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wiki_4300_chunk_0 | Divided differences | In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation.Divided differences is a recursive division process. Given a seq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4301_chunk_0 | Division by infinity | In mathematics, division by infinity is division where the divisor (denominator) is infinity. In ordinary arithmetic, this does not have a well-defined meaning, since infinity is a mathematical concept that does not correspond to a specific number, and moreover, there is no nonzero real number that, when added to itsel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4302_chunk_0 | Division by infinity | The crux of the matter rests in the equation ∞ ÷ y = ∞, where the introduction of y introduces an essential condition. To render the equation coherent, y must assume a magnitude that is sufficiently vast to accommodate the division operation involving infinity. This requirement reflects the conceptual intricacies assoc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4303_chunk_0 | Division by infinity | However, the narrative takes a noteworthy turn as we transition to the equation y × ∞ = ∞. This equation signifies a transformation of the division operation into one of multiplication. In essence, this transition underscores a relationship where division of infinity finds equivalence through multiplication with an app... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4304_chunk_0 | Division by two | In mathematics, division by two or halving has also been called mediation or dimidiation. The treatment of this as a different operation from multiplication and division by other numbers goes back to the ancient Egyptians, whose multiplication algorithm used division by two as one of its fundamental steps. Some mathema... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4305_chunk_0 | Division by zero | Since any number multiplied by zero is zero, the expression 0 0 {\displaystyle {\tfrac {0}{0}}} is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to a 0 {\textstyle {\tfrac {a}{0}}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4306_chunk_0 | Dynamic equation | In mathematics, dynamic equation can refer to: difference equation in discrete time differential equation in continuous time time scale calculus in combined discrete and continuous time | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4307_chunk_0 | Stable roommates problem | In mathematics, economics and computer science, particularly in the fields of combinatorics, game theory and algorithms, the stable-roommate problem (SRP) is the problem of finding a stable matching for an even-sized set. A matching is a separation of the set into disjoint pairs ("roommates"). The matching is stable if... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4308_chunk_0 | Gale–Shapley algorithm | In mathematics, economics, and computer science, the Gale–Shapley algorithm (also known as the deferred acceptance algorithm or propose-and-reject algorithm) is an algorithm for finding a solution to the stable matching problem, named for David Gale and Lloyd Shapley. It takes polynomial time, and the time is linear in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4309_chunk_0 | Lattice of stable matchings | In mathematics, economics, and computer science, the lattice of stable matchings is a distributive lattice whose elements are stable matchings. For a given instance of the stable matching problem, this lattice provides an algebraic description of the family of all solutions to the problem. It was originally described i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4310_chunk_0 | Lattice of stable matchings | The family of all rotations and their partial order can be constructed in polynomial time, leading to polynomial time solutions for other problems on stable matching including the minimum or maximum weight stable matching. The Gale–Shapley algorithm can be used to construct two special lattice elements, its top and bot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4311_chunk_0 | Stable marriage problem | In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem or SMP) is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. A matching is a bijection from the elements of one set to the eleme... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4312_chunk_0 | Stable marriage problem | The stable marriage problem has been stated as follows: Given n men and n women, where each person has ranked all members of the opposite sex in order of preference, marry the men and women together such that there are no two people of opposite sex who would both rather have each other than their current partners. When... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4313_chunk_0 | Elliptic curve primality | In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin the same year. The algori... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4314_chunk_0 | Elliptic curve primality | Primality testing is a field that has been around since the time of Fermat, in whose time most algorithms were based on factoring, which become unwieldy with large input; modern algorithms treat the problems of determining whether a number is prime and what its factors are separately. It became of practical importance ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4315_chunk_0 | Endoscopic group | In mathematics, endoscopic groups of reductive algebraic groups were introduced by Robert Langlands (1979, 1983) in his work on the stable trace formula. Roughly speaking, an endoscopic group H of G is a quasi-split group whose L-group is the connected component of the centralizer of a semisimple element of the L-group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4316_chunk_0 | Clemens conjecture | In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4317_chunk_0 | Equality (mathematics) | For example: x = y {\displaystyle x=y} means that x and y denote the same object. The identity ( x + 1 ) 2 = x 2 + 2 x + 1 {\displaystyle (x+1)^{2}=x^{2}+2x+1} means that if x is any number, then the two expressions have the same value. This may also be interpreted as saying that the two sides of the equals sign repres... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4318_chunk_0 | Equality (mathematics) | This assertion, which uses set-builder notation, means that if the elements satisfying the property P ( x ) {\displaystyle P(x)} are the same as the elements satisfying Q ( x ) , {\displaystyle Q(x),} then the two uses of the set-builder notation define the same set. This property is often expressed as "two sets that h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4319_chunk_0 | Equivalent definitions of mathematical structures | In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry), a notion (for example, ellipse or minimal surface) may have more than one definition. These definitions are equivalent in the context of a given mathematical ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4320_chunk_0 | Equivariant morphism | In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, app... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4321_chunk_0 | Equivariant algebraic topoloy | In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps f: X → Y {\displaystyle f:X\to Y} , and while equivariant topology also considers such maps, there is the additional constraint that each map "resp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4322_chunk_0 | Ergodic flow | In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4323_chunk_0 | Unique ergodicity | Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4324_chunk_0 | Unique ergodicity | Ergodic systems capture the common-sense, every-day notions of randomness, such that smoke might come to fill all of a smoke-filled room, or that a block of metal might eventually come to have the same temperature throughout, or that flips of a fair coin may come up heads and tails half the time. A stronger concept tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4325_chunk_0 | Categorical Algebra | In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4326_chunk_0 | Bass conjecture | In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated. The conjecture was proposed by Hyman Bass. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4327_chunk_0 | Decomposition theorem of Beilinson, Bernstein and Deligne | In mathematics, especially algebraic geometry, the decomposition theorem of Beilinson, Bernstein and Deligne or BBD decomposition theorem is a set of results concerning the cohomology of algebraic varieties. It was originally conjectured by Gelfand and MacPherson. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4328_chunk_0 | Cotangent manifold | In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4329_chunk_0 | Bessel's inequality | In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x {\displaystyle x} in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828.Let H {\displaystyle H} be a Hilbert space, and suppose that e 1 , e 2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4330_chunk_0 | Bessel's inequality | . . {\displaystyle e_{1},e_{2},...} is an orthonormal sequence in H {\displaystyle H} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4331_chunk_0 | Banach ring | In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4332_chunk_0 | Banach ring | Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering A e {\displaystyle A_{e}} and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4333_chunk_0 | Banach ring | The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements. Banach algebras can also be defined over fields of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4334_chunk_0 | Fréchet algebra | In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra A {\displaystyle A} over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation ( a , b ) ↦ a ∗ b {\displaystyle (a,b)\mapsto a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4335_chunk_0 | Fréchet algebra | In that case, by rescaling the seminorms, we may also take C n = 1 {\displaystyle C_{n}=1} for each n {\displaystyle n} and the seminorms are said to be submultiplicative: ‖ a b ‖ n ≤ ‖ a ‖ n ‖ b ‖ n {\displaystyle \|ab\|_{n}\leq \|a\|_{n}\|b\|_{n}} for all a , b ∈ A . {\displaystyle a,b\in A.} m {\displaystyle m} -con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4336_chunk_0 | Vector bornology | In mathematics, especially functional analysis, a bornology B {\displaystyle {\mathcal {B}}} on a vector space X {\displaystyle X} over a field K , {\displaystyle \mathbb {K} ,} where K {\displaystyle \mathbb {K} } has a bornology ℬ F {\displaystyle \mathbb {F} } , is called a vector bornology if B {\displaystyle {\mat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4337_chunk_0 | Bornology | In mathematics, especially functional analysis, a bornology on a set X is a collection of subsets of X satisfying axioms that generalize the notion of boundedness. One of the key motivations behind bornologies and bornological analysis is the fact that bornological spaces provide a convenient setting for homological al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4338_chunk_0 | Hypercyclic operator | In mathematics, especially functional analysis, a hypercyclic operator on a Banach space X is a bounded linear operator T: X → X such that there is a vector x ∈ X such that the sequence {Tn x: n = 0, 1, 2, …} is dense in the whole space X. In other words, the smallest closed invariant subset containing x is the whole s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4339_chunk_0 | Normal operator | In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H → H that commutes with its hermitian adjoint N*, that is: NN* = N*N.Normal operators are important because the spectral theorem holds for them. The class of normal operators is well unders... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4340_chunk_0 | Quasitrace | In mathematics, especially functional analysis, a quasitrace is a not necessarily additive tracial functional on a C*-algebra. An additive quasitrace is called a trace. It is a major open problem if every quasitrace is a trace. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4341_chunk_0 | Normalizer (group theory) | In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring R is a subring of R. This article also deals with centralizers and normalizers in a Lie algebra. The idealizer in a semigroup or ring is another co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4342_chunk_0 | Five Lemma | In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also works in the category of groups, for example. The five lemma can be thought ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4343_chunk_0 | Short five lemma | In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma. It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4344_chunk_0 | Differential graded category | In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded Z {\displaystyle \mathbb {Z} } -module. In detail, this means that Hom ( A , B ) {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4345_chunk_0 | Homotopy fiber | In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f: A → B {\displaystyle f:A\to B} . It acts as a homotopy theoretic kernel of a mapping of topological space... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4346_chunk_0 | En (Lie algebra) | In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and k, with k = n − 4. In some older books and papers, E2 and E4 are used as names for G2 and F4. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4347_chunk_0 | Quasigroup | In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional. A quasigroup with an identity element is called a loop. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4348_chunk_0 | Coherent sheaf cohomology | In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4349_chunk_0 | Quasi-coherent sheaf | In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Cohere... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4350_chunk_0 | Quasi-coherent sheaf | Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in pa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4351_chunk_0 | Adjunction formula (algebraic geometry) | In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4352_chunk_0 | Quartic surface | In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form f ( x , y , z ) = 0 {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4353_chunk_0 | Quartic surface | On the other hand, a projective quartic surface is a surface in projective space P3 of the same form, but now f is a homogeneous polynomial of 4 variables of degree 4, so for example f ( x , y , z , w ) = x 4 + y 4 + x y z w + z 2 w 2 − w 4 {\displaystyle f(x,y,z,w)=x^{4}+y^{4}+xyzw+z^{2}w^{2}-w^{4}} . If the base fiel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4354_chunk_0 | Beilinson regulator | In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology: K n ( X ) → ⊕ p ≥ 0 H D 2 p − n ( X , Q ( p ) ) . {\displaystyle K_{n}(X)\rightarrow \oplus _{p\geq 0}H_{D}^{2p-n}(X,\mathbf {Q} (p)).} Here, X is a complex smooth projective v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4355_chunk_0 | Beilinson regulator | The Beilinson regulator features in Beilinson's conjecture on special values of L-functions. The Dirichlet regulator map (used in the proof of Dirichlet's unit theorem) for the ring of integers O F {\displaystyle {\mathcal {O}}_{F}} of a number field F O F × → R r 1 + r 2 , x ↦ ( log | σ ( x ) | ) σ {\displaystyle {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4356_chunk_0 | V-topology | In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valuation rings. This topology was introduced by Rydh (2010) and studied further by Bhatt & Scholze (2017), who introduced t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4357_chunk_0 | Étale homotopy type | In mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties. Roughly speaking, for a variety or scheme X, the idea is to consider étale coverings U → X {\displaystyle U\rightarrow X} and to replace each connected component o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4358_chunk_0 | Hermite–Minkowski theorem | In mathematics, especially in algebraic number theory, the Hermite–Minkowski theorem states that for any integer N there are only finitely many number fields, i.e., finite field extensions K of the rational numbers Q, such that the discriminant of K/Q is at most N. The theorem is named after Charles Hermite and Hermann... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4359_chunk_0 | Homotopy colimit | In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category Ho ( Top ) {\displaystyle {\text{Ho}}({\textbf {Top}})} . The main idea is this: if we have a diagram F: I → Top {\displaystyle F:I\to {\textbf {Top}}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4360_chunk_0 | Faithful representation | In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g). In more abstract language, this means that the group homo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4361_chunk_0 | Transformation groupoid | In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: Group with a partial function replacing the binary operation; Category in which every morphism is invertibl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4362_chunk_0 | Transformation groupoid | The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed g: A → B {\displaystyle g:A\rightarrow B} , h: B → C {\displaystyle h:B\rightarrow C} , say. Composition is then a total function: ∘: ( B → C ) → ( A → B ) → A → C {\displaystyle \circ :(B\rightarrow C... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4363_chunk_0 | Monoidal closed category | In mathematics, especially in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category and a closed category in such a way that the structures are compatible. A classic example is the category of sets, Set, where the monoidal product of sets A {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4364_chunk_0 | Stirling numbers of the first kind | In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). The Stirling numbers of the first and second k... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4365_chunk_0 | Minimal prime (commutative algebra) | In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4366_chunk_0 | Thom mapping | In mathematics, especially in differential topology, Thom's second isotopy lemma is a family version of Thom's first isotopy lemma; i.e., it states a family of maps between Whitney stratified spaces is locally trivial when it is a Thom mapping. Like the first isotopy lemma, the lemma was introduced by René Thom. (Mathe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4367_chunk_0 | Ambient space (mathematics) | In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object along with the object itself. For example, a 1-dimensional line ( l ) {\displaystyle (l)} may be studied in isolation —in which case the ambient space of l {\displaystyle l} is l {\displaystyle l} , or i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4368_chunk_0 | Double lattice | In mathematics, especially in geometry, a double lattice in ℝn is a discrete subgroup of the group of Euclidean motions that consists only of translations and point reflections and such that the subgroup of translations is a lattice. The orbit of any point under the action of a double lattice is a union of two Bravais ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4369_chunk_0 | Double groupoid | In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4370_chunk_0 | Kunneth formula | In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4371_chunk_0 | Centrosymmetric matrix | In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an n×n matrix A = is centrosymmetric when its entries satisfy Ai,j = An−i + 1,n−j + 1 for i, j ∊{1, ..., n}.If J denotes the n×n exchange matrix with 1 on the antidi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4372_chunk_0 | Commutation matrix | In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K(m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(AT): K(m,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4373_chunk_0 | Elimination matrix | In mathematics, especially in linear algebra and matrix theory, the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizations of matrices into vectorizations or (respectively) vice versa. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4374_chunk_0 | Vectorization (mathematics) | In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a m × n matrix A, denoted vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top of on... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4375_chunk_0 | Galois connection | In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgrou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4376_chunk_0 | Galois connection | A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets. The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as (monotone) Galois connections and antitone Galois connections. A Galois connection... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4377_chunk_0 | Complete Heyting algebra | In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categorie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4378_chunk_0 | Maximal element | {\displaystyle S.} Zorn's lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element. This lemma is equivalent to the well-ordering theorem and the axiom of choice and implies major results in other mathematical areas like the Hahn–Banac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4379_chunk_0 | Cofinality | In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4380_chunk_0 | Bistochastic matrix | In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X = ( x i j ) {\displaystyle X=(x_{ij})} of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., ∑ i x i j = ∑ j x i j = 1 , {\displaystyle \sum _{i}x_{ij}=\s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4381_chunk_0 | Semialgebraic space | In mathematics, especially in real algebraic geometry, a semialgebraic space is a space which is locally isomorphic to a semialgebraic set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4382_chunk_0 | Beilinson–Bernstein localization | In mathematics, especially in representation theory and algebraic geometry, the Beilinson–Bernstein localization theorem relates D-modules on flag varieties G/B to representations of the Lie algebra g {\displaystyle {\mathfrak {g}}} attached to a reductive group G. It was introduced by Beilinson & Bernstein (1981). Ext... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4383_chunk_0 | Order type | In mathematics, especially in set theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f: X → Y {\displaystyle f\colon X\to Y} such that both f and its inverse are monotonic (preser... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4384_chunk_0 | Image functors for sheaves | In mathematics, especially in sheaf theory—a domain applied in areas such as topology, logic and algebraic geometry—there are four image functors for sheaves that belong together in various senses. Given a continuous mapping f: X → Y of topological spaces, and the category Sh(–) of sheaves of abelian groups on a topolo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4385_chunk_0 | Splitting lemma (functions) | In mathematics, especially in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a neighbourhood of a degenerate critical point. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4386_chunk_0 | Hahn embedding theorem | In mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian groups. It is named after Hans Hahn. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4387_chunk_0 | Nielsen transformation | In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non-commutative analogue of row reduction and one of the main tools used in studying free groups, (Fine, Rosenberger &... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4388_chunk_0 | Kuznetsov's theorem | In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element. More precisely, if A is a finite set of generators for G then the word ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4389_chunk_0 | Principal indecomposable module | In mathematics, especially in the area of abstract algebra known as module theory, a principal indecomposable module has many important relations to the study of a ring's modules, especially its simple modules, projective modules, and indecomposable modules. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4390_chunk_0 | Hereditary ring | In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is required only for finitely generated submodules, it is called semihereditary. For a noncommutative ring R, the terms left hered... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4391_chunk_0 | Semisimple module | In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4392_chunk_0 | Semisimple module | An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings. For a group-theory analog of the same notion, see Semisimple representation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4393_chunk_0 | Baer's criterion | In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, giv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4394_chunk_0 | Noncommutative polynomial ring | In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4395_chunk_0 | Virtually nilpotent | In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup H ≤ G {\displaystyle H\leq G}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4396_chunk_0 | Character module | In mathematics, especially in the area of abstract algebra, every module has an associated character module. Using the associated character module it is possible to investigate the properties of the original module. One of the main results discovered by Joachim Lambek shows that a module is flat if and only if the asso... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4397_chunk_0 | Complement (group theory) | In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that G = H K = { h k: h ∈ H , k ∈ K } and H ∩ K = { e } . {\displaystyle G=HK=\{hk:h\in H,k\in K\}{\text{ and }}H\cap K=\{e\}.} Equivalently, every element of G has a unique expre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4398_chunk_0 | Generalized Fitting subgroup | In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable. When G is not solv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4399_chunk_0 | Holomorph (mathematics) | In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group that simultaneously contains (copies of) the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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