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Mathematical function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f ( x ) = exp ( − x 2 ) {\displaystyle f(x)=\exp(-x^{2})}
Gaussian_function
Type of image blur produced by a Gaussian function
In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician
Gaussian_blur
Sigmoid shape special function
of z {\displaystyle z} . The error function at ∞ {\displaystyle \infty } is exactly 1 {\displaystyle 1} (see Gaussian integral). At the real axis, erf
Error_function
Monochrome light beam whose amplitude envelope is a Gaussian function
a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this
Gaussian_beam
Integral of the Gaussian function, equal to sqrt(π)
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f ( x ) = e − x 2 {\displaystyle f(x)=e^{-x^{2}}}
Gaussian_integral
Filter in electronics and signal processing
processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would
Gaussian_filter
Probability distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued
Normal_distribution
Short "burst" or "envelope" of restricted wave action that travels as a unit
frequency within a bandwidth inversely proportional to that width; even a Gaussian function is considered a wave packet because its Fourier transform is a "packet"
Wave_packet
Function used in signal processing
coordinate axes. Only the Gaussian function is both separable and isotropic. The separable forms of all other window functions have corners that depend
Window_function
Mathematical function having a characteristic "bell"-shaped curve
examples include: Gaussian function, the probability density function of the normal distribution. This is the archetypal bell shaped function and is frequently
Bell-shaped_function
Describes the sum of independent normal and exponential random variables
In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal
Exponentially modified Gaussian distribution
Exponentially_modified_Gaussian_distribution
Statistical model
of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with
Gaussian_process
_{0}^{a}{\frac {\varphi (hx)}{1+x^{2}}}\,dx} is Owen's T function. Owen has an extensive list of Gaussian-type integrals; only a subset is given below. ∫ φ (
List of integrals of Gaussian functions
List_of_integrals_of_Gaussian_functions
Mathematical function
chemistry and molecular physics, Gaussian orbitals (also known as Gaussian type orbitals, GTOs or Gaussians) are functions used as atomic orbitals in the
Gaussian_orbital
modified Gaussian distribution or function, used for description of peak shape in many techniques Gauss error function Gaussian process Gaussian filter
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Wavelet proportional to the second derivative of a Gaussian
^{2}=1\right)} second derivative of a Gaussian function, i.e., up to scale and normalization, the second Hermite function. It is a special case of the family
Ricker_wavelet
Feature enhancement algorithm in imaging science
_{t}:\mathbb {R} ^{n}\rightarrow \mathbb {R} } denote the radial Gaussian function Φ t ( x ) = N ( x | 0 , t ) {\displaystyle \Phi _{t}(x)={\mathcal
Difference_of_Gaussians
"Smoothing" integral transform
{\displaystyle f} , weighted with a Gaussian centered at x {\displaystyle x} . Specifically, it is the function F {\displaystyle F} defined by F ( x
Weierstrass_transform
Generalization of the one-dimensional normal distribution to higher dimensions
theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional
Multivariate normal distribution
Multivariate_normal_distribution
Method in approximation theory
definite function. Such functions, including the Gaussian, inverse quadratic, and inverse multiquadric are often used as radial basis functions for this
Radial basis function interpolation
Radial_basis_function_interpolation
Type of noise in signal processing
processing theory, Gaussian noise, named after Carl Friedrich Gauss, is a kind of signal noise that has a probability density function (pdf) equal to that
Gaussian_noise
Volume rendering technique
Gaussian splatting is a volume rendering technique that deals with the direct rendering of volume data without converting the data into surface or line
Gaussian_splatting
Mathematical transform that expresses a function of time as a function of frequency
distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced sine and cosine transforms
Fourier_transform
Set of functions used to represent the electronic wave function
addition to basis sets is the addition of diffuse functions. These are extended Gaussian basis functions with a small exponent, which give flexibility to
Basis_set_(chemistry)
Probability distribution
distribution given by a convolution of a Cauchy-Lorentz distribution and a Gaussian distribution. It is often used in analyzing data from spectroscopy or diffraction
Voigt_profile
Mathematical description of quantum state
eigenvalues ±1, ±i. The eigenvectors are "Hermite functions", i.e. Hermite polynomials multiplied by a Gaussian function. See Byron & Fuller (1992) for a description
Wave_function
Number, approximately 3.14
uncertainty principle only for the Gaussian function. Equivalently, π is the unique constant making the Gaussian normal distribution e−πx2 equal to its
Pi
Smoothing filler for images
geodesic editing, guided filtering, and domain transforms. Gaussian filter Gaussian function Gaussian blur Convolution Banterle, F.; Corsini, M.; Cignoni, P
Bilateral_filter
Statistical distribution for dependence between random variables
limitations of the Gaussian copula and of copula functions more generally, specifically the lack of dependence dynamics. The Gaussian copula is lacking
Copula_(statistics)
Type of mathematical function
log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function. Similarly
Logarithmically concave function
Logarithmically_concave_function
Special case of the short-time Fourier transform
The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then
Gabor_transform
Approximation of the production rate of a resource over time
symmetric logistic distribution curve, often confused with the "normal" gaussian function. It first appeared in "Nuclear Energy and the Fossil Fuels," geologist
Hubbert_curve
Mathematical function
function or Dawson integral (named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussian function. The Dawson function is
Dawson_function
Product of the principal curvatures of a surface
In differential geometry, the Gaussian curvature or Gauss curvature (symbol Κ, named after Carl Friedrich Gauss) of a smooth surface in three-dimensional
Gaussian_curvature
Type of mathematical function
{\displaystyle \varepsilon } Gaussian: Inverse quadratic: Inverse multiquadric: Other Infinitely Smooth RBFs These radial basis functions are also from C ∞ ( R
Radial_basis_function
Constant a such that af(x) is a probability measure
reduce any nonnegative function whose integral is finite to a probability density function. For example, a Gaussian function can be normalized into a
Normalizing_constant
Family of continuous probability distributions
In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions
Inverse_Gaussian_distribution
Generalized function whose value is zero everywhere except at zero
of Gaussian distributions centered at the origin with variance tending to zero. (However, even in some applications, highly oscillatory functions are
Dirac_delta_function
Mathematical functions
to diameter of a circle. As complex functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods {
Lemniscate_elliptic_functions
Concept in statistics and wave theory
inverse hyperbolic secant. Beam diameter § Full width at half maximum Gaussian function Cutoff frequency Spatial resolution This article incorporates public
Full_width_at_half_maximum
Family of power series in mathematics
just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent
Generalized hypergeometric function
Generalized_hypergeometric_function
Concept in laser optics
the Gaussian beam model is no longer accurate and a physical optics analysis is required. Beam divergence Beam parameter product Gaussian function Electromagnetic
Rayleigh_length
Machine learning kernel function
sample of the training set. Gaussian function Kernel (statistics) Polynomial kernel Radial basis function Radial basis function network Obst kernel network
Radial_basis_function_kernel
Functions such that f(–x) equals f(x) or –f(x)
cosh , {\displaystyle \cosh ,} Gaussian function x ↦ exp ( − x 2 ) . {\displaystyle x\mapsto \exp(-x^{2}).} A real function f is odd if, for every x in
Even_and_odd_functions
In physics, a non-Gaussianity is the correction that modifies the expected Gaussian function estimate for the measurement of a physical quantity. In physical
Non-Gaussianity
Complex number whose real and imaginary parts are both integers
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition
Gaussian_integer
Extension of the factorial function
the complete gamma function for contrast.) An important category of exponentially decaying functions is that of Gaussian functions a e − ( x − b ) 2 c
Gamma_function
Mathematical function, denoted exp(x) or e^x
exponential function – Exponential function of an exponential function Exponential field – Mathematical field with an extra operation Gaussian function Half-exponential
Exponential_function
(also known as Gaussian map or mouse map), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function: x n + 1 = exp
Gauss_iterated_map
Special class of quantum states
discontinuous in certain points, the Wigner function of a Gaussian state is positive and continuous. Gaussian states are particulary relevant in photonic
Gaussian_state
Color space defined by the CIE in 1931
color matching functions can be approximated by a sum of Gaussian functions, as follows: Let g(x) denote a piecewise-Gaussian function, defined by g (
CIE_1931_color_space
Concept in statistical mechanics
statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). The discrete version
Gaussian_free_field
Digital modulation scheme
by noise. The probability density function of having an error of a given size can be modelled by a Gaussian function; the mean value will be the relative
Amplitude-shift_keying
Function defined by a hypergeometric series
In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes
Hypergeometric_function
Complex complementary error function
frequencies, with a Gaussian distribution. The integrated response can be written in terms of the Faddeeva function. the Faddeeva function is also used in
Faddeeva_function
Mathematical theorem
The Gaussian correlation inequality (GCI), formerly known as the Gaussian correlation conjecture (GCC), is a mathematical theorem in the fields of mathematical
Gaussian correlation inequality
Gaussian_correlation_inequality
Artificial neural network node function
take many forms, but they are usually found as one of the following functions: Gaussian: ϕ ( v ) = exp ( − ‖ v − c ‖ 2 2 σ 2 ) {\displaystyle \,\phi (\mathbf
Activation_function
Limit of the tangent line at a point that tends to infinity
Other common functions that have one or two horizontal asymptotes include x ↦ 1/x (that has an hyperbola as it graph), the Gaussian function x ↦ exp (
Asymptote
Random matrix with gaussian entries
the Gaussian ensembles are specific probability distributions over self-adjoint matrices whose entries are independently sampled from the gaussian distribution
Gaussian_ensemble
Convergence in distribution of binomial to normal distribution
coin is tossed 3600 times. This is one derivation of the particular Gaussian function used in the normal distribution. It is a special case of the central
De_Moivre–Laplace_theorem
Concept in statistics
integration in the inversion formula to [−1/h, 1/h], or the Gaussian function ψ(t) = e−πt2. Once the function ψ has been chosen, the inversion formula may be applied
Kernel_density_estimation
Self-perpetuating scientific consensus via exclusion of negative evidence
idea, the gaussian curve starts to concentrate more around the center. By the end, the gaussian function will have become a delta function, representing
Gold_effect
Function in discrete mathematics
Fourier transform, of which the most famous is the Gaussian function. Since periodic summation of the function means discretizing its frequency spectrum and
Discrete_Fourier_transform
truncating the Gaussian are usually negligible. For larger values of ε, however, there are many better alternatives to a rectangular window function. For example
Scale_space_implementation
Function used in quantum chemistry
can approximate the STO using a linear combination of Gaussian functions (or GTO above). Gaussians tend to underestimate the values close to the nucleus
Slater-type_orbital
Statistics function
cumulative distribution function of the standard normal Gaussian distribution. The Q-function can be expressed in terms of the error function, or the complementary
Q-function
Algorithm for edge detection in digital images
convolving the image with the Laplacian of the Gaussian function, or, as a fast approximation by difference of Gaussians. Then, zero crossings are detected in
Marr–Hildreth_algorithm
Approximating an arbitrary function with a well-behaved one
In general, a function approximation problem asks us to select a function that closely matches ("approximates") a function in a task-specific way.[better source needed]
Function_approximation
Topics referred to by the same term
distribution, which uses the Gaussian function. Bell curve may also refer to: Bell-shaped function, any mathematical function having a bell-shaped curve
Bell_curve_(disambiguation)
Feature detection algorithm in computer vision
locations are defined as maxima and minima of the result of difference of Gaussians function applied in scale space to a series of smoothed and resampled images
Scale-invariant feature transform
Scale-invariant_feature_transform
Approximation of the definite integral of a function
In numerical analysis, an n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result
Gaussian_quadrature
Smooth and compactly supported function
for the related function discussed in the Non-analytic smooth function article. This function can be interpreted as the Gaussian function exp ( − y 2
Bump_function
Periodic distribution ("function") of "point-mass" Dirac delta sampling
{\displaystyle s_{\tau }(x)} is a convergent series of Gaussian functions, and Gaussians transform into Gaussians, each of their respective Fourier transforms S
Dirac_comb
Synonym for, and visual description of bimodial distribution in statistics
In statistics, an inverted bell curve is a term used loosely or metaphorically to refer to a bimodal distribution that falls to a trough between two peaks
Inverted_bell_curve
Variant Fourier transforms
functions into a sum of sine waves representing the odd component of the function plus cosine waves representing the even component of the function.
Sine_and_cosine_transforms
Function space of all functions whose derivatives are rapidly decreasing
mathematics, Schwartz space S {\displaystyle {\mathcal {S}}} is the function space of all functions whose derivatives of all orders are rapidly decreasing. This
Schwartz_space
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Fundamental theorem in probability theory and statistics
The polytope Kn is called a Gaussian random polytope. A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges
Central_limit_theorem
{\pi \over a}}e^{\tfrac {b^{2}}{4a}}\quad (a>0)} (see Integral of a Gaussian function) ∫ − ∞ ∞ e − ( a x 2 + b x + c ) d x = π a e b 2 4 a − c ( a > 0 )
List of integrals of exponential functions
List_of_integrals_of_exponential_functions
In statistics, Sheppard's corrections are approximate corrections to estimates of moments computed from binned data. The concept is named after William
Sheppard's_correction
Technique for reducing low-resolution image distortion
domain), as shown. Functions based on the Gaussian function are natural choices, because convolution with a Gaussian gives another Gaussian whether applied
Spatial_anti-aliasing
Type of signal in signal processing
normal distribution with zero mean, the signal is said to be additive white Gaussian noise. The samples of a white noise signal may be sequential in time, or
White_noise
Polynomial sequence
theory in connection with nonlinear operations on Gaussian noise. random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon
Hermite_polynomials
Method for approximate evaluation of integrals
more step is needed to get a Gaussian distribution. Since x 0 {\displaystyle x_{0}} is a global maximum of the function f {\displaystyle f} it can be
Laplace's_method
Family of polynomials
In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian numbers, Gaussian polynomials, or q-binomial coefficients)
Gaussian_binomial_coefficient
Linear filter used for texture analysis
discrimination. In the spatial domain, a 2D Gabor filter is a Gaussian kernel function modulated by a sinusoidal plane wave (see Gabor transform). Some
Gabor_filter
Recognition (OCR) Conversion Adaptive Classifier or Cube Conversion Gaussian function Raster Processing Thresholding Raster Processing Edge detection/Edge
Scan2CAD
Variation of artificial neural networks
only sigmoid functions and sometimes Gaussian functions, CPPNs can include both types of functions and many others. The choice of functions for the canonical
Compositional pattern-producing network
Compositional_pattern-producing_network
Continuous probability distribution
The normal-inverse Gaussian distribution (NIG, also known as the normal-Wald distribution) is a continuous probability distribution that is defined as
Normal-inverse Gaussian distribution
Normal-inverse_Gaussian_distribution
Formula in X-ray diffraction and crystallography
{\displaystyle p_{1}(\Delta x)} s, etc. As the convolution of two Gaussians is just another Gaussian, we have that p m ( Δ x ) = 1 ( 2 π m σ 2 2 ) 1 / 2 exp
Scherrer_equation
Quick, temporary change in amplitude of electrical signals
information about the system. A Gaussian pulse is shaped as a Gaussian function and is produced by the impulse response of a Gaussian filter. It has the properties
Pulse_(signal_processing)
Concept in wavelet theory
Since ψ ( t ) {\displaystyle \psi (t)} is the second derivative of Gaussian function, its Fourier transform is: F . T . { ψ ( t ) } = F . T . { C d 2 d
Vanishing_moment
Form of error in digital signals; spurious signals near sharp transitions
the Gaussian filter, whose magnitude Bode plot is a downward opening parabola (quadratic roll-off), as its Fourier transform is again a Gaussian, hence
Ringing_artifacts
Distribution over functions corresponding to an infinitely wide Bayesian neural network
A Neural Network Gaussian Process (NNGP) is a Gaussian process (GP) obtained as the limit of a certain type of sequence of neural networks. Specifically
Neural network Gaussian process
Neural_network_Gaussian_process
Feature observed in spectroscopy
intensity in the spectrum. Ideal line shapes include Lorentzian, Gaussian and Voigt functions, whose parameters are the line position, maximum height and half-width
Spectral_line_shape
Concept in statistics
In statistics, a Gaussian random field (GRF) is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF
Gaussian_random_field
Inverse functions of sin, cos, tan, etc.
trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under
Inverse trigonometric functions
Inverse_trigonometric_functions
machine learning, Gaussian process approximation is a computational method that accelerates inference tasks in the context of a Gaussian process model, most
Gaussian process approximations
Gaussian_process_approximations
Multivalued function in mathematics
{\displaystyle \Gamma } denotes the gamma function. The first identity can be found by writing the Gaussian integral in polar coordinates. The second
Lambert_W_function
Type of continuous-phase frequency-shift keying
{\displaystyle Q(*)} denotes the Q-function and erfc {\displaystyle \operatorname {erfc} } denotes the complementary error function. Gaussian minimum-shift keying,
Minimum-shift_keying
GAUSSIAN FUNCTION
GAUSSIAN FUNCTION
Male
Russian
Variant spelling of Russian Afanasiy, AFANASY means "immortal."
Male
Russian
Variant spelling of Russian Afanasiy, AFANASII means "immortal."
Male
Russian
Variant spelling of Russian Faddei, FADEI means "courageous."
Female
Russian
(Russian Ева): Armenian and Russian form of Greek Eva, YEVA means "life."Â
Male
Russian
Variant spelling of Russian Vasiliy, VASILY means "king."
Male
Russian
Variant spelling of Russian Vikentiy, VIKENTI means "conquering."
Male
Russian
(РоÑÑ) Russian pet form of Czech/Russian Rostislav, ROSTYA means "usurp-glory."
Male
Russian
Variant spelling of Russian Vasiliy, VASSILY means "king."
Boy/Male
Australian, French, German, Irish
Curly-headed
Male
Russian
Variant spelling of Russian Arseniy, ARSENI means "virile."
Male
Russian
Variant spelling of Russian Irinei, IRINEY means "peaceful."
Male
Russian
Variant spelling of Russian Gennadiy, GENNADY means "noble."
Male
Russian
Variant spelling of Russian Afanasiy, AFANASEI means "immortal."
Male
Russian
(Паша) Russian pet form of Czech/Russian Pavel, PASHA means "small."
Male
Russian
Variant spelling of Russian Vasiliy, VASILI means "king."
Female
Russian
(Людмила) Russian feminine form of Czech/Russian Ludmil, LUDMILA means "people's favor."Â
Male
Russian
Variant spelling of Russian Arseniy, ARSENIY means "virile."
Male
Russian
(Russian ИÑидор): Russian form of Greek Isidoros, ISIDOR means "gift of Isis."
Male
Russian
Variant spelling of Russian Aleksey, ALEXEY means "defender."
Male
Russian
Variant spelling of Russian Gennadiy, GENNADI means "noble."
GAUSSIAN FUNCTION
GAUSSIAN FUNCTION
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
King; Ruler
Boy/Male
Hindu
Wiz kid
Boy/Male
Biblical
Who rests himself, who is now captive'.
Surname or Lastname
English
English : unexplained; perhaps a habitational name from a lost or unidentified place. In England it occurs mainly in Bedfordshire, Cambridgeshire, and Norfolk.French : unexplained; possibly a habitational name for someone from a lost or unidentified place.
Girl/Female
Hindu, Indian
Goddess
Boy/Male
Muslim
Gold lamp
Boy/Male
Muslim
Killer, Blood shedder
Boy/Male
Irish
Prudent.
Girl/Female
Hindu
Sacred, Dignified
Surname or Lastname
English
English : variant spelling of Paine. This is also a well-established surname in Ireland.
GAUSSIAN FUNCTION
GAUSSIAN FUNCTION
GAUSSIAN FUNCTION
GAUSSIAN FUNCTION
GAUSSIAN FUNCTION
n.
A Russian weight, equal to forty Russian pounds or about thirty-six English pounds avoirdupois.
n.
A native or inhabitant of Russia; the language of Russia.
n.
A Russian measure of length containing 3,500 English feet.
n.
One who, not being a Russian, favors Russian policy and aggrandizement.
n.
A Russian drink distilled from rye.
n.
A Russian river craft used for transporting freight.
n.
A Russian village community.
a.
Of or pertaining to Lithuania (formerly a principality united with Poland, but now Russian and Prussian territory).
n.
The Russian variety of bagatelle.
n. sing. & pl.
A Russian, or the Russians.
a.
Of or pertaining to Prussia.
n.
A Russian measure of length = 2 ft. 4.246 inches.
n.
A native or inhabitant of Prussia.
n.
Prussian leather.
a.
Prussian; -- applied to certain astronomical tables published in the sixteenth century, founded on the principles of Copernicus, a Prussian.
n.
A kind of carp (Cyprinus gibelio); -- called also Prussian carp.
a.
Of or pertaining to Russia, its inhabitants, or language.
n.
A Russian copper coin. See Kopeck.
n.
Morbid dread of Russia or of Russian influence.