text stringlengths 2.13k 184k | source stringlengths 31 108 |
|---|---|
In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value
$$
A^\ast = \lim_{h\to 0} A(h)
$$
. In essence, given the value of
$$
A(h)
$$
for several values of
$$
h
$$
, we can estimate
$$
A^\ast
$$
by extrapola... | https://en.wikipedia.org/wiki/Richardson_extrapolation |
In physics, a spherical pendulum is a higher dimensional analogue of the pendulum. It consists of a mass moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity.
Owing to the spherical geometry of the problem, spherical coordinates are used to... | https://en.wikipedia.org/wiki/Spherical_pendulum |
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel, who proved it in 1826.
## Theorem
Let the Taylor series
$$
G (x) = \sum_{k=0}^\infty a_k x^k
$$
be a power series with real coefficients
$$
a_k... | https://en.wikipedia.org/wiki/Abel%27s_theorem |
A cryptographic protocol is an abstract or concrete protocol that performs a security-related function and applies cryptographic methods, often as sequences of cryptographic primitives. A protocol describes how the algorithms should be used and includes details about data structures and representations, at which point ... | https://en.wikipedia.org/wiki/Cryptographic_protocol |
In mathematics, a pseudogroup is a set of homeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a transformation group, originating however from the geometric approach of Sophus Lie to investigate symmetries of differential equations, rath... | https://en.wikipedia.org/wiki/Pseudogroup |
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
A special ca... | https://en.wikipedia.org/wiki/Open_mapping_theorem_%28functional_analysis%29 |
In computer graphics, photon mapping is a two-pass global illumination rendering algorithm developed by Henrik Wann Jensen between 1995 and 2001 that approximately solves the rendering equation for integrating light radiance at a given point in space. Rays from the light source (like photons) and rays from the camera a... | https://en.wikipedia.org/wiki/Photon_mapping |
In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality
$$
x \cdot (y + z) = x \cdot y + x \cdot z
$$
is always true in elementary algebra.
For example, in elementary arithmetic, one has
$$
2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3).
$... | https://en.wikipedia.org/wiki/Distributive_property |
The dynamic design analysis method (DDAM) is a US Navy-developed analytical procedure for evaluating the design of equipment subject to dynamic loading caused by underwater explosions (UNDEX). The analysis uses a form of shock spectrum analysis that estimates the dynamic response of a component to shock loading caused ... | https://en.wikipedia.org/wiki/Dynamic_design_analysis_method |
In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined asPaulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory (2000), p. 216.
$$
q_p(a) = \frac{a^{p-1}-1}{p},
$$
or
$$
\delta_p(a) = \frac{a - a^p }{p}
$$
.
This article is about the former; for the latter se... | https://en.wikipedia.org/wiki/Fermat_quotient |
In mathematics, the Leibniz formula for , named after Gottfried Wilhelm Leibniz, states that
$$
\frac{\pi}{4} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots =
# \sum_{k
=0}^{\infty} \frac{(-1)^{k}}{2k + 1},
$$
an alternating series.
It is sometimes called the Madhava–Leibniz series as it was first discover... | https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80 |
In statistics and physics, multicanonical ensemble (also called multicanonical sampling or flat histogram) is a Markov chain Monte Carlo sampling technique that uses the Metropolis–Hastings algorithm to compute integrals where the integrand has a rough landscape with multiple local minima. It samples states according t... | https://en.wikipedia.org/wiki/Multicanonical_ensemble |
A mesh is a representation of a larger geometric domain by smaller discrete cells. Meshes are commonly used to compute solutions of partial differential equations and render computer graphics, and to analyze geographical and cartographic data. A mesh partitions space into elements (or cells or zones) over which the equ... | https://en.wikipedia.org/wiki/Types_of_mesh |
In number theory, the home prime HP(n) of an integer n greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The mth intermediate stage in the process of determining HP(n) is designated HPn(m). For instance, HP(10) = 773, as 10 factors... | https://en.wikipedia.org/wiki/Home_prime |
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases.
There are several other (equivalent) approaches to f... | https://en.wikipedia.org/wiki/Probability_axioms |
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to... | https://en.wikipedia.org/wiki/Repeating_decimal |
Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency (or resonance frequency) of the system, defined as a frequency that generates a maximum amplitude response in the system. When this happens, the object or system a... | https://en.wikipedia.org/wiki/Resonance |
Computer audition (CA) or machine listening is the general field of study of algorithms and systems for audio interpretation by machines. Since the notion of what it means for a machine to "hear" is very broad and somewhat vague, computer audition attempts to bring together several disciplines that originally dealt wit... | https://en.wikipedia.org/wiki/Computer_audition |
In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real value... | https://en.wikipedia.org/wiki/Function_of_several_real_variables |
In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967. This variational principle is for incompressible and inviscid potential f... | https://en.wikipedia.org/wiki/Luke%27s_variational_principle |
In number theory, Khinchin's constant is a mathematical constant related to the simple continued fraction expansions of many real numbers. In particular Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, the coefficients ai of the continued fraction expansion of x have a finite geometric mean tha... | https://en.wikipedia.org/wiki/Khinchin%27s_constant |
In Euclidean geometry, two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.
## Process
A rotation in the plane can be formed by composing a pair of reflections. First reflect a point to its image on the other side of line . Then reflect to its imag... | https://en.wikipedia.org/wiki/Rotations_and_reflections_in_two_dimensions |
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of ... | https://en.wikipedia.org/wiki/Flux |
In computer science, best, worst, and average cases of a given algorithm express what the resource usage is at least, at most and on average, respectively. Usually the resource being considered is running time, i.e. time complexity, but could also be memory or some other resource.
Best case is the function which perfor... | https://en.wikipedia.org/wiki/Best%2C_worst_and_average_case |
Vector field reconstruction is a method of creating a vector field from experimental or computer-generated data, usually with the goal of finding a differential equation model of the system.
## Definition
A differential equation model is one that describes the value of dependent variables as they evolve in time or spac... | https://en.wikipedia.org/wiki/Vector_field_reconstruction |
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
$$
x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0
$$
for an arbitrary complex number
$$
\alpha
$$
, which represents the order ... | https://en.wikipedia.org/wiki/Bessel_function |
In mathematics, the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the non singular domain.
## Formulation... | https://en.wikipedia.org/wiki/Cauchy_principal_value |
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are onl... | https://en.wikipedia.org/wiki/Platonic_solid |
In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as
$$
a^p \equiv a \pmod p.
$$
For example, if and , then , and is an integer multiple of .
If is not divisible by , tha... | https://en.wikipedia.org/wiki/Fermat%27s_little_theorem |
10.84147120.958851...100.998334...1000.999983
As the positive integer becomes larger and larger, the value becomes arbitrarily close to . We say that "the limit of the sequence equals ."
In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the
$$
\l... | https://en.wikipedia.org/wiki/Limit_of_a_sequence |
In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. In other words,... | https://en.wikipedia.org/wiki/Exchangeable_random_variables |
The Canny edge detector is an edge detection operator that uses a multi-stage algorithm to detect a wide range of edges in images. It was developed by John F. Canny in 1986. Canny also produced a computational theory of edge detection explaining why the technique works.
## Development
Canny edge detection is a techniqu... | https://en.wikipedia.org/wiki/Canny_edge_detector |
In mathematics, the support of a real-valued function
$$
f
$$
is the subset of the function domain of elements that are not mapped to zero. If the domain of
$$
f
$$
is a topological space, then the support of
$$
f
$$
is instead defined as the smallest closed set containing all points not mapped to zero. This conc... | https://en.wikipedia.org/wiki/Support_%28mathematics%29 |
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections (see below ).
The set of Euclidean plane isometries f... | https://en.wikipedia.org/wiki/Euclidean_plane_isometry |
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some po... | https://en.wikipedia.org/wiki/Betti_number |
This article deals with a component of numerical methods. For coarse space in topology, see coarse structure.
In numerical analysis, coarse problem is an auxiliary system of equations used in an iterative method for the solution of a given larger system of equations. A coarse problem is basically a version of the same... | https://en.wikipedia.org/wiki/Coarse_space_%28numerical_analysis%29 |
In quantum mechanics, indistinguishable particles (also called identical or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (su... | https://en.wikipedia.org/wiki/Indistinguishable_particles |
+Hyperbolic quaternion multiplication×1ijk11ijkii +1k −jjj −k +1ikkj −i +1
In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form
$$
q = a + bi + cj + dk, \quad a,b,c,d \in \mathbb{R} \!
$$
where the squares of i, j, and k are +1 and distin... | https://en.wikipedia.org/wiki/Hyperbolic_quaternion |
The Chebotarev density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field
$$
\mathbb{Q}
$$
of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only fi... | https://en.wikipedia.org/wiki/Chebotarev_density_theorem |
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module M over a ring R is flat if taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces... | https://en.wikipedia.org/wiki/Flat_module |
In numerical analysis, Chebyshev nodes (also called Chebyshev points or a Chebyshev grid) are a set of specific algebraic numbers used as nodes for polynomial interpolation and numerical integration. They are the projection of a set of equispaced points on the unit circle onto the real interval
$$
[-1, 1]
$$
, the cir... | https://en.wikipedia.org/wiki/Chebyshev_nodes |
In statistics, the backfitting algorithm is a simple iterative procedure used to fit a generalized additive model. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the Gauss–Seidel method, an algorithm used for... | https://en.wikipedia.org/wiki/Backfitting_algorithm |
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer l... | https://en.wikipedia.org/wiki/Complex_multiplication |
In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hilbert spaces), but for general Banach spaces or locally convex topologi... | https://en.wikipedia.org/wiki/Topological_tensor_product |
In applied mathematics, polyharmonic splines are used for function approximation and data interpolation. They are very useful for interpolating and fitting scattered data in many dimensions. Special cases include thin plate splinesJ. Duchon: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. Construct... | https://en.wikipedia.org/wiki/Polyharmonic_spline |
The fast marching method is a numerical method created by James Sethian for solving boundary value problems of the Eikonal equation:
$$
|\nabla u(x)|=1/f(x) \text{ for } x \in \Omega
$$
$$
u(x) = 0 \text{ for } x \in \partial \Omega
$$
Typically, such a problem describes the evolution of a closed surface as a function ... | https://en.wikipedia.org/wiki/Fast_marching_method |
A goniometer is an instrument that either measures an angle or allows an object to be rotated to a precise angular position. The term goniometry derives from two Greek words, γωνία (gōnía) 'angle' and μέτρον (métron) 'measure'. The protractor is a commonly used type in the fields of mechanics, engineering, and geometry... | https://en.wikipedia.org/wiki/Goniometer |
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelih... | https://en.wikipedia.org/wiki/Probability_density_function |
An adaptive filter is a system with a linear filter that has a transfer function controlled by variable parameters and a means to adjust those parameters according to an optimization algorithm. Because of the complexity of the optimization algorithms, almost all adaptive filters are digital filters. Adaptive filters ar... | https://en.wikipedia.org/wiki/Adaptive_filter |
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
## Motivation
Consider a topological space, that is, a space with some notion... | https://en.wikipedia.org/wiki/Mapping_class_group |
In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, Siegfried Heinrich Aronhold, Alfred Clebsch, and Paul Gordan in the 19th century for computing invariants of algebraic forms. It is based on treating the form as if it were a power of a degree one form, which correspond... | https://en.wikipedia.org/wiki/Symbolic_method |
In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph.
Formally, given a graph , a vertex labeling is a function of to a set of labels; a graph with such a function defined is called a vertex-labeled grap... | https://en.wikipedia.org/wiki/Graph_labeling |
In 3D computer graphics, popping refers to an undesirable visual effect that occurs when the transition of a 3D object to a different pre-calculated level of detail (LOD) is abrupt and obvious to the viewer. The LOD-ing algorithm reduces the geometrical complexity of a 3D object the further it is from the viewer and re... | https://en.wikipedia.org/wiki/Popping_%28computer_graphics%29 |
The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing property of conditional expectation, among other names, states that if
$$
X
$$
is a random variable whose expected value
$$
\operatorname{E}(X)
$$
is de... | https://en.wikipedia.org/wiki/Law_of_total_expectation |
In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically checking all possible candidates for whether or not each candidate satisfies the problem's statement.
A brute-force algorith... | https://en.wikipedia.org/wiki/Brute-force_search |
In topology, a subbase (or subbasis, prebase, prebasis) for the topology of a topological space is a subcollection
$$
B
$$
of
$$
\tau
$$
that generates
$$
\tau,
$$
in the sense that
$$
\tau
$$
is the smallest topology containing
$$
B
$$
as open sets. A slightly different definition is used by some authors, ... | https://en.wikipedia.org/wiki/Subbase |
In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on
$$
\mathfrak{g}
$$
defined by the formula
$$
\kappa(u,v)=\opera... | https://en.wikipedia.org/wiki/Cartan%27s_criterion |
Guided local search is a metaheuristic search method. A meta-heuristic method is a method that sits on top of a local search algorithm to change its behavior.
Guided local search builds up penalties during a search. It uses penalties to help local search algorithms escape from local minima and plateaus. When the given... | https://en.wikipedia.org/wiki/Guided_local_search |
In abstract algebra, a skew lattice is an algebraic structure that is a non-commutative generalization of a lattice. While the term skew lattice can be used to refer to any non-commutative generalization of a lattice, since 1989 it has been used primarily as follows.
## Definition
A skew lattice is a set S equipped wit... | https://en.wikipedia.org/wiki/Skew_lattice |
In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot. It is a knot invariant.
## Examples
By way of example, the unknot has crossing number zero, the trefoil knot three and the figure-eight knot four. There are no other knots with a crossi... | https://en.wikipedia.org/wiki/Crossing_number_%28knot_theory%29 |
In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem sh... | https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem |
In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function.
## Definition
Let
$$
f : X \to S
$$
be a function from a topological space
... | https://en.wikipedia.org/wiki/Locally_constant_function |
A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations
$$
x=A\sin(at+\delta),\quad y=B\sin(bt),
$$
which describe the superposition of two perpendicular oscillations in x and y directions of different angular frequency (a and b). The resulting family of c... | https://en.wikipedia.org/wiki/Lissajous_curve |
The Kane quantum computer is a proposal for a scalable quantum computer proposed by Bruce Kane in 1998, who was then at the University of New South Wales. Often thought of as a hybrid between quantum dot and nuclear magnetic resonance (NMR) quantum computers, the Kane computer is based on an array of individual phospho... | https://en.wikipedia.org/wiki/Kane_quantum_computer |
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time domain (if applicable) are discretized, or broken into a finite number of intervals, and the values of the ... | https://en.wikipedia.org/wiki/Finite_difference_method |
In mathematical logic and philosophy, Skolem's paradox is the apparent contradiction that a countable model of first-order set theory could contain an uncountable set. The paradox arises from part of the Löwenheim–Skolem theorem; Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem... | https://en.wikipedia.org/wiki/Skolem%27s_paradox |
A mind map is a diagram used to visually organize information into a hierarchy, showing relationships among pieces of the whole. It is often based on a single concept, drawn as an image in the center of a blank page, to which associated representations of ideas such as images, words and parts of words are added. Major... | https://en.wikipedia.org/wiki/Mind_map |
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called
### Calabi–Yau manifolds
. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory... | https://en.wikipedia.org/wiki/Mirror_symmetry_%28string_theory%29 |
Littlewood's law states that a person can expect to experience events with odds of one in a million (referred to as a "miracle") at the rate of about one per month. It is named after the British mathematician John Edensor Littlewood.
It seeks, among other things, to debunk one element of supposed supernatural phenomeno... | https://en.wikipedia.org/wiki/Littlewood%27s_law |
In mathematics, a Lindelöf spaceWillard, Def. 16.5, p. 110 is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover.
A is a topological space such that every subspac... | https://en.wikipedia.org/wiki/Lindel%C3%B6f_space |
A ternary search algorithm is a technique in computer science for finding the minimum or maximum of a unimodal function.
## The function
Assume we are looking for a maximum of
$$
f(x)
$$
and that we know the maximum lies somewhere between
$$
A
$$
and
$$
B
$$
. For the algorithm to be applicable, there must be some... | https://en.wikipedia.org/wiki/Ternary_search |
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.
Spelled out in more detail, this means that a category C is pre-abelian if:
1. C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets i... | https://en.wikipedia.org/wiki/Pre-abelian_category |
In mathematics, the Champernowne constant is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933. The number is defined by concatenating the base-10 representations of the posit... | https://en.wikipedia.org/wiki/Champernowne_constant |
In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). The surface created by this revolution and which bounds the solid is the surface of revolution.
Assuming that the ... | https://en.wikipedia.org/wiki/Solid_of_revolution |
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits, i.e., decomposes into linear factors.
## Definition
A splitting field of a polynomial p(X) over a field K is a field extension L of K over which p factors i... | https://en.wikipedia.org/wiki/Splitting_field |
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation
$$
(\rho, V)
$$
or irrep of an algebraic structure
$$
A
$$
is a nonzero representation that has no proper nontrivial subrepresentation
$$
(\rho|_W,W)
$$
, with
$$
W \subset V
$$
closed under the action... | https://en.wikipedia.org/wiki/Irreducible_representation |
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint.
... | https://en.wikipedia.org/wiki/Inscribed_angle |
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used in number theory, mathematical statistics, and the theory of asymptotic e... | https://en.wikipedia.org/wiki/Mellin_transform |
For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. Where it is asserted that some list of integers is finite, the question is whethe... | https://en.wikipedia.org/wiki/Effective_results_in_number_theory |
The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds:
By computing a formula for how the scalar curvature of relates to that of , this statement can be rephrased in the followi... | https://en.wikipedia.org/wiki/Yamabe_problem |
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus g > 1, given by an equation of the form
$$
y^2 + h(x)y = f(x)
$$
where f(x) is a polynomial of degree n = 2g + 1 > 4 or n = 2g + 2 > 4 with n distinct roots, and h(x) is a polynomial of degree < g + 2 (if the characteristic of the ground field ... | https://en.wikipedia.org/wiki/Hyperelliptic_curve |
In statistics and econometrics, the multinomial probit model is a generalization of the probit model used when there are several possible categories that the dependent variable can fall into. As such, it is an alternative to the multinomial logit model as one method of multiclass classification. It is not to be confus... | https://en.wikipedia.org/wiki/Multinomial_probit |
Non-photorealistic rendering (NPR) is an area of computer graphics that focuses on enabling a wide variety of expressive styles for digital art, in contrast to traditional computer graphics, which focuses on photorealism. NPR is inspired by other artistic modes such as painting, drawing, technical illustration, and ani... | https://en.wikipedia.org/wiki/Non-photorealistic_rendering |
In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation-complete theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically applying the resolution rule acts as a decision procedure for formula unsa... | https://en.wikipedia.org/wiki/Resolution_%28logic%29 |
In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.
The point from which the object is viewed is called the apex of the solid angle, a... | https://en.wikipedia.org/wiki/Solid_angle |
In computer science, a mask or bitmask is data that is used for bitwise operations, particularly in a bit field. Using a mask, multiple bits in a byte, nibble, word, etc. can be set either on or off, or inverted from on to off (or vice versa) in a single bitwise operation. An additional use of masking involves predica... | https://en.wikipedia.org/wiki/Mask_%28computing%29 |
In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation
$$
x^2 - y^2 = 1 .
$$
In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length
$$
r = \sqrt {x^2 - y^2} .
$$
Whereas the unit circle surrounds i... | https://en.wikipedia.org/wiki/Unit_hyperbola |
PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number s... | https://en.wikipedia.org/wiki/PrimeGrid |
An operator is a function over a space of physical states onto another space of states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are useful tools in classical mechanics. Operators are even more important ... | https://en.wikipedia.org/wiki/Operator_%28physics%29 |
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence
$$
( x_n )_{n=1}^{\infty}
$$
of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.
Like the other axioms of countabilit... | https://en.wikipedia.org/wiki/Separable_space |
In probability theory, the
## Gram–Charlier A series
(named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution over the real line
$$
(-\infty,\infty)
$$
in terms of its cumulants. The serie... | https://en.wikipedia.org/wiki/Edgeworth_series |
In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called non-conjunction, alternative denial (since it says in effect that at least one of its operands ... | https://en.wikipedia.org/wiki/Sheffer_stroke |
The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in an article ... | https://en.wikipedia.org/wiki/AKS_primality_test |
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by repl... | https://en.wikipedia.org/wiki/Cramer%27s_rule |
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as does . Furthermore, itself is both a... | https://en.wikipedia.org/wiki/Coset |
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties t... | https://en.wikipedia.org/wiki/Compact_group |
In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 ... | https://en.wikipedia.org/wiki/Borel_functional_calculus |
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826.
## Statement
... | https://en.wikipedia.org/wiki/Summation_by_parts |
In the mathematical theory of probability, a Doob martingale (named after Joseph L. Doob, also known as a Levy martingale) is a stochastic process that approximates a given random variable and has the martingale property with respect to the given filtration. It may be thought of as the evolving sequence of best approxi... | https://en.wikipedia.org/wiki/Doob_martingale |
In geometry, an altitude of a triangle is a line segment through a given vertex (called apex) and perpendicular to a line containing the side or edge opposite the apex. This (finite) edge and (infinite) line extension are called, respectively, the base and extended base of the altitude. The point at the intersection o... | https://en.wikipedia.org/wiki/Altitude_%28triangle%29 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.