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wiki_4900_chunk_0 | Portmanteau theorem | In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures μn on a space, sharing a common collection of measurable sets. Such a sequence might represent an attemp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4901_chunk_0 | Jacobson density theorem | In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R.The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4902_chunk_0 | Levitzky's theorem | In mathematics, more specifically ring theory and the theory of nil ideals, Levitzky's theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent. Levitzky's theorem is one of the many results suggesting the veracity of the Köthe conjecture, and indee... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4903_chunk_0 | Atomic domain | In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain in which every non-zero non-unit can be written in at least one way as a finite product of irreducible elements. Atomic domains are different from unique factorization domains in that this decomposition of an e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4904_chunk_0 | Nilpotent ideal | In mathematics, more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that I k = 0. By I k, it is meant the additive subgroup generated by the set of all products of k elements in I. Therefore, I is nilpotent if and only if there is a natural numbe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4905_chunk_0 | Exceptional inverse image functor | In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4906_chunk_0 | Local homeomorphism | In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f: X → Y {\displaystyle f:X\to Y} is a local homeomorphism, X {\displaystyle X} is said to be an étale space over Y . {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4907_chunk_0 | Mixed volume | In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex bodies in R n {\displaystyle \mathbb {R} ^{n}} . This number depends on the size and shape of the bodies, and their relative orientation to each other. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4908_chunk_0 | Motivic L-function | In mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields. The local L-factor at a finite place v is similarly given by the characteristic polynomial of a Frobenius element at v acting on the v-inertial invariants of the v-adic realization of the motive. Fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4909_chunk_0 | Multipliers and centralizers (Banach spaces) | In mathematics, multipliers and centralizers are algebraic objects in the study of Banach spaces. They are used, for example, in generalizations of the Banach–Stone theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4910_chunk_0 | Near sets | The underlying assumption with descriptively close sets is that such sets contain elements that have location and measurable features such as colour and frequency of occurrence. The description of the element of a set is defined by a feature vector. Comparison of feature vectors provides a basis for measuring the close... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4911_chunk_0 | Near sets | Near set theory provides a formal basis for the observation, comparison, and classification of elements in sets based on their closeness, either spatially or descriptively. Near sets offer a framework for solving problems based on human perception that arise in areas such as image processing, computer vision as well as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4912_chunk_0 | Negacyclic convolution | In mathematics, negacyclic convolution is a convolution between two vectors a and b. It is also called skew circular convolution or wrapped convolution. It results from multiplication of a skew circulant matrix, generated by vector a, with vector b. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4913_chunk_0 | Negative definite | In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: Negative-definite bilinear form Negative-definite quadratic form Negative-definite matrix Negative-definite function | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4914_chunk_0 | Non-Archimedean geometry | In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated. An example of such a geometry is the Dehn plane. Non-Archimedean geometries may, as the example indicates, have properties significantly different from Euclidean geometry. There are two senses i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4915_chunk_0 | Non-Euclidean space | In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4916_chunk_0 | Nonabelian algebraic topology | Cubical omega-groupoids, higher homotopy groupoids, crossed modules, crossed complexes and Galois groupoids are key concepts in developing applications related to homotopy of filtered spaces, higher-dimensional space structures, the construction of the fundamental groupoid of a topos E in the general theory of topoi, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4917_chunk_0 | Nonabelian algebraic topology | A related example is that of van Kampen theorems for categories of covering morphisms in lextensive categories. Other reports of generalisations of the van Kampen theorem include statements for 2-categories and a topos of topoi . Important results in higher-dimensional algebra are also the extensions of the Galois theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4918_chunk_0 | Noncommutative harmonic analysis | In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier tran... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4919_chunk_0 | Noncommutative harmonic analysis | The interesting examples include many Lie groups, and also algebraic groups over p-adic fields. These examples are of interest and frequently applied in mathematical physics, and contemporary number theory, particularly automorphic representations. What to expect is known as the result of basic work of John von Neumann... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4920_chunk_0 | Noncommutative harmonic analysis | He showed that if the von Neumann group algebra of G is of type I, then L2(G) as a unitary representation of G is a direct integral of irreducible representations. It is parametrized therefore by the unitary dual, the set of isomorphism classes of such representations, which is given the hull-kernel topology. The analo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4921_chunk_0 | Noncommutative harmonic analysis | (For Pontryagin duality the Plancherel measure is some Haar measure on the dual group to G, the only issue therefore being its normalization.) For general locally compact groups, or even countable discrete groups, the von Neumann group algebra need not be of type I and the regular representation of G cannot be written ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4922_chunk_0 | Noncommutative projective geometry | In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4923_chunk_0 | Noncommutative residue | In mathematics, noncommutative residue, defined independently by M. Wodzicki (1984) and Guillemin (1985), is a certain trace on the algebra of pseudodifferential operators on a compact differentiable manifold that is expressed via a local density. In the case of the circle, the noncommutative residue had been studied e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4924_chunk_0 | Noncommutative topology | In mathematics, noncommutative topology is a term used for the relationship between topological and C*-algebraic concepts. The term has its origins in the Gelfand–Naimark theorem, which implies the duality of the category of locally compact Hausdorff spaces and the category of commutative C*-algebras. Noncommutative to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4925_chunk_0 | Nonlinear model | In mathematics, nonlinear modelling is empirical or semi-empirical modelling which takes at least some nonlinearities into account. Nonlinear modelling in practice therefore means modelling of phenomena in which independent variables affecting the system can show complex and synergetic nonlinear effects. Contrary to tr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4926_chunk_0 | Nonlinear model | Thus the nonlinear modelling can utilize production data or experimental results while taking into account complex nonlinear behaviours of modelled phenomena which are in most cases practically impossible to be modelled by means of traditional mathematical approaches, such as phenomenological modelling. Contrary to phe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4927_chunk_0 | Non-linear programming | In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) of an objective function over a set of unknown real vari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4928_chunk_0 | Nonstandard calculus | In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic. Non-rigorous calculations with infinitesimals were wide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4929_chunk_0 | Nuclear map | In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4930_chunk_0 | Nuclear operators between Banach spaces | In mathematics, nuclear operators between Banach spaces are a linear operators between Banach spaces in infinite dimensions that share some of the properties of their counter-part in finite dimension. In Hilbert spaces such operators are usually called trace class operators and one can define such things as the trace. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4931_chunk_0 | Nuclear spaces | In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite dimensional Euclidean spaces. They wer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4932_chunk_0 | Nuclear spaces | Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold. All finite-dimensional vector spaces are nuclear. There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4933_chunk_0 | Domain decomposition | The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution. In the engineering practice in the finite element method, continuity of solutions between non-matching subdomains is implemented ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4934_chunk_0 | Scale invariance | In mathematics, one can consider the scaling properties of a function or curve f (x) under rescalings of the variable x. That is, one is interested in the shape of f (λx) for some scale factor λ, which can be taken to be a length or size rescaling. The requirement for f (x) to be invariant under all rescalings is usual... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4935_chunk_0 | Continuous embedding | In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space. Several of the Sobolev embedding theorems are co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4936_chunk_0 | Operator Theory | In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operato... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4937_chunk_0 | Ehresmann's lemma | In mathematics, or specifically, in differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that if a smooth mapping f: M → N {\displaystyle f\colon M\rightarrow N} , where M {\displaystyle M} and N {\displaystyle N} are smooth manifolds, is a surjective submersion, and a proper map (in particu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4938_chunk_0 | Ordinal logic | In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics. The concept was introduced in 1938 by Alan Turing in his PhD dissertation at Princeton in view of Gödel's incompleteness theorems.While Gödel showed that every logic system suffers... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4939_chunk_0 | Orientation reversing | In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is orientable if such a consistent definition exists. In this case, there are two... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4940_chunk_0 | Orthogonal functions | In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: ⟨ f , g ⟩ = ∫ f ( x ) ¯ g ( x ) d x . {\displaystyle \langle f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4941_chunk_0 | Orthogonal functions | As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero. Suppose { f 0 , f 1 , … } {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4942_chunk_0 | Orthogonality (mathematics) | In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vecto... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4943_chunk_0 | Gell-Mann matrices | In this three-dimensional matrix representation, the Cartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices λ 3 {\displaystyle \lambda _{3}} and λ 8 {\displaystyle \lambda _{8}} , which commute with each other. There are three significant SU(2) subalgebras: { λ 1 , λ 2 , λ 3 } ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4944_chunk_0 | P-adic Teichmuller theory | In mathematics, p-adic Teichmüller theory describes the "uniformization" of p-adic curves and their moduli, generalizing the usual Teichmüller theory that describes the uniformization of Riemann surfaces and their moduli. It was introduced and developed by Shinichi Mochizuki (1996, 1999). The first problem is to reform... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4945_chunk_0 | P-adic analysis | In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers. The theory of complex-valued numerical functions on the p-adic numbers is part of the theory of locally compact groups. The usual meaning taken for p-adic analysis is the theory of p-ad... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4946_chunk_0 | P-adic analysis | Some applications have required the development of p-adic functional analysis and spectral theory. In many ways p-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of p-adic numbers is much simpler. Topological vector spaces ov... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4947_chunk_0 | P-adic cohomology | In mathematics, p-adic cohomology means a cohomology theory for varieties of characteristic p whose values are modules over a ring of p-adic integers. Examples (in roughly historical order) include: Serre's Witt vector cohomology Monsky–Washnitzer cohomology Infinitesimal cohomology Crystalline cohomology Rigid cohomol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4948_chunk_0 | Philosophy of cusp forms | In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups. If G is a reductive algebraic group and P = M A N {\displaystyle P=MAN} is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists of tak... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4949_chunk_0 | Even number | That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8. The same idea will work using any even base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1; and it is even if its last digit is 0.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4950_chunk_0 | Stable range condition | In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring R {\displaystyle R} is the smallest integer n {\displaystyle n} such that whenever v 0 , v 1 , . . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4951_chunk_0 | Stable range condition | , v n {\displaystyle v_{0},v_{1},...,v_{n}} in R {\displaystyle R} generate the unit ideal (they form a unimodular row), there exist some t 1 , . . . , t n {\displaystyle t_{1},...,t_{n}} in R {\displaystyle R} such that the elements v i − v 0 t i {\displaystyle v_{i}-v_{0}t_{i}} for 1 ≤ i ≤ n {\displaystyle 1\leq i\le... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4952_chunk_0 | Flexible identity | In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: a ∙ ( b ∙ a ) = ( a ∙ b ) ∙ a {\displaystyle a\bullet \left(b\bullet a\right)=\left(a\bullet b\right)\bullet a} for any two elements a and b of the set. A magma (that is, a set equipped with a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4953_chunk_0 | Separably closed field | In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4954_chunk_0 | Mayer–Vietoris sequence | Because the (co)homology of most spaces cannot be computed directly from their definitions, one uses tools such as the Mayer–Vietoris sequence in the hope of obtaining partial information. Many spaces encountered in topology are constructed by piecing together very simple patches. Carefully choosing the two covering su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4955_chunk_0 | Kan-Thurston theorem | In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group G {\displaystyle G} to every path-connected topological space X {\displaystyle X} in such a way that the group cohomology of G {\displaystyle G} is the same as the cohomology of the space X {\displaystyle X} . The grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4956_chunk_0 | Vertical tangent | In mathematics, particularly calculus, a vertical tangent is a tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4957_chunk_0 | Universal element | In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4958_chunk_0 | Invertible module | In mathematics, particularly commutative algebra, an invertible module is intuitively a module that has an inverse with respect to the tensor product. Invertible modules form the foundation for the definition of invertible sheaves in algebraic geometry. Formally, a finitely generated module M over a ring R is said to b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4959_chunk_0 | Berlekamp's algorithm | In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was invented by Elwyn Berlekamp in 1967. It was the dominant ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4960_chunk_0 | Finsler geometry | In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski functional F(x, −) is provided on each tangent space TxM, that enables one to define the length of any smooth curve γ: → M as L ( γ ) = ∫ a b F ( γ ( t ) , γ ˙ ( t ) ) d t . {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4961_chunk_0 | Secondary vector bundle structure | In mathematics, particularly differential topology, the secondary vector bundle structure refers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smooth vector bundle (E, p, M), induced by the push-forward p∗: TE → TM of the original projection map p: E → M. This give... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4962_chunk_0 | James' theorem | In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space X {\displaystyle X} is reflexive if and only if every continuous linear functional's norm on X {\displaystyle X} attains its supremum on the closed unit ball in X . {\displaystyle X.} A stronger versi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4963_chunk_0 | Topology of uniform convergence | In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The article operator topologies discusses topologies on spaces of linear maps bet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4964_chunk_0 | Dunford–Schwartz theorem | In mathematics, particularly functional analysis, the Dunford–Schwartz theorem, named after Nelson Dunford and Jacob T. Schwartz, states that the averages of powers of certain norm-bounded operators on L1 converge in a suitable sense. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4965_chunk_0 | Unit distance graph | In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these graphs from a broader definition that allows some non-adjacent pairs of vertice... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4966_chunk_0 | Unit distance graph | The number of colors required to color unit distance graphs is also unknown (the Hadwiger–Nelson problem): some unit distance graphs require five colors, and every unit distance graph can be colored with seven colors. For every algebraic number there is a unit distance graph with two vertices that must be that distance... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4967_chunk_0 | Directed Acyclic Graph | In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called arcs), with each edge directed from one vertex to another, such that following those directions will never form a closed lo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4968_chunk_0 | Left exact functor | In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4969_chunk_0 | Zig-zag lemma | In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4970_chunk_0 | Stably finite ring | In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1. This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivial st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4971_chunk_0 | Free semigroup with involution | In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4972_chunk_0 | Free semigroup with involution | An example from linear algebra is the multiplicative monoid of real square matrices of order n (called the full linear monoid). The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the law (AB)T = BTAT, which has the same form of interaction with ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4973_chunk_0 | Free semigroup with involution | Another example, coming from formal language theory, is the free semigroup generated by a nonempty set (an alphabet), with string concatenation as the binary operation, and the involution being the map which reverses the linear order of the letters in a string. A third example, from basic set theory, is the set of all ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4974_chunk_0 | Adjunction (field theory) | In mathematics, particularly in algebra, a field extension is a pair of fields K ⊆ L , {\displaystyle K\subseteq L,} such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4975_chunk_0 | Indeterminate equations | In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. For example, the equation a x + b y = c {\displaystyle ax+by=c} is a simple indeterminate equation, as is x 2 = 1 {\displaystyle x^{2}=1} . Indeterminate equations cannot be solved uniquely. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4976_chunk_0 | Indeterminate equations | In fact, in some cases it might even have infinitely many solutions. Some of the prominent examples of indeterminate equations include: Univariate polynomial equation: a n x n + a n − 1 x n − 1 + ⋯ + a 2 x 2 + a 1 x + a 0 = 0 , {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{2}x^{2}+a_{1}x+a_{0}=0,} which has multip... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4977_chunk_0 | Indeterminate equations | Pell's equation: x 2 − P y 2 = 1 , {\displaystyle \ x^{2}-Py^{2}=1,} where P {\displaystyle P} is a given integer that is not a square number, and in which the variables x {\displaystyle x} and y {\displaystyle y} are required to be integers. The equation of Pythagorean triples: x 2 + y 2 = z 2 , {\displaystyle x^{2}+y... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4978_chunk_0 | Indeterminate system | In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions). In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4979_chunk_0 | Locally free module | In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective mod... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4980_chunk_0 | Injective envelope | In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in (Eckmann & Schopf 1953). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4981_chunk_0 | Principal polarization | In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4982_chunk_0 | Principal polarization | Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4983_chunk_0 | Principal polarization | Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field. Abelian varieties appear naturally as Jacobian varieties (the connected components of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4984_chunk_0 | Tautness (topology) | In mathematics, particularly in algebraic topology, a taut pair is a topological pair whose direct limit of cohomology module of open neighborhood of that pair which is directed downward by inclusion is isomorphic to the cohomology module of original pair. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4985_chunk_0 | Stationary points | In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of seve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4986_chunk_0 | Hom space | In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4987_chunk_0 | Hom space | In category theory, morphism is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative opera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4988_chunk_0 | Stirling numbers of the second kind | In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S ( n , k ) {\displaystyle S(n,k)} or { n k } {\displaystyle \textstyle \left\{{n \atop k}\right\}} . Stirli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4989_chunk_0 | Transversal (combinatorics) | In mathematics, particularly in combinatorics, given a family of sets, here called a collection C, a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4990_chunk_0 | Transversal (combinatorics) | In this case, the transversal is also called a system of distinct representatives (SDR). : 29 The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of C. In this situation, the members of the system of representatives are not necessarily distinct. : 6... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4991_chunk_0 | Compact Riemann surface | The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm. Ev... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4992_chunk_0 | Compact Riemann surface | A two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and real projective plane do not. Geometrical facts about Riemann surfaces are ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4993_chunk_0 | Abramov's algorithm | In mathematics, particularly in computer algebra, Abramov's algorithm computes all rational solutions of a linear recurrence equation with polynomial coefficients. The algorithm was published by Sergei A. Abramov in 1989. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4994_chunk_0 | Zoll surface | In mathematics, particularly in differential geometry, a Zoll surface, named after Otto Zoll, is a surface homeomorphic to the 2-sphere, equipped with a Riemannian metric all of whose geodesics are closed and of equal length. While the usual unit-sphere metric on S2 obviously has this property, it also has an infinite-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4995_chunk_0 | Osculating plane | In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. The word osculate is from the Latin osculatus which is a past participle of osculari, meaning t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4996_chunk_0 | Whitney embedding theorem | In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: The strong Whitney embedding theorem states that any smooth real m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2m-space, R 2 m ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4997_chunk_0 | Phase locking | In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamical system, or other related invariant property thereof, changes according to two or more of its parameters. The regions of constant r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4998_chunk_0 | Bifurcation diagram | In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4999_chunk_0 | Indeterminate (variable) | In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself. It may be used as a placeholder in objects such as polynomials and formal power series. In particular: It does not designate a constant or a parameter of the pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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