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wiki_4600_chunk_0 | Content (measure theory) | In many important applications the A {\displaystyle {\mathcal {A}}} is chosen to be a ring of sets or to be at least a semiring of sets in which case some additional properties can be deduced which are described below. For this reason some authors prefer to define contents only for the case of semirings or even rings. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4601_chunk_0 | Inner measure | In mathematics, in particular in measure theory, an inner measure is a function on the power set of a given set, with values in the extended real numbers, satisfying some technical conditions. Intuitively, the inner measure of a set is a lower bound of the size of that set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4602_chunk_0 | Elliptic complex | In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for performing Hodge theory. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4603_chunk_0 | Hecke operator | In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Erich Hecke (1937a,1937b), is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4604_chunk_0 | Flat map (ring theory) | In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., f P: O Y , f ( P ) → O X , P {\displaystyle f_{P}\colon {\mathcal {O}}_{Y,f(P)}\to {\mathcal {O}}_{X,P}} is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4605_chunk_0 | Bunch–Nielsen–Sorensen formula | In mathematics, in particular linear algebra, the Bunch–Nielsen–Sorensen formula, named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a symmetric matrix A {\displaystyle A} and the outer product, v v T {\displaystyle vv^{T}} , of vector v {\displaystyle v} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4606_chunk_0 | Sherman–Morrison formula | In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix A {\displaystyle A} and the outer product, u v T {\displaystyle uv^{\textsf {T}}} , of vectors u {\displaystyle u} and v {\displaystyle v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4607_chunk_0 | Matrix determinant lemma | In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, u vT, of a column vector u and a row vector vT. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4608_chunk_0 | Somer–Lucas pseudoprime | In mathematics, in particular number theory, an odd composite number N is a Somer–Lucas d-pseudoprime (with given d ≥ 1) if there exists a nondegenerate Lucas sequence U ( P , Q ) {\displaystyle U(P,Q)} with the discriminant D = P 2 − 4 Q , {\displaystyle D=P^{2}-4Q,} such that gcd ( N , D ) = 1 {\displaystyle \gcd(N,D... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4609_chunk_0 | Dedekind–Hasse norm | In mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4610_chunk_0 | Rational mapping | In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4611_chunk_0 | Weyl chamber | In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite refl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4612_chunk_0 | Gödel constructible universe | In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L {\displaystyle L} , is a particular class of sets that can be described entirely in terms of simpler sets. L {\displaystyle L} is the union of the constructible hierarchy L α {\displaystyle L_{\alpha }} . It was ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4613_chunk_0 | Gödel constructible universe | In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4614_chunk_0 | Galois group | In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4615_chunk_0 | Verbal subgroup | In mathematics, in the area of abstract algebra known as group theory, a verbal subgroup is a subgroup of a group that is generated by all elements that can be formed by substituting group elements for variables in a given set of words. For example, given the word xy, the corresponding verbal subgroup is generated by t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4616_chunk_0 | A-group | In mathematics, in the area of abstract algebra known as group theory, an A-group is a type of group that is similar to abelian groups. The groups were first studied in the 1940s by Philip Hall, and are still studied today. A great deal is known about their structure. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4617_chunk_0 | Erdős–Fuchs theorem | In mathematics, in the area of additive number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of elements of a given additive basis, stating that the average order of this number cannot be too close to being a linear function. The theorem is named after ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4618_chunk_0 | CN-group | In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of (Burnside 1911): are all groups of odd order solvable? Progress was made by showing that CA-groups, groups in which the centralizer of a non-identity element is abelian, of odd order are solvab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4619_chunk_0 | Imperfect group | In mathematics, in the area of algebra known as group theory, an imperfect group is a group with no nontrivial perfect quotients. Some of their basic properties were established in (Berrick & Robinson 1993). The study of imperfect groups apparently began in (Robinson 1972).The class of imperfect groups is closed under ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4620_chunk_0 | Monomial group | In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, induced from characters of degree 1 (Isaacs 1994). In this section only finite groups are considered. A monomial group is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4621_chunk_0 | Monomial group | Every supersolvable group (Bray et al. 1982, Cor 2.3.5) and every solvable A-group (Bray et al. 1982, Thm 2.3.10) is a monomial group. Factor groups of monomial groups are monomial, but subgroups need not be, since every finite solvable group can be embedded in a monomial group, as shown by (Dade & ????) and in textboo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4622_chunk_0 | Dirichlet eta function | In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ζ(s) — a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4623_chunk_0 | Dirichlet eta function | This serves to define the eta function as an entire function. (The above relation and the facts that the eta function is entire and η ( 1 ) ≠ 0 {\displaystyle \eta (1)\neq 0} together show the zeta function is meromorphic with a simple pole at s = 1, and possibly additional poles at the other zeros of the factor 1 − 2 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4624_chunk_0 | Q-derivative | In mathematics, in the area of combinatorics and quantum calculus, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration. For other forms of q-derivative, see Chung et al. (1994). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4625_chunk_0 | Tight closure | In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by Melvin Hochster and Craig Huneke (1988, 1990). Let R {\displaystyle R} be a commutative noetherian ring containing a field of characteristic p > 0 {\displaystyle p>0} . He... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4626_chunk_0 | Carlson's theorem | In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers. The theorem may be obtained from the Phragmén–Lind... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4627_chunk_0 | Nachbin resummation | In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4628_chunk_0 | Fractional Fourier transform | In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate doma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4629_chunk_0 | Fractional Fourier transform | An early definition of the FRFT was introduced by Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Wiener on Hermite polynomials. However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4630_chunk_0 | Fractional Fourier transform | A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber as essentially another name for a z-transform, and in particular for the case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linear... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4631_chunk_0 | Director string | In mathematics, in the area of lambda calculus and computation, directors or director strings are a mechanism for keeping track of the free variables in a term. Loosely speaking, they can be understood as a kind of memoization for free variables; that is, as an optimization technique for rapidly locating the free varia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4632_chunk_0 | Gaussian period | In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4633_chunk_0 | Galerkin's method | In mathematics, in the area of numerical analysis, Galerkin methods are named after the Soviet mathematician Boris Galerkin. They convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4634_chunk_0 | Galerkin's method | In an operator formulation of the differential equation, Bubnov–Galerkin method can be viewed as applying an orthogonal projection to the operator. Petrov–Galerkin method (after Georgii I. Petrov) allows using basis functions for orthogonality constraints (called test basis functions) that are different from the basis ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4635_chunk_0 | Antichain | In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set is known as its width. By Dilworth's theorem, this also equals the minimum number of chains (t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4636_chunk_0 | Helstrom metric | In mathematics, in the area of quantum information geometry, the Bures metric (named after Donald Bures) or Helstrom metric (named after Carl W. Helstrom) defines an infinitesimal distance between density matrix operators defining quantum states. It is a quantum generalization of the Fisher information metric, and is i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4637_chunk_0 | Refinable function | In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity. A function φ {\displaystyle \varphi } is called refinable with respect to the mask h {\displaystyle h} if φ ( x ) = 2 ⋅ ∑ k = 0 N − 1 h k ⋅ φ ( 2 ⋅ x − k ) {\displaystyle \varphi (x)=2\cdot \s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4638_chunk_0 | Refinable function | The operator φ ↦ 2 ⋅ D 1 / 2 ( h ∗ φ ) {\displaystyle \varphi \mapsto 2\cdot D_{1/2}(h*\varphi )} is linear. A refinable function is an eigenfunction of that operator. Its absolute value is not uniquely defined. That is, if φ {\displaystyle \varphi } is a refinable function, then for every c {\displaystyle c} the funct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4639_chunk_0 | Lyndon word | In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a nonempty string that is strictly smaller in lexicographic order than all of its rotations. Lyndon words are named after mathematician Roger Lyndon, who investigated them in 1954, calling them standard lexicographic sequences. Anatoly... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4640_chunk_0 | Hall word | In mathematics, in the areas of group theory and combinatorics, Hall words provide a unique monoid factorisation of the free monoid. They are also totally ordered, and thus provide a total order on the monoid. This is analogous to the better-known case of Lyndon words; in fact, the Lyndon words are a special case, and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4641_chunk_0 | Hall word | These are binary trees; taken together, they form the Hall set. This set is a particular totally ordered subset of a free non-associative algebra, that is, a free magma. In this form, the Hall trees provide a basis for free Lie algebras, and can be used to perform the commutations required by the Poincaré–Birkhoff–Witt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4642_chunk_0 | Hall word | Hall sets were introduced by Marshall Hall based on work of Philip Hall on groups. Subsequently, Wilhelm Magnus showed that they arise as the graded Lie algebra associated with the filtration on a free group given by the lower central series. This correspondence was motivated by commutator identities in group theory du... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4643_chunk_0 | Chain decomposition | In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem characterizes the width of any finite partially ordered set in terms of a partition of the order into a minimum number of chains. It is named for the mathematician Robert P. Dilworth (1950). An antichain in a partially ordered set is a s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4644_chunk_0 | Chain decomposition | A chain decomposition is a partition of the elements of the order into disjoint chains. Dilworth's theorem states that, in any finite partially ordered set, the largest antichain has the same size as the smallest chain decomposition. Here, the size of the antichain is its number of elements, and the size of the chain d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4645_chunk_0 | Mirsky's theorem | In mathematics, in the areas of order theory and combinatorics, Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the order into a minimum number of antichains. It is named for Leon Mirsky (1971) and is closely related to Dilworth's theorem on the widths of partial... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4646_chunk_0 | Kadec norm | In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states that any two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are homeomorphic as topological spaces. The theorem was proved by Mikhail Kadets (1966) and Richard Davis Anderson. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4647_chunk_0 | Graded poset | In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N of all natural numbers. ρ must satisfy the following two properties: The rank function is compatible with the ordering, meaning that for all x and y in the order, if x ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4648_chunk_0 | Univalent function | In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4649_chunk_0 | Structure theorem for finitely generated modules over a principal ideal domain | In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4650_chunk_0 | Uniformity norm | In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norms on functions on a finite group or group-like object which quantify the amount of structure present, or conversely, the amount of randomness. They are used in the study of arithmetic progressions in the group. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4651_chunk_0 | Period mapping | In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4652_chunk_0 | Bauerian extension | In mathematics, in the field of algebraic number theory, a Bauerian extension is a field extension of an algebraic number field which is characterized by the prime ideals with inertial degree one in the extension. For a finite degree extension L/K of an algebraic number field K we define P(L/K) to be the set of primes ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4653_chunk_0 | Modulus (algebraic number theory) | In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle, or extended ideal) is a formal product of places of a global field (i.e. an algebraic number field or a global function field). It is used to encode ramification data for abelian extensions of a global field. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4654_chunk_0 | S-unit equation | In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for S-units. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4655_chunk_0 | Manin obstruction | In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction (named after Yuri Manin) is attached to a variety X over a global field, which measures the failure of the Hasse principle for X. If the value of the obstruction is non-trivial, then X may have points over all local fields but not over... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4656_chunk_0 | Q-Vandermonde identity | In mathematics, in the field of combinatorics, the q-Vandermonde identity is a q-analogue of the Chu–Vandermonde identity. Using standard notation for q-binomial coefficients, the identity states that ( m + n k ) q = ∑ j ( m k − j ) q ( n j ) q q j ( m − k + j ) . {\displaystyle {\binom {m+n}{k}}_{\!\!q}=\sum _{j}{\bin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4657_chunk_0 | Holomorphic curve | In mathematics, in the field of complex geometry, a holomorphic curve in a complex manifold M is a non-constant holomorphic map f from the complex plane to M.Nevanlinna theory addresses the question of the distribution of values of a holomorphic curve in the complex projective line. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4658_chunk_0 | Sylvester equation | In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form: A X + X B = C . {\displaystyle AX+XB=C.} It is named after English mathematician James Joseph Sylvester. Then given matrices A, B, and C, the problem is to find the possible matrices X that obey this equation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4659_chunk_0 | Sylvester equation | All matrices are assumed to have coefficients in the complex numbers. For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more generally, A and B must be square matrices of sizes n and m respectively, and then X and C both have n ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4660_chunk_0 | Sylvester equation | A Sylvester equation has a unique solution for X exactly when there are no common eigenvalues of A and −B. More generally, the equation AX + XB = C has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space. In this case, the condition for the uniqueness of a solution X is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4661_chunk_0 | Iwasawa manifold | In mathematics, in the field of differential geometry, an Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup. An Iwasawa manifold is a nilmanifold, of real dimension 6. Iwasawa manifolds give examples where the first two terms E1 and E2 of the Frölicher ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4662_chunk_0 | Yamabe invariant | In mathematics, in the field of differential geometry, the Yamabe invariant, also referred to as the sigma constant, is a real number invariant associated to a smooth manifold that is preserved under diffeomorphisms. It was first written down independently by O. Kobayashi and R. Schoen and takes its name from H. Yamabe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4663_chunk_0 | Minkowski functional | In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K {\displaystyle K} is a subset of a real or complex vector space X , {\displaystyle X,} then the Minkowski functional or gauge o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4664_chunk_0 | Minkowski functional | In fact, every seminorm p {\displaystyle p} on X {\displaystyle X} is equal to the Minkowski functional (that is, p = p K {\displaystyle p=p_{K}} ) of any subset K {\displaystyle K} of X {\displaystyle X} satisfying { x ∈ X: p ( x ) < 1 } ⊆ K ⊆ { x ∈ X: p ( x ) ≤ 1 } {\displaystyle \{x\in X:p(x)<1\}\subseteq K\subseteq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4665_chunk_0 | Minkowski functional | In particular, through these relationships, Minkowski functionals allow one to "translate" certain geometric properties of a subset of X {\displaystyle X} into certain algebraic properties of a function on X . {\displaystyle X.} The Minkowski function is always non-negative (meaning p K ≥ 0 {\displaystyle p_{K}\geq 0} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4666_chunk_0 | Indefinite inner product space | In mathematics, in the field of functional analysis, an indefinite inner product space ( K , ⟨ ⋅ , ⋅ ⟩ , J ) {\displaystyle (K,\langle \cdot ,\,\cdot \rangle ,J)} is an infinite-dimensional complex vector space K {\displaystyle K} equipped with both an indefinite inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4667_chunk_0 | Polar space | In mathematics, in the field of geometry, a polar space of rank n (n ≥ 3), or projective index n − 1, consists of a set P, conventionally called the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms: Every subspace is isomorphic to a projective space Pd(K) with −1 ≤ d ≤ (n −... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4668_chunk_0 | Component type | In mathematics, in the field of group theory, a component of a finite group is a quasisimple subnormal subgroup. Any two distinct components commute. The product of all the components is the layer of the group. For finite abelian (or nilpotent) groups, p-component is used in a different sense to mean the Sylow p-subgro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4669_chunk_0 | Conjugate permutable subgroup | In mathematics, in the field of group theory, a conjugate-permutable subgroup is a subgroup that commutes with all its conjugate subgroups. The term was introduced by Tuval Foguel in 1997 and arose in the context of the proof that for finite groups, every quasinormal subgroup is a subnormal subgroup. Clearly, every qua... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4670_chunk_0 | Malnormal subgroup | Every malnormal subgroup is a special type of C-group called a trivial intersection subgroup or TI subgroup.When G is finite, a malnormal subgroup H distinct from 1 and G is called a "Frobenius complement". The set N of elements of G which are, either equal to 1, or non-conjugate to any element of H, is a normal subgro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4671_chunk_0 | Ascendant subgroup | In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its successor. The series may be infinite. If the series is finite, then the subgroup... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4672_chunk_0 | Conjugacy-closed subgroup | The property of being conjugacy-closed is transitive, that is, every conjugacy-closed subgroup of a conjugacy-closed subgroup is conjugacy-closed.The property of being conjugacy-closed is sometimes also termed as being conjugacy stable. It is a known result that for finite field extensions, the general linear group of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4673_chunk_0 | Descendant subgroup | In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predecessor. The series may be infinite. If the series is finite, then the subgr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4674_chunk_0 | Norm (group) | It contains the center of the group. It is contained inside the second term of the upper central series. It is a Dedekind group, so is either abelian or has a direct factor isomorphic to the quaternion group. If it contains an element of infinite order, then it is equal to the center of the group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4675_chunk_0 | Perfect core | In mathematics, in the field of group theory, the perfect core (or perfect radical) of a group is its largest perfect subgroup. Its existence is guaranteed by the fact that the subgroup generated by a family of perfect subgroups is again a perfect subgroup. The perfect core is also the point where the transfinite deriv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4676_chunk_0 | Oscillatory integral operator | In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form T λ u ( x ) = ∫ R n e i λ S ( x , y ) a ( x , y ) u ( y ) d y , x ∈ R m , y ∈ R n , {\displaystyle T_{\lambda }u(x)=\int _{\mathbb {R} ^{n}}e^{i\lambda S(x,y)}a(x,y)u(y)\,dy,\qquad x\in \mathbb {R} ^{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4677_chunk_0 | Van der Corput lemma (harmonic analysis) | In mathematics, in the field of harmonic analysis, the van der Corput lemma is an estimate for oscillatory integrals named after the Dutch mathematician J. G. van der Corput. The following result is stated by E. Stein:Suppose that a real-valued function ϕ ( x ) {\displaystyle \phi (x)} is smooth in an open interval ( a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4678_chunk_0 | Acyclic object | In mathematics, in the field of homological algebra, given an abelian category C {\displaystyle {\mathcal {C}}} having enough injectives and an additive (covariant) functor F: C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} ,an acyclic object with respect to F {\displaystyle F} , or simply an F {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4679_chunk_0 | Grothendieck spectral sequence | In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors G ∘ F {\displaystyle G\circ F} , from knowledge of the derived functors of F {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4680_chunk_0 | Τ-additivity | In mathematics, in the field of measure theory, τ-additivity is a certain property of measures on topological spaces. A measure or set function μ {\displaystyle \mu } on a space X {\displaystyle X} whose domain is a sigma-algebra Σ {\displaystyle \Sigma } is said to be τ-additive if for any upward-directed family G ⊆ Σ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4681_chunk_0 | Ramanujan–Nagell equation | In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent. The eq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4682_chunk_0 | Sturm separation theorem | In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solutions of such ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4683_chunk_0 | Oscillation theory | In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation F ( x , y , y ′ , … , y ( n − 1 ) ) = y ( n ) x ∈ [ 0 , + ∞ ) {\displaystyle F(x,y,y',\ \dots ,\ y^{(n-1)})=y^{(n)}\quad x\in [0,+\infty )} is called oscillating if it has an infinite number of ro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4684_chunk_0 | Sturm-Picone comparison theorem | In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real doma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4685_chunk_0 | Volkenborn integral | In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4686_chunk_0 | Lattice (module) | In mathematics, in the field of ring theory, a lattice is a module over a ring which is embedded in a vector space over a field, giving an algebraic generalisation of the way a lattice group is embedded in a real vector space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4687_chunk_0 | Pseudocompact space | In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of pseudocompactness. Pseudocompact spaces were defined by Edwin Hewitt in 194... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4688_chunk_0 | Log semiring | In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4689_chunk_0 | Log semiring | If not qualified, the base is conventionally taken to be e or 1/e, which corresponds to e with a negative. The log semiring has the tropical semiring as limit ("tropicalization", "dequantization") as the base goes to infinity b → ∞ {\displaystyle b\to \infty } (max-plus semiring) or to zero b → 0 {\displaystyle b\to 0}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4690_chunk_0 | Invariants of tensors | In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor A {\displaystyle \mathbf {A} } are the coefficients of the characteristic polynomial p ( λ ) = det ( A − λ I ) {\displaystyle \ p(\lambda )=\det(\mathbf {A} -\lambda \mathbf {I} )} ,where I... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4691_chunk_0 | Heteroclinic orbit | In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit. Consider the co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4692_chunk_0 | Radical polynomial | In mathematics, in the realm of abstract algebra, a radical polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, if k {\displaystyle k} is a polynomial ring, the ring of radical polynomials is the subring generated by the polynomial... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4693_chunk_0 | Radical polynomial | The ring of radical polynomials is a graded subalgebra of the ring of all polynomials. The standard separation of variables theorem asserts that every polynomial can be expressed as a finite sum of terms, each term being a product of a radical polynomial and a harmonic polynomial. This is equivalent to the statement th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4694_chunk_0 | SQ-universal group | In mathematics, in the realm of group theory, a countable group is said to be SQ-universal if every countable group can be embedded in one of its quotient groups. SQ-universality can be thought of as a measure of largeness or complexity of a group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4695_chunk_0 | CA-group | In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4696_chunk_0 | Parafree group | In mathematics, in the realm of group theory, a group is said to be parafree if its quotients by the terms of its lower central series are the same as those of a free group and if it is residually nilpotent (the intersection of the terms of its lower central series is trivial). Parafree groups share many properties wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4697_chunk_0 | Power automorphism | In particular, it must be relatively prime to the order of the group, if the group is finite. If the group is abelian, any powering index works. If the powering index 2 or -1 works, then the group is abelian.The group of power automorphisms commutes with the group of inner automorphisms when viewed as subgroups of the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4698_chunk_0 | Quotientable automorphism | In mathematics, in the realm of group theory, a quotientable automorphism of a group is an automorphism that takes every normal subgroup to within itself. As a result, it gives a corresponding automorphism for every quotient group. All family automorphisms are quotientable, and particularly, all class automorphisms and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4699_chunk_0 | Stability group | In mathematics, in the realm of group theory, the stability group of subnormal series is the group of automorphisms that act as identity on each quotient group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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