In mathematical physics , geometricquantization is a mathematical approach to defining a Quantum mechanics quantum theory corresponding to a given classical theory . It attempts to carry out Quantization physics quantization , for which there is in general no exact recipe, in such a way that certain .... The geometricquantization procedure falls into the following three steps prequantization, polarization ... to a vector field X . Geometricquantization of Poisson manifolds and symplectic foliations also is developed ... title GeometricQuantization and Quantum Mechanics publisher Springer isbn 0 387 90496 7 url cite book author N.M.J. Woodhouse year 1991 title GeometricQuantization publisher Clarendon Press isbn ... abs math ph 0208008 William Ritter s review of GeometricQuantization presents a general framework for all problems in physics and fits geometricquantization into this framework http math.ucr.edu home baez quantization.html John Baez s review of GeometricQuantization , by John Baez is short and pedagogical http www.blau.itp.unibe.ch lecturesGQ.ps.gz Matthias Blau s primer on GeometricQuantization ..., N. Roman Roy, Mathematical foundations of geometricquantization, http arxiv.org abs math ph 9904008 arXiv math ph 9904008. Gennadi Sardanashvily G. Sardanashvily , Geometricquantization of symplectic ... equation in classical physics should be built in. One of the earliest attempts at a natural quantization was Weyl quantization , proposed by Hermann Weyl in 1927. Here, an attempt is made to associate ..., this technique leads to deformation quantization , where the product is taken to be a deformation ... quantization scheme, Weyl s map is not satisfactory. For example, the Weyl map of the classical ... on a symplectic manifold math M math . The quantization is simply math f mapsto f cdot i hbar ..., Gennadi Sardanashvily G. Sardanashvily year 2005 title Geometric and Algebraic Topological Methods ... Category Mathematical quantization Category Functional analysis de Geometrische Quantisierung fr Quantification ... more details
Wiktionary quantization NOTOC Quantization is the procedure of constraining something from a relatively large or continuous set of values such as the real number s to a relatively small discrete set such as the integer s . Signal processing Quantization signal processing Quantization image processing Quantization sound processing Quantization music Physics Quantization physics Canonical quantizationGeometricquantization why the set of choices above? There are also Dirac s, Weyl s and Connes quantizations Discrete spectrum , or otherwise discrete quantity Spatial quantization Charge quantization Noncommutative geometry is IMHO largely irrelevant in a dab page. Quantization physics should contain list of approaches including Connes one Quantization noncommutative mathematics Noncommutative geometry Linguistics Quantization linguistics Similar terms Quantification disambig ar ca Quantificaci es Cuantificaci n fr Quantification it Quantizzazione ko pl Kwantowanie ru sv Kvantisering uk ... more details
for quantization of general classical theories Canonical quantization About quantization in quantum field theory Quantization disambiguation Quantization Quantum field theory In physics , quantization ... quantum mechanics from classical mechanics . One also speaks of field quantization , as in the quantization ... , nuclear physics , condensed matter physics , and quantum optics . Quantization methods Quantization ... in various amplitudes. The full specification of a quantization procedure requires methods of performing renormalization. The first method to be developed for quantization of field theories was canonical quantization . While this is extremely easy to implement on sufficiently simple theories, there are many situations where other methods of quantization yield more efficient procedures for computing quantum amplitudes. However, the use of canonical quantization has left its mark on the language and interpretation of quantum field theory. Canonical quantization main canonical quantization Canonical quantization of a field theory is analogous to the construction of quantum mechanics from classical ... field theory Second quantization second quantization . This procedure can be applied to the quantization ... value condensates . Covariant canonical quantization It turns out that there is a way to perform a canonical quantization without having to resort to the noncovariant approach of foliating spacetime and choosing ... way as in canonical quantization. Actually, there is a way to quantize actions with analysis ... . Deformation Quantization Main article Weyl quantization . Also see Moyal bracket , Moyal product Star product , and Wigner quasi probability distribution . Geometricquantization See geometricquantization Loop quantization See Loop quantum gravity Path integral quantization see Feynman path ... http daarb.narod.ru wircq eng.html What is Relativistic Canonical Quantization ? Category Fundamental ... theory Category Mathematical quantization Category Mathematical physics ar ca Quantificaci ... more details
Redshift quantization is the hypothesis that the redshift s of cosmologically distant objects in particular ... of cosmic distance ladder distance and redshift as expressed in Hubble s Law , redshift quantization would either indicate a quantization of the distances of galaxies from the Earth or a problem ... cite book last1 Arp first1 H. year 1998 chapter Quantization of Redshifts url http redshift.vif.com ... of redshift quantization as reason to reject the standard account of the origin and evolution ... of quantization in excess of what is expected due to galaxy cluster ing, ref cite journal last1 ... most cosmologists dispute the existence of redshift quantization beyond a minimal trace due to the distribution of galaxies across voids and filaments. Redshift quantization has also been ... G. Tifft William G. Tifft was the first to investigate possible redshift quantization, or redshift ... Quantization A Review journal Astrophysics and Space Science volume 227 issue 1 2 pages 25 bibcode ... 1980s and early 1990s, four studies on redshift quantization were performed In 1989, Martin R. Croasdale reported finding a quantization of redshifts using a different sample of galaxies in increments ... Image 2dfgrs.png right thumb 300px Rendering of the 2dFGRS data. A strict quantization of redshifts ... quantization of redshifts by some statistical measures. After Tifft made his proposal, discussion ... community became that any quantization was either coincidental or due to so called geometrical ..., E. Sepulveda suggested that a geometric paradigm based on the polytrope theory could account for all ... the universe. Intermediate periodicities are due to other geometric irregularities. These periodicities or apparent quantizations are relics or faithful fossils of a real quantization that occurred in the primordial atom. ref cite journal last1 Sepulveda first1 E. year 1987 title Geometric Paradigm ... quantization in the 2dF survey and found using Napier s own guidelines for testing redshift ... more details
About quantization in music other uses Quantization disambiguation Unreferenced date December 2009 In digital music processing technology, quantization is the process of transforming performed musical notes, which may have some imprecision due to expressive performance, to an underlying musical representation that eliminates this imprecision. The process results in notes being set on beats and on exact fractions of beats. The most difficult problem in quantization is determining which rhythmic fluctuations are imprecise or expressive and should be removed by the quantization process and which should be represented in the output score. For instance, a simple children s song should probably have very coarse quantization, resulting in few different outputted notes. On the other hand, quantizing ... notes, tuplets, etc. The most frequent application of quantization in this context lies within MIDI application software or hardware. MIDI sequencers typically include quantization in their manifest of edit ... note to the closest point on the timing grid. The purpose of quantization in music processing is to provide a more beat accurate timing of sounds. Quantization is frequently applied to a record of MIDI notes created by the use of a musical keyboard or drum machine. Quantization in MIDI is usually ... the entire note by moving both messages together. Sometimes quantization is applied in terms of a percentage, to partially align the notes to a certain beat. Using a percentage of quantization allows for the subtle preservation of some natural human timing nuances. In recent years audio quantization ... on modern day records to tighten the playing of drums, guitar, bass etc. The phrase pitch quantization can refer to pitch correction used in audio production, such as Auto Tune . Programs using quantization ... Note value Auto Tune Quantization signal processing Quantization sound processing Quantization image processing DEFAULTSORT Quantization Music Category Music production Sound tech stub nl Quantize ru ... more details
Unreferenced date December 2009 A first quantization of a physical system is a semi classical mechanics classical treatment of quantum mechanics , in which particles or physical objects are treated using quantum wave function s but the surrounding environment for example a potential well or a bulk electromagnetic field or gravitational field is treated classically. First quantization is appropriate for studying a single quantum mechanical system being controlled by a laboratory Equipment apparatus that is itself large enough that classical mechanics is applicable to most of the apparatus. First quantization is the subject studied in most undergraduate quantum mechanics courses, and in which the Schrodinger Equation and matrix mechanics Heisenberg matrix mechanics together with bra ket notation are most simply applied. It may be contrasted with Second quantization , which includes quantum mechanical uncertainty effects in all aspects of an experiment including the controlling fields and boundary conditions. DEFAULTSORT First Quantization Category Quantum mechanics Quantum stub de Erste Quantisierung ... more details
Trellis quantization is an algorithm that can improve data compression in Discrete cosine transform DCT based encoding methods. It is used to optimize residual Discrete cosine transform DCT coefficients after Motion compensation Motion estimation motion estimation in lossy compression lossy video compression encoders such as Xvid and x264 . Trellis quantization reduces the size of some DCT coefficients while recovering others to take their place. This process can increase quality because coefficients chosen by Trellis have the lowest Rate distortion rate distortion ratio. Trellis quantization effectively finds the optimal quantization for each block to maximize the PSNR relative to bitrate . It has varying effectiveness depending on the input data and compression method. References http www.doom9.org index.html? xvid vdub final.htm VirtualDub Xvid guide mentioning Trellis quantization http ffmpegx.com options.html FFMPEGx option documentation http akuvian.org src x264 trellis.txt Trellis explanation and pseudocode by the x264 author Category MPEG Category Data compression Category Video compression compu sci stub ru ... more details
value is called quantization error or quantization distortion . This error is either due to rounding or truncation. The error signal is sometimes considered as an additional random signal called quantization noise because of its stochastic behaviour. Quantization error models In the typical case, the original signal is much larger than one Least significant bit LSB . When this is the case, the quantization ... distribution continuous uniform distribution . In the rounding case, the quantization error has a mean ... represents 0.113 of the full signal range. At lower amplitudes the quantization error becomes dependent ... band. In order to make the quantization error independent of the input signal, noise with an amplitude ..., completely eliminates the distortion. It is known as dither . Quantization noise model Image quanterr.png thumb 300px Quantization noise for a 2 bit ADC operating at infinite sample rate . The difference between the blue and red signals in the upper graph is the quantization error, which is added to the quantized signal and is the source of noise. Quantization noise is a Model abstract model of quantization error introduced by quantization signal processing quantization in the Analog .... It can be modelled in several different ways. In an ideal analog to digital converter, where the quantization ... distribution covering all quantization levels, the Signal to quantization noise ratio SQNR can be calculated ... of quantization bits. The most common test signals that fulfill this are full amplitude triangle ... mathrm dB , math Here, the quantization noise is once again assumed to be uniformly distributed. When .... Quantization noise power can be derived from math mathrm N frac delta mathrm v 2 12 mathrm dBW ... of quantization, and to imperfections of the ADC circuitry. On the other hand, specifications often ... ADCs, low level signals in high resolution ADCs, and for simple waveforms the quantization noise is not uniformly ... the quantization noise distribution is strongly affected by the exact amplitude of the signal ... more details
to BRST BRST construction, ref J. M. Figueroa O Farrill and T. Kimura, Geometric BRST Quantization ...Confusing date December 2006 In theoretical physics , BRST quantization where the BRST refers to Becchi, Rouet, Stora and Tyutin is a relatively rigorous mathematical approach to Quantization physics quantizing a Field theory mathematics field theory with a gauge symmetry . Quantization physics Quantization ..., BRST quantization becomes more than an alternate way to arrive at anomaly cancelling ghosts. It is a different ... whose states are composed of particles according to the rules familiar from the canonical quantization ... formalism . Technical summary BRST quantization or the BRST formalism is a Differential geometry differential geometric approach to performing consistent, Anomaly physics anomaly free Time dependent ... Tyutin . Its significance for rigorous canonical quantization of a Yang Mills theory and its correct ... and Edward Witten , has clarified the geometric significance of the BRST operator and related fields ... theory lives. One then Quantization physics quantizes the theory to obtain a Hamiltonian system in the interaction ... of QFT, one would have to have said that the Quantization physics quantization and renormalization prescriptions ... relativity , although the latter has proven resistant to quantization so far, for reasons related ... group, beginning with Yang Mills theory , became amenable to quantization in the late 1960s and early ... canonical quantization . Because the definition of the Hamiltonian involves a unit time vector field ... invariant Lagrangian density one also needs the quantization and gauge fixing prescriptions that enter .... The perturbative expansion associated with this Lagrangian, using the method of functional quantization ... 1 gauge to the same set of Feynman rules that one obtains in the method of canonical quantization . But there is an important ... the Feynman rules obtained from naive functional quantization, one finds that one s calculations contain ... Abelian gauge field. From the functional quantization perspective, the unphysical perturbations of the field ... more details
Incomplete date February 2009 Vector quantization is a classical Quantization signal processing quantization technique from signal processing which allows the modeling of probability density functions by the distribution of prototype vectors. It was originally used for data compression . It works by dividing a large set of points coordinate vector vector s into groups having approximately the same number of points closest to them. Each group is represented by its centroid point, as in k means and some other clustering algorithms. The density matching property of vector quantization is powerful ... data correction and density estimation . Vector quantization is based on the competitive learning ... for vector quantization is Pick a sample point at random Move the nearest quantization vector ... Pick a sample point at random Find the quantization vector centroid with the smallest distance sensitivity ... correlated over many samples. Applications Vector quantization is used for lossy data compression ... due to the density matching property of the algorithm . Use in data compression Vector quantization, also called block quantization or pattern matching quantization is often used in lossy data compression ... storage space, so the data is compressed. Due to the density matching property of vector quantization ... of the quantized values. This conserves space and achieves more compression. Twin vector quantization VQF is part of the MPEG 4 standard dealing with time domain weighted interleaved vector quantization. Video codecs based on vector quantization Expand list date August 2008 Cinepak and old versions ... by the MPEG family. Audio codecs based on vector quantization Expand list date August 2008 ... diagram rate distortion function data clustering LVQ Learning Vector Quantization Centroidal Voronoi tessellation Neural gas Growing Neural Gas , a neural network like system for vector quantization ... links http www.data compression.com vq.html http qccpack.sourceforge.net QccPack Quantization, Compression ... more details
above In computer graphics , color quantization or color image quantization is a process that reduces ... quantization on bitmaps have been studied since the 1970s. Color quantization is critical for displaying ... quantization is primarily used in computer graphics research literature in applications, terms such as optimized ... the best possible. Algorithms Most standard techniques treat color quantization as a problem of clustering ... clustering algorithm can be applied to color quantization, and vice versa. After the clusters are located ... consistent with perceptual difference. The most popular algorithm by far for color quantization .... Before this time, most color quantization was done using the population algorithm or population ... entry are shown. If the palette is fixed, as is often the case in real time color quantization systems such as those used in operating systems, color quantization is usually done using the straight ... overkill. File Spatial color quantization rainbow, 4 colors.png frame 256px A colorful image reduced to 4 colors using spatial color quantization. Color quantization is frequently combined with dither ... gradients and give the appearance of a larger number of colors. Some modern schemes for color quantization ... color quantization , conceived by Puzicha, Held, Ketterer, Buhmann, and Fellner of the University .... Color quantization helped to justify this tradeoff by making it possible to display many high ... and many other operating systems automatically perform quantization and dithering when viewing high ... hardware. Nowadays, color quantization is mainly used in GIF and Portable Network Graphics PNG images ... , only supports up to 256 colors, necessitating quantization for many images. Some early web browsers ... much smaller in filesize without much visual degradation by application of color quantization, since ... to a digital representation necessarily involves some quantization. In practice, 24 bit color ... colors available on early computers, different quantization algorithms produced very different looking ... more details
Flux quantization is a quantum phenomenon in which the magnetic field is quantized in the unit of math h 2e math , also known variously as flux quanta, fluxoids, vortices or fluxon s. Flux quantization occurs in Type II superconductor s subjected to a magnetic field. Below a critical field H sub c1 sub , all magnetic flux is expulsed according to the Meissner effect and perfect diamagnetism is observed, exactly as in a Type I superconductor . Up to a second critical field value, H sub c2 sub , flux penetrates in discrete units while the bulk of the material remains superconducting. Both critical fields are temperature dependent, and tabulated values are the zero temperature extrapolation unless otherwise noted. See also Flux pinning Magnetic flux quantum Category Superconductivity Category Quantum mechanics ... more details
Quantum field theory In physics , canonical quantization is a procedure for quantization physics quantizing ... to the extent possible in canonical quantization. This method was used in the context ... theory context, it is also called second quantization, in contrast to the semi classical first quantization ... this technique to the Quantization of the electromagnetic field quantization of the electromagnetic ... quantization may have been first coined by Pascual Jordan. Citation needed date May 2010 Quantum ... may be considered a quantum analog of the canonical structure of classical mechanics. Second quantization ... quantization procedure is applied to quantum field theory , the classical field physics ... a theory that was already quantized, leading to the term second quantization in the early literature, which is still used to describe field quantization, even though the modern interpretation is different. One drawback to canonical quantization for a relativistic field is that by relying on the Hamiltonian ... . The details of the canonical quantization depend on the field being quantized, and whether ... quantization procedure. ref This treatment is based primarily on Ch. 1 in cite book last1 ... docs bookwebfinal.pdf ref For simplicity, the quantization can be carried in a 1 ... left frac 1 2 pi 2 frac 1 2 partial x phi 2 frac 1 2 m 2 phi 2 V phi right . math Canonical quantization ... math in one of these vacua may be considered a condensate of the field math phi math . Canonical quantization ... in the Higgs mechanism in the standard model of particle physics . Mathematical quantization The classical ... algebra of operators is an deformation quantization deformation of the algebra of smooth ..., deformations are highly nonunique&mdash quantization is an art , and is specified by the physical ... operators Dirac bracket Moyal bracket Weyl quantization References references Historical References ... Canonical Quantization ? http www.quantumfieldtheory.info Pedagogic Aides to Quantum Field Theory ... more details
mergefrom Mass noun date October 2010 mergefrom Count noun date October 2010 mergefrom Telicity date October 2010 In linguistics , a quantized expression is such that, whenever it is true of some entity, it is not true of any proper subparts of that entity. Example If something is an apple , then no proper subpart of that thing is an apple . If something is water , then many of its subparts will also be water . Hence, apple is quantized, while water is not. Quantization has proven relevant to the proper characterization of grammatical telicity roughly, sentences that present events as bounded unbounded in time and the mass noun mass count distinction for nouns . The notion was first applied to linguistic semantics by the linguist Manfred Krifka . Formally, a quantization predicate QUA can be defined as follows, where math U math is the universe of discourse , and math F math is a variable over set mathematics sets , and math p math is a mereology mereological part mathematical structure structure on math U math with math p math the mereology mereological part of relation mathematics relation . center math forall F subseteq U p QUA F iff forall x,y F x wedge F y Rightarrow neg x p y math center See also Fewer vs. less References Krifka, Manfred 1989. Nominal reference, temporal constitution and quantification in event semantics. In Renate Bartsch, Johan van Benthem and Peter van Emde Boas eds. , Semantics and Contextual Expressions 75 115. Dordrecht Foris. Category Logic Category Semantics Category Grammar nl Kwantisatie taalkunde ... more details
In mathematics and physics , in the area of quantum mechanics , Weyl quantization is a method for systematically associating a quantum mechanical Hermitian operator with a classical kernel function in phase space invertibly. A synonym is phase space quantization . The crucial correspondence map from phase space functions to Hilbert space operator mathematics operators underlying the method is called the Weyl transformation , not to be confused with a different definition of the Weyl transformation , and was first detailed by Hermann Weyl ref Hermann Weyl H.Weyl , Quantenmechanik und Gruppentheorie , Zeitschrift f r Physik , 46 1927 pp. 1 46, doi 10.1007 BF02055756 . ref in 1927. In some contrast to Weyl s original intentions in seeking a consistent quantization scheme, this map merely amounts to a change of representation . It need not connect classical with quantum quantities the starting phase space function may well depend on Planck s constant . Indeed, in some familiar cases involving angular momentum, it does. The inverse of this Weyl transformation is the Wigner quasi probability distribution The Wigner Weyl transformation Wigner map , which reverts from Hilbert space to the phase ... integrable functions . Deformation quantization Intuitively, a Deformation theory deformation of a mathematical ... setup in deformation quantization theory is to start with an algebraic structure say a Lie algebra ..., one may define a Noncommutative geometry noncommutative torus as a deformation quantization through ... quantization . Insofar as the algebra of functions on a space determines the geometry of that space ... brackets . For the general formula on arbitrary Poisson manifold s, cf. the Kontsevich quantization ... Expectation values in phase space quantization are obtained isomorphically to tracing operator observables ... its uncertainty principle. Generalizations In more generality, Weyl quantization is studied in cases ... theorem References references Category Mathematical quantization Category Mathematical physics Category ... more details
Landau quantization in quantum mechanics is the quantization of the cyclotron orbits of charged particles in magnetic fields. As a result, the charged particles can only occupy orbits with discrete energy values, called Landau levels. The Landau levels are Degenerate energy level degenerate , with the number of electrons per level directly proportional to the strength of the applied magnetic field. Landau quantization is directly responsible for oscillations in electronic properties of materials as a function of the applied magnetic field. It is named after the Soviet physicist Lev Landau . Derivation Consider a two dimensional system of non interacting particles with charge math q math and spin math S math confined to an area math A L x L y math in the x y plane. Apply a uniform magnetic field math mathbf B begin pmatrix 0 0 B end pmatrix math along the z axis. Using CGS units, the Hamiltonian quantum mechanics Hamiltonian of this system is math hat H frac 1 2m hat mathbf p q hat mathbf A c 2. math Here, math hat mathbf p math is the Canonical coordinates canonical momentum operator and math hat mathbf A math is the Magnetic potential electromagnetic vector potential , which is related to the magnetic field by math mathbf B mathbf nabla times mathbf A . , math There is some freedom in the choice of vector potential for a given magnetic field. However, the Hamiltonian is gauge invariance gauge invariant , which means that adding the gradient of a scalar field to A changes the overall phase of the wave function by an amount corresponding to the scalar field. Physical properties are not influenced by the specific choice of gauge. For simplicity in calculation, choose the Landau gauge , which is math hat mathbf A begin pmatrix 0 B hat x 0 end pmatrix . math where math B mathbf ... rise to non standard quantization conditions on the magnetic field, involving in principle both solutions ... in energy of the different Landau levels, blurring the effect of the quantization. In any case, the motion ... more details
Distinguish Geohashing In computer science , geometric hashing is originally a method for efficiently finding two dimensional objects represented by discrete points that have undergone an affine transformation , though extensions exist to some other object representations and transformations. In an off line step, the objects are encoded by treating each pairs of points as a geometric Basis linear algebra basis . The remaining points can be represented in an Invariant mathematics invariant fashion with respect to this basis using two parameters. For each point, its Quantization signal processing quantized transformed coordinates are stored in the hash table as a key, and indices of the basis points as a value. Then a new pair of basis points is selected, and the process is repeated. In the on line recognition step, randomly selected pairs of data points are considered as candidate bases. For each candidate basis, the remaining data points are encoded according to the basis and possible correspondences from the object are found in the previously constructed table. The candidate basis is accepted if a sufficiently large number of the data points index a consistent object basis. Geometric hashing was originally suggested in computer vision for object recognition in 2D and 3D, ref name ... to different problems such as structural alignment of protein s. Geometric Hashing in Computer Vision Geometric Hashing is a method used for object recognition. Let s say that we want to check if a model image can be seen in an input image. This can be accomplished with geometric hashing. The method ..., the input Image may contain the object in mirror transform. Therefore, geometric hashing should ... or objects . Actually, using 3 points for the basis is another approach for geometric hashing. Geometric ... courses cs468 01 winter papers wr ghao 97.pdf Geometric Hashing An Overview. IEEE Computational Science and Engineering, 4 4 , 10 21. references DEFAULTSORT Geometric Hashing Category Geometric data ... more details
Geometric integration can refer to Homological integration &ndash a method for extending the notion of integral to manifold s. Geometric integrator , a numerical method that preserves of geometric properties of the exact flow of a differential equation . disambig Category Mathematical disambiguation ... more details
In Classical mechanics classical and quantum mechanics , the geometric phase , Pancharatnam Berry phase named after S. Pancharatnam and Michael Berry physicist Sir Michael Berry , Pancharatnam phase or most commonly Berry phase , is a Phase waves phase acquired over the course of a Period physics cycle , when the system is subjected to cyclic adiabatic process quantum mechanics adiabatic process es, which results from the geometrical properties of the parameter space of the Hamiltonian quantum mechanics Hamiltonian . The phenomenon was first discovered in 1956, ref cite journal author S. Pancharatnam title Generalized Theory of Interference, and Its Applications. Part I. Coherent Pencils journal Proc. Indian Acad. Sci. A volume 44 pages 247 262 year 1956 ref and rediscovered in 1984. ref cite journal author M. V. Berry journal Proc. R. Soc. Lond. A title Quantal Phase Factors Accompanying Adiabatic Changes volume 392 issue 1802 pages 45 57 year 1984 doi 10.1098 rspa.1984.0023 bibcode 1984RSPSA.392...45B ref It can be seen in the Aharonov Bohm effect and in the conical intersection of potential ... analogue of the Berry phase is known as the Hannay angle . Theory In general the geometric phase ... of geometric phases The Foucault pendulum One of the easiest examples is the Foucault pendulum . An easy explanation in terms of geometric phases is given by von Bergmann and von Bergmann ref cite journal ... changes of parameters. The stochastic pump effect can be interpreted in terms of a geometric phase ... Christian and Kazimir Wanelik title Resource Letter GPP 1 Geometric Phases in Physics journal Am ... Geometric Phase in Optics http departments.colgate.edu physics faculty EGalvez articles PreprintRflash.pdf ... of diabetic potentials and the quantization of the nonadiabatic matrix , J. Phys. Chem. A 104 ... paths , J. Chem. Phys. 117, 7405 2002 . Frank Wilczek and Alfred Shapere, Geometric Phases in Physics ... Wiley year 2006 isbn 0471778915 DEFAULTSORT Geometric Phase Category Quantum mechanics Category ... more details
Geometric tomography is a mathematical field that focuses on problems of reconstructing homogeneous often convex objects from tomographic data this might be X rays, projections, sections, brightness functions, or covariograms . More precisely, according to R.J. Gardner who introduced the term , Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections shadows on planes or cross sections by planes. ref name ref1 Gardner, R.J., Geometric Tomography, Cambridge University Press, Cambridge, UK, 2nd ed., 2006 ref Theory A key theorem in this area states that any convex body in math E n math can be determined by parallel, coplanar X rays in a set of four directions whose slopes have a transcendental cross ratio . See also Tomography Discrete tomography References reflist External links http cgm.cs.mcgill.ca godfried research tomography.html Geometric tomography applet I http faculty.wwu.edu gardner GeometricTomography.html Geometric tomography applet II Category Tomography Category Projective geometry ... more details
Geometric analysis is a mathematics mathematical discipline at the interface of differential geometry and differential equations . It includes both the use of geometrical methods in the study of partial differential equation s when it is also known as geometric PDE , and the application of the theory of partial differential equations to geometry. It incorporates problems involving curves and surfaces, or domains with curved boundaries, but also the study of Riemannian manifold s in arbitrary dimension. The calculus of variations is sometimes regarded as part of geometric analysis, because differential equations arising from variational principle s have a strong geometric content. Geometric analysis also includes global analysis , which concerns the study of differential equations on manifolds, and the relationship between differential equations and topology . References cite book title Riemannian geometry and Geometric Analysis first J rgen last Jost edition 4th edition year 2005 publisher Springer isbn 978 3540259077 cite book title Groups and Geometric Analysis Integral Geometry, Invariant Differential Operators and Spherical Functions first Sigurdur last Helgason authorlink Sigurdur Helgason mathematician edition 2nd edition year 2000 publisher American Mathematical Society isbn 978 0821826737 cite book title Geometric Analysis on Symmetric Spaces first Sigurdur last Helgason edition 2nd edition year 2008 publisher American Mathematical Society isbn 978 0821845301 Category Mathematical analysis Category Differential geometry mathanalysis stub ... more details
Geometric calculus extends the geometric algebra to include differentiation and integration including differential geometry and differential forms. ref David Hestenes , Garrett Sobczyk Clifford Algebra to Geometric Calculus, a Unified Language for mathematics and Physics Dordrecht Boston G.Reidel Publ.Co., 1984, ISBN 90 277 2561 6 ref File Geometric Calculus Family Tree.png right 300px thumb Figure 1 from 32 A diagram of the history of Geometric Calculus Given a geometric algebra, the vector derivative is defined as the operator math 1 . Essentially, the vector derivative is defined so that the GA version of Green s theorem is true, math oint A dA nabla f oint dA dx f math and then one can write math nabla f nabla cdot f nabla wedge f math as a geometric product, effectively generalizing Stokes theorem including the differential forms version of it . In math 1D math when A is a curve with endpoints math a math and math b math , then math oint A dA nabla f oint dA dx f math reduces to math int a b dx nabla f int a b dx cdot nabla f int a b df f b f a math or the fundamental theorem of integral calculus. Also developed are the concept of vector manifold and geometric integration theory which generalizes Cartan s differential forms . References reflist differential geometry stub Category Calculus ... more details
. Geometric models can be built for objects of any dimension in any space geometric space . Both 2D geometric model 2D and 3D modeling 3D geometric models are extensively used in computer graphics . 2D geometric model 2D model s are important in computer typography and technical drawing . 3D ... . Geometric models are usually distinguished from procedural modeling procedural and Object Oriented ... for instance, geometric shapes can be represented by obect oriented programming objects a digital image can be interpreted as a collection of color ed Square geometry square s and geometric shapes ... often requires a combination of geometric and procedural techniques. Geometric problems originating ... geometric design, and discrete differential geometry. ref H. Pottmann, S. Brell Cokcan and J. Wallner ... wps find journaldescription.cws home 505604 description description Computer Aided Geometric Design Category Geometric algorithms Category Computational science Category Computer aided design de Geometrische ... more details
Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics the study of Face geometry faces of convex polyhedron convex polyhedra , convex geometry the study of convex set s, in particular combinatorics of their intersections , and discrete geometry , which in turn has many applications to computational geometry . Other important areas include metric geometry of polyhedra , such as the Cauchy s theorem geometry Cauchy theorem on rigidity of convex polytopes. The study of regular polytope s, Archimedean solid s, and kissing number s is also a part of geometric combinatorics. Special polytopes are also considered, such as the permutohedron , associahedron and Birkhoff polytope . Further reading http www.cis.upenn.edu cis610 topics.pdf Topics in Geometric Combinatorics http www.ams.org bookstore?fn 20&arg1 geotopo&item PCMS 13 Geometric Combinatorics , Edited by Ezra Miller and Victor Reiner http scholar.google.co.uk scholar?q 22Combinatorics of Finite Geometries 22 Combinatorics of Finite Geometries Category Combinatorics Category Discrete geometry combin stub bs Geometrijska kombinatorika ... more details
Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes. The shapes studied in geometric modeling are mostly two or three dimension al, although many of its tools and principles can be applied to sets of any finite dimension. Today most geometric modeling is done with computers and for computer based applications. 2D geometric model Two dimensional model s are important in computer typography and technical drawing . 3D geometric model Three dimensional model s are central to computer aided design and computer aided manufacturing manufacturing CAD CAM , and widely used in many applied technical fields such as civil engineering civil and mechanical engineering , architecture , geologic modeling geology and medical image processing . ref Farin, G. A History of Curves and Surfaces in CAGD, http books.google.com books?id 0SV5G8fgxLoC&printsec frontcover&dq Computer Aided GEOMETRIC DESIGN&source gbs summary s&cad 0 Handbook of Computer Aided Geometric Design ref Geometric models are usually distinguished from procedural model procedural and object oriented model s, which define the shape implicitly by an opaque algorithm that generates its appearance. They are also contrasted with digital image s and volumetric model s which represent the shape as a subset of a fine regular partition of space and with fractal models that give an infinitely recursive definition of the shape. However, these distinctions are often blurred for instance, a digital image can be interpreted as a collection of color ed square geometry square s and geometric shapes such as circle s are defined by implicit ... Geometric Modeling and Industrial Geometry http demonstrations.wolfram.com topic.html?topic 3D Graphics&limit .... I. Wu & M. Abdulla, Landmobile Radiowave Multipaths DOA Distribution Assessing Geometric Models ... Geometric algorithms Category Computational science Category Computer aided design de Geometrische ... more details
Encyclopedia
top
