n\nProof : First construct injective maps 𝑓: 𝐂𝐒 ⟶ 𝐒 and 𝑔: 𝐒 ⟶ 𝐂𝐒 between the class of CS\n\nframes and S frames (definitions A.4.1, A.4.2), such that (i) for each CS frame 𝔉, 𝑓(𝔉) is an S frame (lemma A.1.0.1) and (ii) for any S frame 𝔉, 𝑔(𝔉) is a CS frame (lemma A.1.0.2).201\n\nThen, show both 𝑓 and 𝑔 to be truth preserving in the following sense (lemmas A.1.0.4, A.1.0.4):\n\nFor any CS frame 𝔉 = (𝑊, ≲), 𝑖 ∈ 𝑊, 𝜌, and 𝐴, 𝐵 ∈ 𝐹𝑜𝑟: (𝔉, 𝜌), 𝑖 ⊩ 𝐴 > 𝐵 iff (𝑓(𝔉), 𝜌), 𝑖 ⊩ 𝐴 > 𝐵\n\nFor any S frame 𝔉 = (𝑊, \\$), 𝑖 ∈ 𝑊, 𝜌, and 𝐴, 𝐵 ∈ 𝐹𝑜𝑟: (𝔉, 𝜌), 𝑖 ⊩ 𝐴 > 𝐵 iff (𝑔(𝔉), 𝜌), 𝑖 ⊩ 𝐴 > 𝐵\n\nSince the above also hold for the basic modal language, the result follows.\n\n200 For clarity of presentation I should redefine S models in terms of S frames and 𝜌 rather than [.]. Keeping track of that irrelevant, yet nontrivial difference would be an unnecessary distraction.\n\nFor the purposes of the following, let’s recall the (relevant) precise definition of the general notion of arbitrary unions and arbitrary intersections. Given a collection of sets 𝓢:\n\n𝑥 ∈ ⋃ 𝓢 ⟺ ∃𝐴 ∈ 𝓢, 𝑥 ∈ 𝐴. 𝑥 ∈ ⋂ 𝓢 ⟺ ∀𝐴 ∈ 𝓢, 𝑥 ∈ 𝐴.\n\nDefinition A.1.1: Define the following map: 𝑓: 𝐂𝐒 ⟶ 𝐒 as follows:\n\n𝑓(𝑊) = 𝑊.\n\n𝑓((𝑆𝑖, ≲𝑖)) ∶= \\$𝑖≲= {𝑆 ∈ ℘(𝑆𝑖): (∀𝑗, 𝑘 ∈ 𝑊)((𝑗 ∈ 𝑆 ∧ 𝑘 ∉ 𝑆) ⟶ 𝑗 <𝑖 𝑘)}\n\nIt may be helpful to think of elements of \\$𝑖≲ as downward ≲𝑖-closed sets. That is, 𝑆 ∈ \\$𝑖≲ iff\n\n(𝑗 ∈ 𝑆 ∧ 𝑘 ≲𝑖 𝑗) ⟶ 𝑘 ∈ 𝑆 for any ∀𝑗, 𝑘 ∈ 𝑊.\n\nDefinition A.1.2: Define the following map: 𝑔: 𝐒 ⟶ 𝐂𝐒 as follows:\n\n𝑔(𝑊) = 𝑊. 𝑔(\\$𝑖) ∶= (≲𝑖\\$, 𝑆𝑖\\$)\n\n- ≲𝑖\\$ = {(𝑗, 𝑘) ∈ 𝑊 × 𝑊: (∀𝑆 ∈ \\$𝑖)(𝑘 ∈ 𝑆 ⟶ 𝑗 ∈ 𝑆)}\n\n- 𝑆𝑖\\$= ⋃ \\$𝑖\n\nLemma A.1.0.1: For each CS frame 𝔉 = (𝑊, ≲), 𝑓(𝔉) = (𝑊, \\$≲) is an S frame.\n\nProof : Let (𝑊, ≲) be CS frame (as per definition 4.22). We want to show that each (𝑊, \\$≲)\n\nis an S frame, i.e. that \\$𝑖 satisfies nesting and weak centering, for each 𝑖 ∈ 𝑊.\n\nNesting: for all 𝑆, 𝑇 ∈ \\$𝑖≲ either 𝑆 ⊆ 𝑇 or 𝑆 ⊆ 𝑇. This follows from totality of ≲. For the trivial\n\ncase, suppose 𝑆, 𝑇 ∈ \\$𝑖≲, and let 𝑆 = ∅. Hence, 𝑆 ⊆ 𝑇. Now, suppose for contradiction that\n\nthere are nonempty sets 𝑆, 𝑇 ∈ \\$𝑖≲ such that neither 𝑆 ⊆ 𝑇 nor 𝑇 ⊆ 𝑆. First, suppose that it’s not the case that 𝑆 ⊆ 𝑇. So, there is some 𝑥 ∈ 𝑆𝑖 such that 𝑥 ∈ 𝑆 but 𝑥 ∉ 𝑇. Next, also assume that\n\nit’s not the case that 𝑇 ⊆ 𝑆. So, there is some 𝑧 ∈ 𝑆𝑖 such that 𝑧 ∈ 𝑇 but 𝑧 ∉ 𝑆. Hence, from 𝑥 ∈\n\n𝑆, 𝑧 ∉ 𝑆, and the definition of \\$𝑖≲ we infer 𝑥 <𝑖 𝑧, and from 𝑧 ∈ 𝑇, 𝑥 ∉ 𝑇, and the definition of\n\n\\$𝑖≲ and we also infer 𝑧 <𝑖 𝑥. But this is impossible, since 𝑥 <𝑖 𝑧 and 𝑧 <𝑖 𝑥 means (𝑧, 𝑥) ∉ ≲\n\nand (𝑥, 𝑧) ∉ ≲, by definition of <𝑖, which contradicts totality of ≲.\n\nWeak Centering: ∃𝑆 ∈ \\$𝑖≲(𝑆 ≠ ∅) and 𝑖 ∈ ⋂(\\$𝑖≲∖ ∅). First to show ∃𝑆 ∈ \\$𝑖≲(𝑆 ≠ ∅). It suffices\n\nto observe that, by definition, ≲𝑖 satisfies CS5: ∀𝑗, 𝑘 ∈ 𝑊((𝑗 ∈ 𝑆𝑖∧ 𝑘 ∉ 𝑆𝑖) ⟶ 𝑗 <𝑖 𝑘). So, 𝑆𝑖∈\n\nnonempty sphere 𝑆 ∈ \\$𝑖≲ such that 𝑖 ∉ 𝑆. Now 𝑆 ≠ ∅ implies that there is some 𝑗 ∈ 𝑊 such that 𝑗 ∈ 𝑆. Hence, in particular (𝑗 ∈ 𝑆 ∧ 𝑖 ∉ 𝑆) ⟶ 𝑗 <𝑖𝑖 is true, by \\$𝑖≲ membership. But since the\n\nantecedent is true by hypothesis, it follows that 𝑗 <𝑖 𝑖 for some 𝑗 ∈ 𝑊. But this is impossible,\n\nsince it contradicts CS3, i.e. the ≲𝑖-minimality of 𝑖. This completes the proof. □\n\nLemma A.1.0.2: For each S frame 𝔉 = (𝑊, \\$), 𝑔(𝔉) = (𝑊, ≲\\$) is a CS frame.\n\nProof : Let (𝑊, \\$) be an S frame (as per definition 2.17). We want to show that (𝑊, ≲\\$) is a\n\nCS frame. First, to show that each (𝑆𝑖\\$, ≲𝑖\\$) is a total preorder for each 𝑖 ∈ 𝑊, i.e. it satisfies CS1.\n\nTransitivity: ∀𝑥, 𝑦, 𝑧 ∈ 𝑊 ((𝑥 ≲𝑖\\$𝑦 ∧ 𝑦 ≲𝑖\\$𝑧) ⟶ 𝑥 ≲𝑖\\$ 𝑧). Suppose (𝑥, 𝑦) ∈ ≲𝑖\\$ and (𝑦, 𝑧) ∈ ≲𝑖\\$ for any 𝑥, 𝑦, 𝑧 ∈ 𝑆𝑖\\$. From the definition of ≲𝑖\\$, this implies 𝑦 ∈ 𝑆 ⟶ 𝑥 ∈ 𝑆 and 𝑧 ∈ 𝑆 ⟶ 𝑦 ∈ 𝑆 for all 𝑆 ∈ \\$𝑖. Hence, 𝑧 ∈ 𝑆 ⟶ 𝑥 ∈ 𝑆 for all 𝑆 ∈ \\$𝑖, by hypothetical syllogism, and (𝑥, 𝑧) ∈ ≲𝑖\\$\n\nby definition of ≲𝑖\\$.\n\nTotality: ∀𝑥, 𝑦 ∈ 𝑊 (𝑥 ≲𝑖\\$𝑦 ∨ 𝑦 ≲𝑖\\$ 𝑥). Suppose for contradiction that there are 𝑥, 𝑦 ∈ 𝑆𝑖\\$ such that (𝑥, 𝑦) ∉ ≲𝑖\\$ and (𝑥, 𝑦) ∉ ≲𝑖\\$. From the definition of ≲𝑖\\$, this implies that there exist spheres 𝑆, 𝑇 ∈ \\$𝑖 such that 𝑦 ∈ 𝑆 ∧ 𝑥 ∉ 𝑆 and 𝑥 ∈ 𝑇 ∧ 𝑦 ∉ 𝑇. Now, 𝑆 ⊆ 𝑇 is impossible, because 𝑦 ∈ 𝑆\n\nbut 𝑦 ∉ 𝑇. By nesting, the only other possibility is 𝑇 ⊊ 𝑆. Suppose 𝑇 ⊊ 𝑆, but that’s also impossible because 𝑥 ∈ 𝑇 but 𝑥 ∉ 𝑆.\n\nNext, to show that each ≲𝑖\\$ satisfies the remaining conditions CS2-CS5.\n\n(CS2) The world 𝑖 is self-accessible: 𝑖 ∈ 𝑆𝑖. To show 𝑖 ∈ 𝑆𝑖\\$. Since \\$𝑖 is weakly centered,\n\nthere is a sphere ∅ ≠ 𝑆 ∈ \\$𝑖, such that 𝑆 = ⋂\\$𝑖 and 𝑖 ∈ ⋂\\$𝑖 ⊆ ⋃ \\$𝑖 = 𝑆𝑖\\$, as required.\n\n(CS3) The element 𝑖 is ≲𝑖-minimal: ∀𝑗 ∈ 𝑊(𝑖 ≲𝑖 𝑗 ). Suppose for contradiction that there\n\nexists 𝑖 ≠ 𝑗 ∈ 𝑆𝑖\\$ such that (𝑖, 𝑗) ∉ ≲𝑖\\$. So, by definition of ≲𝑖\\$ there’s a 𝑇 ∈ \\$𝑖 such that 𝑗 ∈ 𝑇 ∧\n\n𝑖 ∉ 𝑇. This contradicts weak centering, which requires that 𝑖 is included in every nonempty sphere in \\$𝑖.\n\n(CS4) Inaccessible worlds are ≲𝑖\\$-maximal: ∀𝑗, 𝑘 ∈ 𝑊(𝑘 ∉ 𝑆𝑖\\$⟶ 𝑗 ≲𝑖\\$𝑘). It suffices to\n\nobserve that each inaccessible world 𝑘 ∉ ⋃ \\$𝑖 = 𝑆𝑖\\$ satisfies the above condition, by definition\n\nof ≲𝑖\\$. That is, (𝑗, 𝑘) ∈ ≲𝑖\\$ for each 𝑘 ∉ 𝑆𝑖\\$ and any 𝑗 ∈ 𝑊, since (∀𝑆 ∈ \\$𝑖)(𝑘 ∈ 𝑆 ⟶ 𝑗 ∈ 𝑆) is\n\nsatisfied vacuously for all inaccessible worlds (worlds outside of ⋃ \\$𝑖), i.e. the antecedent is\n\n(CS5) Accessible worlds are more similar to 𝑖 than inaccessible worlds: ∀𝑗, 𝑘 ∈ 𝑊 ((𝑗 ∈ 𝑆𝑖\\$∧ 𝑘 ∉ 𝑆𝑖\\$) ⟶ 𝑗 <𝑖\\$ 𝑘)\n\nSuppose for contradiction that 𝑘 ∉ 𝑆𝑖\\$ but (𝑘, 𝑗) ∈ ≲𝑖\\$ for some 𝑗 ∈ 𝑆𝑖\\$. This means that 𝑘 ∉ 𝑆 for all 𝑆 ∈ \\$𝑖 and that there is some 𝑇 ∈ \\$𝑖 such that 𝑗 ∈ 𝑇. Now, (𝑘, 𝑗) ∈ ≲𝑖\\$ implies 𝑗 ∈ 𝑆 ⟶\n\n𝑘 ∈ 𝑆 for all 𝑆 ∈ \\$𝑖, by definition of ≲𝑖\\$. In particular 𝑗 ∈ 𝑇 ⟶ 𝑘 ∈ 𝑇, implying 𝑘 ∈ 𝑇, which is\n\nimpossible.\n\nThis completes the proof. □\n\nLemma A.1.0.3: Functions 𝑓 and 𝑔, as given in definitions A.1.1 and A.1.2, are injections.\n\nProof : It is immediate from definitions.\n\nThere’s a pattern to all the proof directions in lemmas A.1.0.4 and A.1.0.5.\n\nRegarding the non-trivial cases, we’re dealing with two kinds of conditions (ii) and (II), that vary slightly, but have the same general form. Hence the proofs follow a similar pattern. Below I give formulations that aims to emphasize the similarity of the forms of the non- vacuous conditions.\n\nOrdering frames (**): (∃𝑥) (𝑃(𝑥) ∧ (∀𝑦)(𝑅(𝑦, 𝑥) ⟶ 𝑄(𝑦))) Similarity spheres (##): (∃𝑋) (𝑃′(𝑋) ∧ (∀𝑦)(𝑅′(𝑦, 𝑋) ⟶ 𝑄(𝑦))) All the proofs (with some variation) follow a general pattern:\n\n(**) ⟶ (##):\n\n(1) First, I show that (∃𝑥)𝑃(𝑥) gives (∃𝑋)𝑃′(𝑋). (2) Next, I show that (∀𝑦)(𝑅′(𝑦, 𝑋) ⟶ 𝑅(𝑦, 𝑥)).\n\nSteps (1) and (2) generally require the most work, and I use various methods.\n\n(3) I use (2) in conjunction with the second conjunct of the hypothesis (∀𝑦)(𝑅(𝑦, 𝑥) ⟶ 𝑄(𝑦)) to show that (∀𝑦)𝑄(𝑦), by hypothetical syllogism, thus proving (∀𝑦)(𝑅′(𝑦, 𝑋) ⟶ 𝑄(𝑦)), by conditional proof.\n\nFor the (##) ⟶ (**) direction, I use the same proof pattern as for (**) ⟶ (##).\n\nLemma A.1.0.4: For any CS frame 𝔉 = (𝑊, ≲), 𝑖 ∈ 𝑊, 𝜌, and 𝐴, 𝐵 ∈ 𝐹𝑜𝑟:\n\nProof : Suppose ((𝑊, ≲), 𝜌), 𝑖 ⊩ 𝐴 > 𝐵 for arbitrary (𝑊, ≲) ∈ 𝐂𝐒, 𝑖 ∈ 𝑊, 𝜌, and 𝐴, 𝐵 ∈ 𝐹𝑜𝑟 Then by definition, either\n\n(i) 𝑆𝑖∩ [𝐴] = ∅, or\n\n(ii) (∃𝑘)(𝑘 ∈ 𝑆𝑖∩ [𝐴] ∧ (∀𝑗 ∈ 𝑊)(𝑗 ≲𝑖 𝑘 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵]))\n\nWe need to show that either (I) ⋃ \\$𝑖≲∩ [𝐴] = ∅, or\n\n(II) (∃𝑆 ∈ \\$𝑖≲)(𝑆 ∩ [𝐴] ≠ ∅ ∧ (∀𝑗 ∈ 𝑊)(𝑗 ∈ 𝑆 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵]))\n\nThat is, we need to show that (i or ii) iff (I or II). The entire argument applies for any 𝑖 ∈ 𝑊. (i ⟷ I) The vacuous case: ⋃ \\$𝑖≲∩ [𝐴] = ∅ iff 𝑆𝑖∩ [𝐴] = ∅, since ⋃ \\$𝑖≲⊆ 𝑆𝑖, by definition of\n\n\\$𝑖≲, and 𝑆𝑖∈ \\$𝑖≲, by definition of ≲𝑖 and \\$𝑖≲, i.e. CS5 implies 𝑆𝑖 ∈ \\$𝑖≲.\n\n(ii ⟶ II) Assume there is a 𝑘 ∈ 𝑆𝑖∩ [𝐴]. I’ll now define a subset of 𝐾 ⊆ 𝑊, such that 𝑘 ∈ 𝐾,\n\nand show that 𝐾 ∈ \\$𝑖≲. In other words, I’ll define a set 𝐾 ⊆ 𝑊 whose existence is implied by the existence of 𝑘. Let 𝐾 ∶= {𝑗 ∈ 𝑊: 𝑗 ≲𝑖 𝑘}. Observe that 𝐾 ⊆ 𝑆𝑖, since 𝑘 ∈ 𝑆𝑖 and 𝑗 ≲𝑖 𝑘\n\njointly imply 𝑗 ∈ 𝑆𝑖 for all 𝑗 ∈ 𝑊 (𝑆𝑖 is downward ≲𝑖-closed). Denying this would contradict\n\nCS5, i.e. suppose there’s some 𝑗 ∈ 𝑊 such that 𝑗 ≲𝑖 𝑘 yet 𝑗 ∉ 𝑆𝑖. But 𝑘 ∈ 𝑆𝑖 and 𝑗 ∉ 𝑆𝑖 implies\n\n𝑘 <𝑖𝑗, by CS5, contradicting 𝑗 ≲𝑖 𝑘. Now I’ll show that 𝐾 ∈ \\$𝑖≲. It suffices to note that 𝐾 ∈ \\$𝑖≲\n\nfollows from 𝐾 being downward ≲𝑖-closed, i.e. for any 𝑥, 𝑦 ∈ 𝑊, if 𝑥 ∈ 𝐾 and 𝑦 ≲𝑖 𝑥, then 𝑦 ∈\n\n𝐾, which can be easily checked as being equivalent to (𝑥 ∈ 𝐾 ∧ 𝑦 ∉ 𝐾) ⟶ 𝑥 <𝑖 𝑦 for any\n\n𝑥, 𝑦 ∈ 𝑊. Hence, 𝐾 ∈ \\$𝑖≲, by definition of \\$𝑖≲. Now I’ll show that 𝑗 ∈ 𝐾 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵] for all 𝑗 ∈ 𝑊. Suppose 𝑗 ∈ 𝐾 for arbitrary 𝑗 ∈ 𝑊. Hence, 𝑗 ≲𝑖𝑘, by construction of 𝐾. Next, from\n\nhypothesis 𝑗 ≲𝑖 𝑘 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵] for all 𝑗 ∈ 𝑊, we conclude 𝑗 ∈ [𝐴 ⊃ 𝐵]. Hence, 𝑗 ∈ 𝐾 ⟶ 𝑗 ∈\n\n[𝐴 ⊃ 𝐵] for all 𝑗 ∈ 𝑊, by conditional proof, as required.\n\n(II ⟶ ii) Assume that there is a sphere 𝑆 ∈ \\$𝑖≲ such that 𝑆 ∩ [𝐴] ≠ ∅, i.e. there’s some 𝑘 ∈ 𝑆 ∩\n\n[𝐴]. So, there is a 𝑘 ∈ 𝑆𝑖∩ [𝐴], since ⋃ \\$𝑖≲⊆ 𝑆𝑖, by definition of \\$𝑖≲. Now I’ll show that\n\n𝑗 ≲𝑖 𝑘 ⟶ 𝑗 ∈ 𝑆 for all 𝑗 ∈ 𝑊. To that end it suffices to note that 𝑥 ≲𝑖 𝑦 ⟶ (𝑦 ∈ 𝑇 ⟶ 𝑥 ∈ 𝑇) for\n\nany 𝑥, 𝑦 ∈ 𝑊 and 𝑇 ∈ \\$𝑖≲ is the contraposed condition for \\$𝑖≲ membership. Now, assume 𝑗 ≲𝑖 𝑘\n\nfor arbitrary 𝑗 ∈ 𝑊. Hence, 𝑘 ∈ 𝑇 ⟶ 𝑗 ∈ 𝑇 for any 𝑇 ∈ \\$𝑖≲ and 𝑗 ∈ 𝑊. In particular 𝑘 ∈ 𝑆 ⟶ 𝑗 ∈ 𝑆 for any 𝑗 ∈ 𝑊, by hypothesis 𝑆 ∈ \\$𝑖≲. Therefore 𝑗 ∈ 𝑆 for all 𝑗 ∈ 𝑊 that satisfy 𝑗 ≲𝑖 𝑘, by\n\nhypothesis 𝑘 ∈ 𝑆. Hence, 𝑗 ≲𝑖 𝑘 ⟶ 𝑗 ∈ 𝑆 for all 𝑗 ∈ 𝑊, by conditional proof. This, in\n\nfor all 𝑗 ∈ 𝑊, by hypothetical syllogism. Hence, finally 𝑗 ≲𝑖 𝑘 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵] for all 𝑗 ∈ 𝑊, by\n\nconditional proof, as required. □\n\nLemma A.1.0.5: For any S frame 𝔊 = (𝑊, \\$), 𝑖 ∈ 𝑊, 𝜌, and 𝐴, 𝐵 ∈ 𝐹𝑜𝑟: (𝔉, 𝜌), 𝑖 ⊩ 𝐴 > 𝐵 iff (𝑔(𝔉), 𝜌), 𝑖 ⊩ 𝐴 > 𝐵\n\nProof : Suppose ((𝑊, \\$), 𝜌), 𝑖 ⊩ 𝐴 > 𝐵 for arbitrary (𝑊, \\$) ∈ 𝐒, 𝑖 ∈ 𝑊, 𝜌, and 𝐴, 𝐵 ∈ 𝐹𝑜𝑟 Then by definition, either\n\n(i) ⋃ \\$𝑖∩ [𝐴] = ∅, or\n\n(ii) (∃𝑆 ∈ \\$𝑖)(𝑆 ∩ [𝐴] ≠ ∅ ∧ (∀𝑗 ∈ 𝑊)(𝑗 ∈ 𝑆 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵]))\n\nWe need to show that either (I) 𝑆𝑖\\$∩ [𝐴] = ∅, or\n\n(II) (∃𝑘) (𝑘 ∈ 𝑆𝑖\\$∩ [𝐴] ∧ (∀𝑗 ∈ 𝑊)(𝑗 ≲𝑖\\$𝑘 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵]))\n\nThat is, we need to show that (i or ii) iff (I or II). The entire argument applies for any 𝑖 ∈ 𝑊. (i ⟷ II) The vacuous case is immediate, since 𝑆𝑖\\$= ⋃ \\$𝑖 by definition of 𝑔.\n\n(ii ⟶ II) Assume that there is a 𝑆 ∈ \\$𝑖 such that 𝑆 ∩ [𝐴] ≠ ∅, i.e. there is a 𝑘 ∈ 𝑆 ∩ [𝐴]. So,\n\nthere’s a 𝑘 ∈ 𝑆𝑖\\$∩ [𝐴], since 𝑆 ⊆ 𝑆\n\n𝑖\\$= ⋃ \\$𝑖 by definition of 𝑔. Now I’ll show that 𝑗 ≲𝑖\\$𝑘 ⟶ 𝑗 ∈\n\n𝑆 for any 𝑗 ∈ 𝑊. Suppose 𝑗 ≲𝑖\\$𝑘 for arbitrary 𝑗 ∈ 𝑊. Hence, 𝑘 ∈ 𝑇 ⟶ 𝑗 ∈ 𝑇 for any and 𝑇 ∈ \\$𝑖\n\nand 𝑗 ∈ 𝑊, by definition of ≲𝑖\\$, so in particular 𝑘 ∈ 𝑆 ⟶ 𝑗 ∈ 𝑆 for any 𝑗 ∈ 𝑊, by hypothesis 𝑆 ∈ \\$𝑖. So, 𝑗 ∈ 𝑆 for all 𝑗 ∈ 𝑊 that satisfy 𝑗 ≲𝑖\\$𝑘, by hypothesis 𝑘 ∈ 𝑆. Hence, 𝑗 ≲𝑖\\$𝑘 ⟶ 𝑗 ∈\n\n𝑆 for all 𝑗 ∈ 𝑊, by conditional proof. This, in conjunction with main hypothesis (∀𝑗 ∈ 𝑊)(𝑗 ∈ 𝑆 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵]) in (ii) gives 𝑗 ∈ [𝐴 ⊃ 𝐵] for all 𝑗 ∈ 𝑊, by hypothetical syllogism. Hence 𝑗 ≲𝑖\\$𝑘 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵] for all 𝑗 ∈ 𝑊, by conditional proof, as required. (ii ⟵ II) Assume there is a 𝑘 ∈ 𝑆𝑖\\$∩ [𝐴], so ⋃ \\$𝑖∩ [𝐴] ≠ ∅, by definition of 𝑔, and there is a\n\nsphere 𝑆 ∈ \\$𝑖, such that 𝑘 ∈ 𝑆 ∩ [𝐴], by definition of set union. Now we need to show that if\n\n𝑗 ∈ 𝑆, then 𝑗 ≲𝑖\\$ 𝑘 for any 𝑗 ∈ 𝑊. By definition of 𝑗 ≲𝑖\\$𝑘, 𝑘 ∈ 𝑇 ⟶ 𝑗 ∈ 𝑇 for all 𝑇 ∈ \\$𝑖, so in\n\nparticular 𝑘 ∈ 𝑆 ⟶ 𝑗 ∈ 𝑆. Hence 𝑗 ∈ 𝑆. Hence 𝑗 ∈ 𝑆 ⟶ 𝑗 ≲𝑖\\$ 𝑘, for any 𝑗 ∈ 𝑊, by conditional proof. Finally, in conjunction with (∀𝑗 ∈ 𝑊)(𝑗 ≲𝑖\\$𝑘 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵]) from main hypothesis (II), we conclude that 𝑗 ∈ [𝐴 ⊃ 𝐵] for any 𝑗 ∈ 𝑊. Hence (∀𝑗 ∈ 𝑊)(𝑗 ∈ 𝑆 ⟶ 𝑗 ∈ [𝐴 ⊃ 𝐵]), by\n\nReferences\n\nBerto, F. (2010). Impossible worlds and propositions: Against the parity thesis. The j\n\nPhilosophical Quarterly, 40, 471–86.\n\nBerto, F. (2017). Impossible worlds and the logic of imagination. Erkenntnis, 82, 1277- j\n\n1297.\n\nBerto, F. (2014). On conceiving the inconsistent. Proceedings of the Aristotelian Society, j\n\n114, 103-21.\n\nBerto, F., & Jago, M. (2019). Impossible worlds. Manuscript submitted for publication.\n\nBjerring, J.C. (2014). On counterpossibles. Philosophical Studies, 168, 327-353.\n\nBlackburn, P., Rijke, M., & Venema, Y. (2001). Modal logic: Cambridge tracts in theoretical\n\nj computer science. Cambridge: Cambridge University Press.\n\nBlackburn P., van Benthem J., Wolter F. (2006). Handbook of modal Logic, Elsevier.\n\nBrogaard, B., & Salerno, J. (2013). Remarks on counterpossibles. Synthese, 190, 639-660.\n\nCarnap, R. (1947). Meaning and necessity. Chicago: Chicago University Press.\n\nChalmers, D. (2008). Hyperintensionality and impossible worlds: An introduction. J\n\nHyperintensionality and Impossible Worlds. Presented at ANU School of j\n\nPhilosophy, Canberra.\n\nChang, C.C., Keisler, H.J. (2013). Model theory: Third edition. Courier Corporation.\n\nCresswell, M. J. (1970). Classical intensional logics. Theoria, 36, 347–372.\n\nDivers, J. (2002). Possible worlds. London: Routledge.\n\nDunn, J. M. (1976). Intuitive semantics for first-degree entailment and “coupled trees”. J j j j j Philosophical Studies, 29, 149–168.\n\nField, H. (1989). Realism, mathematics and modality. Oxford: Oxford University Press.\n\nFine, K. (1975). Critical notice of Lewis, Counterfactuals. Mind, 8, 451 - 458.\n\nFitting, M. (2015). Intensional logic. The Stanford Encyclopedia of Philosophy. Retreived j j j\n\nj July 19, 2015, from<https://plato.stanford.edu/archives/sum2015/entries/logic-R\n\nintensional/>.\n\nFrege, G. (1884). Grundlagen der Arithmetik. Breslau: Wilhelm Koebner.\n\nGabbay, Dov M. (1972). A general theory of the conditional in terms of a ternary operator. H\n\nh Theoria, 38, 97-104.\n\nGirard, P., & Triplett M. A. (2018). Prioritised ceteris paribus logic for counterfactual ggg ggg reasoning. Synthese, 195, 1681-1703.\n\nGoodman, N. (1983). Fact, fiction, and forecast: Fourth edition. Cambrige, MA: Harvard g g\n\ng University Press.\n\nGoodman, N. (1972). Problems and projects. Indianapolis: Bobbs-Merrill.\n\nHájek, A. (2014). Most counterfactuals are false. Retrieved June 12, 2014, from h j j j j j j j j j j http://Philrsss.Anu.Edu.Au/People-Defaults/Alanh/Papers/Mcf.Pdf.\n\nHintikka, J. (1975). Impossible possible worlds vindicated. Journal of Philosophical Logic, 4,\n\nh 475 - 484.\n\nJago, M. (2009). Logical information and epistemic space. Synthese, 167, 327-341.\n\nJago, M., (2012). Constructing worlds. Synthese, 189, 59-74.\n\nJago, M. (2013). The content of deduction. Journal of Philosophical Logic, 42, 317-334.\n\nJech, T. (2004). Set theory: Third millennium edition, revised and expanded. Springer g ggg g Monographs in Mathematics.\n\nKiourti, I. (2010). Real impossible worlds: the bounds of possibility, (Doctoral dissertation). Retreived from\n\nKripke, S. (1980). Naming and necessity. Cambridge, MA: Harvard University Press.\n\nLewis, D. (1968). Counterpart theory and quantified modal logic. Journal of Philosophy, hh\n\n65, 113–26.\n\nLewis, D. (1970). Anselm and actuality. Noûs, 4, 175–88.\n\nLewis, D. (1971). Completeness and decidability of three logics of counterfactual gggg gg g g conditionals. Theoria, 37, 74-85.\n\nLewis, D. (1973). Counterfactuals. Oxford: Blackwell.\n\nLewis, D. (1979). Counterfactual dependence and time’s arrow. Noûs, 13, 455-476.\n\nLewis, D. (1980). A subjectivist's guide to objective chance. In Richard C. Jeffrey g g g g g g g (Ed.), Studies in Inductive Logic and Probability (pp.83-132). University of California\n\nLewis, D. (1981). Ordering semantics and premise semantics for counterfactuals. Journal of g\n\nIn document Context-indexed counterfactuals and non-vacuous counterpossibles (Page 165-184)" ]

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[ "", null, "Hey there! Ever wondered how many dimes it takes to make a dollar? Well, you’re in the right place! In this article, I’ll dive into the fascinating world of dimes and dollars and break down the math for you. Whether you’re a curious learner or just want to impress your friends with your newfound knowledge, stick around because we’re about to uncover the secret behind this common currency conversion.\n\n## How Many Dimes Make A Dollar\n\n### Definition of a Dime\n\nLet’s begin by understanding what a dime is. A dime is a ten-cent coin that is widely used in the United States. It is the smallest and thinnest coin in circulation, but its value should not be underestimated. With its distinctive silver appearance, the dime holds a significant position in the world of currency.\n\n### History of the Dime\n\nThe dime has a fascinating history that dates back to the late 18th century. It was first introduced in 1796, making it one of the oldest denominations of coins in the United States. Over the years, the design of the dime has evolved, showcasing different American icons and symbols. From the Mercury dime to the Roosevelt dime, each iteration has held a unique place in the hearts of collectors and citizens alike.\n\nNow that we have explored the definition and history of the dime, let’s dive into the math behind how many dimes make a dollar.\n\n## What is a Dollar?\n\n### Definition of a dollar\n\nA dollar is a unit of currency that is widely used in the United States. It is represented by the symbol “\\$” and is the standard monetary unit for commerce and trade in the country. A dollar consists of 100 smaller units called cents, and it is the most commonly used form of payment in everyday transactions.\n\nThe word “dollar” originated from the German word “thaler,” which was a large silver coin used in Europe during the 16th century. The term eventually made its way to the American colonies, where the Continental Congress adopted the dollar as the official currency in 1785.\n\n### History of the dollar\n\nThe history of the dollar in the United States is quite fascinating. It dates back to the late 18th century when the country was striving to establish its own form of currency. Initially, various foreign coins, such as Spanish dollars and British pounds, were used as a means of exchange.\n\nIn 1792, the United States Congress passed the Coinage Act, which established the U.S. Mint and authorized the production of the first official U.S. coins. One of these coins was the silver dollar, which was equivalent to 100 cents. Over time, the dollar evolved and went through several modifications in terms of design and composition.\n\nToday, the dollar is issued in both coin and banknote form. The coin, commonly known as the “dollar coin,” is larger and heavier than the dime but has a higher face value. On the other hand, the dollar bill, or simply the “dollar,” is a paper currency widely circulated and accepted in daily transactions.\n\nThe dollar holds significant value not only within the United States but also globally. It is recognized as one of the world’s reserve currencies and is widely used in international trade and finance.\n\nNext, let’s delve into the mathematics of how many dimes make a dollar.\n\n## Conversion rate of dimes to dollars\n\nNow that we have explored the fascinating history and significance of both the dime and the dollar, it’s time to delve into the mathematics of how many dimes make a dollar.\n\nAs we know, a dime is worth 10 cents, while a dollar consists of 100 cents. Therefore, to determine how many dimes make a dollar, we simply divide the value of a dollar by the value of a dime.\n\nMathematically, this can be represented as:\n\nNumber of dimes = Value of a dollar / Value of a dime\n\nSubstituting the values, we get:\n\nNumber of dimes = 100 cents / 10 cents\n\nSimplifying the equation, we find that:\n\nNumber of dimes = 10 dimes\n\nSo, the conversion rate of dimes to dollars is 10 dimes to 1 dollar.\n\nKnowing this conversion rate can be useful when counting or exchanging coins, especially if you have a large number of dimes and want to determine their equivalent value in dollars.\n\nUnderstanding the conversion rate of dimes to dollars allows us to easily calculate the value of dimes in terms of dollars, providing a useful tool for everyday transactions and financial planning." ]

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[ "# Chemistry\n\nIf the methane contained in 3.50L of a saturated solution at 25 ∘C was extracted and placed under STP conditions, what volume would it occupy?\n\nSolubility of methane is 1.3 millimolar\n\n1. 👍\n2. 👎\n3. 👁\n4. ℹ️\n5. 🚩\n1. M = mols/L\n0.0013 M = mols/3.50L\nmols = 0.0013 mols/L x 3.50L = estimated 0.005 mols but you need to do that more accurately.\nThen substitute for n in PV = nRT with other conditions and solve for volume.\n\n1. 👍\n2. 👎\n3. ℹ️\n4. 🚩\n\n## Similar Questions\n\n1. ### chemistry\n\nA saturated solution of potassium chlorate is formed from 100 grams of water. if the saturated solution is cooled from 80 degrees celsius to 50 degrees celsius, how many grams of precipate are formed? Do you have a table that\n\n2. ### chemistry\n\nA KNO3 solution is made using 88.4g of KNO3 and diluting to a total soultion volume of 1.50l. Calculate the molarity and mass percent of the solution. Assume a density of 1.05 g/mL for the solution.\n\n3. ### Chemistry\n\nA nitric acid solution is found to have a pH of 2.70. Determine [H3O+],[OH-], and the number of moles of HNO3 required to make 5.50L of the solution.\n\n4. ### Chemistry\n\n1.33dm³ of water at 70°c are saturated by 2.25moles of Pb(NO3)2, and 1.33dm³ of water at 18°c are saturated by 0.53mole of the same salt. If 4.50dm³ of the saturated solution are cooled from 70°c to 18°c.calculate the\n\n1. ### soran university\n\n1. Which salt is LEAST soluble at 0 ºC? 2. How many grams of sodium nitrate, NaNO3, are soluble in 100 g of water at 10 ºC? 3. When 50 grams of potassium chloride, KCl, is dissolved in 100 grams of water at 50 ºC, the solution\n\n2. ### Chemistry\n\nI'm trying to find the solubility of Ca(OH)2 in .0125 mol/L aqueous NaOH. Here is my data: Vol of solution used per titration: 20mL Conc of HCl: .1201 mol/L Average Titre: 4.71 I need to calculte: 1. the TOTAL [OH-] in the\n\n3. ### chemistry\n\nHow do I make a 6M NaOH solution? (25mL) and is there a certain way to make NaCl saturated? or is it already saturated?\n\n4. ### Chemistry\n\nMethane has a heat of combustion of about 50 kJ/g. About how much heat would be produced if 12 moles of methane were burned? Methane has a molecular formula of CH4.\n\n1. ### chemistry\n\nThe titration of 5.40 mL of a saturated solution of sodium oxalate, Na2C2O4, at 25∘C requires 29.4 mL of 2.200×10−2 M KMnO4 in acidic solution. What mass of Na2C2O4 in grams would be present in 1.00L of this saturated\n\n2. ### Chemistry\n\n1. Imagine you have a saturated solution of Ca(OH)2, at equilibrium, that has some undissolved Ca(OH)2 precipitated on the bottom of the test tube. If you add an additional scoop of Ca(OH)2 to the solution will concentration of\n\n3. ### Chemistry\n\nThe solubility of potassium chlorate at 20'c is 7.1g per 100g.calculate 1.The mass which dissolved in 240g of water to make a saturated solution 2.Mass of potassium chlorate in 20g of saturated solution at 20'c\n\n4. ### Chem\n\nSuppose solute A has a distribution coefficient of 1.0 between water and diethyl ether. Demonstrate that of 100mL of a solution of 5.0g of A in water were extracted with two 25-mL portions of ether, a smaller amount of A would" ]

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[ "# Chains of null sets\n\nNoam Elkies made a cute observation on his site, which is that there are chains of null subsets of [0, 1] whose union is not null. Proof: Take a maximal chain. The union of this can’t be null or it would contradict the maximality of the chain.\n\nHe then went on to ask how ‘large’ their union can be: Under the continuum hypothesis it can be all of [0, 1]. Do you need the continuum hypothesis for this? Without it how large can the union be?\n\nI tinkered around with applying some cardinal invariants to the problem and came up with a nearly complete solution. I’m going to be posting a pdf of it at some point, but here are the highlights:\n\nThe solution is for all intents and purposes purely combinatorial. I used essentially no measure theory in proving it except for one lemma which has a comparably easy category analogue.\n\nIt basically depends on three cardinal invariants. add, cov and non. The additivity is the smallest cardinal k such that there are k many null sets whose union is not null. cov is the smallest cardinality of a covering of [0, 1] by null sets. non is the smallest cardinality of a non-null set.\n\nWe have aleph_1 <= add <= non, cov <= 2^aleph_0. add is regular, the other two don't have to be, and no inequalities between non and cov are provable in ZFC. (I screwed up and claimed in my email to Elkies that in fact non <= cov and they were both regular. This is false, but it only broke a minor part of my conclusions). It’s relatively easy to see that if non = 2^aleph_0 or add = cov then you can get a chain whose union is [0, 1]. What’s slightly harder is that if such a chain exists then you can find some regular cardinal k (the cofinality of the chain) with cov <= k <= non. So, because you can have non < cov, you can't always find such a chain. So, in the absence of one with union [0, 1], how big can it be? Can it be measurable? (If it is measurable then of course it has measure > 0). Turns out not, and this is where the tiny bit of measure theory comes in. If A is measurable and m(A) > 0 then mu(A + Q) = 1 (addition is mod 1). So, if we have our chain L_t of null sets whose union is measurable and of positive measure, then replacing each L_t with L_t + Q gives a chain of null sets whose union is of full measure. Then adding in the complement we get it to be all of [0, 1].\n\nThe only question I have remaining which I’m not sure about is whether or not the existence of such a regular cardinal is equivalent to the existence of such a chain. I doubt it, but I’m not really sure and probably lack the knowledge of forcing needed to understand a proof either way.\n\nThis entry was posted in Numbers are hard on by ." ]

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[ "", null, "# Diode Q-Point\n\n## Our Take\n\nAs diodes are non-linear, it helps simplify things by figuring out approximately the voltage and current you're expecting to be passing through the diode, the Q-point. Once you have the Q-point, then you can assume that any variations around that point are linear, making your life easier. This can be found in a variety of ways but it needs to be understood that these Q-points are approximations around which we assume things are linear.\n\nIn summary, the way the Q-Point of a diode can be found is one of the following techniques:\n\n1. Load line analysis (need real data to do this and then basically graphing)\n2. Mathematical modeling (iterative techniques)\n3. Ideal Diode Model (assume 0V drop across a forward biased diode and no impedance, calculate the current)\n4. Constant Voltage Drop Model (assume typically .7V drop across a forward biased diode, calculate the current)\n\n## Book Definition\n\nThe designation Q-point is derived from the word quiescent, which means “still or unvarying.”\n\nElectronic Devices and Circuit Theory, 11th Edition by Robert L. Boylestad & Louis Nashelsky\n\nThe Q-point of a diode consists of the dc current and voltage (ID, VD) that define the point of operation on the diode’s i-v characteristic.\n\nMicroelectronic Circuit Design, 4th Edition by Richard C. Jaeger & Travis N. Blalock", null, "Get the latest tools and tutorials, fresh from the toaster.\n\n### What are you looking for?", null, "" ]

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[ "# JavaScript Program for Finding a Triplet from Three Linked Lists with a Sum Equal to a Given Number\n\nIn this article, we are going to implement a JavaScript program for finding a triplet from three linked lists with a sum equal to a given number. This problem is a kind of variation of the standard and famous three-sum problem but in a linked list manner. Let’s see the problem and implement its code along the key points of the problem.\n\n## Introduction to Problem\n\nThis problem is the variation of the standard problem of three sums where we are given three arrays and we have to find if there is any triplet present in the array with a sum exactly equal to the given number.\n\nHere in this problem, we are given three linked lists and a number and we have to pick exactly one element from all the linked lists and the sum of the picked number must be equal to the given number.\n\nIf the sum of the numbers that are picked is equal to the given number, then we have to return yes otherwise false.\n\nFor example\n\nIf the given number is 10 and the linked lists are as follows −\n\nList 1: 1 -> 2 -> 3 -> 4 -> 5\nList 2: 3 -> 7 -> 8 -> 13\nList 3: 9 -> 7 -> 2 -> 8\n\n\nFrom the above given linked list, we can pick the following sets which have a sum equal to 10.\n\nIf we pick 1 from the first list, seven from the second list, and 2 from the last list.\n\nAlso, we can pick 2 from the last list, 3 from the second, and 5 from the first list.\n\nSo, in this case, we will return true.\n\nNow imagine if the given number will be high like 25, then there is no set of three numbers present that will satisfy the condition.\n\nNote − We must have to pick exactly one number from all three given lists.\n\n## Approach\n\nWe have seen the example of the problem, now let’s move to the steps to be followed for the implementation of the code −\n\nBefore moving to the main approach here are some assumptions that we are going to make are −\n\nThe second linked list will be in the sorted way and increasing order while the third linked list will be in the sorted way but in descending order because we are going to use the two-pointer technique which will be only applicable if the above assumption is true.\n\nIf the above-made assumption is not true then we have another method which is to make the linked list second in sorted order by using the merge sort technique which will take a total of O(N*log(N))) time complexity where N is the size of the linked list.\n\nSimilar to the third linked list we can sort that and reverse that to make it a strictly decreasing linked list that will also take O(N*log(N)) time complexity for size N.\n\nMain approach −\n\n### Example\n\nFirst, we will use the while loop to traverse over the linked list first and at each step we will use the two pointers approach to get the sum of all the current elements equals to given number.\n\n// class for linked list node\nclass Node{\nconstructor(data){\nthis.value = data;\nthis.next = null;\n}\n}\n// function to find the triple or return false if the required number is not found\nfunction fun(lista,listb,listc,k){\n// creating temporary value of\nvar tempa = lista\n// Traverse all nodes of list A\nwhile (tempa != null) {\nvar tempb = listb\nvar tempc = listc\n// Using two pointer approach for the listb and listc\nwhile (tempb != null && tempc!=null) {\nvar current_sum = tempb.value + tempc.value + tempa.value;\nif (current_sum == k){\nconsole.log(\"Triplet found: \" + tempa.value + \" \" + tempb.value + \" \" + tempc.value);\nreturn true;\n}\n// If the current sum is smaller then look for a greater value of b\nelse if (current_sum < k)\ntempb = tempb.next;\nelse\ntempc = tempc.next;\n}\ntempa = tempa.next;\n}\nconsole.log(\"No Triplet found in the given lists\");\nreturn false;\n}\n// push function to create the linked list\nlet new_node = new Node(data);\n}\n// creating an unsorted linked list\n\n// create a sorted linked list b consisting of 4 10 15 20\n\n// create another sorted list in descending order 10 9 4 2\n\n//given number\nvar k = 25\n// calling the function\n\n\n### Time and Space Complexity\n\nThe time complexity of the above is O(N*N) where N is the size of the linked list because we are traversing over the first linked list that will cost use N iterations and at each iteration, we are applying a two-pointers approach that will cost us O(N).\n\nWe are not using any extra space which means the space complexity of the above code is O(1).\n\n## Conclusion\n\nIn this article, we have implemented a JavaScript program for finding a triplet from three linked lists with a sum equal to a given number. This problem is a kind of variation of the standard and famous three-sum problem but in a linked list manner. The time complexity of the above is O(N*N) where N is the size of the linked list and the space complexity is O(1).\n\nUpdated on: 24-Mar-2023\n\n71 Views", null, "" ]

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[ "Taylor series of arctan(x) (Spivak)\n\nAt p. 388 of Calculus, Spivak gives a formula:\n\n$$\\frac{1}{1+t^2} = 1 - t^2 + t^4 - ... + (-1)^nt^{2n} + \\frac{(-1)^{n+1}t^{2n+2}}{1+t^2}$$\n\nWhich can be integrated to find $\\arctan(x)$.\n\nI don't understand where this formula comes from, but I found it up to $(-1)^nt^{2n}$ by considering the geometric series for $\\frac{1}{1-x}$ and replacing $x$ by $-x^2$ to get the series for $\\frac{1}{1+x^2}$. I don't see the term $\\frac{(-1)^{n+1}x^{2n+2}}{1+x^2}$ though, because the series I got this way is $\\frac{1}{1+x^2} = \\sum_{n=0}^{\\infty}(-1)^nx^{2n}$.\n\n• In fact, Taylor series has a remainder, this remainder can be written in an integration form or big O form, and as in this case it is written in a fraction form . – Nizar Oct 5 '15 at 9:14\n\nConsider $$1+x+x^2+\\dots+x^n=\\frac{1-x^{n+1}}{1-x}=\\frac{1}{1-x}-\\frac{x^{n+1}}{1-x}$$ that can also be written $$\\frac{1}{1-x}=1+x+x^2+\\dots+x^n+\\frac{x^{n+1}}{1-x}$$ Now substitute $x=-t^2$, that gives $$\\frac{1}{1+t^2}=1+(-t^2)+(-t^2)^2+\\dots+(-t^2)^n+\\frac{(-t^2)^{n+1}}{1+t^2}$$ and not it's just a matter of observing that $(-t^2)^k=(-1)^kt^{2k}$.\n\n• This is a simple derivation since it builds from elementary series. – mavavilj Oct 5 '15 at 20:38\n\nThe term $(-x^2)^{n+1}/(1+x^2)$ is just the rest term:\n\n$${1\\over 1 - (-x^2)} = \\sum_0^\\infty (-x^2)^k = \\sum_0^n (-x^2)^k + \\sum_{n+1}^\\infty (-x^2)^k$$\n\nwhere\n\n$$\\sum_{n+1}^\\infty (-x^2)^k = (-x^2)^{n+1} \\sum_0^\\infty (-x^2)^k = {(-x^2)^k\\over1 - (-x^2)}$$\n\nAlternately you can do it the other way from the RHS and use the formula for geometric series (so you won't have to resort to infinite series):\n\n$$\\sum_0^n(-x^2)^k + {(-x^2)^{n+1}\\over 1 - (-x^2)} = {1 - (-x^2)^{n+1}\\over 1- (-x^2)} + {(-x^2)^{n+1}\\over 1 - (-x^2)} = {1\\over 1+x^2}$$\n\n• Nicely done. The equations would be easier to read in display mode; use \\$\\$ instead of \\$at the start and end of each large equation. – David K Oct 5 '15 at 13:57 • I believe technically one cannot claim$\\sum_0^\\infty (-x^2)^k = \\sum_0^n (-x^2)^k + \\sum_{n+1}^\\infty (-x^2)^k$without knowing first that$\\sum_0^\\infty (-x^2)^k$has a sum / converges. Since if it doesn't (e.g. diverges to$\\infty$) then$\\sum_0^n (-x^2)^k + \\sum_{n+1}^\\infty (-x^2)^k$does not convey anything, since the sums may be arbitrary (e.g.$\\infty$). – mavavilj Oct 5 '15 at 20:36 • @mavavilj That's correct, you have to make sure that the sum actually converges, which is true if$|x| < 1$. – skyking Oct 5 '15 at 21:13 This is a finite geometric sum: $$\\sum_{k=0}^n (-1)^k t^{2k} = \\sum_{k=0}^n (-t^2)^k = \\frac{1-(-1)^{n+1}t^{2n+2}}{1-(-t^2)}$$ • I know the geometric sum formula, but what about the other terms preceding$\\frac{(-1)^{n+1}t^{2n+2}}{1+t^2}$in Spivak's formula? – mavavilj Oct 5 '15 at 9:09 • @mavavilj What do you mean? The other terms are$\\sum_{k=0}^n (-t^2)^k\\$. Just rearrange the above equality to get exactly what you have written. – mrf Oct 5 '15 at 9:31" ]

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[ "### We found 201 resources with the concept dividing fractions\n\nLesson Planet\n\n#### Dividing Fractions to Twelfths (F)\n\nFor Students 5th\nIn this fraction division activity, 5th graders enhance their division skills by dividing the fractions to twelfths. Students solve 10 division problems.\nLesson Planet\n\n#### Dividing Fractions to Ninths\n\nFor Students 5th\nIn this fraction division worksheet, 5th graders enhance their division skills by solving the 10 fraction division problems that range by fractions to the ninths.\nLesson Planet\n\n#### Finding Parts with Division\n\nFor Students 4th - 5th\nFor this fraction division worksheet, students learn to find parts to complete division problems. Students write division sentences to help them find the parts.\nLesson Planet\n\n#### Pizza Match: Enrichment\n\nFor Students 6th - 7th\nIn this fraction division worksheet, students complete a pizza matching game using a deck of cards that represent fractions. Students must match up the cards that represent fractions to equal a whole without the other team knowing....\nLesson Planet\n\n#### Estimate Fraction Products and Quotients: Problem Solving\n\nFor Students 6th - 7th\nIn this estimating fractions worksheet, students use the chart for daily feeding estimates for dogs to answer the five word problems. Students estimate by multiplying and dividing the fractions.\nLesson Planet\n\n#### Accelerated Arithmetic; Dividing Fractions\n\nFor Teachers 5th - 6th\nIn this math worksheet, young scholars divide fractions. Fractions are single digit fractions, both the numerator and denominator are single digits. Students use the method they have learned in class to perform the computation.\nLesson Planet\n\n#### Math Is Fun Worksheet\n\nFor Students 7th - 10th\nIn this dividing fractions activity, students problem solve and calculate the answers to fifteen equations. Students share their answers with their classmates.\nLesson Planet\n\n#### Math Is Fun Worksheet\n\nFor Students 7th - 10th\nIn this dividing fractions worksheet, students problem solve and calculate the answers to fifteen equations. Students check their answers in class.\nLesson Planet\n\n#### Math Is Fun Worksheet\n\nFor Students 7th - 10th\nIn this dividing fractions worksheet, students problem solve and calculate the answers to fifteen equations. Students simplify their answers where needed." ]

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[ "# New to QlikView\n\nDiscussion board where members can get started with QlikView.\n\nAnnouncements\nThe #1 reason QlikView customers adopt Qlik Sense is a desire for a modern BI experience. Read More\ncancel\nShowing results for\nDid you mean:\nHighlighted", null, "Creator III\n\n## Expression is not populating for all project numbers\n\nHello,\n\nI have a table as shown below", null, "Here the POC is correct which is 74%, but POC Expression is not populating for all rows (as shown above) what is wrong in my expression here? Attached is file for reference.\n\n```=\nnum(\naggr((Sum({1<[SCH Task Discipline],[ORA DAY VERSION]={\">=\\$(vCurrentMonthStart)<=\\$(vCurrentMonthEnd)\"}>}[ORA Actual])\n/\nSum({1<[SCH Task Discipline],[ORA DAY VERSION]={\">=\\$(vCurrentMonthStart)<=\\$(vCurrentMonthEnd)\"}>}[ORA Projected])\n), [ORA Project Number])\n,'##0%')```\n\nthanks\n\nLabels (1)\n• ### Expression\n\n1 Solution\n\nAccepted Solutions\nHighlighted", null, "MVP\n\nTry this\n\n```=\nnum(\naggr(NODISTINCT(Sum({1<[SCH Task Discipline],[ORA DAY VERSION]={\">=\\$(vCurrentMonthStart)<=\\$(vCurrentMonthEnd)\"}>}[ORA Actual])\n/\nSum({1<[SCH Task Discipline],[ORA DAY VERSION]={\">=\\$(vCurrentMonthStart)<=\\$(vCurrentMonthEnd)\"}>}[ORA Projected])\n), [ORA Project Number])\n,'##0%')\n\n```\n2 Replies\nHighlighted", null, "MVP\n\nTry this\n\n```=\nnum(\naggr(NODISTINCT(Sum({1<[SCH Task Discipline],[ORA DAY VERSION]={\">=\\$(vCurrentMonthStart)<=\\$(vCurrentMonthEnd)\"}>}[ORA Actual])\n/\nSum({1<[SCH Task Discipline],[ORA DAY VERSION]={\">=\\$(vCurrentMonthStart)<=\\$(vCurrentMonthEnd)\"}>}[ORA Projected])\n), [ORA Project Number])\n,'##0%')\n\n```\nHighlighted", null, "Creator III\n\nthank you @sunny_talwar  for your valuable time and help", null, "", null, "" ]

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[ "This is what I came up with for reading a csv file with multiple types. It seems to get the job done in all cases but 1,2,,\"a\". Where there is a blank space. Can I please have some ideas on how to fix this?\n\n``````const char* getfield(char* line, int num)\n{\nconst char* tok;\nfor (tok = strtok(line, \",\");\ntok && *tok;\ntok = strtok(NULL, \",\\n\"))\n{\nif (!--num)\n}\nreturn NULL;\n}\n\nwhile(fgets(line, 80, inputfp1) != NULL)\n{\nprintf(\" line is %s \\n\", line);\n\nchar* tmp1 = strdup(line);\nchar* tmp2 = strdup(line);\nchar* tmp3 = strdup(line);\nchar* tmp4 = strdup(line);\nprintf(\"Field 1 would be %s\\n\", getfield(tmp1, 1));\nprintf(\"Field 2 would be %s\\n\", getfield(tmp2, 2));\nprintf(\"Field 3 would be %s\\n\", getfield(tmp3, 3));\nprintf(\"Field 4 would be %s\\n\", getfield(tmp4, 4));\n// NOTE strtok clobbers tmp\nfree(tmp1);\nfree(tmp2);\nfree(tmp3);\nfree(tmp4);\n\n//sscanf(line, \"%d, %d, %d, %d\", &column1[i], &column2[i], &column3[i], &column4[i]);\n//printf(\" column1[i] is %d column2[i] is %d column3[i] is %d column4[i] is %d \\n\", column1[i], column2[i], column3[i], column4[i]);\ni++;\nmemset(line, 0, 80);\n}``````\n\nYes, use a library. Writing robust CSV parsers is very difficult because of the difficulty of nicely dealing with \"bad\" data.\n\nCan you tell me what I did wrong here then?\n\nhttps://ideone.com/RylYp1\n\n``````#include <stdio.h>\n#include <stdlib.h>\n#include <string.h>\n\n/*\n* Given a string which might contain unescaped newlines, split it up into\n* lines which do not contain unescaped newlines, returned as a\n* NULL-terminated array of malloc'd strings.\n*/\nchar **split_on_unescaped_newlines(const char *txt) {\nconst char *ptr, *lineStart;\nchar **buf, **bptr;\nint fQuote, nLines;\n\n/* First pass: count how many lines we will need */\nfor ( nLines = 1, ptr = txt, fQuote = 0; *ptr; ptr++ ) {\nif ( fQuote ) {\nif ( *ptr == '\\\"' ) {\nif ( ptr == '\\\"' ) {\nptr++;\ncontinue;\n}\nfQuote = 0;\n}\n} else if ( *ptr == '\\\"' ) {\nfQuote = 1;\n} else if ( *ptr == '\\n' ) {\nnLines++;\n}\n}\n\nbuf = malloc( sizeof(char*) * (nLines+1) );\n\nif ( !buf ) {\nreturn NULL;\n}\n\n/* Second pass: populate results */\nlineStart = txt;\nfor ( bptr = buf, ptr = txt, fQuote = 0; ; ptr++ ) {\nif ( fQuote ) {\nif ( *ptr == '\\\"' ) {\nif ( ptr == '\\\"' ) {\nptr++;\ncontinue;\n}\nfQuote = 0;\ncontinue;\n} else if ( *ptr ) {\ncontinue;\n}\n}\n\nif ( *ptr == '\\\"' ) {\nfQuote = 1;\n} else if ( *ptr == '\\n' || !*ptr ) {\nsize_t len = ptr - lineStart;\n\nif ( len == 0 ) {\n*bptr = NULL;\nreturn buf;\n}\n\n*bptr = malloc( len + 1 );\n\nif ( !*bptr ) {\nfor ( bptr--; bptr >= buf; bptr-- ) {\nfree( *bptr );\n}\nfree( buf );\nreturn NULL;\n}\n\nmemcpy( *bptr, lineStart, len );\n(*bptr)[len] = '\\0';\n\nif ( *ptr ) {\nlineStart = ptr + 1;\nbptr++;\n} else {\nbptr = NULL;\nreturn buf;\n}\n}\n}\n}\n\nint main(void) {" ]

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[ "", null, "# Create a slice with make\n\nIn the Go programming language you can create arrays: collections.\n\nYou can create slices with the built-in make() function; you can create dynamically-sized arrays this way.\n\nRemember that the usual array has a fixed size, that you would define in one of these two ways:\n\nvar a int\nvar a = []int64{ 1,2,3,4 }\n\n\nWhat if you want an array that contains zeroes?\n\n## Make() in Go\n\nThe make function allocates a zeroed array and returns a slice that refers to that array. The syntax of the make() function is:\n\nyour_array := make([]type, length)\n\n\nSo if you'd have a program like this:\n\npackage main\n\nimport \"fmt\"\n\nfunc main() {\na := make([]int, 5)\nprintSlice(\"a\", a)\n\n}\n\nfunc printSlice(s string, x []int) {\nfmt.Printf(\"%s len=%d cap=%d %v\\n\",\ns, len(x), cap(x), x)\n}\n\n\nIt would output the array (contains a lot of zeros, the make() function does this):\n\na len=5 cap=5 [0 0 0 0 0]\n\n\nTo change its size, change the second parameter\n\na := make([]int, 50)\n\n\n### Discussion", null, "", null, "", null, "" ]

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[ "", null, "# ANKTRAIN - Editorial\n\n#1\n\n### PROBLEM LINK:\n\nPractice\n\nContest\n\nAuthor: Ankit Srivastava\n\nTester: Kevin Charles Atienza\n\nEditorialist: Vaibhav Tulsyan\n\nCAKEWALK\n\nNone\n\n### PROBLEM:\n\nGiven a pattern of arrangement of berths in a train, find the train partner of a given berth number. The pattern repeats for every 8 berths.\n\n### QUICK EXPLANATION:\n\nMaintain a map that stores neighbouring berth of the first 8 berths. For a given berth number N, find it’s neighbour M that lies in the same compartment, say C.\nIn order to do this, find the berth equivalent to N in the 1^{st} compartment and it’s neighbour M'. Add appropriate offset to find equivalent\nberth of M' in the compartment C. The berth number of the number is: N - N \\% 8 + M'.\n\n### EXPLANATION:\n\nSubtask 1:\n\nThe approach used for this subtask will be extended and used for subtask 2.\nFrom the constraints of Subtask 1, we know that 1 \\le N \\le 8. This means that we are dealing with only 1 compartment.\n\nLet’s store the neighbours of each berth - this can be hard-coded in the program, as there are only 8 berths.\n\n``` neighbours = { 0 -> \"4LB\", 1 -> \"5MB\", 2 -> \"6UB\", 3 -> \"1LB\", 4 -> \"2MB\", 5 -> \"3UB\", 6 -> \"8SU\", 7 -> \"7SL\" } ```\n\nFor a given value of N, we just need to fetch the value from the neighbours table for (N - 1) since our table has 0-indexed keys.\nThis can be implemented using a list/array/hashmap.\n\nSubtask 2:\n\nNote that the pattern repeats after every 8 berths - hence, group of berths [9…16] is identical to group [1…8],\nand [17…24] is also identical to [1…8].\nThus, all we have to do is find the equivalent neighbour (M') of N in the 1^{st} compartment and add an offset to this neighbour.\nLet’s say N was present in compartment C. The first berth of that compartment would have the number N - (N \\% 8).\nHence, the berth number of the neighbour would be: (N - (N \\% 8) + M').\n\nNote: Since we’re working with integers under a given Modulo, we are using 0-indexing for our neighbours map for simplicity.\n\n### AUTHOR’S AND TESTER’S SOLUTIONS:\n\nSetter’s solution can be found here.\n\nTester’s solution can be found here.\n\nEditorialist’s solution can be found here." ]

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[ "Scatterplots and other such graphs can help you visualize relationships between variables. Once you have visualized relationships, the next step is to analyze those relationships so that you can describe them numerically. That numerical description of the relationship between variables is called a model. Even more importantly, a model also predicts the average value of one variable (Y) from the value of another variable (X). The X variable is also called a predictor. Generally, this model is called a regression model.\nWith JMP, the Fit Y by X platform and the Fit Model platform creates regression models.\nRelationship Types shows the four primary types of relationships.\n • Using Regression with One Predictor\n • Using Regression with Multiple Predictors\n • Comparing Averages for One Variable\n • Comparing Averages for Multiple Variables\nThis example uses the Companies.jmp data table, which contains financial data for 32 companies from the pharmaceutical and computer industries.\n • Discovering the Relationship\n • Fitting the Regression Model\n • Predicting Average Sales\nScatterplot of Sales (\\$M) versus # Employ\nTo predict the sales revenue from the number of employees, fit a regression model. From the red triangle for Bivariate Fit, select Fit Line. A regression line is added to the scatterplot and reports are added to the report window.\nRegression Line\n • the p-value of <.0001\n • the RSquare value of 0.618\n • The p-value is less than the significance level of 0.05. Therefore, including the number of employees in the prediction model significantly improves the ability to predict average sales.\n • Since the RSquare value in this example is large, this confirms that a prediction model based on the number of employees can predict sales revenue. The RSquare value shows the strength of a relationship between variables, also called the correlation. A correlation of 0 indicates no relationship between the variables, and a correlation of 1 indicates a perfect linear relationship.\n 1 Click on the outlier.\n 2 Select Rows > Exclude/Unexclude.\n 3 Fit this model by selecting Fit Line from the red triangle menu for Bivariate Fit.\n • a new regression line\n • a new Linear Fit report, which includes:\n ‒ a new prediction equation\n ‒ a new RSquare value\nComparing the Models\nUsing the results in Comparing the Models, the data analyst can make the following conclusions:\n • The outlier was pulling down the regression line for the larger companies, and pulling the line up for the smaller companies.\n • The new model fits the data better, since the new RSquare value (0.88) is closer to 1 than the first RSquare value (0.618).\nThe prediction for the first model was \\$7499.68, so this model predicts a higher sales total by \\$1461.69.\nThis example uses the Companies.jmp data table, which contains financial data for 32 companies from the pharmaceutical and computer industries.\n • How do the profits of computer companies compare to the profits of pharmaceutical companies?\nTo answer this question, fit Profits (\\$M) by Type.\n 1 Select Help > Sample Data Library and open Companies.jmp.\n 2 If you still have the Companies.jmp sample data table open, you might have rows that are excluded or hidden. To return the rows to the default state (all rows included and none hidden), select Rows > Clear Row States.\n 3 Select Analyze > Fit Y by X.\n 4 Select Profits (\\$M) and click Y, Response.\n 5 Select Type and click X, Factor.\n 6 Click OK.\nProfits by Company Type\n 1 Click on the outlier.\n 2 Select Rows > Exclude/Unexclude. The data point is no longer included in calculations.\n 3 Select Rows > Hide/Unhide. The data point is hidden from all graphs.\n 4 To re-create the plot without the outlier, select Script > Redo Analysis from the red triangle menu for Oneway Analysis. You can close the original Scatterplot window.\nUpdated Plot\n 5 To continue analyzing the relationship, select these options from the red triangle menu for Oneway Analysis:\n ‒ Display Options > Mean Lines. This adds mean lines to the scatterplot.\n ‒ Means and Std Dev. This displays a report that provides averages and standard deviations.\nMean Lines and Report\n • Does a difference exist in the broader population, or is the difference of \\$635 million due to chance?\n • If there is a difference, what is it?\nTo perform the t-test, select Means/Anova/Pooled t from the red triangle for Oneway Analysis.\nt Test Results\nUse the confidence interval limits to determine how much difference exists in the profits of both types of companies. Look at the Upper CL Dif and Lower CL Dif values in t Test Results. The financial analyst concludes that the average profit of pharmaceutical companies is between \\$343 million and \\$926 million higher than the average profit of computer companies.\nIf you have categorical X and Y variables, you can compare the proportions of the levels within the Y variable to the levels within the X variable.\nThis example continues to use the Companies.jmp data table. In Comparing Averages for One Variable, a financial analyst determined that pharmaceutical companies have higher profits on average than do computer companies.\n 1 Select Help > Sample Data Library and open Companies.jmp.\n 2 If you still have the Companies.jmp data file open from the previous example, you might have rows that are excluded or hidden. To return the rows to the default state (all rows included and none hidden), select Rows > Clear Row States.\n 3 Select Analyze > Fit Y by X.\n 4 Select Size Co and click Y, Response.\n 5 Select Type and click X, Factor.\n 6 Click OK.\nCompany Size by Company Type\nThe Contingency Table contains information that is not applicable for this example. From the red triangle menu for Contingency Table deselect Total % and Col % to remove that information. Updated Contingency Table shows the updated table.\nUpdated Contingency Table\nTo answer this question, use the p-value from the Pearson test in the Tests report. See Company Size by Company Type. Since the p-value of 0.011 is less than the significance level of 0.05, the financial analyst concludes the following:\n • The differences in the sample data are not due to chance alone.\n • The percentages differ in the broader population.\nThe section Comparing Averages for One Variable, compared averages across the levels of a categorical variable. To compare averages across the levels of two or more variables at once, use the Analysis of Variance technique (or ANOVA).\n • Type (pharmaceutical or computer)\n • Size (small, medium, big)\n 1 Select Help > Sample Data Library and open Companies.jmp.\n 2 Select Graph > Graph Builder. The Graph Builder window appears.\n 3 Click Profits (\\$M) and drag and drop it into the Y zone.\n 4 Click Size Co and drag and drop it into the X zone.\n 5 Click Type and drag and drop it into the Group X zone.\nGraph of Company Profits\n 6 Right-click on the outlier to select it, and then select Row Exclude. The point is removed, and the scale of the graph automatically updates.\n 7 Click on the Bar", null, "icon. Comparing mean profits is easier with bar charts than with points.\nGraph with Outlier Removed\n • if the differences are limited to this sample and due to chance\n • if the same patterns exist in the broader population\n 1 Return to the Companies.jmp sample data table that has the data point excluded. See Discovering the Relationship.\n 2 Select Analyze > Fit Model.\n 3 Select Profits (\\$M) and click Y.\n 4 Select both Type and Size Co.\n 5 Click the Macros button and select Full Factorial.\n 6 From the Emphasis menu, select Effect Screening.\n 7 Select the Keep dialog open option.\nCompleted Fit Model Window\n 8 Click Run. The report window shows the model results.\nNote: For complete details about all of the Fit Model results, see the ­Fitting Linear Models book.\nThe Effect Tests report (see Effect Tests Report) shows the results of the statistical tests. There is a test for each of the effects included in the model on the Fit Model window: Type, Size Co, and Type*Size Co.\nEffect Tests Report\nFirst, look at the test for the interaction in the model: the Type*Size Co effect. Graph with Outlier Removed showed that the pharmaceutical companies appeared to have different profits between company sizes. However, the effect test indicates that there is no interaction between type and size as it relates to profit. The p-value of 0.218 is large (greater than the significance level of 0.05). Therefore, remove that effect from the model, and re-run the model.\n 1 Return to the Fit Model window.\n 2 In the Construct Model Effects box, select the Type*Size Co effect and click Remove.\n 3 Click Run.\nUpdated Effect Tests Report\n • There is a real difference in profits between computer and pharmaceutical companies in the broader population.\n • There is no correlation between the company’s size and type and its profits.\nThe section Using Regression with One Predictor showed you how to build simple regression models consisting of one predictor variable and one response variable. Multiple regression predicts the average response variable using two or more predictor variables.\nThis example uses the Candy Bars.jmp data table, which contains nutrition information for candy bars.\n • Total fat\n • Carbohydrates\n • Protein\nUse multiple regression to predict the average response variable using these three predictor variables.\n 1 Select Help > Sample Data Library and open Candy Bars.jmp.\n 2 Select Graph > Scatterplot Matrix.\n 3 Select Calories and click Y, Columns.\n 4 Select Total fat g, Carbohydrate g, and Protein g, and click X.\n 5 Click OK.\nScatterplot Matrix Results\nContinue to use the Candy Bars.jmp sample data table.\n 1 Select Analyze > Fit Model.\n 2 Select Calories and click Y.\n 3 Select Total Fat g, Carbohydrate g, and Protein g and click Add.\n 4 Next to Emphasis, select Effect Screening.\nFit Model Window\n 5 Click Run.\n • Using the Actual by Predicted Plot\n • Interpreting the Parameter Estimates\n • Using the Prediction Profiler\nActual by Predicted Plot\nAnother measure of model accuracy is the RSq value (which appears below the plot in Actual by Predicted Plot). The RSq value measures the percentage of variability in calories, as explained by the model. A value closer to 1 means a model is predicting well. In this example, the RSq value is 0.99.\n • The model coefficients\n • P-values for each parameter\nParameter Estimates Report\n • Fat = 11 g\n • Carbohydrate = 43 g\n • Protein = 2 g\nPrediction Profiler\nFactor Values for the Milky Way" ]

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[ "# E306 Report - parallel and series circuits\n\nSeptember 20, 2017 | Author: Abdul Rahman Mariscal | Category: Series And Parallel Circuits, Electrical Resistance And Conductance, Electrical Network, Electric Current, Voltage\n\n#### Short Description\n\nthis paper explains the logic behind the series and parallel circuits....\n\n#### Description\n\nABSTRACT The last experiment the team conducted was all about the two basic types of connecting electrical components: series and parallel circuits. The behavior of the three components of a simple circuit namely the voltage, current, and resistance were carefully analyzed and studied under the two types of electric circuits. An electric circuit is any arrangement of materials that permits electrons to flow. This study is very important in many areas in the industry as all modern machines make use of electricity. In this paper, we verified the relationship that exists between the three fundamental components of an electric circuit. By understanding the principle suggested by the Ohm’s Law, the team gathered numerical evidences by using suitable instruments to measure the quantities directly and compare it with the equation suggested by the Ohm’s law. The team established a model circuit to represent the two types of circuits. This paper limits itself only to the idea of the Ohm’s Law about the relationship between the quantities. Its degree of accuracy is high because the quantities gained are near to the true value. INTRODUCTION We live in a modern world where almost everything is run by electricity. Electricity is governed by the three fundamental components: resistance, current, and voltage. They are always part of simple circuits. A series circuit is one in which the current has only one path to takefrom one side of the source through the load, and back to the other side of the source. On the other hand, a parallel circuit is characterized by all the loads working at the same voltage and the source and independent of one another. The relationship that exists between current, resistance, and voltage is governed by the Ohm’s Law. Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points holding the temperature constant. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:\n\nwhere I is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the current. The law was named after the German physicist Georg Ohm, who described\n\nmeasurements of applied voltage and current through simple electrical circuits containing various lengths of wire. According to Ohm’s law, in series circuits, the current is constant among the loads whereas the total voltage is the summation of all the voltage in each load. The overall resistance is also the sum total of the individual resistance between the loads. According to Ohm’s law, in parallel circuits, the voltage is constant among the loads whereas the total current is the summation of all the current in each load. The reciprocal of the overall resistance is the sum total of the reciprocals of the individual resistance between the loads. The objectives of this experiment are the following: 1. To determine the total current flowing through a series and parallel circuit. 2. To determine the voltage across each resistors and the current flowing through a series and parallel circuit. 3. To investigate the relationship between voltages across each resistor and the total voltage. 4. To investigate the relationship between current flowing through each resistor and total current. MATERIALS AND METHODS Materials:  3 pc Resistance Boxes  5 pcs 1.2 Batteries\n\n  \n\n12 pcs Connecting wires 1 pc VOM 1 pc Ammeter\n\nFig-306-1 Materials Used Methods: A.Resistors in Series 1. Connect the 5 batteries as shown below. Fig-306-2 Primary Set-up 2. Using three resistors, build the circuit by connecting the wires. Fig-306-1 Connecting the wire to the resistor 3. Connect the VOM across the resistors one at a time to measure the voltages: Fig-306-3 Getting the voltage Note: In measuring voltage turn the selector knob so that it points to the desired range of voltage. 4. Connect the VOM to the circuit at the following points: A, B, C, D. Note: In measuring the current turn the knob selector knob so that it points to the desired range of current. Fig-306-4 Measuring the current by ammeter 5. Determine the equivalent resistance. 6. Compute the value of the total current flowing through the circuit and the current flowing through each resistor and voltages across each resistor using equivalent resistance and the measured voltage ( across the batteries. (Use Ohm’s Law and rules for series circuit. B.Resistors in Parallel 1. Connect the 5 batteries as shown below. Fig-306-5 Primary set-up for parallel 2. Using three resistors, build the circuit by connecting the wires as shown. 3. Connect the VOM across the resistors one at a time to measure the voltages: Fig-306-6 Getting the voltage for parallel Note: In measuring voltage turn the selector knob so that it points to the desired range of voltage.\n\n4. Connect the VOM to the circuit at the following points: A, B, C, D. Note: In measuring the current turn the knob selector knob so that it points to the desired range of current. Fig-306-7 Getting the current by ammeter 5. Determine the equivalent resistance. 6. Compute the value of the total current flowing through the circuit and the current flowing through each resistor and voltages across each resistor using equivalent resistance and the measured voltage ( across the batteries. (Use Ohm’s Law and rules for parallel circuit.\n\nRESULTS & GRAPHICAL ANALYSIS A. Resistors in Series Resistance 1 (R1), Ω\n\n100\n\nResistance 2 (R2), Ω\n\n93\n\nResistance 3 (R3), Ω\n\n57\n\nTotal Resistance (RT), Ω\n\n250\n\nTotal Voltage (VDA), V\n\n6.214\n\nTABLE 1: SERIES CIRCUIT Voltage Across Resistance 1 (VAB), V Voltage Across Resistance 2 (VBC), V Voltage Across Resistance 3 (VCD), V Current Flowing through Resistance 1, (iB), A Current Flowing through Resistance 2, (iC), A Current Flowing through Resistance 3, (iD), A Total Current, (iA) Percentage Difference\n\nExperimental\n\nComputed\n\n2.48\n\n2.486\n\n2.32\n\n2.312\n\n1.414\n\n1.417\n\n0.024\n\n0.025\n\n0.024\n\n0.025\n\n0.024\n\n0.025\n\n0.024\n\n0.025\n\n3.504\n\nIf we try to ponder on the results gathered from the experiment, we can see that the voltage from the first resistor is different from the second and of the thirst resistor. We can observe that the voltage is decreasing as we move from one resistor to the next. This implies that the total\n\nvoltage coming out from the source which is the battery is distributed unequally to its loads or resistors. On the other hand, we can see that the current flowing though each load is the same throughout. The team also found out that the total current is the same as the total current of the\n\nsystem.\n\nUsing\n\nthe\n\nequation\n\nthe\n\nload, we can see that the current flowing in each load is different meaning the total current of the system is distributed through the loads. If we try to compare the experimental values from that of the computed values from the equation, we can see an almost negligible difference between the values.\n\ncomputed values were computed and we can see that they are very near from the values gathered from the experiment.\n\nFor SERIES constant)\n\nB. Resistors in Parallel 84\n\nResistance 2 (R2), Ω\n\n27\n\nResistance 3 (R3), Ω\n\n153\n\nTotal Resistance (RT), Ω\n\n100 50 0\n\n5.47\n\n1.414\n\nTABLE 2: PARALLEL CIRCUIT\n\nTotal Current, (iA) Percentage Difference\n\n2.32\n\n2.48\n\nVoltage\n\nExperimental\n\nComputed\n\n5.46\n\n5.47\n\n5.46\n\n5.47\n\n5.47\n\n5.47\n\n0.06\n\n0.065\n\nThe graph above shows the relationship between voltage and resistance if we held the current constant. The voltage and the resistance are directly proportional. If the resistance is higher, the higher will be the voltage needed. For PARALLEL CONNECTION (voltage is constant)\n\n0.19\n\n0.203\n\nCurrent vs Resistance\n\n0.034\n\n0.036\n\n0.29\n\n0.303\n\n4.537\n\nOn the other side of the line, the results gathered from the parallel circuit makes an opposite as the behavior of the parameters in the series circuit. In here, we can see that the voltages in each of the loads or resistors are the same meaning it is constant. If we try to analyze the results gathered from the current flowing through each\n\n200\n\nResistance\n\nVoltage Across Resistance 1 (VAB), V Voltage Across Resistance 2 (VBC), V Voltage Across Resistance 3 (VCD), V Current Flowing through Resistance 1, (iB), A Current Flowing through Resistance 2, (iC), A Current Flowing through Resistance 3, (iD), A\n\nis\n\n150\n\n18.02524544\n\nTotal Voltage (VEA), V\n\n(current\n\nVoltage vs Resistance Resistance\n\nResistance 1 (R1), Ω\n\nCONNECTION\n\n150 100 50 0 0.06\n\n0.19\n\n0.034\n\nCurrent The graph above shows the relationship between current and resistance if we held the voltage constant. . The current and the resistance are\n\ninversely proportional. If the resistance is higher, the lower will be the current needed and vice versa. Thus, we can conclude that the behavior of the parameters current and voltage in series and parallel circuits are exactly opposite with one another. DISCUSSION & SOURCES OF ERRORS The results of the experiment tell us that resistance, current and voltage has a relationship depending on what type of circuit is it either parallel or series circuit.\n\nIn a series circuit it is found that the current here is constant and the voltage and resistance are directly proportional to each other. While in a parallel circuit it is found that voltage is constant and the resistance and current are inversely proportional to each other. The concept of series and parallel circuits shows us the basic electrical engineering concepts showing the difference of the series circuit from the parallel circuit and the relationship between the parameters involved in the Ohm’s Law. A series circuit is a circuit having a constant flow of current throughout the path but having a variable individual voltage that will depend on the number of the resistors present. We can say that the resistance is directly proportional to the voltage while maintaining the flow of current constant. We can also conclude that the resistance is a load when the amount of load increases the flow of current the voltage must also increase to keep the flow of electricity. It was found evident that the series circuit is not good for home used since the total current is equal to the individual. On the other side of the line, the parallel circuit is a circuit where the total voltage used is equal to the individual voltage used and the flow of current is varying depending on the values of the resistors. The total resistance is the reciprocal of the sum of the reciprocal of the individual resistance. In this particular circuit we can say that the resistance is indirectly proportional to the flow of current, as the resistance increases the individual current\n\ndecreases. Comparing the two types of basic circuits we can say that parallel circuit is more convenient and appropriate to use because it uses different current that is why when one of the current is zero ampere other current will not be affected resulting to a continuous flow of electricity. Sources of errors:  The temperature of the working environment may not be constant.  The flow of electricity inside the wire already consumed electricity.  The reading from the ammeter and VOM might be incorrect. To study the basics of electricity are very important to almost all field in the industry especially now that we are at a modern age. The idea of electric circuits make it possible for engineers and designers to build macro and micro engineering systems such as the installation of electric wiring in a building up to the smallest design of a microchip used in computers. ACKNOWLEDGMENT & REFERENCE To the following who had helped me finish this paper, I would like to thank: To God, for giving us the wisdom and to understand more of his fine creations especially the sound that makes life more beautiful. To my friend Bryan, for letting me use his internet connection for the online resources. To my groupmates, for their help and teamwork that made our experiment successful. To my blockmates Alex, Christian, Leif, Marc, and Airyl, for helping me find a source from the internet. To Prof. Ricardo F. De Leon Jr., for assisting and mentoring us during the laboratory hours. To the two laboratory assistants, Mang Gerry and Mang Jose, for being so friendly in lending us the tools and guiding us regarding its proper usage. To my previous Physics highschool teacher, Mrs. Gonzales, for teaching me a lot about electricity. Thank you very much. To God be the Glory! Young, Hugh D., et al. University Physics With Modern Physics. Jurong, Singapore\n\n6297733: Pearson Education South Asis Pte Ltd,,2008. http://en.wikipedia.org/wiki/Ohm's_law http://en.wikipedia.org/wiki/Series_and_parall el_circuits http://www.allaboutcircuits.com/vol_1/chpt_5 /1.html http://en.wikipedia.org/wiki/Ohm's_law" ]

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[ "Cookie Consent by FreePrivacyPolicy.com\nSearch a number\nBaseRepresentation\nbin1101100000100…\n…11100001010111\n321220020120021110\n412300103201113\n5213002414132\n615124550103\n72544153663\noct660234127\n9256216243\n10113326167\n1158a73699\n1231b52333\n131a62b306\n141109d8a3\n159e381cc\nhex6c13857\n\n113326167 has 8 divisors (see below), whose sum is σ = 152659696. Its totient is φ = 74771712.\n\nThe previous prime is 113326163. The next prime is 113326181. The reversal of 113326167 is 761623311.\n\nIt is a sphenic number, since it is the product of 3 distinct primes.\n\nIt is not a de Polignac number, because 113326167 - 22 = 113326163 is a prime.\n\nIt is a Duffinian number.\n\nIt is a congruent number.\n\nIt is not an unprimeable number, because it can be changed into a prime (113326163) by changing a digit.\n\nIt is a pernicious number, because its binary representation contains a prime number (13) of ones.\n\nIt is a polite number, since it can be written in 7 ways as a sum of consecutive naturals, for example, 194428 + ... + 195009.\n\nIt is an arithmetic number, because the mean of its divisors is an integer number (19082462).\n\nAlmost surely, 2113326167 is an apocalyptic number.\n\n113326167 is a deficient number, since it is larger than the sum of its proper divisors (39333529).\n\n113326167 is an equidigital number, since it uses as much as digits as its factorization.\n\n113326167 is an odious number, because the sum of its binary digits is odd.\n\nThe sum of its prime factors is 389537.\n\nThe product of its digits is 4536, while the sum is 30.\n\nThe square root of 113326167 is about 10645.4763632258. The cubic root of 113326167 is about 483.9235231373.\n\nAdding to 113326167 its reverse (761623311), we get a palindrome (874949478).\n\nThe spelling of 113326167 in words is \"one hundred thirteen million, three hundred twenty-six thousand, one hundred sixty-seven\"." ]

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[ "# Why Carnot engine is most efficient?\n\nContents\n\nThe Carnot cycle is reversible representing the upper limit on the efficiency of an engine cycle. … The Carnot cycle achieves maximum efficiency because all the heat is added to the working fluid at the maximum temperature.\n\n## Why Carnot cycle is the most efficient engine?\n\nEfficiency. The Carnot cycle is the most efficient engine possible based on the assumption of the absence of incidental wasteful processes such as friction, and the assumption of no conduction of heat between different parts of the engine at different temperatures.\n\n## How efficient is a Carnot engine?\n\nefficiency =WQH=1−TCTH. These temperatures are of course in degrees Kelvin, so for example the efficiency of a Carnot engine having a hot reservoir of boiling water and a cold reservoir ice cold water will be 1−(273/373)=0.27, just over a quarter of the heat energy is transformed into useful work.\n\n## How do you proof that Carnot engine is most efficient?\n\nIt seems that the only condition used in proving that the Carnot engine is the most efficient is that it is reversible. More specifically, the Carnot engine can be run in reverse as a refrigerator. Furthermore, it is asserted that all reversible engines will have the same efficiency.\n\n## Are Carnot engines 100% efficient?\n\nIn order to achieve 100% efficiency (η=1), Q2 must be equal to 0 which means that all the heat form the source is converted to work. Hence negative temperature of absolute scale is impossible and we cannot reach absolute 0 temperature. …\n\n## Is Carnot cycle the most efficient?\n\nThe Carnot cycle is a theoretical cycle that is the most efficient cyclical process possible. Any engine using the Carnot cycle, which uses only reversible processes (adiabatic and isothermal), is known as a Carnot engine. Any engine that uses the Carnot cycle enjoys the maximum theoretical efficiency.\n\n## Is Carnot engine 100 efficient why why not?\n\nCarnot’s principle states that it is impossible to design a heat engine whose only effect is to absorb heat from a high-temperature region and turn all that heat into work. In other words, it is highly unlikely to design a heat engine that is 100% efficient.\n\n## Which of the following Carnot engines is the most efficient?\n\nHence the Carnot engine operating between 300 K and 0 K has the highest efficiency of 1. This implies that the engine will convert all the heat energy provided to it into doing work.\n\n## What is the efficiency of Carnot cycle?\n\nIt is well known that the Carnot cycle efficiency ( η thermal = 1 − T L T H ) is maximized with the highest possible heat source temperature TH and the lowest possible heat sink temperature TL.\n\n## Is 0% efficiency possible?\n\nEfficiency of a machine is simply defined as the useful work out divided by the heat you had to put in. All you need to do to have a zero efficiency machine is make sure it does not useful work. Friction is an excellent zero-efficiency conversion mechanism.\n\nIMPORTANT:  Can you fit any engine in any car?\n\n## What is the most efficient thermodynamic cycle and why?\n\nClassical thermodynamics indicates that the most efficient thermodynamic cycle operating between two heat reservoirs is the Carnot engine , and a basic theorem expresses that any reversible cycle working between two constant temperature levels should have the same efficiency as a Carnot cycle .\n\n## Which of the following engine has 100% efficiency?\n\nEfficiency of Carnot engine is 100% if. η=1-T2T1 for 100% effeicency η=1 which gives T2=0K.\n\n## What is the maximum value of efficiency?\n\nExplanation: the maximum value of efficiency in a draft tube is 90 percent. it cannot exceed more than 90 percent because of the heat losses due to flow of fluid." ]

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[ "Dear Math Central experts, My name is Paula and I am a student. The questions i am asking is a senior advanced math question. if f(x)= 3x-1 and g(x)= 1/2x + 3 find fog(2) find the values of x for which tanx=0 Hi Paula, The function fog(x) operates in two steps. First you calculate y = g(x) and then you calculate f(y). For example to find fog(6) the steps are: y = g(6) = 1/2(6) + 3 = 3 + 3 = 6. f(y) = f(6) = 3(6) - 1 = 18 - 1 =17. Now try fog(2). For your second problem tan(x) = sin(x)/cos(x) so you can solve this problem if you can find the values of x for which sin(x) = 0 Penny Go to Math Central" ]

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[ "• Sheikh Aman\n\n# Top 7 regression techniques in machine learning you should know.\n\nUpdated: Aug 10\n\n### Topics you will learn.\n\n• What is the regression analysis?\n\n• Why we use regression analysis?\n\n• How many sorts of regression techniques can we have?\n\n1. Linear Regression\n\n2. Logistic Regression\n\n3. Polynomial Regression\n\n4. Stepwise Regression\n\n5. Ridge Regression\n\n6. Lasso Regression\n\n7. ElasticNet Regression\n\n### What is Regression Analysis?\n\nRegression analysis is a sort of predictive modelling technique which investigates the connection between a dependent (target) and experimental variable (s) (predictor). this system is employed for forecasting, statistic modelling and finding the causal effect relationship between the variables. Just like, the connection between rash driving and therefore the number of road accidents by a driver is best studied through regression.\n\nRegression analysis is a crucial tool for modelling and analyzing data. Here, we fit a curve/line to the info points, in such a fashion that the differences between the distances of knowledge points from the curve or line are minimized. I’ll explain this in additional details within the coming sections.\n\n### Why we use Regression Analysis?\n\nAs mentioned above, it estimates the connection between two or more variables. Let’s understand this with a simple example:\n\nLet’s say, you would like to estimate growth in sales of a corporation supported current economic conditions. you've got the recent company data which indicates that the expansion in sales is around two and a half times the expansion within the economy. Using this insight, we will predict future sales of the corporate supported current & past information.\n\nThere are multiple benefits of using multivariate analysis. they're as follows:\n\nIt indicates the many relationships between the variable and the experimental variable.\n\nIt indicates the strength of the impact of multiple independent variables on a variable.\n\nIt also helps us to match the consequences of variables measured on different scales, like the effect of price changes and a number of promotional activities. These benefits help market researchers/data analysts/data scientists to eliminate and evaluate the simplest set of variables to be used for building predictive models.\n\n### How many sorts of regression techniques can we have?\n\nThere are various sorts of regression techniques available to form predictions. These techniques are mostly driven by three metrics (number of independent variables, sort of dependent variables and shape of regression line). We’ll discuss them intimately within the following sections.\n\n### Regression types in machine learning.\n\nFor the creative ones, you'll even cook up new regressions, if you are feeling the necessity to use a mixture of the parameters above, which individuals haven’t used before. But before you begin that, allow us to understand the foremost commonly used regressions:\n\n## 1. Linear regression\n\nIt is one of the foremost widely known modelling technique. rectilinear regression is typically among the primary few topics which individuals pick while learning predictive modelling. during this technique, the variable is continuous, the independent variable(s) are often continuous or discrete, and therefore the nature of the regression curve is linear.\n\nLinear Regression establishes a relationship between the variable (Y) and one or more independent variables (X) employing the best fit line (also referred to as a regression line).\n\nIt is represented by an equation Y=a+b*X + e, where a is that the intercept, b is that the slope of the road and e is that the error term. This equation is often wont to predict the worth of the target variable supported a given predictor variable(s).\n\nHow to obtain best-fit line ( for Value of a and b)?\n\nThis task is often easily accomplished by the smallest amount of Square Method. it's the foremost common method used for fitting a regression curve. It calculates the best-fit line for the observed data by minimizing the sum of the squares of the vertical deviations from each datum to the road. Because the deviations are first squared, when added, there's no cancelling out between positive and negative values.\n\nImportant Points:\n\n• There must be a linear relationship between independent and dependent variables\n\n• Multiple regression suffers from multicollinearity, autocorrelation, heteroskedasticity.\n\n• Linear Regression is extremely sensitive to Outliers. It can terribly affect the regression curve and eventually the forecasted values.\n\n• Multicollinearity can increase the variance of the coefficient estimates and make the estimates very sensitive to minor changes within the model. The result's that the coefficient estimates are unstable\n\n• In the case of multiple independent variables, we will accompany forwarding selection, backward elimination and stepwise approach for selection of the most vital independent variable.\n\n## 2. Logistic Regression\n\nLogistic regression is employed to seek out the probability of event=Success and event=Failure. we should always use logistic regression when the variable is binary (0/ 1, True/ False, Yes/ No) in nature. Here the worth of Y ranges from 0 to 1 and it is often represented by the subsequent equation.\n\n```odd= p/ (1-p) = probability of event_occurrence / probability of not_event_occurrence\nln(odd) = ln(p/(1-p))\nlogit(p) = ln(p/(1-p)) = b0+b1X1+b2X2+b3X3....+bkXk```\n\nAbove, p is the probability of the presence of the characteristic of interest. an issue that you simply should ask here is “why have we used to log within the equation?”.\n\nSince we are working here with a Bernoulli distribution (dependent variable), we'd like to settle on a link function which is best fitted to this distribution. And, it's a logit function. within the equation above, the parameters are chosen to maximise the likelihood of observing the sample values instead of minimizing the sum of squared errors (like in ordinary regression).\n\nImportant Points:\n\n• Logistic regression is widely used for classification problems\n\n• Logistic regression doesn’t require a linear relationship between dependent and independent variables. It handles various sorts of relationships because it applies a non-linear log transformation to the anticipated odds ratio.\n\n• To avoid overfitting and underfitting, we should always include all significant variables. an honest approach to make sure this practice is to use a stepwise method to estimate the logistic regression\n\n• The independent variables shouldn't be correlated with one another i.e. no multicollinearity. However, we have the choices to incorporate interaction effects of categorical variables within the analysis and within the model.\n\n• If the values of the variable are ordinal, then it's called Ordinal logistic regression\n\n• If a variable is multi-class then it's referred to as Multinomial Logistic regression.\n\n## 3. Polynomial Regression\n\nA regression of y on x may be a polynomial regression of y on x if the facility of the experimental variable is quite 1. The equation below represents a polynomial equation:\n\n`y=a+b*x^2`\n\nIn this regression technique, the simplest fit line isn't a line. it's rather a curve that matches into the info points.\n\nImportant Points:\n\n• While there could be a temptation to suit a better degree polynomial to urge lower error, this will end in over-fitting. Always plot the relationships to ascertain the fit and specialise in ensuring that the curve fits the character of the matter. example, how plotting can help:\n\n• Especially look out for curve towards the ends and see whether those shapes and trends add up. Higher polynomials can find yourself producing weird results on extrapolation.\n\n## 4. Stepwise Regression\n\nThis form of regression is employed once we affect multiple independent variables. during this technique, the choice of independent variables is completed with the assistance of an automatic process, which involves no human intervention.\n\nThis feat is achieved by observing statistical values like AIC, T- stats and R-square metric to discern significant variables. Stepwise regression basically fits the regression model by adding/dropping co-variates one at a time supported a specified criterion. a number of the foremost commonly used Stepwise regression methods are listed below:\n\nStandard stepwise regression does two things. It adds and removes predictors as required for every step.\n\nForward selection starts with the most vital predictor within the model and adds variable for every step.\n\nBackward elimination starts with all predictors within the model and removes the smallest amount of significant variable for every step.\n\nThe aim of this modelling technique is to maximise the power of prediction with a minimum number of predictor variables. it's one among the tactic to handle higher dimensionality of knowledge set.\n\n## 5. Ridge Regression\n\nRidge Regression may be a technique used when the info suffers from multicollinearity (independent variables are highly correlated). In multicollinearity, albeit the smallest amount squares estimates (OLS) are unbiased, their variances are large which deviates the observed value faraway from truth value. By adding a degree of bias to the regression estimates, ridge regression reduces the quality errors.\n\nAbove, we saw the equation for rectilinear regression. Remember? It is often represented as:\n\n`Y=a+ b*X`\n\nThis equation also has a mistaken term. the entire equation becomes:\n\n```Y=a+b*Y+e (error term), [error term is that the value needed to correct for a prediction error between the observed and predicted value]\n=> Y=a+Y= a+ b_1 X 1+ b_2 X 2+....+e, for every multiple independent variables.```\n\nIn an equation, prediction errors are often decomposed into two sub-components. First is thanks to the biased and second is thanks to the variance. Prediction error can occur thanks to anybody of those two or both components. Here, we’ll discuss the error caused thanks to variance.\n\nRidge regression solves the multicollinearity problem through a shrinkage parameter called lambda (λ). check out the equation below.\n\nIn this equation, we've two components. First one is a least-square term and the other one is lambda of the summation of β2 (beta- square) where β is that the coefficient. this is often added to least-square term so as to shrink the parameter to possess a really low variance.\n\nImportant Points:\n\n• The assumptions of this regression are the same as least squared regression except normality isn't to be assumed\n\n• Ridge regression shrinks the worth of coefficients but doesn’t reach zero, which suggests no feature selection feature\n\n• This method uses L2 regularizations.\n\n## 6. Lasso Regression\n\nSimilar to Ridge Regression, Lasso (Least Absolute Shrinkage and Selection Operator) also penalizes absolute dimensions of the regression coefficients. additionally, it's capable of reducing the variability and improving the accuracy of rectilinear regression models. check out the equation below: Lasso regression differs from ridge regression during a way that it uses absolute values within the penalty function, rather than squares. This leads to penalizing values which cause a variety of the parameter estimates to point out exactly zero. Larger the penalty applied, further the estimates get shrunk towards temperature. This results in variable selection out of given n variables.\n\nImportant Points:\n\n• The assumptions of lasso regression are the same as least squared regression except normality isn't to be assumed\n\n• This regression shrinks coefficients to zero (exactly zero), which certainly helps in feature selection\n\n• Lasso may be a regularization method and uses L1 regularization\n\n• If a group of predictors are highly correlated, lasso picks just one of them and shrinks the others to zero\n\n## 7. ElasticNet Regression\n\nThis regression technique is a hybrid of Lasso and Ridge Regression techniques. it's trained with L1 and L2 prior as regularizer. Elastic-net is beneficial when there are multiple features which are correlated. Lasso is probably going to select one among these random, while elastic-net is probably going to select both.\n\nA practical advantage of trading-off between Lasso and Ridge regression is that it allows Elastic-Net to inherit a variety of Ridge’s stability under rotation.\n\nImportant Points:\n\n• It encourages group effect just in case of highly correlated variables\n\n• There are not any limitations on the number of selected variables\n\n• It can suffer from double shrinkage\n\n• Beyond these 7 most ordinarily used regression techniques, you'll also check out other models like Bayesian, Ecological and Robust regression.\n\nIf you love this then don't forget to give heart by clicking like and sharing.\n\nSee All" ]

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[ "# Properties\n\n Label 106b Number of curves 1 Conductor 106 CM no Rank 1\n\n# Related objects\n\nShow commands for: SageMath\nsage: E = EllipticCurve(\"106.a1\")\n\nsage: E.isogeny_class()\n\n## Elliptic curves in class 106b\n\nsage: E.isogeny_class().curves\n\nLMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality\n106.a1 106b1 [1, 1, 0, -7, 5] [] 8 $$\\Gamma_0(N)$$-optimal\n\n## Rank\n\nsage: E.rank()\n\nThe elliptic curve 106b1 has rank $$1$$.\n\n## Modular form106.2.a.a\n\nsage: E.q_eigenform(10)\n\n$$q - q^{2} - q^{3} + q^{4} - 4q^{5} + q^{6} - q^{8} - 2q^{9} + 4q^{10} - 4q^{11} - q^{12} + q^{13} + 4q^{15} + q^{16} + 5q^{17} + 2q^{18} - 7q^{19} + O(q^{20})$$" ]

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[ "In this document, we will play with linear regression and generalized linear models in sklearn. We will predict the price of a real estate based on a number of covariates. The dataset is available here.\n\nExploratory Analysis¶\n\nThe variable X1 transaction date seems to be in the form of yyyy.mmdd. Let us change it to just the year number and ignore the month and day. We will also treat transaction year as a factor, rather than numeric. Roughly 70% of the houses are sold in 2013, while the remaining are sold in 2012.\n\nWe can group by X1 transaction date and calculate some summary statistics of other variables. In particular, the most expensive house price of unit area occurs in year 2013. There is nothing else noriceable for other variables.\n\nWe also plot the histogram of house price by the transaction date. It seems that the house price is higher in 2013.\n\nThe next variable to consider is X2 house age. We plot its marginal and joint distribution with the response. Nothing particular stands out in this plot.\n\nSimilar as above, the same figures can be plotted for X3 distance to the nearest MRT station and X4 number of convenience stores. These two variables are negatively and positively, respectively, correlated with the house price.\n\nFinally, we will look at X5 latitude and X6 longitude together. It does not look good to directly plot the house price on a map, so we opt to bin the latitude and longitude. It seems that real estate prices are higher on the southeast region.\n\nSimple Linear Regression¶\n\nNow, we conduct a simple linear regression of Y house price of unit area on all other variables. In terms of preprocessing, we only need to convert X1 transaction date into dummy vectors. We will scale the continuous variables using the min_max_scaler.\n\nThe linear model only has an R2 of 0.5215 on the training set, and 0.5122 on the testing set, which is not ideal. The fitted coefficients are also shown below.\n\nTo visualize the fitted model, let us compare the fitted value with the actual house price.\n\nGamma GLM¶\n\nConsidering the house price can only be nonnegative, the linear model above is actually not adequate as it allows for negative values. A better model is the Gamma Generalized Linear Mode (GLM) with log link, which is also implemented in the sklearn package.\n\nThe score returned by the glm object is based on the null deviance of the model.\n\nAgain, we can make a similar visualization of the fitted values by GLM. Compared with the simple linear model, GLM seems to over-fit the tail." ]

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[ "# Joint Distributions\n\nPage Contents\n\n## Joint Distribution\n\n~X, ~Y random variables, their joint distribution\n\nF_{~X,~Y} ( ~x , ~y ) _ = _ P ( ~X =< ~x , ~Y =< ~y )\n\nGeneralizing: If _ ~#X = ( ~X_1 , ... , ~X_~n) , _ where ~X_~i are random variables,\n\nF_{~#X} ( ~#x ) _ = _ P ( ~X_~i =< ~x_~i ; ~i = 1, ... ~n )\n\n~#X is called a #~{random vector}. The component random variables can be a mix of discrete and continuous, though this is not usually dealt with. #~X is called a #~{continuous random vector} if all the components are continuous or a #~{discrete random vector} if all the components are discrete.\n\n### Joint Discrete Probability Function\n\nIf ~#X is a discrete random vector, the #~{joint probability function} _ ~p ( ~#x ) _ is defined\n\n~p ( ~#x ) _ _ #:= _ _ P ( ~X_1 = ~x_1 , ~X_2 = ~x_2 , ... , ~X_~n = ~x_~n ) , _ _ _ ~x_~i &in. &Omega._{~X_~i} , _ &forall. ~i\n\n### Joint Continuous Density Function\n\nIf _ _ _ F_{~#X} ( ~x_1 , ... , ~x_~n ) _ = _ int{ ... ,-&infty.,~x_1,}int{,-&infty.,~~x_n,} ~f ( ~u_1 , ... , ~u_~n ) d~u_~n ... d~u_1\n\nthen _ ~f ( ~u_1 , ... , ~u_~n ) is called the #~{joint density function} (with respect to integration).\n\n## Marginal Distributions\n\n~#X = ( ~X_1 , ... , ~X_~n ) is a random vector, the #~{marginal distribution} of ~X_~i is defined:\n\nF_{~X_~i} ( ~x_~i ) _ #:= _ F_{~#X} ( &infty. , &infty. , ... , ~x_~i , ... , &infty. )\n\n### Discrete Marginals\n\nFor a discrete random vector\n\nF_{~i} ( ~a ) _ = _ sum{ ... ,~x_1 = -&infty.,&infty.}sum{ ... ,~x_~i = -&infty.,~a}sum{,~x_~n = -&infty.,&infty.} rndb{ {P ( ~X_1 = ~x_1 , ~X_2 = ~x_2 , ... , ~X_~n = ~x_~n )} }\n\n### Continuous Marginals\n\nFor a continuous random vector the ~i^{th} marginal distribution is:\n\nF_{~i} ( ~a ) _ = _ int{ ... ,-&infty.,&infty.,}int{ ... ,~x_~i = -&infty.,~a,}int{,-&infty.,&infty.,} f ( ~u_1 , ... , ~u_~n ) _ d~u_~n ... d~u_1\n\nThe ~i^{th} marginal density is given by\n\nf_{~i} ( ~x ) _ = _ fract{dF_{~i},d ~x} ( ~x )\n\n## Independent Random Variables\n\nRandom variables ~X and ~Y are #~{independent} if\n\nF_{~X,~Y} ( ~x , ~y ) _ = _ F_~X ( ~x ) F_~Y ( ~y ) , _ _ &all. ~x , ~y\n\nIf ~X and ~Y are discrete, then\n\n~X and ~Y independent _ <=> _ P ( ~X = ~x , ~Y = ~y ) _ = _ P ( ~X = ~x ) P ( ~Y = ~y ) , _ &all. ~x , ~y\n\nIf ~X and ~Y are continuous, then\n\n~X and ~Y independent _ <=> _ f_{~X,~Y} ( ~x , ~y ) _ = _ f_~X ( ~x ) f_~Y ( ~y ) , _ _ &all. ~x , ~y\n\n_\n\nGeneralizing: random variables ~X_1 , ... , ~X_~n are #~{independent} if\n\nF_{~#X} ( ~x_1 , ... , ~x_~n ) _ = _ F_1 ( ~x_1 ) F_2 ( ~x_2 ) ... F_~n ( ~n ) , _ _ &all. ~x_1 , ... , ~x_~n\n\nThe resulst for the probability densities extend to the general case." ]

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[ "# [SOLVED]eigenvalue and eigenfunction for Fredholm method\n\n#### dwsmith\n\n##### Well-known member\nGiven\n$f(x) = \\lambda\\int_0^1xy^2f(y)dy$\nAt order $$\\lambda^2$$ and $$\\lambda^3$$, we have repeated zeros so\n$D(\\lambda) = 1 - \\frac{\\lambda}{4}.$\nThen we have\n$\\mathcal{D}(x, y;\\lambda) = xy^2$\nso\n$f(x) = \\frac{\\lambda}{D(\\lambda)}\\int_0^1\\mathcal{D}(x, y;\\lambda)dy.$\nHow do I get the eigenfunction and value from this method?" ]

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[ "index 56e5329..97b7423 100644 (file)\n@@ -1860,7 +1860,7 @@ static int index_stream_convert_blob(struct object_id *oid, int fd,\nstruct strbuf sbuf = STRBUF_INIT;\n\nassert(path);\n-       assert(would_convert_to_git_filter_fd(path));\n+       assert(would_convert_to_git_filter_fd(&the_index, path));\n\nconvert_to_git_filter_fd(&the_index, path, fd, &sbuf,\nget_conv_flags(flags));\n@@ -1950,7 +1950,7 @@ int index_fd(struct object_id *oid, int fd, struct stat *st,\n* Call xsize_t() only when needed to avoid potentially unnecessary\n* die() for large files.\n*/\n-       if (type == OBJ_BLOB && path && would_convert_to_git_filter_fd(path))\n+       if (type == OBJ_BLOB && path && would_convert_to_git_filter_fd(&the_index, path))\nret = index_stream_convert_blob(oid, fd, path, flags);\nelse if (!S_ISREG(st->st_mode))\nret = index_pipe(oid, fd, type, path, flags);" ]

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[ "## Saturday, January 16, 2016\n\n### How to Optimize Json.NET Serialization Performance", null, "Newtonsoft is a pretty fast JSON serializer, but you can make it even faster!\n\nBy default, JsonConvert uses reflection to recursively search through the structure of an object during the serialization process. By implementing a custom JsonConverter that already knows the exact structure of the object, you can significantly increase serialization performance.\n\nHow much faster? That depends! The more complicated the data structure, the larger the performance gain. Below is a simple example...\n\nAction Method Milliseconds Performance Increase\nSerialize Standard 1134 115.59%\nCustom 526\nDeserialize Standard 1488 62.98%\nCustom 913\n\n### Model and Converter\n\n`public class Model`\n`{`\n` public int Int { get; set; }`\n` `\n` public bool Bool { get; set; }`\n` `\n` public string String { get; set; }`\n`}`\n` `\n`public class ModelConverter : JsonConverter`\n`{`\n` public static readonly ModelConverter Instance = new ModelConverter();`\n` `\n` private static readonly Type ModelType = typeof(ICollection<Model>);`\n` `\n` public override bool CanConvert(Type objectType)`\n` {`\n` return ModelType.IsAssignableFrom(objectType);`\n` }`\n` `\n` public override void WriteJson(`\n` JsonWriter writer,`\n` object value,`\n` JsonSerializer serializer)`\n` {`\n` var collection = (ICollection<Model>)value;`\n` `\n` writer.WriteStartArray();`\n` `\n` foreach (var model in collection)`\n` {`\n` writer.WriteStartObject();`\n` `\n` writer.WritePropertyName(\"Int\");`\n` writer.WriteValue(model.Int);`\n` `\n` writer.WritePropertyName(\"Bool\");`\n` writer.WriteValue(model.Bool);`\n` `\n` writer.WritePropertyName(\"String\");`\n` writer.WriteValue(model.String);`\n` `\n` writer.WriteEndObject();`\n` }`\n` `\n` writer.WriteEndArray();`\n` }`\n` `\n` public override object ReadJson(`\n` JsonReader reader,`\n` Type objectType,`\n` object existingValue,`\n` JsonSerializer serializer)`\n` {`\n` var collection = new List<Model>();`\n` `\n` Model model = null;`\n` `\n` while (reader.Read())`\n` {`\n` switch (reader.TokenType)`\n` {`\n` case JsonToken.StartObject:`\n` model = new Model();`\n` collection.Add(model);`\n` break;`\n` `\n` case JsonToken.PropertyName:`\n` SetProperty(reader, model);`\n` break;`\n` `\n` case JsonToken.EndArray:`\n` return collection;`\n` }`\n` }`\n` `\n` return collection;`\n` }`\n` `\n` private static void SetProperty(JsonReader reader, Model model)`\n` {`\n` var name = (string)reader.Value;`\n` `\n` reader.Read();`\n` `\n` switch (name)`\n` {`\n` case \"Int\":`\n` model.Int = Convert.ToInt32(reader.Value);`\n` break;`\n` `\n` case \"Bool\":`\n` model.Bool = (bool)reader.Value;`\n` break;`\n` `\n` case \"String\":`\n` model.String = (string)reader.Value;`\n` break;`\n` }`\n` }`\n`}`\n\n### Unit Tests\n\n`public class JsonConverterPerformanceTests`\n`{`\n` private readonly ITestOutputHelper _output;`\n` `\n` public JsonConverterPerformanceTests(ITestOutputHelper output)`\n` {`\n` _output = output;`\n` }`\n` `\n` [Fact]`\n` public void ModelConverterPerformance()`\n` {`\n` var collection = Enumerable`\n` .Range(1, 100)`\n` .Select(i => new Model`\n` {`\n` Int = i,`\n` Bool = i % 2 == 0,`\n` String = \"Hello World\"`\n` })`\n` .ToList();`\n` `\n` const int iterations = 10000;`\n` `\n` var json = JsonConvert.SerializeObject(collection, ModelConverter.Instance);`\n` `\n` var sw1 = Stopwatch.StartNew();`\n` for (var i = 0; i < iterations; i++)`\n` JsonConvert.SerializeObject(collection);`\n` sw1.Stop();`\n` `\n` _output.WriteLine(\"Standard Serialize: \" + sw1.ElapsedMilliseconds);`\n` `\n` var sw2 = Stopwatch.StartNew();`\n` for (var i = 0; i < iterations; i++)`\n` JsonConvert.SerializeObject(collection, ModelConverter.Instance);`\n` sw2.Stop();`\n` `\n` _output.WriteLine(\"Custom Serialize: \" + sw2.ElapsedMilliseconds);`\n` `\n` var sw3 = Stopwatch.StartNew();`\n` for (var i = 0; i < iterations; i++)`\n` JsonConvert.DeserializeObject<Model[]>(json);`\n` sw3.Stop();`\n` `\n` _output.WriteLine(\"Standard Deserialize: \" + sw3.ElapsedMilliseconds);`\n` `\n` var sw4 = Stopwatch.StartNew();`\n` for (var i = 0; i < iterations; i++)`\n` JsonConvert.DeserializeObject<ICollection<Model>>(json, ModelConverter.Instance);`\n` sw4.Stop();`\n` `\n` _output.WriteLine(\"Custom Deserialize: \" + sw4.ElapsedMilliseconds);`\n` `\n` Assert.True(sw1.ElapsedMilliseconds > sw2.ElapsedMilliseconds);`\n` Assert.True(sw3.ElapsedMilliseconds > sw4.ElapsedMilliseconds);`\n` }`\n`}`\n\nEnjoy,\nTom\n\n1.", null, "take a look at this - https://github.com/kevin-montrose/Jil\n\n1.", null, "This is very interesting, thank you for sharing!\n\nKeep in mind that the performance metrics provided in that repository are using the default Newtonsoft serializers. This means that with the optimizations from this blog post they would be far more comparable, possibly even in favor of Newtonsoft.\n\n2.", null, "This example is very dependent of the model. If you have several models, it doesn't seem to be a best practice, to build a Converter for each Model. To make this conversion more generic by receiving generic objects, probably using Reflection, Would it be possible to achieve the same optimal performance?\n\n1.", null, "You are correct that this implementation is very dependent on the model, which is the whole point and where the performance gain comes from. I could imaging using code generation to optimize the process of creating converters, but I am not sure you could get as good performance otherwise.\n\n3.", null, "But does JSON.Net cache the info in anyway so next time it doesn't use reflection?\n\n1.", null, "Yes, the DefaultContractResolver does cache the contract look ups: https://github.com/JamesNK/Newtonsoft.Json/blob/master/Src/Newtonsoft.Json/Serialization/DefaultContractResolver.cs#L235\n\nRegardless of the cache, Newtonsoft still needs to process those results, and that takes time. By having a hard coded converter the application can just directly process serialization and avoid any overhead (no matter how small) of trying to dynamically serialize properties and values. This is a classic performance problem of dynamic versus static, and static is always going to be faster.\n\nPlease do not misunderstand me, Newtonsoft is absolutely amazing and should be applauded for both it's design and performance! This is just an optimization for extreme performance needs.\n\n4.", null, "I wonder how would you do if there was a really huge array in that model and how would you implement so that the array serialization had to be deferred into a IEnumerable.\n\nNice! If there were a lib that generates code just like XmlSerilizer does for each new type being serialized/deserialized, it would be awesome!", null, "" ]

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[ "0 like 0 dislike\n24 views\nWhy is $\\frac{\\sin{\\theta}}{\\cos{\\theta}} = \\tan{\\theta}$\n| 24 views\n\n0 like 0 dislike\nusing trig ratios we have,\n\n$\\cos{x} = \\frac{A}{H}; \\sin{x}= \\frac{O}{H}$ and $\\tan{x} = \\frac{O}{A}$\n\nHence\n\n$\\frac{\\sin{x}}{\\cos{x}} = (O/H)/(A/H) = O/A= \\tan{x}$ (proven)\nby Silver Status (28.5k points)\n0 like 0 dislike\nSoh CahToa\n\nSine = Opposite\n\nTangent = Hypotenuse\n\nSine is equal to Opposite over Hypotenuse\n\nCosine is equal to Adjacent over Hypotenuse\n\nTangent is equal to Opposite over Adjacent\n\nas follows for tan\n\ntan = opp/adj. which is then the same as, tan = sin/cos\nby Wooden (492 points)\n\n0 like 0 dislike\n0 like 0 dislike\n0 like 0 dislike\n0 like 0 dislike" ]

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[ "Introduction\n\n# Introduction\n\nR was […]\n\nJust an example for the LaTex typesetting, works it also inline?", null, "$3x$ is something.", null, "$\\frac{12}{54} \\sqrt{asd} \\int_{-\\infty}^{+\\infty} aksh\\; dx$\n\nBack to: Introduction to R – Overview" ]

[ null, "http://l.wordpress.com/latex.php", null, "http://l.wordpress.com/latex.php", null ]

[ "# Charged ball hanging in equilibrium between two parallel plate capacitors.\n\n• bitterbilly\nIn summary, an experiment was conducted to test the validity of electrostatic principles using a small ball with a charge of ±1.50 μC suspended between two parallel plate capacitors. The signs of the charges on the plates and ball were determined, and a free body diagram was drawn for the ball. The magnitude of the electric field produced by the capacitor was found to be 3.68 x 10-2N, and the potential difference between the plates was calculated to be 1.00 x 10-4m. The surface area of each plate was determined to be 10.0 C, and the amount of flux passing through one plate was calculated. If the experiment were immersed in non-conductive oil, the\n\n#### bitterbilly\n\n1. An experiment is run to determine the validity of numerous electrostatic principles. A small ball of mass 6.50 x 10-3 kg with a charge of ±1.50 μC is suspended from a non-conductive wire and hangs between two parallel plate capacitors. An angle of 30.0° is measured between the wire and the vertical while the ball is in equilibrium.\na. Determine the sign of the charge on each of the plates and the small ball.\nb. Draw a free body diagram for the small ball.\nc. Determine the magnitude of the electric field produced by the parallel plate capacitor.\nd. Solve for the potential difference between the two plates that are separated by a distance of 1.00 x 10-4m.\ne. A charge of 10.0 C is stored on each plat. Determine the surface area of each plate.\nf. Calculate the amount of flux that passes through one plate.\ng. Discuss the change in the angle of θ if the same experiment were immersed in non-conductive oil. (Assume the dielectric constant for the oil is greater than that of air.)\n\nI am having a lot of trouble trying to figure out what to do.\n\nSo far, for (a) I have the signs as + ---------> -\n+ ---------> - etc.\nFor (b) I have weight going down, the Electric Force, FE going right, and Tension going North/West at 30°.\n\nFor (c) I set Fy = W, and so Fy = 6.37 x 10-2N. Then using trig I found that Fx = 3.68 x 10-2N, which is then equal to the Force.\n\nThis is where I'm stuck, I don't really know how to find the value for the E field. Any help would be much appreciated.\n\nLast edited:\nI agree with your Fy and Fx. Fx is the horizontal electric force on the 1.5μC charge.\nThe equation for the force on a charge, q, in an electric field, E is F = q x E.\nYou should be able to find the Electric field strength E.\nField strength is measured in Volts per metre so if you know the separation of the plates you should be able to find the voltage across them.\nBUT... part (d) in your question states that the plates are 1 x 10^-4 m apart. This is only 1/10 of a mm... are you sure about this?" ]

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[ "## Mathematics\n\n#### Faculty of the Department of Mathematics\n\nClifford Taubes, William Petschek Professor of Mathematics (Chair)\nPaul G. Bamberg, Senior Lecturer on Mathematics\nBret J. Benesh, Preceptor in Mathematics, Contin Ed/Spec Prog Instructor\nLydia Rosina Bieri, Benjamin Peirce Lecturer on Mathematics\nSebastian B. Casalaina-Martin, Lecturer in Mathematics (spring term only)\nJanet Chen, Preceptor in Mathematics\nDanijela Damjanovic, Benjamin Peirce Lecturer on Mathematics\nSamit Dasgupta, Benjamin Peirce Assistant Professor of Mathematics\nAlberto De Sole, Benjamin Peirce Assistant Professor of Mathematics\nJohn F. Duncan, Benjamin Peirce Lecturer on Mathematics\nNoam D. Elkies, Professor of Mathematics, Associate of Lowell House (on leave 2007-08)\nDennis Gaitsgory, Professor of Mathematics (on leave spring term)\nVéronique Godin, Benjamin Peirce Assistant Professor of Mathematics\nThomas Goodwillie, Visiting Professor of Mathematics, Visiting Scholar in Mathematics (Brown University) (spring term only)\nRobin Gottlieb, Professor of the Practice in the Teaching of Mathematics\nBenedict H. Gross, George Vasmer Leverett Professor of Mathematics, Dean of Harvard College\nJohn T. Hall, Preceptor in Mathematics\nJoseph D. Harris, Higgins Professor of Mathematics\nMichael J. Hopkins, Professor of Mathematics\nArthur M. Jaffe, Landon T. Clay Professor of Mathematics and Theoretical Science\nThomas W. Judson, Preceptor in Mathematics, Contin Ed/Spec Prog Instructor\nOliver Knill, Preceptor in Mathematics\nToshiyuki Kobayashi, Visiting Professor of Mathematics (University of Tokyo) (spring term only)\nPeter B. Kronheimer, William Caspar Graustein Professor of Mathematics (Director of Undergraduate Studies)\nThomas Lam, Benjamin Peirce Assistant Professor of Mathematics (on leave spring term)\nMatthew P. Leingang, Preceptor in Mathematics\nBarry C. Mazur, Gerhard Gade University Professor (on leave spring term)\nCurtis T. McMullen, Maria Moors Cabot Professor of the Natural Sciences (on leave spring term)\nBrian Munson, Lecturer in Mathematics\nAndreea C. Nicoara, Benjamin Peirce Assistant Professor of Mathematics, Fellow in the Department of Mathematics (fall term only)\nMartin A. Nowak, Professor of Mathematics and of Biology (on leave fall term)\nRehana Patel, Preceptor in Mathematics\nGerald E. Sacks, Professor of Mathematical Logic (on leave fall term)\nPedram Safari, Lecturer in Mathematics (fall term only)\nWilfried Schmid, Dwight Parker Robinson Professor of Mathematics (on leave 2007-08)\nLior Silberman, Benjamin Peirce Assistant Professor of Mathematics (on leave 2007-08)\nYum Tong Siu, William Elwood Byerly Professor of Mathematics (on leave fall term)\nShlomo Z. Sternberg, George Putnam Professor of Pure and Applied Mathematics\nRobert M. Strain, Benjamin Peirce Assistant Professor of Mathematics\nRichard L. Taylor, Herchel Smith Professor of Mathematics (Director of Graduate Studies)\nBenjamin Weinkove, Benjamin Peirce Assistant Professor of Mathematics\nLauren K. Williams, Benjamin Peirce Assistant Professor of Mathematics (on leave spring term)\nHorng-Tzer Yau, Professor of Mathematics (on leave spring term)\nShing-Tung Yau, William Caspar Graustein Professor of Mathematics\n\n#### Other Faculty Offering Instruction in Mathematics\n\nPeter Koellner, Associate Professor of Philosophy\n\nThe Mathematics Department recommends that all students take mathematics courses. This said, be careful to take only those courses that are appropriate for your level of experience. Incoming students should take advantage of Harvard’s Mathematics Placement Test and of the science advising available in the Science Center the week before classes begin. Members of the Mathematics Department will be available during this period to consult with students. Generally, students with a strong precalculus background and some calculus experience will begin their mathematics education here with a deeper study of calculus and related topics in courses such as Mathematics 1a, 1b, 19a,b, 20, and 21a,b. The Harvard Mathematics Placement Test results recommend the appropriate starting level course, either Mathematics Xa, 1a, 1b, or 21. Recommendation for Mathematics 21 is sufficient qualification for Mathematics 19a,b, 20, 21a, 23a, and 25a.\n\nIn any event, what follows briefly describes these courses: Mathematics 1a introduces the basic ideas and techniques of calculus while Mathematics 1b covers integration techniques, differential equations, sequences and series. Mathematics 21a covers multi-variable calculus while Mathematics 21b covers basic linear algebra with applications to differential equations. Students who do not place into (or beyond) Mathematics 1a can take Mathematics Xa, Xb, a two-term sequence which integrates calculus and precalculus material and prepares students to enter Mathematics 1b.\n\nThere are a number of options available for students whose placement is to Mathematics 21. For example, Mathematics 19a,b are courses that are designed for students concentrating in the life sciences, chemistry, and the environmental sciences. (These course are recommended over Math 21a,b by the various life science, environmental science, and chemistry concentrations). In any event, Math 19a can be taken either before or after Math 21a,b. Math 19b requires some multivariable calculus background, and should not be taken with Math 21b. Math 19a teaches differential equations, related techniques and modeling with applications to the life sciences. Math 19b focuses teaches linear algebra, probability and statistics with a focus on life science examples and applications. Mathematics 20 covers selected topics from Mathematics 21a and 21b for students particularly interested in economic and social science applications.\n\nMathematics 23 is a theoretical version of Mathematics 21 which treats multivariable calculus and linear algebra in a rigorous, proof oriented way. Mathematics 25 and 55 are theory courses that should be elected only by those students who have a particular interest in, and commitment to, mathematics. They assume a solid understanding of one-variable calculus and a willingness to think rigorously and abstractly about mathematics, and to work extremely hard. Both courses study multivariable calculus and linear algebra plus many very deep related topics. Mathematics 25 differs from Mathematics 23 in that the work load in Mathematics 25 is significantly more than in Mathematics 23, but then Mathematics 25 covers more material. Mathematics 55 differs from Mathematics 25 in that the former assumes a very strong proof oriented mathematics background. Mathematics 55, covers the material from Mathematics 25 plus much material from Mathematics 122 and Mathematics 113. Entrance into Mathematics 55 requires the consent of the instructor.\n\nStudents who have had substantial preparation beyond the level of the Advanced Placement Examinations are urged to consult the Director of Undergraduate Studies in Mathematics concerning their initial Harvard mathematics courses. Students should take this matter very seriously. The Mathematics Department has also prepared a pamphlet with a detailed description of all its 100-level courses and their relationship to each other. This pamphlet gives sample lists of courses suitable for students with various interests. It is available at the Mathematics Department Office. Many 100-level courses assume some familiarity with proofs. Courses that supply this prerequisite include Mathematics 23, 25, 55, 101, 112, 121, and 141. Of these, note that Mathematics 101 may be taken concurrently with Mathematics 1, 19, 20, or 21.\n\nMathematics 113, 114, 122, 123, 131, and 132 form the core of the department’s more advanced courses. Mathematics concentrators are encouraged to consider taking these courses, particularly Mathematics 113, 122 and 131. (Those taking 55a,b will have covered the material of Mathematics 113 and 122, and are encouraged to take Mathematics 114, 123, and 132.)\n\nCourses numbered 200-249 are introductory graduate courses. They will include substantial homework and are likely to have a final exam, either in class or take home. Most are taught every year. They may be suitable for very advanced undergraduates. Mathematics 212a, 230a, 231a and 232a will help prepare graduate students for the qualifying examination in Mathematics. Courses numbered 250-299 are graduate topic courses, intended for advanced graduate students.\n\nThe Mathematics Department does not grant formal degree credit without prior approval for taking a course that is listed as a prerequisite of one you have already taken. Our policy is that a student who takes and passes any calculus course is not normally permitted to then take a more elementary course for credit. A student who has passed Mathematics 21a, for example, will normally not be allowed to take Mathematics 1a, or 1b for credit. The Mathematics Department is prepared to make exceptions for sufficient academic reasons; in each case, however, a student must obtain written permission from the Mathematics Director of Undergraduate Studies in advance.\n\nIn the case of students accepting admission as sophomores, this policy is administered as follows: students counting one half course of advanced standing credit in mathematics are deemed to have passed Mathematics 1a, and students counting a full course of advanced standing credit in mathematics are deemed to have passed Mathematics 1a and 1b.\n\nMathematics Xa. Introduction to Functions and Calculus I\nCatalog Number: 1981 Enrollment: Normally limited to 15 students per section.\nBret J. Benesh, Samit Dasgupta, John T. Hall, and members of the Department\nHalf course (fall term). Section meeting times: Section I: M., W., F., at 10; Section II: M., W., F., at 11; Section III: M. W. F., at 12 (with sufficient enrollment). EXAM GROUP: 1\nThe study of functions and their rates of change. Fundamental ideas of calculus are introduced early and used to provide a framework for the study of mathematical modeling involving algebraic, exponential, and logarithmic functions. Thorough understanding of differential calculus promoted by year long reinforcement. Applications to biology and economics emphasized according to the interests of our students.\nNote: Required first meeting: Monday, September 17, 8:30 am, Science Center D. Participation in a one and a half hour workshop is required each week, as well as required participation in a one hour problem session each week. The sequence Xa, Xb gives solid preparation for Mathematics 1b. This course, when taken for a letter grade together with Mathematics Xb, meets the Core area requirement for Quantitative Reasoning.\n\nMathematics Xb. Introduction to Functions and Calculus II\nCatalog Number: 3857 Enrollment: Normally limited to 15 students per section.\nBret J. Benesh, John T. Hall, Brian Munson, and members of the Department\nHalf course (spring term). Section I: M., W., F., at 10; Section II: M. W., F., at 11; Section III: M., W., F., at 12 (with sufficient enrollment); and a twice weekly lab session to be arranged. EXAM GROUP: 1\nContinued investigation of functions and differential calculus through modeling; an introduction to integration with applications; an introduction to differential equations. Solid preparation for Mathematics 1b.\nNote: Participation in a one and a half hour workshop is required each week, as well as required participation in a one hour problem session each week. This course, when taken for a letter grade together with Mathematics Xa, meets the Core area requirement for Quantitative Reasoning.\nPrerequisite: Mathematics Xa.\n\nMathematics 1a. Introduction to Calculus\nCatalog Number: 8434 Enrollment: Normally limited to 30 students per section.\nMatthew P. Leingang, John Duncan, and Rehana Patel (fall term); Matthew P. Leingang (spring term)\nHalf course (fall term; repeated spring term). Fall: Section I, M., W., F., at 9 (with sufficient enrollment); Section II, M., W., F., at 10; Section III, M., W., F., at 11; Section IV, M., W., F., at 12; Section V, Tu., Th., 10–11:30; Section Vl, Tu., Th., 11:30–1. Spring: Section I, M., W., F., at 10; Section II, Tu.Th. 10-11:30 (with sufficient enrollment) and a weekly problem section to be arranged. EXAM GROUP: 1\nThe development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to optimization, graphing, mechanisms, and problems from many other disciplines.\nNote: Required first meeting in fall: Tuesday, September 18, 8:30 am, Science Center B. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\nPrerequisite: A solid background in precalculus.\n\nMathematics 1b. Calculus, Series, and Differential Equations\nCatalog Number: 1804 Enrollment: Normally limited to 30 students per section.\nThomas W. Judson, Danijela Damjanovic, John T. Hall, Brian Munson, and Robert Strain (fall term); Robin Gottlieb, Bret Benesh, and Lydia Bieri (spring term)\nHalf course (fall term; repeated spring term). Section I, M., W., F., at 9 (with sufficient enrollment); Section II, M., W., F., at 10; Section III, M., W., F., at 11; Section IV, M., W., F., at 12 (with sufficient enrollment); Section V: Tu., Th., 10–11:30; Section Vl, Tu., Th., 11:30–1, and a weekly problem section to be arranged. Required exams: Hours to be arranged. EXAM GROUP: 1\nSpeaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.\nNote: Required first meeting in fall: Monday, September 17, 8:30 am, Science Center C. Required first meeting in spring: Wednesday, January 30, 8:30 am, Science Center A. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\nPrerequisite: Mathematics 1a, or Xa and Xb, or equivalent.\n\nMathematics 19a. Modeling and Differential Equations for the Life Sciences\nCatalog Number: 1256\nThomas W. Judson (fall term); John T. Hall (spring term)\nHalf course (fall term; repeated spring term). M., W., F., at 1, and a weekly problem section to be arranged. EXAM GROUP: 6\nConsiders the construction and analysis of mathematical models that arise in the environmental sciences, biology, the ecological sciences, and in earth and atmospheric sciences. Introduces mathematics that include multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and non-linear dynamical systems. Taught via examples from current literature (both good and bad).\nNote: This course is recommended over Math 21a for those planning to concentrate in the life sciences, chemistry, or environmental sciences. Can be taken with or without Mathematics 21a,b. Students with interests in the social sciences and economics might consider Mathematics 20. This course can be taken before or after Mathematics 20. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\n\nMathematics 19b. Linear Algebra, Probability, and Statistics for the Life Sciences\nCatalog Number: 6144\nClifford Taubes\nHalf course (spring term). M., W., F., at 1 and a weekly problem section to be arranged. EXAM GROUP: 6\nProbability, statistics and linear algebra with applications to life sciences, chemistry, and environmental sciences. Linear algebra includes matrices, eigenvalues, eigenvectors, determinants, and applications to probability, statistics, dynamical systems. Basic probability and statistics are introduced, as are standard models, techniques, and their uses including the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis.\nNote: This course is recommended over Math 21b for those planning to concentrate in the life sciences, chemistry, or environmental sciences. Can be taken with Mathematics 21a. Students who have seen some multivariable calculus can take Math 19b before Math 19a. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\n\nMathematics 20. Algebra and Multivariable Mathematics for Social Sciences\nCatalog Number: 0906\nMatthew P. Leingang (fall term); Rehana Patel (spring term)\nHalf course (fall term; repeated spring term). M., W., F., at 9, and a weekly problem section to be arranged. EXAM GROUP: 2\nIntroduction to linear algebra, including vectors, matrices, and applications. Calculus of functions of several variables, including partial derivatives, constrained and unconstrained optimization, and applications. Covers the topics from Mathematics 21a,b which are most important in applications to economics, the social sciences, and some other fields.\nNote: Should not ordinarily be taken in addition to Mathematics 21a,b. Examples drawn primarily from economics and the social sciences though Mathematics 20 may be useful to students in certain natural sciences. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\nPrerequisite: Mathematics 1b or equivalent, or an A or A- in Mathematics 1a, or a 5 on the AB or a 3 or higher on the BC Advanced Placement Examinations in Mathematics.\n\nMathematics 21a. Multivariable Calculus\nCatalog Number: 6760 Enrollment: Normally limited to 30 students per section.\nOliver Knill, Véronique Godin, and Rehana Patel (fall term); Thomas Judson, and Matthew P. Leingang (spring term)\nHalf course (fall term; repeated spring term). Fall: Section I, M., W., F., at 9 (with sufficient enrollment); Section II, M., W., F., at 10; Section III, M., W., F., at 11; Section IV, M., W., F., at 12; Section V, Tu., Th., 10–11:30; Section VI, Tu., Th., 11:30–1; and a weekly problem section to be arranged. . EXAM GROUP: 1\nTo see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives, and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces, and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Green’s, Stokes’, and Divergence Theorems.\nNote: Required first meeting in fall: Tuesday, September 18, 8:30 am, Science Center C. Required first meeting in spring: Wednesday, January 30, 8:30 am, Science Center C. May not be taken for credit by students who have passed Applied Mathematics 21a. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning. Activities using computers to calculate and visualize applications of these ideas will not require previous programming experience. Special sections for students interested in physics are offered each term.\nPrerequisite: Mathematics 1b or equivalent.\n\nMathematics 21b. Linear Algebra and Differential Equations\nCatalog Number: 1771 Enrollment: Normally limited to 30 students per section.\nYum Tong Siu (fall term); Oliver Knill, Samit Dasgupta, and Rehana Patel (spring term)\nHalf course (fall term; repeated spring term). Fall: Section I: M., W., F., at 10; Section II: M., W., F., at 11; Section III: M., W., F., at 12 (with sufficient enrollment); Spring: Section I: M., W., F., at 10; Section II: M., W., F., at 11; Section III: M., W., F., at 12 (with sufficient enrollment); Section IV: Tu., Th., 10–11:30; Section V: Tu., Th., 11:30–1 and a weekly problem section to be arranged. EXAM GROUP: 1\nMatrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as vectors, Euclidean spaces, linear transformations, determinants, eigenvalues, and eigenvectors. Of applications given, a regular section considers dynamical systems and both ordinary and partial differential equations plus an introduction to Fourier series.\nNote: Required first meeting in fall: Monday, September 17, 8:30 am, Science Center A. Required first meeting in spring: Wednesday, January 30, 8:30 am, Science Center D. May not be taken for credit by students who have passed Applied Mathematics 21b. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\nPrerequisite: Mathematics lb or equivalent. Mathematics 21a is commonly taken before Mathematics 21b, but is not a prerequisite, although familiarity with partial derivatives is useful.\n\nMathematics 23a. Linear Algebra and Real Analysis I\nCatalog Number: 2486\nPaul G. Bamberg\nHalf course (fall term). M., W., F., at 11, and a weekly conference section to be arranged. EXAM GROUP: 4\nA rigorous, integrated treatment of linear algebra and multivariable differential calculus, emphasizing topics that are relevant to fields such as physics and economics. Topics: fields, vector spaces and linear transformations, scalar and vector products, elementary topology of Euclidean space, limits, continuity, and differentiation in n dimensions, eigenvectors and eigenvalues, inverse and implicit functions, manifolds, and Lagrange multipliers. Students are expected to master twenty important proofs.\nNote: Course content overlaps substantially with Mathematics 21a,b, 25a,b, so students should plan to continue in Mathematics 23b. See the description in the introductory paragraphs in the Mathematics section of the catalog about the differences between Mathematics 23 and Mathematics 25. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\nPrerequisite: Mathematics 1b or a grade of 4 or 5 on the Calculus BC Advanced Placement Examination, plus an interest both in proving mathematical results and in using them.\n\nMathematics 23b. Linear Algebra and Real Analysis II\nCatalog Number: 8571\nPaul G. Bamberg\nHalf course (spring term). M., W., F., at 11, and a weekly conference section to be arranged. EXAM GROUP: 4\nA rigorous, integrated treatment of linear algebra and multivariable calculus. Topics: Riemann and Lebesgue integration, determinants, change of variables, volume of manifolds, differential forms, and exterior derivative. Applications of linear algebra to differential equations and Fourier analysis. Introduction to infinite-dimensional vector spaces. Stokes’s theorem is presented both in the language of vector analysis (div, grad, and curl) and in the language of differential forms. Students are expected to master twenty important proofs.\nNote: This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\nPrerequisite: Mathematics 23a.\n\nMathematics 25a. Honors Linear Algebra and Real Analysis I\nCatalog Number: 1525\nBenjamin Weinkove\nHalf course (fall term). M., W., F., at 10. EXAM GROUP: 3\nA rigorous treatment of linear algebra. Topics include: Construction of number systems; fields, vector spaces and linear transformations; eigenvalues and eigenvectors, determinants and inner products. Metric spaces, compactness and connectedness.\nNote: Only for students with a strong interest and background in mathematics. There will be a heavy workload. May not be taken for credit after Mathematics 23. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\nPrerequisite: 5 on the Calculus BC Advanced Placement Examination and some familiarity with writing proofs, or the equivalent as determined by the instructor.\n\nMathematics 25b. Honors Linear Algebra and Real Analysis II\nCatalog Number: 1590\nBenjamin Weinkove\nHalf course (spring term). M., W., F., at 10. EXAM GROUP: 3\nA rigorous treatment of basic analysis. Topics include: convergence, continuity, differentiation, the Riemann integral, uniform convergence, the Stone-Weierstrass theorem, Fourier series, differentiation in several variables. Additional topics, including the classical results of vector calculus in two and three dimensions, as time allows.\nNote: There will be a heavy workload. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\nPrerequisite: Mathematics 23a or 25a or 55a.\n\n*Mathematics 55a. Honors Abstract Algebra\nCatalog Number: 4068\nDennis Gaitsgory\nHalf course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16\nA rigorous treatment of abstract algebra including linear algebra and group theory.\nNote: Mathematics 55a is an intensive course for students having significant experience with abstract mathematics. Instructor’s permission required. Every effort will be made to accommodate students uncertain of whether the course is appropriate for them; in particular, Mathematics 55a and 25a will be closely coordinated for the first three weeks of instruction. Students can switch between the two courses during the first three weeks without penalty. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\n\nMathematics 55b. Honors Real and Complex Analysis\nCatalog Number: 3312\nSamit Dasgupta\nHalf course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16\nA rigorous treatment of real and complex analysis.\nNote: Mathematics 55b is an intensive course for students having significant experience with abstract mathematics. Instructor’s permission required. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.\n\n*Mathematics 60r. Reading Course for Senior Honors Candidates\nCatalog Number: 8500\nPeter B. Kronheimer\nHalf course (fall term; repeated spring term). Hours to be arranged.\nNote: Limited to candidates for honors in Mathematics who obtain the permission of both the faculty member under whom they want to work and the Director of Undergraduate Studies. May not count for concentration in Mathematics without special permission from the Director of Undergraduate Studies. Graded Sat/Unsat only.\n\n*Mathematics 91r. Supervised Reading and Research\nCatalog Number: 2165\nPeter B. Kronheimer\nHalf course (fall term; repeated spring term). Hours to be arranged.\nPrograms of directed study supervised by a person approved by the Department.\nNote: May not ordinarily count for concentration in Mathematics.\n\n*Mathematics 99r. Tutorial\nCatalog Number: 6024\nPeter B. Kronheimer and members of the Department\nHalf course (fall term; repeated spring term). Hours to be arranged.\nTopics for 2007-08: (1) Random graphs (fall), prerequisite: Math 122 or familiarity with abstract linear algebra and group theory. Knowledge of elementary probability theory helpful, but not required. (2) Clifford algebras and spinors (spring), prerequisite: Math 122 or familiarity with abstract linear algebra, groups and group actions.\nNote: May be repeated for course credit with permission from the Director of Undergraduate Studies. Only one tutorial may count for concentration credit.\n\nMathematics 101. Sets, Groups and Topology\nCatalog Number: 8066\nJohn F. Duncan\nHalf course (fall term). M., W., F., at 11. EXAM GROUP: 4\nAn introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, and low-dimensional topology.\nNote: Familiarity with algebra, geometry and/or calculus is desirable. Students who have already taken Mathematics 23a,b, 25a,b or 55a,b should not take this course for credit.\nPrerequisite: An interest in mathematical reasoning.\n\nMathematics 106. Ordinary Differential Equations\nCatalog Number: 3377\nThomas W. Judson\nHalf course (spring term). M., W., F., at 12. EXAM GROUP: 5\nAnalytic, numerical, and qualitative analysis of ordinary differential equations. Linear equations, linear and non-linear systems. Applications to mechanics, biology, physics, and the social sciences. Existence and uniqueness of solutions and visual analysis using computer graphics. Topics selected from Laplace transforms, power series solutions, chaos, and numerical solutions.\nPrerequisite: Mathematics 19a,b, 20 or 21a.\n\nMathematics 112. Introductory Real Analysis\nCatalog Number: 1123\nDanijela Damjanovic\nHalf course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14\nAn introduction to mathematical analysis and the theory behind calculus. An emphasis on learning to understand and construct proofs. Covers limits and continuity in metric spaces, uniform convergence and spaces of functions, the Riemann integral.\nPrerequisite: Mathematics 21a,b and either an ability to write proofs or concurrent enrollment in Mathematics 101. Should not ordinarily be taken in addition to Mathematics 23a,b, 25a,b or 55a,b.\n\nMathematics 113. Analysis I: Complex Function Theory\nCatalog Number: 0405\nRobert M. Strain\nHalf course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13\nAnalytic functions of one complex variable: power series expansions, contour integrals, Cauchy’s theorem, Laurent series and the residue theorem. Some applications to real analysis, including the evaluation of indefinite integrals. An introduction to some special functions.\nPrerequisite: Mathematics 23a,b, 25a,b, or 112. Not to be taken after Mathematics 55b.\n\nMathematics 114. Analysis II: Measure, Integration and Banach Spaces - (New Course)\nCatalog Number: 9111\nCurtis T. McMullen\nHalf course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13\nLebesgue measure and integration; general topology; introduction to L p spaces, Banach and Hilbert spaces, and duality.\nPrerequisite: Mathematics 23, 25, 55, or 112.\n\n[Mathematics 115. Methods of Analysis]\nCatalog Number: 1871\nWilfried Schmid\nHalf course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14\nComplex functions; Fourier analysis; Hilbert spaces and operators; Laplace’s equations; Bessel and Legendre functions; symmetries; Sturm-Liouville theory.\nNote: Expected to be given in 2008–09. Mathematics 115 is especially for students interested in physics.\nPrerequisite: Mathematics 23a,b, 25a,b, 55a,b, or 112.\n\nMathematics 116. Convexity and Optimization with Applications\nCatalog Number: 5253\nPaul G. Bamberg\nHalf course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17\nIntroduction to real and functional analysis through topics such as convex programming, duality theory, linear and non-linear programming, calculus of variations, and the maximum principle of optimal control theory.\nPrerequisite: At least one course beyond Mathematics 21.\n\nMathematics 121. Linear Algebra and Applications\nCatalog Number: 7009\nLydia R. Bieri\nHalf course (fall term). M., W., F., at 12. EXAM GROUP: 5\nReal and complex vector spaces, dual spaces, linear transformations and Jordan normal forms. Inner product spaces. Applications to differential equations, classical mechanics, and optimization theory. Emphasizes learning to understand and write proofs.\nPrerequisite: Mathematics 21b or equivalent. Should not ordinarily be taken in addition to Mathematics 23a, 25a, or 55a.\n\nMathematics 122. Algebra I: Theory of Groups and Vector Spaces\nCatalog Number: 7855\nAlberto DeSole\nHalf course (fall term). M., W., F., at 1. EXAM GROUP: 6\nGroups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups.\nPrerequisite: Mathematics 23a, 25a, 121; or 101 with the instructor’s permission. Should not be taken in addition to Mathematics 55a.\n\nMathematics 123. Algebra II: Theory of Rings and Fields\nCatalog Number: 5613\nAlberto De Sole\nHalf course (spring term). M., W., F., at 1. EXAM GROUP: 6\nRings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules.\nPrerequisite: Mathematics 122 or 55a.\n\nMathematics 124. Number Theory\nCatalog Number: 2398\nJoseph D. Harris\nHalf course (fall term). M., W., F., at 2. EXAM GROUP: 7\nFactorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; Pell’s equation; selected Diophantine equations; theory of integral quadratic forms.\nPrerequisite: Mathematics 122 (which may be taken concurrently) or equivalent.\n\nMathematics 129. Number Fields\nCatalog Number: 2345\nRichard L. Taylor\nHalf course (spring term). M., W., F., at 9. EXAM GROUP: 2\nAlgebraic number theory: number fields, unique factorization of ideals, finiteness of class group, structure of unit group, Frobenius elements, local fields, ramification, weak approximation, adeles, and ideles.\nPrerequisite: Mathematics 123.\n\nMathematics 130 (formerly Mathematics 138). Classical Geometry\nCatalog Number: 5811\nPeter B. Kronheimer\nHalf course (spring term). M., W., F., at 11. EXAM GROUP: 4\nEuclidean, spherical and hyperbolic geometry. No prior experience with proofs required.\nNote: Not expected to be given 2008-09.\nPrerequisite: Mathematics 21a,b, 23a, 25a or 55a (may be taken concurrently).\n\nMathematics 131. Topology I: Topological Spaces and the Fundamental Group\nCatalog Number: 2381\nVéronique Godin\nHalf course (fall term). M., W., F., at 12. EXAM GROUP: 5\nAbstract topological spaces; compactness, connectedness, continuity. Homeomorphism and homotopy, fundamental groups, covering spaces. Introduction to combinatorial topology.\nPrerequisite: Some acquaintance with metric space topology (Mathematics 23a,b, 25a,b, 55a,b, 101, or 112) and with groups (Mathematics 101, 122 or 55a).\n\nMathematics 132. Topology II: Smooth Manifolds - (New Course)\nCatalog Number: 7725\nVéronique Godin\nHalf course (spring term). M., W., F., at 12. EXAM GROUP: 5\nDifferential manifolds, smooth maps and transversality. Winding numbers, vector fields, index and degree. Differential forms, Stokes’ theorem, introduction to cohomology.\nPrerequisite: Mathematics 23a,b, 25a,b, 55a,b or 112.\n\nMathematics 136. Differential Geometry\nCatalog Number: 1949\nShlomo Z. Sternberg\nHalf course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13\nThe exterior differential calculus and its application to curves and surfaces in 3-space and to various notions of curvature. Introduction to Riemannian geometry in higher dimensions and to symplectic geometry.\nPrerequisite: Advanced calculus and linear algebra.\n\nMathematics 137. Algebraic Geometry\nCatalog Number: 0556\nJohn F. Duncan\nHalf course (spring term). M., W., F., at 2. EXAM GROUP: 7\nAffine and projective spaces, plane curves, Bezout’s theorem, singularities and genus of a plane curve, Riemann-Roch theorem.\nPrerequisite: Mathematics 123.\n\n[Mathematics 141. Introduction to Mathematical Logic]\nCatalog Number: 0600\nGerald E. Sacks\nHalf course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14\nAn introduction to mathematical logic with applications to computer science and algebra. Formal languages. Completeness and compactness of first order logic. Definability and interpolation. Decidability. Unsolvable problems. Computable functions and Turing machines. Recursively enumerable sets. Transfinite induction.\nNote: Expected to be given in 2008–09.\nPrerequisite: Any mathematics course at the level of Mathematics 21a,b or higher, or permission of instructor.\n\nMathematics 143. Set Theory\nCatalog Number: 6005\nGerald E. Sacks\nHalf course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14\nAxioms of set theory. Gödel’s constructible universe. Consistency of the axiom of choice and of the generalized continuum hypothesis. Cohen’s forcing method. Independence of the AC and GCH.\nNote: Not expected to be given 2008-09.\nPrerequisite: Any mathematics course at the level of 21a or higher, or permission of instructor.\n\nMathematics 152. Discrete Mathematics\nCatalog Number: 8389\nBret J. Benesh\nHalf course (fall term). M., W., F., at 11. EXAM GROUP: 4\nAn introduction to finite groups, finite fields, finite geometry, discrete probability, and graph theory. A unifying theme of the course is the symmetry group of the regular icosahedron, whose elements can be realized as permutations, as linear transformations of vector spaces over finite fields, as collineations of a finite plane, or as vertices of a graph. Taught in a seminar format, and students will gain experience in presenting proofs at the blackboard.\nNote: Students who have taken Mathematics 23a,b, 25a,b or 55a,b should not take this course for credit.\nPrerequisite: Mathematics 21b or equivalent.\n\n[Mathematics 153. Mathematical Biology-Evolutionary Dynamics]\nCatalog Number: 3004 Enrollment: Limited to 30.\nMartin A. Nowak\nHalf course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16\nIntroduces basic concepts of mathematical biology and evolutionary dynamics: evolution of genomes, quasi-species, finite and infinite population dynamics, chaos, game dynamics, evolution of cooperation and language, spatial models, evolutionary graph theory, infection dynamics, somatic evolution of cancer.\nNote: Expected to be given in 2008–09.\nPrerequisite: Mathematics 21a,b, Biological Sciences 50 and 53 or equivalent.\n\nMathematics 154 (formerly Mathematics 191). Probability Theory\nCatalog Number: 4306\nPaul G. Bamberg\nHalf course (spring term). Tu., Th., 4–5:30. EXAM GROUP: 18\nAn introduction to probability theory. Discrete and continuous random variables; distribution and density functions for one and two random variables; conditional probability. Generating functions, weak and strong laws of large numbers, and the central limit theorem. Geometrical probability, random walks, and Markov processes.\nPrerequisite: Any mathematics course at the level of Mathematics 19a,b, or 21a,b or higher, or knowledge of multivariable calculus as demonstrated on the online placement test.\n\nMathematics 155r (formerly Mathematics 192r). Combinatorics\nCatalog Number: 6612\nLauren K. Williams\nHalf course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14\nTopics include enumerative and algebraic combinatorics related to representations of the symmetric group, symmetric functions, and Young tableaux.\nPrerequisite: Mathematics 122 (or equivalent). Knowledge of representation theory of finite groups will be helpful.\n\n#### Cross-listed Courses\n\nApplied Mathematics 105a. Complex and Fourier Analysis\nApplied Mathematics 105b. Ordinary and Partial Differential Equations\nApplied Mathematics 107. Graph Theory and Combinatorics\n*Freshman Seminar 26k. Euclidean Lattices and Sphere Packings\n*Freshman Seminar 26s. Mathematical Structures - (New Course)\n[Philosophy 144. Logic and Philosophy]\n\nMathematics 212a (formerly Mathematics 212ar). Real Analysis\nCatalog Number: 5446\nHorng-Tzer Yau\nHalf course (fall term). M., W., F., at 10. EXAM GROUP: 3\nBanach spaces, Hilbert spaces and functional analysis. Distributions, spectral theory and the Fourier transform.\nPrerequisite: Mathematics 114 or equivalent.\n\nCatalog Number: 7294\nShlomo Z. Sternberg\nHalf course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13\nContinuation of Mathematics 212ar. The spectral theorem for self-adjoint operators in Hilbert space. Applications to partial differential equations.\nPrerequisite: Mathematics 212ar and 213a.\n\nMathematics 213a. Complex Analysis\nCatalog Number: 1621\nAndreea C. Nicoara\nHalf course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13\nA second course in complex analysis: elliptic functions, canonical products, conformal mapping, extremal length, harmonic measure and capacity.\nPrerequisite: Mathematics 55b or 113.\n\nCatalog Number: 2641\nShing-Tung Yau\nHalf course (spring term). M., W., F., at 12. EXAM GROUP: 5\nFundamentals of Riemann surfaces. Topics may include sheaves and cohomology, potential theory, uniformization, and moduli.\nPrerequisite: Mathematics 213a.\n\nMathematics 221. Commutative Algebra - (New Course)\nCatalog Number: 8320\nThomas Lam\nHalf course (fall term). M., W., F., at 12. EXAM GROUP: 5\nA first course in commutative algebra: Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, discrete valuation rings, filtrations, completions and dimension theory.\nPrerequisite: Mathematics 123.\n\n[Mathematics 222. Lie Groups and Lie Algebras]\nCatalog Number: 6738\nWilfried Schmid\nHalf course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14\nLie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.\nNote: Expected to be given in 2008–09.\nPrerequisite: Mathematics 114, 123 and 132.\n\n[Mathematics 223a. (formerly Mathematics 251a.) Algebraic Number Theory]\nCatalog Number: 8652\n----------\nHalf course (fall term). Hours to be arranged.\nA graduate introduction to algebraic number theory. Topics: the structure of ideal class groups, groups of units, a study of zeta functions and L-functions, local fields, Galois cohomology, local class field theory, and local duality.\nNote: Expected to be given in 2008–09.\nPrerequisite: Mathematics 129.\n\n[Mathematics 223b. (formerly Mathematics 251b.) Algebraic Number Theory]\nCatalog Number: 2783\n----------\nHalf course (spring term). Hours to be arranged.\nContinuation of Mathematics 223a. Topics: adeles, global class field theory, duality, cyclotomic fields. Other topics may include: Tate’s thesis or Euler systems.\nNote: Expected to be given in 2008–09.\nPrerequisite: Mathematics 223a.\n\nMathematics 230a. Differential Geometry\nCatalog Number: 0372\nShing-Tung Yau\nHalf course (fall term). M., W., F., at 2. EXAM GROUP: 7\nElements of differential geometry: Riemannian geometry, symplectic and Kaehler geometry, geodesics, Riemann curvature, Darboux’s theorem, moment maps and symplectic quotients, complex and Kaehler manifolds, Dolbeault and de Rham cohomology.\nPrerequisite: Mathematics 132 or equivalent.\n\nCatalog Number: 0504\nShing-Tung Yau\nHalf course (spring term). M., W., F., at 2. EXAM GROUP: 7\nA continuation of Mathematics 230a. Topics in global Riemannian geometry: Ricci curvature and volume comparison; sectional curvature and distance comparison; Toponogov’s theorem and applications; sphere theorems; Gromov’s betti number bounds; Gromov-Hausdorff convergence; Cheeger’s finiteness theorem, and convergence theorems.\nPrerequisite: Mathematics 230a.\n\nMathematics 231a. (formerly Mathematics 272a.) Algebraic Topology\nCatalog Number: 7275\nDanijela Damjanovic\nHalf course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16\nCovering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincaré duality.\nPrerequisite: Mathematics 131 and 132.\n\nMathematics 231br. (formerly Mathematics 272b.) Advanced Algebraic Topology\nCatalog Number: 9127\nMichael J. Hopkins\nHalf course (spring term). M., W., F., at 11. EXAM GROUP: 4\nContinuation of Mathematics 231a. Spectral sequences and techniques of computation. Vector bundles and characteristic classes. Bott periodicity. K-theory, cobordism and stable cohomotopy as examples of cohomology theories.\nPrerequisite: Mathematics 231a.\n\nMathematics 232a. (formerly Mathematics 260a.) Introduction to Algebraic Geometry I\nCatalog Number: 6168\nPeter B. Kronheimer\nHalf course (fall term). M., W., F., at 1. EXAM GROUP: 6\nIntroduction to complex algebraic curves, surfaces, and varieties.\nPrerequisite: Mathematics 123 and 132.\n\nMathematics 232br. (formerly Mathematics 260b.) Introduction to Algebraic Geometry II\nCatalog Number: 9205\nSebastian B. Casalaina-Martin\nHalf course (spring term). M., W., F., at 1. EXAM GROUP: 6\nThe course will cover the classification of complex algebraic surfaces.\nPrerequisite: Mathematics 232a.\n\nMathematics 233a. (formerly Mathematics 261a.) Theory of Schemes I\nCatalog Number: 6246\nBarry C. Mazur\nHalf course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14\nAn introduction to the theory and language of schemes. Textbooks: Algebraic Geometry by Robin Hartshorne and Geometry of Schemes by David Eisenbud and Joe Harris. Weekly homework will constitute an important part of the course.\nPrerequisite: Mathematics 221 and 232a or permission of instructor.\n\nMathematics 233br. (formerly Mathematics 261b.) Theory of Schemes II\nCatalog Number: 3316\nSebastian B. Casalaina-Martin\nHalf course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16\nA continuation of Mathematics 233a. Will cover the theory of schemes, sheaves, and sheaf cohomology.\nPrerequisite: Mathematics 233a.\n\nMathematics 243 (formerly Mathematics 234). Evolutionary Dynamics\nCatalog Number: 8136\nMartin A. Nowak\nHalf course (spring term). Tu., 1–4. EXAM GROUP: 15, 16, 17\nAdvanced topics of evolutionary dynamics. Seminars and research projects.\nPrerequisite: Experience with mathematical biology at the level of Mathematics 153.\n\nMathematics 244. Advanced Set Theory - (New Course)\nCatalog Number: 3138\nPeter Koellner\nHalf course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16\nInner models of large cardinal axioms, focusing on recent work on inner models for large cardinals at the level of supercompact and beyond. Topics include: Continuum Hypothesis and Omega Conjecture.\nPrerequisite: Course in Set Theory or permission of the instructor.\n\nMathematics 262. Manifolds and Homotopy Theory\nCatalog Number: 5564\nThomas Goodwillie (Brown University)\nHalf course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13\nPossible topics: Whitney’s embedding theorem and generalizations, the h-cobordism theorem and generalizations, surgery theory, and calculus of functors. Pace and emphasis of course may depend on the background and interest of the participants.\n\nMathematics 265. Infinite Dimensional Lie Algebras\nCatalog Number: 3191\nAlberto De Sole\nHalf course (fall term). M., W., F., at 11. EXAM GROUP: 4\nA detailed introduction to the structure and the representation theory of some important infinite-dimensional Lie algebras: the Heisenberg algebra, the Virasoro algebra and the affine Kac-Moody algebras.\n\nMathematics 271. Introduction to the Mathematics of General Relativity\nCatalog Number: 2400\nLydia Bieri\nHalf course (spring term). M., W., F., at 12. EXAM GROUP: 5\nA focus on the Einstein field equations within GR. Brief review of SR and fundamentals from differential geometry. Discussion of Schwarzschild solution, black holes, energy-momentum tensor, non-localizability of gravitational energy, isolated gravitating systems.\n\nMathematics 273. Topics in Analysis and Mathematical Physics\nCatalog Number: 7810\nShlomo Z. Sternberg\nHalf course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14\nPossible choices: exact models in statistical mechanics, asymptotic analysis, or supersymmetry.\n\nMathematics 275. Multiplicity-Free Representations: Complex Geometric Methods in Representation Theory\nCatalog Number: 0818\nToshiyuki Kobayashi (Research Institute for Mathematical Sciences, Kyoto)\nHalf course (spring term). M., W., F., at 11. EXAM GROUP: 4\nExplanation of complex geometric methods such as reproducing kernels and \"visible actions\" for the study of infinite dimensional representations. From this viewpoint, various examples of multiplicity-free representations of Lie groups will be discussed.\n\nMathematics 282. Introduction to Seiberg-Witten Theory\nCatalog Number: 8399\nPedram Safari\nHalf course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16\nRudiments of Seiberg-Witten moduli spaces and invariants, including a review of gauge theory techniques. Selected applications to the geometry and topology of 4-manifolds.\n\nMathematics 283. Topics in Knot Theory\nCatalog Number: 7877\nBrian Munson\nHalf course (fall term). M., W., F., at 12. EXAM GROUP: 5\nThe Conway polynomial, the Jones polynomial, Khovanov homology, Vassiliyev and finite type invariants, and configuration space methods. A discussion of links and their generalizations to higher dimensions.\n\nMathematics 287. Algebraic Curves\nCatalog Number: 7465\nJoseph D. Harris\nHalf course (spring term). M., W., F., at 10. EXAM GROUP: 3\nA development of the theory of algebraic curves/Riemann surfaces, touching on many of the classical aspects of their geometry. A focus on developing current research topics.\n\nMathematics 288. Algebraic K-Theory\nCatalog Number: 5052\nMichael J. Hopkins\nHalf course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13\nBegins with some of the classical invariants of algebraic topology (like the finiteness obstruction, simple homotopy, Reidemeister torsion) and locate them in \"algebraic K-theory.\" More advanced topics will include Quillen K-groups, computations, Waldhausen K-theory.\n\nMathematics 299r. Graduate Tutorial in Geometry - (New Course)\nCatalog Number: 8799\nClifford Taubes and members of the Department\nHalf course (spring term). Section 1: TTh 1-2:30; Section 2: MWF at 11. EXAM GROUP: 15, 16\nTutorial 1: Applications of Seiberg-Witten Theory. Applications of Seiberg-Witten theory, Floer homology and refinements to geometry and topology of 3-and 4-manifolds, including algebraic and symplectic manifolds and contact manifolds.Tutorial 2: Elliptic Surfaces. Self-contained introduction to the modern theory of elliptic surfaces for ground fields of characteristic zero and of positive characteristic. An in-depth analysis will be devoted to rational elliptic surfaces and elliptic K3 surfaces.\n\n[*Mathematics 301. Theory and Practice of Teaching in the Mathematical Sciences]\nCatalog Number: 4344\n----------\nNote: Expected to be given in 2008–09.\n\n*Mathematics 302. Topics in Dynamics of Group Actions\nCatalog Number: 5763\nDanijela Damjanovic 5583\n\n*Mathematics 304. Topics in Algebraic Topology\nCatalog Number: 0689\nMichael J. Hopkins 4376\n\n*Mathematics 307. Topics in Differential Geometry and Partial Differential Equations\nCatalog Number: 5133\nBenjamin Weinkove 4942\n\n*Mathematics 308. Topics in Number Theory and Modular Forms\nCatalog Number: 0464\nBenedict H. Gross 1112\n\n*Mathematics 310. Topics in Number Theory\nCatalog Number: 3874\nSamit Dasgupta 5030\n\n*Mathematics 314. Topics in Differential Geometry and Mathematical Physics\nCatalog Number: 2743\nShlomo Z. Sternberg 1965\n\n*Mathematics 318. Topics in Number Theory\nCatalog Number: 7393\nBarry C. Mazur 1975 (on leave spring term)\n\n*Mathematics 319. Topics in Representation Theory\nCatalog Number: 9591\nJohn F. Duncan 5505\n\n*Mathematics 321. Topics in Mathematical Physics\nCatalog Number: 2297\nArthur M. Jaffe 2095\n\n*Mathematics 327. Topics in Several Complex Variables\nCatalog Number: 0409\nYum Tong Siu 7550 (on leave fall term)\n\n*Mathematics 328. Topics in Lie Algebra\nCatalog Number: 7003\nAlberto De Sole 4627\n\n*Mathematics 333. Topics in Complex Analysis, Dynamics and Geometry\nCatalog Number: 9401\nCurtis T. McMullen 3588 (on leave spring term)\n\n*Mathematics 335. Topics in Differential Geometry and Analysis\nCatalog Number: 5498\nClifford Taubes 1243\n\n*Mathematics 342. Topics in Combinatorics\nCatalog Number: 0751\nThomas Lam 5322 (on leave spring term)\n\n*Mathematics 345. Topics in Geometry and Topology\nCatalog Number: 4108\nPeter B. Kronheimer 1759\n\n*Mathematics 346y. Topics in Analysis: Quantum Dynamics\nCatalog Number: 1053\nHorng-Tzer Yau 5260 (on leave spring term)\n\n*Mathematics 347. Topics in Floer Homology and Low Dimensional Topology\nCatalog Number: 7227\nEaman Eftekhary 5045\n\n*Mathematics 350. Topics in Mathematical Logic\nCatalog Number: 5151\nGerald E. Sacks 3862 (on leave fall term)\n\n*Mathematics 351. Topics in Algebraic Number Theory\nCatalog Number: 3492\nRichard L. Taylor 1453\n\n*Mathematics 356. Topics in Harmonic Analysis\nCatalog Number: 6534\nWilfried Schmid 5097 (on leave 2007-08)\n\n*Mathematics 365. Topics in Differential Geometry\nCatalog Number: 4647\nShing-Tung Yau 1734\n\n*Mathematics 371. Topics in Partial Differential Equations and Mathematical Physics\nCatalog Number: 0777\nRobert M. Strain 5323\n\n*Mathematics 372. Topics in Mathematical Relativity\nCatalog Number: 1150\nLydia Rosina Bieri 5794\n\n*Mathematics 379. Topics in Combinatorics\nCatalog Number: 3390\nLauren K. Williams 5499 (on leave spring term)\n\n*Mathematics 381. Introduction to Geometric Representation Theory\nCatalog Number: 0800\nDennis Gaitsgory 5259 (on leave spring term)\n\n*Mathematics 382. Topics in Algebraic Geometry\nCatalog Number: 2037\nJoseph D. Harris 2055\n\n*Mathematics 383. Topics in Algebraic Geometry\nCatalog Number: 7736\nIlia Zharkov 4631\n\n*Mathematics 384. Topics in Automorphic Forms\nCatalog Number: 8009\nLior Silberman 5506 (on leave 2007-08)\n\n*Mathematics 388. Topics in Mathematics and Biology\nCatalog Number: 4687\nMartin A. Nowak 4568 (on leave fall term)\n\n*Mathematics 389. Topics in Number Theory\nCatalog Number: 6851\nNoam D. Elkies 2604 (on leave 2007-08)\n\n*Mathematics 398. Topics in Algebraic and Geometric Topology\nCatalog Number: 0863\nVéronique Godin 5311" ]

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[ "# Abstract:\n\nThis investigation was initiated to develop analytic procedures for estimating aerodynamic coefficient derivatives for missiles. The analytic estimates will depend primarily on the geometrical configurations of the missiles. The problem of determining the coefficient derivatives becomes reasonably tractable for thin airfoil-body combinations with moderate finite aspect ratios and flying at sonic speeds. Starting with the equations for a perfect gas, a linearization of them is achieved by assuming flow over a thin profile. A further assumption of a speed of Mach 1 gives rise to slender body theory. The problem is thus reduced to a potential boundary value problem in a cross-flow plane. Upon consideration of the total momentum in a cross-flow slab, it is found that the resultant lateral force may be expressed as a countour integral of the velocity potential. The effects of missile angle of attack and control-surface angle are incorporated by way of Neumann-type conditions on the boundary contour. For the special case where the missile cross section is a circle with midwing, there is an analytic solution for the potential-flow problem. For the case where there is an arbitrary missile cross section, a computer program has been developed which addresses the problem using a source distribution approach. Results are given for several sample cross sections. It is shown how the cross-flow results may be applied to a typical missile configuration to obtain the aerodynamic coefficient derivatives. Author\n\n# Subject Categories:\n\n• Guided Missiles\n• Fluid Mechanics" ]

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[ "Division table for N = 7004 / 5999÷6000\n\n7004 / 5999 = 1.1675 [+]\n7004 / 5999.01 = 1.1675 [+]\n7004 / 5999.02 = 1.1675 [+]\n7004 / 5999.03 = 1.1675 [+]\n7004 / 5999.04 = 1.1675 [+]\n7004 / 5999.05 = 1.1675 [+]\n7004 / 5999.06 = 1.1675 [+]\n7004 / 5999.07 = 1.1675 [+]\n7004 / 5999.08 = 1.1675 [+]\n7004 / 5999.09 = 1.1675 [+]\n7004 / 5999.1 = 1.1675 [+]\n7004 / 5999.11 = 1.1675 [+]\n7004 / 5999.12 = 1.1675 [+]\n7004 / 5999.13 = 1.1675 [+]\n7004 / 5999.14 = 1.1675 [+]\n7004 / 5999.15 = 1.1675 [+]\n7004 / 5999.16 = 1.1675 [+]\n7004 / 5999.17 = 1.1675 [+]\n7004 / 5999.18 = 1.1675 [+]\n7004 / 5999.19 = 1.1675 [+]\n7004 / 5999.2 = 1.1675 [+]\n7004 / 5999.21 = 1.1675 [+]\n7004 / 5999.22 = 1.1675 [+]\n7004 / 5999.23 = 1.1675 [+]\n7004 / 5999.24 = 1.1675 [+]\n7004 / 5999.25 = 1.1675 [+]\n7004 / 5999.26 = 1.1675 [+]\n7004 / 5999.27 = 1.1675 [+]\n7004 / 5999.28 = 1.1675 [+]\n7004 / 5999.29 = 1.1675 [+]\n7004 / 5999.3 = 1.1675 [+]\n7004 / 5999.31 = 1.1675 [+]\n7004 / 5999.32 = 1.1675 [+]\n7004 / 5999.33 = 1.1675 [+]\n7004 / 5999.34 = 1.1675 [+]\n7004 / 5999.35 = 1.1675 [+]\n7004 / 5999.36 = 1.1675 [+]\n7004 / 5999.37 = 1.1675 [+]\n7004 / 5999.38 = 1.1675 [+]\n7004 / 5999.39 = 1.1675 [+]\n7004 / 5999.4 = 1.1675 [+]\n7004 / 5999.41 = 1.1674 [+]\n7004 / 5999.42 = 1.1674 [+]\n7004 / 5999.43 = 1.1674 [+]\n7004 / 5999.44 = 1.1674 [+]\n7004 / 5999.45 = 1.1674 [+]\n7004 / 5999.46 = 1.1674 [+]\n7004 / 5999.47 = 1.1674 [+]\n7004 / 5999.48 = 1.1674 [+]\n7004 / 5999.49 = 1.1674 [+]\n7004 / 5999.5 = 1.1674 [+]\n7004 / 5999.51 = 1.1674 [+]\n7004 / 5999.52 = 1.1674 [+]\n7004 / 5999.53 = 1.1674 [+]\n7004 / 5999.54 = 1.1674 [+]\n7004 / 5999.55 = 1.1674 [+]\n7004 / 5999.56 = 1.1674 [+]\n7004 / 5999.57 = 1.1674 [+]\n7004 / 5999.58 = 1.1674 [+]\n7004 / 5999.59 = 1.1674 [+]\n7004 / 5999.6 = 1.1674 [+]\n7004 / 5999.61 = 1.1674 [+]\n7004 / 5999.62 = 1.1674 [+]\n7004 / 5999.63 = 1.1674 [+]\n7004 / 5999.64 = 1.1674 [+]\n7004 / 5999.65 = 1.1674 [+]\n7004 / 5999.66 = 1.1674 [+]\n7004 / 5999.67 = 1.1674 [+]\n7004 / 5999.68 = 1.1674 [+]\n7004 / 5999.69 = 1.1674 [+]\n7004 / 5999.7 = 1.1674 [+]\n7004 / 5999.71 = 1.1674 [+]\n7004 / 5999.72 = 1.1674 [+]\n7004 / 5999.73 = 1.1674 [+]\n7004 / 5999.74 = 1.1674 [+]\n7004 / 5999.75 = 1.1674 [+]\n7004 / 5999.76 = 1.1674 [+]\n7004 / 5999.77 = 1.1674 [+]\n7004 / 5999.78 = 1.1674 [+]\n7004 / 5999.79 = 1.1674 [+]\n7004 / 5999.8 = 1.1674 [+]\n7004 / 5999.81 = 1.1674 [+]\n7004 / 5999.82 = 1.1674 [+]\n7004 / 5999.83 = 1.1674 [+]\n7004 / 5999.84 = 1.1674 [+]\n7004 / 5999.85 = 1.1674 [+]\n7004 / 5999.86 = 1.1674 [+]\n7004 / 5999.87 = 1.1674 [+]\n7004 / 5999.88 = 1.1674 [+]\n7004 / 5999.89 = 1.1674 [+]\n7004 / 5999.9 = 1.1674 [+]\n7004 / 5999.91 = 1.1674 [+]\n7004 / 5999.92 = 1.1673 [+]\n7004 / 5999.93 = 1.1673 [+]\n7004 / 5999.94 = 1.1673 [+]\n7004 / 5999.95 = 1.1673 [+]\n7004 / 5999.96 = 1.1673 [+]\n7004 / 5999.97 = 1.1673 [+]\n7004 / 5999.98 = 1.1673 [+]\nNavigation: Home | Addition | Substraction | Multiplication | Division       Tables for 7004: Addition | Substraction | Multiplication | Division\n\nOperand: 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 5991 5992 5993 5994 5995 5996 5997 5998 5999 6000 6001 6002 6003 6004 6005 6006 6007 6008 6009 7000 8000 9000\n\nDivision for: 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 7001 7002 7003 7004 7005 7006 7007 7008 7009 8000 9000" ]

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[ "# Markov chain monte carlo methods for switching diffusion models\n\nJohn C. Liechty, Gareth O. Roberts\n\nResearch output: Contribution to journalArticle\n\n20 Citations (Scopus)\n\n### Abstract\n\nReversible jump Metropolis-Hastings updating schemes can be used to analyse continuous-time latent models, sometimes known as state space models or hidden Markov models. We consider models where the observed process X can be represented as a stochastic differential equation and where the latent process D is a continuous-time Markov chain. We develop Markov chain Monte Carlo methods for analysing both Markov and non-Markov versions of these models. As an illustration of how these methods can be used in practice we analyse data from the New York Mercantile Exchange oil market. In addition, we analyse data generated by a process that has linear and mean reverting states.\n\nOriginal language English (US) 299-315 17 Biometrika 88 2 https://doi.org/10.1093/biomet/88.2.299 Published - Jan 1 2001\n\n### Fingerprint\n\nMonte Carlo Method\nMarkov Chains\nMonte Carlo method\nMarkov Chain Monte Carlo Methods\nDiffusion Model\nMarkov processes\nMonte Carlo methods\nSpace Simulation\nReversible Jump\nLatent Process\nMetropolis-Hastings\nOils\nContinuous-time Markov Chain\nState-space Model\nMarkov Model\nUpdating\nStochastic Equations\nContinuous Time\nHidden Markov models\nModel\n\n### All Science Journal Classification (ASJC) codes\n\n• Statistics and Probability\n• Mathematics(all)\n• Agricultural and Biological Sciences (miscellaneous)\n• Agricultural and Biological Sciences(all)\n• Statistics, Probability and Uncertainty\n• Applied Mathematics\n\n### Cite this\n\nLiechty, John C. ; Roberts, Gareth O. / Markov chain monte carlo methods for switching diffusion models. In: Biometrika. 2001 ; Vol. 88, No. 2. pp. 299-315.\n@article{2280042dba39448f9b9fe02b9dc786e3,\ntitle = \"Markov chain monte carlo methods for switching diffusion models\",\nabstract = \"Reversible jump Metropolis-Hastings updating schemes can be used to analyse continuous-time latent models, sometimes known as state space models or hidden Markov models. We consider models where the observed process X can be represented as a stochastic differential equation and where the latent process D is a continuous-time Markov chain. We develop Markov chain Monte Carlo methods for analysing both Markov and non-Markov versions of these models. As an illustration of how these methods can be used in practice we analyse data from the New York Mercantile Exchange oil market. In addition, we analyse data generated by a process that has linear and mean reverting states.\",\nauthor = \"Liechty, {John C.} and Roberts, {Gareth O.}\",\nyear = \"2001\",\nmonth = \"1\",\nday = \"1\",\ndoi = \"10.1093/biomet/88.2.299\",\nlanguage = \"English (US)\",\nvolume = \"88\",\npages = \"299--315\",\njournal = \"Biometrika\",\nissn = \"0006-3444\",\npublisher = \"Oxford University Press\",\nnumber = \"2\",\n\n}\n\nMarkov chain monte carlo methods for switching diffusion models. / Liechty, John C.; Roberts, Gareth O.\n\nIn: Biometrika, Vol. 88, No. 2, 01.01.2001, p. 299-315.\n\nResearch output: Contribution to journalArticle\n\nTY - JOUR\n\nT1 - Markov chain monte carlo methods for switching diffusion models\n\nAU - Liechty, John C.\n\nAU - Roberts, Gareth O.\n\nPY - 2001/1/1\n\nY1 - 2001/1/1\n\nN2 - Reversible jump Metropolis-Hastings updating schemes can be used to analyse continuous-time latent models, sometimes known as state space models or hidden Markov models. We consider models where the observed process X can be represented as a stochastic differential equation and where the latent process D is a continuous-time Markov chain. We develop Markov chain Monte Carlo methods for analysing both Markov and non-Markov versions of these models. As an illustration of how these methods can be used in practice we analyse data from the New York Mercantile Exchange oil market. In addition, we analyse data generated by a process that has linear and mean reverting states.\n\nAB - Reversible jump Metropolis-Hastings updating schemes can be used to analyse continuous-time latent models, sometimes known as state space models or hidden Markov models. We consider models where the observed process X can be represented as a stochastic differential equation and where the latent process D is a continuous-time Markov chain. We develop Markov chain Monte Carlo methods for analysing both Markov and non-Markov versions of these models. As an illustration of how these methods can be used in practice we analyse data from the New York Mercantile Exchange oil market. In addition, we analyse data generated by a process that has linear and mean reverting states.\n\nUR - http://www.scopus.com/inward/record.url?scp=0242546942&partnerID=8YFLogxK\n\nUR - http://www.scopus.com/inward/citedby.url?scp=0242546942&partnerID=8YFLogxK\n\nU2 - 10.1093/biomet/88.2.299\n\nDO - 10.1093/biomet/88.2.299\n\nM3 - Article\n\nAN - SCOPUS:0242546942\n\nVL - 88\n\nSP - 299\n\nEP - 315\n\nJO - Biometrika\n\nJF - Biometrika\n\nSN - 0006-3444\n\nIS - 2\n\nER -" ]

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[ "Breadth-first search (BFS) is an algorithm for traversing or searching tree or graph data structures. It starts at the tree root (or some arbitrary node of a graph, sometimes referred to as a 'search key'), and explores all of the neighbor nodes at the present depth prior to moving on to the nodes at the next depth level.\n\nClass", null, "Order in which the nodes are expanded Search algorithm Graph $O(|V|+|E|)=O(b^{d})$", null, "$O(|V|)=O(b^{d})$", null, "It uses the opposite strategy as depth-first search, which instead explores the node branch as far as possible before being forced to backtrack and expand other nodes.\n\nBFS and its application in finding connected components of graphs were invented in 1945 by Konrad Zuse, in his (rejected) Ph.D. thesis on the Plankalkül programming language, but this was not published until 1972. It was reinvented in 1959 by Edward F. Moore, who used it to find the shortest path out of a maze, and later developed by C. Y. Lee into a wire routing algorithm (published 1961).\n\n## Pseudocode\n\nInput: A graph G and a starting vertex root of G\n\nOutput: Goal state. The parent links trace the shortest path back to root\n\n1 procedure BFS(G, root) is\n2 let Q be a queue\n3 label root as discovered\n4 Q.enqueue(root)\n5 while Q is not empty do\n6 v := Q.dequeue()\n7 if v is the goal then\n8 return v\n9 for all edges from v to w in G.adjacentEdges(v) do\n10 if w is not labeled as discovered then\n11 label w as discovered\n12 w.parent := v\n13 Q.enqueue(w)\n\n\n### More details\n\nThis non-recursive implementation is similar to the non-recursive implementation of depth-first search, but differs from it in two ways:\n\n1. it uses a queue (First In First Out) instead of a stack and\n2. it checks whether a vertex has been discovered before enqueueing the vertex rather than delaying this check until the vertex is dequeued from the queue.\n\nIf G is a tree, replacing the queue of this breadth-first search algorithm with a stack will yield a depth-first search algorithm. For general graphs, replacing the stack of the iterative depth-first search implementation with a queue would also produce a breadth-first search algorithm, although a somewhat nonstandard one.\n\nThe Q queue contains the frontier along which the algorithm is currently searching.\n\nNodes can be labelled as discovered by storing them in a set, or by an attribute on each node, depending on the implementation.\n\nNote that the word node is usually interchangeable with the word vertex.\n\nThe parent attribute of each node is useful for accessing the nodes in a shortest path, for example by backtracking from the destination node up to the starting node, once the BFS has been run, and the predecessors nodes have been set.\n\nBreadth-first search produces a so-called breadth first tree. You can see how a breadth first tree looks in the following example.\n\n### Example\n\nThe following is an example of the breadth-first tree obtained by running a BFS on German cities starting from Frankfurt:\n\n## Analysis\n\n### Time and space complexity\n\nThe time complexity can be expressed as $O(|V|+|E|)$ , since every vertex and every edge will be explored in the worst case. $|V|$  is the number of vertices and $|E|$  is the number of edges in the graph. Note that $O(|E|)$  may vary between $O(1)$  and $O(|V|^{2})$ , depending on how sparse the input graph is.\n\nWhen the number of vertices in the graph is known ahead of time, and additional data structures are used to determine which vertices have already been added to the queue, the space complexity can be expressed as $O(|V|)$ , where $|V|$  is the number of vertices. This is in addition to the space required for the graph itself, which may vary depending on the graph representation used by an implementation of the algorithm.\n\nWhen working with graphs that are too large to store explicitly (or infinite), it is more practical to describe the complexity of breadth-first search in different terms: to find the nodes that are at distance d from the start node (measured in number of edge traversals), BFS takes O(bd + 1) time and memory, where b is the \"branching factor\" of the graph (the average out-degree).:81\n\n### Completeness\n\nIn the analysis of algorithms, the input to breadth-first search is assumed to be a finite graph, represented explicitly as an adjacency list, adjacency matrix, or similar representation. However, in the application of graph traversal methods in artificial intelligence the input may be an implicit representation of an infinite graph. In this context, a search method is described as being complete if it is guaranteed to find a goal state if one exists. Breadth-first search is complete, but depth-first search is not. When applied to infinite graphs represented implicitly, breadth-first search will eventually find the goal state, but depth-first search may get lost in parts of the graph that have no goal state and never return.\n\n## BFS ordering\n\nAn enumeration of the vertices of a graph is said to be a BFS ordering if it is the possible output of the application of BFS to this graph.\n\nLet $G=(V,E)$  be a graph with $n$  vertices. Recall that $N(v)$  is the set of neighbors of $v$ . For $\\sigma =(v_{1},\\dots ,v_{m})$  be a list of distinct elements of $V$ , for $v\\in V\\setminus \\{v_{1},\\dots ,v_{m}\\}$ , let $\\nu _{\\sigma }(v)$  be the least $i$  such that $v_{i}$  is a neighbor of $v$ , if such a $i$  exists, and be $\\infty$  otherwise.\n\nLet $\\sigma =(v_{1},\\dots ,v_{n})$  be an enumeration of the vertices of $V$ . The enumeration $\\sigma$  is said to be a BFS ordering (with source $v_{1}$ ) if, for all $1 , $v_{i}$  is the vertex $w\\in V\\setminus \\{v_{1},\\dots ,v_{i-1}\\}$  such that $\\nu _{(v_{1},\\dots ,v_{i-1})}(w)$  is minimal. Equivalently, $\\sigma$  is a BFS ordering if, for all $1\\leq i  with $v_{i}\\in N(v_{k})\\setminus N(v_{j})$ , there exists a neighbor $v_{m}$  of $v_{j}$  such that $m .\n\n## Applications\n\nBreadth-first search can be used to solve many problems in graph theory, for example:" ]

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[ "# Journal of Symplectic Geometry\n\n## Volume 19 (2021)\n\n### Holomorphic disks and the disk potential for a fibered Lagrangian\n\nPages: 143 – 239\n\nDOI: https://dx.doi.org/10.4310/JSG.2021.v19.n1.a4\n\n#### Author\n\nDouglas Schultz (Department of Mathematics, Humboldt Universität zu Berlin, Germany)\n\n#### Abstract\n\nWe consider a fibered Lagrangian $L$ in a compact symplectic fibration with small monotone fibers, and develop a strategy for lifting $J$-holomorphic disks with Lagrangian boundary from the base to the total space. In case $L$ is a product, we use this machinery to give a formula for the leading order potential and formulate an unobstructedness criteria for the $A_\\infty$ algebra. We provide some explicit computations, one of which involves finding an embedded $2n + k$ dimensional submanifold of Floer-non-trivial tori in a $2n + 2k$ dimensional fiber bundle." ]

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[ "# www.campusplace.co.in\n\nOne stop blog for all competitive examinations\n\nAptitude Questions with Answers – Problems on Ages 11 11. The ratio of the present age of A and B is 1:2 and 5 years ago, the ratio of t...\n\n# GMAT word problems - Problems on Ages\n\nAptitude Questions with Answers – Problems on Ages 11\n11. The ratio of the present age of A and B is 1:2 and 5 years ago, the ratio of their ages was 1:3. What will be the ratio of their ages after 5 years?\na)2:3            b)4:7            c)3:5            d)7:8\nExplanation: Let the present ages of A and B be x and 2x years respectively.\n5 years ago, their ages were x-5 and 2x-5.\n5 years ago, their ages were in the ratio of 1:3.\nA’s age 5 years ago/ B’s age 5 years ago =\n1 / 3\n=>\nx-5 / 2x-5\n=\n1 / 3\n=>3x-15=2x-5 =>x=10.\nTheir present ages are 10 and 20.\nAfter 5 years, their ages will be 10+5 and 20+5 = 15 and 25\nRatio of their ages after 5 years =15:25 = 3:5\n\nQuantitative Aptitude for CAT – Age related problems\n12.The sum of the ages of 5 girls each both at 3 years intervals is 60 years. What is the age of eldest girl?\na)12 years  b)16 years      c)18 years     d)20 years\nExplanation : Let the age of the youngest girl be x.\nThen each both at 3 years interval, the ages will be x ,x+3, x+6, x+9 ,x+12 .\n=>5x+30=60 >x=6.\nTherefore, age of the eldest girl= x+12 =6+12=18.\n\nAptitude Problems with Answers – Problems on Ages\n13. A’s grandfather was 8 times older to him 16 years ago. His grandfather would be 3 times of his age 8 years from now. 8 years ago, what was the ratio of A ‘s age to that of his grandfather’s age?\na)53:11      b)12:47       c)11:53      d)19:21\nExplanation : Let 16 years ago, A’s age be x and his grandfathers be 8x.\nPresent age of A is x+16 and and grandfathers is 8x+16\nAfter 8 years,  A’s age will be  x+24 and his grandfather’s age will be 8x+24.\nAfter 8 years, grandfather will be 3 times of A.\n\nGrandfather’s age after 8 years / A’s age after 8 years\n=\n3 / 1\n\n8x+24 / x+24\n=\n3 / 1\n=>8x+24=3x+72=> x=\n48 / 5\nRatio of their ages 8 years ago\n=>x+8: 8x+8 =\n48 / 5\n+8  : 8 x\n48 / 5\n+ 8 = 11:53\n\nGMAT Word Problems – Problems on Ages\n14. At present age of the mother is 5 times that of the age of her daughter. After 3 years, age of the mother would be 4 times the age of her daughter. What is the age of the daughter after 3 years?\na)12 years     b)15 years      c)17 years      d)20 years\nExplanation : Mother’s age is 5 times the age of the daughter.\nLet age of the daughter be x and age of the mother be 5x.\nAfter 3 years, their ages will be x+3 and 5x+3.\nGiven that after 3 years, age of mother will be 4 times the age of daughter.\nThe ratio of the ages of mother and daughter after 3 years = 4:1\nTherefore,\n5x+3 / x+3\n=\n4 / 1\n=>5x+3 = 4x+12\n=>x=9\nPresent age of the daughter is 9 years.\nAfter 3 years, age of the daughter will be  x+3 = 9+3 = 12 years\n\nAptitude Questions with Explanations – Problems on Ages\n15. A teacher is 3 times as old as his student. 5 years hence, teacher will be 2 ½ times as old as his student. What is the difference between the ages of teacher and student?\na)40 years    b)50 years      c)60 years     d)5 years\nExplanation:  Let the age of the student be x years . Teacher’s age = 3x\nAfter 5 years, their ages will be x+5 and 3x+ 5.\nAfter 5 years, their ages will be in ration: 1: 2 ½ =2:5\n\nx+5 / 3x+5\n=\n2 / 5\n=>5 (x+5)= 2(3x+5) =>5x+25=x+10 =>x=15 years.\nThe difference between their ages is=>3x-x=2x =2 x 15 = 30 years\n\nProblems on Ages solved problems for Competitive Exams\n16.  6 years ago, the ratio of ages of A and B was 5:6 and 4 years hence, the ratio of their ages will be 10:11. What is the present age of B?\na) 10 years    b) 12 years      c)15 years       d)16 years\nExplanation:\n6 years ago, the ages of A and B were in the ratio of 5:6.\n6 years ago, let the age of A and B were 5x and 6x .\nTheir present ages = 5x+6 and 6x+6\nAfter 4 years , their ages will be in the ratio 10:11\nAfter 4 years , age of A and B will be 5x+6+4  and  6x+ 6 +4 => 5x+10 and 6x+10\n5x+10 / 6x+10\n=\n10 / 11\n=>55x+110 = 60x+100 => x=2.\nPresent age of B= 5x+6 = 5 x 2 +6 =16 years\n\nAptitude for Campus Placements : Problems on Ages solved questions 17.\n17. The sum of the ages of a father and son is 50 years. 5 years ago, the age of the father was 7 times the age of the son. What will be age of the father after 7 years?\na)42 years     b)35 years    c)45 years       d)49 years\nExplanation: 5 years ago, the age of the father was 7 times the age of the son.\n5 years ago , the age of the son was x and father was 7x.\nPresent sum of the ages of father and son = 50 years\n5 years ago, their sum of the ages was 50 – 2x5 = 40 years\nAge of the father , 5 years ago= 40 x\n7 / 8\n=35 years\nPresent age of the father= 35+5 = 40 years\nAfter 9 years, age of the father will be = 40 +7 = 47 years.\n\nGMAT Quantitative Aptitude study material – Problems on Ages\n18. The ratio of the ages of A and B is 1:4 and the product of their ages is 256. What will be the ratio of their ages after 8 years?\na) 3:5    b)10:3        c)3:10   d)None of these\nExplanation:  let the age of A and B be x and 4x.\nProduct of their ages = x x 4x =4x2 =256 =>x=8\nPresent age of A and B = x and 4x = 8 years and 32  years\nRatio of their ages after 8 years will be = 8+8 : 32+8\n=16:40= 2:5\n\nArithmetic free study material- Age related Problems\n19. At the time of marriage Siva was 6 years elder to his wife and 12 years after the  marriage  his age was\n6 / 5\ntimes the age of his wife. What was the age of his wife at the time of marriage?\na) 20 years    b)24 years     c)28 years     d)32 years\nExplanation : Let the age of Siva at the time of marriage be x years.\nThen, age of Siva’s wife at the time of marriage = x-6 years\nAfter 12 years , ratio of their ages was 6:5.\nThe difference between their ages = 6-5= 1 part = 6 years\nTheir ages 12 years after their marriage were 6x6 and 5x6 = 36 and 30.\nAt the time of marriage, their ages were, 36-12 and 30-12 = 24 and 18\nTherefore, at the time of marriage, age of Siva's wife was =18 years\n\nAptitude Questions with Explanations – Problems on Ages\n20. After 15 years the age of Murthy will be 4 times the age which was 15 years ago. Find the present age of Murthy?\na) 25 years    b)30 years     c)35 years     d)40 years" ]

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[ "", null, "The derivative oscillator boosts the benefits of the double smoothed relative strength index (RSI) and the moving average convergence divergence (MACD).\n\nThe derivative oscillator atempts to boost the individual predicting capabilities of the RSI and MACD and it’s good for trend following and pointing to price reversals.\n\n## The Derivative Oscillator Calculation\n\nThere are quite a few steps to this. The first part involves working out the RSI, which is then levelled off by taking an exponential moving average of itself. An exponential moving average is next taken of the already smoother result which adds additional smoothing. A moving average of the RSI is taken next to give us the signal line. We find the derivative oscillator by taking away the signal line value from that of the double smoothed RSI.\n\nMathematically it looks like this:\n\n1. RSI = 100 – 100/(1 + RS)\n\nRS = Average Gains / Average Losses\n\nAverage Gains are worked out as the average price change for the positive price bars in the included sequence of data.\n\nAverage Losses are worked out as the average price change for the negative price bars.\n\nA simplified RSI calculation looks like this:\n\nSimplified RSI = Gains / (Gains + Losses)*100\n\nThe positive price bars are added together to show gains. The negative price bars are added together to show losses.\n\n2. An EMA is now added to the RSI.\n\nSmoothed RSI = EMA(RSI)\n\nDouble Smoothed RSI = EMA(Smoothed RSI)\n\n3. Next, we create a signal line which is the same as a simple moving average (SMA) or the double smoothed RSI.\n\nSignal Line = SMA(Double Smoothed RSI)\n\n4. The final step with this is to take away the signal line from the double smoothed RSI to give us the derivative oscillator.\n\nDerivative Oscillator = Double Smoothed RSI – Signal Line\n\nHow to Use the Derivative Oscillator\n\nSimply enough, a positive oscillator (one where there’s an upslope) is a bullish sign and a negative one (downslope present) is bearish. Zero crossings in either direction point to a possible trade signal.\n\nYou can work out how strong some trends are from the sizes of the different bars. A bar with a higher positive or negative magnitude means a stronger uptrend or downtrend respectively.\n\n## Conclusion\n\nThe derivative oscillator builds on the relative strength index (RSI) and shows some visual resemblance to a MACD histogram.\n\nThe fact that it combines trend following and trend reversal features means that it’s found favor with some traders.\n\nZero crossings are bullish signals if they’re heading upwards, and bearish if they are heading downwards. Bar size in either direction is telling you how strong the trend is.\n\nThe derivative oscillator is an excellent tool that should nevertheless only be used as part of a comprehensive investment toolkit. Using it on its own is potentially risky and inadvisable.", null, "" ]

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[ "Microeconomics Assignment", null, "Words: 1228\n\nWhy do economists use percentages rather than absolute amounts in measuring the responsiveness of consumers to changes in price? Why do economists use percentages rather than absolute amounts in measuring the responsiveness of consumers to changes in price? Economists use percentages rather than absolute amounts for two different reasons. The first reason for using percentages rather than absolute amounts has to do with the affect a particular amount can have on demand.\n\nThe example in our book refers to using dollars or pennies, in one instance the dollar amount dads to a demand that is elastic, however that same dollar amount in pennies would lead one to see that demand is inelastic. The amount is the same, regardless of the currency, therefore the demand should be the same too. This is one reason why economists use percentages. The second reason deals with comparisons. Percentages help us more accurately compare the consumer response to a change, rather than using dollar amounts.\n\nDon’t waste your time!\n\norder now\n\nThe example in the book illustrates a \\$1 increase on a very expensive item and a very inexpensive item, although the absolute amount increased is the same, he percentages are at opposite ends of the spectrum 100% and . 01% respectively. 3. How do you interpret the coefficient of the price elasticity of demand? Explain when Deed is 1. 5, 0. 7, and 1. 0. How do you interpret the coefficient of the price elasticity of demand? Explain when Deed is 1. 5, 0. 7, and 1. 0. The coefficient Deed is used to measure elasticity or inelastic of demand. The coefficient can be interpreted by performing an equation." ]

[ null, "https://anyassignment.com/wp-content/themes/vihaan-blog-lite/assets/images/download-icon.png", null ]

[ "", null, "Software … powerful tools for your research & development!\n\n# RP Coating – Advanced Software forDesigning Optical Multilayer Structures\n\n## Demo File: Bragg Mirror\n\nWith this demo file, containing a custom form, we can conveniently analyze the properties of a Bragg mirror structure, containing some number of layer pairs with quarter-wave thickness.\n\nBy the way, the underlying script code for defining that structure is rather simple:\n\n```beam from superstrate\nsubstrate: (material_s\\$)\nfor j := 1 to N_Bragg do\nbegin\n* (material1\\$), l/4 at l_Bragg\n* (material2\\$), l/4 at l_Bragg\nend\nsuperstrate: air\n```\n\nWhen executing the calculation, we get the following as the first graphical diagram:\n\nThe next diagram is a color diagram, showing how the reflectivity spectra vary with the angle of incidence for p polarization:\n\nHere is the script code producing that diagram:\n\n```diagram 2:\n\n\"Reflectivity for Different Angles of Incidence\"\n\n[if theta <> 0 then\n(\"(for \" + pol\\$ + \" polarization, \" + str(theta_deg:\"°\") + \")\")\nelse\n\"(for normal incidence)\"]\n\nx: lambda_min, lambda_max\n\"wavelength (nm)\", @x\ny: -90, +90\n\"angle of incidence (°)\", @y\nframe\nhx\nhy\n\nif pol\\$ = \"p\" then\ncp: color_I(R_p(x, y * deg))\nif pol\\$ = \"s\" then\ncp: color_I(R_s(x, y * deg))\n```\n\nAdditional diagrams, not shown here, contain plots of the group delay and the group delay dispersion as functions of the wavelength.\n\nThe next diagram shows how the optical field intensity varies within the mirror structure:\n\nNext we make a color diagram showing the field penetration at different wavelengths. For better orientation, we also show the layer boundaries:\n\n```diagram 6:\n\n\"Field Penetration\"\n\n[if theta <> 0 then\n(pol\\$ + \" polarization, \" + str(theta_deg:\"°\"))\nelse\n\"normal incidence\"]\n\nx: lambda_min, lambda_max\n\"wavelength (nm)\", @x\ny: -500, get_d(0) + 1000\n\"depth in the mirror structure (nm)\", @y\nframe\n\n! begin\nvar x1, x2;\nx1 := CS_x2 + 0.02 * ((CS_x2 - CS_x1));\nx2 := CS_x2 + 0.04 * ((CS_x2 - CS_x1));\nfor y := CS_y1 to CS_y2 step ((CS_y2 - CS_y1)) / 500 do\nbegin\nsetcolor(color_I((y - CS_y1) / ((CS_y2 - CS_y1))));\nline(x1 + i * y, x2 + i * y);\nend;\nsetcolor(black);\nend\n\n! set_dir(1)\n\ncp: color_I((if pol\\$ = \"s\" then\nT_s(x, theta)\nelse\nT_p(x, theta); E2(y) / 4)) { color plot }\n\n; Show the layer boundaries:\n! begin\nline(CS_x1, CS_x2); { zero line }\nline(CS_x1 + i * d_tot, CS_x2 + i * d_tot); { other boundary }\nsetcolor(lightgray);\nfor j := 2 to nolayers() do\nline(CS_x1 + i * get_z(j), CS_x2 + i * get_z(j))\nend\n\n\"substrate\", 150, (-200)c\n\"superstrate\", 150, (d_tot + 200)c\n```\n\nFinally, we want to see how the reflectivity profile evolves during growth of the structure. This requires a little script programming: we need to save the original thickness values and vary them during the plot:\n\n```diagram 7:\n\n\"Development of Reflectivity During Fabrication\"\n\n\"for normal incidence\"\n\nx: lambda_min, lambda_max\n\"wavelength (nm)\", @x, color = labelgray\ny: 0, get_d(0)\n\"thickness (nm)\", @y, color = labelgray\nframe\n\n; Save thickness values in an array:\nN := nolayers()\ndefarray d[1, N]\ncalc\nfor j := 1 to N do\nd[j] := get_d(j)\n\nset_d_g(d) :=\n{ set the layers for growth up to a total thickness d }\nbegin\nglobal N, d[];\nvar d_tot;\nd_tot := 0;\nfor j := 1 to N do\nbegin\nvar d_j;\nd_j := minr(d[j], d - d_tot); { thickness of the layer }\nset_d(j, d_j);\ninc(d_tot, d_j);\nend;\nend\n\nd_l := -1 { last d value set }\nR_g(l, d) := { reflectivity for a total growth thickness d }\nbegin\nglobal d_l;\nif d <> d_l then\nbegin\nset_d_g(d);\nd_l := d;\nend;\nR(l);\nend\n\ncp: color_I(R_g(x, y))\n\n; Indicate layer boundaries:\nf: get_z(j),\ncolor = gray,\nstyle = dashed,\nfor j := 2 to N\n```\n\nWhen only the first two layers are grown (bottom part), we get a weak but broadband reflection. Further layers increase the reflectivity, but only in a limited bandwidth.\n\nYou can see that with a few lines of script code one can create all sorts of diagrams – not only a few types envisaged by the software developer. Of course, you can use the technical support if you need some hints on how to make a new type of diagram.\n\n## How the Custom Form is Made\n\nIn the following, you see the beginning of the code (contained in the script) which defines the custom form:\n\n```Custom form:\n--------------------------------------------------------------\n\\$font: \"Arial\", bold, size = 20\nBragg Mirror\n\\$font: \"Courier New\", size = 11, space = 2.1\n\\$def pwidth := 505\n\\$box \"Inputs\", size = (pwidth, 0):\nSubstrate material: ##################\n\\$input (combobox: \"BK7\", \"CaF2\", \"diamond\", \"fsilica\", \"sapphire_o\", \"sapphire_e\", \"SF5\", \"SF8\", \"SF10\", \"SF11\", \"ZERODUR\") material_s\\$\nMaterial 1: ##################\n\\$input (combobox: \"SiO2\", \"TiO2\", \"HfO2\", \"ZrO2\") material1\\$\nMaterial 2: ##################\n\\$input (combobox: \"SiO2\", \"TiO2\", \"HfO2\", \"ZrO2\") material2\\$\nBragg wavelength: #############\n\\$input l_Bragg_m:d6:\"(n)m\", min = 100e-9, max = 10e-6\nNumber of layer pairs: #############\n\\$input N_Bragg:f0, min = 0, max = 1000\n```\n\nThis is what is needed to define the heading and the first few input fields." ]

[ null, "https://www.rp-photonics.com/img/rp_photonics.png", null ]

[ "", null, "", null, "", null, "", null, "", null, "# Conditional Poisson Stochastic Beam Search\n\nBeam search is the default decoding strategy for many sequence generation tasks in NLP. The set of approximate K-best items returned by the algorithm is a useful summary of the distribution for many applications; however, the candidates typically exhibit high overlap and may give a highly biased estimate for expectations under our model. These problems can be addressed by instead using stochastic decoding strategies. In this work, we propose a new method for turning beam search into a stochastic process: Conditional Poisson stochastic beam search. Rather than taking the maximizing set at each iteration, we sample K candidates without replacement according to the conditional Poisson sampling design. We view this as a more natural alternative to Kool et. al. 2019's stochastic beam search (SBS). Furthermore, we show how samples generated under the CPSBS design can be used to build consistent estimators and sample diverse sets from sequence models. In our experiments, we observe CPSBS produces lower variance and more efficient estimators than SBS, even showing improvements in high entropy settings.\n\n06/14/2021\n\n### Determinantal Beam Search\n\nBeam search is a go-to strategy for decoding neural sequence models. The...\n03/14/2019\n\n### Stochastic Beams and Where to Find Them: The Gumbel-Top-k Trick for Sampling Sequences Without Replacement\n\nThe well-known Gumbel-Max trick for sampling from a categorical distribu...\n10/05/2020\n\n### A Streaming Approach For Efficient Batched Beam Search\n\nWe propose an efficient batching strategy for variable-length decoding o...\n07/08/2020\n\n### Best-First Beam Search\n\nDecoding for many NLP tasks requires a heuristic algorithm for approxima...\n10/10/2020\n\n### An Empirical Investigation of Beam-Aware Training in Supertagging\n\nStructured prediction is often approached by training a locally normaliz...\n10/07/2016\n\n### Diverse Beam Search: Decoding Diverse Solutions from Neural Sequence Models\n\nNeural sequence models are widely used to model time-series data in many...\n03/28/2022\n\n### A Well-Composed Text is Half Done! Composition Sampling for Diverse Conditional Generation\n\nWe propose Composition Sampling, a simple but effective method to genera...\n\n## 1 Introduction\n\nMany NLP tasks require the prediction of structured outputs, such as sentences or parse trees, either during decoding or as part of a training algorithm. For today’s neural architectures, beam search reddy-1977 has become the decoding algorithm of choice due to its efficiency and empirical performance AAAI1714571; edunov-etal-2018-understanding; XLNET; meister-etal-2020-best. Beam search is a deterministic method, which invites a natural question: What is the proper stochastic generalization of beam search? Several recent papers have investigated this question kool2019stochastic; Kool2020Estimating; shi2020uniquerandomizer. Here we build on this line of work and introduce an alternative stochastic beam search that the authors contend is a more faithful stochasticization of the original algorithm in that it recovers standard beam search as a special case. We name our algorithm conditional Poisson stochastic beam search (CPSBS) as we draw on the conditional Poisson sampling scheme hajek1964 in its construction. The relationship between CPSBS and other common decoding strategies is displayed visually in Table 1.\n\nAt every iteration, CPSBS replaces the top- operator in the beam search algorithm with conditional Poisson sampling, resulting in a decoding strategy that generates samples-without-replacement. Importantly, annealing our sampling distribution at each time step turns local sampling into a local top- computation and thereby recovers beam search. We subsequently show that these samples can be used to construct a statistically consistent estimator for the expected value of an arbitrary function of the output.\n\nIn our experiments with neural machine translation models, we observe that CPSBS leads to better estimates of expected\n\nbleu and conditional model entropy than SBS and the sum-and-sample estimator Kool2020Estimating, distinctly outperforming Monte Carlo sampling for both small sample sizes and low temperatures. Furthermore, we find that CPSBS can be used as a diverse sampling strategy. We take these results as confirmation that CPSBS is a useful tool in the newfound arsenal of sampling strategies for neural sequence models.\n\n## 2 Beam Search\n\nIn this section, we overview the necessary background on neural sequence models and beam search in order to motivate our algorithm in § 3.\n\n##### Neural Sequence Models.\n\nWe consider locally normalized probabilistic models over sequences :\n\n p(y)=|y|∏t=1p(yt∣y\n\nwhere is a member of a set of well-formed outputs . In the context of language generation models, well-formed outputs are sequences of tokens from a vocabulary ; all begin and end with special tokens bos and eos, respectively. We use to represent the subsequence . In this work, we consider the setting where the maximum sequence length is upper-bounded; we denote this upper bound . Without loss of generality, we may condition on an input , as is necessary for machine translation and other conditional generation tasks.\n\n##### Beam Search.\n\nBeam search is a commonly used search heuristic for finding an approximate solution to the following optimization problem:\n\n y⋆=argmaxy∈Ylogp(y) (2)\n\nIts most straightforward interpretation is as a pruned version of breadth-first search, where the breadth of the search is narrowed to the top- candidates. However, here we will present beam search in a nonstandard lens meister-etal-2020-beam; meister-etal-2021-determinantal in order to emphasize the connection with our stochastic generalization in § 3. Specifically, we present the algorithm as iteratively finding the highest-scoring set under a specific set function.\n\nUnder this paradigm, the initial beam contains only the bos token. At subsequent steps , beam search selects the highest-scoring candidates from the set that we define below:222Sequences already ending in eos are not extended by and are simply added to the set “as is.”\n\n Yt−1∘V\\tiny def={y∘y∣y∈Yt−1 and y∈V} (3)\n\nwhere\n\nis sequence concatenation. Those candidate sets with collectively higher probability under the model\n\nhave higher score. This process continues until all end in eos, or . For notational ease, we define ; throughout this paper, we will assume and identify the elements of with the integers .\n\nWe can formulate the time-step dependent set function whose beam search finds as\n\n Qt(Yt∣Yt−1)\\tiny def% ∝{∏Nn=1wnif |Y|=K0otherwise (4)\n\nwhere is the weight of the element of . To recover beam search, we set our weights equal to probabilities under a model i.e. . Note that we leave the constraint that implicit in Eq. 4. As should be clear from notation, this set function only assigns positive scores to subsets of of size exactly and the assigned score is proportional to the product of the probability of the candidates under the model . Putting this all together, beam search may be viewed as the following iterative process: [ams align,colback=gray!35!white,colframe=white!5!white,arc=0pt,outer arc=0pt]Y_0 &= {bos }\nY_t & = argmax_Y_t’ ⊆B_t  Q_t(Y_t’ ∣Y_t-1)\n& return   Y_T\n\n## 3 Conditional Poisson Stochastic Beams\n\nOur paper capitalizes on a very simple observation: Rather than taking its , we may renormalize Eq. 4 into a distribution and sample-without-replacement a size set at each iteration:\n\n[ams align,colback=gray!35!white,colframe=white!5!white,arc=0pt,outer arc=0pt] Y_0 &= {bos }\nY_t & ∼Q_t(⋅∣Y_t-1)\n& return   Y_T\n\nThis recursion corresponds to performing conditional Poisson sampling (CPS; hajek1964;see App. A for overview), a common sampling-without-replacement design tille_book,333A sampling design is a probability distribution over sets of samples.\n\nat every time step. Thus we term this scheme conditional Poisson stochastic beam search. CPSBS gives us a probability distribution over\n\nsets of candidates of size , i.e., the final beam . We denote the CPSBS distribution and we write to indicate that is the stochastic beam at the end of a sampling run. We may write as a marginal probability, summing over all sequences of beams that could have resulted in :444This formulation reveals that it is wildly intractable to compute .\n\n P(YT)=∑Y1⋯∑YT−1T∏t=1Qt(Yt∣Yt−1) (5)\n\nNote the structural zeros of prevent any incompatible sequence of beams. We provide a theoretical analysis of the scheme in § 4 and an empirical analysis in § 5.\n\n##### Normalizing Qt(⋅∣Yt−1).\n\nAt each time step , we compute —a distribution over subsets of size of a base set —using the CPS design. The normalizing constant for this distribution is defined as\n\n Zt\\tiny def=∑Yt⊆Bt,|Yt|=KN∏n=1wn (6)\n\nDespite there being exponentially many summands, we can sum over all subsets in time via the following recurrence relation:555The reader may recognize this recurrence as the weighted generalization of Pascal’s triangle, , which is why we chose the notation .\n\n W(nk)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩\\multirowsetup 1if% k=0or n=kW(n−1k)+wnW(n−1k−1)if k∈(0,n)0otherwise\n\nWe give complete pseudocode in App. C. Correctness of this algorithm is shown in taskar_dpp. The normalizing constant can then be efficiently computed as\n\n Zt=W(NK) (7)\n##### Sampling from Qt(⋅∣Yt−1).\n\nWe can efficiently sample sets from using the algorithm below:\n\n1: Initialization\n2:for  :\n3:    Number of remaining elements\n4:   Add the element of to with prob.\n wnW(n−1k−1)W(nk)\n5:return Guaranteed to have size\n\nIn words, the algorithm considers adding each element one at a time until elements have been sampled. Notice that line 4 adjusts the probability of sampling item given that items have already been sampled, which ensures that exactly elements are sampled at termination.\n\n##### Setting wn.\n\nThe weight assigned to the item of directly affects its probability of being included in the sampled set, i.e., , also termed an item’s inclusion probability. In this paper, we write to denote the inclusion probability under the distribution , defined as:\n\n π\\scaletoQt5pt(y(n)≤t ∣Yt−1) (8) \\tiny def=∑YtQt(Yt∣Yt−1)1{y(n)≤t∈Yt}\n\nOne strategy is to choose at time step such that . This choice recovers beam search when we anneal our chosen weights : as the temperature parameter , the CP distribution will assign probability 1 to the set containing the top- elements.666In the event of ties, annealed CP will converge to a distribution that breaks ties uniformly at random.\n\nFinding ’s that result in pre-specified inclusion probabilities is possible, but it requires solving a numerical optimization problem aires1999algorithms; GRAFSTROM20092111. Further, in CPSBS, we will be sampling from a different distribution at each time step and it would be quite slow to solve the numerical optimization problem each iteration. Luckily, the choice of yields a good approximation to the target inclusion probabilities in both theory and practice hajek1981sampling; bondesson; aires1999algorithms.\n\n## 4 Statistical Estimation with Conditional Poisson Stochastic Beam Search\n\nIn this section, we discuss statistical estimation with CPSBS samples. To that end, we construct two estimators with different properties. However, only the second estimator provides good performance in practice, which is discussed later in § 5.\n\n### 4.1 The Horvitz–Thompson Estimator\n\nWe build upon the Horvitz–Thompson (HT) estimator HorvitzThompson1952, which is a common technique for estimation from sampling-without-replacement (SWOR) schemes.\n\nLet be , we seek to approximate its expected value under :\n\n Ey∼p[f(y)]=∑y∈Yp(y)f(y) (9)\n\nThe Monte Carlo estimator of the above quantity is\n\n GMC\\tiny def=1MM∑m=1f(y(m)) (10)\n\nwhere . However, in the special case of sampling from a finite population—which is extremely common in NLP—it can be very wasteful. For example, if a distribution is very peaked, it will sample the same item repeatedly; this could lead to inaccurate approximations for some . As a consequence, the mean square error (MSE) of the estimator with respect to can be quite high for small . Indeed, we see this empirically in Fig. 2.\n\nTaking samples without replacement allows us to cover more of the support of in our estimate of . However, we must take into account that our samples are no longer independent, i.e., . We now define the HT estimator, using notation specifically for the case of CPSBS where :\n\n GHT\\tiny def% =∑y∈YTp(y)π\\scaletoP4pt(y)f(y) (11)\n\nwhere CPSBS’s inclusion probability is\n\n (12) =∑Y1⋯∑YTT∏t=1Qt(Yt∣Yt−1)1{y≤t∈Yt}\n\ni.e., the probability of sampling a set that contains the element . In Eq. 11, the distribution may be viewed as a proposal distribution in the sense of importance sampling mcbook and as the corresponding importance weight corrections. If we can exactly compute , then the HT estimator is unbiased777Note that it is common to normalize Eq. 11 by the sum of importance weights, i.e., divide by the sum . While this leads to a biased estimator, it can significantly reduce variance, which is often worthwhile. (see § B.1 for proof). However, the summation in Eq. 12 is intractable so we resort to estimation.\n\n### 4.2 Estimating Inclusion Probabilities\n\nIn this section, we develop statistical estimators of the inclusion probabilities under conditional Poisson stochastic beam search. An important caveat: the analysis in this section only applies to the estimators of the inverse inclusion probabilities themselves. Further analysis may be undertaken to analyze the variance of the Horvitz–Thompson estimators that make use of these estimators.\n\n#### 4.2.1 Naïve Monte Carlo\n\nIt is not straightforward to estimate the reciprocal inclusion probabilities. Thus, we attempt to estimate the inclusion probabilities directly and take the reciprocal of this estimator. This strategy leads to a consistent, but biased, estimator.888Since by Jensen’s inequality for\n\n, the reciprocal of an unbiased estimate of\n\nis not an unbiased estimate of One obvious way to derive an inclusion probability estimator is the Monte Carlo estimator:\n\n ˆπ\\scaleto\\textscmc3pt\\scaletoP4pt(y)\\tiny def=1MM∑m=11{y∈Y(m)T} (13)\n\nwhere .\n\n###### Proposition 4.1.\n\nEq. 13 has the following two properties:\n\n1. [label=)]\n\n2. is an unbiased estimator of and\n\n V[ˆπ\\scaleto\\textscmc3pt\\scaletoP4pt]=1M(π\\scaletoP4pt(y)−π\\scaletoP4pt(y)2) (14)\n3. is a consistent estimator of with asymptotic variance\n\n (15)\n\nHere denotes the asymptotic variance, which is the variance after the number of samples\n\nis large enough such that the central limit theorem has kicked in\n\nbickel2015mathematical.\n\n###### Proof.\n\nProof given in § B.2. ∎\n\nQualitatively, what this result tells us is that if we are asking about the inverse inclusion probability of a candidate with a low inclusion probability, our estimator may have very high variance. Indeed, it is unlikely that we could derive an estimator without this qualitative property due to the presence of the inverse. Moreover, the estimator given in Eq. 13 is not of practical use: If we are interested in the inverse inclusion probability of a specific candidate , then we may have to sample a very large number of beams until we eventually sample one that actually contains . In practice, what this means is that our estimate of the inclusion probably for a rare will often be zero, which we cannot invert.999One solution would be to smooth our estimates of the inclusion probabilities, adding a small to ensure that we do not divide by zero, but the authors find our next approach to be more methodologically sound. Instead, we pursue an importance sampling strategy for estimating , which we outline in the next section.\n\n#### 4.2.2 Importance Sampling\n\nWe now turn to an inclusion probability estimator that is based on importance sampling. Recall from Eq. 12 that the inclusion probability for is a massive summation over sequences of possible beams that could have generated . Rather than computing the sum, we will estimate the sum through taking samples. Our procedure starts by generating hindsight samples from the following proposal distribution that is conditioned on :\n\n ˜Qt(˜Yt∣˜Yt−1,y)\\tiny def=Qt(˜Yt∣˜Yt−1)π\\scaletoQt5pt(y≤t∣˜Yt−1) (16)\n\nIn words, is conditioned on its sets containing the prefix (thus it is always the case that ).101010This proposal distribution can be realized through a minor modification of our algorithm in § 3, where corresponding to is placed at the beginning and added to deterministically. For brevity, we omit an explicit notational dependence of and on .\n\n###### Lemma 4.1.\n\nThe joint proposal distribution111111We have omitted dependency on for brevity. may be expressed in terms of as follows:\n\n ˜P (˜Y1,…,˜YT)=P(˜Y1,…,˜YT)∏Tt=1π\\scaletoQt5pt(y≤t∣˜Yt−1) (17)\n\nwhere we define as the joint probability of the beams under the original distributions . We omit that both and are conditioned on .\n\n###### Proof.\n\nSee § B.2. ∎\n\nIn terms of computation, Eq. 16 makes use of the fact that the per-time-step inclusion probability for a given can be computed efficiently with dynamic programming using the following identity:\n\n π\\scaletoQt5pt(y(n)≤t∣Yt−1) \\tiny def=∑YQt(Yt)1{y(n)≤t∈Yt} =wnZ∂Z∂wn (18)\n\nFor completeness, we give pseudocode in App. C. Given samples for defined in Eq. 16 with respect to a given , we propose the following unbiased estimator of inclusion probabilities:\n\n ˆπ\\scaleto\\textscis3pt\\scaletoP4pt(y)\\tiny def=1MM∑m=1T∏t=1π\\scaletoQt5pt(y\\scaleto≤t5pt∣˜Y(m)t−1) (19)\n\nwhere is a prefix of . One simple derivation of Eq. 19 is as an importance sampler. We start with the estimator given in Eq. 13 and perform the standard algebraic manipulations witnessed in importance sampling:\n\n ∑YTP(YT)1{y∈YT} (20) =∑YT⋯∑Y1P(Y1,…,YT)1{y∈YT} =∑˜YT⋯∑˜Y1P(˜Y1,…,˜YT)˜P(˜Y1,…,˜YT)˜P(˜Y1,…,˜YT) =∑˜YT⋯∑˜Y1˜P(˜Y1,…,˜YT)P(˜Y1,…,˜YT)˜P(˜Y1,…,˜YT) (i)=∑˜YT⋯∑˜Y1˜P(˜Y1,…,˜YT)T∏t=1π\\scaletoQt5pt(y≤t∣˜Yt−1)\n\nwhere equality (i) above follows from Lemma 4.1. This derivation serves as a simple proof that Eq. 19 inherits unbiasedness from Eq. 13\n\n###### Proposition 4.2.\n\nEq. 19 has the following two properties:\n\n1. [label=)]\n\n2. is an unbiased estimator of ;\n\n3. The estimator of the inverse inclusion probabilities is consistent with the following upper bound on the asymptotic variance:\n\n Va ⎡⎢⎣1ˆπ\\scaleto\\textscis3pt\\scaletoP4pt(y)⎤⎥⎦≤1Mr−1π\\scaletoP4pt(y)2 (21)\n\nwhere we assume that the following bound:\n\n ∏Tt=1π\\scaletoQt5pt(y≤t∣˜Yt−1)π\\scaletoP4pt(y)≤r (22)\n\nholds for all .\n\n###### Proof.\n\nProof given in § B.2. ∎\n\nProposition 4.2 tells us that we can use Eq. 19 to construct a consistent estimator of the inverse inclusion probabilities. Moreover, assuming , then we have that the importance sampling yields an estimate , unlike the Monte Carlo estimator . We further see that, to the extent that approximates , then we may expect the variance of Eq. 19 to be small—specifically in comparison to the naïve Monte Carlo estimator in Eq. 13—which is often the case for estimators built using importance sampling techniques when a proposal distribution is chosen judiciously monte-carlo. Thus, given our estimator in Eq. 19, we can now construct a practically useful estimator for using the HT estimator in Eq. 11. In the next section, we observe that this estimator is quite efficient in the sequence model setting.\n\n## 5 Experiments\n\nWe repeat the analyses performed by kool2019stochastic, running experiments on neural machine translation (NMT) models; for reproducibility, we use the pretrained Transformer model for WMT’14 bojar2014findings English–French made available by ott2019fairseq. We evaluate on the En-Fr newstest2014 set, containing 3003 sentences. Further details can be found in App. D. Our implementation of CPSBS modifies the beam search algorithm from the fairseq library. We additionally consider the beam search, stochastic beam search, diverse beam search, and ancestral sampling algorithms available in fairseq.\n\n### 5.1 Statistical Estimators for Language Generation Models\n\nEstimators have a large number of applications in machine learning. For example, the REINFORCE algorithm\n\nreinforce constructs an estimator for the value of the score function; minimum-Bayes risk decoding kumar-byrne-2004-minimum uses an estimate of risk in its optimization problem. In this section, we compare estimators for sentence-level bleu score and conditional model entropy for NMT models. Notably, NMT models that are trained to minimize cross-entropy with the empirical distribution131313Label-smoothing label_smoothing is typically also employed, which leads to even higher entropy distributions. are not peaky distributions ott2018analyzing; eikema2020map; thus, standard estimation techniques, e.g., Monte Carlo, should generally provide good results. However, we can vary the annealing parameter of our model in order to observe the behavior of our estimator with both high- and low-entropy distributions, making this a comprehensive case study. Here the annealed model distribution is computed as\n\n pτ(yt∣y\n\nwhere we should expect a standard Monte Carlo estimator to provide good results at close to 1 when is naturally high entropy. We test our estimator in this setting so as to give a comparison in a competitive setting. Specifically, we assess the performance of our estimator of given in Eq. 11—using inclusion probability estimates from Eq. 19 with and with importance weight normalization—in comparison to three other approaches: Monte Carlo (MC) sampling, the sum-and-sample (SAS) estimator, and stochastic beam search (SBS).\n\n##### Monte Carlo.\n\nUnder the Monte Carlo sampling scheme with sample size , we estimate the expected value of under our model using Eq. 10 with a sample .", null, "Figure 1: bleu score estimates for three different sentences using estimators for respective decoding methods. τ indicates scaling temperature; τ values and sentences are chosen to mimic kool2019stochastic.\n##### Sum and Sample.\n\nThe sum-and-sample estimator botev17a; liu2019rao; Kool2020Estimating is an unbiased estimator that takes as input a deterministically chosen set of size and samples an additional from the remaining elements, , where we obtain the set using beam search in our experiments. Formally, the SAS estimator can be written as:\n\n G\\textscsas \\tiny def=K−1∑k=1p(y(k))f(y(k)) (24)\n##### Stochastic Beam Search.\n\nStochastic beam search kool2019stochastic; Kool2020Estimating\n\nis a SWOR algorithm likewise built on top of beam search. The algorithm makes use of truncated Gumbel random variables at each iteration, resulting in a sampling design equivalent to performing the Gumbel-top-\n\ntrick vieira2014gumbel on the distribution . Estimators built using SBS likewise follow the Horvitz–Thompson scheme of Eq. 11; we refer the reader to the original work for inclusion probability computations. They suggest normalizing the estimator by the sum of sample inclusion probabilities to help reduce variance; we therefore likewise perform this normalization in our experiments.", null, "(a) RMSE of bleu score estimator for different temperatures. Results are averaged across several sentences.\n\nTo assess the error of our estimator, we compute its root MSE (RMSE) with respect to a baseline result. While computing the exact value of an expectation is typically infeasible in the sequence model setting, we can average our (unbiased) MC estimator in Eq. 10 over multiple runs to create a good baseline. Specifically, we compute our MC estimator 50 times for a large sample size (); variances are reported in App. D.1313footnotetext: We refer the reader to the original work kool2019stochastic for equations for inclusion probability estimates.\n\nProbabilistic models for language generation typically have large vocabularies. In this setting, the computation of Eq. 6 is inefficient due to the large number of items in the set that are assigned very small probability under the model. We experiment with truncating this distribution such that the set of possible extensions of a sequence consist only of the highest probability tokens within the core % of probability mass ( in our experiments), similar to the process in nucleus sampling holtzman2019curious. We compare this approach to the original algorithm design in App. D and see that empirically, results are virtually unchanged; the following results use this method. We also compare the decoding time of different sampling methods in Fig. 7.\n\n##### bleu Score Estimation.\n\nbleu Papineni:2002:BMA:1073083.1073135\n\nis a widely used automatic evaluation metric for the quality of machine-generated translations. Estimates of\n\nbleu score are used in minimum risk training shen-etal-2016-minimum\n\nand reinforcement learning-based approaches\n\nRanzatoCAZ15 to machine translation. As such, accurate and low-variance estimates are critical for the algorithms’ performance. Formally, we estimate the expected value of , whose dependence on we leave implicit, under our NMT model for reference translation . For comparison, we use the same sentences and similar annealing temperatures141414Results for converged rapidly for all estimators, thus not providing an interesting comparison. evaluated by kool2019stochastic\n\n. We repeat the sampling 20 times and plot the value and standard deviation (indicated by shaded region) of different estimators in\n\nFig. 1. From Fig. 1, we can see that CPSBS has lower variance than our baseline estimators across all temperatures and data points.151515The sampling distribution at is not the same across strategies, hence the difference in variances even at . Especially in the low temperature setting, our estimator converges rapidly with minor deviation from the exact values even for small sample sizes. Additionally, in Fig. 1(a) we see that the RMSE is typically quite low except at higher temperatures. In such cases, we observe the effects of biasedness, similar to kool2019stochastic’s observations.\n\n##### Conditional Entropy Estimation.\n\nWe perform similar experiments for estimates of a model’s conditional entropy, i.e., , whose dependence on we again leave implicit. We show results in Fig. 1(b), with plots of the value in App. D since results are quite similar to Fig. 1. We see further confirmation that our estimator built on CPSBS is generally quite efficient.\n\n### 5.2 Diverse Sampling\n\nWe show how CPSBS can be used as a diverse set sampling design for language generation models. We generate a sample of translations , i.e., according to the CPSBS scheme, where weights are set as at each time step, as recommended in § 3. In Fig. 3, we show the trade-off between minimum, average and maximum sentence-level bleu score (as a quality measure) and -gram diversity, where we define -gram diversity as the average fraction of unique vs. total -grams for in a sentence:\n\n D=4∑n=1#unique n-grams in K strings# n-grams in K strings (25)", null, "Figure 3: Average (± min and max) bleu score versus diversity for sample size k=5. Points correspond to different annealing temperatures {0.1, …, 0.8}. Results for k=10,20 show very similar trends.\n\nMetrics are averaged across the corpus. We follow the experimental setup of kool2019stochastic, using the newstest2014 dataset and comparing three different decoding methods: SBS, diverse beam search (DiverseBS; vijayakumar2018diverse) and ancestral sampling. As in their experiments, we vary the annealing temperature in the range as a means of encouraging diversity; for DiverseBS we instead vary the strength parameter in the same range. Interestingly, we see that temperature has virtually no effect on the diversity of the set of results returned by CPSBS. Despite this artifact, for which the authors have not found a theoretical justification,161616While scaling sampling weights by a constant should not change the distribution , an transformation of weights—which is the computation performed by temperature annealing—should. the set returned by CPSBS is still overall more diverse (position on -axis) than results returned by DiverseBS and reflect better min, max, and average bleu in comparison to random sampling. SBS provides a better spectrum for the diversity and bleu tradeoff; we thus recommend SBS when diverse sets are desired.\n\n## 6 Conclusion\n\nIn this work, we present conditional Poisson stochastic beam search, a sampling-without-replacement strategy for sequence models. Through a simple modification to beam search, we turn this mainstay decoding algorithm into a stochastic process. We derive a low-variance, consistent estimator of inclusion probabilities under this scheme; we then present a general framework for using CPSBS to construct statistical estimators for expectations under sequence models. In our experiments, we observe a reduction in mean square error, and an increase in sample efficiency, when using our estimator in comparison to several baselines, showing the benefits of CPSBS.\n\n## Appendix A Conditional Poisson Sampling\n\nHere we provide a brief overview of the sampling design at the core of CPSBS: conditional Poisson sampling. We consider a base set where and we map the elements of to the integers . As a warm up, we first consider (unconditional) Poisson sampling, also known as a Bernoulli point process. To sample a subset , we do as follows: for each element\n\n, we flip a coin where the odds of heads is\n\n. Then, we simply take to be the subset of elements whose coin flips were heads. However, this sampling scheme clearly does not guarantee a sample of items, which can cause problems in our application; sampling more than items would make the stochastic beam search process inefficient while sampling fewer than —or even 0—items may not leave us with a large enough set at the end of our iterative process.\n\nIf instead, we condition on the sets always having a prescribed size , i.e., reject samples where , we arrive at the conditional\n\nPoisson process. Formally, the conditional Poisson distribution is defined over\n\nas follows,\n\n (26)\n\nBy analyzing Eq. 26, we can see that sets with the largest product of weights are the most likely to be sampled; further, this distribution is invariant to rescaling of weights due to the size requirement. This is similar to the conditions under which beam search chooses the set of largest weight, i.e., highest scoring, elements. Indeed, we note the extreme similarity between Eq. 4 and Eq. 26, the only difference being a dependence on a prior set. However, unlike beam search, sets with a lower weight product now have the possibility of being chosen.\n\n## Appendix B Proofs\n\n### b.1 Unbiasedness of the Horvitz–Thompson Estimator\n\n###### Proposition B.1.\n\nGiven a SWOR design over the set with inclusion probabilities , the Horvitz–Thompson estimator (Eq. 11) gives us an unbiased estimator of , where is a function whose expectation under we seek to approximate.\n\n###### Proof.\n\nequationparentequation\n\n EY∼Q[GHT] =EY∼QN∑n=1p(n)π(n)f(n) (27a) =EY∼Q∑n∈Bp(n)π(n)1{n∈Y}f(n) (27b) =∑n∈Bp(n)π(n)f(n)EY∼Q1{n∈Y} (27c) =∑n∈Bp(n)π(n)f(n)π(n) (27d) =∑n∈Bp(n)f(n) (27e) =En∼pf(n) (27f)\n\n### b.2 Proofs of Expected Values and Variances of Inclusion Probability Estimators\n\nSee 4.1\n\n###### Proof.\n\nConsider the probability of sampling according to . Algebraic manipulation reveals: equationparentequation\n\n ˜P(˜Y1,…,˜YT) =Qt(˜Y1∣Y0)π\\scaletoQt5pt(y≤1∣Y0)⋯Qt(˜YT∣˜YT−1)π\\scaletoQt5pt(y≤T∣˜YT−1) (28a) = P(˜Y1,…,˜YT)∏Tt=1π\\scaletoQt5pt(y≤t∣˜Yt−1) (28b)\n\nwhich proves the identity. ∎\n\nSee 4.1\n\n###### Proof.\n\ni) The estimator is easily shown to be unbiased:\n\n (29)\n\nand its variance may be derived as follows: equationparentequation\n\n V[ˆπ\\scaleto\\textscmc3pt\\scaletoP4pt(y)] \\tiny def=V[1MM∑m=11{y∈Y(m)T}] (30a) =1MV[1{y∈Y(m)T}] (30b) (30c) =1M(π\\scaletoP4pt(y)−π\\scaletoP4pt(y)2) (30d)\n\nii) By the strong law of large numbers, we have\n\n limM→∞1MM∑m=11{y∈Y(m)T}=π\\scaletoP4pt(y) (31)\n\nSince is continuous, we may appeal to the continuous mapping theorem to achieve consistency:\n\n limM→∞11M∑Mm=11{y∈Y(m)T}=1limM→∞1M∑Mm=11{y∈Y(m)T}=1π\\scaletoP4pt(y) (32)\n\nWe can compute the asymptotic variance by the delta rule: equationparentequation\n\n Va⎡⎢⎣1ˆπ\\scaleto\\textscmc3pt\\scaletoP4pt(y)⎤⎥⎦ =1MV[ˆπ\\scaleto\\textscis3pt\\scaletoP4pt(y)]π\\scaletoP4pt(y)4\\color[rgb]{.5,.5,.5}\\definecolor[% named]{pgfstrokecolor}{rgb}{.5,.5,.5}\\pgfsys@color@gray@stroke{.5}% \\pgfsys@color@gray@fill{.5} (apply the delta rule) (33a) =1Mπ\\scaletoP4pt(y)−π\\scaletoP4pt(y)2π\\scaletoP4pt(y)4\\color[rgb]{.5,.5,.5}\\definecolor[named]{pgfstrokecolor}{rgb}{% .5,.5,.5}\\pgfsys@color@gray@stroke{.5}\\pgfsys@color@gray@fill{.5} (plugging in% the variance computed above) (33b) =1M(1π\\scaletoP4pt(y)3−1π\\scaletoP4pt(y)2) (33c)\n\nSee 4.2\n\n###### Proof.\n\ni) We first prove that the estimator of the inclusion probabilities is unbiased through the following manipulation: equationparentequation\n\n E[ˆπ\\scaleto\\textscis3pt\\scaletoP4pt(y)] =E[1MM∑m=1T∏t=1π\\scaletoQt5pt(y≤t∣˜Y(m)t−1)] (34a) =∑˜Y1,…,˜YT˜P(˜Y1,…,˜YT)T∏t=1π\\scaletoQt5pt(y≤t∣˜Yt−1) (34b) =∑˜Y1,…,˜YT˜P(˜Y1,…,˜YT)P(˜Y1,…,˜YT)P(˜Y1,…,˜YT)T∏t=1π\\scaletoQt5pt(y≤t∣˜Yt−1) (34c) =∑˜Y1,…,˜YTP(˜Y1,…,˜YT)˜P(˜Y1,…,˜YT)˜P(˜Y1,…,˜YT)\\color[rgb]{.5,.5,.5}\\definecolor[% named]{pgfstrokecolor}{rgb}{.5,.5,.5}\\pgfsys@color@gray@stroke{.5}% \\pgfsys@color@gray@fill{.5} {(\\lx@cref{creftype~refnum}{lem:decomposition})} (34d) =∑˜Y1,…,˜YTP(˜Y1,…,˜YT) (34e) (34f) =π\\scaletoP4pt(y) (34g)\n\nii) To show consistency, we appeal to the strong law of large number and the continuous mapping theorem. By the strong law of large numbers, we have that\n\n limM→∞1MM∑m=1T∏t=1π\\scaletoQt5pt(y≤t∣˜Y(m)t−1)=π\\scaletoP4pt(y) (35)\n\nSince is continuous, we have equationparentequation\n\n limM→∞11M∑Mm=1∏Tt=1π\\scaletoQt5pt(y≤t∣˜Y(m)t−1) =1limM→∞1M∑Mm=1∏Tt=1π\\scaletoQt5pt(y≤t∣˜Y(m)t−1) =1π\\scaletoP4pt(y) (36a)\n\nwhich shows consistency.\n\nNow, we derive a bound on the asymptotic variance of the inverse inclusion probabilities: Suppose,\n\n ∏Tt=1π\\scaletoQt5pt(y≤t∣˜Yt−1)π\\scaletoP4pt(y)≤r,∀˜Y1,…,˜YT (37)\n\nWe start with the variance of importance sampling. This is a standard result monte-carlo. Then we proceed with algebraic manipulation integrating the assumption above: equationparentequation\n\n ∑˜Y1,…,˜YT 1{y∈˜YT}2P(˜Y1,…,˜YT)2˜P(˜Y1,…,˜YT)−π\\scaletoP4pt(y)2 (38a) =∑˜Y1,…,˜YTP(˜Y1,…,˜YT)T∏t=1π\\scaletoQt5pt(y≤t∣˜Yt−1)−π\\scaletoP4pt(y)2 (38b) ≤∑˜Y1,…,˜YTP(˜Y1,…,˜YT)π\\scaletoP4pt(y)r−π\\scaletoP4pt(y)2 (38c) (38d) =π\\scaletoP4pt(y)2r−π\\scaleto" ]

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[ "How many amps does a 40 watt light bulb use?\n\nContents\n\nA 40 Watts bulb is capable of drawing 0.36 Amps to operate.\n\nHow many amps does a 60 watt equivalent LED bulb use?\n\nA 60 Watts bulb draws 0.54 Amps to operate. A 80 Watts bulb draws 0.72 Amps to operate.\n\nHow many amps does a 100 watt light bulb use?\n\nDivide the total number of watts by the system’s volts. For example, a 100-watt bulb in a 12-volt system will draw 8.3 amps.\n\nHow many amps does a LED bulb use?\n\nWhen Dave connects incandescent bulbs to an amp meter, they draw up to 1.6 amps, but when LED lights are used, they only pull about . 26 amps.\n\nHow many amps does a 50 watt bulb use?\n\nA 50 watt bulb should pull about 4 Amps from the transformer secondary.\n\nHow many amps does a 300w LED light use?\n\n300 Watts divided by 120 Volts (residential US standard) = 2.5 amps (assuming 100% efficiency) drawn from the wall. The weakest circuits in most homes can support up to 15 amps, or 1800w each. This light fixture will only draw 2.5 amps maximum when both LED circuits are on at the same time. 1 of 3 found this helpful.\n\nIT IS INTERESTING:  Why does my room get dark when I turn the lights off even if my window is shut?\n\nHow many amps does a light need?\n\nA simple formula for calculating amps is to take the watts and divide that by the volts. So, for instance, if the wattage of the lighting fixture you’re working with is 60 and the volts are 12, divide 60 by 12 and you will get five, which are the amps.\n\nHow many 60 watt bulbs can be on a 15 amp circuit?\n\nAssuming a 60 watt light, you can put up to 24 lights on a 15 amp breaker. If you are using low-wattage LED bulbs, an LED bulb using 10 watts, you can install up to 150 bulbs on a single circuit.\n\nHow many lights can be on a 20 amp circuit?\n\nAmount of Lights You Can Put on a 20-Amp Circuit Breaker\n\nWith that fact, a 50-watt light on a 20-amp breaker can have a total of 38 lights on a circuit.\n\nHow many amps does a 150 watt light bulb use?\n\nCurrent (Amps) drawn would be watts divided by supply voltage. In other words, at 110 volts you’d expect this 150-watt light to draw about 1.36 Amps.\n\nHow many amps is 40 watts?\n\nWatts to amps table (120V)\n\nPower (W) Voltage (V) Current (A)\n40 watts 120 volts 0.333 amps\n50 watts 120 volts 0.417 amps\n60 watts 120 volts 0.500 amps\n70 watts 120 volts 0.583 amps\n\nHow do I calculate amps from Watts?\n\nConverting amps to watts can be done using the power formula, which states that I = P ÷ E, where P is power measured in watts, I is current measured in amps, and E is voltage measured in volts. Thus, the power P in watts is equal to the current I in amps multiplied by the voltage V in volts.\n\nHow much power does a 30 watt LED light use?\n\nPower consumption compared to old light sources\n\nIn the above example, the 30 W LED luminaire consumes 49 kWh of electricity per year. A luminaire of similar brightness with incandescent lamps would have a power draw of about 220 W. Equipped with halogen lamps it would still be about 180 W.\n\nHow many amps is 50w 12V?\n\nEquivalent Watts and Amps at 12V DC\n\nPower Current Voltage\n40 Watts 3.333 Amps 12 Volts\n45 Watts 3.75 Amps 12 Volts\n50 Watts 4.167 Amps 12 Volts\n60 Watts 5 Amps 12 Volts\n\nHow many amps does 50 watts draw?\n\nWatts To Amps At 12V (For Batteries)\n\nWatts: Amps (at 12V):\n50 Watts to amps at 12V: 4.17 Amps\n100 Watts to amps at 12V: 8.33 Amps\n200 Watts to amps at 12V: 16.67 Amps\n300 Watts to amps at 12V: 25.00 Amps\n\nHow many 100 watt bulbs can be on A 15 amp circuit?\n\nFor 100-watt fixtures, you can have a maximum of 12 divided by 0.9A, or 13 fixtures. If the lights will be used less than three hours at a time, you could install 16 fixtures on the circuit.\n\nCategories LED" ]

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[ "'\n\n# Search results\n\nFound 1507 matches\nUniform Circular Motion position (X - coordinate)\n\nIn physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with ... more\n\nGyrofrequency\n\nIf the magnetic field is uniform and all other forces are absent, then the Lorentz force will cause a particle to undergo a constant acceleration ... more\n\nCentripetal Force - angular velocity\n\nCentripetal force (from Latin centrum “center” and petere “to seek”) is a force that makes a body follow a curved path: its ... more\n\nUniform circular motion, that is constant speed along a circular path, is an example of a body experiencing acceleration resulting in velocity of a ... more\n\nArchimedean spiral\n\nThe Archimedean spiral is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed ... more\n\n1st Equation of Motion for Rotation - Angular Velocity\n\nIn mathematical physics, equations of motion are equations that describe the behaviour of a physical system in terms of its motion as a function of ... more\n\n3rd Equation of Motion for Rotation - Final Angle : acceleration independent\n\nIn mathematical physics, equations of motion are equations that describe the behaviour of a physical system in terms of its motion as a function of ... more\n\nSagnac Effect - TIme Difference\n\nThe Sagnac effect (also called Sagnac interference), named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is ... more\n\n2nd Equation of Motion for Rotation - Final Angle\n\nIn mathematical physics, equations of motion are equations that describe the behaviour of a physical system in terms of its motion as a function of ... more\n\nNodal Precession\n\nNodal precession is the precession of an orbital plane around the rotation axis of an astronomical body such as Earth. This precession is due to the ... more\n\n...can't find what you're looking for?\n\nCreate a new formula\n\n### Search criteria:\n\nSimilar to formula\nCategory" ]

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[ "!> @file indoor_model_mod.f90 !--------------------------------------------------------------------------------! ! This file is part of the PALM model system. ! ! PALM is free software: you can redistribute it and/or modify it under the ! terms of the GNU General Public License as published by the Free Software ! Foundation, either version 3 of the License, or (at your option) any later ! version. ! ! PALM is distributed in the hope that it will be useful, but WITHOUT ANY ! WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR ! A PARTICULAR PURPOSE. See the GNU General Public License for more details. ! ! You should have received a copy of the GNU General Public License along with ! PALM. If not, see . ! ! Copyright 2018-2020 Leibniz Universitaet Hannover ! Copyright 2018-2020 Hochschule Offenburg !--------------------------------------------------------------------------------! ! ! Current revisions: ! ----------------- ! ! ! Former revisions: ! ----------------- ! \\$Id: indoor_model_mod.f90 4481 2020-03-31 18:55:54Z suehring \\$ ! Change order of dimension in surface array %frac to allow for better ! vectorization. ! ! 4441 2020-03-04 19:20:35Z suehring ! Major bugfix in calculation of energy demand - floor-area-per-facade was wrongly ! calculated leading to unrealistically high energy demands and thus to ! unreallistically high waste-heat fluxes. ! ! 4346 2019-12-18 11:55:56Z motisi ! Introduction of wall_flags_total_0, which currently sets bits based on static ! topography information used in wall_flags_static_0 ! ! 4329 2019-12-10 15:46:36Z motisi ! Renamed wall_flags_0 to wall_flags_static_0 ! ! 4310 2019-11-26 19:01:28Z suehring ! Remove dt_indoor from namelist input. The indoor model is an hourly-based ! model, calling it more/less often lead to inaccurate results. ! ! 4299 2019-11-22 10:13:38Z suehring ! Output of indoor temperature revised (to avoid non-defined values within ! buildings) ! ! 4267 2019-10-16 18:58:49Z suehring ! Bugfix in initialization, some indices to access building_pars where wrong. ! Introduction of seasonal parameters. ! ! 4246 2019-09-30 09:27:52Z pavelkrc ! ! ! 4242 2019-09-27 12:59:10Z suehring ! Bugfix in array index ! ! 4238 2019-09-25 16:06:01Z suehring ! - Bugfix in determination of minimum facade height and in location message ! - Bugfix, avoid division by zero ! - Some optimization ! ! 4227 2019-09-10 18:04:34Z gronemeier ! implement new palm_date_time_mod ! ! 4217 2019-09-04 09:47:05Z scharf ! Corrected \"Former revisions\" section ! ! 4209 2019-09-02 12:00:03Z suehring ! - Bugfix in initialization of indoor temperature ! - Prescibe default indoor temperature in case it is not given in the ! namelist input ! ! 4182 2019-08-21 14:37:54Z scharf ! Corrected \"Former revisions\" section ! ! 4148 2019-08-08 11:26:00Z suehring ! Bugfix in case of non grid-resolved buildings. Further, vertical grid spacing ! is now considered at the correct level. ! - change calculation of a_m and c_m ! - change calculation of u-values (use h_es in building array) ! - rename h_tr_... to h_t_... ! h_tr_em to h_t_wm ! h_tr_op to h_t_wall ! h_tr_w to h_t_es ! - rename h_ve to h_v ! - rename h_is to h_ms ! - inserted net_floor_area ! - inserted params_waste_heat_h, params_waste_heat_c from building database ! in building array ! - change calculation of q_waste_heat ! - bugfix in averaging mean indoor temperature ! ! 3759 2019-02-21 15:53:45Z suehring ! - Calculation of total building volume ! - Several bugfixes ! - Calculation of building height revised ! ! 3745 2019-02-15 18:57:56Z suehring ! - remove building_type from module ! - initialize parameters for each building individually instead of a bulk ! initializaion with identical building type for all ! - output revised ! - add missing _wp ! - some restructuring of variables in building data structure ! ! 3744 2019-02-15 18:38:58Z suehring ! Some interface calls moved to module_interface + cleanup ! ! 3469 2018-10-30 20:05:07Z kanani ! Initial revision (tlang, suehring, kanani, srissman)! ! ! Authors: ! -------- ! @author Tobias Lang ! @author Jens Pfafferott ! @author Farah Kanani-Suehring ! @author Matthias Suehring ! @author Sascha Rißmann ! ! ! Description: ! ------------ !> !> Module for Indoor Climate Model (ICM) !> The module is based on the DIN EN ISO 13790 with simplified hour-based procedure. !> This model is a equivalent circuit diagram of a three-point RC-model (5R1C). !> This module differ between indoor-air temperature an average temperature of indoor surfaces which make it prossible to determine thermal comfort !> the heat transfer between indoor and outdoor is simplified !> @todo Replace window_area_per_facade by %frac(1,m) for window !> @todo emissivity change for window blinds if solar_protection_on=1 !> @note Do we allow use of integer flags, or only logical flags? (concerns e.g. cooling_on, heating_on) !> @note How to write indoor temperature output to pt array? !> !> @bug !------------------------------------------------------------------------------! MODULE indoor_model_mod USE arrays_3d, & ONLY: ddzw, & dzw, & pt USE control_parameters, & ONLY: initializing_actions USE kinds USE netcdf_data_input_mod, & ONLY: building_id_f, building_type_f USE palm_date_time_mod, & ONLY: get_date_time, northward_equinox, seconds_per_hour, & southward_equinox USE surface_mod, & ONLY: surf_usm_h, surf_usm_v IMPLICIT NONE ! !-- Define data structure for buidlings. TYPE build INTEGER(iwp) :: id !< building ID INTEGER(iwp) :: kb_min !< lowest vertical index of a building INTEGER(iwp) :: kb_max !< highest vertical index of a building INTEGER(iwp) :: num_facades_per_building_h = 0 !< total number of horizontal facades elements INTEGER(iwp) :: num_facades_per_building_h_l = 0 !< number of horizontal facade elements on local subdomain INTEGER(iwp) :: num_facades_per_building_v = 0 !< total number of vertical facades elements INTEGER(iwp) :: num_facades_per_building_v_l = 0 !< number of vertical facade elements on local subdomain INTEGER(iwp) :: ventilation_int_loads !< [-] allocation of activity in the building INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: l_v !< index array linking surface-element orientation index !< for vertical surfaces with building INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: m_h !< index array linking surface-element index for !< horizontal surfaces with building INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: m_v !< index array linking surface-element index for !< vertical surfaces with building INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: num_facade_h !< number of horizontal facade elements per buidling !< and height level INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: num_facade_v !< number of vertical facades elements per buidling !< and height level LOGICAL :: on_pe = .FALSE. !< flag indicating whether a building with certain ID is on local subdomain REAL(wp) :: air_change_high !< [1/h] air changes per time_utc_hour REAL(wp) :: air_change_low !< [1/h] air changes per time_utc_hour REAL(wp) :: area_facade !< [m2] area of total facade REAL(wp) :: building_height !< building height REAL(wp) :: eta_ve !< [-] heat recovery efficiency REAL(wp) :: factor_a !< [-] Dynamic parameters specific effective surface according to Table 12; 2.5 !< (very light, light and medium), 3.0 (heavy), 3.5 (very heavy) REAL(wp) :: factor_c !< [J/(m2 K)] Dynamic parameters inner heatstorage according to Table 12; 80000 !< (very light), 110000 (light), 165000 (medium), 260000 (heavy), 370000 (very heavy) REAL(wp) :: f_c_win !< [-] shading factor REAL(wp) :: fapf !< [m2/m2] floor area per facade REAL(wp) :: g_value_win !< [-] SHGC factor REAL(wp) :: h_es !< [W/(m2 K)] surface-related heat transfer coefficient between extern and surface REAL(wp) :: height_cei_con !< [m] ceiling construction heigth REAL(wp) :: height_storey !< [m] storey heigth REAL(wp) :: params_waste_heat_c !< [-] anthropogenic heat outputs for cooling e.g. 1.33 for KKM with COP = 3 REAL(wp) :: params_waste_heat_h !< [-] anthropogenic heat outputs for heating e.g. 1 - 0.9 = 0.1 for combustion with eta = 0.9 or -2 for WP with COP = 3 REAL(wp) :: phi_c_max !< [W] Max. Cooling capacity (negative) REAL(wp) :: phi_h_max !< [W] Max. Heating capacity (positive) REAL(wp) :: q_c_max !< [W/m2] Max. Cooling heat flux per netto floor area (negative) REAL(wp) :: q_h_max !< [W/m2] Max. Heating heat flux per netto floor area (positive) REAL(wp) :: qint_high !< [W/m2] internal heat gains, option Database qint_0-23 REAL(wp) :: qint_low !< [W/m2] internal heat gains, option Database qint_0-23 REAL(wp) :: lambda_at !< [-] ratio internal surface/floor area chap. 7.2.2.2. REAL(wp) :: lambda_layer3 !< [W/(m*K)] Thermal conductivity of the inner layer REAL(wp) :: net_floor_area !< [m2] netto ground area REAL(wp) :: s_layer3 !< [m] half thickness of the inner layer (layer_3) REAL(wp) :: theta_int_c_set !< [degree_C] Max. Setpoint temperature (summer) REAL(wp) :: theta_int_h_set !< [degree_C] Max. Setpoint temperature (winter) REAL(wp) :: u_value_win !< [W/(m2*K)] transmittance REAL(wp) :: vol_tot !< [m3] total building volume REAL(wp), DIMENSION(:), ALLOCATABLE :: t_in !< mean building indoor temperature, height dependent REAL(wp), DIMENSION(:), ALLOCATABLE :: t_in_l !< mean building indoor temperature on local subdomain, height dependent REAL(wp), DIMENSION(:), ALLOCATABLE :: volume !< total building volume, height dependent REAL(wp), DIMENSION(:), ALLOCATABLE :: vol_frac !< fraction of local on total building volume, height dependent REAL(wp), DIMENSION(:), ALLOCATABLE :: vpf !< building volume volume per facade element, height dependent END TYPE build TYPE(build), DIMENSION(:), ALLOCATABLE :: buildings !< building array INTEGER(iwp) :: num_build !< total number of buildings in domain ! !-- Declare all global variables within the module INTEGER(iwp) :: cooling_on !< Indoor cooling flag (0=off, 1=on) INTEGER(iwp) :: heating_on !< Indoor heating flag (0=off, 1=on) INTEGER(iwp) :: solar_protection_off !< Solar protection off INTEGER(iwp) :: solar_protection_on !< Solar protection on REAL(wp), PARAMETER :: dt_indoor = 3600.0_wp !< [s] time interval for indoor-model application, fixed to 3600.0 due to model requirements REAL(wp) :: a_m !< [m2] the effective mass-related area REAL(wp) :: air_change !< [1/h] Airflow REAL(wp) :: c_m !< [J/K] internal heat storage capacity REAL(wp) :: facade_element_area !< [m2_facade] building surface facade REAL(wp) :: floor_area_per_facade !< [m2/m2] floor area per facade area REAL(wp) :: h_t_1 !< [W/K] Heat transfer coefficient auxiliary variable 1 REAL(wp) :: h_t_2 !< [W/K] Heat transfer coefficient auxiliary variable 2 REAL(wp) :: h_t_3 !< [W/K] Heat transfer coefficient auxiliary variable 3 REAL(wp) :: h_t_wm !< [W/K] Heat transfer coefficient of the emmision (got with h_t_ms the thermal mass) REAL(wp) :: h_t_is !< [W/K] thermal coupling conductance (Thermischer Kopplungsleitwert) REAL(wp) :: h_t_ms !< [W/K] Heat transfer conductance term (got with h_t_wm the thermal mass) REAL(wp) :: h_t_wall !< [W/K] heat transfer coefficient of opaque components (assumption: got all !< thermal mass) contains of h_t_wm and h_t_ms REAL(wp) :: h_t_es !< [W/K] heat transfer coefficient of doors, windows, curtain walls and !< glazed walls (assumption: thermal mass=0) REAL(wp) :: h_v !< [W/K] heat transfer of ventilation REAL(wp) :: indoor_volume_per_facade !< [m3] indoor air volume per facade element REAL(wp) :: initial_indoor_temperature = 293.15 !< [K] initial indoor temperature (namelist parameter) REAL(wp) :: net_sw_in !< [W/m2] net short-wave radiation REAL(wp) :: phi_hc_nd !< [W] heating demand and/or cooling demand REAL(wp) :: phi_hc_nd_10 !< [W] heating demand and/or cooling demand for heating or cooling REAL(wp) :: phi_hc_nd_ac !< [W] actual heating demand and/or cooling demand REAL(wp) :: phi_hc_nd_un !< [W] unlimited heating demand and/or cooling demand which is necessary to !< reach the demanded required temperature (heating is positive, !< cooling is negative) REAL(wp) :: phi_ia !< [W] internal air load = internal loads * 0.5, Eq. (C.1) REAL(wp) :: phi_m !< [W] mass specific thermal load (internal and external) REAL(wp) :: phi_mtot !< [W] total mass specific thermal load (internal and external) REAL(wp) :: phi_sol !< [W] solar loads REAL(wp) :: phi_st !< [W] mass specific thermal load implied non thermal mass REAL(wp) :: q_wall_win !< [W/m2]heat flux from indoor into wall/window REAL(wp) :: q_waste_heat !< [W/m2]waste heat, sum of waste heat over the roof to Palm REAL(wp) :: q_c_m !< [W] Energy of thermal storage mass specific thermal load for internal !< and external heatsources (for energy bilanz) REAL(wp) :: q_c_st !< [W] Energy of thermal storage mass specific thermal load implied non thermal mass (for energy bilanz) REAL(wp) :: q_int !< [W] Energy of internal air load (for energy bilanz) REAL(wp) :: q_sol !< [W] Energy of solar (for energy bilanz) REAL(wp) :: q_trans !< [W] Energy of transmission (for energy bilanz) REAL(wp) :: q_vent !< [W] Energy of ventilation (for energy bilanz) REAL(wp) :: schedule_d !< [-] activation for internal loads (low or high + low) REAL(wp) :: skip_time_do_indoor = 0.0_wp !< [s] Indoor model is not called before this time REAL(wp) :: theta_air !< [degree_C] air temperature of the RC-node REAL(wp) :: theta_air_0 !< [degree_C] air temperature of the RC-node in equilibrium REAL(wp) :: theta_air_10 !< [degree_C] air temperature of the RC-node from a heating capacity !< of 10 W/m2 REAL(wp) :: theta_air_ac !< [degree_C] actual room temperature after heating/cooling REAL(wp) :: theta_air_set !< [degree_C] Setpoint_temperature for the room REAL(wp) :: theta_m !< [degree_C} inner temperature of the RC-node REAL(wp) :: theta_m_t !< [degree_C] (Fictive) component temperature timestep REAL(wp) :: theta_m_t_prev !< [degree_C] (Fictive) component temperature previous timestep (do not change) REAL(wp) :: theta_op !< [degree_C] operative temperature REAL(wp) :: theta_s !< [degree_C] surface temperature of the RC-node REAL(wp) :: time_indoor = 0.0_wp !< [s] time since last call of indoor model REAL(wp) :: total_area !< [m2] area of all surfaces pointing to zone REAL(wp) :: window_area_per_facade !< [m2] window area per facade element REAL(wp), PARAMETER :: h_is = 3.45_wp !< [W/(m2 K)] surface-related heat transfer coefficient between !< surface and air (chap. 7.2.2.2) REAL(wp), PARAMETER :: h_ms = 9.1_wp !< [W/(m2 K)] surface-related heat transfer coefficient between component and surface (chap. 12.2.2) REAL(wp), PARAMETER :: params_f_f = 0.3_wp !< [-] frame ratio chap. 8.3.2.1.1 for buildings with mostly cooling 2.0_wp REAL(wp), PARAMETER :: params_f_w = 0.9_wp !< [-] correction factor (fuer nicht senkrechten Stahlungseinfall !< DIN 4108-2 chap.8, (hier konstant, keine Winkelabhängigkeit) REAL(wp), PARAMETER :: params_f_win = 0.5_wp !< [-] proportion of window area, Database A_win aus !< Datenbank 27 window_area_per_facade_percent REAL(wp), PARAMETER :: params_solar_protection = 300.0_wp !< [W/m2] chap. G.5.3.1 sun protection closed, if the radiation !< on facade exceeds this value ! !-- Definition of seasonal parameters, summer and winter, for different building types REAL(wp), DIMENSION(0:1,1:7) :: summer_pars = RESHAPE( (/ & ! building_type 1 0.5_wp, & ! basical airflow without occupancy of the room 2.0_wp, & ! additional airflow depend of occupancy of the room 0.5_wp, & ! building_type 2: basical airflow without occupancy of the room 2.0_wp, & ! additional airflow depend of occupancy of the room 0.8_wp, & ! building_type 3: basical airflow without occupancy of the room 2.0_wp, & ! additional airflow depend of occupancy of the room 0.1_wp, & ! building_type 4: basical airflow without occupancy of the room 1.5_wp, & ! additional airflow depend of occupancy of the room 0.1_wp, & ! building_type 5: basical airflow without occupancy of the room 1.5_wp, & ! additional airflow depend of occupancy of the room 0.1_wp, & ! building_type 6: basical airflow without occupancy of the room 1.5_wp, & ! additional airflow depend of occupancy of the room 0.1_wp, & ! building_type 7: basical airflow without occupancy of the room 1.5_wp & ! additional airflow depend of occupancy of the room /), (/ 2, 7 /) ) REAL(wp), DIMENSION(0:1,1:7) :: winter_pars = RESHAPE( (/ & ! building_type 1 0.1_wp, & ! basical airflow without occupancy of the room 0.5_wp, & ! additional airflow depend of occupancy of the room 0.1_wp, & ! building_type 2: basical airflow without occupancy of the room 0.5_wp, & ! additional airflow depend of occupancy of the room 0.1_wp, & ! building_type 3: basical airflow without occupancy of the room 0.5_wp, & ! additional airflow depend of occupancy of the room 0.1_wp, & ! building_type 4: basical airflow without occupancy of the room 1.5_wp, & ! additional airflow depend of occupancy of the room 0.1_wp, & ! building_type 5: basical airflow without occupancy of the room 1.5_wp, & ! additional airflow depend of occupancy of the room 0.1_wp, & ! building_type 6: basical airflow without occupancy of the room 1.5_wp, & ! additional airflow depend of occupancy of the room 0.1_wp, & ! building_type 7: basical airflow without occupancy of the room 1.5_wp & ! additional airflow depend of occupancy of the room /), (/ 2, 7 /) ) SAVE PRIVATE ! !-- Add INTERFACES that must be available to other modules PUBLIC im_init, im_main_heatcool, im_parin, im_define_netcdf_grid, & im_check_data_output, im_data_output_3d, im_check_parameters ! !-- Add VARIABLES that must be available to other modules PUBLIC dt_indoor, skip_time_do_indoor, time_indoor ! !-- PALM interfaces: !-- Data output checks for 2D/3D data to be done in check_parameters INTERFACE im_check_data_output MODULE PROCEDURE im_check_data_output END INTERFACE im_check_data_output ! !-- Input parameter checks to be done in check_parameters INTERFACE im_check_parameters MODULE PROCEDURE im_check_parameters END INTERFACE im_check_parameters ! !-- Data output of 3D data INTERFACE im_data_output_3d MODULE PROCEDURE im_data_output_3d END INTERFACE im_data_output_3d ! !-- Definition of data output quantities INTERFACE im_define_netcdf_grid MODULE PROCEDURE im_define_netcdf_grid END INTERFACE im_define_netcdf_grid ! ! ! ! !-- Output of information to the header file ! INTERFACE im_header ! MODULE PROCEDURE im_header ! END INTERFACE im_header ! !-- Calculations for indoor temperatures INTERFACE im_calc_temperatures MODULE PROCEDURE im_calc_temperatures END INTERFACE im_calc_temperatures ! !-- Initialization actions INTERFACE im_init MODULE PROCEDURE im_init END INTERFACE im_init ! !-- Main part of indoor model INTERFACE im_main_heatcool MODULE PROCEDURE im_main_heatcool END INTERFACE im_main_heatcool ! !-- Reading of NAMELIST parameters INTERFACE im_parin MODULE PROCEDURE im_parin END INTERFACE im_parin CONTAINS !------------------------------------------------------------------------------! ! Description: ! ------------ !< Calculation of the air temperatures and mean radiation temperature !< This is basis for the operative temperature !< Based on a Crank-Nicholson scheme with a timestep of a hour !------------------------------------------------------------------------------! SUBROUTINE im_calc_temperatures ( i, j, k, indoor_wall_window_temperature, & near_facade_temperature, phi_hc_nd_dummy ) INTEGER(iwp) :: i INTEGER(iwp) :: j INTEGER(iwp) :: k REAL(wp) :: indoor_wall_window_temperature !< weighted temperature of innermost wall/window layer REAL(wp) :: near_facade_temperature REAL(wp) :: phi_hc_nd_dummy ! !-- Calculation of total mass specific thermal load (internal and external) phi_mtot = ( phi_m + h_t_wm * indoor_wall_window_temperature & + h_t_3 * ( phi_st + h_t_es * pt(k,j,i) & + h_t_1 * & ( ( ( phi_ia + phi_hc_nd_dummy ) / h_v ) & + near_facade_temperature ) & ) / h_t_2 & ) !< [degree_C] Eq. (C.5) ! !-- Calculation of component temperature at factual timestep theta_m_t = ( ( theta_m_t_prev & * ( ( c_m / 3600.0_wp ) - 0.5_wp * ( h_t_3 + h_t_wm ) ) & + phi_mtot & ) & / ( ( c_m / 3600.0_wp ) + 0.5_wp * ( h_t_3 + h_t_wm ) ) & ) !< [degree_C] Eq. (C.4) ! !-- Calculation of mean inner temperature for the RC-node in actual timestep theta_m = ( theta_m_t + theta_m_t_prev ) * 0.5_wp !< [degree_C] Eq. (C.9) ! !-- Calculation of mean surface temperature of the RC-node in actual timestep theta_s = ( ( h_t_ms * theta_m + phi_st + h_t_es * pt(k,j,i) & + h_t_1 * ( near_facade_temperature & + ( phi_ia + phi_hc_nd_dummy ) / h_v ) & ) & / ( h_t_ms + h_t_es + h_t_1 ) & ) !< [degree_C] Eq. (C.10) ! !-- Calculation of the air temperature of the RC-node theta_air = ( h_t_is * theta_s + h_v * near_facade_temperature & + phi_ia + phi_hc_nd_dummy ) / ( h_t_is + h_v ) !< [degree_C] Eq. (C.11) END SUBROUTINE im_calc_temperatures !------------------------------------------------------------------------------! ! Description: ! ------------ !> Initialization of the indoor model. !> Static information are calculated here, e.g. building parameters and !> geometrical information, everything that doesn't change in time. ! !-- Input values !-- Input datas from Palm, M4 ! i_global --> net_sw_in !< global radiation [W/m2] ! theta_e --> pt(k,j,i) !< undisturbed outside temperature, 1. PALM volume, for windows ! theta_sup = theta_f --> surf_usm_h%pt_10cm(m) ! surf_usm_v(l)%pt_10cm(m) !< Air temperature, facade near (10cm) air temperature from 1. Palm volume ! theta_node --> t_wall_h(nzt_wall,m) ! t_wall_v(l)%t(nzt_wall,m) !< Temperature of innermost wall layer, for opaque wall !------------------------------------------------------------------------------! SUBROUTINE im_init USE control_parameters, & ONLY: message_string, time_since_reference_point USE indices, & ONLY: nxl, nxr, nyn, nys, nzb, nzt, wall_flags_total_0 USE grid_variables, & ONLY: dx, dy USE pegrid USE surface_mod, & ONLY: surf_usm_h, surf_usm_v USE urban_surface_mod, & ONLY: building_pars, building_type INTEGER(iwp) :: bt !< local building type INTEGER(iwp) :: day_of_year !< day of the year INTEGER(iwp) :: i !< running index along x-direction INTEGER(iwp) :: j !< running index along y-direction INTEGER(iwp) :: k !< running index along z-direction INTEGER(iwp) :: l !< running index for surface-element orientation INTEGER(iwp) :: m !< running index surface elements INTEGER(iwp) :: n !< building index INTEGER(iwp) :: nb !< building index INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: build_ids !< building IDs on entire model domain INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: build_ids_final !< building IDs on entire model domain, !< multiple occurences are sorted out INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: build_ids_final_tmp !< temporary array used for resizing INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: build_ids_l !< building IDs on local subdomain INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: build_ids_l_tmp !< temporary array used to resize array of building IDs INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: displace_dum !< displacements of start addresses, used for MPI_ALLGATHERV INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: k_max_l !< highest vertical index of a building on subdomain INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: k_min_l !< lowest vertical index of a building on subdomain INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: n_fa !< counting array INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: num_facades_h !< dummy array used for summing-up total number of !< horizontal facade elements INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: num_facades_v !< dummy array used for summing-up total number of !< vertical facade elements INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: receive_dum_h !< dummy array used for MPI_ALLREDUCE INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: receive_dum_v !< dummy array used for MPI_ALLREDUCE INTEGER(iwp), DIMENSION(0:numprocs-1) :: num_buildings !< number of buildings with different ID on entire model domain INTEGER(iwp), DIMENSION(0:numprocs-1) :: num_buildings_l !< number of buildings with different ID on local subdomain REAL(wp) :: u_tmp !< dummy for temporary calculation of u-value without h_is REAL(wp) :: du_tmp !< 1/u_tmp REAL(wp) :: du_win_tmp !< 1/building(nb)%u_value_win REAL(wp) :: facade_area_v !< dummy to compute the total facade area from vertical walls REAL(wp), DIMENSION(:), ALLOCATABLE :: volume !< total building volume at each discrete height level REAL(wp), DIMENSION(:), ALLOCATABLE :: volume_l !< total building volume at each discrete height level, !< on local subdomain CALL location_message( 'initializing indoor model', 'start' ) ! !-- Initializing of indoor model is only possible if buildings can be !-- distinguished by their IDs. IF ( .NOT. building_id_f%from_file ) THEN message_string = 'Indoor model requires information about building_id' CALL message( 'im_init', 'PA0999', 1, 2, 0, 6, 0 ) ENDIF ! !-- Determine number of different building IDs on local subdomain. num_buildings_l = 0 num_buildings = 0 ALLOCATE( build_ids_l(1) ) DO i = nxl, nxr DO j = nys, nyn IF ( building_id_f%var(j,i) /= building_id_f%fill ) THEN IF ( num_buildings_l(myid) > 0 ) THEN IF ( ANY( building_id_f%var(j,i) .EQ. build_ids_l ) ) THEN CYCLE ELSE num_buildings_l(myid) = num_buildings_l(myid) + 1 ! !-- Resize array with different local building ids ALLOCATE( build_ids_l_tmp(1:SIZE(build_ids_l)) ) build_ids_l_tmp = build_ids_l DEALLOCATE( build_ids_l ) ALLOCATE( build_ids_l(1:num_buildings_l(myid)) ) build_ids_l(1:num_buildings_l(myid)-1) = & build_ids_l_tmp(1:num_buildings_l(myid)-1) build_ids_l(num_buildings_l(myid)) = building_id_f%var(j,i) DEALLOCATE( build_ids_l_tmp ) ENDIF ! !-- First occuring building id on PE ELSE num_buildings_l(myid) = num_buildings_l(myid) + 1 build_ids_l(1) = building_id_f%var(j,i) ENDIF ENDIF ENDDO ENDDO ! !-- Determine number of building IDs for the entire domain. (Note, building IDs !-- can appear multiple times as buildings might be distributed over several !-- PEs.) #if defined( __parallel ) CALL MPI_ALLREDUCE( num_buildings_l, num_buildings, numprocs, & MPI_INTEGER, MPI_SUM, comm2d, ierr ) #else num_buildings = num_buildings_l #endif ALLOCATE( build_ids(1:SUM(num_buildings)) ) ! !-- Gather building IDs. Therefore, first, determine displacements used !-- required for MPI_GATHERV call. ALLOCATE( displace_dum(0:numprocs-1) ) displace_dum(0) = 0 DO i = 1, numprocs-1 displace_dum(i) = displace_dum(i-1) + num_buildings(i-1) ENDDO #if defined( __parallel ) CALL MPI_ALLGATHERV( build_ids_l(1:num_buildings_l(myid)), & num_buildings(myid), & MPI_INTEGER, & build_ids, & num_buildings, & displace_dum, & MPI_INTEGER, & comm2d, ierr ) DEALLOCATE( displace_dum ) #else build_ids = build_ids_l #endif ! !-- Note: in parallel mode, building IDs can occur mutliple times, as !-- each PE has send its own ids. Therefore, sort out building IDs which !-- appear multiple times. num_build = 0 DO n = 1, SIZE(build_ids) IF ( ALLOCATED(build_ids_final) ) THEN IF ( ANY( build_ids(n) == build_ids_final ) ) THEN CYCLE ELSE num_build = num_build + 1 ! !-- Resize ALLOCATE( build_ids_final_tmp(1:num_build) ) build_ids_final_tmp(1:num_build-1) = build_ids_final(1:num_build-1) DEALLOCATE( build_ids_final ) ALLOCATE( build_ids_final(1:num_build) ) build_ids_final(1:num_build-1) = build_ids_final_tmp(1:num_build-1) build_ids_final(num_build) = build_ids(n) DEALLOCATE( build_ids_final_tmp ) ENDIF ELSE num_build = num_build + 1 ALLOCATE( build_ids_final(1:num_build) ) build_ids_final(num_build) = build_ids(n) ENDIF ENDDO ! !-- Allocate building-data structure array. Note, this is a global array !-- and all building IDs on domain are known by each PE. Further attributes, !-- e.g. height-dependent arrays, however, are only allocated on PEs where !-- the respective building is present (in order to reduce memory demands). ALLOCATE( buildings(1:num_build) ) ! !-- Store building IDs and check if building with certain ID is present on !-- subdomain. DO nb = 1, num_build buildings(nb)%id = build_ids_final(nb) IF ( ANY( building_id_f%var(nys:nyn,nxl:nxr) == buildings(nb)%id ) ) & buildings(nb)%on_pe = .TRUE. ENDDO ! !-- Determine the maximum vertical dimension occupied by each building. ALLOCATE( k_min_l(1:num_build) ) ALLOCATE( k_max_l(1:num_build) ) k_min_l = nzt + 1 k_max_l = 0 DO i = nxl, nxr DO j = nys, nyn IF ( building_id_f%var(j,i) /= building_id_f%fill ) THEN nb = MINLOC( ABS( buildings(:)%id - building_id_f%var(j,i) ), & DIM = 1 ) DO k = nzb, nzt+1 ! !-- Check if grid point belongs to a building. IF ( BTEST( wall_flags_total_0(k,j,i), 6 ) ) THEN k_min_l(nb) = MIN( k_min_l(nb), k ) k_max_l(nb) = MAX( k_max_l(nb), k ) ENDIF ENDDO ENDIF ENDDO ENDDO #if defined( __parallel ) CALL MPI_ALLREDUCE( k_min_l(:), buildings(:)%kb_min, num_build, & MPI_INTEGER, MPI_MIN, comm2d, ierr ) CALL MPI_ALLREDUCE( k_max_l(:), buildings(:)%kb_max, num_build, & MPI_INTEGER, MPI_MAX, comm2d, ierr ) #else buildings(:)%kb_min = k_min_l(:) buildings(:)%kb_max = k_max_l(:) #endif DEALLOCATE( k_min_l ) DEALLOCATE( k_max_l ) ! !-- Calculate building height. DO nb = 1, num_build buildings(nb)%building_height = 0.0_wp DO k = buildings(nb)%kb_min, buildings(nb)%kb_max buildings(nb)%building_height = buildings(nb)%building_height & + dzw(k+1) ENDDO ENDDO ! !-- Calculate building volume DO nb = 1, num_build ! !-- Allocate temporary array for summing-up building volume ALLOCATE( volume(buildings(nb)%kb_min:buildings(nb)%kb_max) ) ALLOCATE( volume_l(buildings(nb)%kb_min:buildings(nb)%kb_max) ) volume = 0.0_wp volume_l = 0.0_wp ! !-- Calculate building volume per height level on each PE where !-- these building is present. IF ( buildings(nb)%on_pe ) THEN ALLOCATE( buildings(nb)%volume(buildings(nb)%kb_min:buildings(nb)%kb_max) ) ALLOCATE( buildings(nb)%vol_frac(buildings(nb)%kb_min:buildings(nb)%kb_max) ) buildings(nb)%volume = 0.0_wp buildings(nb)%vol_frac = 0.0_wp IF ( ANY( building_id_f%var(nys:nyn,nxl:nxr) == buildings(nb)%id ) ) & THEN DO i = nxl, nxr DO j = nys, nyn DO k = buildings(nb)%kb_min, buildings(nb)%kb_max IF ( building_id_f%var(j,i) /= building_id_f%fill ) & volume_l(k) = volume_l(k) + dx * dy * dzw(k+1) ENDDO ENDDO ENDDO ENDIF ENDIF ! !-- Sum-up building volume from all subdomains #if defined( __parallel ) CALL MPI_ALLREDUCE( volume_l, volume, SIZE(volume), MPI_REAL, MPI_SUM, & comm2d, ierr ) #else volume = volume_l #endif ! !-- Save total building volume as well as local fraction on volume on !-- building data structure. IF ( ALLOCATED( buildings(nb)%volume ) ) buildings(nb)%volume = volume ! !-- Determine fraction of local on total building volume IF ( buildings(nb)%on_pe ) buildings(nb)%vol_frac = volume_l / volume ! !-- Calculate total building volume IF ( ALLOCATED( buildings(nb)%volume ) ) & buildings(nb)%vol_tot = SUM( buildings(nb)%volume ) DEALLOCATE( volume ) DEALLOCATE( volume_l ) ENDDO ! !-- Allocate arrays for indoor temperature. DO nb = 1, num_build IF ( buildings(nb)%on_pe ) THEN ALLOCATE( buildings(nb)%t_in(buildings(nb)%kb_min:buildings(nb)%kb_max) ) ALLOCATE( buildings(nb)%t_in_l(buildings(nb)%kb_min:buildings(nb)%kb_max) ) buildings(nb)%t_in = 0.0_wp buildings(nb)%t_in_l = 0.0_wp ENDIF ENDDO ! !-- Allocate arrays for number of facades per height level. Distinguish between !-- horizontal and vertical facades. DO nb = 1, num_build IF ( buildings(nb)%on_pe ) THEN ALLOCATE( buildings(nb)%num_facade_h(buildings(nb)%kb_min:buildings(nb)%kb_max) ) ALLOCATE( buildings(nb)%num_facade_v(buildings(nb)%kb_min:buildings(nb)%kb_max) ) buildings(nb)%num_facade_h = 0 buildings(nb)%num_facade_v = 0 ENDIF ENDDO ! !-- Determine number of facade elements per building on local subdomain. !-- Distinguish between horizontal and vertical facade elements. ! !-- Horizontal facades buildings(:)%num_facades_per_building_h_l = 0 DO m = 1, surf_usm_h%ns ! !-- For the current facade element determine corresponding building index. !-- First, obtain j,j,k indices of the building. Please note the !-- offset between facade/surface element and building location (for !-- horizontal surface elements the horizontal offsets are zero). i = surf_usm_h%i(m) + surf_usm_h%ioff j = surf_usm_h%j(m) + surf_usm_h%joff k = surf_usm_h%k(m) + surf_usm_h%koff ! !-- Determine building index and check whether building is on PE nb = MINLOC( ABS( buildings(:)%id - building_id_f%var(j,i) ), DIM = 1 ) IF ( buildings(nb)%on_pe ) THEN ! !-- Count number of facade elements at each height level. buildings(nb)%num_facade_h(k) = buildings(nb)%num_facade_h(k) + 1 ! !-- Moreover, sum up number of local facade elements per building. buildings(nb)%num_facades_per_building_h_l = & buildings(nb)%num_facades_per_building_h_l + 1 ENDIF ENDDO ! !-- Vertical facades buildings(:)%num_facades_per_building_v_l = 0 DO l = 0, 3 DO m = 1, surf_usm_v(l)%ns ! !-- For the current facade element determine corresponding building index. !-- First, obtain j,j,k indices of the building. Please note the !-- offset between facade/surface element and building location (for !-- vertical surface elements the vertical offsets are zero). i = surf_usm_v(l)%i(m) + surf_usm_v(l)%ioff j = surf_usm_v(l)%j(m) + surf_usm_v(l)%joff k = surf_usm_v(l)%k(m) + surf_usm_v(l)%koff nb = MINLOC( ABS( buildings(:)%id - building_id_f%var(j,i) ), & DIM = 1 ) IF ( buildings(nb)%on_pe ) THEN buildings(nb)%num_facade_v(k) = buildings(nb)%num_facade_v(k) + 1 buildings(nb)%num_facades_per_building_v_l = & buildings(nb)%num_facades_per_building_v_l + 1 ENDIF ENDDO ENDDO ! !-- Determine total number of facade elements per building and assign number to !-- building data type. DO nb = 1, num_build ! !-- Allocate dummy array used for summing-up facade elements. !-- Please note, dummy arguments are necessary as building-date type !-- arrays are not necessarily allocated on all PEs. ALLOCATE( num_facades_h(buildings(nb)%kb_min:buildings(nb)%kb_max) ) ALLOCATE( num_facades_v(buildings(nb)%kb_min:buildings(nb)%kb_max) ) ALLOCATE( receive_dum_h(buildings(nb)%kb_min:buildings(nb)%kb_max) ) ALLOCATE( receive_dum_v(buildings(nb)%kb_min:buildings(nb)%kb_max) ) num_facades_h = 0 num_facades_v = 0 receive_dum_h = 0 receive_dum_v = 0 IF ( buildings(nb)%on_pe ) THEN num_facades_h = buildings(nb)%num_facade_h num_facades_v = buildings(nb)%num_facade_v ENDIF #if defined( __parallel ) CALL MPI_ALLREDUCE( num_facades_h, & receive_dum_h, & buildings(nb)%kb_max - buildings(nb)%kb_min + 1, & MPI_INTEGER, & MPI_SUM, & comm2d, & ierr ) CALL MPI_ALLREDUCE( num_facades_v, & receive_dum_v, & buildings(nb)%kb_max - buildings(nb)%kb_min + 1, & MPI_INTEGER, & MPI_SUM, & comm2d, & ierr ) IF ( ALLOCATED( buildings(nb)%num_facade_h ) ) & buildings(nb)%num_facade_h = receive_dum_h IF ( ALLOCATED( buildings(nb)%num_facade_v ) ) & buildings(nb)%num_facade_v = receive_dum_v #else buildings(nb)%num_facade_h = num_facades_h buildings(nb)%num_facade_v = num_facades_v #endif ! !-- Deallocate dummy arrays DEALLOCATE( num_facades_h ) DEALLOCATE( num_facades_v ) DEALLOCATE( receive_dum_h ) DEALLOCATE( receive_dum_v ) ! !-- Allocate index arrays which link facade elements with surface-data type. !-- Please note, no height levels are considered here (information is stored !-- in surface-data type itself). IF ( buildings(nb)%on_pe ) THEN ! !-- Determine number of facade elements per building. buildings(nb)%num_facades_per_building_h = SUM( buildings(nb)%num_facade_h ) buildings(nb)%num_facades_per_building_v = SUM( buildings(nb)%num_facade_v ) ! !-- Allocate arrays which link the building with the horizontal and vertical !-- urban-type surfaces. Please note, linking arrays are allocated over all !-- facade elements, which is required in case a building is located at the !-- subdomain boundaries, where the building and the corresponding surface !-- elements are located on different subdomains. ALLOCATE( buildings(nb)%m_h(1:buildings(nb)%num_facades_per_building_h_l) ) ALLOCATE( buildings(nb)%l_v(1:buildings(nb)%num_facades_per_building_v_l) ) ALLOCATE( buildings(nb)%m_v(1:buildings(nb)%num_facades_per_building_v_l) ) ENDIF IF ( buildings(nb)%on_pe ) THEN ALLOCATE( buildings(nb)%vpf(buildings(nb)%kb_min:buildings(nb)%kb_max) ) buildings(nb)%vpf = 0.0_wp facade_area_v = 0.0_wp DO k = buildings(nb)%kb_min, buildings(nb)%kb_max facade_area_v = facade_area_v + buildings(nb)%num_facade_v(k) & * dzw(k+1) * dx ENDDO ! !-- Determine volume per total facade area (vpf). For the horizontal facade !-- area num_facades_per_building_h can be taken, multiplied with dx*dy. !-- However, due to grid stretching, vertical facade elements must be !-- summed-up vertically. Please note, if dx /= dy, an error is made! buildings(nb)%vpf = buildings(nb)%vol_tot / & ( buildings(nb)%num_facades_per_building_h * dx * dy + & facade_area_v ) ! !-- Determine floor-area-per-facade. buildings(nb)%fapf = buildings(nb)%num_facades_per_building_h & * dx * dy & / ( buildings(nb)%num_facades_per_building_h & * dx * dy + facade_area_v ) ENDIF ENDDO ! !-- Link facade elements with surface data type. !-- Allocate array for counting. ALLOCATE( n_fa(1:num_build) ) n_fa = 1 DO m = 1, surf_usm_h%ns i = surf_usm_h%i(m) + surf_usm_h%ioff j = surf_usm_h%j(m) + surf_usm_h%joff nb = MINLOC( ABS( buildings(:)%id - building_id_f%var(j,i) ), DIM = 1 ) IF ( buildings(nb)%on_pe ) THEN buildings(nb)%m_h(n_fa(nb)) = m n_fa(nb) = n_fa(nb) + 1 ENDIF ENDDO n_fa = 1 DO l = 0, 3 DO m = 1, surf_usm_v(l)%ns i = surf_usm_v(l)%i(m) + surf_usm_v(l)%ioff j = surf_usm_v(l)%j(m) + surf_usm_v(l)%joff nb = MINLOC( ABS( buildings(:)%id - building_id_f%var(j,i) ), DIM = 1 ) IF ( buildings(nb)%on_pe ) THEN buildings(nb)%l_v(n_fa(nb)) = l buildings(nb)%m_v(n_fa(nb)) = m n_fa(nb) = n_fa(nb) + 1 ENDIF ENDDO ENDDO DEALLOCATE( n_fa ) ! !-- Initialize building parameters, first by mean building type. Note, !-- in this case all buildings have the same type. !-- In a second step initialize with building tpyes from static input file, !-- where building types can be individual for each building. buildings(:)%lambda_layer3 = building_pars(31,building_type) buildings(:)%s_layer3 = building_pars(44,building_type) buildings(:)%f_c_win = building_pars(119,building_type) buildings(:)%g_value_win = building_pars(120,building_type) buildings(:)%u_value_win = building_pars(121,building_type) buildings(:)%eta_ve = building_pars(124,building_type) buildings(:)%factor_a = building_pars(125,building_type) buildings(:)%factor_c = building_pars(126,building_type) buildings(:)%lambda_at = building_pars(127,building_type) buildings(:)%theta_int_h_set = building_pars(13,building_type) buildings(:)%theta_int_c_set = building_pars(12,building_type) buildings(:)%q_h_max = building_pars(128,building_type) buildings(:)%q_c_max = building_pars(129,building_type) buildings(:)%qint_high = building_pars(130,building_type) buildings(:)%qint_low = building_pars(131,building_type) buildings(:)%height_storey = building_pars(132,building_type) buildings(:)%height_cei_con = building_pars(133,building_type) buildings(:)%params_waste_heat_h = building_pars(134,building_type) buildings(:)%params_waste_heat_c = building_pars(135,building_type) ! !-- Initialize seasonal dependent parameters, depending on day of the year. !-- First, calculated day of the year. CALL get_date_time( time_since_reference_point, day_of_year = day_of_year ) ! !-- Summer is defined in between northward- and southward equinox. IF ( day_of_year >= northward_equinox .AND. & day_of_year <= southward_equinox ) THEN buildings(:)%air_change_low = summer_pars(0,building_type) buildings(:)%air_change_high = summer_pars(1,building_type) ELSE buildings(:)%air_change_low = winter_pars(0,building_type) buildings(:)%air_change_high = winter_pars(1,building_type) ENDIF ! !-- Initialize ventilaation load. Please note, building types > 7 are actually !-- not allowed (check already in urban_surface_mod and netcdf_data_input_mod. !-- However, the building data base may be later extended. IF ( building_type == 1 .OR. building_type == 2 .OR. & building_type == 3 .OR. building_type == 10 .OR. & building_type == 11 .OR. building_type == 12 ) THEN buildings(:)%ventilation_int_loads = 1 ! !-- Office, building with large windows ELSEIF ( building_type == 4 .OR. building_type == 5 .OR. & building_type == 6 .OR. building_type == 7 .OR. & building_type == 8 .OR. building_type == 9) THEN buildings(:)%ventilation_int_loads = 2 ! !-- Industry, hospitals ELSEIF ( building_type == 13 .OR. building_type == 14 .OR. & building_type == 15 .OR. building_type == 16 .OR. & building_type == 17 .OR. building_type == 18 ) THEN buildings(:)%ventilation_int_loads = 3 ENDIF ! !-- Initialization of building parameters - level 2 IF ( building_type_f%from_file ) THEN DO i = nxl, nxr DO j = nys, nyn IF ( building_id_f%var(j,i) /= building_id_f%fill ) THEN nb = MINLOC( ABS( buildings(:)%id - building_id_f%var(j,i) ), & DIM = 1 ) bt = building_type_f%var(j,i) buildings(nb)%lambda_layer3 = building_pars(31,bt) buildings(nb)%s_layer3 = building_pars(44,bt) buildings(nb)%f_c_win = building_pars(119,bt) buildings(nb)%g_value_win = building_pars(120,bt) buildings(nb)%u_value_win = building_pars(121,bt) buildings(nb)%eta_ve = building_pars(124,bt) buildings(nb)%factor_a = building_pars(125,bt) buildings(nb)%factor_c = building_pars(126,bt) buildings(nb)%lambda_at = building_pars(127,bt) buildings(nb)%theta_int_h_set = building_pars(13,bt) buildings(nb)%theta_int_c_set = building_pars(12,bt) buildings(nb)%q_h_max = building_pars(128,bt) buildings(nb)%q_c_max = building_pars(129,bt) buildings(nb)%qint_high = building_pars(130,bt) buildings(nb)%qint_low = building_pars(131,bt) buildings(nb)%height_storey = building_pars(132,bt) buildings(nb)%height_cei_con = building_pars(133,bt) buildings(nb)%params_waste_heat_h = building_pars(134,bt) buildings(nb)%params_waste_heat_c = building_pars(135,bt) IF ( day_of_year >= northward_equinox .AND. & day_of_year <= southward_equinox ) THEN buildings(nb)%air_change_low = summer_pars(0,bt) buildings(nb)%air_change_high = summer_pars(1,bt) ELSE buildings(nb)%air_change_low = winter_pars(0,bt) buildings(nb)%air_change_high = winter_pars(1,bt) ENDIF ! !-- Initialize ventilaation load. Please note, building types > 7 !-- are actually not allowed (check already in urban_surface_mod !-- and netcdf_data_input_mod. However, the building data base may !-- be later extended. IF ( bt == 1 .OR. bt == 2 .OR. & bt == 3 .OR. bt == 10 .OR. & bt == 11 .OR. bt == 12 ) THEN buildings(nb)%ventilation_int_loads = 1 ! !-- Office, building with large windows ELSEIF ( bt == 4 .OR. bt == 5 .OR. & bt == 6 .OR. bt == 7 .OR. & bt == 8 .OR. bt == 9) THEN buildings(nb)%ventilation_int_loads = 2 ! !-- Industry, hospitals ELSEIF ( bt == 13 .OR. bt == 14 .OR. & bt == 15 .OR. bt == 16 .OR. & bt == 17 .OR. bt == 18 ) THEN buildings(nb)%ventilation_int_loads = 3 ENDIF ENDIF ENDDO ENDDO ENDIF ! !-- Calculation of surface-related heat transfer coeffiecient !-- out of standard u-values from building database !-- only amount of extern and surface is used !-- otherwise amount between air and surface taken account twice DO nb = 1, num_build IF ( buildings(nb)%on_pe ) THEN du_win_tmp = 1.0_wp / buildings(nb)%u_value_win u_tmp = buildings(nb)%u_value_win * ( du_win_tmp / ( du_win_tmp - & 0.125_wp + ( 1.0_wp / h_is ) ) ) du_tmp = 1.0_wp / u_tmp buildings(nb)%h_es = 1.0_wp / ( du_tmp - ( 1.0_wp / h_is ) ) ENDIF ENDDO ! !-- Initial room temperature [K] !-- (after first loop, use theta_m_t as theta_m_t_prev) theta_m_t_prev = initial_indoor_temperature ! !-- Initialize indoor temperature. Actually only for output at initial state. DO nb = 1, num_build IF ( buildings(nb)%on_pe ) & buildings(nb)%t_in(:) = initial_indoor_temperature ENDDO CALL location_message( 'initializing indoor model', 'finished' ) END SUBROUTINE im_init !------------------------------------------------------------------------------! ! Description: ! ------------ !> Main part of the indoor model. !> Calculation of .... (kanani: Please describe) !------------------------------------------------------------------------------! SUBROUTINE im_main_heatcool ! USE basic_constants_and_equations_mod, & ! ONLY: c_p USE control_parameters, & ONLY: time_since_reference_point USE grid_variables, & ONLY: dx, dy USE pegrid USE surface_mod, & ONLY: ind_veg_wall, ind_wat_win, surf_usm_h, surf_usm_v USE urban_surface_mod, & ONLY: nzt_wall, t_wall_h, t_wall_v, t_window_h, t_window_v, & building_type INTEGER(iwp) :: i !< index of facade-adjacent atmosphere grid point in x-direction INTEGER(iwp) :: j !< index of facade-adjacent atmosphere grid point in y-direction INTEGER(iwp) :: k !< index of facade-adjacent atmosphere grid point in z-direction INTEGER(iwp) :: kk !< vertical index of indoor grid point adjacent to facade INTEGER(iwp) :: l !< running index for surface-element orientation INTEGER(iwp) :: m !< running index surface elements INTEGER(iwp) :: nb !< running index for buildings INTEGER(iwp) :: fa !< running index for facade elements of each building REAL(wp) :: indoor_wall_window_temperature !< weighted temperature of innermost wall/window layer REAL(wp) :: near_facade_temperature !< outside air temperature 10cm away from facade REAL(wp) :: second_of_day !< second of the current day REAL(wp) :: time_utc_hour !< time of day (hour UTC) REAL(wp), DIMENSION(:), ALLOCATABLE :: t_in_l_send !< dummy send buffer used for summing-up indoor temperature per kk-level REAL(wp), DIMENSION(:), ALLOCATABLE :: t_in_recv !< dummy recv buffer used for summing-up indoor temperature per kk-level ! !-- Determine time of day in hours. CALL get_date_time( time_since_reference_point, second_of_day=second_of_day ) time_utc_hour = second_of_day / seconds_per_hour ! !-- Following calculations must be done for each facade element. DO nb = 1, num_build ! !-- First, check whether building is present on local subdomain. IF ( buildings(nb)%on_pe ) THEN ! !-- Determine daily schedule. 08:00-18:00 = 1, other hours = 0. !-- Residental Building, panel WBS 70 IF ( buildings(nb)%ventilation_int_loads == 1 ) THEN IF ( time_utc_hour >= 8.0_wp .AND. time_utc_hour <= 18.0_wp ) THEN schedule_d = 0 ELSE schedule_d = 1 ENDIF ENDIF ! !-- Office, building with large windows IF ( buildings(nb)%ventilation_int_loads == 2 ) THEN IF ( time_utc_hour >= 8.0_wp .AND. time_utc_hour <= 18.0_wp ) THEN schedule_d = 1 ELSE schedule_d = 0 ENDIF ENDIF ! !-- Industry, hospitals IF ( buildings(nb)%ventilation_int_loads == 3 ) THEN IF ( time_utc_hour >= 6.0_wp .AND. time_utc_hour <= 22.0_wp ) THEN schedule_d = 1 ELSE schedule_d = 0 ENDIF ENDIF ! !-- Initialize/reset indoor temperature buildings(nb)%t_in_l = 0.0_wp ! !-- Horizontal surfaces DO fa = 1, buildings(nb)%num_facades_per_building_h_l ! !-- Determine index where corresponding surface-type information !-- is stored. m = buildings(nb)%m_h(fa) ! !-- Determine building height level index. kk = surf_usm_h%k(m) + surf_usm_h%koff ! !-- Building geometries --> not time-dependent facade_element_area = dx * dy !< [m2] surface area per facade element floor_area_per_facade = buildings(nb)%fapf !< [m2/m2] floor area per facade area indoor_volume_per_facade = buildings(nb)%vpf(kk) !< [m3/m2] indoor air volume per facade area buildings(nb)%area_facade = facade_element_area * & ( buildings(nb)%num_facades_per_building_h + & buildings(nb)%num_facades_per_building_v ) !< [m2] area of total facade window_area_per_facade = surf_usm_h%frac(m,ind_wat_win) * facade_element_area !< [m2] window area per facade element buildings(nb)%net_floor_area = buildings(nb)%vol_tot / ( buildings(nb)%height_storey ) total_area = buildings(nb)%net_floor_area !< [m2] area of all surfaces pointing to zone Eq. (9) according to section 7.2.2.2 a_m = buildings(nb)%factor_a * total_area * & ( facade_element_area / buildings(nb)%area_facade ) * & buildings(nb)%lambda_at !< [m2] standard values according to Table 12 section 12.3.1.2 (calculate over Eq. (65) according to section 12.3.1.2) c_m = buildings(nb)%factor_c * total_area * & ( facade_element_area / buildings(nb)%area_facade ) !< [J/K] standard values according to table 12 section 12.3.1.2 (calculate over Eq. (66) according to section 12.3.1.2) ! !-- Calculation of heat transfer coefficient for transmission --> not time-dependent h_t_es = window_area_per_facade * buildings(nb)%h_es !< [W/K] only for windows h_t_is = buildings(nb)%area_facade * h_is !< [W/K] with h_is = 3.45 W / (m2 K) between surface and air, Eq. (9) h_t_ms = a_m * h_ms !< [W/K] with h_ms = 9.10 W / (m2 K) between component and surface, Eq. (64) h_t_wall = 1.0_wp / ( 1.0_wp / ( ( facade_element_area - window_area_per_facade ) & !< [W/K] * buildings(nb)%lambda_layer3 / buildings(nb)%s_layer3 * 0.5_wp & ) + 1.0_wp / h_t_ms ) !< [W/K] opaque components h_t_wm = 1.0_wp / ( 1.0_wp / h_t_wall - 1.0_wp / h_t_ms ) !< [W/K] emmision Eq. (63), Section 12.2.2 ! !-- internal air loads dependent on the occupacy of the room !-- basical internal heat gains (qint_low) with additional internal heat gains by occupancy (qint_high) (0,5*phi_int) phi_ia = 0.5_wp * ( ( buildings(nb)%qint_high * schedule_d + buildings(nb)%qint_low ) & * floor_area_per_facade ) q_int = phi_ia / total_area ! !-- Airflow dependent on the occupacy of the room !-- basical airflow (air_change_low) with additional airflow gains by occupancy (air_change_high) air_change = ( buildings(nb)%air_change_high * schedule_d + buildings(nb)%air_change_low ) !< [1/h]? ! !-- Heat transfer of ventilation !-- not less than 0.01 W/K to provide division by 0 in further calculations !-- with heat capacity of air 0.33 Wh/m2K h_v = MAX( 0.01_wp , ( air_change * indoor_volume_per_facade * & 0.33_wp * (1.0_wp - buildings(nb)%eta_ve ) ) ) !< [W/K] from ISO 13789 Eq.(10) !-- Heat transfer coefficient auxiliary variables h_t_1 = 1.0_wp / ( ( 1.0_wp / h_v ) + ( 1.0_wp / h_t_is ) ) !< [W/K] Eq. (C.6) h_t_2 = h_t_1 + h_t_es !< [W/K] Eq. (C.7) h_t_3 = 1.0_wp / ( ( 1.0_wp / h_t_2 ) + ( 1.0_wp / h_t_ms ) ) !< [W/K] Eq. (C.8) ! !-- Net short-wave radiation through window area (was i_global) net_sw_in = surf_usm_h%rad_sw_in(m) - surf_usm_h%rad_sw_out(m) ! !-- Quantities needed for im_calc_temperatures i = surf_usm_h%i(m) j = surf_usm_h%j(m) k = surf_usm_h%k(m) near_facade_temperature = surf_usm_h%pt_10cm(m) indoor_wall_window_temperature = & surf_usm_h%frac(m,ind_veg_wall) * t_wall_h(nzt_wall,m) & + surf_usm_h%frac(m,ind_wat_win) * t_window_h(nzt_wall,m) ! !-- Solar thermal gains. If net_sw_in larger than sun-protection !-- threshold parameter (params_solar_protection), sun protection will !-- be activated IF ( net_sw_in <= params_solar_protection ) THEN solar_protection_off = 1 solar_protection_on = 0 ELSE solar_protection_off = 0 solar_protection_on = 1 ENDIF ! !-- Calculation of total heat gains from net_sw_in through windows [W] in respect on automatic sun protection !-- DIN 4108 - 2 chap.8 phi_sol = ( window_area_per_facade * net_sw_in * solar_protection_off & + window_area_per_facade * net_sw_in * buildings(nb)%f_c_win * solar_protection_on ) & * buildings(nb)%g_value_win * ( 1.0_wp - params_f_f ) * params_f_w q_sol = phi_sol ! !-- Calculation of the mass specific thermal load for internal and external heatsources of the inner node phi_m = (a_m / total_area) * ( phi_ia + phi_sol ) !< [W] Eq. (C.2) with phi_ia=0,5*phi_int q_c_m = phi_m ! !-- Calculation mass specific thermal load implied non thermal mass phi_st = ( 1.0_wp - ( a_m / total_area ) - ( h_t_es / ( 9.1_wp * total_area ) ) ) & * ( phi_ia + phi_sol ) !< [W] Eq. (C.3) with phi_ia=0,5*phi_int q_c_st = phi_st ! !-- Calculations for deriving indoor temperature and heat flux into the wall !-- Step 1: Indoor temperature without heating and cooling !-- section C.4.1 Picture C.2 zone 3) phi_hc_nd = 0.0_wp CALL im_calc_temperatures ( i, j, k, indoor_wall_window_temperature, & near_facade_temperature, phi_hc_nd ) ! !-- If air temperature between border temperatures of heating and cooling, assign output variable, then ready IF ( buildings(nb)%theta_int_h_set <= theta_air .AND. theta_air <= buildings(nb)%theta_int_c_set ) THEN phi_hc_nd_ac = 0.0_wp phi_hc_nd = phi_hc_nd_ac theta_air_ac = theta_air ! !-- Step 2: Else, apply 10 W/m2 heating/cooling power and calculate indoor temperature !-- again. ELSE ! !-- Temperature not correct, calculation method according to section C4.2 theta_air_0 = theta_air !< temperature without heating/cooling ! !-- Heating or cooling? IF ( theta_air_0 > buildings(nb)%theta_int_c_set ) THEN theta_air_set = buildings(nb)%theta_int_c_set ELSE theta_air_set = buildings(nb)%theta_int_h_set ENDIF ! !-- Calculate the temperature with phi_hc_nd_10 phi_hc_nd_10 = 10.0_wp * floor_area_per_facade phi_hc_nd = phi_hc_nd_10 CALL im_calc_temperatures ( i, j, k, indoor_wall_window_temperature, & near_facade_temperature, phi_hc_nd ) theta_air_10 = theta_air !< temperature with 10 W/m2 of heating phi_hc_nd_un = phi_hc_nd_10 * (theta_air_set - theta_air_0) & / (theta_air_10 - theta_air_0) !< Eq. (C.13) ! !-- Step 3: With temperature ratio to determine the heating or cooling capacity !-- If necessary, limit the power to maximum power !-- section C.4.1 Picture C.2 zone 2) and 4) buildings(nb)%phi_c_max = buildings(nb)%q_c_max * floor_area_per_facade buildings(nb)%phi_h_max = buildings(nb)%q_h_max * floor_area_per_facade IF ( buildings(nb)%phi_c_max < phi_hc_nd_un .AND. phi_hc_nd_un < buildings(nb)%phi_h_max ) THEN phi_hc_nd_ac = phi_hc_nd_un phi_hc_nd = phi_hc_nd_un ELSE ! !-- Step 4: Inner temperature with maximum heating (phi_hc_nd_un positive) or cooling (phi_hc_nd_un negative) !-- section C.4.1 Picture C.2 zone 1) and 5) IF ( phi_hc_nd_un > 0.0_wp ) THEN phi_hc_nd_ac = buildings(nb)%phi_h_max !< Limit heating ELSE phi_hc_nd_ac = buildings(nb)%phi_c_max !< Limit cooling ENDIF ENDIF phi_hc_nd = phi_hc_nd_ac ! !-- Calculate the temperature with phi_hc_nd_ac (new) CALL im_calc_temperatures ( i, j, k, indoor_wall_window_temperature, & near_facade_temperature, phi_hc_nd ) theta_air_ac = theta_air ENDIF ! !-- Update theta_m_t_prev theta_m_t_prev = theta_m_t q_vent = h_v * ( theta_air - near_facade_temperature ) ! !-- Calculate the operating temperature with weighted mean temperature of air and mean solar temperature !-- Will be used for thermal comfort calculations theta_op = 0.3_wp * theta_air_ac + 0.7_wp * theta_s !< [degree_C] operative Temperature Eq. (C.12) ! surf_usm_h%t_indoor(m) = theta_op !< not integrated now ! !-- Heat flux into the wall. Value needed in urban_surface_mod to !-- calculate heat transfer through wall layers towards the facade !-- (use c_p * rho_surface to convert [W/m2] into [K m/s]) q_wall_win = h_t_ms * ( theta_s - theta_m ) & / ( facade_element_area & - window_area_per_facade ) q_trans = q_wall_win * facade_element_area ! !-- Transfer q_wall_win back to USM (innermost wall/window layer) surf_usm_h%iwghf_eb(m) = q_wall_win surf_usm_h%iwghf_eb_window(m) = q_wall_win ! !-- Sum up operational indoor temperature per kk-level. Further below, !-- this temperature is reduced by MPI to one temperature per kk-level !-- and building (processor overlapping) buildings(nb)%t_in_l(kk) = buildings(nb)%t_in_l(kk) + theta_op ! !-- Calculation of waste heat !-- Anthropogenic heat output IF ( phi_hc_nd_ac > 0.0_wp ) THEN heating_on = 1 cooling_on = 0 ELSE heating_on = 0 cooling_on = -1 ENDIF q_waste_heat = ( phi_hc_nd * ( & buildings(nb)%params_waste_heat_h * heating_on + & buildings(nb)%params_waste_heat_c * cooling_on ) & ) / facade_element_area !< [W/m2] , observe the directional convention in PALM! surf_usm_h%waste_heat(m) = q_waste_heat ENDDO !< Horizontal surfaces loop ! !-- Vertical surfaces DO fa = 1, buildings(nb)%num_facades_per_building_v_l ! !-- Determine indices where corresponding surface-type information !-- is stored. l = buildings(nb)%l_v(fa) m = buildings(nb)%m_v(fa) ! !-- Determine building height level index. kk = surf_usm_v(l)%k(m) + surf_usm_v(l)%koff ! !-- (SOME OF THE FOLLOWING (not time-dependent COULD PROBABLY GO INTO A FUNCTION !-- EXCEPT facade_element_area, EVERYTHING IS CALCULATED EQUALLY) !-- Building geometries --> not time-dependent IF ( l == 0 .OR. l == 1 ) facade_element_area = dx * dzw(kk+1) !< [m2] surface area per facade element IF ( l == 2 .OR. l == 3 ) facade_element_area = dy * dzw(kk+1) !< [m2] surface area per facade element floor_area_per_facade = buildings(nb)%fapf !< [m2/m2] floor area per facade area indoor_volume_per_facade = buildings(nb)%vpf(kk) !< [m3/m2] indoor air volume per facade area buildings(nb)%area_facade = facade_element_area * & ( buildings(nb)%num_facades_per_building_h + & buildings(nb)%num_facades_per_building_v ) !< [m2] area of total facade window_area_per_facade = surf_usm_v(l)%frac(m,ind_wat_win) * facade_element_area !< [m2] window area per facade element buildings(nb)%net_floor_area = buildings(nb)%vol_tot / ( buildings(nb)%height_storey ) total_area = buildings(nb)%net_floor_area !< [m2] area of all surfaces pointing to zone Eq. (9) according to section 7.2.2.2 a_m = buildings(nb)%factor_a * total_area * & ( facade_element_area / buildings(nb)%area_facade ) * & buildings(nb)%lambda_at !< [m2] standard values according to Table 12 section 12.3.1.2 (calculate over Eq. (65) according to section 12.3.1.2) c_m = buildings(nb)%factor_c * total_area * & ( facade_element_area / buildings(nb)%area_facade ) !< [J/K] standard values according to table 12 section 12.3.1.2 (calculate over Eq. (66) according to section 12.3.1.2) ! !-- Calculation of heat transfer coefficient for transmission --> not time-dependent h_t_es = window_area_per_facade * buildings(nb)%h_es !< [W/K] only for windows h_t_is = buildings(nb)%area_facade * h_is !< [W/K] with h_is = 3.45 W / (m2 K) between surface and air, Eq. (9) h_t_ms = a_m * h_ms !< [W/K] with h_ms = 9.10 W / (m2 K) between component and surface, Eq. (64) h_t_wall = 1.0_wp / ( 1.0_wp / ( ( facade_element_area - window_area_per_facade ) & !< [W/K] * buildings(nb)%lambda_layer3 / buildings(nb)%s_layer3 * 0.5_wp & ) + 1.0_wp / h_t_ms ) !< [W/K] opaque components h_t_wm = 1.0_wp / ( 1.0_wp / h_t_wall - 1.0_wp / h_t_ms ) !< [W/K] emmision Eq. (63), Section 12.2.2 ! !-- internal air loads dependent on the occupacy of the room !-- basical internal heat gains (qint_low) with additional internal heat gains by occupancy (qint_high) (0,5*phi_int) phi_ia = 0.5_wp * ( ( buildings(nb)%qint_high * schedule_d + buildings(nb)%qint_low ) & * floor_area_per_facade ) q_int = phi_ia ! !-- Airflow dependent on the occupacy of the room !-- basical airflow (air_change_low) with additional airflow gains by occupancy (air_change_high) air_change = ( buildings(nb)%air_change_high * schedule_d + buildings(nb)%air_change_low ) ! !-- Heat transfer of ventilation !-- not less than 0.01 W/K to provide division by 0 in further calculations !-- with heat capacity of air 0.33 Wh/m2K h_v = MAX( 0.01_wp , ( air_change * indoor_volume_per_facade * & 0.33_wp * (1.0_wp - buildings(nb)%eta_ve ) ) ) !< [W/K] from ISO 13789 Eq.(10) !-- Heat transfer coefficient auxiliary variables h_t_1 = 1.0_wp / ( ( 1.0_wp / h_v ) + ( 1.0_wp / h_t_is ) ) !< [W/K] Eq. (C.6) h_t_2 = h_t_1 + h_t_es !< [W/K] Eq. (C.7) h_t_3 = 1.0_wp / ( ( 1.0_wp / h_t_2 ) + ( 1.0_wp / h_t_ms ) ) !< [W/K] Eq. (C.8) ! !-- Net short-wave radiation through window area (was i_global) net_sw_in = surf_usm_v(l)%rad_sw_in(m) - surf_usm_v(l)%rad_sw_out(m) ! !-- Quantities needed for im_calc_temperatures i = surf_usm_v(l)%i(m) j = surf_usm_v(l)%j(m) k = surf_usm_v(l)%k(m) near_facade_temperature = surf_usm_v(l)%pt_10cm(m) indoor_wall_window_temperature = & surf_usm_v(l)%frac(m,ind_veg_wall) * t_wall_v(l)%t(nzt_wall,m) & + surf_usm_v(l)%frac(m,ind_wat_win) * t_window_v(l)%t(nzt_wall,m) ! !-- Solar thermal gains. If net_sw_in larger than sun-protection !-- threshold parameter (params_solar_protection), sun protection will !-- be activated IF ( net_sw_in <= params_solar_protection ) THEN solar_protection_off = 1 solar_protection_on = 0 ELSE solar_protection_off = 0 solar_protection_on = 1 ENDIF ! !-- Calculation of total heat gains from net_sw_in through windows [W] in respect on automatic sun protection !-- DIN 4108 - 2 chap.8 phi_sol = ( window_area_per_facade * net_sw_in * solar_protection_off & + window_area_per_facade * net_sw_in * buildings(nb)%f_c_win * solar_protection_on ) & * buildings(nb)%g_value_win * ( 1.0_wp - params_f_f ) * params_f_w q_sol = phi_sol ! !-- Calculation of the mass specific thermal load for internal and external heatsources phi_m = (a_m / total_area) * ( phi_ia + phi_sol ) !< [W] Eq. (C.2) with phi_ia=0,5*phi_int q_c_m = phi_m ! !-- Calculation mass specific thermal load implied non thermal mass phi_st = ( 1.0_wp - ( a_m / total_area ) - ( h_t_es / ( 9.1_wp * total_area ) ) ) & * ( phi_ia + phi_sol ) !< [W] Eq. (C.3) with phi_ia=0,5*phi_int q_c_st = phi_st ! !-- Calculations for deriving indoor temperature and heat flux into the wall !-- Step 1: Indoor temperature without heating and cooling !-- section C.4.1 Picture C.2 zone 3) phi_hc_nd = 0.0_wp CALL im_calc_temperatures ( i, j, k, indoor_wall_window_temperature, & near_facade_temperature, phi_hc_nd ) ! !-- If air temperature between border temperatures of heating and cooling, assign output variable, then ready IF ( buildings(nb)%theta_int_h_set <= theta_air .AND. theta_air <= buildings(nb)%theta_int_c_set ) THEN phi_hc_nd_ac = 0.0_wp phi_hc_nd = phi_hc_nd_ac theta_air_ac = theta_air ! !-- Step 2: Else, apply 10 W/m2 heating/cooling power and calculate indoor temperature !-- again. ELSE ! !-- Temperature not correct, calculation method according to section C4.2 theta_air_0 = theta_air !< Note temperature without heating/cooling ! !-- Heating or cooling? IF ( theta_air_0 > buildings(nb)%theta_int_c_set ) THEN theta_air_set = buildings(nb)%theta_int_c_set ELSE theta_air_set = buildings(nb)%theta_int_h_set ENDIF !-- Calculate the temperature with phi_hc_nd_10 phi_hc_nd_10 = 10.0_wp * floor_area_per_facade phi_hc_nd = phi_hc_nd_10 CALL im_calc_temperatures ( i, j, k, indoor_wall_window_temperature, & near_facade_temperature, phi_hc_nd ) theta_air_10 = theta_air !< Note the temperature with 10 W/m2 of heating phi_hc_nd_un = phi_hc_nd_10 * ( theta_air_set - theta_air_0 ) & / ( theta_air_10 - theta_air_0 ) !< Eq. (C.13) ! !-- Step 3: With temperature ratio to determine the heating or cooling capacity !-- If necessary, limit the power to maximum power !-- section C.4.1 Picture C.2 zone 2) and 4) buildings(nb)%phi_c_max = buildings(nb)%q_c_max * floor_area_per_facade buildings(nb)%phi_h_max = buildings(nb)%q_h_max * floor_area_per_facade IF ( buildings(nb)%phi_c_max < phi_hc_nd_un .AND. phi_hc_nd_un < buildings(nb)%phi_h_max ) THEN phi_hc_nd_ac = phi_hc_nd_un phi_hc_nd = phi_hc_nd_un ELSE ! !-- Step 4: Inner temperature with maximum heating (phi_hc_nd_un positive) or cooling (phi_hc_nd_un negative) !-- section C.4.1 Picture C.2 zone 1) and 5) IF ( phi_hc_nd_un > 0.0_wp ) THEN phi_hc_nd_ac = buildings(nb)%phi_h_max !< Limit heating ELSE phi_hc_nd_ac = buildings(nb)%phi_c_max !< Limit cooling ENDIF ENDIF phi_hc_nd = phi_hc_nd_ac ! !-- Calculate the temperature with phi_hc_nd_ac (new) CALL im_calc_temperatures ( i, j, k, indoor_wall_window_temperature, & near_facade_temperature, phi_hc_nd ) theta_air_ac = theta_air ENDIF ! !-- Update theta_m_t_prev theta_m_t_prev = theta_m_t q_vent = h_v * ( theta_air - near_facade_temperature ) ! !-- Calculate the operating temperature with weighted mean of temperature of air and mean !-- Will be used for thermal comfort calculations theta_op = 0.3_wp * theta_air_ac + 0.7_wp * theta_s ! surf_usm_v(l)%t_indoor(m) = theta_op !< not integrated yet ! !-- Heat flux into the wall. Value needed in urban_surface_mod to !-- calculate heat transfer through wall layers towards the facade q_wall_win = h_t_ms * ( theta_s - theta_m ) & / ( facade_element_area & - window_area_per_facade ) q_trans = q_wall_win * facade_element_area ! !-- Transfer q_wall_win back to USM (innermost wall/window layer) surf_usm_v(l)%iwghf_eb(m) = q_wall_win surf_usm_v(l)%iwghf_eb_window(m) = q_wall_win ! !-- Sum up operational indoor temperature per kk-level. Further below, !-- this temperature is reduced by MPI to one temperature per kk-level !-- and building (processor overlapping) buildings(nb)%t_in_l(kk) = buildings(nb)%t_in_l(kk) + theta_op ! !-- Calculation of waste heat !-- Anthropogenic heat output IF ( phi_hc_nd_ac > 0.0_wp ) THEN heating_on = 1 cooling_on = 0 ELSE heating_on = 0 cooling_on = -1 ENDIF q_waste_heat = ( phi_hc_nd * ( & buildings(nb)%params_waste_heat_h * heating_on + & buildings(nb)%params_waste_heat_c * cooling_on ) & ) / facade_element_area !< [W/m2] , observe the directional convention in PALM! surf_usm_v(l)%waste_heat(m) = q_waste_heat ENDDO !< Vertical surfaces loop ENDIF !< buildings(nb)%on_pe ENDDO !< buildings loop ! !-- Determine the mean building temperature. DO nb = 1, num_build ! !-- Allocate dummy array used for summing-up facade elements. !-- Please note, dummy arguments are necessary as building-date type !-- arrays are not necessarily allocated on all PEs. ALLOCATE( t_in_l_send(buildings(nb)%kb_min:buildings(nb)%kb_max) ) ALLOCATE( t_in_recv(buildings(nb)%kb_min:buildings(nb)%kb_max) ) t_in_l_send = 0.0_wp t_in_recv = 0.0_wp IF ( buildings(nb)%on_pe ) THEN t_in_l_send = buildings(nb)%t_in_l ENDIF #if defined( __parallel ) CALL MPI_ALLREDUCE( t_in_l_send, & t_in_recv, & buildings(nb)%kb_max - buildings(nb)%kb_min + 1, & MPI_REAL, & MPI_SUM, & comm2d, & ierr ) IF ( ALLOCATED( buildings(nb)%t_in ) ) & buildings(nb)%t_in = t_in_recv #else IF ( ALLOCATED( buildings(nb)%t_in ) ) & buildings(nb)%t_in = buildings(nb)%t_in_l #endif IF ( ALLOCATED( buildings(nb)%t_in ) ) THEN ! !-- Average indoor temperature. Note, in case a building is completely !-- surrounded by higher buildings, it may have no facade elements !-- at some height levels, which will lead to a divide by zero. DO k = buildings(nb)%kb_min, buildings(nb)%kb_max IF ( buildings(nb)%num_facade_h(k) + & buildings(nb)%num_facade_v(k) > 0 ) THEN buildings(nb)%t_in(k) = buildings(nb)%t_in(k) / & REAL( buildings(nb)%num_facade_h(k) + & buildings(nb)%num_facade_v(k), KIND = wp ) ENDIF ENDDO ! !-- If indoor temperature is not defined because of missing facade !-- elements, the values from the above-lying level will be taken. !-- At least at the top of the buildings facades are defined, so that !-- at least there an indoor temperature is defined. This information !-- will propagate downwards the building. DO k = buildings(nb)%kb_max-1, buildings(nb)%kb_min, -1 IF ( buildings(nb)%num_facade_h(k) + & buildings(nb)%num_facade_v(k) <= 0 ) THEN buildings(nb)%t_in(k) = buildings(nb)%t_in(k+1) ENDIF ENDDO ENDIF ! !-- Deallocate dummy arrays DEALLOCATE( t_in_l_send ) DEALLOCATE( t_in_recv ) ENDDO END SUBROUTINE im_main_heatcool !-----------------------------------------------------------------------------! ! Description: !------------- !> Check data output for plant canopy model !-----------------------------------------------------------------------------! SUBROUTINE im_check_data_output( var, unit ) CHARACTER (LEN=*) :: unit !< CHARACTER (LEN=*) :: var !< SELECT CASE ( TRIM( var ) ) CASE ( 'im_hf_roof') unit = 'W m-2' CASE ( 'im_hf_wall_win' ) unit = 'W m-2' CASE ( 'im_hf_wall_win_waste' ) unit = 'W m-2' CASE ( 'im_hf_roof_waste' ) unit = 'W m-2' CASE ( 'im_t_indoor_mean' ) unit = 'K' CASE ( 'im_t_indoor_roof' ) unit = 'K' CASE ( 'im_t_indoor_wall_win' ) unit = 'K' CASE DEFAULT unit = 'illegal' END SELECT END SUBROUTINE !-----------------------------------------------------------------------------! ! Description: !------------- !> Check parameters routine for plant canopy model !-----------------------------------------------------------------------------! SUBROUTINE im_check_parameters ! USE control_parameters, ! ONLY: message_string END SUBROUTINE im_check_parameters !-----------------------------------------------------------------------------! ! Description: !------------- !> Subroutine defining appropriate grid for netcdf variables. !> It is called from subroutine netcdf. !-----------------------------------------------------------------------------! SUBROUTINE im_define_netcdf_grid( var, found, grid_x, grid_y, grid_z ) CHARACTER (LEN=*), INTENT(IN) :: var LOGICAL, INTENT(OUT) :: found CHARACTER (LEN=*), INTENT(OUT) :: grid_x CHARACTER (LEN=*), INTENT(OUT) :: grid_y CHARACTER (LEN=*), INTENT(OUT) :: grid_z found = .TRUE. ! !-- Check for the grid SELECT CASE ( TRIM( var ) ) CASE ( 'im_hf_roof', 'im_hf_roof_waste' ) grid_x = 'x' grid_y = 'y' grid_z = 'zw' ! !-- Heat fluxes at vertical walls are actually defined on stagged grid, i.e. xu, yv. CASE ( 'im_hf_wall_win', 'im_hf_wall_win_waste' ) grid_x = 'x' grid_y = 'y' grid_z = 'zu' CASE ( 'im_t_indoor_mean', 'im_t_indoor_roof', 'im_t_indoor_wall_win') grid_x = 'x' grid_y = 'y' grid_z = 'zw' CASE DEFAULT found = .FALSE. grid_x = 'none' grid_y = 'none' grid_z = 'none' END SELECT END SUBROUTINE im_define_netcdf_grid !------------------------------------------------------------------------------! ! Description: ! ------------ !> Subroutine defining 3D output variables !------------------------------------------------------------------------------! SUBROUTINE im_data_output_3d( av, variable, found, local_pf, fill_value, & nzb_do, nzt_do ) USE indices USE kinds CHARACTER (LEN=*) :: variable !< INTEGER(iwp) :: av !< INTEGER(iwp) :: i !< INTEGER(iwp) :: j !< INTEGER(iwp) :: k !< INTEGER(iwp) :: l !< INTEGER(iwp) :: m !< INTEGER(iwp) :: nb !< index of the building in the building data structure INTEGER(iwp) :: nzb_do !< lower limit of the data output (usually 0) INTEGER(iwp) :: nzt_do !< vertical upper limit of the data output (usually nz_do3d) LOGICAL :: found !< REAL(wp), INTENT(IN) :: fill_value !< value for the _FillValue attribute REAL(sp), DIMENSION(nxl:nxr,nys:nyn,nzb_do:nzt_do) :: local_pf !< local_pf = fill_value found = .TRUE. SELECT CASE ( TRIM( variable ) ) ! !-- Output of indoor temperature. All grid points within the building are !-- filled with values, while atmospheric grid points are set to _FillValues. CASE ( 'im_t_indoor_mean' ) IF ( av == 0 ) THEN DO i = nxl, nxr DO j = nys, nyn IF ( building_id_f%var(j,i) /= building_id_f%fill ) THEN ! !-- Determine index of the building within the building data structure. nb = MINLOC( ABS( buildings(:)%id - building_id_f%var(j,i) ), & DIM = 1 ) IF ( buildings(nb)%on_pe ) THEN ! !-- Write mean building temperature onto output array. Please note, !-- in contrast to many other loops in the output, the vertical !-- bounds are determined by the lowest and hightest vertical index !-- occupied by the building. DO k = buildings(nb)%kb_min, buildings(nb)%kb_max local_pf(i,j,k) = buildings(nb)%t_in(k) ENDDO ENDIF ENDIF ENDDO ENDDO ENDIF CASE ( 'im_hf_roof' ) IF ( av == 0 ) THEN DO m = 1, surf_usm_h%ns i = surf_usm_h%i(m) !+ surf_usm_h%ioff j = surf_usm_h%j(m) !+ surf_usm_h%joff k = surf_usm_h%k(m) !+ surf_usm_h%koff local_pf(i,j,k) = surf_usm_h%iwghf_eb(m) ENDDO ENDIF CASE ( 'im_hf_roof_waste' ) IF ( av == 0 ) THEN DO m = 1, surf_usm_h%ns i = surf_usm_h%i(m) !+ surf_usm_h%ioff j = surf_usm_h%j(m) !+ surf_usm_h%joff k = surf_usm_h%k(m) !+ surf_usm_h%koff local_pf(i,j,k) = surf_usm_h%waste_heat(m) ENDDO ENDIF CASE ( 'im_hf_wall_win' ) IF ( av == 0 ) THEN DO l = 0, 3 DO m = 1, surf_usm_v(l)%ns i = surf_usm_v(l)%i(m) !+ surf_usm_v(l)%ioff j = surf_usm_v(l)%j(m) !+ surf_usm_v(l)%joff k = surf_usm_v(l)%k(m) !+ surf_usm_v(l)%koff local_pf(i,j,k) = surf_usm_v(l)%iwghf_eb(m) ENDDO ENDDO ENDIF CASE ( 'im_hf_wall_win_waste' ) IF ( av == 0 ) THEN DO l = 0, 3 DO m = 1, surf_usm_v(l)%ns i = surf_usm_v(l)%i(m) !+ surf_usm_v(l)%ioff j = surf_usm_v(l)%j(m) !+ surf_usm_v(l)%joff k = surf_usm_v(l)%k(m) !+ surf_usm_v(l)%koff local_pf(i,j,k) = surf_usm_v(l)%waste_heat(m) ENDDO ENDDO ENDIF ! !< NOTE im_t_indoor_roof and im_t_indoor_wall_win not work yet ! CASE ( 'im_t_indoor_roof' ) ! IF ( av == 0 ) THEN ! DO m = 1, surf_usm_h%ns ! i = surf_usm_h%i(m) !+ surf_usm_h%ioff ! j = surf_usm_h%j(m) !+ surf_usm_h%joff ! k = surf_usm_h%k(m) !+ surf_usm_h%koff ! local_pf(i,j,k) = surf_usm_h%t_indoor(m) ! ENDDO ! ENDIF ! ! CASE ( 'im_t_indoor_wall_win' ) ! IF ( av == 0 ) THEN ! DO l = 0, 3 ! DO m = 1, surf_usm_v(l)%ns ! i = surf_usm_v(l)%i(m) !+ surf_usm_v(l)%ioff ! j = surf_usm_v(l)%j(m) !+ surf_usm_v(l)%joff ! k = surf_usm_v(l)%k(m) !+ surf_usm_v(l)%koff ! local_pf(i,j,k) = surf_usm_v(l)%t_indoor(m) ! ENDDO ! ENDDO ! ENDIF CASE DEFAULT found = .FALSE. END SELECT END SUBROUTINE im_data_output_3d !------------------------------------------------------------------------------! ! Description: ! ------------ !> Parin for &indoor_parameters for indoor model !------------------------------------------------------------------------------! SUBROUTINE im_parin USE control_parameters, & ONLY: indoor_model CHARACTER (LEN=80) :: line !< string containing current line of file PARIN NAMELIST /indoor_parameters/ initial_indoor_temperature ! !-- Try to find indoor model package REWIND ( 11 ) line = ' ' DO WHILE ( INDEX( line, '&indoor_parameters' ) == 0 ) READ ( 11, '(A)', END=10 ) line ENDDO BACKSPACE ( 11 ) ! !-- Read user-defined namelist READ ( 11, indoor_parameters ) ! !-- Set flag that indicates that the indoor model is switched on indoor_model = .TRUE. ! !-- Activate spinup (maybe later ! IF ( spinup_time > 0.0_wp ) THEN ! coupling_start_time = spinup_time ! end_time = end_time + spinup_time ! IF ( spinup_pt_mean == 9999999.9_wp ) THEN ! spinup_pt_mean = pt_surface ! ENDIF ! spinup = .TRUE. ! ENDIF 10 CONTINUE END SUBROUTINE im_parin END MODULE indoor_model_mod" ]

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[ "", null, "# Machine Learning and Data Science: Multinomial (Multiclass) Logistic Regression\n\nWritten on August 18, 2017 by Dr Donald Kinghorn\nShare:\n\n## Logistic Regression: Multi-Class (Multinomial) -- Full MNIST digits classification example¶\n\nThis post will be an implementation and example of what is commonly called \"Multinomial Logistic Regression\". The particular method I will look at is \"one-vs-all\" or \"one-vs-rest\".\n\nWhat's in a name? \"A rose by any other name would smell as sweet\". In my opinion calling this \"Multinomial Logistic Regression\" stinks! A multinomial is a specific mathematical thing and I already used \"multinomial term expansion of feature sets\". I really feel that a more descriptive name would be \"Multi-Class\". ... but Multinomial Logistic Regression is the name that is commonly used.\n\nThis post is heavy on Python code and job runs. It includes the implementation code from the previous post with additional code to generalize that to multi-class. The usage example will be image classification of hand written digits (0-9) using the MNIST dataset.\n\nI've done four earlier posts on Logistic Regression that give a pretty thorough explanation of Logistic Regress and cover theory and insight for what I'm looking at in this post, Logistic Regression Theory and Logistic and Linear Regression Regularization, Logistic Regression Implementation, Logistic Regression: Examples 1 -- 2D data fit with multinomial model and 0 1 digits classification on MNIST dataset.\n\nThis will be a \"calculator\" style implementation using Python in this Jupyter notebook. Everything needed to \"tinker\" with the method is contained in this notebook except the MNIST dataset. I pulled the MNIST training set from Kaggle. For information on the dataset itself see Yann Lecun's site http://yann.lecun.com/exdb/mnist/index.html. To use this notebook for your own experimentation you would need to download that dataset.\n\nThis posts along with all of the others in this series were converted to html from Jupyter notebooks. The notebooks are available at https://github.com/dbkinghorn/blog-jupyter-notebooks\n\n## Understanding Multi-Class (Multinomial) Logistic Regression¶\n\nYou can think of logistic regression as if the logistic (sigmoid) function is a single \"neuron\" that returns the probability that some input sample is the \"thing\" that the neuron was trained to recognize. It is a binary classifier. It just gives the probability that the input it is looking at is the ONE thing that it was trained to recognize. To generalize this to several \"things\" (classes) we can create a collection of these binary \"neurons\" with one for each class of the things the we want to distinguish. You could think of that as a single layer network of these sigmoid neurons.\n\nTo classify the 10 digits 0-9 there would be 10 of these sigmoid neurons in a single layer network. Like this,", null, "The $f_i$ are the features i.e. pixels in an image, $h_i$ are the 10 individual digit models and MAX(P) is the result with the highest probability.\n\nThat is basically what we are going to do. In general the steps are,\n\n• Create a 0,1 vector $y_k$ for each class $k$. Each $y_k$ will have a 1 matching the position of all samples in the training set that match that class and 0 otherwise. (I will put them in a matrix $Y$ where the $k^{th}$ column of $Y$ is $y_k$)\n• Do an optimization loop over all $k$ classes finding an optimal parameter vector $a_k$ to define $k$ models $h_k$\n• To test i.e. classify, an input evaluate it with each $h_k$ to get a probability that it is in class $k$.\n• Pick the class with the highest probability as the \"answer\".\n\nSpecifically for the MNIST digits dataset being used;\n\n• There will be $k=10$ classes with labels {0,1,2,3,4,5,6,7,8,9}.\n• For a set with $m$ samples $Y_{set}$ will be an $(m \\times 10)$ matrix of 0's and 1's corresponding to samples in each class. For example the first column of $Y$ will have a 1 in each row that is a sample image of a \"0\". ... The tenth column of $Y$ will have a 1 in each row that is a sample of a \"9\".\n• The full data set has 42000 samples which will be divided into\n• 29400 training-set samples,\n• 6300 validation-set samples,\n• 6300 test-set samples.\n• The digit images in the MNIST dataset have 28 x 28 pixels. These pixels together with the bias term is the number of features. That means that each sample feature vector will have 784 + 1 = 785 features that we will need to find 785 parameters for.\n• The optimization loop will be over the 10 classes and will produce a matrix $A$ of optimized parameters by minimizing a cost function for each for the 10 classes. Each column of the 10 columns $A$ will be a model parameter vector corresponding to each of the 10 classes (0-9).\n• To test or use the resulting model the input sample will be evaluated for each of the 10 \"class models\" and sorted by highest probability. The result with the highest probability is the prediction from the model.\n\nSimple! Lets do it.\n\n## Core Logistic Regression Functions (Python Code)¶\n\nThis section is the base code for, logistic regression with regularization, that was worked up in the previous posts. You can skip over this section if you have seen the code in the last post and just refer back to it if you need to see how some function was defined.\n\nIn :\nimport pandas as pd # data handeling\nimport numpy as np # numerical computing\nfrom scipy.optimize import minimize # optimization code\nimport matplotlib.pyplot as plt # plotting\nimport seaborn as sns\n%matplotlib inline\nsns.set()\nimport itertools # combinatorics functions for multinomial code\n\nIn :\n#\n# Main Logistic Regression Equations\n#\ndef g(z) : # sigmoid function\nreturn 1.0/(1.0 + np.exp(-z))\n\ndef h_logistic(X,a) : # Model function\nreturn g(np.dot(X,a))\n\ndef J(X,a,y) : # Cost Function\nm = y.size\nreturn -(np.sum(np.log(h_logistic(X,a))) + np.dot((y-1).T,(np.dot(X,a))))/m\n\ndef J_reg(X,a,y,reg_lambda) : # Cost Function with Regularization\nm = y.size\nreturn J(X,a,y) + reg_lambda/(2.0*m) * np.dot(a[1:],a[1:])\n\nm = y.size\nreturn (np.dot(X.T,(h_logistic(X,a) - y)))/m\n\nm = y.size\nreturn gradJ(X,a,y) + reg_lambda/(2.0*m) * np.concatenate((, a[1:])).T\n\nIn :\n#\n# Some model checking functions\n#\ndef to_0_1(h_prob) : # convert probabilites to true (1) or false (0) at cut-off 0.5\nreturn np.where(h_prob >= 0.5, 1, 0)\n\ndef model_accuracy(h,y) : # Overall accuracy of model\nreturn np.sum(h==y)/y.size * 100\n\ndef model_accuracy_pos(h,y) : # Accuracy on positive cases\nreturn np.sum(y[h==1] == 1)/y[y==1].size * 100\n\ndef model_accuracy_neg(h,y) : # Accuracy on negative cases\nreturn np.sum(y[h==0] == 0)/y[y==0].size * 100\n\ndef false_pos(h,y) : # Number of false positives\nreturn np.sum((y==0) & (h==1))\n\ndef false_neg(h,y) : # Number of false negatives\nreturn np.sum((y==1) & (h==0))\n\ndef true_pos(h,y) : # Number of true positives\nreturn np.sum((y==1) & (h==1))\n\ndef true_neg(h,y) : # Number of true negatives\nreturn np.sum((y==0) & (h==0))\n\ndef model_precision(h,y) : # Precision = TP/(TP+FP)\nreturn true_pos(h,y)/(true_pos(h,y) + false_pos(h,y))\n\ndef model_recall(h,y) : # Recall = TP/(TP+FN)\nreturn true_pos(h,y)/(true_pos(h,y) + false_neg(h,y))\n\ndef print_model_quality(title, h, y) : # Print the results of the functions above\nprint( '\\n# \\n# {} \\n#'.format(title) )\nprint( 'Total number of data points = {}'.format(y.size))\nprint( 'Number of Positive values(1s) = {}'.format(y[y==1].size))\nprint( 'Number of Negative values(0s) = {}'.format(y[y==0].size))\nprint( '\\nNumber of True Positives = {}'.format(true_pos(h,y)) )\nprint( 'Number of False Positives = {}'.format(false_pos(h,y)) )\nprint( '\\nNumber of True Negatives = {}'.format(true_neg(h,y)) )\nprint( 'Number of False Negatives = {}'.format(false_neg(h,y)) )\nprint( '\\nModel Accuracy = {:.2f}%'.format( model_accuracy(h,y) ) )\nprint( 'Model Accuracy Positive Cases = {:.2f}%'.format( model_accuracy_pos(h,y) ) )\nprint( 'Model Accuracy Negative Cases = {:.2f}%'.format( model_accuracy_neg(h,y) ) )\nprint( '\\nModel Precision = {}'.format(model_precision(h,y)) )\nprint( '\\nModel Recall = {}'.format(model_recall(h,y)) )\n\nIn :\ndef multinomial_partitions(n, k):\n\"\"\"returns an array of length k sequences of integer partitions of n\"\"\"\nnparts = itertools.combinations(range(1, n+k), k-1)\ntmp = [(0,) + p + (n+k,) for p in nparts]\nsequences = np.diff(tmp) - 1\nreturn sequences[::-1] # reverse the order\n\ndef make_multinomial_features(fvecs,order=[1,2]) :\n'''Make multinomial feature matrix\nfvecs is a matrix of feature vectors (columns)\n\"order\" is a set of multinomial degrees to create\ndefault is [1,2] meaning for example: given f1, f2 in fvecs\nreturn a matrix made up of a [1's column, f1,f2,f1**2,f1*f2,f2**2] '''\nXtmp = np.ones_like(fvecs[:,0])\nfor ord in order :\nif ord==1 :\nfstmp = fvecs\nelse :\npwrs = multinomial_partitions(ord,fvecs.shape)\nfstmp = np.column_stack( ( np.prod(fvecs**pwrs[i,:], axis=1) for i in range(pwrs.shape) ))\n\nXtmp = np.column_stack((Xtmp,fstmp))\nreturn Xtmp\n\ndef mean_normalize(X):\n'''apply mean normalization to each column of the matrix X'''\nX_mean=X.mean(axis=0)\nX_std=X.std(axis=0)\nreturn (X-X_mean)/X_std\n\ndef apply_normalizer(X,X_mean,X_std) :\nreturn (X-X_mean)/X_std\n\n\n## Data setup for the 10 digit classes¶\n\nThe data is the same that was used in the last post but this time I will use all of the 0-9 images. There are 42000 total. Each image has 784 pixels and the first column is the label for what the image is. Let's read that in and look at the first 10 entries, then put that into a matrix called data_full_matrix.\n\nIn :\ndata_full = pd.read_csv(\"./data/kg-mnist/train.csv\")\n\nOut:\nlabel pixel0 pixel1 pixel2 pixel3 pixel4 pixel5 pixel6 pixel7 pixel8 ... pixel774 pixel775 pixel776 pixel777 pixel778 pixel779 pixel780 pixel781 pixel782 pixel783\n0 1 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0\n1 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0\n2 1 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0\n3 4 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0\n4 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0\n5 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0\n6 7 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0\n7 3 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0\n8 5 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0\n9 3 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0\n\n10 rows × 785 columns\n\nIn :\ndata_full_matrix=data_full.as_matrix()\nprint(data_full_matrix.shape)\n\n(42000, 785)\n\n\nYou can show any of the images in that matrix with the following snipit of code. This would show the image in the 4th row (index 3) which is a hand written 4.\n\nIn :\nplt.figure(figsize=(1,1))\nplt.imshow(data_full_matrix[3,1:].reshape((28,28)) )\n\nOut:", null, "### Create matrix $Y$¶\n\nThere is a column in $Y$ for each of the digits 0-9. I print out the first 10 rows so you can see how it is laid out. The first column has a 1 at row 2,5 and 6 (1,4,5 is you count from 0), that means that those rows correspond to the number 0. There are 42000 rows in $Y$.\n\nIn :\nY = np.zeros((data_full_matrix.shape,10))\nfor i in range(10) :\nY[:,i] = np.where(data_full_matrix[:,0]==i, 1,0)\n\nIn :\nY[0:10,:]\n\nOut:\narray([[ 0., 1., 0., 0., 0., 0., 0., 0., 0., 0.],\n[ 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.],\n[ 0., 1., 0., 0., 0., 0., 0., 0., 0., 0.],\n[ 0., 0., 0., 0., 1., 0., 0., 0., 0., 0.],\n[ 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.],\n[ 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.],\n[ 0., 0., 0., 0., 0., 0., 0., 1., 0., 0.],\n[ 0., 0., 0., 1., 0., 0., 0., 0., 0., 0.],\n[ 0., 0., 0., 0., 0., 1., 0., 0., 0., 0.],\n[ 0., 0., 0., 1., 0., 0., 0., 0., 0., 0.]])\n\n### Separate out the label column and remove columns that are all 0's¶\n\nWe got rid of 76 feature columns that were all 0's\n\nIn :\ny_labels, data_09 = data_full_matrix[:,0], data_full_matrix[:,1:]\nprint(data_09.shape)\ndata_09 = data_09[:,data_09.sum(axis=0)!=0]\nprint(data_09.shape)\n\n(42000, 784)\n(42000, 708)\n\n\n### Divide the dataset into sets for Training, Validation and Testing¶\n\nThere will be 29400 images in the Training set and 6300 images in each of the Validation and Test sets. The matrix $Y$ is divided up the same way.\n\nIn :\ndata_train_09,Y_train_09 = data_09[0:29400,:], Y[0:29400,:]\ndata_val_09, Y_val_09 = data_09[29400:35700,:], Y[29400:35700,:]\ndata_test_09, Y_test_09 = data_09[35700:,:], Y[35700:,:]\n\nIn :\ny_labels_train = y_labels[0:29400]\ny_labels_val = y_labels[29400:35700]\ny_labels_test = y_labels[35700:]\n\nIn :\nprint(data_train_09.shape,Y_train_09.shape)\nprint(data_val_09.shape, Y_val_09.shape)\nprint(data_test_09.shape, Y_test_09.shape)\n\n(29400, 708) (29400, 10)\n(6300, 708) (6300, 10)\n(6300, 708) (6300, 10)\n\n\n### Mean normalize the data sets¶\n\nEach of the data sets are normalized using the mean and standard deviation from the whole 42000 element data set. The make_multinomial_features functions is used here simply to add the column of 1's to the data for the bias term.\n\nIn :\nX_mean = data_09.mean(axis=0)\nX_std = data_09.std(axis=0)\nX_std[X_std==0]=1.0 # if there are any 0 values in X_std set them to 1\n\norder = \n\nX_train = make_multinomial_features(data_train_09, order=order)\nX_train[:,1:] = apply_normalizer(X_train[:,1:],X_mean,X_std)\nY_train = Y_train_09\n\nX_val = make_multinomial_features(data_val_09, order=order)\nX_val[:,1:] = apply_normalizer(X_val[:,1:],X_mean,X_std)\nY_val = Y_val_09\n\nX_test = make_multinomial_features(data_test_09, order=order)\nX_test[:,1:] = apply_normalizer(X_test[:,1:],X_mean,X_std)\nY_test = Y_test_09\n\n\n### Find optimal parameters for the 10 models¶\n\nThis is the main training loop. All 10 models are optimized using the columns of $Y$ and the training data set. Each model is fit to it's number (0-9) by evaluation it's cost function against all of the other numbers \"the rest\".\n\nYou can see that some of the models required many more iterations before convergence. There was also some numerical overflow present. I'm not too concerned about this since it is an artifact of the optimization run. The models converged OK and gave reasonably good set of parameters for each of the 10 models. It is possible to work on each model separately to try to get better fits and the regularization term could be adjusted per model. I did play with the optimization somewhat but wont worry about it too much since in teh next post I'll be doing an implementation of \"Stochastic Gradient Descent\" and will likely use this data again as an example.\n\nIn :\nreg =300.0 # Regularization term\nnp.random.seed(42)\naguess = np.random.randn(X_train.shape) # A random guess for the parameters\nA_opt = np.zeros((X_train.shape,10)) # The matrix of optimized parameters\nRes=[] # List to hold the full optimizitaion output of each model\n\nfor i in range(10):\nprint('\\nFitting {} against the rest\\n'.format(i))\ndef opt_J_reg(a) :\nreturn J(X_train,a,Y_train[:,i])\n\nres = minimize(opt_J_reg, aguess, method='CG', jac=opt_gradJ_reg, tol=1e-6, options={'disp': True})\nRes.append(res)\nA_opt[:,i] = res.x\n\nFitting 0 against the rest\n\nWarning: Desired error not necessarily achieved due to precision loss.\nCurrent function value: 0.028295\nIterations: 27\nFunction evaluations: 586\n\nFitting 1 against the rest\n\n\n/home/kinghorn/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:5: RuntimeWarning: overflow encountered in exp\n\"\"\"\n/home/kinghorn/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:12: RuntimeWarning: divide by zero encountered in log\nif sys.path == '':\n\nWarning: Desired error not necessarily achieved due to precision loss.\nCurrent function value: 0.031864\nIterations: 17\nFunction evaluations: 230\n\nFitting 2 against the rest\n\nWarning: Desired error not necessarily achieved due to precision loss.\nCurrent function value: 0.070383\nIterations: 45\nFunction evaluations: 1182\n\nFitting 3 against the rest\n\nWarning: Desired error not necessarily achieved due to precision loss.\nCurrent function value: 0.086118\nIterations: 32\nFunction evaluations: 330\n\nFitting 4 against the rest\n\nWarning: Desired error not necessarily achieved due to precision loss.\nCurrent function value: 0.052558\nIterations: 20\nFunction evaluations: 254\n\nFitting 5 against the rest\n\nWarning: Desired error not necessarily achieved due to precision loss.\nCurrent function value: 0.119611\nIterations: 14\nFunction evaluations: 127\n\nFitting 6 against the rest\n\nWarning: Desired error not necessarily achieved due to precision loss.\nCurrent function value: 0.045620\nIterations: 21\nFunction evaluations: 273\n\nFitting 7 against the rest\n\nWarning: Desired error not necessarily achieved due to precision loss.\nCurrent function value: 0.089702\nIterations: 13\nFunction evaluations: 188\n\nFitting 8 against the rest\n\nWarning: Desired error not necessarily achieved due to precision loss.\nCurrent function value: 0.168584\nIterations: 11\nFunction evaluations: 126\n\nFitting 9 against the rest\n\nWarning: Desired error not necessarily achieved due to precision loss.\nCurrent function value: 0.105018\nIterations: 57\nFunction evaluations: 1374\n\n\nYou can look at the fit quality of each model. The model for the digit 8 has the worst finial value for the cost function and it looks like it had many false negatives. I am using the Validation data set check the quality of fit.\n\nIn :\nnum=8\na_opt = A_opt[:,num]\nh_prob = h_logistic(X_val,a_opt)\nh_predict = to_0_1(h_prob)\nprint_model_quality('Validation-data fit', h_predict, Y_val[:,num])\n\n#\n# Validation-data fit\n#\nTotal number of data points = 6300\nNumber of Positive values(1s) = 611\nNumber of Negative values(0s) = 5689\n\nNumber of True Positives = 341\nNumber of False Positives = 69\n\nNumber of True Negatives = 5620\nNumber of False Negatives = 270\n\nModel Accuracy = 94.62%\nModel Accuracy Positive Cases = 55.81%\nModel Accuracy Negative Cases = 98.79%\n\nModel Precision = 0.8317073170731707\n\nModel Recall = 0.55810147299509\n\n\nThe fit for the \"0\" model has a low cost function and the quality of fit looks much better than that for \"8\".\n\nIn :\nnum=0\na_opt = A_opt[:,num]\nh_prob = h_logistic(X_val,a_opt)\nh_predict = to_0_1(h_prob)\nprint_model_quality('Validation-data fit', h_predict, Y_val[:,num])\n\n#\n# Validation-data fit\n#\nTotal number of data points = 6300\nNumber of Positive values(1s) = 618\nNumber of Negative values(0s) = 5682\n\nNumber of True Positives = 577\nNumber of False Positives = 44\n\nNumber of True Negatives = 5638\nNumber of False Negatives = 41\n\nModel Accuracy = 98.65%\nModel Accuracy Positive Cases = 93.37%\nModel Accuracy Negative Cases = 99.23%\n\nModel Precision = 0.92914653784219\n\nModel Recall = 0.9336569579288025\n\n\n### Use the model to make predictions for untested number images¶\n\nThe following function will return the probabilities predicted by each of the models for some given input image. The probabilities are sorted with the most likely being listed first.\n\nIn :\ndef predict(sample, sample_label):\nprint('\\nTest sample is : {}\\n'.format(sample_label))\nprobs = np.zeros((10,2))\nfor num in range(10):\na_opt = A_opt[:,num]\nprobs[num,0] = num\nprobs[num,1] = h_logistic(sample,a_opt)\n\nprobs = probs[probs[:,1].argsort()[::-1]] # put the best guess at the top\nprint('Model prediction probabilites\\n')\nfor i in range(10):\nprint( \"{} with probability = {:.3f}\".format(int(probs[i,0]), probs[i,1]) )\n\n\n\nFollowing are a few random images picked from the test set.\n\nThe first image is of an \"8\". You can see that the model did not give a very high probability for \"8\" but it was higher than any of the other probabilities so it did give the correct answer!\n\nIn :\nsamp = 23\nsamp_label = y_labels_test[samp]\nsample = X_test[samp,:]\npredict(sample, samp_label)\n\nTest sample is : 8\n\nModel prediction probabilites\n\n8 with probability = 0.253\n3 with probability = 0.097\n9 with probability = 0.081\n5 with probability = 0.000\n2 with probability = 0.000\n7 with probability = 0.000\n4 with probability = 0.000\n1 with probability = 0.000\n6 with probability = 0.000\n0 with probability = 0.000\n\nIn :\nsamp = 147\nsamp_label = y_labels_test[samp]\nsample = X_test[samp,:]\npredict(sample,samp_label)\n\nTest sample is : 2\n\nModel prediction probabilites\n\n2 with probability = 0.994\n6 with probability = 0.016\n3 with probability = 0.004\n1 with probability = 0.000\n9 with probability = 0.000\n0 with probability = 0.000\n5 with probability = 0.000\n8 with probability = 0.000\n4 with probability = 0.000\n7 with probability = 0.000\n\nIn :\nsamp = 6200\nsamp_label = y_labels_test[samp]\nsample = X_test[samp,:]\npredict(sample,samp_label)\n\nTest sample is : 4\n\nModel prediction probabilites\n\n4 with probability = 0.977\n2 with probability = 0.034\n7 with probability = 0.022\n9 with probability = 0.005\n3 with probability = 0.002\n6 with probability = 0.002\n0 with probability = 0.002\n8 with probability = 0.001\n5 with probability = 0.001\n1 with probability = 0.001\n\n\n### Checking how well the model did for each of the datasets¶\n\nThe next function will give a printout of the percentage of correct prediction in a dataset. We first look at the training and validation sets.\n\nIn :\ndef print_num_correct(datasets):\nfor dataset in datasets :\nset_name, yl, Xd = dataset\nyls = yl.size\nprobs = np.zeros((10,2))\ncount = 0\nfor samp in range(yls):\nfor num in range(10):\na_opt = A_opt[:,num]\nprobs[num,0] = num\nprobs[num,1] = h_logistic(Xd[samp,:],a_opt)\n\nprobs = probs[probs[:,1].argsort()[::-1]]\nif probs[0,0] == yl[samp] :\ncount +=1\nprint('\\n{}'.format(set_name))\nprint(\"{} correct out of {} : {}% correct\".format(count, yls, count/yls * 100))\n\n\nIn :\ndatasets = [('Training Set:', y_labels_train, X_train),('Validation Set:',y_labels_val, X_val)]\nprint_num_correct(datasets)\n\nTraining Set:\n26481 correct out of 29400 : 90.07142857142857% correct\n\nValidation Set:\n5590 correct out of 6300 : 88.73015873015872% correct\n\n\n... and now the test set.\n\nIn :\nprint_num_correct([('Test Set:', y_labels_test, X_test)])\n\nTest Set:\n5597 correct out of 6300 : 88.84126984126985% correct\n\n\n### Conclusion¶\n\nThat's not too bad for a simple method like Logistic Regression. It was fairly easy to implement and extend to the multi-class case. The results showing the number correct is fairly consistent across the different datasets and it looks like it has a prediction value of around 88%.\n\nIn the next post I'll do an implementation of Stochastic Gradient Descent (SGD) which is commonly used in machine learning especially for training neural networks. I may be able to add multinomial features to the digits model using SGD for the optimization since it should work well with very large numbers of features. We'll see :-)\n\n#### Happy computing! --dbk¶\n\nTags: Machine Learning, Data Science, Python, Jupyter notebook, Programming\n\nThis is really great-thank you! I was wondering if you could comment on the warning that scipy gives: \"Desired error not necessarily achieved due to precision loss\"\n\nPosted on 2018-08-08 15:43:40\n\nHi Thanks, I believe scipi numpy run in float32 by default i.e. single precision. The optimization method I used is CG, conjugate gradient (you could try others too) This generates \"sort of\" an approximation to the inverse Hession by doing rank-1 updates with information from the gradients. It uses that to generate search directions ... it has a lot of checks for numerical precision tolerances and will warn if anything \"may\" be dropping digits by becoming small (or too large). During an optimization run it's usually not too much of a concern as long as you are still getting descent directions on each step. [and the job isn't \"blowing up\"]\n\nThat whole field is fascinating ... Numerical Analysis. I studied just enough to be aware of problems without going to deep into \"pedantic\" analysis. It's good to take a course or at least have a good book on the subject.\n\nYou could probably get rid of the warnings by changing to float64 for the calcs i.e double precision. Double is usually the default for scientific work. For a problem like this though single should be OK since the results are approximate anyway (i.e. probabilistic) -- best wishes -Don\n\nPosted on 2018-08-09 15:18:26\n\nGreat--thanks for the suggestions. I will take a look!\n\nPosted on 2018-08-09 16:44:35" ]

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[ "Fig. This is generally considered a better result, as it clearly indicates the need for sensor replacement (whereas a metallic strain sensor may give the false impression of continued function following an over-stress event). Modern manufacturing techniques have made possible the construction of strain gauges made of silicon instead of metal. By measuring this change in resistance, we can infer the amount of pressure applied to the diaphragm. This deformation in the shape is both compressive or tensile is called strain, and it is measured by the strain gauge. When it is stretched, its resistance increases and Vice Versa. Thus strain gauges can be used to measure force and related parameters like displacement and stress. Attaching a strain gauge to a diaphragm results in a device that changes resistance with applied pressure. Measurement of other quantities: The principle of change in resistance due to applied force can also be calibrated to measure a number of other quantities like force, pressure, displacement, acceleration etc since all these parameters are related to each other. Strain gauges work on the principle of the conductor’s resistance which gives you the value of Gauge Factor by the formula: GF = [∆R / (RG * ε)] In practice, the change in the strain of an object is a very small quantity which can only be measured using a Wheatstone Bridge. Tie Bar Sensors Clamping Force The majority of strain gauges are foil types, available in a wide choice Measuring System of shapes and … Measure a strain down to 10microstrain over a 50mm gauge length. Now, we know that resistance of the conductor is the inverse function of the length. Although both measure the same physical quantity, they differ fundamentally in their mode of operation. An Introduction to Measurements using Strain Gages. Figure 13.5. Gauge factor (GF) 3. The following formula is valid: = ×ε Δ K R R As the diaphragm bows outward with applied fluid pressure, the strain gauge stretches to a greater length, causing its resistance to increase. This website uses cookies to improve your experience. However, silicon is not chemically compatible with many process fluids, and so pressure must be transferred to the silicon diaphragm/sensor via a non-reactive fill fluid (commonly a silicone-based or fluorocarbon-based liquid). Working principle of strain gauge : Gauge Factor: It is the ration of per unit change in resistance to per unit change in length. strain gauge transducers usually employ four strain gauge elements that are electrically connected to form a Wheatstone bridge circuit. Fig. As mentioned earlier, strain gauges work on the principle of the conductor’s resistance which gives you the value of Gauge Factor by the formula: GF = [ΔR / (RG * ε)] Now, in practice, the change in the strain of an object is a very small quantity which can only be measured using a Wheatstone Bridge. In this way, the silicon sensor experiences the same pressure that it would if it were directly exposed to the process fluid, without having to contact the process fluid. A short summary of this paper. The strain gauge is a classic example of a piezoresistive element, a typical strain gauge element shown here on the tip of my finger: In order to be practical, a strain gauge must be glued (bonded) on to a larger specimen capable of withstanding an applied force (stress): As the test specimen is stretched or compressed by the application of force, the conductors of the strain gauge are similarly deformed. Principle of Working of Strain Gauges. Related posts: Strain gauge A strain gauge is an electrical transducer which is used for measuring mechanical surface strain. Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Two or one strain gauge can work for a half bridge. The strain gauge can be attached to a diaphragm that recognises a change in resistance when the sensor element is deformed. Accept Read More, Differential Pressure Transmitter Working Principle, Pressure Gauges with Capsule Sensor Principle, Programmable Logic Controllers Multiple Choice Questions, Three-valve Manifold on Remote Seal DP Transmitter, Pressure Temperature Compensation Flow Measurement, Communicating Delta PLC Software to Simulator. Types of Strain Gauges 3.1. or. Strain Gauges (source: omega.com) Contents hide 1. The strain gauges are used for two main purposes: Measurement of strain: Whenever any material is subjected to high loads, they come under strain, which can be measured easily with the strain gauges. 22 Full PDFs related to this paper. Mechanical type, 2. When force is applied to any metallic wire its length increases due to the strain. Piezoelectric load cells work on the same principle of deformation as the strain gauge load cells, but a voltage output is generated by the basic piezoelectric material – proportional to the deformation of load cell. As mentioned earlier, strain gauges work on the principle of the conductor’s resistance which gives you the value of Gauge Factor by the formula: GF = … Book: Measurement and Instrumentation Principles by Alan S. Morris, Pro Engineer Surface Modeling Tutorial -The Simplest Command to Start ProE Surface Modeling. Useful for dynamic/frequent measurements of force. Electrical resistance of any conductor is proportional to the ratio of length over cross-sectional area (R ∝ { l / A } ), which means that tensile deformation (stretching) will increase electrical resistance by simultaneously increasing length and decreasing cross-sectional area while compressive deformation (squishing) will decrease electrical resistance by simultaneously decreasing length and increasing cross-sectional area. A strain gauge is a long length of conductor arranged in a zigzag pattern on a membrane.. Strain Gauge Working Principle. A strain gage consists of a small diameter wire (actually an etched metal foil) that is attached to a backing material (usually made of plastic) as Within its elastic limits, many metals exhibit good spring characteristics. This video explains in just 2 minutes the working principle of an electrical strain gauge. When force is applied to any metallic wire its length increases due to the strain. working of strain gauge. Another simplified illustration shows how this works: The isolating diaphragm is designed to be much more flexible (less rigid) than the silicon diaphragm, because its purpose is to seamlessly transfer fluid pressure from the process fluid to the fill fluid, not to act as a spring element. It converts mechanical strain to electrical signal. There is a linear relationship between the strain of the strain gauge and the change in its resistance. The Figure 1 shows a typical strain gauge diagram. 14/02/2010 how sensors work - strain gauge. What is Strain? The main principle of the bridge’s balance is that in strain gauge in the judged arms they must change the values of their resistance in opposite directions. The electrical resistance strain gages very closely meet the requirements stated above. • A thin piece of conductive material is … Strain Gage: Materials material gage factor, G TCR (10-5) Ni80 Cr20 2.1 - 2.6 10 Pt92 W8 3.6 – 4.4 24 Silicon (n type) -100 to -140 70 to 700Germanium (p type) 102TCR = temperature coefficient of resistivity (ºC-1) • Note: • G for semiconductor materials ~ 50-70 x that of metals When force is applied to any metallic wire its length increases due to the strain. A metal isolating diaphragm transfers process fluid pressure to the fill fluid, which in turn transfers pressure to the silicon wafer. Strain Gauge Sensors or Piezoresistive sensors. It is often easy to measure the parameters like length, displacement, weight etc that can be felt easily by some senses. Frequently Asked Questions. B) Johansson Extensometer Torsion tape stretched between knife edges. Here's What You Need to Know, 4 Most Common HVAC Issues & How to Fix Them, Commercial Applications & Electrical Projects, Fluid Mechanics & How it Relates to Mechanical Engineering, Hobbyist & DIY Electronic Devices & Circuits, Naval Architecture & Ship Design for Marine Engineers, Book: Measurement and Instrumentation Principles by Alan S. Morris. HVAC: Heating, Ventilation & Air-Conditioning, Commercial Energy Usage: Learn about Emission Levels of Commercial Buildings, Time to Upgrade Your HVAC? The more is the applied force, more is the strain and more is the increase in length of the wire. A strain gauge based displacement transducer for measurement of the displacement in the range of 0 to 10 mm is reported. As the length of the conductor increases its resistance decreases. This is a common source of error in metallic piezoresistive pressure instruments: if overpressured, they tend to lose accuracy due to damage of the spring and strain gauge elements. Strain Gauge Sensors or Piezoresistive sensors. Which measuring principle is being used? In the case of a bridge system, the strain gauges can work independently in the number of four, two or one. Gauge factor = GF = (∆R/R)/ (∆L/L). Piezoresistive means “pressure-sensitive resistance,” or a resistance that changes value with applied pressure. 13. The strain gage is used universally by stress analysts in the experimental deter-mination of stresses. A strain gauge (also spelled strain gage) is a device used to measure strain on an object. This also affects the application areas of both methods. There are some materials whose resistance changes when strain is applied to them or when they are stretched and this change in resistance can be measured easily. Poisson’s Ratio (ν) 2.2. The change in resistance is converted to an output signal There are three separate effects that contribute to the change in resistance of a conductor. A strain gauge, in mechanical term, is a device for measuring mechanical strain. The input and output relationship of the strain gauges can be expressed by the term gauge factor or gauge gradient, which is defined as the change in resistance R for the given value of applied strain ε. They can be used for measurement of force, strain, stress, pressure, displacement, acceleration etc. In principle, there are two sensor types: Piezoelectric sensors and strain gauges. The more is the applied force, more is the strain and more is the increase in length of the wire. Strain Gauge Working Principle.jpg 603×243 19.8 KB. Invented by Edward E. Simmons and Arthur C. Ruge in 1938, the most common type of strain gauge consists of an insulating flexible backing which supports a metallic foil pattern. Piezoresistive means “pressure-sensitive resistance,” or a resistance that changes value with applied pressure. Stress, Strain, and Strain Gages, Page 2 Strain gage The principle discussed above, namely that a wire’s resistance increases with strain, is key to understanding how a strain gage works. PRINCIPLES OF SENSORS & TRANSDUCERS • Strain gauge • The strain gauge can be considered as an electromechanical transducer used for measuring strain in a structure. gauge factor G f = (∆R/R)/ ( ∆l/l) where, R = nominal gauge resistance, ∆R = change in resistance, l = length of the specimen in an unstressed condition, ∆l = change in specimen length. This paper. Download with Google Download with Facebook. These are: 1. For such cases special devices called strain gauges are very useful. If L1 is the initial length of the wire and L2 is the final length after application of the force, the strain is … Save my name, email, and website in this browser for the next time I comment. When compression applied area thickness and resistance decreases. We'll assume you're ok with this, but you can opt-out if you wish. The basic principle of the piezoresistive pressure sensor is to use a strain gauge made from a conductive material that changes its electrical resistance when it is stretched. Strain gauge: Principle of Working, Materials Used, Applications Strain gauges are devices whose resistance changes under the application of force or strain. Create a free account to download. Types of Strain Gages Types of strain gages are classified into foil strain gages, wire strain gages, and semiconductor strain gages, etc. 2.1. Measuring strain gauge circuits In order to measure strain with a bonded resistance strain gauge, it must be connected to an electric circuit that is capable of measuring the minute changes in resistance corresponding to strain. Commonly used phenomena include changes in capacitance, or changes in ohmic resistance of a strain gauge … A load cell is made by bonding strain gauges to a spring material. The discovery of the principle upon which electrical resistance strain gage is based was made in 1856 by Lord Kelvin, who loaded copper To efficiently detect the strain, strain gauges are bonded to the position on the spring material where the strain will be the largest. Gauge factor is given as the ratio of change in electrical resistance R to the mechanical strain ε Mechanical strain is defined as the ratio of change in dimension to the original dimension in which the strain is applied, in this case length. Since strain always accompanies vibration, the strain gage or the principle by which it works is broadly applicable in the field of shock and vibra-tion measurement.Here it serves to determine not only the magnitude of the strains Credits : Tony R. Kuphaldt – Creative Commons Attribution 4.0 License. The gauge factor of strain gauge is defined as the unit change in resistance per unit change in length. Half end is twisted in one direction, while other half in other direction. The strain gauges can sense the displacements as small as 5 µm. Download Free PDF. The strain gage was invented by Ed Simmons at Caltech in 1936. Thus, the strain gauge works to convert an applied pressure into a measurable voltage signal which may be amplified and converted into a 4-20 mA loop current signal (or into a digital “fieldbus” signal)." ]

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[ "# Modified Warshall's algorithm to find shortest path matrix\n\nIn this example, we will learn about weighted graph with positive and negative weights, shortest path and Warshall’s algorithm.\nSubmitted by Manu Jemini, on January 09, 2018", null, "Image source: http://mathworld.wolfram.com/images/eps-gif/GraphNodesEdges_1000.gif\n\nA weighted graph with positive and negative weights can be understood as a graph with edges having cost. Edge is the line connecting two nodes or a pair of nodes. The cost can be positive or negative.", null, "Image source: https://i.stack.imgur.com/GlrNb.png\n\nThe shortest distance is the distance between two nodes. For Example, to reach a city from another, can have multiple paths with the different number of costs. A path with the minimum possible cost is the shortest distance.", null, "Image source: https://i.stack.imgur.com/tyTz7.png\n\nWarshall’s algorithm is an algorithm which is used to find the shortest path between the source and destination nodes. These types of problems generally solved with BST if the cost of every edge is 1. But here the edges can have different values, even negative values. Hence we need to use this algorithm.\n\nC program\n\n```#include <stdio.h>\n#define infinity 9999\n#define MAX 20\n\nint minimum(int a,int b)\n{\nif(a<=b)\nreturn a;\nelse\nreturn b;\n}/*End of minimum()*/\n\nint display(int matrix[MAX][MAX],int n )\n{\nint i,j;\nfor(i=0;i<n;i++)\n{\nfor(j=0;j<n;j++)\nprintf(\"%7d\",matrix[i][j]);\nprintf(\"\\n\");\n}\n}/*End of display()*/\n\nmain()\n{\nint i,j,k,n;\n\nprintf(\"Enter number of vertices : \");\nscanf(\"%d\",&n);\n\nprintf(\"Enter weighted matrix :\\n\");\nfor(i=0;i<n;i++)\nfor(j=0;j<n;j++)\n\nprintf(\"Weighted matrix is :\\n\");\n\n/*Replace all zero entries of adjacency matrix with infinity*/\nfor(i=0;i<n;i++)\nfor(j=0;j<n;j++)\npath[i][j]=infinity;\nelse\n\nfor(k=0;k<n;k++)\n{\nprintf(\"Q%d is :\\n\",k);\ndisplay(path,n);\nfor(i=0;i<n;i++)\nfor(j=0;j<n;j++)\npath[i][j]=minimum( path[i][j] , path[i][k]+path[k][j] );\n}\nprintf(\"Shortest path matrix Q%d is :\\n\",k);\ndisplay(path,n);\n}/*End of main() */\n```\n\nOutput", null, "" ]

[ null, "https://www.includehelp.com/ds/Images/graph-nodes-edges.jpg", null, "https://www.includehelp.com/ds/Images/weighted-graph.jpg", null, "https://www.includehelp.com/ds/Images/shortest-distance.jpg", null, "https://www.includehelp.com/ds/Images/modified-warshall-algorithm.jpg", null ]

[ "# #265. 【NOIP2016】愤怒的小鸟\n\nKiana 最近沉迷于一款神奇的游戏无法自拔。\n\n### 样例一\n\n#### input\n\n2\n2 0\n1.00 3.00\n3.00 3.00\n5 2\n1.00 5.00\n2.00 8.00\n3.00 9.00\n4.00 8.00\n5.00 5.00\n\n\n\n#### output\n\n1\n1\n\n\n\n### 样例二\n\n#### input\n\n3\n2 0\n1.41 2.00\n1.73 3.00\n3 0\n1.11 1.41\n2.34 1.79\n2.98 1.49\n5 0\n2.72 2.72\n2.72 3.14\n3.14 2.72\n3.14 3.14\n5.00 5.00\n\n\n\n#### output\n\n2\n2\n3\n\n\n\n### 样例三\n\n#### input\n\n1\n10 0\n7.16 6.28\n2.02 0.38\n8.33 7.78\n7.68 2.09\n7.46 7.86\n5.77 7.44\n8.24 6.72\n4.42 5.11\n5.42 7.79\n8.15 4.99\n\n\n\n#### output\n\n6\n\n\n\n### 限制与约定\n\n1$\\leq 2$$=0$$\\leq 10$\n2$\\leq 30$\n3$\\leq 3$$\\leq 10 4\\leq 30 5\\leq 4$$\\leq 10$\n6$\\leq 30$\n7$\\leq 5$$\\leq 10 8\\leq 6 9\\leq 7 10\\leq 8 11\\leq 9$$\\leq 30$\n12$\\leq 10$\n13$\\leq 12$$=1 14=2 15\\leq 15$$=0$$\\leq 15 16=1 17=2 18\\leq 18$$=0$$\\leq 5$\n19$=1$\n20$=2$" ]

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[ "Solutions by everydaycalculation.com\n\n## Subtract 15/35 from 40/12\n\n1st number: 3 4/12, 2nd number: 15/35\n\n40/12 - 15/35 is 61/21.\n\n#### Steps for subtracting fractions\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 12 and 35 is 420\n\nNext, find the equivalent fraction of both fractional numbers with denominator 420\n2. For the 1st fraction, since 12 × 35 = 420,\n40/12 = 40 × 35/12 × 35 = 1400/420\n3. Likewise, for the 2nd fraction, since 35 × 12 = 420,\n15/35 = 15 × 12/35 × 12 = 180/420\n4. Subtract the two like fractions:\n1400/420 - 180/420 = 1400 - 180/420 = 1220/420\n5. After reducing the fraction, the answer is 61/21\n6. In mixed form: 219/21\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:" ]

[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]

[ "# More than one time dimension\n\nWe know that space-time dimensions are 3+1 macroscopically, but what if 2+2? Obviously it is tough to imagine two time dimensions, but mathematically we can always imagine as either having two parameters $t_1$ and $t_2$ or else in Lorentz matrix $$\\eta_{00} = \\eta_{11} = -1$$ and, $$\\eta_{22} = \\eta_{33} = 1.$$\n\nIs there any physical reason for not taking this, like the norms become negative or something else?\n\n• Maybe you are interested the answer to this question too. – Dilaton Nov 7 '12 at 14:41\n• The \"norms\" could become negative anyway with usual Lorentzian signature. – c.p. Nov 8 '12 at 18:23\n• You may want to look at the following paper arxiv.org/abs/hep-ph/9910207 – Newman Nov 9 '12 at 2:20\n• About why there can not be more than one time dimension (I do not know how serious) space.mit.edu/home/tegmark/dimensions.pdf – user126422 Jun 22 '17 at 4:06\n• In 'simulation theory' (yes yes... Wait for it) there would of course be a dimension of time in which our dimension is simulated. What I'm keen to know is... Does the two-time theory make predictions which are testable? Is there a way in which we could possibly know if we're in a simulation? And would this extra time dimension behave as the imagined time dimension of our simulators world would? – Richard Jan 18 '19 at 1:20\n\nAs Cumrun Vafa explains in the video linked to below the picture of him in this article, F-theory works in a total of $10+2$ dimensions. The signature of the last two infinitesimal dimensions is ambiguous, so that they can indeed both be timelike. Since these are only infinitesimal dimensions, any causality issues etc are not a problem in this case.\n\nAnd as Cumrun Vafa nicely explains in his talk, F-theory gives quite a nice phenomenology with an astonishingly realistic CKM-Matrix, coupling constants, etc; so it is NOT true that theories that operate in more than one time dimension are completely off base, as some people claim. There is no reason to dogmatically dismiss every theory that has more than one time dimension.\n\nBTW, the talk is very accessible and enjoyable.\n\nThe late Irving Segal of MIT had a theory where the usual Lorentz group was replaced by SO(4,2) and there were indeed two time dimensions. His book Mathematical Cosmology and Extragalactic Astronomy, Academic Press, 1976, worked out the details. His theory has not been generally accepted, although there may be a few mathematical physicists at Montreal who are still interested in it. One of the consequences of this \"chronometry\" as he called it was that a part of the observed redshift was merely due to the discrepancies between the two times, and was not a Doppler effect, and thus the universe was not expanding. This theory is not currently accepted.\n\nHe was a brilliant mathematician. He understood Physics. He did not understand how to do Physics. He made some great contributions to Mathematical Physics in his theorems about operator algebras, and those theorems were motivated by Physics. In fact, he was only interested in maths that was motivated by Physics.\n\nThe hyperbolicity of the associated classical field equations is lost in $$d$$ space plus $$2$$ time dimensions. One cannot define a locally SO(d,2)-invariant distinction between past an future, no matter how curled up one of the time dimensions is.\n\nAs a result, there is no way to implement causality (i.e., no way to enforce the limiting information transmission to a finite speed), and the resulting models have very little to do with the real world.\n\n• Well, 2+2 dimensions is obviously not compatible with reality, since we observe 3 spacelike dimensions. The paper by Dvali (see the comment by Newman) discusses the possibility that a timelike dimension is curled up. It's far from obvious to me that our universe couldn't act like it had hyperbolicity if it was 3+2-dimensional, but with one of the timelike dimensions curled up. – user4552 Apr 15 '13 at 3:21\n• @BenCrowell: The concept of hyperbolicity is by definition tied to the Lorentzian metric! Curling up the second time direction does not change the signature of the metric. Thus even a curled up 2D time makes the distinction between spacelike and timelike impossible. – Arnold Neumaier Apr 22 '15 at 9:10\n• the point being: that near-hyperbolicity should be recoverable from a 3+2 signature if one of the time dimensions is curled. If not, why not? – lurscher Apr 29 '19 at 22:18\n• @lurscher: In 3+2 dimensions, one cannot define a locally SO(3,2)-invariant distinction between past an future, no matter how curled up the time. – Arnold Neumaier Apr 30 '19 at 9:09\n\nAlthough space-like and time-like coordinates enter in the same foot on a relativistic theory, there are physical differences between them, for example\n\n• One can travel back and forward through space-like coordinated, while it is impossible in time-like coordinates.\n\n• If there exist more than one time-like dimensions, that would mean our time is a linear combination of those. Since one does not see other time-like coordinated, it implies that the other (transverse) time-like coordinates are compact.\n\nThe presence of closed time-like coordinates spoils causality... reason why no more than a single time coordinate is considered in physics.\n\nCheers.\n\n• I do not agree. With a time cylinder model, with $T$ being the (constant) radius of the cylinder, the space-time interval would be (here $c = 1$) : $(\\Delta s)² = (\\Delta t)² + T (\\Delta \\theta)² - (\\Delta r)²$. If $T$ is of the same order as the Planck time, the apparent violation of causality (in r and t) is only appreciable for a length separation of the same order as the Planck length. For standard lengths, the term $T (\\Delta \\theta)²$ is neglectable. – Trimok Nov 14 '12 at 9:47\n\nThere could actually be two time dimensions, though you have to implement those properly. Should you simply replace the domain of time from $\\mathbb R$ to $\\mathbb R^2$ you would completely mess up our notion of cause and effect, therefore giving rise to many logical and temporal paradoxes hard to explain. I don't even know if such a theory would be logically coherent.\n\nHowever, these two time dimensions need not to belong to the same mathematical domain, nor to have an equivalent meaning. Maybe this link can be useful. See also this short review on wikipedia.\n\nThe causality issues relevant to including a second dimension of time owe their incoherence to the controlling assumption that space is contiguous and time is continuous. A formulation that understands two dimensions of space and two of time works well when the spatial dimensions are considered separated by time and the temporal dimensions are separated by space. The resulting construct gives rise to a phenomenology of a continuous form of time and a contiguous form of space. Causality is not violated as causality is a phenomenological issue." ]

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[ "# Synchronous Machine\n\n#### sitting_duck\n\nJoined Apr 18, 2010\n14\nHi,\n\nI'm a mechanical engineering student and I have an exam soon on Electrical Energy Systems. I am pretty rubbish at all things electrical and would really appriciate some help.\n\nI am trying to solve this question on synchronous machines.\n\nThe problem\n\nAttached jpg.\n\nMy attempt;\n\nI know that the rotor speed is found by;\n\nS=120(frequency)/(number of poles)\n\nIf I use the rated frequency I get 1000rpm for my answer.\n\nI am also aware that;\n\nslip speed=(sync speed)-(rotor speed)\n\nHow do I determine the synchronous speed from this?\n\n#### strantor\n\nJoined Oct 3, 2010\n5,633\nHi,\n\nI'm a mechanical engineering student and I have an exam soon on Electrical Energy Systems. I am pretty rubbish at all things electrical and would really appriciate some help.\n\nI am trying to solve this question on synchronous machines.\n\nThe problem\n\nAttached jpg.\n\nMy attempt;\n\nI know that the rotor speed is found by;\n\nS=120(frequency)/(number of poles)\n\nIf I use the rated frequency I get 1000rpm for my answer.\n\nI am also aware that;\n\nslip speed=(sync speed)-(rotor speed)\n\nHow do I determine the synchronous speed from this?\nwrong. that's how you figure out synchronous speed. that's your answer, synchronous speed = 1000rpm\n\n#### t_n_k\n\nJoined Mar 6, 2009\n5,455\nHi sitting_duck,\n\nKeep in mind the fact that synchronous machines don't have any slip. They are generally 'locked' to an infinite bus frequency corresponding to a synchronous mechanical speed. Slip is an important property in induction machines which generally run below (& occasionally above) synchronous speed.\n\n#### sitting_duck\n\nJoined Apr 18, 2010\n14\nThank's a million.\n\nHow do I determine Eao then?\n\nI know that\n\nPbus=(EV/Z)*sin(delta)\n\nQbus=(EV/Z)*(cos(delta))-(v^2/z)\n\nI presume that the last part of the question is important here, That an infinite bus is assumed at zero angle and rated voltage. But I'm not sure how to use this information.\n\n#### t_n_k\n\nJoined Mar 6, 2009\n5,455\nPresumably the equivalent circuit shown is a per-phase representation of the three phase generator.\n\nFor part (b)\n\nAt 220V line-line bus voltage the per-phase bus voltage would be 220/√3=127V\n\nThe total KVA rating is 4kVA so the per-phase value is 4/3=1.333kVA.\n\nIf the machine is supplying rated kVA then the line current would be 1.333kVA/127=10.5A\n\nThe per-phase open circuit voltage at rated load and unity power factor would be 127+10.5*j5 = 137.4V at 22.46° power angle. The generator open circuit line-line voltage would then be 137.4√3=238V.\n\nFor part (c)\n\nStart with the per-phase induced emf as 40*If=160V. Given this open circuit voltage and the power angle of 10° find the difference voltage between the open circuit value and the load infinite per-phase bus voltage of 127V at 0°. Divide the difference by j5Ω to find the 'injected' line current [including the phase angle]. Then deduce the per-phase power flow using the bus voltage and the 'injected' line current, having regard to the injected current phase angle [i.e. the power factor]." ]

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[ "# Why substituting \"for loop\" with direct assignment of Matrix worsen the performance time?\n\nWhy the execution time of “testS()” is faster than “testP()”, although the later has reduced a “for loop” (which is supposed to consume time) by a direct assignment of the Matrix, as shown below:\n\n``````using BenchmarkTools\n\nfunction testS(S,r)\n\nfor i in size(S,1)\n\nif r==1\n\nfor j in size(S,2)\n\nS[i,j] = S[i,j] +1;\n\nend\n\nelseif r==2\n\nfor j in size(S,2)\n\nS[i,j] = S[i,j] +2;\n\nend\n\nend\n\nend\n\nend\n\nfunction testP(S,r)\n\nfor i in size(S,1)\n\nif r==1\n\nS[i,:] = [S[i,1]+1 S[i,2]+1 S[i,3]+1 S[i,4]+1];\n\nelseif r==2\n\nS[i,:] = [S[i,1]+2 S[i,2]+2 S[i,3]+2 S[i,4]+2];\n\nend\n\nend\n\nend\n\ndt = 0.0001;\n\ntmin = 0;\n\ntmax = 1;\n\ntimeSim = tmin:dt:tmax;\n\nS = [0 0 0 0; 0 0 0 0; 0 0 0 0]\n\nr = 1;\n\n@btime begin for t in tmin:dt:tmax testS(S,r) end end\n# It gives 595.700 μs\n\n@btime begin for t in tmin:dt:tmax testP(S,r) end end\n# It gives 949.500 μs\n``````\n\n` [S[i,1]+1 S[i,2]+1 S[i,3]+1 S[i,4]+1];` allocates an array. If you instead wrote `S[i,:] .+= 1`, you would recover the performance.\n\nThe problem is that in my real code the assignment is not the same for the matrix elements i.e.,\n\n``````....\nS[i,:] = [S[i,1]+4 S[i,2]+3 S[i,3]+6 S[i,4]+2];\n...\n``````\n\nPerformance aside, your code is actually doing nothing other than `S[end] += r`. Also the matrix version is doing a different thing. Note: you might be used to Python’s `for i in range(n)` which will loop from `0` to `n-1`, in Julia, you must state explicitly `for i in 0:n-1`. In your code, `for i in size(S,1)` means `i = 3` and `for J in size(S,2)` means `j = 4`, so what you do is actually `S[3,4] += r` and the matrix form is different.\n\n1 Like\n\nYes, but lets say the following:\n\n``````using BenchmarkTools\n\nfunction testS(S,r)\nfor i in 1:size(S,1)\nif r==1\nfor j in 1:size(S,2)\nS[i,j] = S[i,j] +i;\nend\nelseif r==2\nfor j in 1:size(S,2)\nS[i,j] = S[i,j] +j;\nend\nend\nend\nend\n\nfunction testP(S,r)\nfor i in 1:size(S,1)\nif r==1\nS[i,:] = [S[i,1]+1 S[i,2]+4 S[i,3]+5 S[i,4]+6];\nelseif r==2\nS[i,:] = [S[i,1]+4 S[i,2]+2 S[i,3]+5 S[i,4]+2];\nend\nend\nend\n\ndt = 0.0001;\ntmin = 0;\ntmax = 1;\ntimeSim = tmin:dt:tmax;\nS = [0 0 0 0; 0 0 0 0; 0 0 0 0]\nr = 1;\n@btime begin for t in tmin:dt:tmax testS(S,r) end end # It gives 660.100 μs\n@btime begin for t in tmin:dt:tmax testP(S,r) end end # It gives 1.550 ms\n``````\n\n@Seif_Shebl @Oscar_Smith please have a look here!\n\nAgain, what your loop does is this:\n\n``````if r == 1\nS .+= 1:size(S,1)\nelseif r == 2\nS .+= (1:size(S,2))'\nend\n``````\n\nYou should take care of the dot in `.+=` as @Oscar_Smith said.\nAnd the matrix form does:\n\n``````if r == 1\nS .+= [4 3 6 2]\nelseif r == 2\nS .+= [4 2 5 2]\nend\n``````\n1 Like\n\nIn my real code, the situation is as below:\n\n``````using BenchmarkTools\n\nfunction testS(S,r)\nfor i in 1:size(S,1)\nif r==1\nfor j in 1:size(S,2)\nS[i,j] = S[i,j] +i;\nend\nelseif r==2\nfor j in 1:size(S,2)\nS[i,j] = S[i,j] +j;\nend\nend\nend\nend\n\nfunction testP(S,r)\nfor i in 1:size(S,1)\nif r==1\nS[i,:] = [S[i,1]+1 S[i,2]+4 S[i,3]+5 S[i,4]+6];\nelseif r==2\nS[i,:] = [S[i,1]+4 S[i,2]+2 S[i,3]+5 S[i,4]+2];\nend\nend\nend\n\ndt = 0.0001;\ntmin = 0;\ntmax = 1;\ntimeSim = tmin:dt:tmax;\nS = [0 0 0 0; 0 0 0 0; 0 0 0 0]\nr = 1;\n@btime begin for t in tmin:dt:tmax testS(S,r) end end # It gives 660.100 μs\n@btime begin for t in tmin:dt:tmax testP(S,r) end end # It gives 1.550 ms\n``````\n\nAs I said, you’re comparing two different things, here is the same code with the matrix form actually faster:\n\n``````function testS(S,r)\ns1 = [1, 4, 5, 6]\ns2 = [4, 2, 5, 2]\nfor i in 1:size(S,1)\nif r==1\nfor j in 1:size(S,2)\nS[i,j] = S[i,j] + s1[j];\nend\nelseif r==2\nfor j in 1:size(S,2)\nS[i,j] = S[i,j] + s2[j];\nend\nend\nend\nend\n\nfunction testP(S,r)\nif r==1\nS .+= [1 4 5 6]\nelseif r==2\nS .+= [4 2 5 2]\nend\nend\n\n@btime begin for t in tmin:dt:tmax testS(\\$S,\\$r) end end\n@btime begin for t in tmin:dt:tmax testP(\\$S,\\$r) end end\n1.320 ms (49495 allocations: 2.43 MiB)\n1.191 ms (39494 allocations: 1.52 MiB)\n``````\n\n@Seif_Shebl @Oscar_Smith thank you for your reply. Below, is my real code, in which I believe I am comparing between two similar things, but again the “for loop” gives faster performance:\n\n``````using BenchmarkTools\nusing Parameters, Base;\n\nBase.@kwdef mutable struct SSs\nPP::Vector{Float64} = [0,0,0]\nLL::Vector{Float64} = [0,0,0]\nend #mutable struct RLC\n\nfunction testS(S,r)\nfor i in 1:size(S,1)\nif r==1\nfor j in 1:3\nS[i].LL[j] = S[i].PP[j]*S[i].PP[j]+1;\nend\nelseif r==2\nfor j in 1:3\nS[i].LL[j] = S[i].PP[j]*S[i].PP[j]+2;\nend\nend\nend\nend\n\nfunction testP(S,r)\nfor i in 1:size(S,1)\nif r==1\nS[i].LL = S[i].PP.*S[i].PP.+1;\nelseif r==2\nS[i].LL = S[i].PP.*S[i].PP.+2;\nend\nend\nend\n\ndt = 0.0001;\ntmin = 0;\ntmax = 1;\ntimeSim = tmin:dt:tmax;\nS = SSs[];\npush!(S, SSs());\nS.PP = [2,3,1];\npush!(S, SSs());\nS.PP = [4,5,6];\nr = 1;\n@btime begin for t in tmin:dt:tmax testS(S,r) end end\n640.000 μs\n@btime begin for t in tmin:dt:tmax testP(S,r) end end\n1.166 ms\n``````\n\nYou’re defining a new matrix here (which allocates more memory) and assigning it to the mutable slot `LL`:\n\n``````S[i].LL = S[i].PP.*S[i].PP.+1\n``````\n\nYou need to instead use dot-assignment:\n\n``````S[i].LL .= S[i].PP.*S[i].PP.+1\n``````\n\nAn abbreviated way to insert all those dots is the `@.` broadcast macro:\n\n``````@. S[i].LL = S[i].PP^2 + 1\n``````\n\nFor such small arrays, you’re likely to get an additional speedup by substituting static/mutable arrays from StaticArrays.jl.\n\n2 Likes\n\nActually, I didtn get why in my way, I am defining a new matrix. In other words, what does the dot is doing here?\n\nCan you give me an example here?\n\nTake a look at this example, to understand what is going on:\n\n``````julia> Base.@kwdef mutable struct SSs\nPP::Vector{Float64} = [0,0,0]\nLL::Vector{Float64} = [0,0,0]\nend\nSSs\n\njulia> pp1 = [1.,1.,1.]; ll1 = [2.,2.,2.];\n\njulia> s1 = SSs(pp1,ll1)\nSSs([1.0, 1.0, 1.0], [2.0, 2.0, 2.0])\n\njulia> s1.PP = [0.,0.,0.] # this replaces the array PP\n3-element Vector{Float64}:\n0.0\n0.0\n0.0\n\njulia> pp1 # the original pp1 was not modified\n3-element Vector{Float64}:\n1.0\n1.0\n1.0\n\njulia> s1.LL .= [0.,0.,0.]; # note the . : this mutates s1.LL\n\njulia> ll1 # the original array was modified\n3-element Vector{Float64}:\n0.0\n0.0\n0.0\n\n``````\n\nThat said, only by writting `s1.LL = [0.,0.,0.]` (for example), you create the array `[0.,0.,0.]` which is a mutable object, and allocates memory. For this specific case you could use `s1.LL .= (0,0,0)`, which creates a tuple on the right side, but being immutable, tuples do not allocate memory (the compiler can optimize them out, it knows that their values or length won’t change).\n\nThe StaticArrays version of that would be:\n\n``````julia> using StaticArrays\n\njulia> Base.@kwdef mutable struct S\nPP::SVector{3,Float64} = @SVector [0,0,0]\nLL::SVector{3,Float64} = @SVector [0,0,0]\nend\nS\n\njulia> s = S()\nS([0.0, 0.0, 0.0], [0.0, 0.0, 0.0])\n\njulia> s.PP = @SVector [1.,1.,1.]; # one way\n\njulia> s.LL = SVector{3,Float64}(2,2,2); # another way\n\njulia> s\nS([1.0, 1.0, 1.0], [2.0, 2.0, 2.0])\n\n``````\n1 Like\n\nOr just `s1.LL .= 0`\n\n1 Like\n\nTo the OP: unless you are performing matrix algebra, you can normally assume that ordinary loops will give the best performance.\n\nAlso, remember that Julia Arrays are column major, so try to let your inner loop iterate over the first index, not the second, as shown in your MWE.\n\nOf course", null, "@lmiq Thank you very much for your explanation.\nIs the below correct for (3x3 Matrix{Float64, 2} )in the StaticArrays version?\n\n``````Base.@kwdef mutable struct AAs\nLss::SArray{Float64, 2} =@SArray [0 0 0; 0 0 0; 0 0 0]\nend\n``````\n\n`@SMatrix` I think\n\n1 Like\n\nSince you’re mutating your matrix, you’ll want to use `MMatrix` instead of `SMatrix`, with this type signature:\n\n``````MMatrix{3, 3, Float64, 9}\n``````\n\nWith a mutable matrix, you don’t need the container struct to be mutable itself.\n\n1 Like\n\n@DNF thank you for your explanation.\nWhat does “Julia Arrays are column major,” mean. Can you give me an example for \" inner loop iterate over the first index, not the second\"?\n\nArrays are stored in contiguous, linear blocks of memory:\n\n``````julia> v = [1:6...]\n6-element Vector{Int64}:\n1\n2\n3\n4\n5\n6\n``````\n\nColumn-major matrices are contiguous along the first dimension, while row-major matrices are contiguous along the second:\n\n``````julia> reshape(v, 2, 3) # column-major\n2×3 Matrix{Int64}:\n1 3 5\n2 4 6\n\njulia> reshape(v, 2, 3)' # transposed to row-major\n1 2\n3 4\n5 6\n``````\n\nThe CPU is faster when it can operate on contiguous blocks of memory, so it’s better for the inner loop (with the fastest-changing index) to correspond to those contiguous blocks. For column-major languages like Julia, that means `w[1,1], w[2,1], w[3,1]` rather than `w[1,1], w[1,2], w[1,3]`. For example,\n\n``````julia> function alongcolumns(w)\nfor j in axes(w, 2)\nfor i in axes(w, 1)\nw[i, j] += 1\nend\nend\nend\nalongcolumns (generic function with 1 method)\n\njulia> function alongrows(w)\nfor i in axes(w, 2)\nfor j in axes(w, 1)\nw[i, j] += 1\nend\nend\nend\nalongrows (generic function with 1 method)\n\njulia> w = rand(100, 100);\n\njulia> @btime alongcolumns(\\$w)\n710.000 ns (0 allocations: 0 bytes)\n\njulia> @btime alongrows(\\$w)\n2.889 μs (0 allocations: 0 bytes)\n``````\n\nThat’s a 4x difference, just by switching the loop order!\n\n3 Likes" ]

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[ "# How to use Excel DAYS Function\n\n## Introduction\n\nDAYS function in Excel is a type of DATE/TIME function. It gives value of the number of days falling between two dates. This function can be applied to find the number of the DAYS from the date present in the cell. This can also be used to extract and put the number of the DAYS into other functions, for example in DATE function.\n\n## Syntax\n\n=DAYS(end_date,start_date)\n\n## Arguments\n\n• End_Date- The end date to which one wants to calculate the number of days.\n• Start_Date- The start date from which one wants to calculate the number of days.\n\n## Keynotes\n\n• In General formatting of Excel, Dates are in form of serial numbers. Serial number 1 is for 1 JAN, 1900 and the number increases from thereon. For getting it in readable form use the formatting tab.\n• The argument used must represent a valid date in Excel format.\n• If the arguments used have number values which are out of range of valid excel date, #NUM error is displayed\n• If the arguments used have strings which cannot be converted into valid dates, #VALUE error is displayed.\n\n## Examples\n\nIn this example, dates are given in column A and B, number of days between these dates are calculated using the DAYS function in column C." ]

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[ "", null, "# Two functions $f : R \\to R, \\; g: R \\to R$ are defined as follows <br> $f(x) = \\begin{cases} 0 , & \\quad \\text{x } \\text{ is rational}\\\\ 1, & \\quad \\text{x} \\text{ is irrational} \\end{cases}$ <br> $g(x) = \\begin{cases} -1 , & \\quad \\text{x } \\text{ is rational}\\\\ 0, & \\quad \\text{x} \\text{ is irrational} \\end{cases}$ <br> Then $(fog) (\\pi) + (gof)(e)$ is equal to :\n( A ) $-1$\n( B ) $0$\n( C ) $2$\n( D ) $1$" ]

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[ "# Android屏幕重力感应\n\n2011年3月21日10:25:17 发表评论 15 views\n\n1. 内核重力感应器驱动部分，如 MMA7660\n\n``vi drivers/input/gsensor/mma7660.c``\n\n``````axis.y = mma7660_convert_to_int(buffer[MMA7660_REG_X_OUT]);\naxis.x = mma7660_convert_to_int(buffer[MMA7660_REG_Y_OUT]);\naxis.z = mma7660_convert_to_int(buffer[MMA7660_REG_Z_OUT]);``````\n\n``````#ifdef CONFIG_SWAP_XY\n{\ntypeof(x) __tmp;\n__tmp = x;\nx = y;\ny = __tmp;\n}\n#endif\n#ifdef CONFIG_REVERSE_X\nx = -x;\n#endif\n#ifdef CONFIG_REVERSE_Y\ny = -y;\n#endif\n/* X Y 感应方向有问题，这很容易看出来。 */\n#ifdef CONFIG_REVERSE_Z\nz = -z;\n/* 按正常情况下拿着设备，屏幕斜向上，如果不灵敏，把屏幕朝下试试，如果灵敏了，一般是 Z 反了。 */\n#endif``````\n\n2. Android部分，如果出现菜单显示正常，但是玩重力感应游戏时有问题，这部分就要修改(2.1)。\n\n``vim frameworks/base/core/java/android/view/WindowOrientationListener.java``\n``````public void onSensorChanged(SensorEvent event) {\nfloat[] values = event.values;\nfloat X = values[_DATA_X];\nfloat Y = values[_DATA_Y];\nfloat Z = values[_DATA_Z];\nfloat OneEightyOverPi = 57.29577957855f;\nfloat gravity = (float) Math.sqrt(X*X+Y*Y+Z*Z);\n}``````\n• yiisaa\n• 这是我的微信扫一扫\n•", null, "• zhengweiqiangcom\n• 我的微信公众号扫一扫\n•", null, "", null, "" ]

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[ "Advertisement\n\n# Singularities in the differential geometry of submanifolds\n\nConference paper\nPart of the Lecture Notes in Mathematics book series (LNM, volume 209)\n\n## Keywords\n\nPlane Curve Simple Closed Curve Space Curf Projective Connection Parabolic Point\nThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.\n\n## Preview\n\nUnable to display preview. Download preview PDF.\n\n## References\n\n1. \nM. Barner, Über die Mindestanzahl stationärer Schmiegebenen beigeschlossen strengkonvexen Raumkurven. Abh. Math. Sem. Univ. Hamburg, 20, 196–215 (1956).\n2. \nE.A. Feldman, The geometry of immersions, I, Trans. Amer. Math. Soc. 120, 185–224 (1965). II, same Trans. 125, 181–215 (1966).\n3. \n_____, On parabolic and umbilic points of immersed hypersurfaces. Trans. Amer. Math. Soc. 127, 1–28 (1967).\n4. \n_____, Deformations of closed space curves. Jour. Diff. Geom. 2, 67–75 (1968).\n5. \n_____, Non-degenerate curves on a Riemannian manifold. Mimeographed preprint.Google Scholar\n6. \nH. Guggenheimer, Differential Geometry, McGraw-Hill, New York, (1963).\n7. \nA. Haefliger, Sur les self-intersections des applications différentiables. Bull. Soc. Math. France. 87, 351–359. (1959).\n8. \n_____, Points multiples d’une application et produit cyclique reduit. Amer. J. Math. 83, 57–70,(1961)\n9. \nB. Halpern, Global theorems for closed plane curves. Bull. Amer. Math. Soc. 76, 96–100 (1970).\n10. \nO. Haupt and H. Künneth, Geometrische Ordnungen. Springer-Verlag, Berlin-Heidelberg-New York, 1967.\n11. \nE.S. Jones, A generalization of the two-vertex theorem for space curves. Thesis, University of Minnesota, 1970.Google Scholar\n12. \nJ.A. Little, On singularities of submanifolds of higher dimensional Euclidean spaces. Annali di Matematica (4) 83, 261–336 (1969).\n13. \n______, Non-degenerate homotopies of curves on the unit 2-sphere. J. Diff. Geom. 4 (1970)Google Scholar\n14. \n_____, Third order nondegenerate homotopies of space curves. J. Diff. Geom. to appear.Google Scholar\n15. \nI.G. Macdonald, Some enumerative formulas for algebraic curves. Proc.Cambridge Phil. Soc. 54, 399–416 (1958).\n16. \nI.G. Macdonald, Symmetric products of an algebraic curve, Topology 1, 319–343 (1962).\n17. \nJ. Mather, Stability of C Mappings: V, Transversality, to appear.Google Scholar\n18. \nW.F. Pohl, Connexions in differential geometry of higher order. Trans. Amer. Math. Soc. 125, 310–325 (1966).\n19. \n_____, Extrinsic complex projective geometry. Proceedings of the Conference on Complex Analysis (Minneapolis, 1964). Springer-Verlag, Berlin-Heidelberg-New York, 18–29 (1965).Google Scholar\n20. \n_____, On a theorem related to the four-vertex theorem. Annals of Math. 84, 356–367 (1966).\n21. \n_____, The self-linking number of a closed space curve. J. Math. Mech. 17, 975–986 (1968).\n22. \nS. Sasaki, The minimum number of points of inflexion of closed curves in the projective plane. Tôhoku Math. J. (2) 9, 113–117 (1957).\n23. \nR.L.E. Schwarzenberger, The secant bundles of a projective variety. Proc. London Math Soc. (3) 14, 369–384 (1964).\n24. \nB. Segre, Sulle coppie di tangenti fra loro parallele relative ad una curva chuisa sghemba. Hommage au Professeur Lucien Godeaux, 141–167. Libraire Universitaire, Louvain, 1968.Google Scholar\n25. \n_____, Alcune proprietà differenziali in grande della curve chuise sghemba. Rend. Mat. (6) 1, 237–297 (1968).\n26. \n_____, Global differential properties of closed twisted curves. Rend. Sem. Mat. Fie. Milano 38, 256–263 (1968).\n27. \nH. Suzuki, Characteristic classes of some higher order tangent bundles of complex projective spaces. J. Math. Soc. Japan, 18,386–393 (1966).\n28. \n_____, Bounds for dimensions of odd order nonsingular immersions of RPn. Trans. Amer. Math. Soc. 121, 269–275 (1966).\n29. \n_____, Higher order non-singular immersions in projective spaces. Quart. J. Math. Oxford 2nd Series 20, 33–44 (1969).\n30. \nJ.H. White, Self-linking and the Gauss integral in higher dimensions. Amer. J. Math. 91, 693–728 (1969).\n31. \nJ.H. White, Some differential invariants of submanifolds of euclidean space. J. Diff. Geom. 4, 207–224 (1970).\n32. \n_____, Self-linking and the directed secant span of a differentiable manifold, to appear.Google Scholar\n33. \nC. Yoshioka, On the higher order non-singular immersions. Sci. Rep.Niigata Univ. Ser. A No.5, 23–30 (1967).Google Scholar\n\n## Copyright information\n\n© Springer-Verlag Berlin · Heidelberg 1971\n\n## Authors and Affiliations\n\nThere are no affiliations available" ]

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[ "# Quick Statistics question\n\nHello, I was wondering if someone could help me with a question my homework has invoked. In statistics, if a probability is greater than 1 would it just be equal to 1? The problem arose quite a few times when working with Discrete Random Variables.\n\nFor example, 80% of a 25 person survey is considered a success. The probability that more than 15 of the people will be a success is...\n\np(16)+p(17)+p(18)+p(19)+p(20)+p(21)+p(22)+p(23)+p(24)+p(25)\n\nI worked this specific problem out with Minitab as instructed, and after summing the column that I stored the probabilities in it yielded 5.97511. Clearly, this seems wrong. I would think that the probability couldn't be greater than 1, or ever greater than .999999(repeating).\n\nI also encountered the problem a few times when using the Appendix charts which we were asked to use to solve a few of the problems.\n\nThanks for any help.\n\nLast edited:\n\nmatt grime\nHomework Helper\nYou probably worked it out wrongly. The probability of an event is a number between 0 and 1 inclusive.\n\nOkay, thank you. I dont guess you could help me work out the example I gave so I could see what Im doing wrong could you?\n\nSorry, I figured it out. It seems I was using the wrong command in my calculator, and looking at the table wrong. Thanks again for letting me know that my assumption was correct, and that I was in fact making a mistake.\n\nHallsofIvy\nHomework Helper\nnew324 said:\nOkay, thank you. I dont guess you could help me work out the example I gave so I could see what Im doing wrong could you?\nIf you gave us the data you were working on, we might.\n\n\"For example, 80% of a 25 person survey is considered a success. The probability that more than 15 of the people will be a success is...\"\n\nDoesn't make sense. 80% of 25 is 20. I think that by \"80% of a 25 person survey\", you mean getting responsed from 80% of the people: 20 people. I don't know what you mean by \"more than 15 of the people will be a success\".\n\nWhat you have calculated would be the probability of getting more than 15 people to respond to the survey IF the numbers you are using, P(16), P(17), etc. are the probabilities of exactly that number of people responding. I'm wondering if your P(16), etc. are not the probability of 16 or fewer people responding (i.e. the cumulative probability). That might account for you getting such a high answer. What, for example is P(25)?\n\nWell I appreciate the input, but as I stated I figured out my problem. But I figured I would go ahead and explain, I should have said SAMPLE survey rather than survey. It is known that 80% of the total population is a success. The question is what the probability is that the random sample survey of 25 people (out of the total population) would yield a more than 15 successus. Which would be the probability of 16 successes + the probability of 17 successes all the way up to 25.\n\nmatt grime" ]

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[ "", null, "# ANALYSIS OF BENTS WITH TENSION-ONLY X-BRACING\n\n15 January 2019 Off\n\n# ANALYSIS OF BENTS WITH TENSION-ONLY X-BRACING\n\nXBRACING is a spreadsheet program written in MS-Excel for the purpose of analyzing X-braced bents with tension-only systems. From 1-story up to 10-story bents may be analyzed. Specifically, given the bent width, the story heights, and the lateral story loads, all the member (segment) forces are determined, as well as the horizontal and vertical reactions at the supports, and the individual story drifts (horizontal deflections).\n\nThis program is a workbook consisting of eleven (11) worksheets, described as follows:\n\n• Doc – Documentation sheet\n• X(1) – Analysis of 1-story bent with tension-only X-bracing\n• X(2) – Analysis of 2-story bent with tension-only X-bracing\n• X(3) – Analysis of 3-story bent with tension-only X-bracing\n• X(4) – Analysis of 4-story bent with tension-only X-bracing\n• X(5) – Analysis of 5-story bent with tension-only X-bracing\n• X(6) – Analysis of 6-story bent with tension-only X-bracing\n• X(7) – Analysis of 7-story bent with tension-only X-bracing\n• X(8) – Analysis of 8-story bent with tension-only X-bracing\n• X(9) – Analysis of 9-story bent with tension-only X-bracing\n• X(10) – Analysis of 10-story bent with tension-only X-bracing\n\nProgram Assumptions and Limitations:\n\n1. This program assumes that the vertical bent is fully braced, between every story, utilizing a tension-only X-bracing system.\n\n2. In a tension-only bracing system, one brace is assumed effective in tension while the other brace is assumed to buckle (in compression), requiring the tension brace to take all of the load.\n\n3. This program uses the “Method of Virtual Work” to determine the horizontal deflections at each of the story levels. The horizontal deflection at a particular story level is determined by first applying a “dummy” unit load at that level. Then, the member forces from the unit load are determined. With the member forces due to the applied lateral loads already having been calculated, the horizontal deflection at that story level is:\n\nDn = S F*u*L/(A*E)\n\nwhere:Dn = horizontal deflection at particular story level number considered (inches)\n\nF = force in each member due to all applied lateral story loads (kips)\n\nu = force in each member due to unit load applied at paricular story, ‘n’\n\nL = length of each member in bent (inches)\n\nA = area of each member in bent (in.^2)\n\nE = modulus of elasticity of members in bent, all assumed equal (ksi)\n\nThe horizontal deflections (story drifts) determined are for the joints at the left side of the bent, with all members (segments) assumed pinned at both ends." ]

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[ "# Computational electromagnetics\n\nComputational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment.\n\nIt typically involves using computationally efficient approximations to Maxwell's equations and is used to calculate antenna performance, electromagnetic compatibility, radar cross section and electromagnetic wave propagation when not in free space.\n\nA specific part of computational electromagnetics deals with electromagnetic radiation scattered and absorbed by small particles.\n\n## Background\n\nSeveral real-world electromagnetic problems like electromagnetic scattering, electromagnetic radiation, modeling of waveguides etc., are not analytically calculable, for the multitude of irregular geometries found in actual devices. Computational numerical techniques can overcome the inability to derive closed form solutions of Maxwell's equations under various constitutive relations of media, and boundary conditions. This makes computational electromagnetics (CEM) important to the design, and modeling of antenna, radar, satellite and other communication systems, nanophotonic devices and high speed silicon electronics, medical imaging, cell-phone antenna design, among other applications.\n\nCEM typically solves the problem of computing the E (electric) and H (magnetic) fields across the problem domain (e.g., to calculate antenna radiation pattern for an arbitrarily shaped antenna structure). Also calculating power flow direction (Poynting vector), a waveguide's normal modes, media-generated wave dispersion, and scattering can be computed from the E and H fields. CEM models may or may not assume symmetry, simplifying real world structures to idealized cylinders, spheres, and other regular geometrical objects. CEM models extensively make use of symmetry, and solve for reduced dimensionality from 3 spatial dimensions to 2D and even 1D.\n\nAn eigenvalue problem formulation of CEM allows us to calculate steady state normal modes in a structure. Transient response and impulse field effects are more accurately modeled by CEM in time domain, by FDTD. Curved geometrical objects are treated more accurately as finite elements FEM, or non-orthogonal grids. Beam propagation method (BPM) can solve for the power flow in waveguides. CEM is application specific, even if different techniques converge to the same field and power distributions in the modeled domain.\n\n## Overview of methods\n\nOne approach is to discretize the space in terms of grids (both orthogonal, and non-orthogonal) and solving Maxwell's equations at each point in the grid. Discretization consumes computer memory, and solving the equations takes significant time. Large-scale CEM problems face memory and CPU limitations. As of 2007, CEM problems require supercomputers,[citation needed] high performance clusters,[citation needed] vector processors and/or parallelism. Typical formulations involve either time-stepping through the equations over the whole domain for each time instant; or through banded matrix inversion to calculate the weights of basis functions, when modeled by finite element methods; or matrix products when using transfer matrix methods; or calculating integrals when using method of moments (MoM); or using fast fourier transforms, and time iterations when calculating by the split-step method or by BPM.\n\n## Choice of methods\n\nChoosing the right technique for solving a problem is important, as choosing the wrong one can either result in incorrect results, or results which take excessively long to compute. However, the name of a technique does not always tell one how it is implemented, especially for commercial tools, which will often have more than one solver.\n\nDavidson gives two tables comparing the FEM, MoM and FDTD techniques in the way they are normally implemented. One table is for both open region (radiation and scattering problems) and another table is for guided wave problems.\n\n## Maxwell's equations in hyperbolic PDE form\n\nMaxwell's equations can be formulated as a hyperbolic system of partial differential equations. This gives access to powerful techniques for numerical solutions.\n\nIt is assumed that the waves propagate in the (x,y)-plane and restrict the direction of the magnetic field to be parallel to the z-axis and thus the electric field to be parallel to the (x,y) plane. The wave is called a transverse magnetic (TM) wave. In 2D and no polarization terms present, Maxwell's equations can then be formulated as:\n\n${\\frac {\\partial }{\\partial t}}{\\bar {u}}+A{\\frac {\\partial }{\\partial x}}{\\bar {u}}+B{\\frac {\\partial }{\\partial y}}{\\bar {u}}+C{\\bar {u}}={\\bar {g}}$", null, "where u, A, B, and C are defined as\n\n${\\bar {u}}=\\left({\\begin{matrix}E_{x}\\\\E_{y}\\\\H_{z}\\end{matrix}}\\right),$", null, "$A=\\left({\\begin{matrix}0&0&0\\\\0&0&{\\frac {1}{\\epsilon }}\\\\0&{\\frac {1}{\\mu }}&0\\end{matrix}}\\right),$", null, "$B=\\left({\\begin{matrix}0&0&{\\frac {-1}{\\epsilon }}\\\\0&0&0\\\\{\\frac {-1}{\\mu }}&0&0\\end{matrix}}\\right),$", null, "$C=\\left({\\begin{matrix}{\\frac {\\sigma }{\\epsilon }}&0&0\\\\0&{\\frac {\\sigma }{\\epsilon }}&0\\\\0&0&0\\end{matrix}}\\right).$", null, "In this representation, ${\\bar {g}}$", null, "is the forcing function, and is in the same space as ${\\bar {u}}$", null, ". It can be used to express an externally applied field or to describe an optimization constraint. As formulated above:\n\n${\\bar {g}}=\\left({\\begin{matrix}E_{x,constraint}\\\\E_{y,constraint}\\\\H_{z,constraint}\\end{matrix}}\\right).$", null, "${\\bar {g}}$", null, "may also be explicitly defined equal to zero to simplify certain problems, or to find a characteristic solution, which is often the first step in a method to find the particular inhomogeneous solution.\n\n## Integral equation solvers\n\n### The discrete dipole approximation\n\nThe discrete dipole approximation is a flexible technique for computing scattering and absorption by targets of arbitrary geometry. The formulation is based on integral form of Maxwell equations. The DDA is an approximation of the continuum target by a finite array of polarizable points. The points acquire dipole moments in response to the local electric field. The dipoles of course interact with one another via their electric fields, so the DDA is also sometimes referred to as the coupled dipole approximation. The resulting linear system of equations is commonly solved using conjugate gradient iterations. The discretization matrix has symmetries (the integral form of Maxwell equations has form of convolution) enabling Fast Fourier Transform to multiply matrix times vector during conjugate gradient iterations.\n\n### Method of moments element method\n\nThe method of moments (MoM) or boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in boundary integral form). It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, fracture mechanics, and plasticity.\n\nMoM has become more popular since the 1980s. Because it requires calculating only boundary values, rather than values throughout the space, it is significantly more efficient in terms of computational resources for problems with a small surface/volume ratio. Conceptually, it works by constructing a \"mesh\" over the modeled surface. However, for many problems, BEM are significantly computationally less efficient than volume-discretization methods (finite element method, finite difference method, finite volume method). Boundary element formulations typically give rise to fully populated matrices. This means that the storage requirements and computational time will tend to grow according to the square of the problem size. By contrast, finite element matrices are typically banded (elements are only locally connected) and the storage requirements for the system matrices typically grow linearly with the problem size. Compression techniques (e.g. multipole expansions or adaptive cross approximation/hierarchical matrices) can be used to ameliorate these problems, though at the cost of added complexity and with a success-rate that depends heavily on the nature and geometry of the problem.\n\nBEM is applicable to problems for which Green's functions can be calculated. These usually involve fields in linear homogeneous media. This places considerable restrictions on the range and generality of problems suitable for boundary elements. Nonlinearities can be included in the formulation, although they generally introduce volume integrals which require the volume to be discretized before solution, removing an oft-cited advantage of BEM.\n\n### Finite integration technique\n\n− The finite integration technique (FIT) is a spatial discretization scheme to numerically solve electromagnetic field problems in time and frequency domain. It preserves basic topological properties of the continuous equations such as conservation of charge and energy. FIT was proposed in 1977 by Thomas Weiland and has been enhanced continually over the years. This method covers the full range of electromagnetics (from static up to high frequency) and optic applications and is the basis for commercial simulation tools.[failed verification][failed verification]\n\nThe basic idea of this approach is to apply the Maxwell equations in integral form to a set of staggered grids. This method stands out due to high flexibility in geometric modeling and boundary handling as well as incorporation of arbitrary material distributions and material properties such as anisotropy, non-linearity and dispersion. Furthermore, the use of a consistent dual orthogonal grid (e.g. Cartesian grid) in conjunction with an explicit time integration scheme (e.g. leap-frog-scheme) leads to compute and memory-efficient algorithms, which are especially adapted for transient field analysis in radio frequency (RF) applications.\n\n### Fast multipole method\n\nThe fast multipole method (FMM) is an alternative to MoM or Ewald summation. It is an accurate simulation technique and requires less memory and processor power than MoM. The FMM was first introduced by Greengard and Rokhlin and is based on the multipole expansion technique. The first application of the FMM in computational electromagnetics was by Engheta et al.(1992). FMM can also be used to accelerate MoM.\n\n### Plane wave time-domain\n\nWhile the fast multipole method is useful for accelerating MoM solutions of integral equations with static or frequency-domain oscillatory kernels, the plane wave time-domain (PWTD) algorithm employs similar ideas to accelerate the MoM solution of time-domain integral equations involving the retarded potential. The PWTD algorithm was introduced in 1998 by Ergin, Shanker, and Michielssen.\n\n### Partial element equivalent circuit method\n\nThe partial element equivalent circuit (PEEC) is a 3D full-wave modeling method suitable for combined electromagnetic and circuit analysis. Unlike MoM, PEEC is a full spectrum method valid from dc to the maximum frequency determined by the meshing. In the PEEC method, the integral equation is interpreted as Kirchhoff's voltage law applied to a basic PEEC cell which results in a complete circuit solution for 3D geometries. The equivalent circuit formulation allows for additional SPICE type circuit elements to be easily included. Further, the models and the analysis apply to both the time and the frequency domains. The circuit equations resulting from the PEEC model are easily constructed using a modified loop analysis (MLA) or modified nodal analysis (MNA) formulation. Besides providing a direct current solution, it has several other advantages over a MoM analysis for this class of problems since any type of circuit element can be included in a straightforward way with appropriate matrix stamps. The PEEC method has recently been extended to include nonorthogonal geometries. This model extension, which is consistent with the classical orthogonal formulation, includes the Manhattan representation of the geometries in addition to the more general quadrilateral and hexahedral elements. This helps in keeping the number of unknowns at a minimum and thus reduces computational time for nonorthogonal geometries.\n\n## Differential equation solvers\n\n### Finite-difference time-domain\n\nFinite-difference time-domain (FDTD) is a popular CEM technique. It is easy to understand. It has an exceptionally simple implementation for a full wave solver. It is at least an order of magnitude less work to implement a basic FDTD solver than either an FEM or MoM solver. FDTD is the only technique where one person can realistically implement oneself in a reasonable time frame, but even then, this will be for a quite specific problem. Since it is a time-domain method, solutions can cover a wide frequency range with a single simulation run, provided the time step is small enough to satisfy the Nyquist–Shannon sampling theorem for the desired highest frequency.\n\nFDTD belongs in the general class of grid-based differential time-domain numerical modeling methods. Maxwell's equations (in partial differential form) are modified to central-difference equations, discretized, and implemented in software. The equations are solved in a cyclic manner: the electric field is solved at a given instant in time, then the magnetic field is solved at the next instant in time, and the process is repeated over and over again.\n\nThe basic FDTD algorithm traces back to a seminal 1966 paper by Kane Yee in IEEE Transactions on Antennas and Propagation. Allen Taflove originated the descriptor \"Finite-difference time-domain\" and its corresponding \"FDTD\" acronym in a 1980 paper in IEEE Trans. Electromagn. Compat.. Since about 1990, FDTD techniques have emerged as the primary means to model many scientific and engineering problems addressing electromagnetic wave interactions with material structures. An effective technique based on a time-domain finite-volume discretization procedure was introduced by Mohammadian et al. in 1991. Current FDTD modeling applications range from near-DC (ultralow-frequency geophysics involving the entire Earth-ionosphere waveguide) through microwaves (radar signature technology, antennas, wireless communications devices, digital interconnects, biomedical imaging/treatment) to visible light (photonic crystals, nanoplasmonics, solitons, and biophotonics). Approximately 30 commercial and university-developed software suites are available.\n\n### Discontinuous time-domain method\n\nAmong many time domain methods, discontinuous Galerkin time domain (DGTD) method has become popular recently since it integrates advantages of  both the finite volume time domain (FVTD) method and the finite element time domain (FETD) method. Like FVTD, the numerical flux is used to exchange information between neighboring elements, thus all operations of DGTD are local and easily parallelizable. Similar to FETD, DGTD employs unstructured mesh and capable of high-order accuracy if the high-order hierarchical basis function is adopted. With above merits, DGTD method is widely implemented for the transient analysis of multiscale problems involving large number of unknowns.\n\n### Multiresolution time-domain\n\nMRTD is an adaptive alternative to the finite difference time domain method (FDTD) based on wavelet analysis.\n\n### Finite element method\n\nThe finite element method (FEM) is used to find approximate solution of partial differential equations (PDE) and integral equations. The solution approach is based either on eliminating the time derivatives completely (steady state problems), or rendering the PDE into an equivalent ordinary differential equation, which is then solved using standard techniques such as finite differences, etc.\n\nIn solving partial differential equations, the primary challenge is to create an equation which approximates the equation to be studied, but which is numerically stable, meaning that errors in the input data and intermediate calculations do not accumulate and destroy the meaning of the resulting output. There are many ways of doing this, with various advantages and disadvantages. The finite element method is a good choice for solving partial differential equations over complex domains or when the desired precision varies over the entire domain.\n\n### Finite integration technique\n\nThe finite integration technique (FIT) is a spatial discretization scheme to numerically solve electromagnetic field problems in time and frequency domain. It preserves basic topological properties of the continuous equations such as conservation of charge and energy. FIT was proposed in 1977 by Thomas Weiland and has been enhanced continually over the years. This method covers the full range of electromagnetics (from static up to high frequency) and optic applications and is the basis for commercial simulation tools.[failed verification][failed verification]\n\nThe basic idea of this approach is to apply the Maxwell equations in integral form to a set of staggered grids. This method stands out due to high flexibility in geometric modeling and boundary handling as well as incorporation of arbitrary material distributions and material properties such as anisotropy, non-linearity and dispersion. Furthermore, the use of a consistent dual orthogonal grid (e.g. Cartesian grid) in conjunction with an explicit time integration scheme (e.g. leap-frog-scheme) leads to compute and memory-efficient algorithms, which are especially adapted for transient field analysis in radio frequency (RF) applications.\n\n### Pseudo-spectral time domain\n\nThis class of marching-in-time computational techniques for Maxwell's equations uses either discrete Fourier or discrete Chebyshev transforms to calculate the spatial derivatives of the electric and magnetic field vector components that are arranged in either a 2-D grid or 3-D lattice of unit cells. PSTD causes negligible numerical phase velocity anisotropy errors relative to FDTD, and therefore allows problems of much greater electrical size to be modeled.\n\n### Pseudo-spectral spatial domain\n\nPSSD solves Maxwell's equations by propagating them forward in a chosen spatial direction. The fields are therefore held as a function of time, and (possibly) any transverse spatial dimensions. The method is pseudo-spectral because temporal derivatives are calculated in the frequency domain with the aid of FFTs. Because the fields are held as functions of time, this enables arbitrary dispersion in the propagation medium to be rapidly and accurately modelled with minimal effort. However, the choice to propagate forward in space (rather than in time) brings with it some subtleties, particularly if reflections are important.\n\n### Transmission line matrix\n\nTransmission line matrix (TLM) can be formulated in several means as a direct set of lumped elements solvable directly by a circuit solver (ala SPICE, HSPICE, et al.), as a custom network of elements or via a scattering matrix approach. TLM is a very flexible analysis strategy akin to FDTD in capabilities, though more codes tend to be available with FDTD engines.\n\n### Locally one-dimensional\n\nThis is an implicit method. In this method, in two-dimensional case, Maxwell equations are computed in two steps, whereas in three-dimensional case Maxwell equations are divided into three spatial coordinate directions. Stability and dispersion analysis of the three-dimensional LOD-FDTD method have been discussed in detail.\n\n## Other methods\n\n### EigenMode expansion\n\nEigenmode expansion (EME) is a rigorous bi-directional technique to simulate electromagnetic propagation which relies on the decomposition of the electromagnetic fields into a basis set of local eigenmodes. The eigenmodes are found by solving Maxwell's equations in each local cross-section. Eigenmode expansion can solve Maxwell's equations in 2D and 3D and can provide a fully vectorial solution provided that the mode solvers are vectorial. It offers very strong benefits compared with the FDTD method for the modelling of optical waveguides, and it is a popular tool for the modelling of fiber optics and silicon photonics devices.\n\n### Physical optics\n\nPhysical optics (PO) is the name of a high frequency approximation (short-wavelength approximation) commonly used in optics, electrical engineering and applied physics. It is an intermediate method between geometric optics, which ignores wave effects, and full wave electromagnetism, which is a precise theory. The word \"physical\" means that it is more physical than geometrical optics and not that it is an exact physical theory.\n\nThe approximation consists of using ray optics to estimate the field on a surface and then integrating that field over the surface to calculate the transmitted or scattered field. This resembles the Born approximation, in that the details of the problem are treated as a perturbation.\n\n### Uniform theory of diffraction\n\nThe uniform theory of diffraction (UTD) is a high frequency method for solving electromagnetic scattering problems from electrically small discontinuities or discontinuities in more than one dimension at the same point.\n\nThe uniform theory of diffraction approximates near field electromagnetic fields as quasi optical and uses ray diffraction to determine diffraction coefficients for each diffracting object-source combination. These coefficients are then used to calculate the field strength and phase for each direction away from the diffracting point. These fields are then added to the incident fields and reflected fields to obtain a total solution.\n\n## Validation\n\nValidation is one of the key issues facing electromagnetic simulation users. The user must understand and master the validity domain of its simulation. The measure is, \"how far from the reality are the results?\"\n\nAnswering this question involves three steps: comparison between simulation results and analytical formulation, cross-comparison between codes, and comparison of simulation results with measurement.\n\n### Comparison between simulation results and analytical formulation\n\nFor example, assessing the value of the radar cross section of a plate with the analytical formula:\n\n${\\text{RCS}}_{\\text{Plate}}={\\frac {4\\pi A^{2}}{\\lambda ^{2}}},$", null, "where A is the surface of the plate and $\\lambda$", null, "is the wavelength. The next curve presenting the RCS of a plate computed at 35 GHz can be used as reference example.\n\n### Cross-comparison between codes\n\nOne example is the cross comparison of results from method of moments and asymptotic methods in their validity domains.\n\n### Comparison of simulation results with measurement\n\nThe final validation step is made by comparison between measurements and simulation. For example, the RCS calculation and the measurement of a complex metallic object at 35 GHz. The computation implements GO, PO and PTD for the edges.\n\nValidation processes can clearly reveal that some differences can be explained by the differences between the experimental setup and its reproduction in the simulation environment.\n\n## Light scattering codes\n\nThere are now many efficient codes for solving electromagnetic scattering problems. They are listed as:\n\nSolutions which are analytical, such as Mie solution for scattering by spheres or cylinders, can be used to validate more involved techniques." ]

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[ "Percentage Calculator\n\n# How calculate 6% of 24?\n\nA simple way to calculate percentages of X\n\n 6% of 24 = 1.44 24 + 6% = 25.44 24 - 6% = 22.56\n What is Calculate the percentage: %\n\nIn the store, the product costs 24, you were given a discount 6 and you want to understand how much you saved.\n\nSolution:\n\nAmount saved = Product price * Percentage Discount/ 100\n\nAmount saved = (6 * 24) / 100\n\nMore random interest calculations:\n2% от 24 = 0.48\n24 + 2% = 24.48\n24 - 2% = 23.52\n14% от 24 = 3.36\n24 + 14% = 27.36\n24 - 14% = 20.64\n22% от 24 = 5.28\n24 + 22% = 29.28\n24 - 22% = 18.72\n37% от 24 = 8.88\n24 + 37% = 32.88\n24 - 37% = 15.12\n46% от 24 = 11.04\n24 + 46% = 35.04\n24 - 46% = 12.96\n53% от 24 = 12.72\n24 + 53% = 36.72\n24 - 53% = 11.28\n67% от 24 = 16.08\n24 + 67% = 40.08\n24 - 67% = 7.92\n71% от 24 = 17.04\n24 + 71% = 41.04\n24 - 71% = 6.96\n87% от 24 = 20.88\n24 + 87% = 44.88\n24 - 87% = 3.12\n97% от 24 = 23.28\n24 + 97% = 47.28\n24 - 97% = 0.72\n\nAnd what if the percentage is more than 100? Then the resulting result will be greater than the sum itself24. For example:\n300% от 24 = 72\n500% от 24 = 120\n1000% от 24 = 240" ]

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[ "How many dimes are in 82 quarters?", null, "Here, we will show you how to calculate how many dimes there are in 82 quarters.\n\nFirst, calculate how many cents there are in 82 quarters by multiplying 82 by 25, and then divide that result by 10 cents to get the answer.\n\nHere is the math to illustrate better:\n\n82 quarters x 25 cents\n= 2,050 cents\n\n2,050 cents / 10 cents\n= 205 dimes\n\nThus, the answer to the question \"How many dimes are in 82 quarters?\" is as follows:\n\n205 dimes\n\nNote: We multiplied 82 by 25, because there are 25 cents in a quarter, and we divided 2,050 by 10, because there are 10 cents in a dime.\n\nCoin Converter\nFill out the form below or go here if you need to convert another coin denomination.\n\nHow many\n\nare in\n\nHow many dimes are in 84 quarters?\nHere is the next number of coins we converted.\n\nCopyright  |   Privacy Policy  |   Disclaimer  |   Contact" ]

[ null, "http://grinebiter.com/Images/CoinConverter.jpg", null ]

[ "Research ArticlePHYSICS\n\n# Coherent virtual absorption of elastodynamic waves\n\nSee allHide authors and affiliations\n\nVol. 5, no. 8, eaaw3255\n\n## Abstract\n\nAbsorbers suppress reflection and scattering of an incident wave by dissipating its energy into heat. As material absorption goes to zero, the energy impinging on an object is necessarily transmitted or scattered away. Specific forms of temporal modulation of the impinging signal can suppress wave scattering and transmission in the transient regime, mimicking the response of a perfect absorber without relying on material loss. This virtual absorption can store energy with large efficiency in a lossless material and then release it on demand. Here, we extend this concept to elastodynamics and experimentally show that longitudinal motion can be perfectly absorbed using a lossless elastic cavity. This energy is then released symmetrically or asymmetrically by controlling the relative phase of the impinging signals. Our work opens previously unexplored pathways for elastodynamic wave control and energy storage, which may be translated to other phononic and photonic systems of technological relevance.\n\n## INTRODUCTION\n\nEfficient absorbers are of great importance in a wide variety of technological fields, from energy harvesting and radar detection in electromagnetics to sound proofing in acoustics and vibration isolation in mechanical systems (14). Common to these systems is the notion that efficient absorption can be achieved when the material loss is balanced by the impedance of the impinging wave. In other words, a proper amount of material loss can push one of the complex scattering zeros of the system onto the real frequency axis (5, 6), and, as a result, the impinging energy at this frequency is all lost into heat or other chemical processes. Therefore, the system is not conservative, and its scattering matrix is not unitary (for a multiport linear network, the scattering and transmission toward the ports are governed by the scattering matrix, which maps the incident fields to the outgoing fields).\n\nConsidering simultaneous excitation provides an additional degree of freedom to control the location of the scattering zeros of the system and to move one of them onto the real frequency axis. Coherent perfect absorption (CPA) is achieved through the interference of multiple incident waves impinging on the absorber, enabling a way to control the absorption mechanism in real time through the proper choice of the relative intensities and phases of the input beams. The dependence of CPAs on the input waveforms therefore provides the opportunity to flexibly control light scattering and absorption (7). The demonstration of an acoustic CPA has been presented in (8), opening a path toward several applications of practical interest, including highly sensitive detection and amplification of small variations in the incident signals or in the properties of the involved materials to realize mass or temperature transducers and for the efficient control and conversion of energy in harvesting applications (9, 10). These opportunities suggest that coherent absorption can also be of great interest in the context of elastodynamic waves.\n\nIn lossless systems, the scattering matrix cannot admit zeros on the real frequency axis, given its unitarity; therefore, the impinging wave needs to be transmitted, reflected, or scattered at all real frequencies. The zeros are necessarily confined to the upper half of the complex frequency plane (5, 6, 11, 12), above the real axis. It has been recently suggested that the time evolution of an incoming signal can be tailored to efficiently engage these complex zeros, implying that a specific choice of nonmonochromatic signals oscillating at a complex frequency can totally eliminate transmission, reflections, and scattering, thus realizing a virtual absorber with zero material loss (1315). As long as the input signal illuminates the structure with the right evolution in time, the impinging energy is neither scattered nor transmitted, but instead, it is captured and stored within the system with unitary efficiency. By varying the impinging signals, the stored energy can then be released through its scattering channels in a controllable fashion. Having such a coherent control over the stored energy in the cavity enables unprecedented functionalities, such as flexible energy storage and memory.\n\nThis concept is in some ways connected with the time-reversal excitation technique (16, 17), based on which, under the assumptions of linearity and time-reversal symmetry, the excitation of a cavity with a time-reversed replica of its decaying fields should be accepted by the cavity without scattering or reflections. However, one cannot draw a strict analogy between the two phenomena because they deal with different objectives. The time-reversal technique is based on the idea that an arbitrarily radiated pulse can be made to converge toward the source, provided that an array of sensors can time reverse it with the required accuracy. In contrast to coherent virtual absorption, the time-reversal approach does not therefore deal with eigenmodes of a structure. In the complex zero approach, the incident pulse is determined by the open cavity geometry. Exciting the structure with any of the complex zeros of the structure enables zero scattering and ideal wave capturing, which is a nontrivial conclusion. In analogy with the CPA operation, in the following, we experimentally demonstrate coherent virtual absorption of elastodynamic waves traveling along a solid bar, controlling storage and release of impinging longitudinal waves by tailoring their time evolution.\n\n## RESULTS\n\nConsider a two-port lossless elastic waveguide with a circular cross section supporting longitudinal motion (Fig. 1A). The system can be divided into three domains, with stepwise constant cross section Aj=πrj2 for each j-th domain. A cavity of length L connects two identical side channels, excited by input signals Î+(0,ω) and Î(L,ω), as shown in Fig 1A. Because of the mechanical impedance mismatch of the central section, such a structure produces scattered fields Ô+(L,ω) and Ô(0,ω) at the ports, linearly related to the input fields through the frequency-based scattering matrix (18)[Ô+(L,ω)Ô(0,ω)]=[Ŝ11(ω)Ŝ12(ω)Ŝ21(ω)Ŝ22(ω)][Î+(0,ω)Î(L,ω)](1)", null, "Fig. 1 Elastic coherent virtual absorber.(A) Illustration of a mirror-symmetric waveguide with stepwise constant cross-sectional area A and wave velocity c. The system, whose central domain has length L, supports incoming fields I±(x, t) and outgoing fields O±(x, t). (B) The scattering properties of the waveguide are described by the scattering matrix Ŝ(Ω). When Ω is analytically continued to the complex plane as Ω = ΩRE + iΩIM, Ŝ(Ω) has a countable infinite set of zeros, divided into symmetric and antisymmetric ones. The contour plot shows the quantity λA(Ω)=T̂(Ω)−R̂(Ω) and the location of the antisymmetric zeros (black dots), as well as the location of the symmetric zero at ΩRE = 2π (blue dot). The inputs of the system can be designed to be equal to one of those zeros, thus canceling the scattered fields. (C) Incident and scattered fields for Ω = π + i0.51 such that λA(Ω) = 0. In this case, the scattered fields are identically zero for τ < 0. a.u., arbitrary unit. (D) Incident and scattered fields for Ω = 2π + i0.51 such that λA(Ω) ≠ 0. In this case, the scattered fields appear as soon as the excitations hit the structure.\n\nHere and in the following, the hat symbol denotes a Fourier transform, while ω is the radial frequency. We assume that the waveguide is made of a single material of wave velocity c=E/ρ, where E and ρ are the material modulus of elasticity and density, respectively. Note that the impedance mismatch at the boundary of the outer and inner cores not only relates to the impedance mismatch of the materials on different sides of each interface but also is imparted by the cross-sectional areas of the rods on different sides of the interface (see the Supplementary Materials). Because of symmetry, Ŝ11(ω)=Ŝ22(ω)=R̂(ω), and because of time-reversal symmetry, Ŝ12(ω)=Ŝ21(ω)=T̂(ω). As a result, the components of the scattering matrix can be derived as (see the Supplementary Materials)R̂(ω)=R0+R1ei2ωLc1+R0R1ei2ωLc, T̂(ω)=T0T1eiωLc1+R0R1ei2ωLc(2)where R0=R1=(r02r12)/(r02+r12) and T0=r02r12T1=2r02/(r02+r12) are the local reflection and transmission coefficients at the two interfaces. Here, r0 and r1 are the radius of the outer and inner rods, respectively. By analyzing the scattering matrix, we investigate the conditions under which the system can efficiently absorb the incident energy impinging at its interfaces at x = 0 and x = L. For symmetric or antisymmetric excitations Î+(0,ω)=±Î(L,ω), because of symmetry, the outputs follow Ô+(L,ω)=±Ô(0,ω)=λ(ω)Î+(0,ω). The eigenvalue λ(ω) for the symmetric case is λS(ω)=T̂(ω)+R̂(ω), while that for the antisymmetric case is λA(ω)=T̂(ω)R̂(ω). Zeros of the scattering matrix, associated to perfectly absorbing modes, are found when λS(ω) = 0 or λA(ω) = 0, at frequenciesωz=cL[πn+iln(r12+r02r12r02)], n=1,2,3,(3)where even and odd n values correspond respectively to symmetric and antisymmetric excitations. In lossy systems, these zeros can correspond to real frequencies, yielding what is known in the optics literature as CPA (6, 7). This corresponds to a system with loss balanced to the outer impedance, for which coherent excitation on both sides with same or opposite phase is fully absorbed without transmission or scattering. If the system is lossless, however, then all these zeros lie in the upper half of the complex frequency plane, with ωz = ωRE + iωIM, ωIM > 0 (throughout the paper we assume an eiωt time convention). In this case, no real frequency excitation can be absorbed in the system, as expected from energy conservation. If instead we excite the structure coherently from the two input ports with time-growing waves oscillating at the complex frequency ωz and the proper relative phase, then we engage the corresponding complex zero of the system and achieve coherent virtual absorption, i.e., absence of scattering and transmission, and energy storage in the cavity with unitary efficiency. In practice, these exponentially growing inputs cannot be sustained indefinitely, and the stored energy is released once the excitation is stopped or modified. While the signals are growing at the required rate eωIMt, the system stores energy at a rate proportional to eIMtcos2(2ωREt), which can be tailored with large flexibility by controlling the position of the complex zero in the frequency plane.\n\nTo investigate the dynamics of the virtual absorption process, consider a coherent antisymmetric excitation where I+(0, τ) = − I(L, τ) = f(τ), where τ = t/tL and tL = L/c is the time needed for the wave to travel through the cavity at speed c. To determine the required excitation signal f(τ), we first find the complex zeros for antisymmetric excitation by setting λA(Ω) = 0, where Ω = ωL/c = ΩRE + iΩIM. Figure 1B shows two of these zeros in the upper half of the complex frequency plane, indicated by black dots. We choose to engage the first zero, at Ω = π + i0.51. We therefore excite the two ports with input signals f(τ) to oscillate at the complex frequency Ω for τ < 0 and modulated by a fast-decaying exponential for τ > 0f(τ)=[eΩIMτΘ(τ)+e(Dτ)2Θ(τ)]cos(ΩREτ)(4)where Θ(τ) is the step function and D is a decay factor of choice. The excitation signals at the two ports are shown in the upper panel of Fig. 1C, while the lower panel shows the time domain output signals. As long as the input signals engage the complex zero of the system, for τ < 0, all the impinging energy is virtually absorbed and stored in the system. As soon as the input signals diverge from the virtual absorption condition, for τ > 0, the system releases its stored energy. In general, the system can release the stored energy at τ = 0 through all its complex poles, which are symmetrically located to its zeros in the complex frequency plane in the case of lossless systems because of time-reversal symmetry. Depending on the transient region around τ = 0, different eigenmodes of the system may be excited with different amplitudes. In the example at hand, the system releases its stored energy mostly into the first (dominant) eigenmode, which is consistent with a time-reversed replica of the input signals, but, for different truncation schemes of the input signal, the outgoing fields may be substantially different than the input signals. Figure 1D shows a scenario in which we excite the system antisymmetrically but at the complex frequency Ω = 2π + i0.51, which would correspond to a zero for even excitation. In this case, the incorrect relative phase of the incoming signals produces strong reflections at the port, and virtual absorption is not achieved. Simply flipping the phase of one of the two input signals would completely suppress all output fields for any τ < 0, underlining the importance of the coherent excitation at the two ports to achieve this phenomenon.\n\nTo validate our theoretical results, we performed a proof-of-principle experiment in the setup shown in Fig. 2A. The geometry consists of a 0.6-m aluminum bar with a circular cross section and design parameters L = 0.2 m and r1 = 2r0 = 0.01 m. The system is excited antisymmetrically with piezoelectric actuators [lead-zirconate-titanate (PZT)] at its fifth zero [i.e., f = ω/2π = (64.85 + i2.11) kHz]. A scanning laser Doppler vibrometer (see Materials and Methods) measures the radial velocity field vr(x, t) along the waveguide in response to the external excitation. The radial contraction of the waveguide due to the Poisson effect, albeit not accounted for in our waveguide model, may slightly affect the dynamic properties of the system for slender structures at low frequencies; however, this effect is negligible here (see the Supplementary Materials). In parallel, to validate our experimental results, we developed a finite-difference time-domain (FDTD)–based tool to perform realistic numerical simulations of the same geometry (see Materials and Methods). Numerical simulations shown in Fig. 2B represent the radial velocity of incident (top) and scattered (bottom) fields, each normalized with respect to the peak velocity value of the incident fields. As it is seen from these figures, the incident energy is perfectly absorbed and stored in the middle portion of the bar, which acts as the resonating cavity, as long as the incident waveform is tailored to excite the complex zero of the system (in this example, this is the case for t < 270 μs). For the example considered here, the excited zero is not close to the real frequency axis; hence, the corresponding Q-factor is limited. For this reason, the stored energy inside the cavity is not much larger than the incident one. For a complex zero closer to the real frequency axis, i.e., for a larger Q-factor cavity, at any instant in time, the stored energy may be much larger, roughly equal to Q times the instantaneous impinging energy.", null, "Fig. 2 Experimental wavefield measurements.(A) Waveguide with resonator and side channels of length L = 0.2 m and circular cross section with radii r0 = 0.005 m and r1 = 0.010 m. The excitation is provided by piezoelectric actuators (PZT) placed at the two ends of the system. (B) Radial velocity fields simulated with the finite-difference time-domain (FDTD) method: incident waves (top) and scattered waves (bottom). (C) Similarly, the measured radial velocity: incident waves (top) and scattered waves (bottom). The incoming energy is first stored in the resonator (i.e., 0.2 m < x < 0.4 m) and then released through scattering roughly at t = 270 μs. (D) Time history for both numerical and experimental fields at x = 0.5 m [red dashed line in (B) and (C)] shows that the incoming energy is released through scattering only after the incident fields stop growing exponentially. [Photo credit for (A): Giuseppe Trainiti, Georgia Institute of Technology].\n\nAs soon as the incident wave starts decaying (i.e., for t > 270 μs), the cavity releases its energy. The standing wave pattern inside the cavity (middle rod) is due to its resonance. The experimental results shown in Fig. 2C are in very good agreement with our simulation results. Figure 2D shows a cut of the results shown in panels B and C of Fig. 2 at x = 0.5 m. The top panel in Fig. 2D compares the incident signals from numerical simulations and experiments, while the output signals are compared in the lower panel, confirming the evidence of coherent virtual absorption in the rod. In principle, one can excite the resonator with any of the complex zeros of the system. In our realization, we chose the fifth zero because of the limited dimensions of the symmetric side channels of length L. If we used one of the first zeros, then the incident pulses would be wider, meaning that after releasing the energy from the resonator, the signal will reach the end of the side rods (i.e., the position of source) and reflect back toward the middle resonator. This reflected signal would distort the measured signal at the probe point. However, by choosing longer side rods, one may avoid this issue and also excite the system with lower-order zeros. On the other hand, zeros of very high order may become challenging, as the slender rod assumption may not hold for waves having wavelengths comparable to the rod cross section.\n\nHaving verified that the incident energy can be virtually absorbed and stored in a lossless resonator, we explore the degree of control over the release of stored energy, exploiting the coherence of the two impinging signals. Figure 3 illustrates a scenario in which we repeatedly excite the system at the complex zero, release its energy, and pump and release it again. The top panel shows the input signal, while the middle and bottom panels show the output signals and the stored energy in the middle section of the rod, respectively. The structure can coherently capture the impinging pulses with high efficiency and then release it at will as the exciting pulses are stopped. The system is then ready to store the next pulse. The release of stored energy can also be controlled by changing the relative phase ϕ of the excitation signals at the opposite ports, exploiting the coherence of the storage process. To investigate this scenario, we consider an exponentially modulated variation of the relative phase between the two input signals, ϕ(τ) = π[ehΩIMτΘ(−τ) + Θ(τ)], with h being a control parameter. We compare the case of ideal excitation of the complex zero (Fig. 4A) to the case in which the relative phase is slowly changed as ϕ(τ) (Fig. 4B). The middle panels show the instantaneous power at the input and output ports and the net power flow into the resonator [i.e., PR(τ) = PI+(0, τ) + PI(L, τ) − PO(L, τ) − PO+(0, τ)]. We also integrate these quantities in time to obtain a measure of the total energy entering and exiting the resonator, as well as the net energy stored in the system up to time τ [i.e., ER(τ) = EI(τ) − EO(τ)]. Figure 4B shows that, as soon as we start deviating the relative phase from the required value (in this example, we assume that h = 5), the resonator starts releasing its stored energy, again highlighting the effect of coherence in the storage process. In the case without phase variation (Fig. 4A), because of mirror symmetry, the resonator releases its energy equally through its scattering channels [PO(L, τ) = PO+(0, τ)]. On the contrary, in Fig. 4B, the asymmetry in excitation enables additional control on the port through which the stored energy is released. Note that the transition time over which the relative phase change is applied can markedly affect the redistribution of released energy between outputs. Figure 5 shows how the amount of released energy difference between outputs ΔE0 can be controlled by how quickly the relative phase ϕ(τ) varies from 0 to π, changing the value of h. For relatively small values of h, we break the symmetry between scattering ports and maximize ΔE0.", null, "Fig. 3 Control of scattering and energy storage by changing the complex frequency of the excitation signals.(A) Excitation signals. (B) Outgoing signals. (C) Stored energy in the system.", null, "Fig. 4 Scattering and energy storage control through input relative phase variation.(A) Response for zero relative phase ϕ(τ) = 0 (top) between the inputs at Ω = 5π + i0.51 is represented in terms of the normalized power inputs PI+(0, τ) and PI−(L, τ), the power outputs PO−(L, τ) and PO+(0, τ), and the power stored into the resonator PR(τ) (middle), as well as the associated integrals evaluated between −∞ and τ, with EI(τ) and EO(τ) as the energy that entered and exited the system up to time τ, respectively (bottom). (B) Imposing an exponentially increasing relative phase law ϕ = ϕ(τ) between the inputs (top) enables the dynamic control of the scattering process, with scattering onset anticipated at τ < 0 (middle), a different stored energy profile (bottom). The imposed relative phase also induces different energy redistribution between the two outputs of the system, with EO+(L, τ) − EO−(0, τ) = ΔE0 ≠ 0.", null, "Fig. 5 Output energy redistribution due to input relative phase variation.(A) Exponential relative phase law ϕ(τ) = π[ehΩIMτΘ(−τ) + Θ(τ)] for different values of the parameter h. (B) Effect of the relative phase variation on the total output energies EO+(L, τ) and EO−(0, τ) and their difference ΔE0.\n\nIn practice, growing the input signals for a long period of time may be impractical, and most of the stored energy at any given instant would, in any case, be contributed by the last part of the excitation transient, inversely proportional to the quality factor of the cavity. In this sense, cavities with higher-quality factors have complex zeros closer to the real frequency axis, implying that in this case, the excitation becomes quasi-harmonic. As the intensity grows, we may also incur into nonlinearities, which change the picture and break the temporal symmetry between the storage and release processes. Suitably tailored nonlinear cavities may be envisioned to accept an incoming signal with unitary efficiency but then trap it in an embedded eigenstate, as recently envisioned in the context of quantum optics (19).\n\n## DISCUSSION\n\nHere, we have introduced and experimentally demonstrated the concept of coherent virtual absorption, storage, and release of energy on demand in elastodynamics. We have shown that the impinging displacement energy can be absorbed and stored in a lossless resonator with unitary efficiency for a desired period of time. We have also shown that we can control the release of the elastic energy with large flexibility and its directionality, exploiting the coherence of the storage process. Our results open interesting opportunities for applications in elastodynamics and structural mechanics, including sensors that may be able detect small changes in the resonance characteristics of a cavity, for example, as a result of applied mechanical strain, temperature, or material changes. These may also be of interest for energy storage and release. In addition, the control of the rate and time evolution of energy release may be beneficial for efficient conversion of mechanical energy into electrical or for the implementation of a broad range of memory, amplification, and computational functionalities within mechanical substrates. Efficient excitation of an elastodynamic or acoustic cavity; a phase-dependent, nonlinear amplification of the input signals; and controlled storage and release of acoustic energy are also relevant for focused sound generation as part of arrays, loudspeakers, and ultrasonic transducers. More broadly, this proof of concept may be directly translated to other phononic or photonic setups, enabling a large degree of control of phonons and photons using the coherence of specifically tailored nonmonochromatic signals.\n\n## MATERIALS AND METHODS\n\nFor the experimental validation of the concept of coherent virtual absorption in elastodynamics, the elastic waveguides were realized by a slender 1566 carbon steel rod (E = 210 GPa and ρ = 7800 kg/m3) with stepwise constant circular cross section (Fig. 6). The rod is 600 mm long, and it is made of a 200-mm resonant element with sectional radius r0 = 10 mm and two symmetric side channels with radii r0 = 5 mm. The slenderness of the system guaranteed waves below approximately 118 kHz to be considered purely longitudinal (20). Elastic waves were excited at the ends of the system by two separate cylindrical piezoelectric actuators, which were glued to the rod through a thin epoxy layer. The excitation signal was first provided by a function generator (i.e., Agilent 33220A), then sent to an amplifier (E&I 1040L), and, lastly, sent to the actuators. The transient response of the system is measured as the radial component of the velocity field through a scanning laser Doppler vibrometer (Polytec PSV-400M2). A grid of 648 points was defined across the entire length of the system. The data collection process consisted of exciting the system and then collecting the time response one grid point at a time. This approach assumed that the experiment is repeatable and required that both the system’s excitation and its response measurement are repeated for each of the grid points. Once all the individual grid point’s time domain responses were collected, they were combined to produce a representation of the entire system’s response in the space-time domain as a transient wavefield. The data sampling was performed at 512 kHz for 1 ms. For each point in the measurement grid, 10 averages were performed to improve the signal-to-noise ratio. Upon acquisition, the collected data were filtered in the frequency domain with a band-pass filter between 30 and 90 kHz. The incident and scattered fields were identified on the basis of their propagation direction. The forward and backward propagating waves were isolated by Fourier-transforming the measured wavefield from the space-time domain to the frequency-wavenumber domain. Here, the frequency axis divides the domain into two subdomains, corresponding the components of the wavefield traveling in either the forward or backward directions. By setting one of the two components to zero and inverse Fourier–transforming the information in the space-time domain, it was possible to isolate the other component. On the basis of which component was filtered, we retained either the incident or the scattered field.\n\n## SUPPLEMENTARY MATERIALS\n\nScattering matrix\n\nZeros of the scattering matrix\n\nThis is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.\n\n## REFERENCES AND NOTES\n\nAcknowledgments: Funding: This work was supported by the Air Force Office of Scientific Research through MURI grant No. FA9550-17-1-0002, the National Science Foundation through EFRI grant 1641069 and EFRI grant 1741685 and the Simons Foundation. Author contributions: G.T. conducted the experiments and developed the numerical codes in collaboration with Y.R., who also contributed to the formulation of the concept. A.A. formulated the idea and supervised the project, while M.R. contributed to the theoretical formulation and supervised the experimental demonstrations. All authors contributed to writing the paper. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.\nView Abstract" ]

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[ "# how does the ICC calculate the EM problems??\n\nStatus\nNot open for further replies.\n\n#### devop\n\n##### Full Member level 1", null, "how to calculate the EM\n\nHi :\nhow does the ICC calculate the EM problems??\ngo to lef to find the max current density limite,\nthen ....how to calculate a max density of a certain path??\n\nthanks a lot\n\nAdded after 42 minutes:\n\ndevop said:\nHi :\nhow does the ICC calculate the EM problems??\ngo to lef to find the max current density limite,\nthen ....how to calculate a max density of a certain path??\n\nthanks a lot\nTo calculate current density, Astro first needs to run timing analysis to determine the transition time for each net. Once the current density is calculated, it is checked against the constraint to determine whether it is a violation. The constraint depends on the width of each metal segment and the temperature specified.\n\nI think it's easy to calculate the current flow through VDD to GND(based on the lib),but how to calculate the current flow through a signal net, calculate base on the extracted RC model of net?I think it's a huge data.\n\ncan anyone help?\n\n#### aliputa\n\n##### Newbie level 2", null, "Re: how to calculate the EM\n\nas i known, there is hardly a EM violation on the signal nets. It more likely happens on the P/G nets, so, if we discussed some EM violation, it is often on the P/G grid.\nAs others friends said, the EM violation threshold will be determined by the PVT(Process, Voltage, temperature), and it just like a look-up table, and its Axis is the PVT.\n\nif you want to calculate the signal EM with some information, there is a typical threshold value--\"1mA/1um\", but it is maybe old, you should know the information about the latest advanced process, it's a pity that i don't know these data.\n\nso, i wish these could help you.\nmaybe there is some error, it‘s pleasure to be correct.\n\nAdded after 10 minutes:\n\ndevop said:\nI think it's easy to calculate the current flow through VDD to GND(based on the lib),but how to calculate the current flow through a signal net, calculate base on the extracted RC model of net?I think it's a huge data.\n\ncan anyone help?\n\nsorry , i noted these words just now.\ni have heard some information from a engineer from a EDA software company that \"EM could be calculated by the metal width, switch activities, voltage(on this net)\", you said the RC, yes ,if there is a constant voltage, the current will be determined by the R.\n\nAfter calculate ,the value which have been computed will be compared with the EM violation Threshold(on some PVT).\n\nand you say that is a huge work, i think you should know your chip clearly, you should know where is the path which could have a EM violation most likely. and you choose these path to be checked instead of all those path.(just like everyone should check the P/G nets for EM violation ,because we know the PG could have a EM violation most likely)\n\nStatus\nNot open for further replies." ]

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[ "# Eye's optical characteristic measuring system\n\nAn eye's optical characteristic measuring system, comprising a target projecting system for projecting a target image on an ocular fundus of an eye under test, a photodetection optical system for guiding the target image to a photoelectric detector, a calculating unit for calculating an optical transmission function of the eye under test based on light amount intensity distribution of the target image detected by the photoelectric detector, and an arithmetic unit for calculating visual acuity of a person under test from intersection of the optical transmission function and a predetermined threshold value.\n\nDescription\nBACKGROUND OF THE INVENTION\n\n The present invention relates to a eye's optical characteristic measuring system capable of estimating and calculating a visual acuity of an eye under test based on light amount intensity distribution characteristic of a target image projected on a fundus of the eye.\n\n In the past, the present applicant has already filed a patent application for a system, which comprises a target projecting system for projecting a target image to an ocular fundus of an eye under test, and a photodetection optical system for guiding the target image to a photoelectric detector. Based on light amount intensity distribution of the target image detected by the photoelectric detector, the system calculates a simulation image on the fundus, which would be formed when the target image is projected to the fundus of the eye. Then, the system can identify what kind of image is formed on the fundus of the eye under test.\n\n The system as described above provides an effect such that it is possible to calculate and identify in what condition various types of target images are projected to the fundus of the eye without actually projecting the various types of target images.\n\n However, in the above system already in application, an image itself obtained by simulation can be observed, while, with respect to the visual acuity value, a visual acuity value of the eye under test must be estimated by the tester himself based on the result of observation. In this respect, there has been problem that it is difficult to find accurate visual acuity value.\n\nSUMMARY OF THE INVENTION\n\n To solve the above problems of the conventional type eye's optical characteristic measuring system used in the past, it is an object of the present invention to provide a system, by which it is possible to obtain an accurate visual acuity value objectively from measurement data without asking the result of the observation to a person under test.\n\n To attain the above object, the eye's optical characteristic measuring system according to the present invention comprises a target projecting system for projecting a target image on an ocular fundus of an eye under test, a photodetection optical system for guiding the target image to a photoelectric detector, a calculating unit for calculating an optical transmission function of the eye under test based on light amount intensity distribution of the target image detected by the photoelectric detector, and an arithmetic unit for calculating visual acuity of a person under test from intersection of the optical transmission function and a predetermined threshold value. Also, the present invention provides an eye's optical characteristic measuring system as described above, wherein the optical transmission function is square wave frequency characteristics. Further, the present invention provides an eye's optical characteristic measuring system as described above, wherein the threshold value is a modulation threshold. Also, the present invention provides an eye's optical characteristic measuring system as described above, wherein a plurality of the modulation threshold are prepared to correspond to age of the persons under test, and a modulation threshold suitable for each person under test is used.\n\nBRIEF DESCRIPTION OF THE DRAWINGS\n\n FIG. 1 is a schematical drawing of an ocular fundus of a human eye;\n\n FIG. 2 is a basic block diagram of an eye's optical characteristic measuring system according to an embodiment of the present invention;\n\n FIG. 3(A) and FIG. 3(B) each represents a drawing to show a condition of reflection at an ocular fundus of an eye under test in the eye's optical characteristic measuring system; and\n\n FIG. 4 is a diagram showing relation between optical transmission function of en eye under test and a threshold value corresponding to each age group.\n\nDETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT\n\n Description will be given below on an embodiment of the present invention referring to the drawings.\n\n First, brief description will be given on tissues of an ocular fundus of a human eye.\n\n FIG. 1 is a schematical drawing of tissues of an ocular fundus of a human eye. Reference numeral 31 denotes a visual cell layer, 32 is a retinal pigment epithelial layer, 33 is a choroidal membrane, and 34 is a sclera.\n\n The visual cell layer 31 is an aggregation of fibrous visual cells aligned perpendicularly to the retinal pigment epithelial layer 32. A light beam passing through the visual cell layer 31 (visual cell) is reflected with mirror reflection by the retinal pigment epithelial layer 32. On the other hand, a part of the light beam passes through the retinal pigment epithelial layer 32 and is reflected with scattering reflection by the choroidal membrane 33 and the sclera 34 positioned behind. However, the light reflected by scattering reflection exerts almost no influence on an image to be observed and recognized by a person.\n\n It is demonstrated in the experiment that, when the light beam entering the visual cell layer 31 passes through the visual cell, the light beam passes through it by repeating the reflection almost similar to total reflection in the visual cell.\n\n FIG. 2 shows a basic block diagram of an eye's optical characteristic measuring system according to an embodiment of the present invention.\n\n In this figure, reference numeral 1 is an eye under test, 2 is a target projecting optical system for projecting a target image, and 3 is a photodetection optical system for receiving the light beam reflected from the eye under test.\n\n The projecting optical system 2 comprises a light source 5, a projection lens 6 for converging a projected light beam emitted from the light source 5, a half-mirror 7 arranged on an optical axis of the projection lens 6, a polarization beam splitter 8 for directing the projected light beam passing through the half-mirror 7, for reflecting and projecting a linear polarization component (P linearly polarized light) in a first direction of polarization toward the eye 1 under test and for allowing S linearly polarized light having a direction of polarization deviated by 90° from P linearly polarized light to pass, a relay lens 9 arranged on a projection optical axis of the polarization beam splitter 8 closer to the polarization beam splitter 8 side, an objective lens 11, a correction optical system 12 arranged between the objective lens 11 and the eye 1 under test and comprising a spherical lens, and a ¼ wave plate 13. Further, a gaze target system 17 is arranged to face to the half-mirror 7 and comprises a gaze target 15 and a condenser lens 16. The light source 5 and the gaze target 15 are placed at positions conjugate to the ocular fundus of the eye 1 under test. As to be described later, the light source 5 and the gaze target 15 form an image on the ocular fundus. The light source 5 is integrated with the projection lens 6, and these can be moved in direction of the optical axis in linkage with a focusing lens 19 (to be described later).\n\n The photodetection optical system 3 shares the following components with the projecting optical system 2: the polarization beam splitter 8, the relay lens 9 arranged on the projection optical axis of the polarization beam splitter 8, the objective lens 11, the correction optical system 12, and the ¼ wave plate.\n\n On an optical axis of the reflected light passing through the polarization beam splitter 8, there are provided the focusing lens 19 movable along the reflection light optical axis and an image forming lens 20. The image forming lens 20 focus the reflection light beam on a photoelectric detector 21, which is arranged at a position conjugate to the ocular fundus of the eye 1 under test.\n\n A photodetection signal from the photoelectric detector 21 is stored in a storage unit 27 via a signal processing unit 26. The writing of data from the signal processing unit 26 to the storage unit 27 is controlled by a control unit 28. The control unit 28 has a visual acuity calculating unit and calculates an estimated visual acuity value by required calculation based on the data stored in the storage unit 27. The result of the calculation is displayed on a display unit 29.\n\n Now, description will be given on operation of the optical system.\n\n The focusing lens 19 is positioned at a reference position, and a person with the eye 1 under test is instructed to gaze at the gaze target 15. In this case, the correction optical system 12 is set to a correction amount 0.\n\n With the eye 1 gazing at the gaze target 15, a projecting light beam is projected to the ocular fundus of the eye 1 by the projecting optical system 2, and an image of a point light source is formed on the ocular fundus of the eye 1 under test. Visual light is used for the gaze target 15, and infrared light is used for the projected light beam.\n\n The projected light beam (infrared light) from the light source 5 passes through the projection lens 6 and the half-mirror 7 and reaches the polarization beam splitter 8. At the polarization beam splitter 8, a P linearly polarized light component is reflected. This passes through the relay lens 9, and is projected to the ocular fundus of the eye 1 under test by the objective lens 11, and the correction optical system 12 via the ¼ wave plate 13, and a first target image is formed on the ocular fundus.\n\n When the P linearly polarized light passes through the ¼ wave plate 13, it is turned to a right circularly polarized light. The projected light beam is totally reflected by the ocular fundus of the eye 1, and the totally reflected light beam is turned to a left circularly polarized light when it is reflected by the ocular fundus. Further, when the totally reflected light beam passes through the ¼ wave plate 13, it is turned to an S linearly polarized light, which has a direction of polarization deviated by 90° from a direction of polarization of the P linearly polarized light.\n\n The S linearly polarized light is guided to the polarization beam splitter 8 via the correction optical system 12, the objective lens 11 and the relay lens 9. The polarization beam splitter 8 reflects the P linearly polarized light and allows the S linearly polarized light to pass. Thus, the totally reflected light beam passes through the polarization beam splitter 8 and forms an image as a second target image on the photoelectric detector 21 by the focusing lens 19 and the image forming lens 20.\n\n Incidentally, the projected light beam projected to the ocular fundus of the eye 1 under test is not totally reflected by a surface of the fundus with mirror reflection. A part of the light beam enters into a superficial layer through the surface of the fundus and is reflected with scattering reflection, i.e. the so-called bleeding reflection occurs. When the light beam reflected with scattering reflection is received by the photoelectric detector 21 at the same time as the light beam reflected with mirror reflection, it is turned to noise in light amount intensity distribution of the second target image, and the eye's optical characteristic of the optical system of the eye cannot be accurately measured.\n\n The condition of polarization of the light beam reflected by scattering reflection is in random status. For this reason, when the light beam passes through the ¼ wave plate 13 and is turned to a linearly polarized light, the component matching with the S linearly polarized light is restricted to a limited part. The components other than the components matching with the S linearly polarized light in the light beam reflected by scattering light are reflected by the polarization beam splitter 8. Therefore, the ratio of A to B is negligibly low, where A is the S linearly polarized light component of the light beam reflected with scattering reflection and B is the S linearly polarized light component reflected with mirror reflection at the ocular fundus of the eye 1.\n\n Accordingly, the light received by the photoelectric detector 21 is the reflected light beam with mirror reflection, which substantially removes the reflected light component with scattering reflection. By adding the ¼ wave plate 13 as a component element of the projecting optical system 2 and the photodetection optical system 3, eye's optical characteristic of the optical system of the eye can be accurately measured. The control unit 28 calculates the light amount intensity distribution characteristic, and an optical transmission function of the optical system of the eye based on a photodetection signal from the photoelectric detector 21 and also on the data stored in the storage unit 27. Further, the estimated visual acuity value of the eye 1 under test is calculated according to the optical transmission function.\n\n The optical characteristic of the ocular fundus can be measured by the following procedure:\n\n FIG. 3(A) shows a condition when the light beam is focused on the ocular fundus, and FIG. 3(B) shows a condition when the light beam is not focused on the ocular fundus. Because of the influence of the detailed structure of the ocular fundus as described above, the following relationship exists under both conditions:\n\nP(x,y)&Asteriskpseud; R(x,y)&Asteriskpseud; R(x,y)&Asteriskpseud; P(x,y)=I(x,y)  (1)\n\n where P (x,y) is amplitude transmittance of the eye's optical system of the eye 1 under test, R (x,y) is amplitude transmittance of the visual cells including reflection characteristics at the retinal pigment epithelial layer 32, and I (x,y) is 2-demensional light amount intensity distribution to be measured on a 2-dimensional detector calculated from the photodetection signal from the 2-dimensional detector (photoelectric detector 21).\n\n The mark &Asteriskpseud; indicates convolution integration.\n\n Next, both sides of the equation (1) are processed by Fourier transform.\n\n Here, if it is assumed that p (u,v) is an optical transmission function of the optical system of the eye, r (u,v) is an optical transmission function of the visual cell, and i (u,v) is a 2-dimensional optical transmission function on the 2-dimensional detector, the following relationship exists:\n\n FT[P(x,y)]=p(u,v)\n\n FT[R(x,y)]=r(u,v)\n\n FT[I(x,y)]=i(u,v)\n\n By Fourier transform of the equation (1):\n\np(u,v)×[r(u,v)]2×p(u,v)=i(u,v)  (2)\n\n Therefore, the following equation is approximately established:\n\n[p(u,v)r(u,v)]2=i(u,v)  (3)\n\n Then,\n\np(u,v)r(u,v)={square root}{square root over ([i(u,v)])}  (4)\n\n Because:\n\n|FT[I(x,y)]|=i(u,v)  (5),\n\n the 2-dimensional light amount intensity distribution I (x,y) on the 2-dimensional detector to be measured is processed by Fourier transform. The i (u,v) is obtained by the equation (5). This is substituted in the equation (4), and optical transmission function p (u,v) r (u,v) of the optical system of the eye and the visual cell are calculated.\n\n Hereunder, description will be given below on calculating procedure to estimate the visual acuity value of the eye under test based on the optical transmission function.\n\n The optical transmission function p (u,v) r (u,v) obtained from calculation by the above equation relate to light amount intensity distribution of sinusoidal wave. Frequency characteristics of the eye to a sinusoidal wave chart is the so-called MTF. Using the optical transmission function as a mono-dimensional function, it is determined as:\n\nMTF(u)=p(u)r(u)  (6)\n\n On the other hand, when the visual acuity value is to be judged, in case of a black-white Landolt ring used as a target for visual acuity test, for example, it is measured to which size of the Landolt ring the person under test can recognize the gap. The light amount intensity distribution of square wave corresponds to the size of the gap.\n\n Here, the frequency characteristics (optical transmission function) of the eye to a square wave chart is called square-MTF (square frequency characteristics) (hereinafter referred as “S-MTF”). When visual acuity is measured, measurement is made according to the frequency characteristics with respect to the square wave chart, and the frequency characteristics of the eye to the sinusoidal wave chart (MTF) as given above must be converted to S-MTF.\n\n S-MTF (u) as given above can be calculated from the following equation (7) based on the equation (6) as given above:\n\nS-MTF(u)=4/&pgr;{MTF(u)−(MTF(3u))/3+(MTF(5u)) /5−(MTF(7u))/7+ . . . }  (7)\n\n The result of calculation by the equation (7) is indicated as S-MTF curve shown in FIG. 4. In the diagram, the value of S-MTF is represented on the axis of ordinate, and the frequency u is represented on the axis of abscissa.\n\n When the visual acuity value of the person under test is estimated from the S-MTF curve, it is possible to determine a constant threshold value, and to estimate the visual acuity from an intersection of the threshold value and S-MTF curve. In case the person under test has small S-MTF value, the procedure may be likely to cause error in the estimated value. For this reason, in the present invention, the so-called modulation threshold (hereinafter referred as “MT (u)”), indicating the threshold value of a nervous system in the visual system, is used instead of a constant threshold value.\n\n The value of MT (u) can be experimentally obtained by entering two light beams to the eye to directly form interference fringes on the retina of the fundus and by instructing a person to observe the condition of the interference fringes. The MT (u) thus obtained is sinusoidal wave MT (u). The value used as the threshold value is square-MT to square wave (hereinafter referred as “S-MT”), and this is converted from the sinusoidal wave MT (u).\n\n The conversion can be made by the following equation similarly to the procedure to obtain S-MTF:\n\nS-MT(u)=4/&pgr;{MT(u)−(MT(3u))/3+(MT(5u))/5−(MT(7u))/7+ . . . }  (8)\n\n The S-MT (u) obtained here indicates a boundary value, which can be identified by the eye under test. If the value is higher than S-MT (u), it can be identified by the eye under test. If the value is lower than S-MT (u), it cannot be identified by the eye under test.\n\n Further, the S-MT (u) generally varies according to age of the person under test. There are prepared the function in the teen-agers defined as S-MT10 (u), that of the twenties defined as S-MT20 (u), and that of the thirties as S-MT30 (u) . . . Using the modulation-threshold (S-MTa (u)) corresponding to the age of the person under test, S-MTa(u) is superimposed on the graph shown in FIG. 4 which shows S-MTF of the person under test. Then, the frequency u corresponding to the intersection of S-MTF curve and S-MTa (u) curve is calculated.\n\n For instance, in case the person under test is in the teen-age, the frequency u10 corresponding to the intersection of S-MT10 and S-MTF in FIG. 4 is obtained.\n\n The relation between the so-called decimal visual acuity value (D.V.A.) and u is given by: D.V.A.=u/100. Based on the frequency u10 obtained above, the decimal visual acuity value (D.V.A.) can be obtained. Visual acuity value is obtained by using the threshold value, which corresponds to MT (u) of the age of the person under test, and the error can be reduced in the estimated value.\n\n Thus, without relying on the answer from the person under test, the visual acuity value of the eye under test can be measured.\n\n In the present embodiment, the position of the focusing lens 19 is regarded as the reference position, and the correction optical system 12 is set to correction amount 0, and the measurement is performed. Based on the result of the measurement, the visual acuity value of the naked eye of the person under test is estimated. The present invention is not limited to this case, and a visual acuity value after correcting the refraction of a certain amount can be also estimated if the correction optical system is adjusted or the focusing lens is moved, and measurement is made after correction of refraction of a certain amount and by performing similar calculation.\n\n The eye's optical characteristic measuring system of the present invention comprises a target projecting system for projecting a target image on an ocular fundus of an eye under test, a photodetection optical system for guiding the target image to a photoelectric detector, a calculating unit for calculating an optical transmission function of the eye under test based on light amount intensity distribution of the target image detected by the photoelectric detector, and an arithmetic unit for calculating visual acuity of a person under test from intersection of the optical transmission function and a predetermined threshold value. As a result, visual acuity of an eye under test can be accurately estimated without relying on the so-called subjective optometric procedure, by which visual acuity is measured according to the answer from the person under test after showing various sizes of targets for visual acuity test. The measurement can be accomplished by simply projecting a predetermined target image to the ocular fundus of the eye under test and by measuring light amount intensity distribution of the target image.\n\n## Claims\n\n1. An eye's optical characteristic measuring system, comprising a target projecting system for protecting a target image on an ocular fundus of an eye under test, a photodetection optical system for guiding the target image to a photoelectric detector, a calculating unit for calculating an optical transmission function of the eye under test based on light amount intensity distribution of the target image detected by said photoelectric detector, and an arithmetic unit for calculating visual acuity of a person under test from intersection of the optical transmission function and a predetermined threshold value.\n\n2. An eye's optical characteristic measuring system according to claim 1, wherein said optical transmission function is square wave frequency characteristics.\n\n3. An eye's optical characteristic measuring system according to claim 1, wherein said threshold value is a modulation threshold.\n\n4. An eye's optical characteristic measuring system according to claim 3, wherein a plurality of said modulation threshold are prepared to correspond to age of the persons under test, and a modulation threshold suitable for each person under test is used.\n\nPatent History\nPublication number: 20030156256\nType: Application\nFiled: Feb 11, 2003\nPublication Date: Aug 21, 2003\nInventors: Gaku Takeuchi (Tokyo-to), Katsuhiko Kobayashi (Tokyo-to), Masahiro Shibutani (Tokyo-to), Yumi Kubotera (Tokyo-to)\nApplication Number: 10364771\nClassifications\nCurrent U.S. Class: Objective Type (351/205)\nInternational Classification: A61B003/10;" ]

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[ "## MET – METABOLIC EQUIVALENT & MATHS\n\nThe MET concept represents a simple, practical, and easily understood procedure for expressing the energy cost of physical activities as a multiple of the resting metabolic rate. The cells in your muscles use oxygen to help create the energy needed to move your muscles. One MET is approximately 3.5 milliliters of oxygen consumed per kilogram (kg) of body weight per minute.\n\nCalories burned in one hour= METs x Weight (in Kg). For  example: If your weight is 70 kg  and running 12kmph (means with a pace of 5 min/km) your MET value is about 11.5 from Table.And  Calorie Expenditure = 11,5 x 70= 805 calorie\n\nEnergy expenditure may differ from person to person based on several factors, including  age and fitness level. For example, a young athlete who exercises daily won’t need to expend the same amount of energy during a brisk walk as an older, sedentary person.\n\nTable 1: MET-Values\n\nExample: Comparing two person want to loose weight and spend only half an hour for physical activities, five days a week. One prefer walking at speed of  4 kmph, and other can run at a speed of 12.8 kmph for half an hour. Both has the same weight:  80 kg.\n\nCalorie Expenditure for half an hour:the walker =  3 x 0,5 x 80=120 calories and the runner = 11,8 x 0,5 x 80=472 calories\n\nFor one  week-5 days  exercise,   annually calorie expenditures.\n\nTable 2: Calorie Expenditure runner vs walker\n\nIt is well known that one kg of fat=7.700 calories. But according to the recent research: For one kg of fat =15.400 calories. Let’s take the worst case scenario for those who want to loose weight. For one year of activity of 5 days a week and 30 minutes per day the walker might loose 2 Kg    versus the runner 8 Kg , 1 over 4.\n\nOf course these are straight mathematics formula and human does not fit the formulas one-to-one; but these calculation might give a rough  idea.\n\nAccording to the latest trend HIIT- High Intensity Interval Training, hill running  type exercises even offer more benefit for the same amount of time spent.\n\nBut the trick here is that all those calorie expenditure versus weight lost is valid if only your other daily calorie expenditure be the same amount your calorie intake, so the numbers emanate from above calculation would be a bonus calorie expenditure which lead weight lost.\n\nYou may calculate your daily calorie expenditure from MET value times your weight for one hour activity or laziness, and your calorie intake from what you eat and diet-calorie list on internet for remaining 23 and half hours.\n\nEverything hides in the numbers and maths. Very popular elemantary school math water tank problem: A water tank can be filled by tap A in 3 hours and by tap B in 5 hours. When the tank is full, it can be drained by tap C in 4 hours. if the tank is initially empty and all three pipes are open, how many hours will it take to fill up the tank?\n\nLet’s change the famous pool problem in elementary school a little bit. A pool can be filled in “x-hours” with tap A and “in y-hours” with tap B alone; It can be emptied “in z-hours” with a C-tap located under the pool. If all three taps are opened at the same time while the pool is initially full, will the pool fill or empty? In what time frame?\n\nIf we accept a full pool of 1 unit and open all taps at the same time, the formula for the occupancy load of the pool after a certain period of time:\n\nPool’s occupancy load = 1+ (1/tA+1/tB)-1/tC\n\nIf the emptying time of the C-tap is 1 hour and the filling time of A and B alone is 2 hours, and everything else is kept constant, the occupancy state of the pool remains the same, which is 1. If the C tap is opened to discharge more water, that is, if the discharge time is shortened, or if the A and B-taps are throttled, that is, the filling time is prolonged, the pool starts to empty. If the opposite happens, the pool will overflow. Here, the A-tap represents our daily normal diet, the B-tap represents the snacks in between, and the C-tap represents calorie expenditure, such as exercise, jogging, etc.\n\nCalorie intake is the number of calories you eat and drink each day, and calorie expenditure is your resting metabolic rate plus spending with any physical activity. If the calorie intake exceeds the calorie burning, the excess calories are deposited in your body as fat and body weight increases. If calorie intake is less than caloric expenditure, your body fat stores provide the necessary calories and weight loss occurs.\n\nOf course, not every pool can be expected to fit exactly mathematical formulas presented here. Some may be leaking, the exact volume may be different from the supplied, algae, some rust and dirt may obstruct the flow of water, etc.\n\nHowever, if we can consume more calories than we take in a certain period, just like in the pool problem, we are losing weight. Of course, this rate may occur at different rates depending on personal and genetic characteristics and metabolism.\n\nSo for example say  if you want to lose 6 Kg: For the worst case which is “for one kg of fat =15.400 calories”,  you need to 6×15.000=90.000 calorie deficit approximately. If you are planning 6 month for this, you should reach 500 calorie deficit per day; by reducing calorie intake, increasing expenditure, or both. A calorie deficit of 500 calories or more per day is a common initial goal for weight loss for adults." ]

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[ "top of page\n\n### Program Highlights\n\nWorkshop_highlights\n\no Basic principles of traffic signal control and traffic flow theory, and limitations of current practice\n\no Signal control logic including time budget and delay estimation\n\no A brief introduction of cell transmission model (CTM) and microsimulation software VISSIM that are capable of capturing dynamic traffic, spatial queuing, and other properties for deriving performance indices, such as delay, queue length, and vehicle throughput.\n\no DISCO (Dynamic Intersection Signal Control Optimisation) software demonstration for signal control optimization and coordination between multiple junctions\n\no VISSIM micro-simulation software demonstration", null, "", null, "### DISCO Software demonstration with hands-on example\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)\n\na. Simulation and Visualisation\n\nb. Optimization (Single junction to multiple junctions)\n\n### VISSIM software demonstration\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)\n\na. Demonstration of VISSIM inputs and outputs\n\nb. Visualization\n\n### Basic principles and limitations of current practice\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)\n\n### Traffic Signal Control fundamentals\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)\n\na. Time budget, delay vs throughput (long cycle vs short cycle)\n\nb. Delay estimation: D/D/1 formula, Webster\n\nc. Optimisation for minimum delay using D/D/1\n\n### Advanced model for traffic control: Cell Transmission Model (CTM)\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)\n\na. Signalised Intersection\n\ni. Webster Formula and Reserve Capacity (RC)\n\ni. Design Flow to Capacity (DFC)\n\nii. Lane Flow Diagram (LFD)" ]

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[ "# matlab julia python cheat sheet\n\nA - 2M> A * 2[,1] [,2] [,3][1,] 2 4 6[2,] 8 10 12[3,] 14 A .+ AM> multivariate normaldistribution given mean and covariance Array{Int64,2}:1 4 7 2 5 8 3 6 9, M> Noteworthy differences from C/C++. c = [a' b']c =   1   4   2   – The cheat sheet for MATLAB, Python NumPy, R, and Julia. A / A, R> While Julia can also be used as an interpreted language with dynamic types from the command line, it aims for high-performance in scientific computing that is superior to the other dynamic programming languages for technical computing thanks to its LLVM-based just-in-time (JIT) compiler. A = matrix(1:6,nrow=2,byrow=T)R> Matrices (or multidimensional arrays) are not only presenting the fundamental elements of many algebraic equations that are used in many popular fields, such as pattern classification, machine learning, data mining, and math and engineering in general. MATLAB (stands for MATrix LABoratory) is the name of an application and language that was developed by MathWorks back in 1984. This is indeed a huge distinction—for some, a dispositive one–but I want to consider the technical merits. A .+ A; J> total_elements=length(A)9J>B=reshape(A,1,total_elements)1x9 7]])P> np.c_[a,b]array([[1, 4],       [2, 0.38959   0.69911   0.15624   0.65637, P> A[,1] [,2] [,3][1,] 6 1 1[2,] 4 -2 5[3,] 2 8 7R> a = matrix(c(1,2,3), nrow=3, byrow=T)R> A = matrix(1:9, nrow=3, byrow=T)R> 8 9# use '.==' for# element-wise checkJ> install.packages('MASS')
R> total_elements = dim(A) * dim(A)R> Matlab-Julia-Python cheat sheet. b = np.array([1, 2, 3])P> b=vec([1 2 3])3-element Array{Int64,1}:123, Reshaping  4   5   6M> 3   4   5   9   7   8   A = np.array([[4, 7], [2, 6]])P> 64 81

R> 18M> 0.70711   0.70711   0.70711eig_val eye(3)3x3 Array{Float64,2}:1.0 0.0 0.00.0 1.0 0.00.0 Alex Rogozhnikov, Log-likelihood benchmark, September 2015. (last updated: June 22, 2018) Libraries such as NumPy and matplotlib provide Python with matrix operations and plotting. A[:,1:2] 3x2 Array{Int64,2}:1 24 57 8, Extracting A / 2, # [102, 126, 150]]), R> 9M> MATLAB. = [1 2 3; 4 5 6; 7 8 9]M> A ^ 23x3 Array{Int64,2}:30 36 4266 81 96102 126 as column vector
R> cov = [2 0; 0 2]cov =   2   0   0 Although similar tools exist for other languages, I found myself to be most productive doing my research and data analyses in IPython notebooks. t(A)[,1] [,2] [,3][1,] 1 4 7[2,] 2 5 8[3,] 3 6 9, J> 1, J> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])P> 7M> A + 2M> shortcut:# A.reshape(1,-1)P> eye(3)ans =Diagonal Matrix   1   0   A - AP> ],       [16, 25, 36],       [49, 64, 81]])P> http://octave.sourceforge.net/packages.php% pkg install % A=[1 2 3; 4 5 6; 7 8 9]3x3 Array{Int64,2}:1 2 34 5 67 ])P> C = rbind(A,B)R> = [1 2 3; 4 5 6; 7 8 9]M> np.diag(a)array([[1, 0, 0],       [0, 0.7751204[2,] 0.3439412 0.5261893[3,] 0.2273177 0.223438, J> 3   4   5   6   7   8   You signed in with another tab or window. 3

R> t(b %*% A)[,1][1,] 14[2,] 32[3,] 50, J> A = matrix(1:9,nrow=3,byrow=T)

R>     ])# 1st 2 columnsP> 1, P> b = np.array([ , , ])P> -0.20000   0.40000, P> It is the example of high-level scripting and also named as 4th generation language. Develop Machine Learning project with MATLAB, Simulink, … np.ones((3,2))array([[ 1.,  1. 7.5000e-03   1.7500e-03   7.5000e-03   All four languages, MATLAB/Octave, Python, R, and Julia are dynamically typed, have a command line interface for the interpreter, and come with great number of additional and useful libraries to support scientific and technical computing. (2012), “Julia: A fast dynamic language for technical computing”. At its core, this article is about a simple cheat sheet for basic operations on numeric matrices, which can be very useful if you working and experimenting with some of the most popular languages that are used for scientific computing, statistics, and data analysis. a=[1; 2; 3]3-element Array{Int64,1}: 123, P> Octave’s syntax is mostly compatible with MATLAB syntax, so it provides a short learning curve for MATLAB developers who want to use open-source software. A = [3 1; 1 3]A =   3   1   1 A=[6 1 1; 4 -2 5; 2 8 7]3x3 Array{Int64,2}:6 1 14 -2 A + 2P> b=[4 5 6];J> to power n(here: matrix-matrix multiplication with A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])# 1st rowP> 0.7071068 -0.7071068[2,] 0.7071068 0.7071068, J> Btw., if someone is interested, I made a cheat sheet for Python vs. R. vs. Julia vs. Matplab some time ago. 1-D # arrays, R> x2 = matrix(c(2, 2.1, 2, 2.1, 2.2), ncol=5)R> A = [1 2 3; 4 5 6; 7 8 9]% 1st rowM> It is also worth mentioning that MATLAB is the only language in this cheat sheet which is not free and open-sourced. matlab-to-julia Translates MATLAB source code into Julia. 0.0 1.0, M> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])P> [matlab logo](../Images/matcheat_matlab_logo.png), ! http://sebastianraschka.com/Articles/2014_matlab_vs_numpy.html, !  [-1.37031244, -1.18408792]]), # It provides a high-performance multidimensional array object, and tools for working with these arrays. a = [1 2 3]M> Most people recommend the usage of the NumPy array type over NumPy matrices, since arrays are what most of the NumPy functions return. A[,1] [,2] [,3][1,] 1 2 3[2,] 4 5 9[3,] 7 8 [Julia benchmark](../Images/matcheat_julia_benchmark.png), http://octave.sourceforge.net/packages.php, https://github.com/JuliaStats/Distributions.jl. requires the ‘mass’ package
R> MIT 2007 basic functions Matlab cheat sheet; Statistics and machine learning Matlab cheat sheet; Cheat sheets for Cross Reference between languages. A = np.array([ [1,2,3], [4,5,6] ])P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])P> vector
R> np.cov([x1, x2, x3])Array([[ 0.025  ,  0.0075 , A[1:2,][,1] [,2] [,3][1,] 1 2 3[2,] 4 5 6, J> Python For Data Science Cheat Sheet NumPy Basics Learn Python for Data Science Interactively at www.DataCamp.com NumPy DataCamp Learn Python for Data Science Interactively The NumPy library is the core library for scienti c computing in Python. Aarray([[ 6,  1,  1],       diagm(a)3x3 Array{Int64,2}:1 0 00 2 00 0 3, Getting A = [1 2 3; 4 5 6; 7 8 9]% 1st columnM> Numeric matrix manipulation - The cheat sheet for MATLAB, Python NumPy, R, and Julia. columnarJ> rand(3,2)3x2 Array{Float64,2}:0.36882 0.2677250.571856 1 1, J>   [ 0.70710678,  0.70710678]]), R> 0.1303697[6,] 0.8413189 -0.1623758[7,] -1.0495466 A = matrix(c(4,7,2,6), nrow=2, byrow=T)R> ],     See this reference on NumPy and info on matplotlib (links open in new tab). np.zeros((3,2))array([[ 0.,  0. a=[1 2 3];J> A = np.array([[1,2,3],[4,5,6],[7,8,9]])P> np.linalg.det(A)-306.0, R> A[0,:]array([1, 2, 3])# 1st 2 rowsP> B = [7 8 9; 10 11 12]M> t(b)[,1][1,] 1[2,] 2[3,] 3, J> x2=[2. Julia, MATLAB, Numpy Cheat Sheet October 19, 2016 October 19, 2016 I mostly use Python for my data analysis, but I’ve been playing around with Julia some, and I find these kinds of side-by-side comparisons to be quite valuable! A * Aarray([[ 1,  4,  9],       and Edelman, A. det(A)-306.0, M> A = matrix(1:9,nrow=3,byrow=T)

# 1st row

R> A .- A; J>   6M> ],     cov([x1 x2 x3])3x3 Array{Float64,2}:0.025 0.0075 These cheat sheets let you find just the right command for the most common tasks in your workflow: Automated Machine Learning (AutoML): automate difficult and iterative steps of your model building; MATLAB Live Editor: create an executable notebook with live scripts; Importing and Exporting Data: read and write data in many forms 4   5   6, P> size(A)(2,3), M> A * 2ans =    2    4    6  size(A)ans =   2   3, P> A=[1 2 3; 4 5 6; 7 8 9]3x3 Array{Int64,2}:1 2 34 5 67 rand( MvNormal(mean, cov), 5)2x5 Array{Float64,2}:-0.527634 A .+ 2;J> 4   5   6, P> note that numpy doesn't have # explicit “row-vectors”, but 42    66    81    96   a = np.array([1,2,3])P>   5   8   3   6   9, P> = It allows me to easily combine Python code (sometimes optimized by compiling it via the Cython C-Extension or the just-in-time (JIT) Numba compiler if speed is a concern) with different libraries from the Scipy stack including matplotlib for inline data visualization (you can find some of my example benchmarks in this GitHub repository). A ./ A; Matrix pkg load statisticsM> 8],       [3, 6, 9]]), R> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])P> x3 = np.array([ 0.6, 0.59, 0.58, 0.62, 0.63])P> a = np.array([1,2,3])P> [7, 8, 9]]), R> C = [A; B]    1    2    3  Credits This cheat sheet … Please enter your username or email address to reset your password. Tags: Cheat Sheet, Data Science, Python, R, SQL. A = matrix(1:9, nrow=3, byrow=T)
R> x2 = [2.0000 2.1000 2.0000 2.1000 2.2000]'M> Such multidimensional data structures are also very powerful performance-wise thanks to the concept of automatic vectorization: instead of the individual and sequential processing of operations on scalars in loop-structures, the whole computation can be parallelized in order to make optimal use of modern computer architectures. A = matrix(1:6,nrow=2,byrow=T)R> C = np.concatenate((A, B), axis=0)P> A(1:2,:)ans =   1   2   3   requires the Distributions package from rowsM> Matrices(here: 3x3 matrix to row vector), M> mean=[0., 0. b=[1 2 3] 1x3 Array{Int64,2}:1 2 3# note that this A = matrix(c(3,1,1,3), ncol=2)R> -0.4161082[8,] -1.3236339 0.7755572[9,] 0.2771013 A = matrix(1:9,nrow=3,byrow=T)

R>   0.686977, P> A(A(:,3) == 9,:)ans =   4   5   9   squared), M> A 7% 1st 2 columnsM> A[,1] 1 4 7

# 1st 2 columns
R> Python: Cheat sheet (free PDF) ... the mathematical prowess of MatLab, ... Python was named as the number one language that developers would be using if they weren't using Julia, with Python … This MATLAB-to-Julia translator begins to approach the problem starting with MATLAB, which is syntactically close to Julia. 8 9J> det(A) -306, J> A . [ 1.,  1. Hot news about happenings in NIGERIA generally with special focus on political developments and News around the world. b = matrix(c(1,2,3), ncol=3)R> A = matrix(1:9,nrow=3,byrow=T)


# 1st column as row covariances of the means of x1, x2, and x3), M> (eig_vec,eig_val)=eig(a)([2.0,4.0],2x2 A = [4 7; 2 6]A =   4   7   2 b = np.array([4,5,6])P> 0.; 0. Python NumPy is my personal favorite since I am a big fan of the Python programming language. 5   7   8, P> c = [a; b]c =   1   2   3   2. A.shape(2, 3), R> cov=[2. A * A[,1] [,2] [,3][1,] 1 4 9[2,] 16 25 36[3,] 49 0   0   1   0   0   0   np.random.multivariate_normal(mean, cov, 5)Array([[ 11 12, M> B = np.array([[7, 8, 9],[10,11,12]])P> In this sense, GNU Octave has the same philosophical advantages that Python has around code reproducibility and access to the software. r/compsci: Computer Science Theory and Application. 10 11 12, J> But in context of scientific computing, they also come in very handy for managing and storing data in an more organized tabular form. 8    9   10   11   12, P> vector)P>      [10, 11, 12]]), R>  0.00135,  0.00043]]), R> Keep this #Python Cheat Sheet handy when learning to code; Is #BigData The Most Hyped Technology Ever?     [ 0.51615758,  0.64593471],     Python's NumPy library also has a dedicated \"matrix\" type with a syntax that is a little bit closer to the MATLAB matrix: For example, the \" * \" operator would perform a matrix-matrix multiplication of NumPy matrices - same operator performs element-wise multiplication on NumPy arrays. A[A[:,2] == 9]array([[4, 5, 9],       This Wikibook is a place to capture information that could be helpful for people interested in migrating code from MATLAB™ to Julia, and also those who are familiar with MATLAB and would like to learn Julia. A[,1] [,2] [,3][1,] 1 2 3[2,] 4 5 6, J> Matrix functions MATLAB/Octave Python NumPy, R, Julia; Related: 50+ Data Science and Machine Learning Cheat Sheets; Guide to Data Science Cheat Sheets; Top 20 R packages by popularity = A %*% A[,1] [,2] [,3][1,] 30 36 42[2,] 66 81 96[3,] 5],       [3, 6]])P> GitHub Gist: instantly share code, notes, and snippets. it in Octave:% download the package from: % (Source: http://julialang.org/benchmarks/, with permission from the copyright holder), If you are interested in downloading this cheat sheet table for your references, you can find it here on GitHub, M> A = [1 2 3; 4 5 6; 7 8 9]A =   1   2   b = matrix(1:3, nrow=3)

R> = [1 2 3; 4 5 6; 7 8 9]M> Explore our solutions on Machine Learning with MATLAB [Cheat sheet] MATLAB basic functions reference. A * bans =   14   32   save filename Saves all variables currently in workspace to file filename.mat. A[,1] [,2][1,] 3 1[2,] 1 3R> A %^% 2[,1] [,2] [,3][1,] 30 66 102[2,] 36 81 126[3,]  [-0.2, 0.4]]), R> Some of the fields that could most benefit from parallelization primarily use programming languages that were not designed with parallel computing in mind. diag(a)ans =Diagonal Matrix   1   0   16 18. Using such a complex environment can prove daunting at first, but this Cheat Sheet can help: Get to know common […] View All Result .   [16, 25, 36],       [49, 64, 81]]), R> Although R has great in-built functions for performing all sorts statistics, as well as a plethora of freely available libraries developed by the large R community, I often hear people complaining about its rather unintuitive syntax. 150, M> A     ~/Desktop/statistics-1.2.3.tar.gzM> 3, P> b = [1 2 3]
M> A = np.array([[1, 2, 3], [4, 5, 6]])P> Comment block %{Comment block %} # Block # comment # following PEP8 #= Comment block =# For loop. Personally, I haven't used Julia that extensively, yet, but there are some exciting benchmarks that look very promising: Bezanson, J., Karpinski, S., Shah, V.B.   [ 0.,  0.  [-2.11810813, 1.45784216],       = 0, variance = 2), % This cheat sheet provides the equivalents for four different languages – MATLAB/Octave, Python and NumPy, R, and Julia. A=[1 2 3; 4 5 6; 7 8 9];#1st columnJ> mvnrnd(mean,cov,5)   2.480150  -0.559906  1   4  -2   5   2   8   Aarray([[1, 2, 3],       [4, 5, 6],   Aarray([[3, 1],       [1, 3]])P> A * A3x3 Array{Int64,2}:30 36 4266 81 96102 126 Matlab Cheat sheet. 0   0   0, P> Sebastian Raschka, Numeric matrix manipulation - The cheat sheet for MATLAB, Python Nympy, R and Julia… np.linalg.matrix_power(A,2)array([[ 30,  36,  A[:,1] 3-element Array{Int64,1}:147#1st 2 mat.or.vec(3, 2) + 1[,1] [,2][1,] 1 1[2,] 1 1[3,] A[,1:2][,1] [,2][1,] 1 2[2,] 4 5[3,] 7 8, J> Python. [ 8, 10, 12],       [14, 16, 18]])P> Combined with interactive notebook interfaces or dynamic report generation engines (MuPAD for MATLAB, IPython Notebook for Python, knitr for R, and IJulia for Julia based on IPython Notebook) data analysis and documentation has never been easier. value 9 in column 3), M> A=[1 2 3; 4 5 6; 7 8 9]3x3 Array{Int64,2}:1 2 34 5 67 A = [6 1 1; 4 -2 5; 2 8 7]A =   6   1   102   126   150, P> A - 2P> A(:,1)ans =   1   4   A + AR> Noteworthy differences from R. Noteworthy differences from Python. =Diagonal Matrix   2   0   0     ~/Desktop/io-2.0.2.tar.gz  % pkg install % A ./ A, P> ]2-element Array{Float64,1}:0.00.0J> MATLAB commands in numerical Python (NumPy) 3 Vidar Bronken Gundersen /mathesaurus.sf.net 2.5 Round off Desc. This Python Cheat Sheet will guide you to interactive plotting and statistical charts with Bokeh. = [1 2 3; 4 5 6; 7 8 9]M> ones(3,2)ans =   1   1   1   0.0, M> mean = [0 0]M> is a 2D array. x3=[0.6 .59 .58 .62 .63]';J> 3   4   5   6   7   8   2, 0],       [0, 0, 3]]), R> A[1,:] 1x3 Array{Int64,2}:1 2 3#1st 2 rowsJ> [back to article] The Matrix Cheatsheet by Sebastian Raschka is licensed under a Creative Commons Attribution 4.0 International License. A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])P>   2M> cov( [x1,x2,x3] )ans =   2.5000e-02   Joy as Nigerian man gets job in America after bagging his master’s degree in this US school (photos). MATLAB, unlike Python and Julia, is neither beer-free nor speech-free. A=[1 2 3; 4 5 6; 7 8 9]; #semicolon suppresses output#1st x3 = matrix(c(0.6, 0.59, 0.58, 0.62, 0.63), ncol=5)

R> # vectors in Julia are columns, M> Noteworthy differences from Matlab.   4, P> b = [4 5 6]M> A = [1 2 3; 4 5 6]A =   1   2   3  Python Bokeh Cheat Sheet is a free additional material for Interactive Data Visualization with Bokeh Course and is a handy one-page reference for those who need an extra push to get started with Bokeh.. One of its strengths is the variety of different and highly optimized \"toolboxes\" (including very powerful functions for image and other signal processing task), which makes suitable for tackling basically every possible science and engineering task. A .- 2;J> C=[A; B]4x3 Array{Int64,2}:1 2 34 5 67 8 910 A = [1 2 3; 4 5 6; 7 8 9]M> The Mandalorian season 2 episode 7 recap: Mando goes undercover – Bioreports, Virgin Galactic aborts first powered-flight attempt from Spaceport America – Bioreports, Delta police nabs three suspects, PDP chief over communal clash, NPC kicks off census enumeration exercise in Katsina, Katsina compiles register of CBOs, CSOs and NGOS, Police burnt house, abducted two friends in Abia, victim tells panel, 9 great reads from Bioreports this week – Bioreports, HomePod Mini vs. Echo Dot vs. Nest Mini: Picking the best mini smart speaker – Bioreports, Solar eclipse 2020: A history of eclipses and bizarre responses to them – Bioreports, Pfizer-BioNTech Covid-19 Vaccines Are Prepped for Shipment, NFL Ratings Drop Leaves Networks Scrambling to Make Advertisers Whole, AstraZeneca Agrees to Buy Alexion for \\$39 Billion, The Best-Managed Companies of 2020—and How They Got That Way, Despite his very little beginning, this man succeeds, becomes a lawyer, check out his throwback photo as poor kid, In the spirit of Christmas, kind Nigerian man offers to distribute free chicken to people of these areas, many react, 3 years after starting business, man expands, shares photos of how his company grew, 28-year-old lady who hawked to send herself to school now pursues PhD in US after obtaining 2 master’s degrees, He’s not coming back home! A + AP> It is meant to supplement existing resources, for instance the noteworthy differences from other languagespage from the Julia manual. rows and columns by criteria(here: get rows that have A=[1 2 3; 4 5 6; 7 8 9];J> A .- AM> ]]), R> -0.1882706[2,] 0.8496822 -0.7889329[3,] -0.1564171 0.00175 0.00135 0.00043, J> A=[3 1; 1 3]2x2 Array{Int64,2}:3 11 3J> If you look for further online resources, please ensure that they are for Julia … A=[1 2 3; 4 5 6; 7 8 9];# elementwise operatorJ> A = matrix(1:9, nrow=3, byrow=T)
R> [eig_vec,eig_val] = eig(A)eig_vec =  -0.70711   J> A = [1 2 3; 4 5 6]M> A[:,]array([,       ,   Aarray([[1, 2, 3],       [4, 5, Comparing Numpy and Matlab array summation speed (2) I recently converted a MATLAB script to Python with Numpy, and found that it ran significantly slower.     [102, 126, 150]]), R> A ^ 2[,1] [,2] [,3][1,] 1 4 9[2,] 16 25 36[3,] 49 install.packages('expm')
R> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])P> [python logo](../Images/matcheat_numpy_logo.png), ! Julia. a[,1][1,] 1[2,] 2[3,] 3, J> b = np.array([1,2,3])P> using DistributionsJ> Comment one line % This is a comment # This is a comment # This is a comment. For a given MA… save filename x y z Saves x, y, and z to file filename.mat. b = b[np.newaxis].T# alternatively # b = A[1,] 1 2 3

# 1st 2 rows

R> I expected similar performance, so I'm wondering if I'm doing something wrong. Barray([[1, 2, 3, 4, 5, 6, 7, 8, 9]]), R> library(MASS)
R> t(A[,1])[,1] [,2] [,3][1,] 1 4 7

# 1st column x1 = matrix(c(4, 4.2, 3.9, 4.3, 4.1), ncol=5)R> 7 8 9, J> With its first release in 2012, Julia is by far the youngest of the programming languages mentioned in this article. python for matlab users cheat sheet . A * Aans =    30    36    A .^ 23x3 Array{Int64,2}:1 4 916 25 3649 64 81, Matrix 8 9, P> 0.02500 0.00750 0.00175[2,] 0.00750 0.00700 0.00135[3,] Vice versa, the \".dot()\" method is used for element-wise multiplication of NumPy matrices, wheras the equivalent operation would for NumPy arrays would be achieved via the \" * \"-operator. 42    66    81    96   Matplotlib Cheat Sheet: Plotting in Python This Matplotlib cheat sheet introduces you to the basics that you need to plot your data with Python and includes code samples. A_inv = inv(A)A_inv =   0.60000  -0.70000    4    5    6    7    A[,1] [,2][1,] 4 7[2,] 2 6R> A[0,0]1, R> 5 8 3 6 9, J> 64   81M> ],       [ 0.,  Jean Francois Puget, A Speed Comparison Of C, Julia, Python, Numba, and Cython on LU Factorization, January 2016. However this wiki intends to be more comprehensive, and to be structured in such a way as to make it easy for one to find answers to questions like: 1. ],       [ 1.,  1. Cheat sheet: Using MATLAB & Python together Complete the form to get the free e-Book.   [ 0.01067605,  0.09692771]]), R> A = matrix(c(1,2,3,4,5,6,7,8,9),nrow=3,byrow=T)
# A(:,1:2)ans =   1   2   4   the dimensionof a matrix(here: 2D, rows x cols), M> A[0:2,:]array([[1, 2, 3], [4, 5, 6]]), R> MATLAB Cheat Sheet Basic Commands % Indicates rest of line is commented out.     [7, 8, 9]]), R> 5   3   6M> [python logo](../Images/matcheat_julia_logo.png), ! 1.55432624, -1.17972629],       Like the other languages, which will be covered in this article, it has cross-platform support and is using dynamic types, which allows for a convenient interface, but can also be quite \"memory hungry\" for computations on large data sets. np.eye(3)array([[ 1.,  0.,  0. 3.055316  -0.985215  -0.990936   1.122528 64 81, J> Aarray([[1, 2, 3],       [4, 5, 9],   =   1   4   7   2   5   8   30-Day Trial . 0 3, J> np.array([1,2,3]).reshape(1,3), R> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])P> A=[1 2 3; 4 5 6]2x3 Array{Int64,2}:1 2 34 5 6J> 0.6015240.848084 0.858935, M> Alternative data structures: NumPy matrices vs. NumPy arrays. 7   8   9, P> 2.1 2.2]';J> A[ A[:,3] .==9, :] 2x3 Array{Int64,2}:4 5 97 8 9, M> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])P> https://github.com/JuliaStats/Distributions.jlJ> diag(1:3)[,1] [,2] [,3][1,] 1 0 0[2,] 0 2 0[3,] 0 0.370725 -0.761928 -3.91747 1.47516-0.448821 2.21904 2.24561 1.4900494[10,] -1.3536268 0.2338913, # -2.933047   0.560212   0.098206   requires statistics toolbox package% how to install and load     [ 66,  81,  96],       and eigenvalues, M> rowJ> a = matrix(1:3, ncol=3)R> 0.,  0.,  1. solve(A)[,1] [,2][1,] 0.6 -0.7[2,] -0.2 0.4, J> Let us look at the differences between Python and Matlab: MATLAB is the programming language and it is the part of commercial MATLAB software that is often employed in research and industry. ones(3,2)3x2 Array{Float64,2}:1.0 1.01.0 1.01.0 cov(matrix(c(x1, x2, x3), ncol=3))[,1] [,2] [,3][1,] 0   0   2   0   0   0   J>     [7, 8, 9]])P> = np.array([[6,1,1],[4,-2,5],[2,8,7]])P> M atlab > M atlab vs. other languages > Comparison of Python and MATLAB . A=[1 2 3; 4 5 6; 7 8 9];J> Aarray([[1, 2, 3],       [4, 5, 9],   A=[1 2 3; 4 5 6; 7 8 9];J> x2 = np.array([ 2, 2.1, 2, 2.1, 2.2])P> e.g., A += A instead of # A = A + A, R> equivalent to
# A = matrix(1:9,nrow=3,byrow=T)

R> Cannot retrieve contributors at this time. A=[4 7; 2 6]2x2 Array{Int64,2}:4 72 6J> a Mando and Boba Fett (who's cleaned up his armor) make an excellent team, even if they aren't together much in... Affirm Holdings Inc. is postponing its initial public offering, according to people familiar with the matter, the second company in... - A young boy from the Bono Region of Ghana named Prince Benson Mankotam has succeeded in becoming a lawyer... © 2020 Bioreports - Hot news about happenings in NIGERIA generally with special focus on political developments and News around the world. 9

R> B=[7 8 9; 10 11 12];J> rbind(A,B)[,1] [,2] [,3][1,] 1 2 3[2,] 4 5 6, J>   [ 0.,  1.,  0.     ]), R> A = matrix(c(1,2,3,4,5,9,7,8,9),nrow=3,byrow=T)

R> eig_valarray([ 4.,  2. = [1 2 3; 4 5 6; 7 8 9]M> 0.001750.0075 0.007 0.001350.00175 0.00135 0.00043, Calculating eigenvectors If used within matrix definitions it indicates the end of a row. A'3x3 Array{Int64,2}:1 4 72 5 83 6 9, M> A = matrix(1:9, ncol=3)R> 6]])P> x1 = [4.0000 4.2000 3.9000 4.3000 4.1000]’M> Key Differences Between Python and Matlab. A=[1 2 3; 4 5 9; 7 8 9]3x3 Array{Int64,2}:1 2 34 5 97 matlab/Octave Python R Round round(a) around(a) or math.round(a) round(a) A[1,1] 1, J> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])# 1st column Before we jump to the actual cheat sheet, I wanted to give you at least a brief overview of the different languages that we are dealing with. [matlab logo](../Images/matcheat_octave_logo.png), ! A.Tarray([[1, 4, 7],       [2, 5, for i = 1: N % do something end. A = matrix(c(6,1,1,4,-2,5,2,8,7), nrow=3, byrow=T)R> eig_val, eig_vec = np.linalg.eig(A)P> A = np.array([ [1,2,3], [4,5,9], [7,8,9]])P> A[1,1]1, M> 42 96 150, J> Since it makes use of pre-compiled C code for operations on its \"ndarray\" objects, it is considerably faster than using equivalent approaches in (C)Python. mvrnorm(n=10, mean, cov)[,1] [,2][1,] -0.8407830 A = matrix(1:9, nrow=3, byrow=T)R> 150, M> x3 = [0.60000 0.59000 0.58000 0.62000 0.63000]’M> b = b'b =   1   2   A[,1] [,2] [,3][1,] 1 2 3[2,] 4 5 6[3,] 7 8 9
R> On each far left-hand and the right-hand side of the document, there are task descriptions. The general logic is the same but the syntax is different. mean = np.array([0,0])P> A[:,0]array([1, 4, 7])# 1st column (as column rand(3,2)ans =   0.21977   0.10220   matrix(runif(3*2), ncol=2)[,1] [,2][1,] 0.5675127 [ 4, -2,  5],       [ 2,  8,  matrix(here: 5 random vectors with mean 0, covariance 3, P> eig_vecArray([[ 0.70710678, -0.70710678],         [7, 8, 9]])P> np.random.rand(3,2)array([[ 0.29347865,  0.17920462],   MATLAB/Octave Python Description a(2:end) a[1:] miss the first element a([1:9]) miss the tenth element a(end) a[-1] last element a(end-1:end) a[-2:] last two elements Maximum and minimum MATLAB/Octave Python Description max(a,b) maximum(a,b) pairwise max max([a b]) concatenate((a,b)).max() max of all values in two vectors [v,i] = max(a) v,i = a.max(0),a.argmax(0) a ]]), R> I have used it quite extensively a couple of years ago before I discovered Python as my new favorite language for data analysis. 42],       [ 66,  81,  96],   A = [1 2 3; 4 5 9; 7 8 9]A =   1   2   np.r_[a,b]array([[1, 2, 3],       [4,   8   10   12   14   16   5, 6],        [ 7, 8, 9],   0.692063 0.390495, (Thanks to Keith C. Campbell for providing me with the syntax for the Julia language.). ],       [ A / 2, P> 3   6   9, P> R was also the first language which kindled my fascination for statistics and computing. A . mat.or.vec(3, 2)[,1] [,2][1,] 0 0[2,] 0 0[3,] 0 0, J> People from all … C[,1] [,2] [,3][1,] 1 2 3[2,] 4 5 6[3,] 7 8 9[4,] ; If used at end of command it suppresses output. MATLAB is an incredibly flexible environment that you can use to perform all sorts of math tasks. Jun 19, 2014 by Sebastian Raschka. np.dot(A,b) # or A.dot(b)array([, , ]), R> = [1 2 3; 4 5 6; 7 8 9]M> A*b3-element Array{Int64,1}:143250, M> A B = matrix(A, ncol=total_elements)R> b = matrix(c(1,2,3), ncol=3)R> * This image is a freely usable media under public domain and represents the first eigenfunction of the L-shaped membrane, resembling (but not identical to) MATLAB's logo trademarked by MathWorks Inc. b = [ 1; 2; 3 ]M> Even today, MATLAB is probably (still) the most popular language for numeric computation used for engineering tasks in academia as well as in industry." ]

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[ "", null, "How to Measure with a Light Microscope\n\nMeasurement with the Light Microscope\n\nYour microscope may be equipped with a scale (called a reticule, eyepiece reticle, eyepiece micrometer) that is built into one eyepiece. Please note that every eyepiece is different and some have a reticle mount and others do not. You can check with our office what reticle size your Meiji Techno Accepts. The reticule can be used to measure any planar dimension in a microscope field since the ocular can be turned in any direction and the object of interest can be repositioned with the stage manipulators. To measure the length of an object, please note the number of ocular divisions spanned by the object. Then multiply by the conversion factor for the magnification used. The conversion factor is different at each magnification. Therefore, when using a reticule for the first time, it is necessary to calibrate the scale by focusing on a second micrometer scale (a stage micrometer) placed directly on the stage.\n\nConversion Factor\n\nIdentify the ocular eyepiece micrometer. A typical scale consists of 50 - 100 divisions. You may have to adjust the focus of your eyepiece in order to make the scale as sharp as possible. If you do that, also adjust the other eyepiece to match the focus. Any ocular scale must be calibrated, using a device called a stage micrometer ( some are Glass for Biological Applications and some are Metal for Metallurgical Applications. A stage micrometer is simply a microscope slide with a scale etched on the surface. A typical micrometer scale is 2 mm long and at least part of it should be etched with divisions of 0.01 mm (10 µm).", null, "Suppose that a stage micrometer scale has divisions that are equal to 0.1 mm, which is 100 micrometers (µm). Suppose that the scale is lined up with the ocular scale, and at 100x it is observed that each micrometer division covers the same distance as 10 ocular divisions. Then one ocular division (smallest increment on the scale) = 10 µm at 100 power. The conversion to other magnifications is accomplished by factoring in the difference in magnification. In the example, the calibration would be 25 µm at 40x, 2.5 µm at 400x, and 1 µm at 1000x.\n\nSome stage micrometers are finely divided only at one end. These are particularly useful for determining the diameter of a microscope field. One of the larger divisions is positioned at one edge of the field of view, so that the fine part of the scale ovelaps the opposite side. The field diameter can then be determined to the maximum available precision.", null, "Estimating and reporting dimensions\n\nBe aware that even under the best of circumstances the limit of resolution of your microscope is 1 or 2 µm (or worse) at any dry magnification, and 0.5 µm or so using oil immersion. No directly measured linear dimension or value that is calculated from a linear dimension should be reported with implied accuracy that is better than that. That includes means, surface areas, volumes, and any other derived values. For example, suppose you measure the length of a flagellum on a Chlamydomonas cell at 400x, and determine that it covered 3 1/2 ocular divisions. The length is directly calculated as 3.5 divisions times 2.5 µm per division, which comes out to 8.75 µm. You know, however, that at 400x the absolute best you can do is to estimate to the nearest µm, so before reporting this measurement round it to 9 micrometers (not 9.0, which would imply an accuracy to the nearest 0.1 µm). For more information on reporting uncertain quantities see our Resources section (analytical resources).\n\nThe calculation of a volume is subject to error propagation, namely the magnification of an error when deriving a figure from one or more measured variables. For example, suppose you measure the length and diameter of an object to be 65 and 30 micrometers, respectively, assuming a cylindrical shape. The volume is given by the formula v = ¼r2l, where r = radius and l = length. The formula gives a volume of 45, 946 µm3. The volume isn't accurate to the nearest cubic micrometer, however.\n\nLet's make the very optimistic assumption that the measurement of 65 micrometers is indeed accurate to the nearest 1 µm. Then the number 65 means \"greater than 64.5 and less than 65.5.\" The number 30 really means \"greater than or equal to 29.5 and less than or equal to 30.5.\" The smaller set of measurements yields a volume of 44,085 µm3, while the larger yields a volume of 47,855 µm3. False precision would be implied even if one reported a volume of 46,000 µm3, obtained by rounding the middle measurement. It would probably be better to report a range in this case, of 44,000 to 48,000 µm3. By the way, 46,000µm3 is 0.046 mm3, which probably represents a better choice of units in this case.\n\nMaking assumptions\n\nIn many areas of experimental science, including biosciences, the ability to estimate and make reasonable assumptions is a valuable skill. In order to make some quantitative estimates, particularly of volumes, you will have to make assumptions regarding the shape of some organisms. For example, if a specimen appears round, you would likely make your volume calculation based on the assumption that the specimen is a perfect sphere. For something like a Paramecium you might assume a cylindrical shape in order to simplify your estimate, while remaining aware that you could be way off the mark.\n\nA specimen such as Chaos (Pelomyxa) carolinensis represents a real challenge. Ameoboid organisms are irregularly shaped most of the time. Is it flat on the slide, or does it extend up toward the coverslip? Perhaps it is attached to both. What model do you use as a basis for volume estimation? Is it best to assume a particular shape and take measurements at different times? Is it best to estimate a maximum and minimum for each possible dimension and obtain a range of possible volumes? Remember, you are only asked to estimate. Sometimes the best estimates have a potential error of more than an order of magnitude." ]

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[ "# Quotient Group is Group\n\n## Theorem\n\nLet $G$ be a group.\n\nLet $N$ be a normal subgroup of $G$.\n\nThen the quotient group $G / N$ is indeed a group.\n\n### Corollary\n\nLet $G$ be a group.\n\nLet $N$ be a normal subgroup of $G$.\n\nIf $G$ is finite, then:\n\n$\\index G N = \\order {G / N}$\n\n## Proof\n\nBy Subgroup is Normal iff Left Cosets are Right Cosets, the set of left cosets for $N$ equals the set of right cosets.\n\nIt follows that $G / N$ does not depend on whether left cosets are used to define it or right cosets.\n\nWithout loss of generality, we will work with the left cosets.\n\nBy definition of quotient group, the elements of $G / N$ are the cosets of $N$ in $G$, where the group operation is defined as:\n\n$\\paren {a N} \\paren {b N} = \\paren {a b} N$\n\nThe operation has been shown in Coset Product is Well-Defined to be a well-defined operation.\n\nNow we need to demonstrate that $G / N$ is a group.\n\n### $\\text G 0$: Closure\n\nThis follows from Coset Product is Well-Defined.\n\nAs $a b \\in G$, it follows that $\\paren {a b} N$ is a left coset.\n\nThus $G / N$ is closed.\n\n$\\Box$\n\n### $\\text G 1$: Associativity\n\nThe associativity of coset product follows directly from Subset Product within Semigroup is Associative:\n\n $\\displaystyle a N \\paren {b N c N}$ $=$ $\\displaystyle a N \\paren {b c N}$ $\\displaystyle$ $=$ $\\displaystyle a \\paren {b c} N$ $\\displaystyle$ $=$ $\\displaystyle \\paren {a b} c N$ $\\displaystyle$ $=$ $\\displaystyle \\paren {a b} N c N$ $\\displaystyle$ $=$ $\\displaystyle \\paren {a N b N} c N$\n\nThus $G / N$ is associative.\n\n$\\Box$\n\n### $\\text G 2$: Identity\n\nThe left coset $e N = N$ serves as the identity:\n\n $\\displaystyle \\forall x \\in G: \\ \\$ $\\displaystyle N \\paren {x N}$ $=$ $\\displaystyle \\paren {e N} \\paren {x N}$ Coset by Identity $\\displaystyle$ $=$ $\\displaystyle \\paren {e x} N$ $\\displaystyle$ $=$ $\\displaystyle x N$\n\nSimilarly $\\paren {x N} N = x N$.\n\n$\\Box$\n\n### $\\text G 3$: Inverses\n\nWe have $\\paren {x N}^{-1} = x^{-1} N$:\n\n $\\displaystyle \\paren {x N} \\paren {x^{-1} N}$ $=$ $\\displaystyle \\paren {x x^{-1} } N$ $\\displaystyle$ $=$ $\\displaystyle e N$ $\\displaystyle$ $=$ $\\displaystyle N$\n\nSimilarly $\\paren {x^{-1} N} \\paren {x N} = N$.\n\nThus $x^{-1} N$ is the inverse of $x N$.\n\n$\\Box$\n\nAll the group axioms are seen to be fulfilled, and $G / N$ has been shown to be a group.\n\n$\\blacksquare$\n\n## Also see\n\nFrom Subgroup is Normal iff Left Cosets are Right Cosets, the left coset space of a normal subgroup is equal to its right coset space.\n\nIt follows that $G / N$ does not depend on whether left cosets are used to define it or right cosets.\n\nThus we do not need to distinguish between the left quotient group and the right quotient group - the two are one and the same." ]

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[ "# What Do You Know About Polyhedron Models?\n\n10 Questions | Total Attempts: 103", null, "", null, "Settings", null, "", null, "A polyhedron model is described as a physical representation of a polyhedron, made out of a cardboard, panel material, wood board, plastic board or, occasionally, with a solid material. The construction of polyhedron models has now become a common mathematical practice. To learn more, take this short quiz.\n\n• 1.\nWhen constructing the model, what does the size depicts?\n• A.\n\nDurability\n\n• B.\n\nMaterial\n\n• C.\n\nHeight\n\n• D.\n\nStrength\n\n• 2.\nWhat is the folding of a pattern called?\n• A.\n\nNet\n\n• B.\n\nKite\n\n• C.\n\nFolder\n\n• D.\n\nTower\n\n• 3.\nFrom what is the folding of a pattern formed?\n• A.\n\nSingle sheet of cardboard\n\n• B.\n\nSingle sheet of paper\n\n• C.\n\nDouble sheets of cardboard\n\n• D.\n\nMultiple sheets of cardboard\n\n• 4.\nWhich factors determine the selection of colors?\n• A.\n\nPhysical appearance of the polyhedron\n\n• B.\n\nStructure of the polyhedron\n\n• C.\n\nGeometric understanding of the polyhedron\n\n• D.\n\nAlgebraic representation of the polyhedron\n\n• 5.\nWhich of these is an example of complex models?\n• A.\n\nGyrations\n\n• B.\n\nUniformity\n\n• C.\n\nStellations\n\n• D.\n\nCryptography\n\n• 6.\nHow many polygons are inside the complex model?\n• A.\n\nHundreds\n\n• B.\n\nThousands\n\n• C.\n\nTens of thousands\n\n• D.\n\nDozens\n\n• 7.\nWhich of these is an example of stellated octahedron?\n• A.\n\nMerkaba religious symbol\n\n• B.\n\nMerkaba architectural design\n\n• C.\n\nRaymond design\n\n• D.\n\nRhiwana religious symbol\n\n• 8.\nHow many polyhedra do we have?\n• A.\n\n75\n\n• B.\n\n80\n\n• C.\n\n85\n\n• D.\n\n90\n\n• 9.\nWhich software is used to print nets?\n• A.\n\n• B.\n\nStella\n\n• C.\n\nAspen Hysis\n\n• D.\n\n• 10.\nHow many polyhedral compounds do we have?\n• A.\n\n5\n\n• B.\n\n6\n\n• C.\n\n7\n\n• D.\n\n8\n\nRelated Topics", null, "Back to top" ]

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[ "Request a call back\n\nf r of a wire is increased by 50 and potential difference decreased by 50% then find the percentage change in current\nAsked by imayushanand223 | 29 Dec, 2021, 08:49: AM", null, "Expert Answer\nResistance of wire , R = ρ × ( L / A)  = ρ × [ L / ( π r2 ) ]\n\nwhere ρ is resistivity of material , L is length of wire , A is area of crs section and r is radius of wire\n\nhence, resistance of wire is iversly proportional to quare of radus.\n\nR = C / r2\n\nwhere C is constant if material of wire and length of wire are fixed .\n\nIf radius is increased by 50%  , then new radius r' = 1.5r\n\nNew resistance = R '  = C / (1.5 r )2  = ( 4 / 9 ) R\n\nIf voltage is decreased by 50% , then current with new resistance R ' is determined as\n\nI '  = [ (1/2) V ] / [ ( 4 / 9) R ]  = ( 9 / 8 ) ( V / R ) = 1.125 I\n\nwhere I is current with initial voltage V and initial radius of wire r .\n\nHence we see that current increases by 12.5 %\nAnswered by Thiyagarajan K | 29 Dec, 2021, 10:25: AM\nCBSE 11-science - Physics\nif the momentum of body decreases by 30% its kinetic energy decreased by", null, "Asked by bhargavchary19 | 23 Dec, 2022, 07:33: AM", null, "ANSWERED BY EXPERT\nCBSE 11-science - Physics", null, "ANSWERED BY EXPERT\nCBSE 11-science - Physics\nAsked by kdimple765 | 09 Jun, 2022, 05:07: PM", null, "ANSWERED BY EXPERT\nCBSE 11-science - Physics\nAsked by rajveermundel7 | 18 May, 2022, 09:01: PM", null, "ANSWERED BY EXPERT\nCBSE 11-science - Physics\nAsked by duritanaik | 03 May, 2022, 08:59: AM", null, "ANSWERED BY EXPERT\nCBSE 11-science - Physics\nAsked by imayushanand223 | 29 Dec, 2021, 08:49: AM", null, "ANSWERED BY EXPERT\nCBSE 11-science - Physics\nAsked by vithalsawant119 | 24 Nov, 2021, 09:06: PM", null, "ANSWERED BY EXPERT\nCBSE 11-science - Physics\nAsked by anupchan2005 | 03 Sep, 2021, 10:20: PM", null, "ANSWERED BY EXPERT\nCBSE 11-science - Physics\nAsked by bamanetushar48 | 29 May, 2021, 02:16: PM", null, "ANSWERED BY EXPERT\nCBSE 11-science - Physics\nAsked by aartiingole27 | 31 Mar, 2021, 02:26: PM", null, "ANSWERED BY EXPERT" ]

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[ "# Digital Clock Matches Puzzle\n\nThere are many puzzles with matches that involve adding, removing, or moving a certain number of matches to create new numbers or shapes. This is like that with a digital clock.\n\nGiven a valid time on a 12-hour digital clock, output the digit that requires moving the fewest lines to make it so every visible digit on the clock becomes that digit. If more than one digit is the minimum, output them all. If it is impossible to make every digit the same, output -1 or a falsy value other than 0 (you'll get a lot of these).\n\nThe clock digits look like this:\n\n |\n|\n_\n_|\n|_\n_\n_|\n_|\n\n|_|\n|\n_\n|_\n_|\n_\n|_\n|_|\n_\n|\n|\n_\n|_|\n|_|\n_\n|_|\n_|\n_\n| |\n|_|\n\n\n## Test Cases:\n\nInput: 123\n\nClock Display:\n\n _ _\n| : _| _|\n| : |_ _|\n\n\nOutput: 4\n\nExplanation: The display for 1:23 requires a total of 12 lines to be drawn. Therefore, for every digit to be the same, each digit would have to have 4 lines. The only digit that has 4 lines is 4. Therefore, the answer has to be 4.\n\nInput: 1212\n\nClock Display:\n\n _ _\n| _| : | _|\n| |_ : | |_\n\n\nOutput: -1\n\nExplanation: The display for 12:12 requires 14 lines. 14 divided by 4 is not an integer, therefore it is impossible for every digit to be the same.\n\nInput: 654\n\nClock Display:\n\n _ _\n|_ : |_ |_|\n|_| : _| |\n\n\nOutput: 5\n\nExplanation: The total number of lines is 15. 15 divided by 3 is 5, so each digit must have 5 lines. The only digits that have 5 lines are 2,3, and 5. The answer is 5 because it only requires 2 moves to make every digit 5. Simply move the line at the bottom left of the 6 to the bottom of the 4, then you have:\n\n _ _\n|_ : |_ |_|\n_| : _| _|\n\n\nThen, as you can see, all you need to do is move the line at the top right of the digit that was originally 4 to the top, and you get 5:55. To make every digit a 2 or 3 would require more than 2 moves.\n\nInput: 609\n\nClock Display:\n\n _ _ _\n|_ : | | |_|\n|_| : |_| _|\n\n\nOutput: 609 (6,0,9 or [6,0,9] also ok).\n\nExplanation: 6, 0, and 9 are the only digits that have 6 lines. As such, they are also the only possible solutions. It's not hard to see that it would take two moves to make any of these the only digit. Therefore, you output all three digits.\n\n## Notes:\n\n• Although the input time must be valid, the output time does not (e.g. 999 as an output is OK.)\n• I am very flexible with input. You can require a leading 0. You can use a number with a decimal point. You can use a string. You can use an array. You can have a parameter for every digit.\n\n# Julia, 160157 154\n\nx->(c=count_ones;l=[119;36;93;109;46;107;123;37;127;111];m=l[x+1];n=map(a->c(a)==mean(map(c,m))?sum(map(b->c(a$b),m)):1/0,l);find(n.==minimum(n).!=1/0)-1) This is a lambda function. Assign it to a variable to call it. Accepts a vector of integers in range 0-9 of any length and returns a (possibly empty) vector of results. ## Test cases julia> clock = x->(c=co... # assign function to variable (anonymous function) julia> clock([1 2 3]) 1-element Array{Int64,1}: 4 julia> clock([1 2 1 2]) 0-element Array{Int64,1} julia> clock([6 5 4]) 1-element Array{Int64,1}: 5 clock([6 0 9]) 3-element Array{Int64,1}: 0 6 9 ## Explanation Enumerate the seven segments and represent them as a bit vector. +---+ +-0-+ | | Enumerate 1 2 +---+ > the seven > +-3-+ | | segments 4 5 +---+ +-6-+ Example: 1 (segments 2 + 5 enabled) becomes 36 (bits 2 + 5 set). Here are the representations for digits 0-9. l=[119;36;93;109;46;107;123;37;127;111]; m=l[x+1]; We can use the digit as index to get it's bit vector representation. +1 because of 1-based indexing in julia. The function c=count_ones; counts the number of 1-bits in an integer. We assign an alias because we need it more often. The full program, somewhat ungolfed: x->( c=count_ones; l=[119;36;93;109;46;107;123;37;127;111]; m=l[x+1]; n=map(a->c(a)==mean(map(c,m))?sum(map(b->c(a$b),m)):1/0,l);\nfind(n.==minimum(n).!=1/0)-1\n)\n\n\nNow, the last two lines in detail:\n\nmean(map(c,m)) calculates the average number of lines per input digit.\n\nn=map(a->...,l) loops over the vector representation of all digits.\n\nIf the number of lines of our current digit a is unequal to the average linecount of the input, return inf.\n\nc(a)==mean(map(c,m))?...:1/0\n\n\nIf not, return the sum of the Hamming Distances between our current and all input digits.\n\nsum(map(b->c(a\\$b),m))\n\n\nWe now have a vector n of length 10 representing the numbers 0-9 that gives us the total number of additions/deletions we have to perform to tranform all input digits to that number, or inf, if such a transformation is impossible without changing the number of lines.\n\nfind(n.==minimum(n).!=1/0)-1\n\n\nFinally, output the locations (0-based) of all minima that are not inf." ]

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[ "7:33 am Science\n\n# How Many Gallons Is 3000 Ml?\n\nA gallon is a unit of measurement that is used to measure volume. It is equal to 4 quarts, or 128 fluid ounces. The name “gallon” comes from the Latin word “galleo,” which means “to be full.”\n\nA gallon is equal to 3.785411784 liters and weighs about 8.34 pounds.\n\nA liter is a volume unit that is defined as 1000 milliliters (mL). A milliliter (ml) is equal to 1 cubic centimeter (cm3) or 1/1000 liter.\n\nThere are 31.5 milliliters in one fluid ounce and 437.5 milliliters in one cup.\n\n(Visited 12 times, 1 visits today)" ]

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[ "", null, "RELATEED CONSULTING", null, "", null, "PC端+移动端+微信端+App端+线下全面集成，客户池管控...", null, "• 核心功能\n• 总包商\n\n• 主包商\n\n• 承包商\n\n• 分包商\n\n• 供应商\n\n• 运营商\n\n• 投资商\n\n• 咨询商\n\n• 服务商\n• 应用工具\n•  座机\n\n•  手机\n\n•  传真\n\n• 邮件\n\n•  短信\n\n•  微信\n\n•  QQ\n\n•  MSN\n\n•  微博\n\n• 客户集中管控\n• 项目集成管理\n• 项目全局视图\n• 项目多方协作\n• 项目过程控制\n• 项目执行轨迹\n\n•\n门店客户\n•\n商铺客户\n•\n商城会员\n•\n网站会员\n•\n微信用户\n•\n终端客户\n•\n经销商\n•\n分销商\n\n•\n项目池\n•\n父项目\n•\n子项目\n•\n线上项目\n•\n线下项目\n•\n内部项目\n•\n外部项目\n•\n合作项目\n\n•\n客户信息\n•\n洽谈进展\n•\n执行轨迹\n•\n产品明细\n•\n报价信息\n•\n合同信息\n•\n采购信息\n•\n费用信息\n\n•\n关联客户\n•\n多联系人\n•\n多参与方\n•\n父子项目\n•\n总包商\n•\n主包商\n•\n承包商\n•\n分包商\n\n•\n项目策略\n•\n项目状态\n•\n项目审批\n•\n项目阶段\n•\n项目流程\n•\n甘特图\n•\n工作台\n•\n分析中心\n\n•\n立项评估\n•\n方案制定\n•\n商务谈判\n•\n合同签订\n•\n初期调研\n•\n利润预估\n•\n上门安装\n•\n售后回访\n\n0533-3182878/3188155" ]

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[ "# Why Does a Comet Have a Tail?\n\nA comet has two tails. One is a dust tail pushed by light from the sun. Wired Science blogger Rhett Allain uses physics to explain how light can push on matter.\n\nActually, comets have two tails. So, this is the tale of two tails. OK, that was a poor pun -- I'm sorry. But comets are a hot item now. First, there is the comet Pan-STARRS as seen above. This isn't the only comet of importance. Hopefully, in the fall of 2013 we will have a super awesome comet to look at -- ISON. It might be the best comet since I don't know when.\n\nSo let's look at some interesting things about these comet tails. Be warned, I am not an astrophysicist. Instead, I am going to use some fundamental principles to try to explain why comets do what comets do. Oh, sure I could just look this stuff up. However, speculation is quite entertaining (at least for me).\n\n## What Is a Comet?\n\nNot every comet is the same, but it wouldn't be terrible to say that a comet is a dirty-icy object in the solar system. When they come near the sun, they melt (I'm not sure \"melt\" is the most appropriate term here) and produce gas and dust. The gas and dust form both a coma and a tail (or two tails). If the comet is large enough and close enough to Earth, you can see the comet from the sunlight that reflects off this gas and dust.\n\n## Why Two Tails?\n\nThere are two tails because there are two ways the comet can interact with the sun. Everyone thinks about light coming from the sun. However, there is also the solar wind. The solar wind is really just charged particles (like electrons and protons) that escape from the sun due to their high velocities. These charged particles then interact with the ionized gas produced from the comet.\n\nThe other tail is due to an interaction with the dust produced by the comet and the light from the sun. Really, it is this interaction that I want to talk about.\n\n## How Does Light Push on Matter?\n\nImportant idea number 1: Matter is made of positive and negative charges. If you have anything with structure (like dust particles) then it has to have atoms in it. Basically, dust is made of a combination of electrons, protons and neutrons. That's it.\n\nImportant idea number 2: Light is an electromagnetic wave. What does this even mean? It can mean lots of things. For this discussion, the important thing is that if you have a region of space moving at the speed of light an electric and magnetic field can move in accordance with a set of rules we call Maxwell's equations. Here is a typical representation of a sinusoidal EM wave from the awesome textbook Matter and Interactions.\n\nThe electric field and magnetic field in this light must both be perpendicular to each other and to the direction the wave moves. That's important.\n\nImportant idea number 3: If you have a charged particle in an electric field, it will experience a force. For a positive charge, this force will be in the same direction as the electric field. For negative charges, the force is in the opposite direction as the electric field.", null, "In the above diagram, I am using the yellow arrows to represent a region with a constant electric field. The red ball is a positive charge and the blue is a negative charge. The red and blue arrows represent the forces on these charges.\n\nImportant idea number 4: A moving electric charge will experience a force when moving in a magnetic field. The force will be perpendicular to both the magnetic field and the direction the charge is moving.", null, "Just to make things a little bit more confusing, I am now using the yellow arrows to represent a magnetic field. In this diagram, the positive and negative charges are moving in opposite directions but both have a magnetic force in the same direction. Yes, I used red arrows to represent both the velocity of the charge and the magnetic force. Maybe that was a bad idea.\n\nHere is a super short video demo of this magnetic force. The current in the wire is the same as a moving charge. I put the wire over a magnet and you can see the magnetic force pushes the wire to the side.\n\nThat's all the important ideas. Now back to light. Suppose there is a positive charge sitting all by itself in empty space - not bothering anybody. Along comes some light - an electromagnetic wave. Here is an electromagnetic wave moving towards the charge.", null, "When the EM wave first gets to the charge, there is no interaction with the magnetic field since the charge isn't moving. However, the electric field interacts with the charge, it will exert a force and change it's momentum. Once the charge is moving (say up in the diagram), there will be a magnetic force on that charge that pushes it in the same direction as the propagation of the EM wave.\n\nWhat if it's a negative charge? In that case, the electric field would make the negative charge move down in the diagram above. However, the magnetic force would still be in the same direction.\n\nBut isn't the charge moving quite slowly? Yes - and that means the magnetic force is tiny. Light interacting with matter does not have a strong effect.\n\nOk, you know I cheated here, right? Of course this simplifies the interaction with light and matter quite a bit. However, I can at least show some possible way that light can push on matter. The pressure that light pushes on stuff can be written as:", null, "What kind of pressure does the sun push on stuff? Wikipedia has a nice page on radiation pressure. At the distance of the orbit of Mercury, the pressure is about 43.3 x 10-6 N/m2. That's not much.\n\nCould you use this radiation pressure for some type of solar sail? If so, what would you call it? The answer is yes. It would be called a solar sail.\n\nThe basic idea is to create a large surface area so that even a small pressure could produce a significant force. Even a force of 1 or 2 Newtons would be good enough since it wouldn't require any fuel and it would always be pushing. Of course the problem is making these sails that are big but don't add much mass to the spacecraft. Oh - and there is the problem of getting into space. A solar sail would only be useful after the spacecraft is off the planet's surface.\n\n## If Light Pushes on Dust, Wouldn't It Push On the Comet?\n\nThe short answer is that light DOES push on the comet. Let's look at two different pieces of dust in orbit near Mercury.", null, "Let me call the radiation pressure at this point P. If the big dust has a radius twice that of the small dust, then I can calculate the force from the light on these two particles.", null, "So, the bigger dust has a greater force. Just as expected. However, force doesn't tell you everything. What about the acceleration? Let's assume that both dust particles have the same density (ρ). Since there is just one force, the acceleration would be the force divided by mass. Oh, remember that the volume of a sphere is proportional to the radius cubed.\n\nSo, the dust that is twice as big has half the acceleration. Although the force on the bigger dust is bigger, so is the mass. In fact if you double the radius of the dust, you triple the mass but only double the force from the light. Smaller dust has a greater acceleration. And this is why the dust gets pushed away from the comet, but the comet doesn't get pushed to have the same trajectory.\n\n## Why Do the Two Tails Point in Different Directions?\n\nI am going to have to make a simulation showing this dust trail - and trust me, I shall. The force on the dust is small. You can't just look at the force from the light pressure, you have to still consider the gravitational force from the interaction with the sun. However, for the solar wind, this is a collision (well, an electrostatic interaction) between two masses. The charged particles from the sun are moving fast enough that this collision with the ionized gas results in the gas moving directly away from the sun. So, the interactions with the gas and dust result in different trajectories and tails pointing in different directions." ]

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[ "# Solution sets of systems of equations over finite lattices and semilattices\n\nEndre Tóth Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H–6720 Szeged, Hungary  and  Tamás Waldhauser Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H–6720 Szeged, Hungary\n###### Abstract.\n\nSolution sets of systems of homogeneous linear equations over fields are characterized as being subspaces, i.e., sets that are closed under linear combinations. Our goal is to characterize solution sets of systems of equations over arbitrary finite algebras by a similar closure condition. We show that solution sets are always closed under the centralizer of the clone of term operations of the given algebra; moreover, the centralizer is the only clone that could characterize solution sets. If every centralizer-closed set is the set of all solutions of a system of equations over a finite algebra, then we say that the algebra has Property . Our main result is the description of finite lattices and semilattices with Property : we prove that a finite lattice has Property  if and only if it is a Boolean lattice, and a finite semilattice has Property  if and only if it is distributive.\n\n###### Key words and phrases:\nsystem of equations, solution set, clone, lattice, semilattice, distributive lattice, distributive semilattice, Boolean lattice, primitive positive formula, quantifier elimination\n###### 2010 Mathematics Subject Classification:\n08A40, 06A12, 06B99, 06D99, 06E99, 03C10\n\n## 1. Introduction\n\nIn universal algebra, investigations of systems of equations usually focus on either finding a solution, the complexity of finding a solution or deciding if there is a solution at all. For us the main interest is the “shape” of the solution sets, just like in the following basic result of linear algebra: solution sets of systems of homogeneous linear equations in variables over a field are precisely the subspaces of the vector space , i.e., sets of -tuples that are closed under linear combinations. Our goal is to give a similar characterization (i.e., a kind of closure condition) for solution sets of systems of equations over arbitrary finite algebras.\n\nLet us fix a nonempty set and a set of operations on ; then we obtain the algebra . Any equation over is of the form , where and are -ary term functions. We can also say that and are from the set of operations generated by by means of compositions. After this observation we can see that in every equation, the operations on both sides are from , which we will call the clone generated by (Definition 2.1). We will investigate solution sets of systems of equations over finite algebras in this view. The algebraic sets studied by B. I. Plotkin in his universal algebraic geometry are essentially the same as our solution sets; the only difference being that we consider only finite systems of equations. Recently A. Di Nola, G. Lenzi and G. Vitale characterized the solution sets of certain systems of equations over lattice ordered abelian groups (see ).\n\nIn our previous paper we proved that for any system of equations over a clone , the solution set is closed under the centralizer of the clone (see Definition 2.2). We also proved that for clones of Boolean functions this condition is sufficient as well. We will say that a clone (or the associated algebra) has Property  if closure under the centralizer characterizes the solution sets (here SDC stands for “Solution sets are Definable by closure under the Centralizer”). Thus clones of Boolean functions (i.e., two-element algebras) always have Property , and in we gave an example of a three-element algebra that does not have Property . In this paper we describe all finite lattices and semilattices with Property . In Section 2 we present the necessary notations and definitions. In Section 3 we give a connection between Property  and quantifier elimination of certain primitive positive formulas. Also we show that for systems of equations over a clone , if all solution sets can be described by closure under a clone , then must be the centralizer of . Section 4 contains the full description of finite lattices with Property : a finite lattice has Property  if and only if it is a Boolean lattice. In Section 5 finite semilattices having Property  are described as semilattice reducts of distributive lattices.\n\n## 2. Preliminaries\n\n### 2.1. Operations and clones\n\nLet be an arbitrary set with at least two elements. By an  operation on we mean a map ; the positive integer is called the arity of the operation . The set of all operations on is denoted by . For a set of operations, by we mean the set of -ary members of . In particular, stands for the set of all -ary operations on .\n\nWe will denote tuples by boldface letters, and we will use the corresponding plain letters with subscripts for the components of the tuples. For example, if , then denotes the -th component of , i.e., . In particular, if , then is a short form for . If and , then denotes the -tuple obtained by applying to the tuples componentwise:\n\n f(t(1),…,t(m))=(f(t(1)1,…,t(m)1),…,f(t(1)n,…,t(m)n)).\n\nWe say that is closed under , if for all and for all we have .\n\nLet and . By the composition of by we mean the operation defined by\n\n h(x)=f(g1(x),…,gn(x))% for all x∈Ak.\n\nNow we present the precise definition of clones.\n\n###### Definition 2.1.\n\nIf is closed under composition and contains the projections for all , then is said to be a clone (notation: ).\n\nFor an arbitrary set of operations on , there is a least clone containing , called the clone generated by . The elements of this clone are those operations that can be obtained from members of and from projections by finitely many compositions. In other words, is the set of term operations of the algebra .\n\nThe set of all clones on is a lattice under inclusion; the greatest element of this lattice is , and the least element is the trivial clone consisting of projections only. There are countably infinitely many clones on the two-element set; these have been described by Post , hence the lattice of clones on is called the Post lattice. If is a finite set with at least three elements, then the clone lattice on is of continuum cardinality , and it is a very difficult open problem to describe all clones on even for .\n\n### 2.2. Centralizer clones\n\nWe say that the operations and commute (notation: ) if\n\n f(g(a11,a12,…,a1m),…,g(an1,an2,…,anm))=g(f(a11,a21,…,an1),…,f(a1m,a2m,…,anm))\n\nholds for all . This can be visualized as follows: for every matrix , first applying to the rows of and then applying to the resulting column vector yields the same result as first applying to the columns of and then applying to the resulting row vector (see Figure 1).\n\n###### Definition 2.2.\n\nFor any , the set for all is called the centralizer of .\n\nIt is easy to verify that if all commute with an operation , then the composition also commutes with . This implies that is a clone for all (even if itself is not a clone).\n\nClones arising in this form are called primitive positive clones; such clones seem to be quite rare: there are only finitely many primitive positive clones over any finite set .\n\n###### Example 2.3.\n\nLet be a field, and let be the clone of all operations over that are represented by a linear polynomial:\n\n L:={a1x1+⋯+akxk+c∣k≥0,a1,…,ak,c∈K}.\n\nSince is generated by the operations , and the constants , the centralizer consists of those operations over that commute with and (i.e., is additive and homogeneous), and also commute with the constants (i.e., for all ):\n\n L∗:={a1x1+⋯+akxk∣k≥1,a1,…,ak∈K and a1+⋯+ak=1}.\n\nSimilarly, one can verify that for the clone\n\n L0:={a1x1+⋯+akxk∣k≥0,a1,…,ak∈K}.\n\n### 2.3. Equations and solution sets\n\nLet us fix a finite set , a clone and a natural number . By an -ary equation over (-equation for short) we mean an equation of the form , where . We will often simply write this equation as a pair . A system of -equations is a finite set of -equations of the same arity:\n\n E:={(f1,g1),…,(ft,gt)}, where fi,gi∈C(n) (i=1,…,t).\n\nNote that we consider only systems consisting of a finite number of equations. This does not restrict generality, since we are dealing only with finite algebras. We define the set of solutions of as the set\n\n Sol(E):={a∈An∣fi(a)=gi(a) for i=1,…,t}.\n\nFor we denote by the set of -equations satisfied by :\n\n EqC(a):={(f,g)∣f,g∈C(n) and f(a)=g(a)}.\n\nLet be an arbitrary set of tuples. We denote by the set of -equations satisfied by :\n\n EqC(T):=⋂a∈TEqC(a).\n###### Remark 2.4.\n\nFor any given and , the operators and give rise to a Galois connection between sets of -tuples and systems of -ary equations. In particular, if is the solution set of a system of equations (i.e., is Galois closed), then ; moreover, is the largest system of equations with .\n\n###### Example 2.5.\n\nConsidering the “linear” clones of Example 2.3, -equations are linear equations and -equations are homogeneous linear equations.\n\nIn a previous paper we proved that for any clone, the solution sets are closed under the centralizer of the clone. Furthermore, we proved the following theorem, which characterizes solution sets of systems of equations over clones of Boolean functions.\n\n###### Theorem 2.6 ().\n\nFor any clone of Boolean functions and , the following conditions are equivalent:\n\n1. there is a system of -equations such that ;\n\n2. is closed under .\n\nThus for two-element algebras, closure under the centralizer characterizes solution sets. We will say that a clone has Property , if this is true for the clone:\n\n###### Property (SDC).\n\nThe following are equivalent for all and :\n\n1. there exists a system of -equations such that ;\n\n2. the set is closed under .\n\nHere SDC is an abbreviation for “Solution sets are Definable by closure under the Centralizer”. In we presented a clone on a three-element set that does not have Property , showing that in general this is not a trivial property.\n\n### 2.4. The Pol-Inv Galois connection\n\nFor a positive integer , a set is called an -ary relation on ; let denote the set of all relations on . For any , let denote the -ary part of , i.e., the set of -ary members of .\n\nFor a relation and operation , if for arbitrary tuples we have , then we say that is a polymorphism of , or is an invariant relation of (or we also say that preserves ). We will denote this as . Note that is equivalent to being closed under (see Subsection 2.1). Preservation induces the so-called - Galois connection. For any and for any , let\n\n Inv(F) :={ρ∈RA∣∀f∈F:f⊳ρ}, and Pol(R) :={f∈OA∣∀ρ∈R:f⊳ρ}.\n\nIt is easy to verify that is a clone for all . Moreover, for every set of operations on a finite set, the clone generated by is by the results of Bodnarčuk, Kalužnin, Kotov, Romov and Geiger [1, 5].\n\nGiven a set of relations , a primitive positive formula over (pp. formula for short) is a formula of the form\n\n (2.1) Φ(x1,…,xn)=∃y1∃y2…∃ym\\bigwithtj=1ρj(z(j)1,…,z(j)rj),\n\nwhere , and are variables from the set . The set\n\n rel(Φ):={(a1,…,an)∣Φ(a1,…,an) is true}\n\nis an -ary relation, which is the relation defined by . If , then let denote the set of all relations that can be defined by a primitive positive formula over , and let denote the set of all relations that can be defined by a quantifier-free primitive positive formula over . If contains the equality relation and is closed under primitive positive definability, then we say that is a relational clone. The relational clone generated by is [1, 5].\n\nFor , we define the following relation on , called the graph of :\n\n f∙={(a1,…,an,b)∣f(a1,…,an)=b}⊆An+1.\n\nFor , let . It is not hard to see that for any and the function commutes with if and only if preserves the graph of (or equivalently, if and only if preserves the graph of ). Therefore for any we have .\n\n## 3. Quantifier elimination\n\nLet , then let denote the set of all relations that are solution sets of some equation over :\n\n F∘={Sol(f,g)∣∣n∈N, f,g∈F(n)}⊆RA.\n\nThe following remark shows that the graph of an operation also belongs to .\n\n###### Remark 3.1.\n\nLet , and define by . Then we have\n\n Sol(˜f,e(n+1)n+1) ={(a1,…,an,b)∈An+1∣∣˜f(a1,…,an,b)=e(n+1)n+1(a1,…,an,b)} ={(a1,…,an,b)∈An+1∣∣f(a1,…,an)=b}=f∙.\n\nThe following three lemmas prepare the proof of Theorem 3.6, which gives us an equivalent condition to Property  that we will use in sections 4 and 5.\n\n###### Lemma 3.2.\n\nFor every clone , we have and .\n\n###### Proof.\n\nIn accordance with Remark 3.1, for all we have . Therefore , which implies that . To prove the reversed containment, let us consider an arbitrary relation with . Then, for any , we have\n\n (x1,…,xn)∈ρ ⟺f(x1,…,xn)=g(x1,…,xn) ⟺∃y:f(x1,…,xn)=y\\withg(x1,…,xn)=y ⟺∃y:(x1,…,xn,y)∈f∙\\with(x1,…,xn,y)∈g∙.\n\nThis means that can be defined by a pp. formula over , therefore . Thus, we obtain , and this implies that . Therefore . ∎\n\n###### Lemma 3.3.\n\nFor every clone and , there is a system of -equations such that if and only if .\n\n###### Proof.\n\nLet be an arbitrary quantifier-free pp. formula over . By definition, is of the form\n\n Φ(x1,…,xn)=\\bigwithtj=1Sol(fj,gj)=\\bigwithtj=1(fj(z(j)1,…,z(j)rj)=gj(z(j)1,…,z(j)rj)),\n\nwhere and for all . We define the operations and (by identifying variables and by adding fictitious variables) for all . Then is equivalent to the formula\n\n Ψ(x1,…,xn)=\\bigwithtj=1(˜fj(x1,…,xn)=˜gj(x1,…,xn)),\n\nand for all . Since and are equivalent, they define the same set , and it is obvious that the set defined by is the solution set of the system . Conversely, it is clear that every solution set can be defined by a quantifier-free pp. formula of the form of . ∎\n\n###### Lemma 3.4.\n\nFor every clone , we have . Consequently, a set is closed under if and only if .\n\n###### Proof.\n\nFrom Section 2 using that and that , we have\n\n Inv(C∗)=Inv(Pol(C∙))=⟨C∙⟩∃.\n\nThe second statement of the lemma follows immediately from Lemma 3.2 by observing that is closed under if and only if . ∎\n\n###### Theorem 3.5 ().\n\nFor every clone and , if there is a system of -equations such that , then is closed under .\n\n###### Proof.\n\nLet , and let be a system of -equations and . By Lemma 3.3 we have . Using Lemma 3.4, this means that is closed under . ∎\n\nThe previous theorem shows that in Property , condition (a) implies (b). Therefore, for all clones , it suffices to investigate the implication . As a consequence of lemmas 3.2, 3.3 and 3.4, we obtain the promised equivalent reformulation of Property  in terms of quantifier elimination.\n\n###### Theorem 3.6.\n\nFor every clone , the following five conditions are equivalent:\n\n1. has Property ;\n\n2. ;\n\n3. ;\n\n4. every primitive positive formula over is equivalent to a quantifier-free primitive positive formula over ;\n\n5. is a relational clone.\n\n###### Proof.\n\n(i)(ii): By Lemma 3.3, is the solution set of some system of equations over if and only if .\n\n(ii)(iii): This follows from (the proof of) Lemma 3.4.\n\n(iii)(iv): This is trivial.\n\n(iii)(v): This follows from the fact that the relational clone generated by is . ∎\n\nIn the following corollary we will see that Theorem 3.6 implies that is the only clone that can describe solution sets over (if there is such a clone at all). Thus, the abbreviation SDC can also stand for “Solution sets are Definable by closure under any Clone”.\n\n###### Corollary 3.7.\n\nLet be a clone, and assume that there is a clone such that for all and the following equivalence holds:\n\n T is the solution set of a system of C-equations⟺T is % closed under D.\n\nThen we have .\n\n###### Proof.\n\nThe condition in the corollary gives us by Lemma 3.3 that for all , we have if and only if . This means that , thus is a relational clone. Therefore, by Theorem 3.6 this is equivalent to the condition . Applying the operator to the last equality we get that\n\n C∗=Pol(Inv(C∗))=Pol(Inv(D))=D.\n\n## 4. Systems of equations over lattices\n\nIn this, and in the following section denotes a finite lattice, with meet operation and join operation . Furthermore, denotes the least and denotes the greatest element of (that is, and ).\n\nThe following lemma shows that Property  does not hold for non-distributive lattices, i.e., solution sets of systems of equations over a non-distributive lattice can not be characterized via closure conditions.\n\n###### Lemma 4.1.\n\nLet be a finite lattice. If Property  holds for , then is a distributive lattice.\n\n###### Proof.\n\nLet be a non-distributive finite lattice and . By Lemma 3.4, the set\n\n T={(x,y)∣∃u∈L:u∧x=u∧y and u∨x=u∨y}⊆L2\n\nis closed under . We prove that is not the solution set of a system of equations over , hence Property  does not hold for . Suppose that there exists a system of -equations such that . Since is not distributive, by Birkhoff’s theorem we know that there is a sublattice of , which is isomorphic either to or . Now neither of the equations\n\n x=y (⟺x∧y=x∨y),x=x∧y,x=x∨y,y=x∧y,y=x∨y\n\nbelong to ; we prove this by presenting a counterexample for each equation. These counterexamples are shown in Figure 2, where we choose the elements and as presented in the figure. (Note that an element , chosen like on the figure, shows that . In the table, the entry in the line starting with the term and column starting with the term witnesses that is not a solution of .)", null, "Figure 2. Counterexamples showing that these equations do not belong to E.\n\nThere are no other non-trivial 2-variable equations over , therefore we get that satifies only trivial equations, hence . This is a contradiction, since . ∎\n\nThe following lemma will help us prove that Property  can only hold for Boolean lattices. Before the lemma, for a distributive lattice we define the median of the elements as\n\n m(x,y,z)=(x∧y)∨(x∧z)∨(y∧z)=(x∨y)∧(x∨z)∧(y∨z).\n###### Lemma 4.2.\n\nLet be a distributive lattice, and for all let\n\n p(x,y,z,u)=(x∧y)∨(x∧z)∨(y∧z)∨(u∧x)∨(u∧y)∨(u∧z).\n\nThen for all we have\n\n p(x,y,z,u)=x∨y∨z∨u⟺m(x,y,z)∨u=x∨y∨z.\n###### Proof.\n\nLet be arbitrary elements. Let us denote simply by and by for better readability.\n\nFirst let us suppose that . It is easy to see that always holds (since every meet in is less than or equal to ). Since , we get that , hence . Observe that by the distributivity of , can be rewritten as , and from the previous chain of inequalities we can see that , therefore we have . Thus .\n\nFor the other direction suppose that . Using that is distributive, we get that , and by the assumption this implies that . Our assumption also implies that , therefore we have . ∎\n\n###### Theorem 4.3.\n\nLet be a finite distributive lattice. If Property  holds for , then is a Boolean lattice.\n\n###### Proof.\n\nLet be a finite distributive lattice and let . Since is distributive, by Birkhoff’s representation theorem can be embedded into a Boolean lattice , hence we may suppose without loss of generality that is already a sublattice of . We can also assume that and . Let us denote the complement of an element by . We define the dual of (from Lemma 4.2) as , i.e.,\n\n q(x,y,z,u)=(x∨y)∧(x∨z)∧(y∨z)∧(u∨x)∧(u∨y)∧(u∨z).\n\nLet be the following set:\n\n T={(x,y,z)∈L3∣∣∃u∈L: p(x,y,z,u)=x∨y∨z∨u and q(x,y,z,u)=x∧y∧z∧u}.\n\nBy Lemma 3.4, the set is closed under . Let be arbitrary with an element witnessing that . From Lemma 4.2 it follows that if and only if . Meeting both sides of the latter equality by , we get\n\n (4.1) u∧m′=(m∧m′)∨(u∧m′)=(m∨u)∧m′=(x∨y∨z)∧m′.\n\nBy the dual of Lemma 4.2, we know that if and only if . Then joining the last equality and (4.1), we get that\n\n u =u∧1L=u∧(m′∨m)=(u∧m′)∨(u∧m) =((x∨y∨z)∧m′)∨(x∧y∧z).\n\nIt is not hard to derive from the defining identities of Boolean algebras that the latter formula is in fact the symmetric difference in . Alternatively, using Stone’s representation theorem for Boolean algebras, we may assume that , and are sets, and that the operations are the set-theoretic intersection, union and complementation. Then corresponds to the set of elements that belong to at least two of the sets , and . Thus consists of those elements that belong to exactly one of , and , and contains those elements that belong to one or three of the sets , and , and this is indeed in .\n\nWe have proved that the element witnessing that can only be :\n\n (4.2) ∀x,y,z∈L:(x,y,z)∈T⟺∃u∈L:u=x△y△z⟺x△y△z∈L.\n\nIt is easy to see that , and using the main theorem of , we get that if , then must hold. (In our case this theorem says that every term function of is uniquely determined by its restriction to .) Therefore only trivial equations can appear in , hence . Then (4.2) implies that is closed under the ternary operation . In particular, for any we have , which means that is a Boolean lattice. ∎\n\nWe will need the following lemmas for the proof of Theorem 4.7, which states that Boolean lattices have Property . This will complete the determination of lattices with Property .\n\n###### Lemma 4.4.\n\nLet be a finite distributive lattice and let . Then every system of -equations is equivalent to a system of inequalities , where and ().\n\n###### Proof.\n\nLet be a finite distributive lattice, let and let\n\n E={f1=g1,…,ft=gt}\n\nbe a system of -equations. For arbitrary we have if and only if and , therefore is equivalent to the system of inequalities\n\n E′={f1≤g1,g1≤f1,…,ft≤gt,gt≤ft}.\n\nDenote the disjunctive normal forms of the left hand sides of the inequalities in as , and denote the conjunctive normal forms of the right hand sides of the inequalities in as (). Then is equivalent to the system of inequalities\n\n {DNF1≤CNF1,…,DNF2t≤CNF2t}.\n\nEach is a join of some meets, and each is a meet of some joins. Therefore, for every the inequality holds if and only if every meet in is less than or equal to every join in . This means that there exists a system of inequalities equivalent to , such that and (). ∎\n\n###### Lemma 4.5.\n\nLet be a Boolean algebra. Then for every , we have\n\n1. ;\n\n2. ;\n\n3. .\n\n###### Proof.\n\nLet be arbitrary elements. For the proof of (i) let us first suppose that . Joining both sides of the inequality by , we get\n\n a′∨(a∧u)=(a′∨a)∧(a′∨u)=1B∧(a′∨u)=a′∨u≤a′∨b,\n\nand from this, follows. For the other direction, if holds, then meeting both sides by , we get that\n\n a∧u≤a∧(a′∨b)=(a∧a′)∨(a∧b)=0B∨(a∧b)=a∧b,\n\nand from this, follows.\n\nThe second statement is the dual of (i).\n\nFor the proof of (iii) let us use (i) with , and then we get that\n\n a∧b′≤c′∨d⟺c∧(a∧b′)=(c∧a)∧b′≤d.\n\nThen using (ii) with , we get\n\n (c∧a)∧b′≤d⟺c∧a≤b∨d,\n\nwhich proves (iii). ∎\n\nHelly’s theorem from convex geometry states that if we have convex sets in , such that any of them have a nonempty intersection, then the intersection of all sets is nonempty as well. The following lemma says something similar for intervals in lattices (with ).\n\n###### Lemma 4.6.\n\nLet be a lattice, (). Then we have\n\n k⋂i=1[ci,di]≠∅⟺∀i,j∈{1,…,k}:ci≤dj.\n###### Proof.\n\nLet be a lattice, and (). Then obviously,\n\n k⋂i=1[ci,di]=[c1∨⋯∨ck,d1∧⋯∧dk],\n\nwhich is nonempty if and only if , which holds if and only if for all . ∎\n\nThe last step in the characterization of finite lattices having Property  is to show that Boolean lattices do indeed have Property . For proving this, we will use the equivalence of this property with the quantifier\\hypeliminability for pp. formulas over (see Theorem 3.6).\n\n###### Theorem 4.7.\n\nIf is a finite Boolean lattice, then Property  holds for .\n\n###### Proof.\n\nLet be a finite Boolean lattice, and let . Let us denote the complement of an element by . By Theorem 3.6, Property  holds for if and only if any pp. formula over is equivalent to a quantifier-free pp. formula. Let us consider a pp. formula with a single quantifier:\n\n (4.3) Φ(x1,…,xn)=∃u\\bigwithtj=1ρ<" ]

[ null, "https://media.arxiv-vanity.com/render-output/4526103/x1.png", null ]

[ "### Merge branch 'theo-cli' into 'theo'\n\n```generic plotting script for bob measure\n\nSee merge request !52```\nparents a2ebcf46 2b95ebd9\nPipeline #19184 passed with stage\nin 42 minutes and 18 seconds\nThis diff is collapsed.\nThis diff is collapsed.\nThis diff is collapsed.\n [Min. criterion: EER] Threshold on Development set `dev-1`: -8.025286e-03 ==== =================== .. Development dev-1 ==== =================== FMR 6.263% (31/495) FNMR 6.208% (28/451) FAR 5.924% FRR 11.273% HTER 8.599% ==== ===================\n [Min. criterion: EER] Threshold on Development set `dev-1`: -8.025286e-03 ==== =================== =============== .. Development dev-1 Test test-1 ==== =================== =============== FMR 6.263% (31/495) 5.637% (27/479) FNMR 6.208% (28/451) 6.131% (29/473) FAR 5.924% 5.366% FRR 11.273% 10.637% HTER 8.599% 8.001% ==== =================== =============== [Min. criterion: EER] Threshold on Development set `dev-2`: 1.652567e-02 ==== =================== =============== .. Development dev-2 Test test-2 ==== =================== =============== FMR 4.591% (23/501) 3.333% (16/480) FNMR 4.484% (20/446) 7.006% (33/471) FAR 4.348% 3.170% FRR 9.547% 11.563% HTER 6.947% 7.367% ==== =================== ===============\n ... ... @@ -41,6 +41,34 @@ def split(filename): the first column containing -1 or 1 (i.e. negative or positive) and the second the scores (float).'''.format(filename)) return None, None raise return (scores[numpy.where(neg_pos == -1)], scores[numpy.where(neg_pos == 1)]) def split_files(filenames): \"\"\"split_files Parse a list of files using :py:func:`split` Parameters ---------- filenames : :any:`list`: A list of file paths Returns ------- :any:`list`: A list of tuples, where each tuple contains the ``negative`` and ``positive`` scores for one probe of the database. Both ``negatives`` and ``positives`` can be either an 1D :py:class:`numpy.ndarray` of type ``float``, or ``None``. \"\"\" if filenames is None: return None res = [] for file_path in filenames: try: res.append(split(file_path)) except: raise return res\n ... ... @@ -89,9 +89,9 @@ def roc(negatives, positives, npoints=100, CAR=False, **kwargs): from . import roc as calc out = calc(negatives, positives, npoints) if not CAR: return pyplot.plot(100.0 * out[0, :], 100.0 * out[1, :], **kwargs) return pyplot.plot(out[0, :], out[1, :], **kwargs) else: return pyplot.semilogx(100.0 * out[0, :], 100.0 * (1 - out[1, :]), **kwargs) return pyplot.semilogx(out[0, :],(1 - out[1, :]), **kwargs) def roc_for_far(negatives, positives, far_values=log_values(), **kwargs): ... ... @@ -142,7 +142,7 @@ def roc_for_far(negatives, positives, far_values=log_values(), **kwargs): from matplotlib import pyplot from . import roc_for_far as calc out = calc(negatives, positives, far_values) return pyplot.semilogx(100.0 * out[0, :], 100.0 * (1 - out[1, :]), **kwargs) return pyplot.semilogx(out[0, :], (1 - out[1, :]), **kwargs) def precision_recall_curve(negatives, positives, npoints=100, **kwargs): ... ... @@ -260,7 +260,7 @@ def epc(dev_negatives, dev_positives, test_negatives, test_positives, return pyplot.plot(out[0, :], 100.0 * out[1, :], **kwargs) def det(negatives, positives, npoints=100, axisfontsize='x-small', **kwargs): def det(negatives, positives, npoints=100, **kwargs): \"\"\"Plots Detection Error Trade-off (DET) curve as defined in the paper: Martin, A., Doddington, G., Kamm, T., Ordowski, M., & Przybocki, M. (1997). ... ... @@ -381,9 +381,9 @@ def det(negatives, positives, npoints=100, axisfontsize='x-small', **kwargs): pticks = [ppndf(float(v)) for v in desiredTicks] ax = pyplot.gca() # and finally we set our own tick marks ax.set_xticks(pticks) ax.set_xticklabels(desiredLabels, size=axisfontsize) ax.set_xticklabels(desiredLabels) ax.set_yticks(pticks) ax.set_yticklabels(desiredLabels, size=axisfontsize) ax.set_yticklabels(desiredLabels) return retval ... ... @@ -481,9 +481,9 @@ def cmc(cmc_scores, logx=True, **kwargs): out = calc(cmc_scores) if logx: pyplot.semilogx(range(1, len(out) + 1), out * 100, **kwargs) pyplot.semilogx(range(1, len(out) + 1), out, **kwargs) else: pyplot.plot(range(1, len(out) + 1), out * 100, **kwargs) pyplot.plot(range(1, len(out) + 1), out, **kwargs) return len(out) ... ...\n ''' Click commands for ``bob.measure`` ''' import click from .. import load from . import figure from . import common_options from bob.extension.scripts.click_helper import (verbosity_option, open_file_mode_option) @click.command() @common_options.scores_argument(nargs=-1) @common_options.eval_option() @common_options.table_option() @common_options.output_log_metric_option() @common_options.criterion_option() @common_options.thresholds_option() @common_options.far_option() @common_options.titles_option() @open_file_mode_option() @verbosity_option() @click.pass_context def metrics(ctx, scores, evaluation, **kwargs): \"\"\"Prints a table that contains FtA, FAR, FRR, FMR, FMNR, HTER for a given threshold criterion (eer or hter). You need to provide one or more development score file(s) for each experiment. You can also provide evaluation files along with dev files. If only dev scores are provided, you must use flag `--no-evaluation`. Resulting table format can be changed using the `--tablefmt`. Examples: \\$ bob measure metrics dev-scores \\$ bob measure metrics -l results.txt dev-scores1 eval-scores1 \\$ bob measure metrics {dev,eval}-scores1 {dev,eval}-scores2 \"\"\" process = figure.Metrics(ctx, scores, evaluation, load.split_files) process.run() @click.command() @common_options.scores_argument(nargs=-1) @common_options.title_option() @common_options.titles_option() @common_options.sep_dev_eval_option() @common_options.output_plot_file_option(default_out='roc.pdf') @common_options.eval_option() @common_options.points_curve_option() @common_options.axes_val_option(dflt=[1e-4, 1, 1e-4, 1]) @common_options.x_rotation_option() @common_options.x_label_option() @common_options.y_label_option() @common_options.lines_at_option() @common_options.const_layout_option() @common_options.figsize_option() @common_options.style_option() @verbosity_option() @click.pass_context def roc(ctx, scores, evaluation, **kwargs): \"\"\"Plot ROC (receiver operating characteristic) curve: The plot will represent the false match rate on the horizontal axis and the false non match rate on the vertical axis. The values for the axis will be computed using :py:func:`bob.measure.roc`. You need to provide one or more development score file(s) for each experiment. You can also provide evaluation files along with dev files. If only dev scores are provided, you must use flag `--no-evaluation`. Examples: \\$ bob measure roc dev-scores \\$ bob measure roc dev-scores1 eval-scores1 dev-scores2 eval-scores2 \\$ bob measure roc -o my_roc.pdf dev-scores1 eval-scores1 \"\"\" process = figure.Roc(ctx, scores, evaluation, load.split_files) process.run() @click.command() @common_options.scores_argument(nargs=-1) @common_options.output_plot_file_option(default_out='det.pdf') @common_options.title_option() @common_options.titles_option() @common_options.sep_dev_eval_option() @common_options.eval_option() @common_options.axes_val_option(dflt=[0.01, 95, 0.01, 95]) @common_options.x_rotation_option(dflt=45) @common_options.x_label_option() @common_options.y_label_option() @common_options.points_curve_option() @common_options.lines_at_option() @common_options.const_layout_option() @common_options.figsize_option() @common_options.style_option() @verbosity_option() @click.pass_context def det(ctx, scores, evaluation, **kwargs): \"\"\"Plot DET (detection error trade-off) curve: modified ROC curve which plots error rates on both axes (false positives on the x-axis and false negatives on the y-axis) You need to provide one or more development score file(s) for each experiment. You can also provide evaluation files along with dev files. If only dev scores are provided, you must use flag `--no-evaluation`. Examples: \\$ bob measure det dev-scores \\$ bob measure det dev-scores1 eval-scores1 dev-scores2 eval-scores2 \\$ bob measure det -o my_det.pdf dev-scores1 eval-scores1 \"\"\" process = figure.Det(ctx, scores, evaluation, load.split_files) process.run() @click.command() @common_options.scores_argument(eval_mandatory=True, nargs=-1) @common_options.output_plot_file_option(default_out='epc.pdf') @common_options.title_option() @common_options.titles_option() @common_options.points_curve_option() @common_options.const_layout_option() @common_options.x_label_option() @common_options.y_label_option() @common_options.figsize_option() @common_options.style_option() @verbosity_option() @click.pass_context def epc(ctx, scores, **kwargs): \"\"\"Plot EPC (expected performance curve): plots the error rate on the eval set depending on a threshold selected a-priori on the development set and accounts for varying relative cost in [0; 1] of FPR and FNR when calculating the threshold. You need to provide one or more development score and eval file(s) for each experiment. Examples: \\$ bob measure epc dev-scores eval-scores \\$ bob measure epc -o my_epc.pdf dev-scores1 eval-scores1 \"\"\" process = figure.Epc(ctx, scores, True, load.split_files) process.run() @click.command() @common_options.scores_argument(nargs=-1) @common_options.output_plot_file_option(default_out='hist.pdf') @common_options.eval_option() @common_options.n_bins_option() @common_options.criterion_option() @common_options.thresholds_option() @common_options.const_layout_option() @common_options.show_dev_option() @common_options.print_filenames_option() @common_options.title_option() @common_options.titles_option() @common_options.figsize_option() @common_options.style_option() @verbosity_option() @click.pass_context def hist(ctx, scores, evaluation, **kwargs): \"\"\" Plots histograms of positive and negatives along with threshold criterion. You need to provide one or more development score file(s) for each experiment. You can also provide evaluation files along with dev files. If only dev scores are provided, you must use flag `--no-evaluation`. By default, when eval-scores are given, only eval-scores histograms are displayed with threshold line computed from dev-scores. If you want to display dev-scores distributions as well, use ``--show-dev`` option. Examples: \\$ bob measure hist dev-scores \\$ bob measure hist dev-scores1 eval-scores1 dev-scores2 eval-scores2 \\$ bob measure hist --criter hter --show-dev dev-scores1 eval-scores1 \"\"\" process = figure.Hist(ctx, scores, evaluation, load.split_files) process.run() @click.command() @common_options.scores_argument(nargs=-1) @common_options.titles_option() @common_options.sep_dev_eval_option() @common_options.table_option() @common_options.eval_option() @common_options.output_log_metric_option() @common_options.output_plot_file_option(default_out='eval_plots.pdf') @common_options.points_curve_option() @common_options.n_bins_option() @common_options.lines_at_option() @common_options.const_layout_option() @common_options.figsize_option() @common_options.style_option() @verbosity_option() @click.pass_context def evaluate(ctx, scores, evaluation, **kwargs): '''Runs error analysis on score sets \\b 1. Computes the threshold using either EER or min. HTER criteria on development set scores 2. Applies the above threshold on evaluation set scores to compute the HTER, if a eval-score set is provided 3. Reports error rates on the console 4. Plots ROC, EPC, DET curves and score distributions to a multi-page PDF file You need to provide 2 score files for each biometric system in this order: \\b * development scores * evaluation scores Examples: \\$ bob measure evaluate dev-scores \\$ bob measure evaluate scores-dev1 scores-eval1 scores-dev2 scores-eval2 \\$ bob measure evaluate /path/to/sys-{1,2,3}/scores-{dev,eval} \\$ bob measure evaluate -l metrics.txt -o my_plots.pdf dev-scores eval-scores ''' # first time erase if existing file ctx.meta['open_mode'] = 'w' click.echo(\"Computing metrics with EER...\") ctx.meta['criter'] = 'eer' # no criterion passed to evaluate ctx.invoke(metrics, scores=scores, evaluation=evaluation) # second time, appends the content ctx.meta['open_mode'] = 'a' click.echo(\"Computing metrics with HTER...\") ctx.meta['criter'] = 'hter' # no criterion passed in evaluate ctx.invoke(metrics, scores=scores, evaluation=evaluation) if 'log' in ctx.meta: click.echo(\"[metrics] => %s\" % ctx.meta['log']) # avoid closing pdf file before all figures are plotted ctx.meta['closef'] = False if evaluation: click.echo(\"Starting evaluate with dev and eval scores...\") else: click.echo(\"Starting evaluate with dev scores only...\") click.echo(\"Computing ROC...\") # set axes limits for ROC ctx.forward(roc) # use class defaults plot settings click.echo(\"Computing DET...\") ctx.forward(det) # use class defaults plot settings if evaluation: click.echo(\"Computing EPC...\") ctx.forward(epc) # use class defaults plot settings # the last one closes the file ctx.meta['closef'] = True click.echo(\"Computing score histograms...\") ctx.meta['criter'] = 'eer' # no criterion passed in evaluate ctx.forward(hist) click.echo(\"Evaluate successfully completed!\") click.echo(\"[plots] => %s\" % (ctx.meta['output']))\n '''Stores click common options for plots''' import logging import click from click.types import INT, FLOAT import matplotlib.pyplot as plt from matplotlib.backends.backend_pdf import PdfPages from bob.extension.scripts.click_helper import (bool_option, list_float_option) LOGGER = logging.getLogger(__name__) def scores_argument(eval_mandatory=False, min_len=1, **kwargs): \"\"\"Get the argument for scores, and add `dev-scores` and `eval-scores` in the context when `--evaluation` flag is on (default) Parameters ---------- eval_mandatory : If evaluation files are mandatory min_len : The min lenght of inputs files that are needed. If eval_mandatory is True, this quantity is multiplied by 2. Returns ------- callable A decorator to be used for adding score arguments for click commands \"\"\" def custom_scores_argument(func): def callback(ctx, param, value): length = len(value) min_arg = min_len or 1 ctx.meta['min_arg'] = min_arg if length < min_arg: raise click.BadParameter( 'You must provide at least %d score files' % min_arg, ctx=ctx ) else: ctx.meta['scores'] = value step = 1 if eval_mandatory or ctx.meta['evaluation']: step = 2 if (length % (min_arg * 2)) != 0: pref = 'T' if eval_mandatory else \\ ('When `--evaluation` flag is on t') raise click.BadParameter( '%sest-score(s) must ' 'be provided along with dev-score(s). ' 'You must provide at least %d score files.' \\ % (pref, min_arg * 2), ctx=ctx ) for arg in range(min_arg): ctx.meta['dev_scores_%d' % arg] = [ value[i] for i in range(arg * step, length, min_arg * step) ] if step > 1: ctx.meta['eval_scores_%d' % arg] = [ value[i] for i in range((arg * step + 1), length, min_arg * step) ] ctx.meta['n_sys'] = len(ctx.meta['dev_scores_0']) if 'titles' in ctx.meta and \\ len(ctx.meta['titles']) != ctx.meta['n_sys']: raise click.BadParameter( '#titles not equal to #sytems', ctx=ctx ) return value return click.argument( 'scores', type=click.Path(exists=True), callback=callback, **kwargs )(func) return custom_scores_argument def eval_option(**kwargs): '''Get option flag to say if eval-scores are provided''' return bool_option( 'evaluation', 'e', 'If set, evaluation scores must be provided', dflt=True ) def sep_dev_eval_option(dflt=True, **kwargs): '''Get option flag to say if dev and eval plots should be in different plots''' return bool_option( 'split', 's','If set, evaluation and dev curve in different plots', dflt ) def cmc_option(**kwargs): '''Get option flag to say if cmc scores''' return bool_option('cmc', 'C', 'If set, CMC score files are provided') def semilogx_option(dflt=False, **kwargs): '''Option to use semilog X-axis''' return bool_option('semilogx', 'G', 'If set, use semilog on X axis', dflt) def show_dev_option(dflt=False, **kwargs): '''Option to tell if should show dev histo''' return bool_option('show-dev', 'D', 'If set, show dev histograms', dflt) def print_filenames_option(dflt=True, **kwargs): '''Option to tell if filenames should be in the title''' return bool_option('show-fn', 'P', 'If set, show filenames in title', dflt) def const_layout_option(dflt=True, **kwargs): '''Option to set matplotlib constrained_layout''' def custom_layout_option(func): def callback(ctx, param, value): ctx.meta['clayout'] = value plt.rcParams['figure.constrained_layout.use'] = value return value return click.option( '-Y', '--clayout/--no-clayout', default=dflt, show_default=True, help='(De)Activate constrained layout', callback=callback, **kwargs)(func) return custom_layout_option def axes_val_option(dflt=None, **kwargs): ''' Option for setting min/max values on axes ''' return list_float_option( name='axlim', short_name='L', desc='min/max axes values separated by commas (e.g. ``--axlim ' ' 0.1,100,0.1,100``)', nitems=4, dflt=dflt, **kwargs ) def thresholds_option(**kwargs): ''' Option to give a list of thresholds ''' return list_float_option( name='thres', short_name='T', desc='Given threshold for metrics computations, e.g. ' '0.005,0.001,0.056', nitems=None, dflt=None, **kwargs ) def lines_at_option(**kwargs): '''Get option to draw const far line''' return list_float_option( name='lines-at', short_name='la', desc='If given, draw veritcal lines at the given axis positions', nitems=None, dflt=None, **kwargs ) def x_rotation_option(dflt=0, **kwargs): '''Get option for rotartion of the x axis lables''' def custom_x_rotation_option(func): def callback(ctx, param, value): value = abs(value) ctx.meta['x_rotation'] = value return value return click.option( '-r', '--x-rotation', type=click.INT, default=dflt, show_default=True, help='X axis labels ration', callback=callback, **kwargs)(func) return custom_x_rotation_option def cost_option(**kwargs): '''Get option to get cost for FAR''' def custom_cost_option(func): def callback(ctx, param, value): if value < 0 or value > 1: raise click.BadParameter(\"Cost for FAR must be betwen 0 and 1\") ctx.meta['cost'] = value return value return click.option( '-C', '--cost', type=float, default=0.99, show_default=True, help='Cost for FAR in minDCF', callback=callback, **kwargs)(func) return custom_cost_option def points_curve_option(**kwargs): '''Get the number of points use to draw curves''' def custom_points_curve_option(func): def callback(ctx, param, value): if value < 2: raise click.BadParameter( 'Number of points to draw curves must be greater than 1' , ctx=ctx ) ctx.meta['points'] = value return value return click.option( '-n', '--points', type=INT, default=100, show_default=True, help='The number of points use to draw curves in plots', callback=callback, **kwargs)(func) return custom_points_curve_option def n_bins_option(**kwargs): '''Get the number of bins in the histograms''' def custom_n_bins_option(func): def callback(ctx, param, value): if value is None: value = 'auto' elif value < 2: raise click.BadParameter( 'Number of bins must be greater than 1' , ctx=ctx ) ctx.meta['n_bins'] = value return value return click.option( '-b', '--nbins', type=INT, default=None," ]

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[ "# Reading A Metric Ruler Worksheet Pdf\n\nRead SI Metric Ruler abcteach.com. Measurement worksheets using metric (international unit) rulers. Each printable PDF includes an answer key and there are measurement worksheets of centimeter and millimeter positions as well as length measurements of objects placed either the start or the middle positions of the ruler., Worksheet: Metric Measurement. 1. A section of a metric ruler is shown below. There are letters along the bottom of the ruler. Each number on the ruler represents centimeters..\n\n### Reading a Metric Ruler Worksheets thermometer and\n\nReading Ruler Worksheets Worksheet Metric albertcoward.co. Title: Reading a Ruler Metric Measurement Worksheets Author: http://www.k12mathworksheets.com Keywords: measurement worksheets, how to read a ruler, ruler worksheets, Reading a Metric Ruler Worksheets. This Measurement Worksheet is great for practicing reading a Metric Ruler. The measurement worksheet will produce eight Metric Ruler problems per page. First select the increment you wish to use and then next you have the option to add a memo line that will appear on the worksheet for additional instructions. The worksheet's answer page will be created if ….\n\nHow to Read a Tape Measure Our thanks to Johnson Level for allowing us to reprint the following. Understanding Tape Measures A tape measure, also called measuring tape, is a type of flexible ruler. Tape measures are made from a variety of materials, including fiber glass, plastic and cloth. They are among the most common measuring tools used today. Generally speaking, the term “tape measure Title: Reading a Ruler Metric Measurement Worksheets Author: http://www.k12mathworksheets.com Keywords: measurement worksheets, how to read a ruler, ruler worksheets\n\nAutomotive technicians use micrometers in order to ensure that the work they do is extremely accurate. Note: Worksheets • Micrometer Measurement Exercise • Reading a Micrometer • Measurement Quiz • Micrometer Test Evaluation Guidelines Included is a worksheet that students can be given to record their micrometer measurements. Automotive micrometers SkillS Exploration 10–12 3 Automotive technicians use micrometers in order to ensure that the work they do is extremely accurate. Note: Worksheets • Micrometer Measurement Exercise • Reading a Micrometer • Measurement Quiz • Micrometer Test Evaluation Guidelines Included is a worksheet that students can be given to record their micrometer measurements. Automotive micrometers SkillS Exploration 10–12 3\n\nTitle: Measuring lengths in inches - grade 1 measurement worksheet Author: K5 Learning Subject: Grade 1 Measurement Worksheets - lengths, weights and capacities Measurement worksheets using metric (international unit) rulers. Each printable PDF includes an answer key and there are measurement worksheets of centimeter and millimeter positions as well as length measurements of objects placed either the start or the middle positions of the ruler.\n\nThis is the Free Worksheets For Teachers Social Studies section. Here you will find all we have for Free Worksheets For Teachers Social Studies. For instance there are many worksheet that you can print here, and if you want to preview the Free Worksheets For Teachers Social Studies simply click the link or image and you will take to save page section. Look closely at the difference between the marks on a standard ruler compared to the marks on a metric ruler. (17) How do you think you can tell if a ruler is marked in metric or standard units if it doesn’t say inches,\n\nReading A Metric Ruler. Showing top 8 worksheets in the category - Reading A Metric Ruler. Some of the worksheets displayed are Reading a ruler metric measurement work, Read si metric ruler, Measuring centimeters, Reading marking ruler whole centimeter s1, Score teacher date reading a metric ruler, How to read a ruler metric measurement work Look closely at the difference between the marks on a standard ruler compared to the marks on a metric ruler. (17) How do you think you can tell if a ruler is marked in metric or standard units if it doesn’t say inches,\n\nThese free Printable Rulers are easy to print. Each is available in PDF format: just download one, open it in a PDF reader, and print. Please make sure you are printing at 100% or actual size to the rulers … Reading a Metric Ruler Worksheets. This Measurement Worksheet is great for practicing reading a Metric Ruler. The measurement worksheet will produce eight Metric Ruler problems per page. First select the increment you wish to use and then next you have the option to add a memo line that will appear on the worksheet for additional instructions. The worksheet's answer page will be created if …\n\nReading ruler worksheet free worksheets library download and measuring length worksheets. Measurement worksheets dynamically created reading a tape measure worksheets. Measurement read a ruler 20 printable worksheets by wilbert worksheets. Measurement worksheets dynamically created reading a metric ruler worksheets. Measuring one step worksheet downloads less thing downloads. Reading a metric We feel it carry something new for Using A Metric Ruler Worksheet With Measuring In Cm Worksheet Activity Sheets Measuring Cm. Hopefully this graphic will likely be certainly one of great reference for Using A Metric Ruler Worksheet With Measuring In Cm Worksheet Activity Sheets Measuring Cm.\n\nReading Metric Ruler. Showing top 8 worksheets in the category - Reading Metric Ruler. Some of the worksheets displayed are Reading a ruler metric measurement work, Reading marking ruler whole centimeter s1, Kindergarten measurement work, Practice with reading measuring devices work part 1, How to read a ruler metric measurement work, Measuring Reading a metric ruler worksheet keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website\n\nAutomotive technicians use micrometers in order to ensure that the work they do is extremely accurate. Note: Worksheets • Micrometer Measurement Exercise • Reading a Micrometer • Measurement Quiz • Micrometer Test Evaluation Guidelines Included is a worksheet that students can be given to record their micrometer measurements. Automotive micrometers SkillS Exploration 10–12 3 How to Read a Tape Measure Our thanks to Johnson Level for allowing us to reprint the following. Understanding Tape Measures A tape measure, also called measuring tape, is a type of flexible ruler. Tape measures are made from a variety of materials, including fiber glass, plastic and cloth. They are among the most common measuring tools used today. Generally speaking, the term “tape measure\n\nReading a tape measure pdf Begin with the Green Carpentry Math Book, Worksheet M3-1, Pages 11-12. reading rassenkunde des deutschen volkes pdf a tape measure powerpoint Introduces the student on how to read a tape measure. reading a tape measure for dummies One exercise in inches.the numbers on a tape measure or ruler. This hand-out is meant to help. Reading a ruler is one of the … Reading a ruler pdf HOW TO READ A RULER. Ruler reading 12 14 pdf. Ruler reading 18 2 pdf. reading a ruler quiz File Type: pdf.How to Read a Ruler Postera valuable resource for helping students learn to read a ruler and to understand equivalent", null, "Reading a Ruler (Metric System) 5-Pack Math Worksheets Land. Rulers, graduated cylinders, and thermometers on “Practice With Reading Measuring Devices Worksheet” are from math-aids.com Here are a couple of examples of graduated cylinders:, Name : Teacher : Date : Score : Math-Aids.Com Reading a Metric Ruler How many Centimeters ? 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50.\n\nReading a Metric Ruler Worksheets thermometer and. Reading a metric ruler worksheet keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website, Title: Reading a Ruler Metric Measurement Worksheets Author: http://www.k12mathworksheets.com Keywords: measurement worksheets, how to read a ruler, ruler worksheets.\n\n### Grade 3 Measurement Worksheet k5learning.com", null, "Using A Metric Ruler Worksheet With Measuring In Cm. Grade 3 Measurement Worksheets - lengths, weights, capacities and temperatures Keywords Grade 3 measurement worksheets length weight capacity temperature metric customary measuring cups scales rulers Reading Metric Ruler. Showing top 8 worksheets in the category - Reading Metric Ruler. Some of the worksheets displayed are Reading a ruler metric measurement work, Reading marking ruler whole centimeter s1, Kindergarten measurement work, Practice with reading measuring devices work part 1, How to read a ruler metric measurement work, Measuring.", null, "• How To Read A Ruler Name Mrs. Rannikko's 8th Grade Science\n• Reading a Ruler (Metric System) 5-Pack Math Worksheets Land\n\n• Rulers, graduated cylinders, and thermometers on “Practice With Reading Measuring Devices Worksheet” are from math-aids.com Here are a couple of examples of graduated cylinders: 27/08/2018В В· In this Article: Article Summary Reading an Inch Ruler Reading a Metric Ruler Community Q&A 11 References. There are two types of rulers: the inch ruler with fractional division, and the metric ruler with decimal division. Reading a ruler can seem daunting with all the little lines, but it …\n\nReading Metric Ruler. Showing top 8 worksheets in the category - Reading Metric Ruler. Some of the worksheets displayed are Reading a ruler metric measurement work, Reading marking ruler whole centimeter s1, Kindergarten measurement work, Practice with reading measuring devices work part 1, How to read a ruler metric measurement work, Measuring reading ruler worksheets worksheet metric. reading a metric ruler worksheet answer key math aidscom free worksheets library download and print on,reading ruler worksheets inches measurement measurements a practice,ruler reading exercises worksheets inches a metric worksheet answer key free library download and print on,reading a metric ruler\n\nThis Measurement Worksheet is great for practicing reading a Metric Ruler. The measurement worksheet will produce eight Metric Ruler problems per page. First select the increment you wish to use and then next you have the option to add a memo line that will appear on the worksheet … A metric ruler shows markings for millimeters and centimeters. A millimeter (1 mm) is about as wide as the thickness of a dime. A centimeter (1 cm) is about as wide as your fingernail. Some rulers have inches on one side and centimeters (metric system) on the other. Always remember to measure with the metric side of your ruler for science. Notice the lines on this ruler: The longest lines are\n\nMass, Volume, and Density Metric System Notes for interactive notebooks (pdf) (blog entry) Mass, Volume, or Length? Practice using the correct units (pdf) Reading a metric ruler practice worksheet (pdf) Measuring liquid - volume/graduated cylinder practice (Blog entry) Pour to Score - an interactive website from PBS to practice determining Grade 3 Measurement Worksheets - lengths, weights, capacities and temperatures Keywords Grade 3 measurement worksheets length weight capacity temperature metric customary measuring cups scales rulers\n\nLook closely at the difference between the marks on a standard ruler compared to the marks on a metric ruler. (17) How do you think you can tell if a ruler is marked in metric or standard units if it doesn’t say inches, Measuring metric length the basic unit in the metric system for measuring length is the meter. a meter is about the distance from the floor to a doorknob.\n\nRead SI Metric Ruler Author: www.abcteach.com Created Date: 1/13/2006 11:53:50 AM Science Skill Sheet—Reading a metric ruler From your science teacher to help you out! You will be asked to use pieces of lab equipment carefully and accurately.\n\nReading A Metric Ruler. Showing top 8 worksheets in the category - Reading A Metric Ruler. Some of the worksheets displayed are Reading a ruler metric measurement work, Read si metric ruler, Measuring centimeters, Reading marking ruler whole centimeter s1, Score teacher date reading a metric ruler, How to read a ruler metric measurement work Title: Measuring lengths in inches - grade 1 measurement worksheet Author: K5 Learning Subject: Grade 1 Measurement Worksheets - lengths, weights and capacities\n\nAt the end of each application worksheet you will find a list of other worksheets contained in this packet that may be helpful for a fuller understanding of the application worksheet. Measuring metric length the basic unit in the metric system for measuring length is the meter. a meter is about the distance from the floor to a doorknob.\n\nWe feel it carry something new for Using A Metric Ruler Worksheet With Measuring In Cm Worksheet Activity Sheets Measuring Cm. Hopefully this graphic will likely be certainly one of great reference for Using A Metric Ruler Worksheet With Measuring In Cm Worksheet Activity Sheets Measuring Cm. Measuring metric length the basic unit in the metric system for measuring length is the meter. a meter is about the distance from the floor to a doorknob.\n\nAt the end of each application worksheet you will find a list of other worksheets contained in this packet that may be helpful for a fuller understanding of the application worksheet. These measurement worksheets are great for practicing reading a Metric Ruler. These measurement worksheets will produce eight Metric Ruler problems per worksheet. These measurement worksheets will produce eight Metric Ruler problems per worksheet.\n\nMeasuring metric length the basic unit in the metric system for measuring length is the meter. a meter is about the distance from the floor to a doorknob. This is the Free Worksheets For Teachers Social Studies section. Here you will find all we have for Free Worksheets For Teachers Social Studies. For instance there are many worksheet that you can print here, and if you want to preview the Free Worksheets For Teachers Social Studies simply click the link or image and you will take to save page section.\n\n## Measuring Centimeters SuperTeacherWorksheets", null, "Free Worksheets for Teachers social Studies Pdmdentalcollege. Reading A Metric Ruler. Showing top 8 worksheets in the category - Reading A Metric Ruler. Some of the worksheets displayed are Reading a ruler metric measurement work, Read si metric ruler, Measuring centimeters, Reading marking ruler whole centimeter s1, Score teacher date reading a metric ruler, How to read a ruler metric measurement work, measuring with a metric ruler worksheets how to read worksheet for all download and share. conversion worksheets class number 1 point unit worksheet energy measuring length with a metric ruler answers system,reading a metric ruler worksheet pdf printable centimeter number names worksheets o measurements using,metric math worksheets measurement.\n\n### Reading a Metric Ruler Worksheets thermometer and\n\nMeasuring lengths in inches k5learning.com. Reading Metric Ruler. Showing top 8 worksheets in the category - Reading Metric Ruler. Some of the worksheets displayed are Reading a ruler metric measurement work, Reading marking ruler whole centimeter s1, Kindergarten measurement work, Practice with reading measuring devices work part 1, How to read a ruler metric measurement work, Measuring, Science Skill Sheet—Reading a metric ruler From your science teacher to help you out! You will be asked to use pieces of lab equipment carefully and accurately..\n\nThese measurement worksheets are great for practicing reading a Metric Ruler. These measurement worksheets will produce eight Metric Ruler problems per worksheet. These measurement worksheets will produce eight Metric Ruler problems per worksheet. Name : Teacher : Date : Score : Math-Aids.Com Reading a Metric Ruler How many Centimeters ? 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50\n\nTitle: Measuring lengths in inches - grade 1 measurement worksheet Author: K5 Learning Subject: Grade 1 Measurement Worksheets - lengths, weights and capacities A metric ruler shows markings for millimeters and centimeters. A millimeter (1 mm) is about as wide as the thickness of a dime. A centimeter (1 cm) is about as wide as your fingernail. Some rulers have inches on one side and centimeters (metric system) on the other. Always remember to measure with the metric side of your ruler for science. Notice the lines on this ruler: The longest lines are\n\nRulers, graduated cylinders, and thermometers on “Practice With Reading Measuring Devices Worksheet” are from math-aids.com Here are a couple of examples of graduated cylinders: Reading a Triple Beam Balance Worksheet pdf and Ohaus website link. The correct units pdf Reading a The correct units pdf Reading a metric ruler practice worksheet pdf Measuring.\n\nAutomotive technicians use micrometers in order to ensure that the work they do is extremely accurate. Note: Worksheets • Micrometer Measurement Exercise • Reading a Micrometer • Measurement Quiz • Micrometer Test Evaluation Guidelines Included is a worksheet that students can be given to record their micrometer measurements. Automotive micrometers SkillS Exploration 10–12 3 Title: Measuring lengths in inches - grade 1 measurement worksheet Author: K5 Learning Subject: Grade 1 Measurement Worksheets - lengths, weights and capacities\n\nGrade 3 Measurement Worksheets - lengths, weights, capacities and temperatures Keywords Grade 3 measurement worksheets length weight capacity temperature metric customary measuring cups scales rulers Reading a tape measure pdf Begin with the Green Carpentry Math Book, Worksheet M3-1, Pages 11-12. reading rassenkunde des deutschen volkes pdf a tape measure powerpoint Introduces the student on how to read a tape measure. reading a tape measure for dummies One exercise in inches.the numbers on a tape measure or ruler. This hand-out is meant to help. Reading a ruler is one of the …\n\nReading a Metric Ruler Worksheets, thermometer, and graduated cylinder. Reading a Metric Ruler Worksheets, thermometer, and graduated cylinder . Reading a Metric Ruler Worksheets, thermometer, and graduated cylinder. Visit. Discover ideas about Measurement Worksheets. We provide dynamically created measurement worksheets that allows you to select different objects to practice measuring … A metric ruler shows markings for millimeters and centimeters. A millimeter (1 mm) is about as wide as the thickness of a dime. A centimeter (1 cm) is about as wide as your fingernail. Some rulers have inches on one side and centimeters (metric system) on the other. Always remember to measure with the metric side of your ruler for science. Notice the lines on this ruler: The longest lines are\n\nThese free Printable Rulers are easy to print. Each is available in PDF format: just download one, open it in a PDF reader, and print. Please make sure you are printing at 100% or actual size to the rulers … Reading Metric Ruler. Showing top 8 worksheets in the category - Reading Metric Ruler. Some of the worksheets displayed are Reading a ruler metric measurement work, Reading marking ruler whole centimeter s1, Kindergarten measurement work, Practice with reading measuring devices work part 1, How to read a ruler metric measurement work, Measuring\n\nReading a metric ruler worksheet keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website Measurement worksheets using metric (international unit) rulers. Each printable PDF includes an answer key and there are measurement worksheets of centimeter and millimeter positions as well as length measurements of objects placed either the start or the middle positions of the ruler.\n\nReading a Metric Ruler Worksheets, thermometer, and graduated cylinder. Reading a Metric Ruler Worksheets, thermometer, and graduated cylinder . Reading a Metric Ruler Worksheets, thermometer, and graduated cylinder. Visit. Discover ideas about Measurement Worksheets. We provide dynamically created measurement worksheets that allows you to select different objects to practice measuring … Mass, Volume, and Density Metric System Notes for interactive notebooks (pdf) (blog entry) Mass, Volume, or Length? Practice using the correct units (pdf) Reading a metric ruler practice worksheet (pdf) Measuring liquid - volume/graduated cylinder practice (Blog entry) Pour to Score - an interactive website from PBS to practice determining\n\nReading a ruler pdf HOW TO READ A RULER. Ruler reading 12 14 pdf. Ruler reading 18 2 pdf. reading a ruler quiz File Type: pdf.How to Read a Ruler Postera valuable resource for helping students learn to read a ruler and to understand equivalent Metric Measurement Worksheets: cm and mm Metric linear measurement worksheets for measuring centimeters and millimeters with a ruler. Worksheets with …\n\n28/05/2016В В· Topic: Reading a Ruler (Metric System) - Worksheet 1. Write the measure on each ruler. 1. = _ cm _ mm. 2. = _ cm _ mm. 3. = _ cm _ mm. 4. = _ cm _ mm. 5. grade3-24readrulermetric.pdf. Read/Download File Report Abuse. PD Ruler - Glasses.com Fold the PD ruler in half along the indicated line. 3. You may measure it yourself in the mirror or have a friend help you for more … These free Printable Rulers are easy to print. Each is available in PDF format: just download one, open it in a PDF reader, and print. Please make sure you are printing at 100% or actual size to the rulers …\n\nTitle: Measuring lengths in inches - grade 1 measurement worksheet Author: K5 Learning Subject: Grade 1 Measurement Worksheets - lengths, weights and capacities Rulers, graduated cylinders, and thermometers on “Practice With Reading Measuring Devices Worksheet” are from math-aids.com Here are a couple of examples of graduated cylinders:\n\nGrade 3 Measurement Worksheets - lengths, weights, capacities and temperatures Keywords Grade 3 measurement worksheets length weight capacity temperature metric customary measuring cups scales rulers These free Printable Rulers are easy to print. Each is available in PDF format: just download one, open it in a PDF reader, and print. Please make sure you are printing at 100% or actual size to the rulers …\n\nThese measurement worksheets are great for practicing reading a Metric Ruler. These measurement worksheets will produce eight Metric Ruler problems per worksheet. These measurement worksheets will produce eight Metric Ruler problems per worksheet. This Measurement Worksheet is great for practicing reading a Metric Ruler. The measurement worksheet will produce eight Metric Ruler problems per page. First select the increment you wish to use and then next you have the option to add a memo line that will appear on the worksheet …\n\nThis Measurement Worksheet is great for practicing reading a Metric Ruler. The measurement worksheet will produce eight Metric Ruler problems per page. First select the increment you wish to use and then next you have the option to add a memo line that will appear on the worksheet … Reading ruler worksheet free worksheets library download and measuring length worksheets. Measurement worksheets dynamically created reading a tape measure worksheets. Measurement read a ruler 20 printable worksheets by wilbert worksheets. Measurement worksheets dynamically created reading a metric ruler worksheets. Measuring one step worksheet downloads less thing downloads. Reading a metric\n\nMeasuring With A Metric Ruler Worksheets How To Read. Worksheet: Metric Measurement. 1. A section of a metric ruler is shown below. There are letters along the bottom of the ruler. Each number on the ruler represents centimeters., This Measurement Worksheet is great for practicing reading a Metric Ruler. The measurement worksheet will produce eight Metric Ruler problems per page. First select the increment you wish to use and then next you have the option to add a memo line that will appear on the worksheet ….\n\n### Reading A Metric Ruler Worksheets Printable Worksheets", null, "Measuring With A Metric Ruler Worksheets How To Read. Reading A Metric Ruler. Showing top 8 worksheets in the category - Reading A Metric Ruler. Some of the worksheets displayed are Reading a ruler metric measurement work, Read si metric ruler, Measuring centimeters, Reading marking ruler whole centimeter s1, Score teacher date reading a metric ruler, How to read a ruler metric measurement work, Look closely at the difference between the marks on a standard ruler compared to the marks on a metric ruler. (17) How do you think you can tell if a ruler is marked in metric or standard units if it doesn’t say inches,.\n\n### Johnson Level How to Read a Tape Measure - Techni-Tool", null, "", null, "Rulers, graduated cylinders, and thermometers on “Practice With Reading Measuring Devices Worksheet” are from math-aids.com Here are a couple of examples of graduated cylinders: Reading a Metric Ruler Worksheets. This Measurement Worksheet is great for practicing reading a Metric Ruler. The measurement worksheet will produce eight Metric Ruler problems per page. First select the increment you wish to use and then next you have the option to add a memo line that will appear on the worksheet for additional instructions. The worksheet's answer page will be created if …\n\nThis Measurement Worksheet is great for practicing reading a Metric Ruler. The measurement worksheet will produce eight Metric Ruler problems per page. First select the increment you wish to use and then next you have the option to add a memo line that will appear on the worksheet … Measuring metric length the basic unit in the metric system for measuring length is the meter. a meter is about the distance from the floor to a doorknob.\n\nTitle: Measuring lengths in inches - grade 1 measurement worksheet Author: K5 Learning Subject: Grade 1 Measurement Worksheets - lengths, weights and capacities These free Printable Rulers are easy to print. Each is available in PDF format: just download one, open it in a PDF reader, and print. Please make sure you are printing at 100% or actual size to the rulers …\n\n28/05/2016В В· Topic: Reading a Ruler (Metric System) - Worksheet 1. Write the measure on each ruler. 1. = _ cm _ mm. 2. = _ cm _ mm. 3. = _ cm _ mm. 4. = _ cm _ mm. 5. grade3-24readrulermetric.pdf. Read/Download File Report Abuse. PD Ruler - Glasses.com Fold the PD ruler in half along the indicated line. 3. You may measure it yourself in the mirror or have a friend help you for more … Rulers, graduated cylinders, and thermometers on “Practice With Reading Measuring Devices Worksheet” are from math-aids.com Here are a couple of examples of graduated cylinders:\n\nHow to Read a Tape Measure Our thanks to Johnson Level for allowing us to reprint the following. Understanding Tape Measures A tape measure, also called measuring tape, is a type of flexible ruler. Tape measures are made from a variety of materials, including fiber glass, plastic and cloth. They are among the most common measuring tools used today. Generally speaking, the term “tape measure Measuring metric length the basic unit in the metric system for measuring length is the meter. a meter is about the distance from the floor to a doorknob.\n\nHow to Read a Tape Measure Our thanks to Johnson Level for allowing us to reprint the following. Understanding Tape Measures A tape measure, also called measuring tape, is a type of flexible ruler. Tape measures are made from a variety of materials, including fiber glass, plastic and cloth. They are among the most common measuring tools used today. Generally speaking, the term “tape measure Reading a tape measure pdf Begin with the Green Carpentry Math Book, Worksheet M3-1, Pages 11-12. reading rassenkunde des deutschen volkes pdf a tape measure powerpoint Introduces the student on how to read a tape measure. reading a tape measure for dummies One exercise in inches.the numbers on a tape measure or ruler. This hand-out is meant to help. Reading a ruler is one of the …\n\nworksheet how to read a ruler image of reading tape worksheets free measurements metric,reading ruler worksheets inches worksheet metric how to read a math and practice,reading a ruler practice worksheets free library download and print on metric worksheet math aidscom,reading ruler worksheets free best photos of printable worksheet metric a answer key,reading a ruler worksheet … measuring with a metric ruler worksheets how to read worksheet for all download and share. conversion worksheets class number 1 point unit worksheet energy measuring length with a metric ruler answers system,reading a metric ruler worksheet pdf printable centimeter number names worksheets o measurements using,metric math worksheets measurement\n\nAutomotive technicians use micrometers in order to ensure that the work they do is extremely accurate. Note: Worksheets • Micrometer Measurement Exercise • Reading a Micrometer • Measurement Quiz • Micrometer Test Evaluation Guidelines Included is a worksheet that students can be given to record their micrometer measurements. Automotive micrometers SkillS Exploration 10–12 3 27/08/2018В В· In this Article: Article Summary Reading an Inch Ruler Reading a Metric Ruler Community Q&A 11 References. There are two types of rulers: the inch ruler with fractional division, and the metric ruler with decimal division. Reading a ruler can seem daunting with all the little lines, but it …\n\nName : Teacher : Date : Score : Math-Aids.Com Reading a Metric Ruler How many Centimeters ? 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 A metric ruler shows markings for millimeters and centimeters. A millimeter (1 mm) is about as wide as the thickness of a dime. A centimeter (1 cm) is about as wide as your fingernail. Some rulers have inches on one side and centimeters (metric system) on the other. Always remember to measure with the metric side of your ruler for science. Notice the lines on this ruler: The longest lines are\n\nmeasuring with a metric ruler worksheets how to read worksheet for all download and share. conversion worksheets class number 1 point unit worksheet energy measuring length with a metric ruler answers system,reading a metric ruler worksheet pdf printable centimeter number names worksheets o measurements using,metric math worksheets measurement Grade 3 Measurement Worksheets - lengths, weights, capacities and temperatures Keywords Grade 3 measurement worksheets length weight capacity temperature metric customary measuring cups scales rulers\n\nReading ruler worksheet free worksheets library download and measuring length worksheets. Measurement worksheets dynamically created reading a tape measure worksheets. Measurement read a ruler 20 printable worksheets by wilbert worksheets. Measurement worksheets dynamically created reading a metric ruler worksheets. Measuring one step worksheet downloads less thing downloads. Reading a metric measuring with a metric ruler worksheets how to read worksheet for all download and share. conversion worksheets class number 1 point unit worksheet energy measuring length with a metric ruler answers system,reading a metric ruler worksheet pdf printable centimeter number names worksheets o measurements using,metric math worksheets measurement\n\nRead SI Metric Ruler Author: www.abcteach.com Created Date: 1/13/2006 11:53:50 AM This Measurement Worksheet is great for practicing reading a Metric Ruler. The measurement worksheet will produce eight Metric Ruler problems per page. First select the increment you wish to use and then next you have the option to add a memo line that will appear on the worksheet …\n\nReading A Metric Ruler Worksheet. Measuring worksheets kindergarten measure the line cm 1 learning printable measurement sheet a in using ruler going up in. Teach students how to read a ruler the nearest one fourth inch with this. Reading A Metric Ruler Worksheet. Measuring worksheets kindergarten measure the line cm 1 learning printable measurement sheet a in using ruler going up in. Teach students how to read a ruler the nearest one fourth inch with this.\n\nReading Metric Ruler. Showing top 8 worksheets in the category - Reading Metric Ruler. Some of the worksheets displayed are Reading a ruler metric measurement work, Reading marking ruler whole centimeter s1, Kindergarten measurement work, Practice with reading measuring devices work part 1, How to read a ruler metric measurement work, Measuring Reading A Metric Ruler Worksheet. Measuring worksheets kindergarten measure the line cm 1 learning printable measurement sheet a in using ruler going up in. Teach students how to read a ruler the nearest one fourth inch with this.\n\nWorksheet: Metric Measurement. 1. A section of a metric ruler is shown below. There are letters along the bottom of the ruler. Each number on the ruler represents centimeters. Reading a tape measure pdf Begin with the Green Carpentry Math Book, Worksheet M3-1, Pages 11-12. reading rassenkunde des deutschen volkes pdf a tape measure powerpoint Introduces the student on how to read a tape measure. reading a tape measure for dummies One exercise in inches.the numbers on a tape measure or ruler. This hand-out is meant to help. Reading a ruler is one of the …\n\nAutomotive technicians use micrometers in order to ensure that the work they do is extremely accurate. Note: Worksheets • Micrometer Measurement Exercise • Reading a Micrometer • Measurement Quiz • Micrometer Test Evaluation Guidelines Included is a worksheet that students can be given to record their micrometer measurements. Automotive micrometers SkillS Exploration 10–12 3 worksheet how to read a ruler image of reading tape worksheets free measurements metric,reading ruler worksheets inches worksheet metric how to read a math and practice,reading a ruler practice worksheets free library download and print on metric worksheet math aidscom,reading ruler worksheets free best photos of printable worksheet metric a answer key,reading a ruler worksheet …\n\nReading ruler worksheet free worksheets library download and measuring length worksheets. Measurement worksheets dynamically created reading a tape measure worksheets. Measurement read a ruler 20 printable worksheets by wilbert worksheets. Measurement worksheets dynamically created reading a metric ruler worksheets. Measuring one step worksheet downloads less thing downloads. Reading a metric measuring with a metric ruler worksheets how to read worksheet for all download and share. conversion worksheets class number 1 point unit worksheet energy measuring length with a metric ruler answers system,reading a metric ruler worksheet pdf printable centimeter number names worksheets o measurements using,metric math worksheets measurement\n\nHow to Read a Tape Measure Our thanks to Johnson Level for allowing us to reprint the following. Understanding Tape Measures A tape measure, also called measuring tape, is a type of flexible ruler. Tape measures are made from a variety of materials, including fiber glass, plastic and cloth. They are among the most common measuring tools used today. Generally speaking, the term “tape measure Measurement worksheets using metric (international unit) rulers. Each printable PDF includes an answer key and there are measurement worksheets of centimeter and millimeter positions as well as length measurements of objects placed either the start or the middle positions of the ruler.\n\nRead SI Metric Ruler Author: www.abcteach.com Created Date: 1/13/2006 11:53:50 AM Grade 3 Measurement Worksheets - lengths, weights, capacities and temperatures Keywords Grade 3 measurement worksheets length weight capacity temperature metric customary measuring cups scales rulers", null, "Reading a metric ruler worksheet keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website These free Printable Rulers are easy to print. Each is available in PDF format: just download one, open it in a PDF reader, and print. Please make sure you are printing at 100% or actual size to the rulers …" ]

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[ "# How does Conserv calculate averages?\n\n## There are many ways to slice and dice environmental data, here's how we do it\n\nThe way averages are calculated is slightly different depending on the context.  Is the average part of a report based on spaces, or is it a single sensor in the analytics view?\n\n### Reports\n\nConserv offers several types of reports that can create aggregate statistics for spaces.  This is a bit different from looking at data from an individual sensor.  When running a report, the averages in the report are the averages for all sensors in that space.  For example, if you have three sensors in a space, the average temperature shown for that space is the mean of the readings across all three sensors in the space over the time period covered by the report.\n\n### Analytics\n\nAverages in the analytics view are simpler.  Since the current analytics view shows you data based on sensors, not spaces, the averages shown are the mean values for that specific sensor over a specific time period.  For example, when viewing a temperature data series for a sensor, aggregated by day, the average shown in the pop up on the graph is the mean temperature for that specific sensor for the selected day.  If the data is aggregated by hour, then you'll see the mean temperature for the selected sensor for that hour.\n\n### Metrics\n\nOne of the features of Conserv Cloud that is present in both the reports and the analytics view is the \"Metrics\", such as the percentage of time that a space or sensor was in / out of range.  These metrics do not use averages.  Instead, they look at the percentage of readings taken that are in and out of the ranges defined in your \"Levels\" in Conserv Cloud.  For a single sensor, this means if 100 readings were taken over the selected time period, and 95 of them were in range, the \"in range\" percentage would be 95.  When metrics are calculated for spaces (such as in reports, or in the weekly summary), we look at all of the readings in that space over the selected time period.  For example, if you ran a report for a space that had three sensors, and each sensor had taken 100 readings, we would calculate the percentage of the 300 readings (100 readings per sensor x 3 sensors) that were in range, and display the result:\n\n Sensor 1 Sensor 2 Sensor 3 80 readings in range 100 readings in range 60 readings in range\n\n80 + 100 + 60 = 240 in range readings, out of 300 taken.\n\n240 / 300 * 100 = 80% of readings in range" ]

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[ "• Last Updated : 18 Jan, 2019\n\nThis method is used to return a new DateTime that adds the specified number of months to the value of this instance.\n\nSyntax:\n\n`public DateTime AddMonths (int months);`\n\nHere, months is the number of months. The months parameter can be negative or positive.\n\nReturn Value: This method returns an object whose value is the sum of the date and time represented by this instance and months.\n\nException: This method will throw ArgumentOutOfRangeException if the resulting DateTime is less than MinValue or greater than MaxValue` or` months is less than -120, 000 or greater than 120, 000.\n\nBelow programs illustrate the use of the above-discussed method:\n\nExample 1:\n\n `// C# program to demonstrate the``// DateTime.AddMonths(Int32) Method``using` `System;`` ` `class` `GFG {`` ` `// Main Method``public` `static` `void` `Main()``{`` ` `    ``// Creating a DateTime object``    ``DateTime d1 = ``new` `DateTime(2018, 4, 17);`` ` `    ``for` `(``int` `i = 0; i <= 10; i++)``    ``{`` ` `        ``// using the method``        ``Console.WriteLine(d1.AddMonths(i).ToString(``\"d\"``));``         ` `    ``}``     ` `    ``Console.WriteLine(``\"In Leap Years:\"``);``     ` `    ``// Creating a DateTime object``    ``// by taking a leap year``    ``// It is 31st March 2016``    ``DateTime d2 = ``new` `DateTime(2016, 03, 31);``     ` `    ``// taking a month value``    ``int` `m = 1;``     ` `    ``// using the method``    ``// Result will be 30 April 2016``    ``Console.WriteLine(d2.AddMonths(m).ToString(``\"d\"``));``     ` `     ` `}``}`\n\nOutput:\n\n```04/17/2018\n05/17/2018\n06/17/2018\n07/17/2018\n08/17/2018\n09/17/2018\n10/17/2018\n11/17/2018\n12/17/2018\n01/17/2019\n02/17/2019\nIn Leap Years:\n04/30/2016\n```\n\nExample 2:\n\n `// C# program to demonstrate the``// DateTime.AddMonths(Int32) Method``using` `System;`` ` `class` `GFG {`` ` `// Main Method``public` `static` `void` `Main()``{`` ` `    ``// Creating a DateTime object``    ``// taking MaxValue``    ``DateTime d1 = DateTime.MaxValue;`` ` `    ``// taking a month MaxValue``    ``int` `m = 12005;`` ` `    ``// using the method will ``    ``// give an runtime error``    ``// as months parameter is``    ``// greater than 12000``    ``Console.WriteLine(d1.AddMonths(m).ToString(``\"d\"``));``}``}`\n\nRuntime Error:\n\nUnhandled Exception:\nSystem.ArgumentOutOfRangeException: The added or subtracted value results in an un-representable DateTime.\nParameter name: months\n\nNote:\n\n• This method does not change the value of this DateTime object. Instead, it returns a new DateTime object whose value is the result of this operation.\n• This calculates the resulting month and year, taking into account leap years and the number of days in a month, then adjusts the day part of the resulting DateTime object.\n• The time-of-day part of the resulting DateTime object remains the same as this instance.\n\nReference:\n\nMy Personal Notes arrow_drop_up" ]

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[ "# What is a T Test: An Explanation for Statistics\n\n## What is a T Test?\n\nA T Test, also known as Student’s T Test, is a statistical hypothesis test that is used to determine whether there is a significant difference between the means of two groups. It is a type of inferential statistic which allows researchers to make inferences about the population based on a sample.\n\n## Types of T Tests\n\nThere are three main types of T Tests: Independent Samples T Test, Paired Sample T Test, and One Sample T Test. Each of these tests has a specific use case and is applied based on the nature of the data and the research question.\n\n### Independent Samples T Test\n\nThe Independent Samples T Test is used when comparing the means of two independent groups. For example, comparing the average scores of two different groups of students.\n\n### Paired Sample T Test\n\nThe Paired Sample T Test is used when comparing the means of the same group at two different times. For example, measuring the performance of a group of students before and after a specific training.\n\n### One Sample T Test\n\nThe One Sample T Test is used when comparing the mean of a single group against a known mean. For example, comparing the average score of a class to the known average score of the entire school.\n\n## How Does a T Test Work?\n\nThe T Test works by comparing the means of two groups and determining whether the difference between these means is statistically significant. The test calculates a T-value, which is the ratio of the difference between the group means to the difference within the groups. The larger the absolute value of the T-value, the greater the evidence against the null hypothesis, which is the assumption that there is no significant difference between the group means.\n\n## Interpreting T Test Results\n\nThe results of a T Test are usually reported as a T-value and a p-value. The T-value is a measure of the size of the difference relative to the variation in your sample data. The p-value is the probability that you would observe the effect seen in your sample data if the null hypothesis were true.\n\nIf the p-value is less than the chosen significance level (usually 0.05), then the null hypothesis is rejected and the difference between the groups is considered statistically significant.\n\n## Applications of T Tests\n\nT Tests are widely used in both research and business. In research, they are used to test hypotheses and draw conclusions about populations based on sample data. In business, they can be used to compare performance metrics, customer satisfaction levels, product quality, and many other applications.\n\n## Limitations of T Tests\n\nWhile T Tests are a powerful tool, they do have limitations. They assume that the data is normally distributed, that the samples are independent, and that the variances of the two groups being compared are equal. If these assumptions are not met, the results of the T Test may not be valid." ]

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[ "Contents\n\n### Context\n\n#### Algebra\n\nhigher algebra\n\nuniversal algebra\n\n## Theorems\n\n#### 2-Category theory\n\n2-category theory\n\n# Contents\n\n## Idea\n\nA monad with arities is a monad that admits a generalized nerve construction. This allows us to view its algebras as presheaves-with-properties in a canonical way.\n\nThis generalized nerve construction also generalizes the construction of the syntactic category of a Lawvere theory.\n\n## Definition\n\nLet $\\mathcal{C}$ be a category, and $i_A : \\mathcal{A} \\subset \\mathcal{C}$ a subcategory. As explained at dense functor, for any object $X$ of $\\mathcal{C}$, there is a canonical cocone over the forgetful functor $(\\mathcal{A} \\downarrow X) \\to \\mathcal{C}$, which we call the canonical $\\mathcal{A}$-cocone at $X$. The subcategory $\\mathcal{A} \\subset \\mathcal{C}$ is called dense if this cocone is colimiting for every object $X$ of $C$.\n\nIf $\\mathcal{C}$ be a category and $i_A : \\mathcal{A} \\subset \\mathcal{C}$ is a dense subcategory, then the $\\mathcal{A}$-nerve functor is given by\n\n\\begin{aligned} \\nu_{\\mathcal{A}} : \\mathcal{C} &\\to [\\mathcal{A}^{op}, \\mathrm{Set}] \\\\ X &\\mapsto \\mathcal{C}(i_A, X) \\end{aligned} \\,.\n\nA monad $(T,\\mu,\\eta)$ on $\\mathcal{C}$ is said to have arities $\\mathcal{A}$ if $\\nu_{\\mathcal{A}} \\circ T$ sends canonical $\\mathcal{A}$-cocones to colimiting cocones.\n\n## Nerve Theorem\n\nThe nerve theorem consists of two statements:\n\nI. If $\\mathcal{A}$ is dense in $\\mathcal{C}$ and if $T$ is a monad with arities $\\mathcal{A}$ on $\\mathcal{C}$, then $\\mathcal{C}^T$ has a dense subcategory $\\Theta_T$ given by the free $T$-algebras on objects of $\\mathcal{A}$.\n\nBy definition of density, this means that the nerve functor $\\nu_{\\Theta_T} : \\mathcal{C}^T \\to [\\Theta_T^{op}, \\mathrm{Set}]$ is full and faithful. This allows us to view $T$-algebras as presheaves (on $\\Theta_T$) with a certain property. The second part of the nerve theorem tells us what this property is.\n\nII. Let $j: \\mathcal{A} \\to \\Theta_T$ be the restricted free algebra functor. A presheaf $P : \\Theta_T^{op} \\to \\mathrm{Set}$ is in the essential image of $\\nu_{\\Theta}$ if and only if the restriction along $j$,\n\n$P\\circ j : A^{op} \\to \\Set$\n\nis in the essential image of $\\nu_A$.\n\nThe proof of the nerve theorem, following BMW, is fairly straightforward. Consider the adjunction $j_! : [\\mathcal{A}^{op},Set] \\rightleftarrows [\\Theta_T^{op},Set] : j^*$ given by restriction and left Kan extension. The assumption that $T$ has arities $\\mathcal{A}$ can be reformulated to say that the nerve $\\nu_{\\mathcal{A}} : \\mathcal{C} \\to [\\mathcal{A}^{op},Set]$ is a strong monad morphism? from $T$ to $j^* j_!$, i.e. there is a coherent isomorphism $\\nu_{\\mathcal{A}} T \\cong j^* j_! \\nu_{\\mathcal{A}}$. Since $\\nu_{\\mathcal{A}}$ is fully faithful, this means that if we identify $\\mathcal{C}$ with the image of $\\nu_{\\mathcal{A}}$, then the monad $T$ gets identified with $j^* j_!$. But the adjunction $j_! \\dashv j^*$ is also monadic (since $j$ is bijective on objects), so the category of $T$-algebras gets identified with the full subcategory of $j^* j_!$-algebras, i.e. presheaves on $\\Theta_T$, whose underlying presheaf on $\\mathcal{A}$ is in the image of $\\nu_{\\mathcal{A}}$. This is exactly the two statements of the nerve theorem.\n\n## Examples\n\nSee BMW for more.\n\nSee the discussion at\n\nThe associated paper is\n\n• Mark Weber, Familial 2-functors and parametric right adjoints (2007) (tac)\n\nThese ideas are clarified and expanded on in" ]

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[ "## Tuesday, February 22, 2011\n\n### Gravitational waves remain still undetected\n\nBoth Sean Carroll and Lubos report that the LIGO has not detected gravitational waves from black holes with masses in the range 25-100 solar masses. This conforms with theoretical predictions. Earlier searches from Super Novae give also null result: in this case the searches are already at the boundaries of resolution so that one can start to worry.\n\nThe reduction of the spinning rate of Hulse-Taylor binary is consistent with the emission of gravitational waves with the predicted rate so that it seems that gravitons are emitted. One can however ask whether gravitational waves might remain undetected for some reason.\n\nMassive gravitons is the first possibility. For a nice discussion see the article of Goldhaber and Nieto giving in their conclusions a table summarizing upper bounds on graviton mass coming from various arguments involving model dependent assumptions. The problem is that it is not at all clear what massive graviton means and whether a simple Yukawa like behavior (exponential damping) for Newtonian gravitational potential is consistent with the general coordinate invariance. In the case of massive photons one has similar problem with gauge invariance. One can of course naiively assume Yukawa like behavior for the Newtonian gravitational potential and derive lower bounds for the Compton wave length of gravitons. The bound is given by λc> 100 Mpc (parsec (pc) is about four light years).\n\nSecond bound comes from the pulsar timing measurements. The photons emitted by the pulsar are assume to surf in the sea of gravitational waves created by the pulsar. If gravitons are massive in Yukawa sense they arrive with velocities which are below light velocity, a dispersion of both graviton and photon arrival times is predicted. This gives a much weaker lower bound λc> 1 pc. Note that the distance of Hulse-Taylor binary is 6400 pc so that this upper bound for graviton mass could explain the possible absence of gravitational waves from Hulse-Taylor binary. There are also other bounds on graviton mass but all are plagued by model dependent assumptions.\n\nAlso in TGD framework one can imagine explanations for the possible absence of gravitational waves. I have discussed the possibility that gravitons are emitted as dark gravitons with gigantic value of hbar, which decay eventually to bunches of ordinary gravitons meaning that continous stream of gravitons is replaced with bursts which would not be interpreted in terms of gravitons but as noise (see this).\n\nOne of the breakthroughs of the last year was related to the twistor approach to TGD in zero energy ontology (ZEO).\n\n1. This approach leads to the vision that all building blocks (light-like wormhole throats) of physical particles -including also virtual particles and also string like objects- are massless. On mass shell particles are bound states of massless particles but virtual states do not satisfy bound state constraint and because negative energies are possible, also space-like virtual momenta are possible.\n\n2. Massive physical particles are identified as bound states of massless wormhole throats: since the three momenta can have different (as a special case opposite) directions, the bound states of light-like wormhole throats can be indeed massive.\n\n3. Masslessness of the fundamental objects saves from problems with gauge invariance and general coordinate invariance. It also makes it possible to apply twistor formalism, implies the absence of UV divergences, and yields an enormous simplification of generalized Feynman diagrammatics since mass shell constraints are satisfied at lines besides momentum conservation at vertices.\n\n4. A simple argument forces to conclude that all spin one and spin two particles- in particular graviton- identified in terms of multi-wormhole throat states must have arbitrary small but non-vanishing mass. The resulting physical IR cutoff guarantees the absence of IR divergences. This allows to preserve the exact Yangian symmetry of the M-matrix. One implication is that photon eats the TGD counterpart of the neutral Higgs and that only pseudoscalar counterpart of Higgs survives. The scalar counterparts of gluons suffer the same fate whereas their pseudoscalar partners would survive.\n\nIs the massivation of gauge bosons and gravitons in this sense consistent with the Yukawa type behavior?\n\n1. The first thing to notice is that this massivation would be essentially a non-local quantal effect since both emitter and receiver both emit and receive light-like momenta. Therere the description of the massivation in terms of Yukawa potential and using ordinary QFT might well be impossible or be a good approximation at best.\n\n2. If the massive gauge bosons (gravitons) correspond to wormhole throat pair (pair of these) such that the three-momenta are light-like but in exactly opposite directions, no Yukawa type screening and velocity dispersion should take place.\n\n3. If the three momenta are not exactly opposite as is possible in quantum theory, Yukawa screening could take place since the classical cm velocity calculated from the total momentum for a massive particle is smaller than maximal signal velocity. The massivation of intermediate gauge bosons and the fact that Yukawa potential description works for them satisfactorily supports this interpretation.\n\n4. If the space-time sheets mediating gravitational interaction have gigantic values of gravitational Planck constant Compton length of graviton is scaled up dramatically so that screening would be absent but velocity dispersion would remain. This leaves open the possibility that gravitons from Hulse-Taylor binary could reveal the velocity dispersion if they are detected some day.\n\nFor details about large hbar gravitons see the chapter Quantum Astro-Physics of \"Physics in Many-Sheeted Space-time\". For the twistor approach to TGD see the chapter Yangian Symmetry, Twistors, and TGD of \"Towards M-Matrix\".\n\nAt 3:35 AM,", null, "Ulla said...\n\nOff topic, but\nhttp://www.insidescience.org/research/the-hunt-for-earth-s-missing-carbon\n\nAt 4:11 AM,", null, "Anonymous said...\n\nIt's perfectly reasonable to wonder why GWs might not be detected, but you can't use recent reports of non-detection to support your stance. Gravitational-wave detectors aren't yet sensitive enough to actually detect GWs with any regularity. At current sensitivity, it's not surprising that there have been no detections. Wait a few years, and if the advanced configurations of the detectors still don't have a detection, then I would start seriously considering new theories of gravitation.\n\nAt 4:55 AM,", null, "Matti Pitkanen said...\n\nAs a matter fact, I stated just the same in the beginning of posting concerning the sources studied.\n\nIn the case of other sources we are however already now at the limits. My purpose was to just mention briefly the notion of dark gravitons: in the case that gravitons are not detected this notion might explain the non-detection. The Allais effect could be actually seen as a support for quantal interference effect for dark gravitons\n\nMy intention was also to demonstrate that small mass for gravitons/photons does not mean loss of general coordinate invariance/gauge invariance in TGD framework and to discuss what massivation could actually mean physically. In GRT framework it is far from obvious whether the notion of massive graviton makes sense." ]

[ null, "https://img2.blogblog.com/img/b16-rounded.gif", null, "https://img2.blogblog.com/img/anon16-rounded.gif", null, "https://img2.blogblog.com/img/b16-rounded.gif", null ]

[ "Ph.D Student Rawitz Dror Combinatorial and LP-Based Methods for Designing Approximation Algorithms Department of Computer Science Professor Emeritus Reuven Bar-Yehuda\n\nAbstract\n\nThis thesis focuses on two approximation paradigms: the local ratio technique and the primal-dual schema.  We present two relatively simple approximation frameworks for minimization problems, one for each approach, which extend known approximation frameworks for covering problems, and we show that the two frameworks are equivalent.  We also present two equivalent frameworks for maximization problems.  We demonstrate our generic algorithms on a variety of problems.\n\nWhile all applications of the local ratio technique to date use non-negative weight functions, we present local ratio analyses to several known algorithms that use weight functions with some negative coefficients.  In order to present equivalent analyses that are based on the primal-dual schema we define new integer programming formulations with constraints that have negative coefficients.\n\nWe present two new combinatorial approximation frameworks that are not based on LP-duality, or even on linear programming.  Instead, they are based on weight manipulation in the spirit of the local ratio technique.  We show that the first framework is equivalent to the method of dual fitting.  We define a method called primal fitting, and show it is tantamount to our second framework.  We demonstrate both frameworks on a variety of problems including the metric uncapacitated facility location problem.\n\nA k-partite graph is a graph G=(V1,…,Vk,E), where V1,…,Vk are k non-empty disjoint independent sets of vertices.  Such a graph is called complete k-partite if E contains all possible edges.  We discuss three variants of the following optimization problem: given a graph and a non-negative weight function on the vertices and edges, find a minimum weight set of vertices and incident edges whose removal from the graph leaves a complete k-partite graph.  We use the local-ratio technique to develop 2-approximation algorithms for the first two variants of the problem.  We use approximation preserving reductions in order to (4-4/k)-approximate the third variant." ]

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[ "", null, "A dressmaker is a person who makes clothes. To make clothes, the dressmaker uses a body tape measure to measure someone’s body for body measurements. So when starting off as a dressmaker, you require the ability to read body measurements.\n\nMyself as a dressmaker, in this content I will share with you how to read body measurements using both inch and centimeter tape measures. Feel free to jump to the item that interest you:\n\n## A tape measure\n\nA tape measure, also known as a measuring tape, is a flexible tape used to measure length. Tape measures are made from different materials like cloth, plastic, fibre glass or metal strip. It is light weight which allows it to be easily carried in a pocket or toolkit, and its flexibility allows one to measure around curves or corners. It is one of the most effective measuring tools used today.\n\n## A body tape measure\n\nA body tape measure is a kind of tape measure that is used for measuring the body. It is usually a soft and flexible strip or ribbon made mostly from plastic or fibre glass. It has linear-measuring markings of both Customary System of measurement and the Metric System of measurement. The Customary System of measurement, also known as the U.S Customary System, is based on the English System of measurement. Length and distance in the Customary System are measured in inches, feet, yards and miles. While the Metric System is a system of measurement that uses the meter liter and gram as base units of length ( distance), capacity ( volume), and weight ( mass) respectively. The inch and centimeter units are the measuring units commonly on the body tape measure and are used for body measurements.\n\n## What is an inch?\n\nAn inch is a unit of length in the Customary System of measurement. Length in inches is represented by ‘in’ or ‘” ‘ symbol. For example 8 inches can be written as ‘8 in’ or ‘8″ ‘. An inch is equal to 1/36 yard or 1/12 of a foot. An inch in Metric system is equal to 2.54 cm.\n\n## How to read a body tape measure in inches\n\nOn the tape measure marked with customary units you will find the most significant marks are commonly the inch marks. These are usually marked by long thin lines and represented by a large numeric value. There are two half inches, four quarter inches, eight eighth inches, and sixteen sixteenth inches in one inch.\n\n#### Sum up the distribution of inches to evaluate the total length:\n\nIf you are going to calculate a specific length, firstly you need to highlight the spot where the measuring tape touches with the outline of the thing you are measuring. Afterward, calculate the closest inch before this specific point. Then, calculate the nearest half-inch before this point. Afterward, calculate the closest quarter inch before this point, and so on. Sum up all the inches and fractions of inches until you find the correct result.\n\nTherefore, when you take someone’s measurement, for example waist measurement, you place the measuring tape around that person’s waist, and the waist length is determined by the starting point of the tape measure to where the tape touches its starting point.\n\nFor example, let’s say we measured past the 26-inches mark and the 1/4 -inch mark and stopped at the 1/2-inch mark. Our reading will be, 26 ( inches) + 1/2 ( inches) = 26 1/2 inches (26.5 inches)\n\nAnother example, let’s say we took shoulder measurements which is from the joint of the shoulder and neck to the end of the shoulder, and we measured past 4-inches mark, past 1/2-inch mark, past 3/4 inches mark and stopped at 7/8 inches mark. Our reading will be, 4 ( inches) + 7/8 ( inches) = 4 7/8 inches (4.875 inches)\n\n## What is a centimeter?\n\nA centimeter is a unit of length in the Metric system. It is equal to one hundredth of a meter, it is also equal to 10 millimeters. In the Customary system it is equal to 0.3937 inch ( 4/10 inch). An example of a centimeter is exactly the width of an adult’s small fingernail. Length in centimeter is represented by a ‘cm’ symbol. For example 16 centimeters can be written as ’16 cm’.\n\n## How to read a body tape measure in centimeters\n\nOn the Metric measuring tape the most significant markings are the centimeters. Centimeters are generally presented by the large lines, and you will see a number next to each line. The line represents each centimeter. There are two half centimeters and ten tenth centimeters in one centimeter." ]

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[ "July 14, 2020", null, "### Gold Set to Test 50% Fibonacci Retracement – Brace for a\n\nWhile the 50% ratio is often used in Fibonacci analysis, it is not a Fibonacci ratio. Some say that the 50% level is a Gann ratio, created by W.D. Gann in the early 1900’s. Others call the 50% level an inverse of a “sacred ratio.” Just like the Fibonacci ratios, many people will either take the inverse or …", null, "### 50 Pips A Day Forex Strategy - FXN Trading\n\nFibonacci Retracement Lines are a used as a predictive technical indicator in forex and CFD trading. Learn to use Fibonacci to locate potential retracement points, swing highs and swing lows to …", null, "### Top 4 Fibonacci Retracement Mistakes to Avoid\n\nChapter 7 of the FX Leaders trading course. The Fibonacci technical trading strategy is still the most popular technical indicator among Forex traders. Learn about Fibonacci with support and resistance, Fibonacci with trend lines and Fibonacci with candlesticks.", null, "### Fibonacci Retracement in MT4 / MT5 Indicators - Page 1 of 6\n\nSimple Easy Forex Auto Fibo Trade Zone Trading Strategy (Fibonacci 50% Retracement System).. This Auto Fibo Trade Zone forex trading indicator is designed to draw a Fibonacci retracement and trading zone, using as a basis the ZigZag indicator.", null, "### Technical Tools for Traders | Fibonacci\n\n2019/09/11 · Want to learn how to use Fibonacci retracements? In this video, I show you step-by-step how I use the Fibonacci retracement tool with Forex price …", null, "### Fibonacci Retracement Trading Strategy With Price Action Forex\n\nRather, we use a 50% ratio instead. This is a direct Fibonacci number but because of its ability to create large success with this method. This is specific to the forex markets due to to the market retracing around half a major movement before continuing the trend. So we altogether we have the following ratios: 61.8%, 38.2& and 50%", null, "### Best Fibonacci Retracement Channel Trading Strategy?\n\n2020/01/31 · EURSEK is attempting to push over the 10.671 hurdle, which is the 50.0% Fibonacci retracement of the down leg from 10.932 to 10.410 after violating the Ichimoku cloud and the 200-day simple moving average (SMA) to the upside.", null, "Fibonacci Trend Strategy is an strategy suitable for day trader and swing trader based on Finacci indicators bur following the direction of retracement.Time Frame 15 min, 30 min, 60 min, 240 min.Currency pairs: major, minor, Gold and Indices.", null, "### Fibonacci Retracement | Know When to Enter a Forex Trade\n\n- Advertisement - In my time trading I have read many articles and books which state the 50% Fibonacci retracement is the level in which the market has the highest probability of reversing at in the event of a retracement taking place, there are some reasons people give as to why the 50% level is […]", null, "### Forex Fibonacci Retracement Strategy For Beginners\n\nImprove Your Forex Trading Strategy With 3 Best Fibonacci Trading Systems. 50%, 61.8% and 100%. Fibonacci retracement is a very popular tool used by many technical traders to help identify strategic places for transactions to be placed, target prices or stop losses.", null, "### Fibonacci Trend Strategy - Forex Strategies\n\n2018/07/16 · Chapter 6: Three Simple Fibonacci Trading Strategies #1 – Pullback Trades. First, you want to identify a security in a strong trend. A strong trend can be defined as a stock with successive highs with pullbacks of less than 50%. If you are day trading, you will want to identify this setup on a 5-minute chart 20 to 30 minutes after the market", null, "### Fibonacci EA - Best Forex EA's | Expert Advisors | FX Robots\n\nWe’ll see how these ratios are determined and how they can be used in forex trading. Fibonacci retracements are calculated by using the ratios of 23.6%, 38.2%, 50%, and 61.8%. How the Fibonacci trading ratios are calculated. In an earlier section of this post, we …", null, "### The Best Target in the Forex Market: the -61.8% Fibonacci\n\n- This is the second installment of our series on Support and Resistance in the Forex Market. In part one, we looked at psychological levels. In this article, we delve down the rabbit hole of", null, "### Transcend Fibonacci PRO \"Beyond The Limit\"\n\nTraders will also commonly plot the 50% level – although that is not a true Fibonacci number. The picture below will outline these levels: Each of these levels are relative to the prior trend.", null, "### Fibonacci EA 38% & 50% Buy/Sell Limit @ Forex Factory\n\n2018/07/23 · It appears that the 50% Fibonacci Retracement confirmation occurs after it has been broken. However, the resulting Action Action formation remains valid because the pin bar signal is in the 50% level zone, so its Fibonacci Retracement level is also valid as the support level.", null, "### What is the Fibonacci Retracement? - Elite Forex Trading\n\n2019/11/07 · Learn how to use Fibonacci retracements as part of a forex trading strategy. Fibonacci levels are watched to identify support and resistance levels. 50 percent and 61.8 percent by drawing", null, "### Fibonacci 50% level @ Forex Factory\n\nWell, seeing as how Fibonacci levels are used to find support and resistance levels, this also applies to Fibonacci! Fibonacci retracements do NOT always work! They are not foolproof. Let’s go through an example when the Fibonacci retracement tool fails.", null, "### Mastering Fibonacci Retracement Levels - YouTube\n\nA Fibonacci Forex trading strategy. We have already established that the price of a market can often turn, or find support or resistance, at different Fibonacci levels. Within a Fibonacci trading strategy, traders can go one step further and add in more technical analysis to help confirm whether the market will actually turn or not.", null, "", null, "" ]

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[ "## You are looking at 1-10 of 14,263 entries  for:\n\n• Type: Overview Page\n• Chemistry\nclear all", null, "", null, "", null, "View:", null, "## 0.5\n\norsymbol (in enzyme kinetics) for the value of the concentration of a substrate, A, in mol dm−3, at which the velocity of the reaction, v, is half the maximum velocity, V; i.e. when v = 0.5V.[...]", null, "## 50\n\nsymbol for the molar concentration of an agonist that produces 50% of the maximal possible effect of that agonist. Other percentage values ([A]20, [A]40, etc.) can be specified. The action of the ...", null, "## 2′-5′A\n\nsymbol for any member of a series of oligonucleotides of the general formula pa A[2′p5′A]n, where p and A are phosphoric and adenosine residues, respectively, and a and n are ...", null, "## A\n\nsymbol for1 acid‐catalysed (of a reaction mechanism).2 a residue of the α‐amino acid l‐alanine (alternative to Ala).3 a residue of the base adenine in a nucleic‐acid sequence.[...]", null, "## a\n\n1 abbr. for adsorbed.2 symbol for atto+ (SI prefix denoting 10−18).3 axial.4 year.", null, "## a\n\nsymbol for1 absorption coefficient.2 acceleration (in vector equations it is printed in bold italic (a).3 activity (def. 3).4 van der Waals coefficient.5 as subscript, denotes affinity.[...]", null, "## A\n\nsymbol for1 absorbance.2 activity (def. 2).3 affinity.4 Helmholtz function.5 mass number/nucleon number.", null, "## Å\n\nsymbol for ångström (unit of length equal to 10−10 m).", null, "## (A + T)/(G + C) ratio\n\nThe ratio of the sum of the adenine plus thymine bases to the sum of the guanine plus cytosine bases in a DNA molecule or preparation. The ratio is to ...", null, "## A 1 cell\n\n(formerly) an alternative name for D cell.", null, "", null, "", null, "View:" ]

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[ "# HackyHour Würzburg 52\n\n## Virtual!\n\n• When: December 17th, 2020 at 5:00pm :warning: it is the 3rd Thursday this month\n• Where: :exclamation: Virtual! :exclamation:\n• Zoom:\n• https://uni-wuerzburg.zoom.us/j/91208057434?pwd=aE9XMng1bzBiRVpMaW5OdTBiZHo0dz09\n• Meeting ID: 912 0805 7434\n• Info: HackyHour Website\n• :vertical_traffic_light: As long as the covid numbers are high in Würzburg we can not meet in person :cry: That won’t keep us from having fun, though! :computer:\n\n## Topic Suggestions\n\nAdd your :+1: to the end of a line you are interested in\n\n## Maybe another time\n\n• 3D Modelling, Blender, VR programming (maybe invite Annika K.)\n• GenForce - An efficient PyTorch library for deep generative modeling. May the Generative Force (GenForce) be with You.\n• Tidy Time Series Analysis in R: tidyverts\n• A challenge platform for data science\n\n## Participants\n\nPlease add your name and indicate if you prefer to join remotely :computer: and if you want to order pizza :pizza:\n\n• Markus :computer: :pizza:\n• Ludmilla\n• Frank (partially) :computer: or :desktop_computer:; no :pizza: for me\n• Rick :computer: :microscope:\n• Franzi (I will join around 5:30 pm)\n• Hannah\n\n## Advent of Code Coding Dojo\n\n### Day 1\n\n#### Part I\n\nimport numpy as np\n\nfor i in range(len(input)):\nfor j in range((i + 1), len(input)):\ns = input[i] + input[j]\nif s == 2020:\nprint(input[i]*input[j])\n\n\n#### Part II\n\nimport numpy as np\n\nfor i in range(len(input)):\nfor j in range((i + 1), len(input)):\nfor k in range((j+1), len(input)):\ns = input[i] + input[j] + input[k]\nif s == 2020:\nprint(input[i]*input[j]*input[k])\n\n\n### Day 2\n\n#### Part I\n\nwith open('input') as entry:\nvalid = 0\nfor l in lines:\nparts = l.split()\nprint(parts)\noccurences = parts.split('-')\nletter = parts\nprint(occurences, letter)\nif letter in parts:\nc = parts.count(letter)\nif c >= int(occurences) and c <= int(occurences):\nvalid += 1\n\nprint(valid)\n\n\n#### Part II\n\nwith open('input') as entry:\nvalid = 0\nfor l in lines:\nparts = l.split()\nprint(parts)\noccurences = parts.split('-')\nletter = parts\nprint(occurences, letter)\nif letter == parts[int(occurences)-1]:\nif letter != parts[int(occurences)-1]:\nvalid += 1\nif letter == parts[int(occurences)-1]:\nif letter != parts[int(occurences)-1]:\nvalid += 1\nprint(valid)\n\n\n### Day 3\n\n#### Part I\n\nwith open('input') as entry:\nlines = [l.strip() for l in lines]\nx = 0\ny = 0\ntree = 0\nwhile y < len(lines)-1:\nx += 3\nx = x%len(lines)\ny += 1\nif lines[y][x] == '#':\ntree += 1\nprint(tree)\n\n• Right 1, down 1.\n• Right 3, down 1. (This is the slope you already checked.)\n• Right 5, down 1.\n• Right 7, down 1.\n• Right 1, down 2.\n\n#### Part II\n\nwith open('input') as entry:\nlines = [l.strip() for l in lines]\ntreeproduct = 1\nxsteps = [1,3,5,7,1]\nysteps = [1,1,1,1,2]\nfor i in range(5):\nx = 0\ny = 0\ntree = 0\nwhile y < len(lines)-1:\nx += xsteps[i]\nx = x%len(lines)\ny += ysteps[i]\nif lines[y][x] == '#':\ntree += 1\ntreeproduct *= tree\nprint(treeproduct)\n\n\n### Day 4\n\n#### Part I\n\nbyr (Birth Year)\niyr (Issue Year)\neyr (Expiration Year)\nhgt (Height)\nhcl (Hair Color)\necl (Eye Color)\npid (Passport ID)\ncid (Country ID)\n\nwith open('input') as entry:\ncount = 0\nline = line.strip()\nwhile line:\npass_dict = {}\nwhile line != '':\npairs = line.split()\nfor i in range(len(pairs)):\npair = pairs[i].split(':')\npass_dict[pair] = pair\nline = line.strip()\nif 'byr' in pass_dict:\nif 'iyr' in pass_dict:\nif 'eyr' in pass_dict:\nif 'hgt' in pass_dict:\nif 'hcl' in pass_dict:\nif 'ecl' in pass_dict:\nif 'pid' in pass_dict:\ncount = count +1\nline = line.strip()\nprint(count)\n# print()\n# pass_dict = {}\n# pass_dict[''] =\n\n\n#### Part II\n\nbyr (Birth Year) - four digits; at least 1920 and at most 2002.\niyr (Issue Year) - four digits; at least 2010 and at most 2020.\neyr (Expiration Year) - four digits; at least 2020 and at most 2030.\nhgt (Height) - a number followed by either cm or in:\nIf cm, the number must be at least 150 and at most 193.\nIf in, the number must be at least 59 and at most 76.\nhcl (Hair Color) - a # followed by exactly six characters 0-9 or a-f.\necl (Eye Color) - exactly one of: amb blu brn gry grn hzl oth.\npid (Passport ID) - a nine-digit number, including leading zeroes.\ncid (Country ID) - ignored, missing or not.\n\nimport re\n\nwith open('input') as entry:\ncount = 0\nline = line.strip()\nwhile line:\npass_dict = {}\nwhile line != '':\npairs = line.split()\nfor i in range(len(pairs)):\npair = pairs[i].split(':')\npass_dict[pair] = pair\nline = line.strip()\nif 'byr' in pass_dict and int(pass_dict['byr'])>=1920 and int(pass_dict['byr'])<=2002:\nif 'iyr' in pass_dict and int(pass_dict['iyr'])>=2010 and int(pass_dict['iyr'])<=2020:\nif 'eyr' in pass_dict and int(pass_dict['eyr'])>=2020 and int(pass_dict['eyr'])<=2030:\nif 'hgt' in pass_dict:\nhgt = pass_dict['hgt']\nnumber = int(hgt[:-2])\nunit = hgt[-2:]\nif (unit == \"cm\" and number >= 150 and number <= 193) or (unit == 'in' and 59 <= number <= 76):\nif 'hcl' in pass_dict and re.match('^#[0-9a-f]{6}\\$',pass_dict['hcl']):\nif 'ecl' in pass_dict and (pass_dict['ecl'] == 'amb' or pass_dict['ecl'] == 'blu' or pass_dict['ecl'] == 'brn' or pass_dict['ecl'] == 'gry' or pass_dict['ecl'] == 'grn' or pass_dict['ecl'] == 'hzl' or pass_dict['ecl'] == 'oth'):\nif 'pid' in pass_dict:\nif len(pass_dict['pid']) == 9 and pass_dict['pid'].isnumeric():\ncount = count +1" ]

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