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In mathematics, Gaussian brackets are a special notation invented by Carl Friedrich Gauss to represent the convergents of a simple continued fraction
Gaussian_brackets
algorithm Gaussian brackets – described on WolframMathWorld Gaussian's modular arithmetic Gaussian integer, usually written as Z[i] Gaussian prime Gaussian logarithms
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
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
Theorem in probability theory
5\times 7=105} terms. We can evaluate the characteristic function of gaussians by the Isserlis theorem: E [ e − i X ] = ∑ k ( − i ) k k ! E [ X k ] =
Isserlis's_theorem
Family of polynomials
In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian numbers, Gaussian polynomials, or q-binomial coefficients)
Gaussian_binomial_coefficient
Special triangle in geometry
denominators k1, k2, ... then the relevant recursive relation is that of Gaussian brackets: hn = anhn − 1 + hn − 2, kn = ankn − 1 + kn − 2. The successive convergents
Calabi_triangle
Number represented as a0+1/(a1+1/...)
denominators k1, k2, ... then the relevant recursive relation is that of Gaussian brackets: h n = a n h n − 1 + h n − 2 , k n = a n k n − 1 + k n − 2 . {\displaystyle
Simple_continued_fraction
Matrix-valued random variable
components per matrix element. The Gaussian unitary ensemble GUE ( n ) {\displaystyle {\text{GUE}}(n)} is described by the Gaussian measure with density 1 Z GUE
Random_matrix
Situations involving imperfect or unknown information
uncertainty is often the standard uncertainty, which assumes an approximately Gaussian distribution, with the uncertainty expressing one standard deviation. This
Uncertainty
Mathematics of smooth surfaces
curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that
Differential geometry of surfaces
Differential_geometry_of_surfaces
Image-generating machine learning model
are trained with the objective of removing successive applications of Gaussian noise on training images, which can be thought of as a sequence of denoising
Stable_Diffusion
Relativistic vector field
indices for more details on notation. Formulae are given in SI units and Gaussian-cgs units. The contravariant electromagnetic four-potential can be defined
Electromagnetic four-potential
Electromagnetic_four-potential
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
Equations describing classical electromagnetism
opposite "magnetic charges". Precisely, the total magnetic flux through a Gaussian surface is zero, and the magnetic field is a solenoidal vector field. The
Maxwell's_equations
Formulation of quantum mechanics
product is a Gaussian as a function of x(t + ε) centered at x(t) with variance ε. The multiple integrals are a repeated convolution of this Gaussian Gε with
Path-integral_formulation
Assignment of an initial value for variable
to the constructor parameters: Example: class GaussianInteger { private: int re; int im; public: GaussianInteger(int re = 0, int im = 0): re{re}, im{im}
Initialization (computer programming)
Initialization_(computer_programming)
Concept in combinatorics (part of mathematics)
one can move on to define the q-binomial coefficients, also known as the Gaussian binomial coefficients, as [ n k ] q = [ n ] ! q [ n − k ] ! q [ k ] ! q
Q-Pochhammer_symbol
Tensor field in Riemannian geometry
the bracket ⟨ , ⟩ {\displaystyle \langle ,\rangle } refers to the inner product on the tangent space induced by the metric tensor and the brackets and
Riemann_curvature_tensor
In mathematics, invariant of square matrices
determinant as a linear combination of determinants of submatrices, or with Gaussian elimination, which allows computing a row echelon form with the same determinant
Determinant
Type of mathematical generalization
define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients: ( n k ) q = [ n ] q ! [
Q-analog
Concept in information theory
under constraints of mean and variance is the Gaussian. Let g ( x ) {\displaystyle g(x)} be a Gaussian PDF with mean μ and variance σ 2 {\displaystyle
Differential_entropy
Signal-processing operation
If the imaging beam has a Gaussian distribution, then when the truncation ratio (the ratio of the diameter of the Gaussian beam to the diameter of the
Apodization
Power transferred per unit area
monochromatic propagating electromagnetic wave such as a plane wave or a Gaussian beam travelling in a non-magnetic medium, the time-averaged Poynting vector
Intensity_(physics)
Approach used in computer vision systems
in blob detection. The scale-normalized Laplacian of the Gaussian and difference-of-Gaussian features (Lindeberg 1994, 1998; Lowe 2004) ∇ n o r m 2 L
Corner_detection
Basic concepts of algebra
linear combinations of the others. History of algebra Binary operation Gaussian elimination Mathematics education Number line Polynomial Cancelling out
Elementary_algebra
Aesthetic quality of blur in the out-of-focus parts of an image
uniform disk, a more computationally intensive operation than the "standard" Gaussian blur; the former produces sharp circles around highlights whereas the latter
Bokeh
Correlation as a function of distance
their correlation functions; the most notable example is the class of Gaussian processes. Probability distributions defined on a finite number of points
Correlation_function
this system, then one need promote the canonical Dirac brackets, not the canonical Poisson brackets to commutation relations. Examination of the above Hamiltonian
First-class_constraint
Wigner distribution function in physics as opposed to in signal processing
through a filter of size larger than ħ (e.g., convolving with a phase-space Gaussian, a Weierstrass transform, to yield the Husimi representation, below), results
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
inequalities for Gaussian processes and applications". Israel Journal of Mathematics. 50 (4): 265–289. doi:10.1007/bf02759761. Gordon, Y. (1988). "Gaussian processes
Dvoretzky's_theorem
Notion in geometry
connection (or covariant differentiation) ∇ {\displaystyle \nabla } and Lie bracket [ ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ]} by the following formula:
Curvature of Riemannian manifolds
Curvature_of_Riemannian_manifolds
Generalization of the concept from statistical mechanics
then partition function can be understood to be a sum or integral over Gaussians. The correlation function C ( x j , x k ) {\displaystyle C(x_{j},x_{k})}
Partition function (mathematics)
Partition_function_(mathematics)
Method of data analysis
independent identically distributed Gaussian noise, then the columns of T will also contain similarly identically distributed Gaussian noise (such a distribution
Principal_component_analysis
Chemical species structure notation
standard abbreviation of the chemical elements, in square brackets, such as [Au] for gold. Brackets may be omitted in the common case of atoms which: are
Simplified Molecular Input Line Entry System
Simplified_Molecular_Input_Line_Entry_System
Objective collapse theory in quantum mechanics
\psi _{x}^{i}|\psi _{x}^{i}\rangle } . The localization operator has a Gaussian form: L ^ x i = ( 1 π r C 2 ) 3 4 e − ( q ^ i − x ) 2 2 r C 2 , {\displaystyle
Ghirardi–Rimini–Weber_theory
Notion in statistics
where the Fisher information appears as the covariance of the fitted Gaussian. Statistical systems of a scientific nature (physical, biological, etc
Fisher_information
Branch of mathematics
Gauss map, Gaussian curvature, first and second fundamental forms, proved the Theorema Egregium showing the intrinsic nature of the Gaussian curvature
Differential_geometry
Operation in mathematical calculus
extrapolate to T(0). Gaussian quadrature evaluates the function at the roots of a set of orthogonal polynomials. An n-point Gaussian method is exact for
Integral
Photographic lens
profiles of the circles of confusion in such a way as to become truly Gaussian. Thereby, it is also deemed to improve the "bokeh" of the lens, that is
Minolta_STF_135mm_f/2.8_T4.5
Foundational principle in quantum physics
respectively. The minimum is attained for a Gaussian-shaped pulse (Gabor wavelet) [For the un-squared Gaussian (i.e. signal amplitude) and its un-squared
Uncertainty_principle
Stochastic differential equation
fluid. The force η ( t ) {\displaystyle {\boldsymbol {\eta }}(t)} has a Gaussian probability distribution with correlation function ⟨ η i ( t ) η j ( t
Langevin_equation
French mathematician and physicist (1781–1840)
predicted in 1857 that Poisson brackets would eventually supplant those of Lagrange. Jacobi's identity for Poisson's brackets became the basis for the study
Siméon_Denis_Poisson
Filters used in signal processing that are optimal in some sense
interpreted as a maximum likelihood method in the context of a (coloured) Gaussian noise model and the associated Whittle likelihood. If the transmitted signal
Matched_filter
Superimposition of two or more exposures to create a single image
averaging also permits there to be a time-windowing function, such as a Gaussian, that weights time periods near the center of the exposure time more strongly
Multiple_exposure
{\displaystyle {n \choose k}} is the binomial coefficient. Thus, e.g., Gaussians compose hyperbolically, exp ( − a ( q 2 + p 2 ) ) ⋆ exp ( − b
Deformation_quantization
Quadratic form related to curvatures of surfaces
u {\displaystyle L=-1,\,M=0,\,N=-\sin ^{2}u} . First fundamental form Gaussian curvature Gauss–Codazzi equations Shape operator Third fundamental form
Second_fundamental_form
Vector graphics editor from Adobe Inc
Improved path simplification; faster drop shadow, inner/outer glow, and Gaussian blur effects; improved file save/open on networks and removable media;
Adobe_Illustrator
Array of numbers
computing its inverse. One of the oldest, which is still in common use is Gaussian elimination. A symmetric real matrix A is called positive-definite if the
Matrix_(mathematics)
Description of a quantum-mechanical system
its energy is called the zero-point energy, and the wave function is a Gaussian. The harmonic oscillator, like the particle in a box, illustrates the generic
Schrödinger_equation
Integration over the space of functions
integrals that can be evaluated exactly usually start with the following Gaussian integral: ∫ exp { − 1 2 ∫ R [ ∫ R f ( x ) K ( x ; y ) f ( y ) d y + J
Functional_integration
Model of optics describing light as geometric rays
systems to be described by simple matrices. This leads to the techniques of Gaussian optics and paraxial ray tracing, which are used to find basic properties
Geometrical_optics
Imperial dynasty in China (202 BC – 220 AD)
mathematical proof of the Pythagorean theorem, use of the decimal fraction, Gaussian elimination to solve linear equations, and continued fractions to find
Han_dynasty
Calculates a confidence interval
first head of the Statistics Section at the National Physical Laboratory. Gaussian ratio distribution Fieller, EC. (1954). "Some problems in interval estimation"
Fieller's_theorem
Concept in crystallography
approximation, in which the probability density function is modeled as a Gaussian. Under this approximation, static displacive disorder is ignored and it
Debye–Waller_factor
Method of representing variables in Bayesian inference
in brackets in the middle of the node. Variables that are actually random matrices are similarly indicated by putting the matrix size in brackets in the
Plate_notation
Smooth manifold with an inner product on each tangent space
sub-Riemannian manifolds. In 1827, Carl Friedrich Gauss discovered that the Gaussian curvature of a surface embedded in 3-dimensional space only depends on
Riemannian_manifold
Mathematical sequences in combinatorics
promoted later by Donald Knuth, though the bracket notation conflicts with a common notation for Gaussian coefficients. The mathematical motivation for
Stirling_number
1093/mnrasl/slu076. hdl:2299/19219. S2CID 67807856. Bortle, Anna; et al. (2021). "A Gaussian Process Regression Reveals No Evidence for Planets Orbiting Kapteyn's Star"
List_of_nearest_stars
Measure of covariance of components of a random vector
a number of other dualities between marginalizing and conditioning for Gaussian random variables. For K X X = var ( X ) = E [ ( X − E [ X ] ) ( X
Covariance_matrix
Difference between logarithm and harmonic series
of a disk in the complex plane containing at least k {\displaystyle k} Gaussian integers. The following bounds have been established: 1.819776 < δ < 1
Euler's_constant
{R}}_{n}-{\vec {R}}_{n-1})} . This conformation is known as the Gaussian chain. The Gaussian approximation for ψ ( r → ) {\displaystyle \psi ({\vec {r}})}
Path integrals in polymer science
Path_integrals_in_polymer_science
Example of a phase-space star product in mathematics
phase space) is given in the article on the Wigner–Weyl transform: two Gaussians compose with this ★-product according to a hyperbolic tangent law: exp
Moyal_product
How many times curves wind around each other
{\displaystyle \epsilon } is the antisymmetric symbol. Since the theory is just Gaussian, no ultraviolet regularization or renormalization is needed. Therefore
Linking_number
Imaging technique
gating effect of OCT the complex degree of coherence is represented as a Gaussian function expressed as γ ( τ ) = exp [ − ( π Δ ν τ 2 ln 2 ) 2 ] ⋅ exp
Optical_coherence_tomography
Type of optimization problem
\theta } as x , y {\displaystyle x,y} changes. If the surface has nonzero Gaussian curvature, then the constraint is nonholonomic, because any change of θ
Nonholonomic_system
Inverse functions of sin, cos, tan, etc.
developed by Leonhard Euler, the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series. For real and complex values of z: ∫ arcsin ( z
Inverse trigonometric functions
Inverse_trigonometric_functions
as partition function, integrals of loop diagrams, etc. The following Gaussian integrals are useful in calculating path integrals appearing in path integral
Common integrals in quantum field theory
Common_integrals_in_quantum_field_theory
Mathematical theorem
holomorphic functions on Cn that are square-integrable with respect to a Gaussian measure. Fock observed in 1920s that the operators a j = ∂ ∂ z j , a j
Stone–von_Neumann_theorem
Quantum field theory with four-point interactions
^{2}]-{\frac {\lambda }{4!}}\varphi ^{4}.} The first term between the brackets is the energy related to the four-momentum of the particle, the second
Quartic_interaction
Regions of an electromagnetic field
source or aperture that is closer than the Rayleigh length. (Presuming a Gaussian beam, which is appropriate for fiber optics.) The most convenient practice
Near_and_far_field
Formulation of quantum mechanics
\hbar )^{-1}W} . Suppose a particle is initially in a minimally uncertain Gaussian state, with the expectation values of position and momentum both centered
Phase-space_formulation
Statistical measure of the magnitude of a phenomenon
for multi-factorial designs has been provided. Provided that the data is Gaussian distributed a scaled Hedges' g, n 1 n 2 / ( n 1 + n 2 ) g {\textstyle {\sqrt
Effect_size
DNA locus associated with variation in a quantitative trait
the more the distribution of the genotypes will resemble a normal, or Gaussian distribution. This shows that multifactorial inheritance is polygenic,
Quantitative_trait_locus
Special kind of square matrix
transformations; the 3×3 upper unitriangular matrices form the Heisenberg group. Gaussian elimination QR decomposition Cholesky decomposition Hessenberg matrix Tridiagonal
Triangular_matrix
3D printing technique
fixtures, and jigs Conformal cooling channels Rotors and impellers Complex bracketing Laser melting can produce chemical structures (pure metals, their oxides
Selective_laser_melting
Mathematical group
Ferraro, Alessandro; Olivares, Stefano; Paris, Matteo G. A. (March 2005), "Gaussian states in continuous variable quantum information", arXiv:quant-ph/0503237
Symplectic_group
Oscillatory error in Fourier series
(weights) the values of the kernel. If a kernel is non-negative, such as for a Gaussian kernel, then the value of the filtered signal will be a convex combination
Gibbs_phenomenon
carry both brine and natural gas for many miles to towns and villages. Gaussian elimination: The Chinese solved a system of linear equations. The rectangle
List_of_Chinese_inventions
Representation theory of the symplectic group
contraction operators, determined only up to a sign, have kernels that are Gaussian functions. On an infinitesimal level the semigroup is described by a cone
Oscillator_representation
Spectroscopic technique
Besides Lorentzian broadening, photoemission spectra are also affected by a Gaussian broadening, whose contribution can be expressed by I G ( E ) = I 0 σ 2
X-ray photoelectron spectroscopy
X-ray_photoelectron_spectroscopy
Family of probability distributions related to the normal distribution
enclosed statement is false, the Iverson bracket is zero. There are many variant notations, e.g. wavey brackets: ⧙a=b⧘ is equivalent to the [a=b] notation
Exponential_family
Topological space that locally resembles Euclidean space
century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature. Investigations of Niels Henrik Abel and Carl Gustav Jacobi on
Manifold
Mixing of fluids due to eddy currents
Ornstein-Uhlenbeck process) is also a Gaussian process. Thus, the mean scalar field predicted by the Langevin equation is the Gaussian distribution ⟨ ϕ ( x → , t
Eddy_diffusion
Method for bounding the errors of numerical computations
replaces the numerical operations, in that the linear algebra method known as Gaussian elimination becomes its interval version. However, since this method uses
Interval_arithmetic
equations Gauss–Seidel method: solves systems of linear equations iteratively Gaussian elimination Levinson recursion: solves equation involving a Toeplitz matrix
List_of_algorithms
algorithms to the end-users. Some examples are the introduction of blur and Gaussian pyramid to reduce data complexity, and multi-image and multi-metric framework
Elastix_(image_registration)
Intrinsic geometric structures in mathematics
loops about a point give rise to the holonomy group at that point. The Gaussian curvature at a point can be recovered from parallel transport around increasingly
Riemannian connection on a surface
Riemannian_connection_on_a_surface
Art and practice of creating images by recording light
Electromagnetic spectrum Fourier optics Focal length 35mm equivalent focal length Gaussian optics Lens flare Newton's rings Orb (optics) Optical transfer function
Outline_of_photography
Distortion of an object in photography
engage in the active process of interpretation. Consider an idealised Gaussian optical system, with the image and the object in the same medium. Thus
Perspective_distortion
Distance between the nearest and the furthest objects that are in focus in an image
simplifying assumptions: for example, they assume the paraxial approximation of Gaussian optics. They are suitable for practical photography, lens designers would
Depth_of_field
mortgage consumers. Formulas for calculating aggregate risk were based on the gaussian copula which wrongly assumed that individual components of mortgages were
Causes_of_the_Great_Recession
Origin and evolution of the symbols used to write equations and formulas
theorem of algebra and the quadratic reciprocity law. Gauss developed the Gaussian elimination method of solving linear systems, which was initially listed
History of mathematical notation
History_of_mathematical_notation
Technique pioneered by Nico van Kampen
approximation implies that fluctuations around the mean are Gaussian distributed. Non-Gaussian features of the distributions can be computed by taking into
System_size_expansion
General relativity model near spacetime singularities
statistical distribution of the exact values of sτ around its average is also Gaussian with the variance D s τ = 3.5 s τ ¯ 3 τ 2 = 0.26 τ {\displaystyle D_{s_{\tau
BKL_singularity
Digital imaging circuit
input electrons, this complex distribution function converges towards a Gaussian. Because of the lower costs and better resolution, EMCCDs are capable of
Charge-coupled_device
Statistics concept
of n independent observations x1,..., xn of a p-dimensional zero-mean Gaussian random variable X with covariance R, the maximum likelihood estimator of
Estimation of covariance matrices
Estimation_of_covariance_matrices
Probability distribution on a hyper-sphere of arbitrary dimension
variances decrease, the distributions of all three variables become more Gaussian, and the final approximation gets better as the dimensionality, p {\displaystyle
Von_Mises–Fisher_distribution
Nonlinear differential operator used to study conformal mappings
resulting Q is sometimes called the Q-value of the equation. Note that the Gaussian hypergeometric differential equation can be brought into the above form
Schwarzian_derivative
include the discovery of square roots, cube roots, the Pythagorean theorem, Gaussian elimination, the Horner scheme, improved calculations of pi, and negative
Science and technology of the Han dynasty
Science_and_technology_of_the_Han_dynasty
Technique for the generative modeling of a discrete probability distribution
standard framework of continuous diffusion does not apply, since it uses gaussian noise, which is continuous. Nevertheless, an analogous theory can be produced
Discrete_diffusion_model
GAUSSIAN BRACKETS
GAUSSIAN BRACKETS
Male
Russian
Variant spelling of Russian Vikentiy, VIKENTI means "conquering."
Female
Russian
(Russian Ева): Armenian and Russian form of Greek Eva, YEVA means "life."Â
Male
Russian
Variant spelling of Russian Afanasiy, AFANASY means "immortal."
Male
Russian
(Паша) Russian pet form of Czech/Russian Pavel, PASHA means "small."
Male
Russian
Variant spelling of Russian Vasiliy, VASSILY means "king."
Male
Russian
Variant spelling of Russian Afanasiy, AFANASEI means "immortal."
Male
Russian
(Russian ИÑидор): Russian form of Greek Isidoros, ISIDOR means "gift of Isis."
Male
Russian
Variant spelling of Russian Faddei, FADEI means "courageous."
Male
Russian
Variant spelling of Russian Arseniy, ARSENI means "virile."
Female
Russian
(Людмила) Russian feminine form of Czech/Russian Ludmil, LUDMILA means "people's favor."Â
Male
Russian
Variant spelling of Russian Aleksey, ALEXEY means "defender."
Male
Russian
Variant spelling of Russian Gennadiy, GENNADY means "noble."
Boy/Male
Australian, French, German, Irish
Curly-headed
Male
Russian
Variant spelling of Russian Vasiliy, VASILY means "king."
Male
Russian
Variant spelling of Russian Irinei, IRINEY means "peaceful."
Male
Russian
Variant spelling of Russian Gennadiy, GENNADI means "noble."
Male
Russian
(РоÑÑ) Russian pet form of Czech/Russian Rostislav, ROSTYA means "usurp-glory."
Male
Russian
Variant spelling of Russian Vasiliy, VASILI means "king."
Male
Russian
Variant spelling of Russian Afanasiy, AFANASII means "immortal."
Male
Russian
Variant spelling of Russian Arseniy, ARSENIY means "virile."
GAUSSIAN BRACKETS
GAUSSIAN BRACKETS
Girl/Female
Hindu, Indian, Malayalam
New; Expert; Wisdom
Surname or Lastname
English and Scottish
English and Scottish : variant of Ingram, influenced by Graham.
Boy/Male
Tamil
Deep insight
Boy/Male
Armenian, Australian
Nomadic Cart
Girl/Female
Muslim
Well shaped. Beautiful.
Girl/Female
Muslim/Islamic
Creation origination
Girl/Female
Arabic, Muslim
Left Hand Side
Girl/Female
Arabic, Muslim
Facilitated; Wealthy; Successful
Boy/Male
Tamil
Jayendra | ஜயேநà¯à®¤à¯à®°
Lord of victory
Boy/Male
Russian
Fighter.
GAUSSIAN BRACKETS
GAUSSIAN BRACKETS
GAUSSIAN BRACKETS
GAUSSIAN BRACKETS
GAUSSIAN BRACKETS
a.
Of or pertaining to Prussia.
n.
A Russian copper coin. See Kopeck.
n.
A native or inhabitant of Russia; the language of Russia.
n.
A Russian drink distilled from rye.
a.
Of or pertaining to Russia, its inhabitants, or language.
n.
A Russian measure of length containing 3,500 English feet.
n.
The Russian variety of bagatelle.
n.
A Russian river craft used for transporting freight.
n.
A native or inhabitant of Prussia.
a.
Of or pertaining to Lithuania (formerly a principality united with Poland, but now Russian and Prussian territory).
n.
A Russian measure of length = 2 ft. 4.246 inches.
n. sing. & pl.
A Russian, or the Russians.
n.
Morbid dread of Russia or of Russian influence.
n.
A Russian weight, equal to forty Russian pounds or about thirty-six English pounds avoirdupois.
n.
Prussian leather.
n.
One who, not being a Russian, favors Russian policy and aggrandizement.
n.
A Russian village community.
n.
A kind of carp (Cyprinus gibelio); -- called also Prussian carp.
a.
Prussian; -- applied to certain astronomical tables published in the sixteenth century, founded on the principles of Copernicus, a Prussian.