In mathematics , approximationtheory is concerned with how function mathematics function s can best be approximation approximated with simpler function mathematics functions , and with Quantitative property quantitative ly characterization mathematics characterizing the approximation error errors introduced ... journals Journal of ApproximationTheory Constructive Approximation East Journal on Approximations ... Fourier series Schauder basis Pad approximant References N.I.Achiezer Akhiezer , Theory of approximation ..., Theory of approximation of functions of a real variable , 1963 ISBN 048667830X C. Hastings, Jr ... related topic is the approximation of functions by generalized Fourier series , that is, approximations ... or Rational function rational ratio of polynomials approximations. The objective is to make the approximation ... Image Logerror.png thumb 300px Error between optimal polynomial and log x red , and Chebyshev approximation ... optimal polynomial and exp x red , and Chebyshev approximation and exp x blue over the interval 1 ... approximation to f than P . In particular, Q is closer to f than P for each value x sub i sub where ... for a polynomial of degree N . Chebyshev approximation One can obtain polynomials very close to the optimal ... for the log function. Chebyshev approximation is the basis for Clenshaw Curtis quadrature , a numerical ... math and math x n 2 math are presumably the end points of the interval of approximation , these equations ... Remez , Sur le calcul effectif des polynomes d approximation de Tschebyscheff 1934 C. R. Acad. Sci., Paris, 199 , 337 340, K. G. Steffens, The History of ApproximationTheory From Euler to Bernstein Birkhauser, Boston 2006 ISBN 0817643532 External links http www.math.technion.ac.il hat History of ApproximationTheory HAT http www.emis.de journals SAT Surveys in ApproximationTheory SAT Category Approximationtheory Category Numerical analysis bs Teorija aproksimacije es Teor a de la aproximaci n fr Th orie de l approximation hi it Teoria dell approssimazione he ... more details
Infobox Journal title Journal of ApproximationTheory cover File Journal of Approximation Theory.gif discipline Approximationtheory abbreviation J. Approx. Theory publisher Elsevier country United States U.S. ISSN 0021 9045 CODEN JAXTAZ history 1968 to present website http www.math.ohio state.edu JAT The Journal of ApproximationTheory is devoted to advances in pure and applied approximationtheory and related areas. ref cite web url http www.elsevier.com wps find journaldescription.cws home 622853 description description title Journal of ApproximationTheory Elsevier accessdate 2007 04 30 ref External links http www.math.ohio state.edu JAT Journal of ApproximationTheory web site References references Category Mathematics journals Category Approximationtheory Category Publications established in 1968 Category Elsevier academic journals Category English language journals Category Monthly journals math journal stub sci journal stub ... more details
laws theory and empirical measurement s, which are constantly compared to one another. The approximation ... using a small number of Significant figures significant digits . Approximationtheory is a branch of mathematics, a quantitative part of functional analysis . Diophantine approximation deals with approximations of real number s by rational number s. Related to approximation of functions is the Asymptotic ...Certainty Unreferenced date September 2007 An approximation is an Accuracy and precision inexact representation of something that is still close enough to be useful. Although approximation is most often ... representation is known, an approximation may yield a sufficiently accurate solution while reducing ... may be repeated until a satisfactorily precise solution is obtained. The use of Perturbation theory ... the convergence rate of evolutionary algorithms, fitness approximation that leads to build model of the fitness function to choose smart search steps is a good solution. The type of approximation ... to this data, and the savings usually in time and effort that can be achieved by approximation ... . Mathematics class wikitable align right Symbols representing approximation style font size xx large & 8784 general approximation style font size xx large asymptotic analysis Approximation usually occurs ... key and typing X. See also col begin col break Approximation error Congruence relation Estimation Fitness approximation Fermi estimate Linear approximation Newton s method col break Numerical analysis Orders of approximation Runge Kutta methods Successive Approximation ADC Taylor series Least squares col break Wiktionarypar approximation col end Category Approximations Category Numerical analysis br Tostadur bg cs Aproximace da Approksimation de Approximation es Aproximaci n eo Proksimuma kalkulado fr Approximation ko hi is N mundun it Approssimazione he nl Benadering ja pl Aproksymacja pt Aproxima o ru simple Approximation fi Approksimaatio ... more details
Refimprove date December 2009 In particle physics , the quenched approximation is an approximation often used in lattice gauge theory in which the quantum loops of fermion s in Feynman diagram s are neglected. Equivalently, the corresponding one loop determinants are set to one. This approximation is often forced upon the physicists because the calculation with the Grassmann number s is computationally very difficult in lattice gauge theory. Recent calculations ref name urlRev. Mod. Phys. 82, 1349 2010 Nonperturbative QCD simulations with 2 1 flavors of improved staggered quarks cite web url http link.aps.org doi 10.1103 RevModPhys.82.1349 author A. Bazavov et. al title Rev. Mod. Phys. 82, 1349 2010 Nonperturbative QCD simulations with 2 1 flavors of improved staggered quarks format work accessdate doi 10.1103 RevModPhys.82.1349 ref typically avoid the quenched approximation. References Reflist DEFAULTSORT Quenched Approximation Category Lattice models Quantum stub pt Aproxima o temperada ... more details
In computer science , hardness of approximation is a field that studies the complexity of finding near optimal solutions to optimization problems. It complements the study of approximation algorithm s by proving, for certain problems, a limit on the factors with which their solution can be efficiently approximated. Typically such limits show a factor of approximation beyond which a problem becomes NP hard, implying that finding a polynomial time approximation for the problem is impossible unless P NP . Some hardness of approximation results, however, are based on other hypotheses, a notable one among which is the unique games conjecture . Since the early 1970s it was known that many optimization problems could not be solved in polynomial time unless NP P , but in many of these problems the optimal solution could be efficiently approximated to a certain degree. In the early 1990s, with the development of PCP complexity PCP theory, it became clear that there is a limit to the approximability of many of these optimization problems for many optimization problems there is a threshold beyond which they are NP hard to approximate. Hardness of approximationtheory deals with studying the approximation threshold of such problems. Examples For an example of an NP hard optimization problem that is hard to approximate, see set cover . See also PCP theorem Further reading http www.cs.berkeley.edu luca pubs inapprox.ps Survey, a good starter by Luca Trevisan External links http www.cs.washington.edu education courses 533 05au CSE 533 The PCP Theorem and Hardness of Approximation, Autumn 2005 , syllabus from the University of Washington , Venkatesan Guruswami and Ryan O Donnell Category Computational complexity theory Category Mathematical optimization comp sci theory stub ... more details
Rational approximation may refer to Diophantine approximation , the approximation of real numbers by rational numbers. Pad approximation , the approximation of functions by rational functions. See also Simple rational approximation Mathdab ... more details
Infobox Journal cover discipline mathematics abbreviation Constr. Approx. publisher Springer Science Business Media Springer country United States U.S. ISSN 0176 4276 eISSN 1432 0940 CODEN CONSA? history 1985 to present website http www.math.vanderbilt.edu ca Constructive Approximation is an international mathematics journal dedicated to Approximations and Expansions and related research in computation, function theory, functional analysis, interpolation spaces and interpolation of operators, numerical analysis, space of functions, special functions, and applications. ref cite web url http www.springer.com west home?SGWID 4 102 70 1056700 detailsPage journal description&changeHeader true&SHORTCUT www.springer.com journal 00365 about title Constructive Approximation Numerical Analysis Journals, Books & Online Media ¦ Springer accessdate 2007 04 30 ref References references External links http www.math.vanderbilt.edu ca Constructive Approximation web site Category Mathematics journals Category Approximation theory Category English language journals Category Publications established in 1985 Category Springer academic journals Category Bimonthly journals math journal stub sci journal stub ... more details
In number theory , the field of Diophantine approximation , named after Diophantus of Alexandria , deals with the approximation of real number s by rational number s. The absolute value of the difference ... approximation because its denominator is smaller. Approximation to algebraic numbers The theory ... course by Michel Waldschmidt. DEFAULTSORT Diophantine Approximation Category Number theory Category ... measure of how good the approximation is. However, since the rational numbers are dense in the real .... So this measure tells us nothing about the quality of the approximation. A better measure of the quality of the approximation is by comparison of the difference to the size of the denominator ... numbers. This link between diophantine approximation and transcendental number transcendence theory continues to the present day. Many of the proof techniques are shared between the two areas ... generalised this to the case of simultaneous approximation. The proofs were difficult, and not effective results in number theory effective . This means that we cannot use the results or their proofs .... Duffin and A. C. Schaeffer, Khintchine s problem in metric Diophantine approximation , Duke Mathematical ... not zero. Uniform distribution Another topic that has seen a thorough development is the theory of equidistributed ... that Diophantine approximation results were closely related to the general problem of cancellation in exponential sums, which occurs throughout analytic number theory in the bounding of error terms ... remaining in Diophantine approximation, for example the Littlewood conjecture . Recent developments ... outlined a broad program rooted in ergodic theory that allows one to prove number theoretic results ... to classical problems in Diophantine approximation. Among its notable successes are the proof of the decades ... Khinchin in metric Diophantine approximation have also been obtained within this framework. See also ... Cassels authorlink J. W. S. Cassels title An introduction to Diophantine approximation series Cambridge ... more details
The need for function approximations arises in many branches of applied mathematics , and computer science in particular. In general, a function approximation problem asks us to select a function mathematics function among a well defined class that closely matches approximates a target function in a task specific way. One can distinguish two major classes of function approximation problems First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions for example, special function s can be approximated by a specific class of functions for example, polynomial s or rational function s that often have desirable properties inexpensive computation, continuity, integral and limit values, etc. . Second, the target function, call it g , may be unknown instead of an explicit formula, only a set of points of the form x , g x is provided. Depending on the structure of the domain mathematics domain and codomain of g , several techniques for approximating g may be applicable. For example, if g is an operation on the real number s, techniques of interpolation , extrapolation , regression analysis , and curve fitting can be used. If the codomain clarify date December 2010 of g is a finite set, one is dealing with a statistical classification classification problem instead. To some extent the different problems regression, classification, fitness approximation have received a unified treatment in statistical learning theory , where they are viewed as supervised learning problems. See also Radial basis function network Fitness approximation DEFAULTSORT Function Approximation Category Statistical models Category Machine learning Category Statistical approximations mathanalysis stub statistics stub hr Aproksimacija funkcija sr ... more details
algorithms Category Computational complexity theory Category Approximation algorithms cs Aproxima n ...In computer science and operations research , approximation algorithms are algorithm s used to find approximate solutions to optimization problem s. Approximation algorithms are often associated with NP ... wants provable solution quality and provable run time bounds. Ideally, the approximation is optimal up to a small constant factor for instance within 5 of the optimal solution . Approximation algorithms ... expensive due to the input size. A typical example for an approximation algorithm is the one ... twice as large as the optimal one. This is a constant factor approximation algorithm with a factor ... be approximated within any factor greater than 1 such a family of approximation algorithms is often called a polynomial time approximation scheme or PTAS . Others are impossible to approximate within ... approximation algorithms emerge from the linear programming relaxation of the integer program. Not all approximation algorithms are suitable for all practical applications. They often use IP LP semidefinite ... which lead to difficult implementation problems. Also, some approximation algorithms have ... complexity theory since the 1990 result of Feige, Goldwasser, Lovasz, Safra and Szegedy ..., it has now been shown that Johnson s 1974 approximation algorithms for Max SAT, Set Cover, Independent Set and Coloring all achieve the optimal approximation ratio, assuming P NP. Performance guarantees For some approximation algorithms it is possible to prove certain properties about the approximation of the optimum result. For example, in the case of a approximation algorithm A it has been ... the relative performance guarantee . An approximation algorithm has an absolute performance guarantee ... to return solutions with a performance guarantee of at most r n , then A is said to be an r n approximation algorithm and has an approximation ratio of r n . Likewise, a problem with an r n approximation ... more details
The Kirkwood superposition approximation was introduced by Matsuda 2000 as a means of representing a discrete probability distribution . The name apparently refers to a 1942 paper by John Gamble Kirkwood John G. Kirkwood . ref Kirkwood, J. G. & Boggs, E. M. 1942 . The radial distribution function in liquids . J. Chem. Phys. 10, 394. ref The Kirkwood approximation for a discrete probability density function math P x 1 ,x 2 , ldots ,x n math is given by math P prime x 1,x 2, ldots ,x n frac frac frac prod mathcal T n 1 subseteq mathcal V p mathcal T n 1 prod mathcal T n 2 subseteq mathcal V p mathcal T n 2 vdots prod mathcal T 1 subseteq mathcal V p mathcal T 1 math where math prod mathcal T i subseteq mathcal V p mathcal T i math is the product of probabilities over all subsets of variables of size i in variable set math scriptstyle mathcal V math . This kind of formula has been considered by Watanabe 1960 and, according to Watanabe, also by Robert Fano. For the three variable case, it reduces to simply math P prime x 1,x 2,x 3 frac p x 1,x 2 p x 2,x 3 p x 1,x 3 p x 1 p x 2 p x 3 math The Kirkwood approximation does not generally produce a valid probability distribution the normalization condition is violated . Watanabe claims that for this reason informational expressions of this type are not meaningful, and indeed there has been very little written about the properties of this measure. The Kirkwood approximation is the probabilistic counterpart of the interaction information . Judea Pearl 1988 3.2.4 indicates that an expression of this type can be exact in the case of a decomposable model, that is, a probability distribution that admits a graph mathematics graph structure whose clique graph theory cliques form a tree graph theory tree . In such cases, the numerator contains the product of the intra clique joint distributions and the denominator contains the product of the clique ... Journal of Research and Development 4 , 66&ndash 82. Category probability theory Category Discrete ... more details
In mathematics , a Banach space is said to have the approximation property AP in short , if every compact operator is a limit of finite rank operator s. The converse is always true. Every Hilbert space has this property. There are, however, Banach space s which do not Enflo published the first counterexample in an 1973 article. However, a lot of work in this area was done by Grothendieck 1955 . Later many other counterexamples were found. The space of bounded operator s on math ell 2 math does not have the approximation property Szankowski . The spaces math ell p math for math p neq 2 math and math c 0 math see Sequence space have closed subspaces that do not have the approximation property. Definition A Banach space math X math is said to have the approximation property, if, for every compact set math K subset X math and every math varepsilon 0 math , there is an operator mathematics operator math T colon X to X math of finite rank so that math Tx x leq varepsilon math , for every math x in K math . Some other flavours of the AP are studied Let math X math be a Banach space and let math 1 leq lambda infty math . We say that math X math has the math lambda math approximation property math lambda math AP , if, for every compact set math K subset X math and every math varepsilon 0 math ... is said to have bounded approximation property BAP , if it has the math lambda math AP for some math lambda math . A Banach space is said to have metric approximation property MAP , if it is 1 AP. A Banach space is said to have compact approximation property CAP , if in the definition of AP an operator ... Robert G. Bartle title MR0402468 53 6288 Review of Per Enflo s A counterexample to the approximation ... year 1977 MR 402468 Per Enflo Enflo, P. A counterexample to the approximation property in Banach spaces ... s example of a Banach space without the approximation property . S minaire Goulaouic Schwartz 1972 ... York, 1981. viii 880 pp. ISBN 3 540 10394 5. MR 610799 Category Operator theory Category Banach ... more details
The eikonal approximation is a method of approximation useful in wave scattering equations within the realms of quantum mechanics , optics , quantum electrodynamics , and partial wave expansion Informal description The main advantage the eikonal approximation offers is that the equations reduce to a differential equation in a single variable. This reduction into a single variable is the result of the straight line approximation or the eikonal approximation which allows us to choose the straight line as a special direction. Relation to the WKB approximation The early steps involved in the eikonal approximation in quantum mechanics are very closely related to the WKB approximation . The WKB approximation involves an expansion in terms of Planck s constant . It, like the eikonal approximation, reduces the equations into a differential equation in a single variable. But the difficulty with the WKB approximation is that this variable is described by the trajectory of the particle which, in general, is complicated. The advantage of the eikonal approximation is that the classical trajectory is a straight line. Formal description Empty section date July 2010 References http www.nhn.ou.edu shajesh eikonal sp.pdf Eikonal Approximation K. V. Shajesh Department of Physics and Astronomy, University of Oklahoma Category Mathematical analysis ... more details
approximation . IEEE Trans. Inform. Theory, 21 pp.263 271 1975 . ref See also Stochastic gradient descent Stochastic optimization Simultaneous perturbation stochastic approximation References reflist ...Stochastic approximation methods are a family of iterative stochastic optimization algorithm s that attempt to find zeroes or extrema of functions which cannot be computed directly, but only estimated via noisy observations. The first, and prototypical, algorithms of this kind were the Robbins Monro and Kiefer Wolfowitz algorithms. NOTOC Robbins Monro algorithm In the Herbert Robbins Robbins Monro algorithm, introduced in 1951 ref name rm A Stochastic Approximation Method, Herbert Robbins and Sutton Monro, Annals of Mathematical Statistics 22 , 3 September 1951 , pp. 400 407. ref , one has a function math M x math for which one wishes to find the value of math x math , math x 0 math , satisfying math M x 0 alpha math . However, what is observable is not math M x math , but rather a random variable math N x math such that math E N x x M x math . The algorithm is then to construct a sequence math x 1, x 2, dots math which satisfies math x n 1 x n a n alpha N x n math . Here, math a 1, a 2, dots math is a sequence of positive step sizes. Herbert Robbins Robbins and Monro proved ref name rm sup , Theorem 2 sup that math x n math convergence of random variables converges in math L 2 math and hence also in probability to math x 0 math provided that math N x math is uniformly bounded, math ... models, and so on. ref name kushneryin Stochastic Approximation Algorithms and Applications , Harold ... Approximation and Recursive Algorithms and Applications , 2003, ISBN 0387008942. ref ref Stochastic Approximation and Recursive Estimation , Mikhail Borisovich Nevel son and Rafail Zalmanovich Has ... theory , in which case the unknown function which we wish to optimize or find the zero of may vary ... and R. Douglas Martin were the first to use stochastc approximation in 1975 when dealing with Robust ... more details
distinguish2 the Born Oppenheimer approximation In scattering theory and, in particular in quantum mechanics , the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. It is the perturbation theory perturbation method applied to scattering by an extended body. It is accurate if the scattered field is small, compared to the incident field, in the scatterer. For example, the radar scattering of radio waves by a light styrofoam column can be approximated by assuming that each part of the plastic is polarized by the same electric field that would be present at that point without the column, and then calculating the scattering as a radiation integral over that polarization distribution. Born approximation to the Lippmann Schwinger equation The Lippmann Schwinger equation for the scattering state math vert Psi mathbf p pm rangle math with a momentum p and out going or in going &minus boundary condition s is math vert Psi mathbf p pm rangle vert Psi mathbf p circ rangle G circ E p pm i0 V vert Psi mathbf p pm rangle math where math G circ math is the free particle Green s function , 0 is a positive infinitesimal quantity, and V the interaction potential. math vert Psi mathbf p circ rangle math is the corresponding free scattering solution sometimes called incident field. The factor math vert Psi mathbf p pm rangle math on the right hand side is sometimes called driving field . This equation becomes within Born approximation math vert Psi mathbf p pm rangle vert Psi mathbf p circ rangle G circ E p pm ... be solved by some other method. In the distorted wave Born approximation DWBA to nuclear reactions ... p 1 pm rangle math and the Born approximation math vert Psi mathbf p pm rangle vert Psi mathbf p ... Mechanics publisher Addison Wesley year 1994 isbn 0 201 53929 2 Wu and Ohmura, Quantum Theory of Scattering .... 1173 82. Category Scattering theory de Bornsche N herung ru uk ... more details
Boussinesq approximation may refer to Boussinesq approximation buoyancy for buoyancy driven flows for small density differences in the fluid Boussinesq approximation water waves for long waves propagating on the surface of a fluid layer under the action of gravity Turbulence modeling and Viscosity Eddy viscosity eddy viscosity for the Boussinesq approximation resulting in the use of an eddy viscosity to model the turbulence Reynolds stresses disambig ... more details
The GW approximation GWA is an approximation made in order to calculate the self energy of a Many body problem many body system of electrons. The approximation is that the expansion of the self energy in terms of the single particle Green s function Green function G and the screened Coulomb interaction W in units of math hbar 1 math math Sigma iGW GWGWG cdots math can be truncated after the first term math Sigma approx iG W math Another way to say the same thing is that that self energy is expanded in a formal Taylor series in powers of the screened interaction W and the lowest order term is kept in the expansion in GWA. The above formulae are schematic in nature and show the overall idea of the approximation. More precisely, if we label an electron coordinate with its position, spin, and time and bundle all three into a composite index the numbers 1, 2, etc. , we have math Sigma 1,2 iG 1,2 W 1 ,2 int d3 int d4 , G 1,3 G 3,4 G 4,2 W 1,4 W 3,2 ... math where the superscript means the time index is shifted forward by an infinitessimal amount. The GWA is then math Sigma 1,2 approx iG 1,2 W 1 ,2 math To put this in context, if one replaces W by the bare Coulomb interaction i.e. the usual 1 r interaction , one generates the standard perturbative series for the self energy found in most many body textbooks. The GWA with W replaced by the bare Coulomb yields nothing other than the Hartree Fock exchange potential self energy . Therefore, loosely speaking, the GWA represents a type of dynamically screened Hartree Fock self energy. In a solid state system, the series for the self energy ... supporting the GW approximation ABINIT plane wave pseudopotential method http www.flapw.de spex Spex ..., Norman H. March editor , World Scientific Publishing Company Many Body Perturbation Theory The GW Approximation, Christoph Friedrich and Arno Schindlmayr http www.fzj.helmholtz.de nic series volume31 friedrich.pdf full version in pdf Category Quantum field theory physics stub it Approssimazione ... more details
unreferenced date March 2008 In science , engineering , and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations in increasing order of precision, a zeroth order approximation, a first order approximation, a second order approximation, and so forth. Formally, an n th order approximation is one where the order of magnitude of the error is at most math x n math , or in terms of big O notation , the error is math O x n . math In suitable circumstances, approximating a function by a Taylor polynomial of degree n yields an n th order approximation, by Taylor s theorem a first order approximation is a linear approximation , and so forth. The term is also used more loosely, as detailed below. Usage in science and engineering Zeroth order approximation also 0 sup th ... it has 3,914 people in actuality. This is also sometimes referred to as an order of magnitude approximation. A zeroth order approximation of a function mathematics function that is, mathematics mathematically ... approximation also 1 sup st sup order is the term scientists use for a further educated guess at an answer ... order approximation of a function that is, mathematically determining a formula to fit multiple data points will be a linear approximation , straight line with a slope a polynomial of degree 1. For example ... to the data, obtained by simply averaging the y values. Other methods for selecting a constant approximation can be used. Second order approximation also 2 sup nd sup order is the term scientists use ... is generally given. A second order approximation of a function that is, mathematically determining ... of reality, they are not typically referred to by number. A third order approximation ... about at the annual calibration. In this usage, the ordinality of the approximation is not exact, but is used .... See also Linearization Taylor approximation Category Numerical analysis ... more details
The Wien approximation was originally proposed as a description of the complete spectrum of thermal ..., it was soon superseded by Planck s law , developed by Max Planck . Unlike the Wien approximation ... The Wien approximation may be derived from Planck s law by assuming math h nu gg kT ... ref name rybickilightman1979 and so Planck s law approximately equals the Wien approximation at high ... more details
In algebraic topology , in the cellular approximation theorem , a Map mathematics map between CW complex es can always be taken to be of a specific type. Concretely, if X and Y are CW complexes, and f X Y is a continuous map, then f is said to be cellular , if f takes the n skeleton n skeleton of X to the n skeleton of Y for all n , i.e. if math f X n subseteq Y n math for all n . The content of the cellular approximation theorem is then that any continuous map f X Y between CW complexes X and Y is Homotopy homotopic to a cellular map, and if f is already cellular on a subcomplex A of X , then we can furthermore choose the homotopy to be stationary on A . From an algebraic topological viewpoint, any map between CW complexes can thus be taken to be cellular. Idea of proof The proof can be given by Mathematical induction induction after n , with the statement that f is cellular on the skeleton X sup n sup . For the base case n 0, notice that every Path connected Path connectedness path component of Y must contain a 0 cell. The Image mathematics image under f of a 0 cell of X can thus be connected to a 0 cell of Y by a path, but this gives a homotopy from f to a map, which is cellular on the 0 skeleton of X. Assume inductively that f is cellular on the n &minus 1 skeleton of X , and let e sup n sup be an n cell of X . The Closure topology closure of e sup n sup is compact ... that any compact subspace of a CW complex meets that is, Intersection set theory intersects non trivial ... Some homotopy groups The cellular approximation theorem can be used to immediately calculate some Homotopy ... base point preserving map f math S n , math math S k , math is by the cellular approximation theorem homotopic to a constant map, whence math pi n S k 0 , math . Cellular approximation for pairs Let f ... f to A and use cellular approximation to obtain a homotopy of f to a cellular map on A . Use the homotopy extension property to extend this homotopy to all of X , and apply cellular approximation ... more details
In geometric optics , the paraxial approximation is a small angle approximation used in Gaussian optics and Ray tracing physics ray tracing of light through an optical system such as a lens optics lens . ref name Greivenkamp cite book first John E. last Greivenkamp year 2004 title Field Guide to Geometrical Optics publisher SPIE others SPIE Field Guides vol. FG01 isbn 0 8194 5294 7 pages 19 20 ref A paraxial ray is a Ray optics ray which makes a small angle to the optical axis of the system, and lies close to the axis throughout the system. ref name Greivenkamp Generally, this allows three important approximations for in radian s for calculation of the ray s path ref name Greivenkamp math begin align sin theta & approx theta tan theta & approx theta end align math and math cos theta approx 1 math The paraxial approximation is used in Gaussian optics and first order raytracing. ref name Greivenkamp Ray transfer matrix analysis is one method that uses the approximation. In some cases, the second order approximation is also called paraxial . To second order, the approximations above for sine and tangent do not change the next term in their Taylor series expansion is zero , while for cosine the second order approximation is math cos theta approx 1 theta 2 over 2 . math The paraxial approximation is fairly accurate for angles under about 10 , but is inaccurate for larger angles. For larger angles it is often necessary to distinguish between meridional ray s, which lie in a plane containing the optical axis , and sagittal ray s, which do not. References reflist External links http demonstrations.wolfram.com ParaxialApproximationAndTheMirror Paraxial Approximation and the Mirror by David Schurig, The Wolfram Demonstrations Project . Category Geometrical optics ca Aproximaci paraxial de Paraxiale Optik es Aproximaci n paraxial fr Approximation de Gauss it Approssimazione parassiale hu Paraxi lis k zel t s nl Paraxiale benadering ja ru uk ... more details
In mathematics , an approximation to the identity refers to a sequence or net that converges to the identity in some algebra. Specifically, it can mean Nascent delta function , most commonly Mollifier , more narrowly Approximate identity , more abstractly disambig ... more details
is almost the approximation by the trapezoid rule of the integral math int 1 n ln x , rm d x n ln n n 1, math and the error in this approximation is given by the Euler Maclaurin formula math begin align ... notation , combining the equations above yields the approximation formula in its logarithmic form math ... 1 12n 1 lambda n frac 1 12n . math Stirling s formula is in fact the first approximation to the following ... approximation may now be written math Gamma z sqrt frac 2 pi z left frac z e right z ... The approximation math Gamma z approx sqrt frac 2 pi z left frac z e sqrt z sinh frac 1 z frac ... expansion of the hyperbolic sine function. This approximation is good to more than 8 decimal digits ... . Gerg Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler ref name Nemes2010 Citation last Nemes first Gerg year ... versions are due to Jacques Binet . See also Lanczos approximation Spouge s approximation Notes ... Dan Romik, Stirling s Approximation for n The Ultimate Short Proof? , The American Mathematical Monthly ... http www.luschny.de math factorial approx SimpleCases.html Peter Luschny, Approximation formulas ... and the Gamma Function 2006 MathWorld urlname StirlingsApproximation title Stirling s Approximation PlanetMath urlname StirlingsApproximation title Stirling s approximation Category Approximations Category Asymptotic analysis Category Analytic number theory Category Gamma and related functions ... more details
Otheruses WKB disambiguation In mathematical physics , the WKB approximation or WKB method is a method for finding approximate solutions to linear partial differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation See Old quantum theory in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be slowly changing. The name is an acronym for Wentzel Kramers Brillouin . It is also known as the LG or Liouville Green method. Other often used acronyms for the method include JWKB and WKBJ , where the J stands for Jeffreys. Brief history This method ... WKBJ WKB ref ref cite book chapter A Survey on the Liouville Green WKB approximation for linear difference ..., WKB theory is a method for approximating the solution of a differential equation whose highest derivative is multiplied by a small parameter . The method of approximation is as follows For a differential ... S n x math in the expansion. WKB theory is a special case of multiple scale analysis . ref cite book title Acoustics basic physics, theory and methods first Paul last Filippi publisher Academic Press ... a pair because math S 0 math can take two signs the first order WKB approximation will be a linear ... x 2 frac 2m hbar 2 left V x E right , math math B x 2 A x B x 0. math Next, the semiclassical approximation ... emission Langer correction Method of steepest descent Laplace Method Old quantum theory Perturbation methods Quantum tunneling Slowly varying envelope approximation References Reflist Modern references cite book author Razavy, Moshen title Quantum Theory of Tunneling publisher World Scientific year ... url http farside.ph.utexas.edu teaching jk1 lectures node70.html title The W.K.B. Approximation year 2002 An application of the WKB approximation to the scattering of radio waves from the ionosphere ... de WKB N herung es Aproximaci n WKB fr Approximation BKW ko WKB it Approssimazione WKB nl Wentzel ... more details
Orphan date December 2008 Cleanup rewrite date May 2009 A fundamental principle in the field of acoustics , the acoustic approximation states that an acoustic wave is created by a small, adiabatic, pressure ripple riding on a comparatively large equilibrium bias pressure . Typically, the acoustic pressure is on the order of a few ppm of the equilibrium pressure. By extension, the acoustic approximation also guarantees that an acoustic wave travels at a speed exactly equal to the local speed of sound , such that the Mach number math M 1 math See also Sound References reflist External links Category Acoustics physics stub ... more details
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