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wiki_5200_chunk_0 | Principal value | In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. A simple case arises in taking the square root of a positive real number. For example, 4 has two square roots: 2 and −2; of these the positi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5201_chunk_0 | Computable ordinal | In mathematics, specifically computability and set theory, an ordinal α {\displaystyle \alpha } is said to be computable or recursive if there is a computable well-ordering of a computable subset of the natural numbers having the order type α {\displaystyle \alpha } . It is easy to check that ω {\displaystyle \omega } ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5202_chunk_0 | Computable ordinal | The Church–Kleene ordinal is a limit ordinal. An ordinal is computable if and only if it is smaller than ω 1 C K {\displaystyle \omega _{1}^{CK}} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5203_chunk_0 | Normal fan | In mathematics, specifically convex geometry, the normal fan of a convex polytope P is a polyhedral fan that is dual to P. Normal fans have applications to polyhedral combinatorics, linear programming, tropical geometry and other areas of mathematics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5204_chunk_0 | Milnor's sphere | In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnorpg 14 was trying to understand the structure of ( n − 1 ) {\displaystyle (n-1)} -connected manifolds of dimension 2 n {\displaystyle 2n} (since n {\displaystyle n} -connected 2 n {\displaystyle 2n} -manifolds are homeomor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5205_chunk_0 | Inverse function theorem | In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the invers... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5206_chunk_0 | Riemann curvature | In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5207_chunk_0 | Virtual fundamental class | In mathematics, specifically enumerative geometry, the virtual fundamental class E ∙ vir {\displaystyle _{E^{\bullet }}^{\text{vir}}} of a space X {\displaystyle X} is a replacement of the classical fundamental class ∈ A ∗ ( X ) {\displaystyle \in A^{*}(X)} in its chow ring which has better behavior with respect to t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5208_chunk_0 | Mercer's theorem | In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in (Mercer 1909), is one of the most notable results of the work of James Mercer (1883–1932). It is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5209_chunk_0 | Approximation property | In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true. Every Hilbert space has this property. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5210_chunk_0 | Schatten class operator | In mathematics, specifically functional analysis, a pth Schatten-class operator is a bounded linear operator on a Hilbert space with finite pth Schatten norm. The space of pth Schatten-class operators is a Banach space with respect to the Schatten norm. Via polar decomposition, one can prove that the space of pth Schat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5211_chunk_0 | Schatten class operator | {\displaystyle \|ST\|_{S_{1}}\leq \|S\|_{S_{p}}\|T\|_{S_{q}}\ {\mbox{if}}\ S\in S_{p},\ T\in S_{q}{\mbox{ and }}1/p+1/q=1.} If we denote by S ∞ {\displaystyle S_{\infty }} the Banach space of compact operators on H with respect to the operator norm, the above Hölder-type inequality even holds for p ∈ {\displaystyle p\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5212_chunk_0 | Unconditional convergence | In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge to that same value. Unconditional convergence is equivalent ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5213_chunk_0 | Trace class operator | In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear alg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5214_chunk_0 | Trace class operator | In quantum mechanics, mixed states are described by density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5215_chunk_0 | Von Neumann bicommutant theorem | In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory. The formal state... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5216_chunk_0 | Von Neumann bicommutant theorem | Then the closures of M in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant M′′ of M.This algebra is called the von Neumann algebra generated by M. There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5217_chunk_0 | Von Neumann bicommutant theorem | One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5218_chunk_0 | Compact space | One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5219_chunk_0 | Compact space | Compactness was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to spaces of functions. The Arzelà–Ascoli theorem and the Peano existence theorem exemplify applications of this notion of compactness to classical analysis. Following its initi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5220_chunk_0 | Classification of manifolds | In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5221_chunk_0 | Cauchy's theorem (group theory) | In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive integer with xp = e, where e is the identity ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5222_chunk_0 | Descendant tree (group theory) | Consequently, descendant trees play a fundamental role in the classification of finite p-groups. By means of kernels and targets of Artin transfer homomorphisms, descendant trees can be endowed with additional structure. An important question is how the descendant tree T ( R ) {\displaystyle {\mathcal {T}}(R)} can actu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5223_chunk_0 | Nilpotent Lie group | In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {1}. Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motiva... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5224_chunk_0 | Nilpotent Lie group | It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Analogous terms ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5225_chunk_0 | Right coset | The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by . Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group G, the numbe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5226_chunk_0 | Noetherian group | In mathematics, specifically group theory, a subgroup series of a group G {\displaystyle G} is a chain of subgroups: 1 = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G {\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G} where 1 {\displaystyle 1} is the trivial subgroup. Subgroup series can simplify the study of a group to the study of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5227_chunk_0 | P-group generation algorithm | In mathematics, specifically group theory, finite groups of prime power order p n {\displaystyle p^{n}} , for a fixed prime number p {\displaystyle p} and varying integer exponents n ≥ 0 {\displaystyle n\geq 0} , are briefly called finite p-groups. The p-group generation algorithm by M. F. Newman and E. A. O'Brien is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5228_chunk_0 | Free product | In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5229_chunk_0 | Free product | Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial group. Therefore, the free product is not the coproduct in the category of abelian groups. The free product is important in algebraic topology because of van Kampen's theorem, which states that the fundamental gro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5230_chunk_0 | Free product | In particular, the fundamental group of the wedge sum of two spaces (i.e. the space obtained by joining two spaces together at a single point) is, under certain conditions given in the Seifert van-Kampen theorem, the free product of the fundamental groups of the spaces. Free products are also important in Bass–Serre th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5231_chunk_0 | Component group | In mathematics, specifically group theory, the identity component of a group G refers to several closely related notions of the largest connected subgroup of G containing the identity element. In point set topology, the identity component of a topological group G is the connected component G0 of G that contains the ide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5232_chunk_0 | Finite index | In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted | G: H | {\displaystyle |G:H|} or {\displaystyle } or ( G: H ) {\displaystyle (G:H)} . Because G is the disjoint union of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5233_chunk_0 | Finite index | More generally, | Z: n Z | = n {\displaystyle |\mathbb {Z} :n\mathbb {Z} |=n} for any positive integer n. When G is finite, the formula may be written as | G: H | = | G | / | H | {\displaystyle |G:H|=|G|/|H|} , and it implies Lagrange's theorem that | H | {\displaystyle |H|} divides | G | {\displaystyle |G|} . When G i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5234_chunk_0 | H-object | In mathematics, specifically homotopical algebra, an H-object is a categorical generalization of an H-space, which can be defined in any category C {\displaystyle {\mathcal {C}}} with a product × {\displaystyle \times } and an initial object ∗ {\displaystyle *} . These are useful constructions because they help export ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5235_chunk_0 | Prime element | In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept which is the same in UFD... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5236_chunk_0 | Torsion-free abelian group | In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order. While finitely generated abelian groups are completel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5237_chunk_0 | Power associative | In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5238_chunk_0 | Complete homogeneous symmetric polynomial | In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete homogeneous symmetric polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5239_chunk_0 | Weak Lefschetz theorem | In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5240_chunk_0 | Formal algebraic geometry | In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5241_chunk_0 | Formal algebraic geometry | A locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally Noetherian formal schemes contains all locally Noetherian schemes. Formal schemes were motivated by and generalize Zariski's theory of formal holomorphic fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5242_chunk_0 | Grothendieck-Riemann-Roch theorem | In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5243_chunk_0 | Grothendieck-Riemann-Roch theorem | The Grothendieck–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves. The theorem has been very influential, not least for the developme... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5244_chunk_0 | Grothendieck-Riemann-Roch theorem | Conversely, complex analytic analogues of the Grothendieck–Riemann–Roch theorem can be proved using the index theorem for families. Alexander Grothendieck gave a first proof in a 1957 manuscript, later published. Armand Borel and Jean-Pierre Serre wrote up and published Grothendieck's proof in 1958. Later, Grothendieck... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5245_chunk_0 | Fiber product of schemes | In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5246_chunk_0 | Inverse image functor | In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f: X → Y {\displaystyle f:X\to Y} , the inverse image functor is a functor from the category of sheaves on Y to the category of sh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5247_chunk_0 | Eilenberg–Zilber theorem | In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space X × Y {\displaystyle X\times Y} and those of the spaces X {\displaystyle X} and Y {\displaystyle Y} . The theorem first appeared in a 1953 paper... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5248_chunk_0 | Euler class | In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5249_chunk_0 | Hartogs number | In mathematics, specifically in axiomatic set theory, a Hartogs number is an ordinal number associated with a set. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5250_chunk_0 | Logarithmic differential | In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where f ′ {\displaystyle f'} is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely f ′ , {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5251_chunk_0 | Stabilization hypothesis | In mathematics, specifically in category theory and algebraic topology, the Baez–Dolan stabilization hypothesis, proposed in (Baez & Dolan 1995), states that suspension of a weak n-category has no more essential effect after n + 2 times. Precisely, it states that the suspension functor n C a t k → n C a t k + 1 {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5252_chunk_0 | Day convolution | In mathematics, specifically in category theory, Day convolution is an operation on functors that can be seen as a categorified version of function convolution. It was first introduced by Brian Day in 1970 in the general context of enriched functor categories. Day convolution acts as a tensor product for a monoidal cat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5253_chunk_0 | F-algebra | In mathematics, specifically in category theory, F-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor F, the signature... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5254_chunk_0 | Preadditive categories | In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5255_chunk_0 | Pseudo-abelian category | In mathematics, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a kernel. Recall that an idempotent morphism p {\displaystyle p} is an endomorphism of an object with the property that p ∘ p = p {\displaystyle p\circ p=p} . Elementary con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5256_chunk_0 | Exponential object | In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined product... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5257_chunk_0 | Normal polytope | In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting poin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5258_chunk_0 | Elementary symmetric polynomial | In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial P is given by an ex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5259_chunk_0 | Power sum symmetric polynomial | In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5260_chunk_0 | Fatou's theorem | In mathematics, specifically in complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5261_chunk_0 | Kähler quotient | In mathematics, specifically in complex geometry, the Kähler quotient of a Kähler manifold X {\displaystyle X} by a Lie group G {\displaystyle G} acting on X {\displaystyle X} by preserving the Kähler structure and with moment map μ: X → g ∗ {\displaystyle \mu :X\to {\mathfrak {g}}^{*}} (with respect to the Kähler form... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5262_chunk_0 | Robust geometric computation | In mathematics, specifically in computational geometry, geometric nonrobustness is a problem wherein branching decisions in computational geometry algorithms are based on approximate numerical computations, leading to various forms of unreliability including ill-formed output and software failure through crashing or in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5263_chunk_0 | Robust geometric computation | For instance, two-dimensional convex hulls can be computed using predicates that test the sign of quadratic polynomials, and therefore may require twice as many bits of precision within these calculations as the input numbers. When integer arithmetic cannot be used (for instance, when the result of a calculation is an ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5264_chunk_0 | Equilibrium points | In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5265_chunk_0 | Isothermal coordinates | In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form g = φ ( d x 1 2 + ⋯ + d x n 2 ) , {\displaystyle g=\va... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5266_chunk_0 | Isothermal coordinates | Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5267_chunk_0 | Morse–Bott function | In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5268_chunk_0 | Morse–Bott function | Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics (critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5269_chunk_0 | Rule of proportion | In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. The method is also occasionally known as the "cross your heart" method because lines resemb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5270_chunk_0 | Fundamental theorem of Hilbert spaces | In mathematics, specifically in functional analysis and Hilbert space theory, the fundamental theorem of Hilbert spaces gives a necessarily and sufficient condition for a Hausdorff pre-Hilbert space to be a Hilbert space in terms of the canonical isometry of a pre-Hilbert space into its anti-dual. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5271_chunk_0 | Vector-valued Hahn–Banach theorems | In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are always valued in the real numbers R {\displaystyle \mathbb {R} } or the complex numbers C {\displaystyle \mathbb {C} } ) to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5272_chunk_0 | Topological vector lattice | In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X {\displaystyle X} that has a partial order ≤ {\displaystyle \,\leq \,} making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5273_chunk_0 | Ordered topological vector space | In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) X that has a partial order ≤ making it into an ordered vector space whose positive cone C := { x ∈ X: x ≥ 0 } {\displaystyle C:=\left\{x\in X:x\geq 0... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5274_chunk_0 | Amenable Banach algebra | In mathematics, specifically in functional analysis, a Banach algebra, A, is amenable if all bounded derivations from A into dual Banach A-bimodules are inner (that is of the form a ↦ a . x − x . a {\displaystyle a\mapsto a.x-x.a} for some x {\displaystyle x} in the dual module). An equivalent characterization is that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5275_chunk_0 | C* algebra | In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: A i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5276_chunk_0 | C* algebra | This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras, which culminated in a series of papers on rings of operators. These papers cons... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5277_chunk_0 | C* algebra | Around 1943, the work of Israel Gelfand and Mark Naimark yielded an abstract characterisation of C*-algebras making no reference to operators on a Hilbert space. C*-algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quant... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5278_chunk_0 | Saturated family | In mathematics, specifically in functional analysis, a family G {\displaystyle {\mathcal {G}}} of subsets a topological vector space (TVS) X {\displaystyle X} is said to be saturated if G {\displaystyle {\mathcal {G}}} contains a non-empty subset of X {\displaystyle X} and if for every G ∈ G , {\displaystyle G\in {\mat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5279_chunk_0 | Disk algebra | In mathematics, specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions ƒ: D → C {\displaystyle \mathbb {C} } ,(where D is the open unit disk in the complex plane C {\displaystyle \mathbb {C} } ) that extend to a continuous function on the c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5280_chunk_0 | Disk algebra | By construction the disc algebra is a closed subalgebra of the Hardy space H∞. In contrast to the stronger requirement that a continuous extension to the circle exists, it is a lemma of Fatou that a general element of H∞ can be radially extended to the circle almost everywhere. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5281_chunk_0 | Hydra game | In mathematics, specifically in graph theory and number theory, a hydra game is a single-player iterative mathematical game played on a mathematical tree called a hydra where, usually, the goal is to cut off the hydra's "heads" while the hydra simultaneously expands itself. Hydra games can be used to generate large num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5282_chunk_0 | Elementary abelian | In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number, and the elementary abelian groups in which the common order is p are a particular kind of p-group. A group for which ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5283_chunk_0 | Semi-direct product | In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: an inner semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5284_chunk_0 | Direct product (group theory) | In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5285_chunk_0 | Generalized homology theory | From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5286_chunk_0 | Coherence theorem | In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism". The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5287_chunk_0 | Complete quadrilateral | In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a complete quadrilateral is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5288_chunk_0 | Hasse–Arf theorem | In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of the upper numbering filtration of the Galois group of a finite Galois extension. A special case of it when the residue fields are finite was originally proved by Helmut Hasse, and the general result was prove... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5289_chunk_0 | Newman's conjecture | In mathematics, specifically in number theory, Newman's conjecture is a conjecture about the behavior of the partition function modulo any integer. Specifically, it states that for any integers m and r such that 0 ≤ r ≤ m − 1 {\displaystyle 0\leq r\leq m-1} , the value of the partition function p ( n ) {\displaystyle p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5290_chunk_0 | Cunningham number | In mathematics, specifically in number theory, a Cunningham number is a certain kind of integer named after English mathematician A. J. C. Cunningham. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5291_chunk_0 | Binomial number | In mathematics, specifically in number theory, a binomial number is an integer which can be obtained by evaluating a homogeneous polynomial containing two terms. It is a generalization of a Cunningham number. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5292_chunk_0 | Extremal orders of an arithmetic function | In mathematics, specifically in number theory, the extremal orders of an arithmetic function are best possible bounds of the given arithmetic function. Specifically, if f(n) is an arithmetic function and m(n) is a non-decreasing function that is ultimately positive and lim inf n → ∞ f ( n ) m ( n ) = 1 {\displaystyle \... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5293_chunk_0 | Baum–Connes conjecture | In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5294_chunk_0 | Baum–Connes conjecture | For instance, the surjectivity part implies the Kadison–Kaplansky conjecture for discrete torsion-free groups, and the injectivity is closely related to the Novikov conjecture. The conjecture is also closely related to index theory, as the assembly map μ {\displaystyle \mu } is a sort of index, and it plays a major rol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5295_chunk_0 | Hermitian conjugate | In mathematics, specifically in operator theory, each linear operator A {\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint) operator A ∗ {\displaystyle A^{*}} on that space according to the rule ⟨ A x , y ⟩ = ⟨ x , A ∗ y ⟩ , {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5296_chunk_0 | Fréchet lattice | In mathematics, specifically in order theory and functional analysis, a Fréchet lattice is a topological vector lattice that is also a Fréchet space. Fréchet lattices are important in the theory of topological vector lattices. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5297_chunk_0 | Band (order theory) | In mathematics, specifically in order theory and functional analysis, a band in a vector lattice X {\displaystyle X} is a subspace M {\displaystyle M} of X {\displaystyle X} that is solid and such that for all S ⊆ M {\displaystyle S\subseteq M} such that x = sup S {\displaystyle x=\sup S} exists in X , {\displaystyle X... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5298_chunk_0 | Order convergence | In mathematics, specifically in order theory and functional analysis, a filter F {\displaystyle {\mathcal {F}}} in an order complete vector lattice X {\displaystyle X} is order convergent if it contains an order bounded subset (that is, is contained in an interval of the form := { x ∈ X: a ≤ x and x ≤ b } {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5299_chunk_0 | Locally convex vector lattice | In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in the theory of topological vector lattices. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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