
	Where results make sense

Topic: Gaussian integral

Related Topics

Integral

List of integrals of irrational functions

Integral calculus

List of integrals of rational functions

DeWitt notation

Functional derivative

Integral domain

Gaussian integer

Finitary operation

Perturbation theory (quantum mechanics)

Feynman diagram

Statistical mechanics

Mnemonic

Vertex

Cross section

In the News (Sun 6 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Nonlinear Regression - Fitting the Integral of a Function
		In some nonlinear regression applications, it may be necessary to fit data to the integral of a function rather than to the function itself.
		For example, the dependent variable might correspond to the area under a Gaussian distribution between lower and upper X values, so the function being fitted is the integral of the Gaussian distribution rather than the Gaussian distribution itself.
		In this example, the t work variable ranges from 0 to x as the integral is computed; so the value of npd(t,Mean,StdDev) is the height of the Gaussian for each value of t as t is swept across the interval (0,x); the Integral function computes the area under the curve across that interval.
	www.nlreg.com /integral.htm (636 words)

	Encyclopedia: Integral calculus (Site not responding. Last check: 2007-10-17)
		An integral which can only be evaluated by considering it as the limit of integrals on successively larger and larger intervals is called an improper integral.
		Improper integrals usually turn up when the range of the function to be integrated is infinite or, in the case of the Riemann integral, when the domain of the function is infinite.
		The Perron integral, which is equivalent to the restricted Denjoy integral.
	www.nationmaster.com /encyclopedia/integral-calculus (1477 words)

	Encyclopedia: Gaussian integral (Site not responding. Last check: 2007-10-17)
		Gaussian curves parameterised for statistics A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c.
		In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution).
		While functional integrals have no rigorous definition (or even a nonrigorous computational one in most cases), we can define a Gaussian functional integral in analogy to the finite-dimensional case.
	www.nationmaster.com /encyclopedia/Gaussian-integral (743 words)

	G03 Manual: RUNNING
		Gaussian will generate names for the first two segments, and the third will be given the name my_job.
		If Gaussian is being used on a machine with limited physical memory, so that the default of 48 MB is not available, the default algorithms as well as the default memory allocation should be set appropriately during installation.
		Gaussian may be run using the NQS batch facility on those UNIX systems that support it.
	www.gaussian.com /g_ur/m_running.htm (1906 words)

	Functional integration - Encyclopedia, History, Geography and Biography
		Physicists often refer informally to functional integrals over spaces of paths (or field configurations) as path integrals, which are different from path integrals in the usual sense.
		Functional integration techniques in physics were pioneered by Richard Feynman, who successfully applied his "path integral formulation" to problems in quantum mechanics and quantum field theory, as well as classical and quantum statistical mechanics.
		Functional integrals over manifolds are sometimes approximated by a lattice, but there is no guarantee this will give a good approximation or even converge.
	www.arikah.net /encyclopedia/Functional_integral (956 words)

	ipedia.com: Integral Article (Site not responding. Last check: 2007-10-17)
		The integral value of a real number x is defined as the largest integer which is less than, or equal to, x.
		In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero.
		The Riemann integral was created by Bernhard Riemann and was the first rigorous definition of the integral.
	www.ipedia.com /integral.html (1470 words)

	PlanetMath: multidimensional Gaussian integral
		See Also: Jacobi determinant, area under Gaussian curve, proof of Gaussian maximizes entropy for given covariance
		Cross-references: formula, distributions, Gaussian, independent, product, separated, integral, Jacobian matrix, determinant, variable, vector, eigenvalues, diagonal matrix, eigenvectors, orthonormal, real, matrix, symmetric
		This is version 19 of multidimensional Gaussian integral, born on 2002-02-13, modified 2006-10-14.
	planetmath.org /encyclopedia/MultidimensionalGaussianIntegral.html (118 words)

	Library
		The generalized Huygens-Fresnel diffraction integral for misaligned asymmetric first-order optical systems is derived by using the canonical operator method, which enables us to study propagation properties of anisotropic Gaussian Schell-model (AGSM) beams through misaligned asymmetric first-order optical systems.
		Therefore generalized partially coherent anisotropic Gaussian Schell-model beams called decentered anisotropic Gaussian Schell-model (DAGSM) beams are introduced, and AGSM beams can be regarded as a special case of DAGSM beams.
		We study the influence of third-order spherical aberration on the group velocity dispersion and on the propagation time delay of a plane pulse that is focused by a thin lens.
	r6u.net /diffractive-diffraction-diffractive/diffractive-diffraction-diffractive-23.php (1223 words)

	World War 1 and 2 - Method of steepest descent
		where f(x) is some twice-differentiable function, M is a large number, and the integral endpoints a and b could possibly be infinite.
		This latter integral is a Gaussian integral if the limits of integration go from −∞ to +∞ (which can be assumed so because the exponential decays very fast away from x
		In extensions of this method, complex analysis is used to find a contour of steepest descent for an equivalent integral, expressed as a path integral.
	www.worldwardiary.com /history/Saddle-point_method (346 words)

	Gaussian integral - Encyclopedia, History, Geography and Biography
		Gaussian integral - Encyclopedia, History, Geography and Biography
		Let the value of the integral be $s$ .
		This encyclopedia, history, geography and biography article about Gaussian integral contains research on
	www.arikah.com /encyclopedia/Gaussian_integral (354 words)

	Gaussian kernel correlation integral (Site not responding. Last check: 2007-10-17)
		Between the given values of r, C(r) is interpolated by an exact power law and the integral is evaluated numerically.
		Above the largest given value of r, C(r)=1 is assumed and the corresponding integral is evaluated analytically.
		The derivative is carried out analytically on the above expression and the resulting integral is evaluated in the same manner as described.
	www.if.pwr.wroc.pl /~mirek/Dynamics/docs/wuppertal/c2g.html (134 words)

	Citations: Some new twists to problems involving the Gaussian probability integral - Simon, Divsalar (ResearchIndex) (Site not responding. Last check: 2007-10-17)
		The related problem of calculating the bit error probability (BER) has been addressed, but not to the same extent, in a number of articles resulting in both exact expressions [8, 9] and approximations [10] This paper presents an alternative expression for the SER of coherent M PSK using a....
		a particular form of the Gaussian probability integral developed a number of years ago by Craig [21 was used to simplify and render more accurate a number of performance results related to communication problems dealing with coherent detection, in particular, those where the argument of Q(x) is....
		For uniform signaling, the lower limits in (7) or (11) are both nonnegative, but for non uniform signaling, one or both of these limits can be negative, in which case the....
	citeseer.lcs.mit.edu /context/470384/0 (2053 words)

	Method of steepest descent - Psychology Central (Site not responding. Last check: 2007-10-17)
		The technique is also often referred to as Laplace's method, which in fact concerns the special case of real-valued functions f admitting a maximum at a real point.
		In extensions of Laplace's method, complex analysis, and in particular Cauchy's integral formula, is used to find a contour of steepest descent for an (asymptotically with large M) equivalent integral, expressed as a path integral.
		Again the main idea is to reduce, at least asymptotically, the calculation of the given integral to that of a simpler integral that can be explicitly evaluated.
	psychcentral.com /psypsych/Saddle-point_method (729 words)

	Properties of the Gaussian Distribution (Site not responding. Last check: 2007-10-17)
		In that case the normalized Gaussian distribution is of the form
		is a measure of the width of the Gaussian distribution.
		This integration must be performed numerically, because the indefinite integral of a Gaussian is not analytic
	www.chm.uri.edu /urichm/chm531/walk/node6.html (406 words)

	[No title]
		The left-hand side is a linear combination of integrals that involve finitely many derivatives of $F$.
		Since the $k$ and $\phi $ integrals, by (\ref{eq:shift-by-kappa}), are absolutely convergent, the integral over $k$ may be interchanged with the $\phi $ integrals.
		Since Gaussian random variables are independent if their covariance vanishes, we have the \emph{independence property} \[ \mu _{\Gamma }\ast (F_{X} G_{Y}) = (\mu _{\Gamma }\ast F_{X}) \ (\mu _{\Gamma }\ast G_{Y}), \] whenever the hierarchical distance between $X$ and $Y$ exceeds the range of $\Gamma $.
	www.ma.utexas.edu /mp_arc/e/02-231.latex.mime (7638 words)

	ELibM Book Review: Fermionic functional integrals and the renormalization group
		Functional integrals are notorious for being mathematically undefined and much effort has been made to change this situation.
		Grassmann Gaussian Integrals are introduced and in Section 1.4 their connection to Pfaffians and determinants is reviewed.
		In Section 1.5, the above-mentioned relation of Grassmann integrals and fermionic quantum field theories in second quantized Hamiltonian form is reviewed, and formulas generalizing (2) to a complete set of observables, the time-ordered functions, from which all observables can be reconstructed, are stated without proof.
	siba-sinmemis.unile.it /misc/articles/FKT (2102 words)

	MPQC: Class List
		Int2eCints is an interface to various specializations of two-electron integral evaluators implemented in Cints
		The Integral abstract class acts as a factory to provide objects that compute one and two electron integrals
		This is an abstract base type for classes that compute integrals involving two electrons in three Gaussian functions
	www.mpqc.org /mpqc-html/annotated.html (3591 words)

	NSDL Metadata Record -- Feynman path integral
		The argument is by analogy to the gaussian integral...
		One can bravely trudge onward and hope to come up with something, say \`a la Riemann integral, by partitioning X, picking some representative of each partition, approximating the functional F based on these and calculating a multi-dimensional integral as usual over the sample values of phi.
		The Feynman path integral was constructed as part of a re-formulation of, based on the sum-over-histories postulate of quantum mechanics, and can be thought of as an adaptation of Green's function methods for solving initial/boundary value problems.
	nsdl.org /mr/1034576 (202 words)

	IngentaConnect Counterexample to the Conjecture on Monotonicity of a Gaussian In... (Site not responding. Last check: 2007-10-17)
		IngentaConnect Counterexample to the Conjecture on Monotonicity of a Gaussian In...
		Counterexample to the Conjecture on Monotonicity of a Gaussian Integral
		It is shown that, for the Kantorovich metric \varkappa on probability measures, the integral I(\gamma)=\iint\limits_{F\oplus F}\varkappa(\Cal L(X_1),\Cal L(X_2))(\gamma \otimes \gamma) \,d(X_1,X_2), where \gamma is a centered Gaussian measure on a Euclidean space F of random variables X, is not always monotonic in \gamma.
	www.ingentaconnect.com /content/klu/joth/2002/00000109/00000006/00372041 (134 words)

	Atmol Integw program, Nijmegen version
		(b) To generate the least squares expansion of a Slater orbital of arbitary exponent, given the corresponding expansion for a Slater orbital of unit exponent, it is necessary to multiply the exponents of the Gaussian primitives by (Slater exponents)**2, hence the use of the SCALE parameter in the shown example.
		Because the program is unable to monitor the MXBLK setting during the output of a batch of integrals, it is possible to commence output at a block position beneath MXBLK, but to have exceeded the MXBLK parameter when output of the batch is complete.
		The INTEGRAL program monitors the computer time remaining for a run, and when there is insufficient time to continue, a standard dump procedure is initiated, and execution terminated.
	www.theochem.kun.nl /~pwormer/integw.html (5223 words)

	The stationary phase approximation
		The former may seem a little out-of-place, since there is no phase in the problem, but that is because we formulated it in such a way as to anticipate its application to the path integral.
		The application to the path integral follows via a similar argument.
		The integral over the coefficients becomes a generalized Gaussian integral, which brings down a factor of
	www.nyu.edu /classes/tuckerman/stat.mech/lectures/lecture_16/node2.html (501 words)

	[SciPy-user] Triple integral with Gaussian quadrature (Site not responding. Last check: 2007-10-17)
		It's just quad() run over the function defined by doing the double integral over the remaining dimensions (and the double integral is also implemented with quad() by integrating over the function defined by doing the integral using quad() to finally integrate over the last dimension).
		Exactly which QUADPACK function is called depends on the inputs; infinite bounds require one function, weighted integrals require others, explicit breakpoints require yet another.
		In order to do triple integrals with Gaussian quadrature, you can probably do a similar recursive scheme like tplquad() does.
	www.scipy.net /pipermail/scipy-user/2005-October/005402.html (237 words)

	Fubini's theorem - Psychology Central (Site not responding. Last check: 2007-10-17)
		Tonelli's theorem states that a product measure integral can be evaluated by way of an iterated integral for nonnegative measurable functions, regardless of whether they have finite integral.
		In fact, the existence of the first integral above (the integral of the absolute value), is guaranteed by Tonelli's theorem.
		To see how Fubini's theorem is used to prove this, see Gaussian integral.
	find.psychcentral.com /psypsych/Fubini%27s_theorem (273 words)

	Some New Twists to Problems Involving the Gaussian Probability Integral (Site not responding. Last check: 2007-10-17)
		However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
		Using an alternate form of the Gaussian probability integral discovered a number of years ago, it is shown that the solution to a number of previously considered communication problems can be simplified and, in some cases, made more accurate (i.e., exact rather than bounded).
		Also obtained is a generalization of this new alternate form to the case of a two-dimensional Gaussian probability integral with arbitrary correlation which can be used to evaluate the symbol-error probability of MPSK with I-Q unbalance.
	www.comsoc.org /comm/private/1998/feb/200_46comm02-simon.html (315 words)

	The integral for -infinity to infinity of a Gaussian is given by (Site not responding. Last check: 2007-10-17)
		The integral for -infinity to infinity of a Gaussian is given by
		where the integral in the last step is from Dwight's Integral tables.
		There is a bit of a problem with this in the form of the trigonometric part of the integral
	www.phys.ufl.edu /~coldwell/ligo/gabor/Integrals.html (80 words)

	Math Forum Discussions
		integral, and the value that return int or the oder functions isn't a
		I tried too with another integral which is an approssimation of the
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?threadID=1127467&messageID=3696307 (209 words)

	Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators - Storming Media
		Abstract: Generalized Gaussian quadratures appear to have been introduced by Markov late in the last century, and have been studied in great detail as a part of modern analysis.
		Recently, a numerical scheme was introduced for the design of such quadratures; numerical results presented indicate that such quadratures dramatically reduce the computational cost of the evaluation of integrals under certain conditions.
		In this paper, we modify the approach, improving the stability of the scheme and extending its range of applicability.
	www.stormingmedia.us /17/1769/A176903.html (161 words)

	Gaussian Integral (Site not responding. Last check: 2007-10-17)
		Probability Integral, is the integral of the 1-D Gaussian over
		The integral from 0 to a finite upper limit
		The general class of integrals of the form
	www.math.sdu.edu.cn /mathency/math/g/g090.htm (71 words)

	SISC Volume 20 Issue 2
		Generalized Gaussian quadratures appear to have been introduced by Markov late in the last century and have been studied in great detail as a part of modern analysis.
		Recently, a numerical scheme for the design of such quadratures was introduced by Ma et al.; numerical results presented in their paper indicate that such quadratures dramatically reduce the computational cost of the evaluation of integrals under certain conditions.
		In this paper, we modify their approach, improving the stability of the scheme and extending its range of applicability.
	epubs.siam.org /sam-bin/dbq/article/31077 (211 words)

	Jiangbin Yang's Ph.D. Thesis
		Under the Gaussian assumption, for an ARIMA process or a steady-state state space model subject to a change in process mean level, the residuals are independent and identically distributed (i.i.d.) with zero means before the occurrence of change.
		For computation of the average run lengths (ARLs) of the control chart procedures applied to the residuals whose means are time-varying after change, I derive an explicit formula for Shewhart, establish integral equations for CUSUM and EWMA, and develop efficient numerical procedures for solving the integral equations.
		I develop numerical procedures based on integral equations for computation of the ARLs of these change detection procedures, and numerically study their performance.
	www.sce.carleton.ca /~yang/thesis.htm (591 words)

Try your search on: Qwika (all wikis)