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Topic: Gaussian function

Related Topics

Topics named after Carl Friedrich Gauss

Gaussian stochastic process

Gaussian distribution

Normal distribution

Probability density function

Central limit theorem

Dummy variable

Gaussian

Normally distributed and uncorrelated does not imply independent

Mexican hat wavelet

Method of least squares

Eigenfunction

Complex Mexican hat wavelet

Residue theorem

Stochastic process

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	NationMaster - Encyclopedia: Gaussian orbital
		Gaussians for short, are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method.
		For example, the 3-21G basis set has one contracted Gaussian function that is a linear combination of three primitive Gaussian functions for each inner-shell atomic orbital and two basis functions, one contracted Gaussian function that is a linear combination of two primitive Gaussians and one primitive Gaussian function, for each valence orbital.
		The 6-31G* basis set represents each inner-shell orbital with one contracted Gaussian function that is a linear combination of six primitive Gaussian functions and each valence orbital with two basis functions, one contracted Gaussian function that is a linear combination of three primitive Gaussians and one primitive Gaussian function.
	www.nationmaster.com /encyclopedia/Gaussian-orbital (0 words)

	Gaussian function - Definition, explanation
		Gaussian functions are among those functions that are "elementary" but lack "elementary antiderivatives", i.e., their antiderivatives are not among the functions ordinarily considered in first-year calculus courses.
		In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to the central limit theorem.
		A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator.
	www.calsky.com /lexikon/en/txt/g/ga/gaussian_function.php (0 words)

	Gaussian Blur Effect
		A Gaussian Blur is a general purpose blur filter.
		A Gaussian Blur is distinct from other blurs in that it has a well defined effect on different levels of detail within an image.
		As well as having this well defined and consistent frequency response, certain characteristics of the Gaussian function mean that large blurs can be applied much faster than other similar kinds of filters.
	www.websupergoo.com /helpie/source/2-effects/gaussianblur.htm (156 words)

	Graphs of Functions and Algebra - Interactive Tutorials
		Quadratic functions and the properties of their graphs such as vertex and x and y intercepts are explored interactively using an applet.
		Rational functions and the properties of their graphs such as domain, vertical and horizontal asymptotes, x and y intercepts are explored using an applet.
		The graphs and properties such as domain, range and asymptotes of the 6 hyperbolic functions: sinh(x), cosh(x), tanh(x), coth(x), sech(x) and csch(x) are explored using an applet.
	www.analyzemath.com /precalculus.html (0 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		The a is the height of the Gaussian peak, b is the position of the center of the peak and c is related to the FWHM of the peak according to
		Gaussian functions are among those functions that are elementary but lack elementary antiderivatives.
		Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing -- see the article on scale space representation.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Gaussian_function (533 words)

	Gamasutra - Features - "Using Bitmaps for Automatic Generation of Large-Scale Terrain Models "[04.27.00]
		In the Gaussian filter, the values in the mask are chosen according to the shape of a Gaussian function.
		In other words, the function does not favor any particular direction when it smooths, which is particularly useful when the areas needing smoothing are oriented in an arbitrary direction (not known in advance), and there is no reason to smooth in any specific direction.
		Second, the Gaussian function has a single lobe, which means that the Gaussian filter replaces each pixel with a weighted average of the neighboring pixels around it (like the averaging filter), such that a pixel's weight decreases monotonically with distance from the central pixel.
	www.gamasutra.com /features/20000427/martin_02.htm (0 words)

	Gaussian, Error and Complementary Error function
		The gaussian function goes to zero at plus and minus infinity while all the derivatives of any order evaluated at x = 0 are zero.
		The gaussian function, error function and complementary error function are frequently used in probability theory since the normalized gaussian curve represents the probability distribution with standard deviation
		Gaussian (upper) and the complementary error function on a semi-logarithmic scale.
	ece-www.colorado.edu /~bart/book/gaussian.htm (0 words)

	G03 Manual: GEN
		N is the number of primitive functions composing the basis function, and it is called the degree-of-contraction of the basis function.
		The other sp-shell is a least-squares fit of 3 gaussians to Slater 2s and 2p orbitals with the constraint that the s and p functions have equal exponents.
		Sc is the scale factor and hence the exponent of the slater function being expanded.
	www.gaussian.com /g_ur/k_gen.htm (0 words)

	Mathematical Programming Glossary
		A goal is neither a constraint nor an objective because it is neither a firm requirement nor a function that must be at an extreme value.
		This finds an interval, [a,b], that contains a maximum of a unimodal function whose domain is [0,1], such that the length of the final interval, [a,b], satisfies: b-a < e (where e > 0 is specified).
		In one context, this is the functional value and domain: {(x, z): x in X and z=f(x)}, where f:X-->R. In another context, this is a (maybe undirected) network.
	glossary.computing.society.informs.org /index.php?page=G.html (3717 words)

	Gaussian Process Regression
		A simple implementation of a Gaussian process for regression is provided by the gpr.m program (which can conveniently be used together with minimize.m for optimization of the hyperparameters).
		A commonly used covariance function is the sum of a sqaured exponential (SE) contribution and independent noise.
		The squared exponential (SE) covariance function (also called the radial basis function (RBF) or Gaussian covariance function) is given by in equation (2.31) in section 2.3 page 19 for the scalar input case, and equation (5.1) section 5.1 page 106 for multivariate inputs.
	www.gaussianprocess.org /gpml/code/matlab/doc/regression.html (1672 words)

	Gaussian-Type Functions
		For example, the 3-21G basis set has one contracted Gaussian function that is a linear combination of three primitive Gaussian functions for each inner-shell atomic orbital and two basis functions, one contracted Gaussian function that is a linear combination of two primitive Gaussians and one primitive Gaussian function, for each valence orbital.
		The 6-31G* basis set represents each inner-shell orbital with one contracted Gaussian function that is a linear combination of six primitive Gaussian functions and each valence orbital with two basis functions, one contracted Gaussian function that is a linear combination of three primitive Gaussians and one primitive Gaussian function.
		In addition, six d-type Gaussian functions for each nonhydrogen atom in the second or third period are included in the basis set.
	www.chm.davidson.edu /ronutt/che401/Gaussian/Gaussian.htm (599 words)

	Gamasutra - Features - "Using Bitmaps for Automatic Generation of Large-Scale Terrain Models "[04.27.00]
		In the Gaussian filter, the values in the mask are chosen according to the shape of a Gaussian function.
		In other words, the function does not favor any particular direction when it smooths, which is particularly useful when the areas needing smoothing are oriented in an arbitrary direction (not known in advance), and there is no reason to smooth in any specific direction.
		Second, the Gaussian function has a single lobe, which means that the Gaussian filter replaces each pixel with a weighted average of the neighboring pixels around it (like the averaging filter), such that a pixel's weight decreases monotonically with distance from the central pixel.
	gamasutra.com /features/20000427/martin_02.htm (1287 words)

	Computer image display system and processor therefor - Patent 4156914
		First, the Gaussian function is convolved vertically with computer stored intensities in a vertical slice of interest, that is, multiplied by those intensity values in the vertical slice and summed to derive a resultant intermediate value on a given horizontal line.
		That is, Gaussian function is multiplied by intensity values in the vertical slice and summed to derive resultant intermediate values at each point on a given horizontal line.
		The Gaussian function also provides multiplicative separability of vertical and horizontal components in the matched-filter convolution integral which gives the computer-display brightness B (x,y): ##EQU5## where f (x.sub.i,y.sub.i) represents the computer data-input function, and B.sub.o is the display-brightness scale factor, a display-system constant.
	www.freepatentsonline.com /4156914.html (3513 words)

	ACM Sigplan Notices 28, 11 (Nov 1993), 22-27.
		Once the basic concepts are in hand, we should revel in the polymorphism of Gaussian integers as pairs of integers, as well as atomic elements of the Gaussian ring, much as we revel in the dual role of C "ints" as both integers and bit strings.
		We know that for a Gaussian integer divisor m+ni, we need m^2+n^2 distinct remainders, so the most elegant choice of representative remainders is a square of area m^2+n^2 which is tilted at an angle of atan(n/m);[5] equivalently, we consider the remainder fraction r/d to reside in an upright square of area 1.
		function extended to the complex numbers computes the center of the square[19] pixel in which a point falls.
	home.pipeline.com /~hbaker1/Gaussian.html (6038 words)

	Deconvolve Gaussian Response Function
		Although most instances of Gaussian response function deconvolution involve undoing a symmetric (2-sided) smearing of the signal, it is also possible to deconvolve either of the one-sided cases.
		Both the width of the response function and the extent of the low pass filtration are critical.
		Rather, when a Gaussian is smeared by a Gaussian instrument response function, the result is also a Gaussian with a variance equal to the sum of the individual variances.
	www.clecom.co.uk /science/autosignal/help/Deconvolve_Gaussian_Respons.htm (1687 words)

	Gaussian Distribution (Site not responding. Last check: )
		The Gaussian distribution is a continuous function which approximates the exact binomial distribution of events.
		The nature of the gaussian gives a probability of 0.683 of being within one standard deviation of the mean.
		The Gaussian distribution is also commonly called the "normal distribution" and is often described as a "bell-shaped curve".
	hyperphysics.phy-astr.gsu.edu /hbase/math/gaufcn.html (178 words)

	Spreadsheet Filtering by FFT Gaussian-based Convolution
		It is seen from the right-side graph that the function remains unchanged by the delta function convolution, as expected.
		The kernel is a Gaussian and the function with the sharp edges is a pulse.
		The frequencies passed are determined by the width of the Gaussian multiplying the cosine.
	physics.mercer.edu /hpage/filter/gauss.html (3428 words)

	PlanetMath: Dirac delta function
		can also be defined as a normalized Gaussian function (normal distribution) in the limit of zero width.
		Cross-references: width, limit, Gaussian, dimensions, continuous function, Kronecker delta, argument, function
		This is version 2 of Dirac delta function, born on 2002-01-19, modified 2002-07-04.
	planetmath.org /encyclopedia/DiracDeltaFunction.html (81 words)

	Numerical integration Summary
		Gaussian quadratures allow one to pick the optimal abscissas as which to evaluate the function.
		Some commonly used Gaussian quadratures are the Gauss-Legendre formula and the Gauss-Chebyshev formula, which are used for closed, definite integrals, the Gauss-Hermite formula, which is used for integrals that have - and as the limits of integration, and the Gauss-Laguerre formula, which is used for integrals on the interval [0,).
		The interpolating function may be an affine function (a polynomial of degree 1) which passes through the points (a, f(a)) and (b, f(b)).
	www.bookrags.com /Numerical_integration (2133 words)

	Transfer Functions
		A node’s transfer functions serves the purpose of controlling the output signal strength for the node (except for the input layer which uses the inputs themselves).
		Where the sigmoid function acts as a gate (opened, closed or somewhere in-between) for a node’s output response, the gaussian function acts like a probabilistic output controller.
		The hyperbolic function counterparts to the sigmoid and gaussian functions are the hyperbolic tangent and hyperbolic secant functions.
	www.qnetv2k.com /Qnet2000Manual/html/qnet0n3n.htm (483 words)

	No Title
		DZ sets have two basis functions per orbital, etc. Since valence orbitals of atoms are more affected by forming a bond than the inner (core) orbitals, more basis functions are assigned frequently to describe valence orbitals.
		Gaussian primitives are normalized here since coefficients for basis functions consisting of one primitive (last row) are exactly 1.0.
		The polarization functions are simply functions having higher values of L than those present in occupied atomic orbitals for the corresponding atom.
	www.ccl.net /cca/documents/basis-sets/basis.html (5465 words)

	Figs 7 and 8
		The response of the Gaussian operator to the edge subjected to additive Gaussian noise is much improved with respect to the simple box operator (which is similar to a Roberts or Sobel operator).
		In practice, two dimensional convolution with large Gaussians takes a long time, so that in practice it is common to approximate this by two one dimensional Gaussians, one aligned with the x-axis, the other with the y axis.
		A local maximum occurs at a peak in the gradient function, or alternatively where the derivative of the gradient function is set to zero.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/MARBLE/low/edges/canny.htm (1087 words)

	IRAF Help page for "gauss" (V2.10.4p2)
		If BILINEAR is "yes" and the major axis of the Gaussian kernel is aligned along either the x or y axis, GAUSS uses the fact that the Gaussian function is mathematically separable (bilinear) in x and y to speed up the convolution process.
		Convolve an image with a circular Gaussian function of sigma 2.0, and size 4.0 sigma using nearest neighbour boundary extension and the bilinear kernel.
		Convolve an image with an elliptical Gaussian function whose sigma in the major and minor axis direction is 2.0 and 1.5 respectively, and whose position angle is 45 degrees, using wrap around boundary extension.
	ecf.hq.eso.org /scripts/irafhelp?gauss (700 words)

	Systat Software Inc. - AutoSignal - HTML Help
		Although most instances of Gaussian response function deconvolution involve undoing a symmetric (2-sided) smearing of the signal, it is also possible to deconvolve either of the one-sided cases.
		Both the width of the response function and the extent of the low pass filtration are critical.
		Rather, when a Gaussian is smeared by a Gaussian instrument response function, the result is also a Gaussian with a variance equal to the sum of the individual variances.
	www.systat.com /products/AutoSignal/help/?sec=1106 (1667 words)

	Gaussian Function (Site not responding. Last check: )
		The green area under the function represents the probability of the value falling on the range between squiggle and squiggle + delta squiggle.
		, which corresponds to the area in green under the function, then we can say that the probability of the value falling on the range squiggle to squiggle + delta squiggle is 125/250 = 0.5, or 50%.
		If you do this procedure to a function f(x) that is already normalized, then the integration from negative infiniti to infiniti will produce 1.
	web.umr.edu /~jstoffer/Math/gauss.html (227 words)

	pamgauss (Site not responding. Last check: )
		The values are scaled so that the area under the surface of the two- dimensional Gaussian function is the maxval of the image.
		If you want to be sure you get a whole Gaussian function, make sure that you choose a sigma and image dimen- sions so that if you made it any larger, the sample values at the edges would be zero.
		The higher the number, the more spread out the function is. Normally, you want to make this number low enough that the function reaches zero value before the edge of your image.
	www.linuxcommand.org /man_pages/pamgauss1.html (403 words)

	The convolution theorem and its applications
		A more general property of the delta function is that the integral of a delta function times some other function is equal to the value of that other function at the position of the delta function.
		The integral over all space of a Gaussian is 1, which satisfies one of the properties required for the delta function, and if we take the limit of a Gaussian as the standard deviation tends to zero, it satisfies the other properties.
		The B-factors (or atomic displacement parameters, to be precise) correspond to a Gaussian smearing function.
	www-structmed.cimr.cam.ac.uk /Course/Convolution/convolution.html (2266 words)

	The Gaussian Function
		The Gaussian function has remarkable mathematical properties that are important for understanding the concept of Gaussian scale-space reviewed in Chapter 3.
		The Gaussian function not only allows us to approximate very well the probability distribution of the noise in many measurements of physical processes but it also leads to tractable mathematics.
		The Gauss function is the Green's function (or propagator) of the diffusion equation for an infinite domain.
	www.ensc.sfu.ca /people/grad/brassard/personal/THESIS/node186.html (0 words)

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