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In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution co... | https://en.wikipedia.org/wiki/Viscosity_solution |
A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. Graphical models are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine... | https://en.wikipedia.org/wiki/Graphical_model |
In mathematics, a covering system (also called a complete residue system) is a collection
$$
\{a_1\pmod{n_1},\ \ldots,\ a_k\pmod{n_k}\}
$$
of finitely many residue classes
$$
a_i\pmod{n_i} = \{ a_i + n_ix:\ x \in \Z \},
$$
whose union contains every integer.
## Examples and definitions
The notion of covering system wa... | https://en.wikipedia.org/wiki/Covering_system |
In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form
$$
\ddot x = \frac{d^2 x}{dt^2} = A(x),
$$
or equivalently of the form
$$
\dot v = \frac{dv}{dt} = A(x), \qquad \dot x = \frac{dx}{dt} = v,
$$
particularly in the case of a dynamical system of classica... | https://en.wikipedia.org/wiki/Leapfrog_integration |
In geometry, a deltoid curve, also known as a tricuspoid curve or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three or one-and-a-half times its radius. It is named af... | https://en.wikipedia.org/wiki/Deltoid_curve |
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
- the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field F, the group is , where 0 refe... | https://en.wikipedia.org/wiki/Multiplicative_group |
In telecommunication, a Berger code is a unidirectional error detecting code, named after its inventor, J. M. Berger. Berger codes can detect all unidirectional errors. Unidirectional errors are errors that only flip ones into zeroes or only zeroes into ones, such as in asymmetric channels. The check bits of Berger cod... | https://en.wikipedia.org/wiki/Berger_code |
In probability theory, the stability of a random variable is the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The distributions of random variables having this property are said to be "stable distributions". Results availa... | https://en.wikipedia.org/wiki/Stability_%28probability%29 |
In digital image processing and computer vision, image segmentation is the process of partitioning a digital image into multiple image segments, also known as image regions or image objects (sets of pixels). The goal of segmentation is to simplify and/or change the representation of an image into something that is more... | https://en.wikipedia.org/wiki/Image_segmentation |
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperpla... | https://en.wikipedia.org/wiki/Hyperplane |
In applied statistics, a partial regression plot attempts to show the effect of adding another variable to a model that already has one or more independent variables. Partial regression plots are also referred to as added variable plots, adjusted variable plots, and individual coefficient plots.
## Motivation
When perf... | https://en.wikipedia.org/wiki/Partial_regression_plot |
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.
In the 3-manifold case, a lens space can be visualized as the result of gluing two solid tori together by a homeomorphism of their... | https://en.wikipedia.org/wiki/Lens_space |
In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a ... | https://en.wikipedia.org/wiki/Construction_of_the_real_numbers |
Gaussian splatting is a volume rendering technique that deals with the direct rendering of volume data without converting the data into surface or line primitives. The technique was originally introduced as splatting by Lee Westover in the early 1990s.
This technique is revitalized and explodes in popularity in 2023, ... | https://en.wikipedia.org/wiki/Gaussian_splatting |
In statistics, normality tests are used to determine if a data set is well-modeled by a normal distribution and to compute how likely it is for a random variable underlying the data set to be normally distributed.
More precisely, the tests are a form of model selection, and can be interpreted several ways, depending on... | https://en.wikipedia.org/wiki/Normality_test |
In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space.
Physical events correspond to mathematical points in time and space, the set of a... | https://en.wikipedia.org/wiki/Four-velocity |
In geometry, a straight skeleton is a method of representing a polygon by a topological skeleton. It is similar in some ways to the medial axis but differs in that the skeleton is composed of straight line segments, while the medial axis of a polygon may involve parabolic curves. However, both are homotopy-equivalent t... | https://en.wikipedia.org/wiki/Straight_skeleton |
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also oc... | https://en.wikipedia.org/wiki/Weighted_arithmetic_mean |
A scene graph is a general data structure commonly used by vector-based graphics editing applications and modern computer games, which arranges the logical and often spatial representation of a graphical scene. It is a collection of nodes in a graph or tree structure. A tree node may have many children but only a singl... | https://en.wikipedia.org/wiki/Scene_graph |
{{DISPLAYTITLE:Lp space}}
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Ries... | https://en.wikipedia.org/wiki/Lp_space |
In differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors
$$
X,Y
$$
, that produces an output vector
$$
T(X,Y)
$$
representing the displacement within a tangent space when the tangent space is developed (or "rolled... | https://en.wikipedia.org/wiki/Torsion_tensor |
In information science, an ontology encompasses a representation, formal naming, and definitions of the categories, properties, and relations between the concepts, data, or entities that pertain to one, many, or all domains of discourse. More simply, an ontology is a way of showing the properties of a subject area and ... | https://en.wikipedia.org/wiki/Ontology_%28information_science%29 |
Z-order is an ordering of overlapping two-dimensional objects, such as windows in a stacking window manager, shapes in a vector graphics editor, or objects in a 3D application. One of the features of a typical GUI is that windows may overlap, so that one window hides part or all of another. When two windows overlap, th... | https://en.wikipedia.org/wiki/Z-order |
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, an... | https://en.wikipedia.org/wiki/Zermelo_set_theory |
Swarm intelligence (SI) is the collective behavior of decentralized, self-organized systems, natural or artificial. The concept is employed in work on artificial intelligence. The expression was introduced by Gerardo Beni and Jing Wang in 1989, in the context of cellular robotic systems.
Swarm intelligence systems cons... | https://en.wikipedia.org/wiki/Swarm_intelligence |
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrizes the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it... | https://en.wikipedia.org/wiki/Hyperbolic_angle |
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal elements are the eigenvalues of the original m... | https://en.wikipedia.org/wiki/Schur_decomposition |
The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by French engineer and physicist Augustin-Jean Fresnel () who was the first to understand that ligh... | https://en.wikipedia.org/wiki/Fresnel_equations |
In probability theory, a degenerate distribution on a measure space
$$
(E, \mathcal{A}, \mu)
$$
is a probability distribution whose support is a null set with respect to
$$
\mu
$$
. For instance, in the -dimensional space endowed with the Lebesgue measure, any distribution concentrated on a -dimensional subspace wi... | https://en.wikipedia.org/wiki/Degenerate_distribution |
In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space. More generally, the unit -sphere is an -sphere of unit radius in -dimensional Euclidean space; the unit circle is a special case, the unit -sphere in the plane. An (open... | https://en.wikipedia.org/wiki/Unit_sphere |
The Boy or Girl paradox surrounds a set of questions in probability theory, which are also known as The Two Child Problem, Mr. Smith's Children and the Mrs. Smith Problem. The initial formulation of the question dates back to at least 1959, when Martin Gardner featured it in his October 1959 "Mathematical Games column"... | https://en.wikipedia.org/wiki/Boy_or_girl_paradox |
Pruning is a data compression technique in machine learning and search algorithms that reduces the size of decision trees by removing sections of the tree that are non-critical and redundant to classify instances. Pruning reduces the complexity of the final classifier, and hence improves predictive accuracy by the redu... | https://en.wikipedia.org/wiki/Decision_tree_pruning |
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. Computers ar... | https://en.wikipedia.org/wiki/Finite_element_method |
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
In simple terms, the Hodge conjecture asserts that the basic topological information like the number of holes i... | https://en.wikipedia.org/wiki/Hodge_conjecture |
In number theory, a Wieferich prime is a prime number p such that p2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both ... | https://en.wikipedia.org/wiki/Wieferich_prime |
In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised... | https://en.wikipedia.org/wiki/Haar_wavelet |
In computer vision, the fundamental matrix
$$
\mathbf{F}
$$
is a 3×3 matrix which relates corresponding points in stereo images. In epipolar geometry, with homogeneous image coordinates, x and x′, of corresponding points in a stereo image pair, Fx describes a line (an epipolar line) on which the corresponding point ... | https://en.wikipedia.org/wiki/Fundamental_matrix_%28computer_vision%29 |
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
Two triangles are in perspective axially if and only if they are in perspective centrally.
Denote the three vertices of one triangle by and , and those of the other by and . Axial perspectivity means that lines and meet in a point... | https://en.wikipedia.org/wiki/Desargues%27s_theorem |
In differential geometry, Pu's inequality, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves contained in it.
## Statement
A student of Charles Loewner, Pu proved in his 1950 thesis that every Riemannian surface
$... | https://en.wikipedia.org/wiki/Pu%27s_inequality |
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex.
The formula for the nth hexagonal number
$$
h_n= 2n^2-n = n(2n-1)... | https://en.wikipedia.org/wiki/Hexagonal_number |
Image resolution is the level of detail of an image. The term applies to digital images, film images, and other types of images. "Higher resolution" means more image detail.
Image resolution can be measured in various ways. Resolution quantifies how close lines can be to each other and still be visibly resolved. Resolu... | https://en.wikipedia.org/wiki/Image_resolution |
In 3D computer graphics, normal mapping, or Dot3 bump mapping, is a texture mapping technique used for faking the lighting of bumps and dents – an implementation of bump mapping. It is used to add details without using more polygons. A common use of this technique is to greatly enhance the appearance and details of a l... | https://en.wikipedia.org/wiki/Normal_mapping |
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, whi... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space
$$
X
$$
, the so-called homology groups
$$
H_n(X).
$$
Intuitively, singular homology counts, for each dimension
$$
n
$$
, the
$$
n
$$
-dimensional holes of a space. Singular homology is a ... | https://en.wikipedia.org/wiki/Singular_homology |
In logic, a functionally complete set of logical connectives or Boolean operators is one that can be used to express all possible truth tables by combining members of the set into a Boolean expression.. ("[F]unctional completeness of [a] set of logical operators"). A well-known complete set of connectives is . Each of ... | https://en.wikipedia.org/wiki/Functional_completeness |
Metabolic network modelling, also known as metabolic network reconstruction or metabolic pathway analysis, allows for an in-depth insight into the molecular mechanisms of a particular organism. In particular, these models correlate the genome with molecular physiology. A reconstruction breaks down metabolic pathways (s... | https://en.wikipedia.org/wiki/Metabolic_network_modelling |
In image processing, a Gabor filter, named after Dennis Gabor, who first proposed it as a 1D filter.
The Gabor filter was first generalized to 2D by Gösta Granlund, by adding a reference direction.
The Gabor filter is a linear filter used for texture analysis, which essentially means that it analyzes whether there is ... | https://en.wikipedia.org/wiki/Gabor_filter |
In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order (the torsion elements of A). An abelian group A is called a torsion group (or periodic group) if every element of A has finite order and is called torsion-free if every e... | https://en.wikipedia.org/wiki/Torsion_subgroup |
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if
$$
S \subseteq N \in \Sigma \mbox{ and } \mu(N) = 0\ \Rightarrow\ S \in \S... | https://en.wikipedia.org/wiki/Complete_measure |
In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although the naive algorithm is often better for smaller matrices. The Strassen algor... | https://en.wikipedia.org/wiki/Strassen_algorithm |
In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for many differential equations as oscillatory integrals.
## Definition
An ... | https://en.wikipedia.org/wiki/Oscillatory_integral |
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a ... | https://en.wikipedia.org/wiki/Polynomial |
A partially observable Markov decision process (POMDP) is a generalization of a Markov decision process (MDP). A POMDP models an agent decision process in which it is assumed that the system dynamics are determined by an MDP, but the agent cannot directly observe the underlying state. Instead, it must maintain a sensor... | https://en.wikipedia.org/wiki/Partially_observable_Markov_decision_process |
Action principles lie at the heart of fundamental physics, from classical mechanics through quantum mechanics, particle physics, and general relativity. Action principles start with an energy function called a Lagrangian describing the physical system. The accumulated value of this energy function between two states of... | https://en.wikipedia.org/wiki/Action_principles |
In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being a... | https://en.wikipedia.org/wiki/Progressively_measurable_process |
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisit... | https://en.wikipedia.org/wiki/Modular_arithmetic |
In additive number theory and combinatorics, a restricted sumset has the form
$$
S=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n \ \mathrm{and}\ P(a_1,\ldots,a_n)\not=0\},
$$
where
$$
A_1,\ldots,A_n
$$
are finite nonempty subsets of a field F and
$$
P(x_1,\ldots,x_n)
$$
is a polynomial over F.
If
$$
P
$$
is a... | https://en.wikipedia.org/wiki/Restricted_sumset |
The Vigenère cipher () is a method of encrypting alphabetic text where each letter of the plaintext is encoded with a different Caesar cipher, whose increment is determined by the corresponding letter of another text, the key.
For example, if the plaintext is `attacking tonight` and the key is `oculorhinolaryngology`, ... | https://en.wikipedia.org/wiki/Vigen%C3%A8re_cipher |
Let
$$
\phi:M\to N
$$
be a smooth map between smooth manifolds
$$
M
$$
and
$$
N
$$
. Then there is an associated linear map from the space of 1-forms on
$$
N
$$
(the linear space of sections of the cotangent bundle) to the space of 1-forms on
$$
M
$$
. This linear map is known as the pullback (by
$$
\phi
$$
),... | https://en.wikipedia.org/wiki/Pullback_%28differential_geometry%29 |
In the mathematical field of topology, a uniform space is a set with additional structure that is used to define uniform properties, such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axio... | https://en.wikipedia.org/wiki/Uniform_space |
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.
##... | https://en.wikipedia.org/wiki/Zariski_tangent_space |
Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty principle. Therefore, even at absolute zero, atoms and molecules retain some vibrat... | https://en.wikipedia.org/wiki/Zero-point_energy |
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset
$$
S
$$
of integers and a target-sum
$$
T
$$
, and the question is to decide whether any subset of the integers sum to precisely
$$
T
$$
. The problem is known to be NP-complete. Moreover, so... | https://en.wikipedia.org/wiki/Subset_sum_problem |
In physics, the Heisenberg picture or Heisenberg representation is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which observables incorporate a dependency on time, but the states are time-independent. It stands in contrast to the Schrödinger picture in which observables are constant ... | https://en.wikipedia.org/wiki/Heisenberg_picture |
The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory.
Given a set of elements (henceforth referred to as the universe, specifying all possible elements under consideration) and a collection, referred to as , of a given subsets whose union equals... | https://en.wikipedia.org/wiki/Set_cover_problem |
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields.
In the original case, the ultrametric field of interest was essentially the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of th... | https://en.wikipedia.org/wiki/Newton_polygon |
"A Mathematical Theory of Communication" is an article by mathematician Claude E. Shannon published in Bell System Technical Journal in 1948. It was renamed The Mathematical Theory of Communication in the 1949 book of the same name, a small but significant title change after realizing the generality of this work. It ha... | https://en.wikipedia.org/wiki/A_Mathematical_Theory_of_Communication |
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Stockum in 1937 and later confirmed by Kurt Gödel in 1949, who discovered a s... | https://en.wikipedia.org/wiki/Closed_timelike_curve |
In graph theory and combinatorial optimization, Guan's route problem, the Chinese postman problem, postman tour or route inspection problem is to find a shortest closed path or circuit that visits every edge of an (connected) undirected graph at least once. When the graph has an Eulerian circuit (a closed walk that co... | https://en.wikipedia.org/wiki/Chinese_postman_problem |
In coding theory, the Singleton bound, named after the American mathematician Richard Collom Singleton (1928–2007), is a relatively crude upper bound on the size of an arbitrary block code
$$
C
$$
with block length
$$
n
$$
, size
$$
M
$$
and minimum distance
$$
d
$$
. It is also known as the Joshibound proved by ... | https://en.wikipedia.org/wiki/Singleton_bound |
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(x) and the related function Bi(x), are linearly independent solutions to the differential equation
$$
\frac{d^2y}{dx^2} - xy = 0 ,
$... | https://en.wikipedia.org/wiki/Airy_function |
In the domain of physics and probability, a Markov random field (MRF), Markov network or undirected graphical model is a set of random variables having a Markov property described by an undirected graph. In other words, a random field is said to be a Markov random field if it satisfies Markov properties. The concept or... | https://en.wikipedia.org/wiki/Markov_random_field |
In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten.
## Model equation
In systems d... | https://en.wikipedia.org/wiki/Total_variation_diminishing |
Rotational frequency, also known as rotational speed or rate of rotation (symbols ν, lowercase Greek nu, and also n), is the frequency of rotation of an object around an axis.
Its SI unit is the reciprocal seconds (s−1); other common units of measurement include the hertz (Hz), cycles per second (cps), and revolutions ... | https://en.wikipedia.org/wiki/Rotational_frequency |
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of monodromy comes from "running round singly". It is closely associated with coverin... | https://en.wikipedia.org/wiki/Monodromy |
In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them.
## Connection Laplacian
The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tenso... | https://en.wikipedia.org/wiki/Laplace_operators_in_differential_geometry |
Implementation is the realization of an application, execution of a plan, idea, model, design, specification, standard, algorithm, policy, or the administration or management of a process or objective.
## Industry-specific definitions
### Information technology
In the information technology industry, implementation re... | https://en.wikipedia.org/wiki/Implementation |
In differential geometry, a hyperkähler manifold is a Riemannian manifold
$$
(M, g)
$$
endowed with three integrable almost complex structures
$$
I, J, K
$$
that are Kähler with respect to the Riemannian metric
$$
g
$$
and satisfy the quaternionic relations
$$
I^2=J^2=K^2=IJK=-1
$$
. In particular, it is a hyper... | https://en.wikipedia.org/wiki/Hyperk%C3%A4hler_manifold |
An algebraic Riccati equation is a type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time.
A typical algebraic Riccati equation is similar to one of the following:
the continuous time algebraic Riccati equation (CARE):
$$
A^\top P + P A - ... | https://en.wikipedia.org/wiki/Algebraic_Riccati_equation |
In mathematics, the Morlet wavelet (or Gabor wavelet) is a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope). This wavelet is closely related to human perception, both hearing and vision.
## History
In 1946, physicist Dennis Gabor, applying ideas from quantum physics, intro... | https://en.wikipedia.org/wiki/Morlet_wavelet |
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, ... | https://en.wikipedia.org/wiki/Integer_factorization |
The Schur complement is a key tool in the fields of linear algebra, the theory of matrices, numerical analysis, and statistics.
It is defined for a block matrix. Suppose p, q are nonnegative integers such that p + q > 0, and suppose A, B, C, D are respectively p × p, p × q, q × p, and q × q matrices of complex numbers.... | https://en.wikipedia.org/wiki/Schur_complement |
In cryptography, Triple DES (3DES or TDES), officially the Triple Data Encryption
## Algorithm
(TDEA or Triple DEA), is a symmetric-key block cipher, which applies the DES cipher algorithm three times to each data block. The 56-bit key of the Data Encryption Standard (DES) is no longer considered adequate in the face... | https://en.wikipedia.org/wiki/Triple_DES |
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
## Statement
The... | https://en.wikipedia.org/wiki/Dirichlet%27s_test |
The cocountable topology, also known as the countable complement topology, is a topology that can be defined on any infinite set
$$
X
$$
. In this topology, a set is open if its complement in
$$
X
$$
is either countable or equal to the entire set. Equivalently, the open sets consist of the empty set and all subsets ... | https://en.wikipedia.org/wiki/Cocountable_topology |
In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the orig... | https://en.wikipedia.org/wiki/Conservative_extension |
In mathematics, a sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carl... | https://en.wikipedia.org/wiki/Equidistributed_sequence |
In logic, mathematics and linguistics, and (
$$
\wedge
$$
) is the truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as
$$
\wedge
$$
or
$$
\&
$$
or
$$
K
$$
(prefix) or
$$
\times
$$
or
$$
\cdot
$$
in which
$$
\wedge
$$
is the mos... | https://en.wikipedia.org/wiki/Logical_conjunction |
In the context of live-action and computer animation, interpolation is inbetweening, or filling in frames between the key frames. It typically calculates the in-between frames through use of (usually) piecewise polynomial interpolation to draw images semi-automatically.
For all applications of this type, a set of "key ... | https://en.wikipedia.org/wiki/Interpolation_%28computer_graphics%29 |
Lexicography is the study of lexicons and the art of compiling dictionaries. It is divided into two separate academic disciplines:
- Practical lexicography is the art or craft of compiling, writing and editing dictionaries.
- Theoretical lexicography is the scholarly study of semantic, orthographic, syntagmatic and par... | https://en.wikipedia.org/wiki/Lexicography |
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a compass.
The idealized ruler, known as a straightedge, is assumed... | https://en.wikipedia.org/wiki/Straightedge_and_compass_construction |
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups
$$
A
$$
and
$$
B
$$
is another abelian group
$$
A\oplus B
$$
consisting of the ordered p... | https://en.wikipedia.org/wiki/Direct_sum |
In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function
$$
f
$$
between topological spaces
$$
X
$$
and
$$
Y
$$
is the quotient
$$
M_f = (([0,1]\times X) \amalg Y)\,/\,\sim
$$
where the
$$
\amalg
$$
denotes the disjoint union, and ~ is the equivalence relation generated by
... | https://en.wikipedia.org/wiki/Mapping_cylinder |
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in
$$
\mathbb R^n,
$$
and the study of these lattices provides fundamental information on algebraic numbers. initiated this line of research at the ... | https://en.wikipedia.org/wiki/Geometry_of_numbers |
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.
The Brauer group arose out of attempts to classify division algebras ... | https://en.wikipedia.org/wiki/Brauer_group |
Biological data refers to a compound or information derived from living organisms and their products. A medicinal compound made from living organisms, such as a serum or a vaccine, could be characterized as biological data. Biological data is highly complex when compared with other forms of data. There are many forms o... | https://en.wikipedia.org/wiki/Biological_data |
In Riemannian geometry, a Jacobi field is a vector field along a geodesic
$$
\gamma
$$
in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesic... | https://en.wikipedia.org/wiki/Jacobi_field |
In differential geometry, a discipline within mathematics, a distribution on a manifold
$$
M
$$
is an assignment
$$
x \mapsto \Delta_x \subseteq T_x M
$$
of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle
$$
TM
$$... | https://en.wikipedia.org/wiki/Distribution_%28differential_geometry%29 |
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence
$$
I_1\subseteq I_2... | https://en.wikipedia.org/wiki/Noetherian_ring |
In mathematics, bicubic interpolation is an extension of cubic spline interpolation (a method of applying cubic interpolation to a data set) for interpolating data points on a two-dimensional regular grid. The interpolated surface (meaning the kernel shape, not the image) is smoother than corresponding surfaces obtaine... | https://en.wikipedia.org/wiki/Bicubic_interpolation |
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