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wiki_4100_chunk_0 | Weighted digraph | In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4101_chunk_0 | Multigraph | In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edges), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge. There are 2 distinct notions of multiple edges: Edges without own ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4102_chunk_0 | Oriented tree | In mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree or singly connected network) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4103_chunk_0 | Minimal resolution (algebra) | In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants characterizing the structure of a specific module or obje... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4104_chunk_0 | Minimal resolution (algebra) | Thus one speaks of a P resolution. In particular, every module has free resolutions, projective resolutions and flat resolutions, which are left resolutions consisting, respectively of free modules, projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consist... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4105_chunk_0 | Splitting lemma | In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence 0 ⟶ A ⟶ q B ⟶ r C ⟶ 0. {\displaystyle 0\longrightarrow A\mathrel {\overset {q}{\longrightarrow }} B\mathrel {\overset {r}{\longrightar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4106_chunk_0 | Linear endomorphism | In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V → W {\displaystyle V\to W} between two vector spaces that preserves the operations of vector addition and scalar mult... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4107_chunk_0 | Linear endomorphism | In the case where V = W {\displaystyle V=W} , a linear map is called a linear endomorphism. Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V {\displaystyle V} and W {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4108_chunk_0 | Vector subspace | In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4109_chunk_0 | Householder's method | In mathematics, and more specifically in numerical analysis, Householder's methods are a class of root-finding algorithms that are used for functions of one real variable with continuous derivatives up to some order d + 1. Each of these methods is characterized by the number d, which is known as the order of the method... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4110_chunk_0 | Duhamel's principle | In mathematics, and more specifically in partial differential equations, Duhamel's principle is a general method for obtaining solutions to inhomogeneous linear evolution equations like the heat equation, wave equation, and vibrating plate equation. It is named after Jean-Marie Duhamel who first applied the principle t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4111_chunk_0 | Duhamel's principle | The philosophy underlying Duhamel's principle is that it is possible to go from solutions of the Cauchy problem (or initial value problem) to solutions of the inhomogeneous problem. Consider, for instance, the example of the heat equation modeling the distribution of heat energy u in Rn. Indicating by ut (x, t) the tim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4112_chunk_0 | Duhamel's principle | By contrast, the inhomogeneous problem for the heat equation, corresponds to adding an external heat energy f (x, t) dt at each point. Intuitively, one can think of the inhomogeneous problem as a set of homogeneous problems each starting afresh at a different time slice t = t0. By linearity, one can add up (integrate) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4113_chunk_0 | Goldberg polyhedron | In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4114_chunk_0 | Goldberg polyhedron | A Goldberg polyhedron is a dual polyhedron of a geodesic sphere. A consequence of Euler's polyhedron formula is that a Goldberg polyhedron always has exactly twelve pentagonal faces. Icosahedral symmetry ensures that the pentagons are always regular and that there are always 12 of them. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4115_chunk_0 | Projective frame | In mathematics, and more specifically in projective geometry, a projective frame or projective basis is a tuple of points in a projective space that can be used for defining homogeneous coordinates in this space. More precisely, in a projective space of dimension n, a projective frame is a n + 2-tuple of points such th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4116_chunk_0 | Projective frame | Let P(V) be a projective space of dimension n, where V is a K-vector space of dimension n + 1. Let p: V ∖ { 0 } → P ( V ) {\displaystyle p:V\setminus \{0\}\to \mathbf {P} (V)} be the canonical projection that maps a nonzero vector v to the corresponding point of P(V), which is the vector line that contains v. Every fra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4117_chunk_0 | Projective frame | In particular, the only homography fixing the points of a frame is the identity map. This result is much more difficult in synthetic geometry (where projective spaces are defined through axioms). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4118_chunk_0 | Projective frame | It is sometimes called the first fundamental theorem of projective geometry.Every frame can be written as ( p ( e 0 ) , … , p ( e n ) , p ( e 0 + ⋯ + e n ) ) , {\displaystyle (p(e_{0}),\ldots ,p(e_{n}),p(e_{0}+\cdots +e_{n})),} where ( e 0 , … , e n ) {\displaystyle (e_{0},\dots ,e_{n})} is basis of V. The projective c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4119_chunk_0 | Krull's theorem | In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact is equivalent to Zorn'... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4120_chunk_0 | Algebraic ideal | Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4121_chunk_0 | Algebraic ideal | There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory). The related, but distinct, concept of an ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4122_chunk_0 | Noncommutative torus | In mathematics, and more specifically in the theory of C*-algebras, the noncommutative tori Aθ, also known as irrational rotation algebras for irrational values of θ, form a family of noncommutative C*-algebras which generalize the algebra of continuous functions on the 2-torus. Many topological and geometric propertie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4123_chunk_0 | Crossed product | In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. It is related to the semidirect product construction for groups. (Roughly speaking, crossed product is the expected s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4124_chunk_0 | Spread of a matrix | In mathematics, and more specifically matrix theory, the spread of a matrix is the largest distance in the complex plane between any two eigenvalues of the matrix. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4125_chunk_0 | Hyperfactorial | In mathematics, and more specifically number theory, the hyperfactorial of a positive integer n {\displaystyle n} is the product of the numbers of the form x x {\displaystyle x^{x}} from 1 1 {\displaystyle 1^{1}} to n n {\displaystyle n^{n}} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4126_chunk_0 | Superfactorial | In mathematics, and more specifically number theory, the superfactorial of a positive integer n {\displaystyle n} is the product of the first n {\displaystyle n} factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4127_chunk_0 | Coherence axiom | In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4128_chunk_0 | Escaping set | In mathematics, and particularly complex dynamics, the escaping set of an entire function ƒ consists of all points that tend to infinity under the repeated application of ƒ. That is, a complex number z 0 ∈ C {\displaystyle z_{0}\in \mathbb {C} } belongs to the escaping set if and only if the sequence defined by z n + 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4129_chunk_0 | Diamond principle | In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe (L) and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the axiom of cons... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4130_chunk_0 | Clubsuit | In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principles that are a weaker version of the corresponding ◊S; it was introduced in 1975 by Adam Ostaszewski. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4131_chunk_0 | Fichera's existence principle | In mathematics, and particularly in functional analysis, Fichera's existence principle is an existence and uniqueness theorem for solution of functional equations, proved by Gaetano Fichera in 1954. More precisely, given a general vector space V and two linear maps from it onto two Banach spaces, the principle states n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4132_chunk_0 | Dimension (graph theory) | In mathematics, and particularly in graph theory, the dimension of a graph is the least integer n such that there exists a "classical representation" of the graph in the Euclidean space of dimension n with all the edges having unit length. In a classical representation, the vertices must be distinct points, but the edg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4133_chunk_0 | Primary pseudoperfect number | In mathematics, and particularly in number theory, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation 1 N + ∑ p | N 1 p = 1 , {\displaystyle {\frac {1}{N}}+\sum _{p\,|\;\!N}{\frac {1}{p}}=1,} where the sum is over only the prime divisors of N. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4134_chunk_0 | Dirichlet principle | In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4135_chunk_0 | Universe (mathematics) | In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. In set theory, universes are often classes that contain (as elements) all sets for which one hopes to p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4136_chunk_0 | Universe (mathematics) | Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations. For instance, the canonical motivating example of a category is Set, the category of all sets, which cannot be formalized in a set theory without some notion of a universe. In type theory, a universe is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4137_chunk_0 | Hadamard factorization theorem | In mathematics, and particularly in the field of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving its zeroes and an exponential of a polynomial. It is named for Jacques Hadamard. The theorem may be viewed as an extension ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4138_chunk_0 | Weierstrass factorization theorem | In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4139_chunk_0 | Monodromy operator | In mathematics, and particularly ordinary differential equations (ODEs), a monodromy matrix is the fundamental matrix of a system of ODEs evaluated at the period of the coefficients of the system. It is used for the analysis of periodic solutions of ODEs in Floquet theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4140_chunk_0 | Characteristic multiplier | In mathematics, and particularly ordinary differential equations, a characteristic multiplier is an eigenvalue of a monodromy matrix. The logarithm of a characteristic multiplier is also known as characteristic exponent. They appear in Floquet theory of periodic differential operators and in the Frobenius method. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4141_chunk_0 | Milnor number | In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If f is a complex-valued holomorphic function germ then the Milnor number of f, denoted μ(f), is either a nonnegative integer, or is infinite. It can be considered both a geometric invari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4142_chunk_0 | Trivialization (mathematics) | The space E {\displaystyle E} is known as the total space of the fiber bundle, B {\displaystyle B} as the base space, and F {\displaystyle F} the fiber. In the trivial case, E {\displaystyle E} is just B × F , {\displaystyle B\times F,} and the map π {\displaystyle \pi } is just the projection from the product space to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4143_chunk_0 | Trivialization (mathematics) | Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4144_chunk_0 | Trivialization (mathematics) | Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, and the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself (with the identity mapping as projection) to E {\displaystyle E} is called a section of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4145_chunk_0 | Connection one-form | In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4146_chunk_0 | Connection one-form | In physics, connection forms are also used broadly in the context of gauge theory, through the gauge covariant derivative. A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4147_chunk_0 | Connection one-form | The main tensorial invariant of a connection form is its curvature form. In the presence of a solder form identifying the vector bundle with the tangent bundle, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4148_chunk_0 | Density bundle | In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at x is a function that a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4149_chunk_0 | Sum-of-divisors function | In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4150_chunk_0 | Positive-definite function on a group | In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4151_chunk_0 | Poisson kernel | In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Si... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4152_chunk_0 | Freudenthal suspension theorem | In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and increasing the index of th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4153_chunk_0 | Accessible pointed graph | In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions that allow multiple roots. Rooted graphs may also be known (depending on... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4154_chunk_0 | Real-root isolation | In mathematics, and, more specifically in numerical analysis and computer algebra, real-root isolation of a polynomial consist of producing disjoint intervals of the real line, which contain each one (and only one) real root of the polynomial, and, together, contain all the real roots of the polynomial. Real-root isola... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4155_chunk_0 | Real-root isolation | Moreover, it may be difficult to distinguish the real roots from the non-real roots with small imaginary part (see the example of Wilkinson's polynomial in next section). The first complete real-root isolation algorithm results from Sturm's theorem (1829). However, when real-root-isolation algorithms began to be implem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4156_chunk_0 | Antiholomorphic function | In mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions. A function of the complex variable z defined on an open set in the complex plane is said to be antiholomorphic if its derivative with respect to z exists i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4157_chunk_0 | Antiholomorphic function | This implies that a function is antiholomorphic if and only if it can be expanded in a power series in z in a neighborhood of each point in its domain. Also, a function f(z) is antiholomorphic on an open set D if and only if the function f(z) is holomorphic on D. If a function is both holomorphic and antiholomorphic, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4158_chunk_0 | Gauge symmetry (mathematics) | In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory. A gauge symmetry of a Lagrangian L {\displaystyle L} is defined as a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4159_chunk_0 | Gauge symmetry (mathematics) | For instance, this is the case of gauge symmetries in classical field theory. Yang–Mills gauge theory and gauge gravitation theory exemplify classical field theories with gauge symmetries.Gauge symmetries possess the following two peculiarities. Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy Noet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4160_chunk_0 | Periodic summation | In mathematics, any integrable function s ( t ) {\displaystyle s(t)} can be made into a periodic function s P ( t ) {\displaystyle s_{P}(t)} with period P by summing the translations of the function s ( t ) {\displaystyle s(t)} by integer multiples of P. This is called periodic summation: s P ( t ) = ∑ n = − ∞ ∞ s ( t ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4161_chunk_0 | Topological dual space | In mathematics, any vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V , {\displaystyle V,} together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is def... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4162_chunk_0 | Topological dual space | When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for dual include polarer Raum , espace conjugué, adjoint space , a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4163_chunk_0 | Chebyshev approximation | In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What is meant by best and simpler will depend on the application. A closely related topic is the approximation of functions by gener... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4164_chunk_0 | Chebyshev approximation | This is typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's floating point arithmetic. This is accomplished by using a polynomial of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4165_chunk_0 | Arithmetic combinatorics | In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4166_chunk_0 | Arithmetic Geometry | In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.In more abstract terms, arithmetic geometry can be defined as the study of s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4167_chunk_0 | Arithmetico–geometric series | In mathematics, arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Put plainly, the nth term of an arithmetico-geometric sequence is the product of the nth term of an arithmetic sequence and the nth term of a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4168_chunk_0 | Arithmetico–geometric series | For instance, the sequence 0 1 , 1 2 , 2 4 , 3 8 , 4 16 , 5 32 , ⋯ {\displaystyle {\dfrac {\color {blue}{0}}{\color {green}{1}}},\ {\dfrac {\color {blue}{1}}{\color {green}{2}}},\ {\dfrac {\color {blue}{2}}{\color {green}{4}}},\ {\dfrac {\color {blue}{3}}{\color {green}{8}}},\ {\dfrac {\color {blue}{4}}{\color {green}{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4169_chunk_0 | Asymmetry | In mathematics, asymmetry can arise in various ways. Examples include asymmetric relations, asymmetry of shapes in geometry, asymmetric graphs et cetera. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4170_chunk_0 | Comodule over a Hopf algebroid | In mathematics, at the intersection of algebraic topology and algebraic geometry, there is the notion of a Hopf algebroid which encodes the information of a presheaf of groupoids whose object sheaf and arrow sheaf are represented by algebras. Because any such presheaf will have an associated site, we can consider quasi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4171_chunk_0 | Cerf theory | In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions f: M → R {\displaystyle f\colon M\to \mathbb {R} } on a smooth manifold M {\displaystyle M} , their generic singularities and the topology of the subspaces these singular... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4172_chunk_0 | Auxiliary function | In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high order at some point. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4173_chunk_0 | Axiomatic proof | In mathematics, axiomatization is the process of taking a body of knowledge and working backwards towards its axioms. It is the formulation of a system of statements (i.e. axioms) that relate a number of primitive terms — in order that a consistent body of propositions may be derived deductively from these statements. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4174_chunk_0 | Base change lifting | In mathematics, base change lifting is a method of constructing new automorphic forms from old ones, that corresponds in Langlands philosophy to the operation of restricting a representation of a Galois group to a subgroup. The Doi–Naganuma lifting from 1967 was a precursor of the base change lifting. Base change lifti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4175_chunk_0 | Basic hypergeometric series | In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4176_chunk_0 | Bicubic interpolation | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4177_chunk_0 | Binary splitting | In mathematics, binary splitting is a technique for speeding up numerical evaluation of many types of series with rational terms. In particular, it can be used to evaluate hypergeometric series at rational points. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4178_chunk_0 | Birational automorphism | In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4179_chunk_0 | Calculus on Euclidean space | In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space R n {\displaystyle \mathbb {R} ^{n}} as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the Uni... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4180_chunk_0 | Calculus on finite weighted graphs | In mathematics, calculus on finite weighted graphs is a discrete calculus for functions whose domain is the vertex set of a graph with a finite number of vertices and weights associated to the edges. This involves formulating discrete operators on graphs which are analogous to differential operators in calculus, such a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4181_chunk_0 | Calculus on finite weighted graphs | Differential equations or difference equations on such graphs can be employed to leverage the graph's structure for tasks such as image segmentation (where the vertices represent pixels and the weighted edges encode pixel similarity based on comparisons of Moore neighborhoods or larger windows), data clustering, data c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4182_chunk_0 | Catastrophe theory | In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circum... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4183_chunk_0 | Catastrophe theory | It considers the special case where the long-run stable equilibrium can be identified as the minimum of a smooth, well-defined potential function (Lyapunov function). Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4184_chunk_0 | Catastrophe theory | However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures. In the late 1970s, applications of catastrophe theory to areas outside its scope began to be criticized, especially in biology and social scien... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4185_chunk_0 | Mathematical fallacies | In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4186_chunk_0 | Mathematical fallacies | The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4187_chunk_0 | Mathematical fallacies | Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.Mathematical fallacies exist in many branches of mathematics. In elementary algebra, typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a mul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4188_chunk_0 | Integral manifold | In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a submanifold, and the fact that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4189_chunk_0 | Integral manifold | A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold (which one sets equal to 0 to find solutions to the system). Given a collection of differential 1-forms α i , i = 1 , 2 , … , k... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4190_chunk_0 | Integral manifold | A maximal integral manifold is an immersed (not necessarily embedded) submanifold i: N ⊂ M {\displaystyle i:N\subset M} such that the kernel of the restriction map on forms i ∗: Ω p 1 ( M ) → Ω p 1 ( N ) {\displaystyle i^{*}:\Omega _{p}^{1}(M)\rightarrow \Omega _{p}^{1}(N)} is spanned by the α i {\displaystyle \textsty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4191_chunk_0 | Chiral homology | In mathematics, chiral homology, introduced by Alexander Beilinson and Vladimir Drinfeld, is, in their words, "a “quantum” version of (the algebra of functions on) the space of global horizontal sections of an affine D X {\displaystyle {\mathcal {D}}_{X}} -scheme (i.e., the space of global solutions of a system of non-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4192_chunk_0 | Chromatic tower | In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic le... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4193_chunk_0 | Global class field theory | In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.Hilbert is credited as one of pioneers of the notion of a class field. However, this notion wa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4194_chunk_0 | Global class field theory | The existence theorem states that the reciprocity map can be used to give a bijection between the set of abelian extensions of F and the set of closed subgroups of finite index of C F . {\displaystyle C_{F}.} A standard method for developing global class field theory since the 1930s was to construct local class field t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4195_chunk_0 | Global class field theory | This was first done by Emil Artin and Tate using the theory of group cohomology, and in particular by developing the notion of class formations. Later, Neukirch found a proof of the main statements of global class field theory without using cohomological ideas. His method was explicit and algorithmic. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4196_chunk_0 | Global class field theory | Inside class field theory one can distinguish special class field theory and general class field theory. Explicit class field theory provides an explicit construction of maximal abelian extensions of a number field in various situations. This portion of the theory consists of Kronecker–Weber theorem, which can be used ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4197_chunk_0 | Sweedler notation | In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4198_chunk_0 | Coarse function | In mathematics, coarse functions are functions that may appear to be continuous at a distance, but in reality are not necessarily continuous. Although continuous functions are usually observed on a small scale, coarse functions are usually observed on a large scale. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4199_chunk_0 | Cocompact embedding | In mathematics, cocompact embeddings are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness. Cocompactness has been in use in mathematical analysis since the 1980s, without being referred to by any name (Lemma 6),(Lemma 2.5),(Theorem 1), or by ad-hoc monikers such as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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