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Cyclotomic unit Summary Cyclotomic_unit In mathematics, a cyclotomic unit (or circular unit) is a unit of an algebraic number field which is the product of numbers of the form (ζan − 1) for ζn an nth root of unity and 0 < a < n. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Skorokhod space Summary Skorokhod_space In mathematics, a càdlàg (French: "continue à droite, limite à gauche"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-cont... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cesàro fractal Summary Cesàro_fractal In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham. The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve, and Koch curve are all special cases of the general de Rham curve. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Decomposable measure Summary Decomposable_measure In mathematics, a decomposable measure (also known as a strictly localizable measure) is a measure that is a disjoint union of finite measures. This is a generalization of σ-finite measures, which are the same as those that are a disjoint union of countably many finite ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Negative-definite bilinear form Summary Definite_bilinear_form In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Deformation ring Summary Deformation_ring In mathematics, a deformation ring is a ring that controls liftings of a representation of a Galois group from a finite field to a local field. In particular for any such lifting problem there is often a universal deformation ring that classifies all such liftings, and whose sp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Degeneracy (math) Summary Degenerate_case In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.The definitions of many classes of c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Degeneracy (math) Summary Degenerate_case Equivalently, it becomes a "line segment".Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Degeneracy (math) Summary Degenerate_case As another example, the solution set of a system of equations that depends on parameters generally has a fixed cardinality and dimension, but cardinality and/or dimension may be different for some exceptional values, called degenerate cases. In such a degenerate case, the solut... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Degeneracy (math) Summary Degenerate_case In particular, the class of objects may often be defined or characterized by systems of equations. In most scenarios, a given class of objects may be defined by several different systems of equations, and these different systems of equations may lead to different degenerate cas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Degeneracy (math) Summary Degenerate_case A degenerate case thus has special features which makes it non-generic, or a special case. However, not all non-generic or special cases are degenerate. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Degeneracy (math) Summary Degenerate_case For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. In fact, degenerate cases often correspond to singularities, either in the object or in some configuration space. For example, a conic section is degenerate if and on... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constant random variable Summary Constant_random_variable In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter definition, it is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Delta operator Summary Delta_operator In mathematics, a delta operator is a shift-equivariant linear operator Q: K ⟶ K {\displaystyle Q\colon \mathbb {K} \longrightarrow \mathbb {K} } on the vector space of polynomials in a variable x {\displaystyle x} over a field K {\displaystyle \mathbb {K} } that reduces degrees ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Delta operator Summary Delta_operator To say that an operator reduces degree by one means that if f {\displaystyle f} is a polynomial of degree n {\displaystyle n} , then Q f {\displaystyle Qf} is either a polynomial of degree n − 1 {\displaystyle n-1} , or, in case n = 0 {\displaystyle n=0} , Q f {\displaystyle Qf} is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Delta-matroid Summary Delta-matroid In mathematics, a delta-matroid or Δ-matroid is a family of sets obeying an exchange axiom generalizing an axiom of matroids. A non-empty family of sets is a delta-matroid if, for every two sets E {\displaystyle E} and F {\displaystyle F} in the family, and for every element e {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Delta-matroid Summary Delta-matroid An alternative and equivalent definition is that a family of sets forms a delta-matroid when the convex hull of its indicator vectors (the analogue of a matroid polytope) has the property that every edge length is either one or the square root of two. Delta-matroids were defined by A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Delta-matroid Summary Delta-matroid As a special case, an even delta-matroid is a delta-matroid in which either all sets have even number of elements, or all sets have an odd number of elements. If a constraint satisfaction problem has a Boolean variable on each edge of a planar graph, and if the variables of the edges... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dendrite (mathematics) Summary Dendrite_(mathematics) In mathematics, a dendrite is a certain type of topological space that may be characterized either as a locally connected dendroid or equivalently as a locally connected continuum that contains no simple closed curves. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dendroid (topology) Summary Dendroid_(topology) In mathematics, a dendroid is a type of topological space, satisfying the properties that it is hereditarily unicoherent (meaning that every subcontinuum of X is unicoherent), arcwise connected, and forms a continuum. The term dendroid was introduced by Bronisław Knaster ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dendroid (topology) Summary Dendroid_(topology) In Knaster's original publication on dendroids, in 1961, he posed the problem of characterizing the dendroids which can be embedded into the Euclidean plane. This problem remains open. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dendroid (topology) Summary Dendroid_(topology) Another problem posed in the same year by Knaster, on the existence of an uncountable collection of dendroids with the property that no dendroid in the collection has a continuous surjection onto any other dendroid in the collection, was solved by Minc (2010) and Islas (2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dendroidal set Summary Dendroidal_set In mathematics, a dendroidal set is a generalization of simplicial sets introduced by Moerdijk & Weiss (2007). They have the same relation to (colored symmetric) operads, also called symmetric multicategories, that simplicial sets have to categories. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dense graph Summary Sparse_graph In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dense graph Summary Sparse_graph The graph density of simple graphs is defined to be the ratio of the number of edges |E| with respect to the maximum possible edges. For undirected simple graphs, the graph density is: D = | E | ( | V | 2 ) = 2 | E | | V | ( | V | − 1 ) {\displaystyle D={\frac {|E|}{\binom {|V|}{2}}}={\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dense graph Summary Sparse_graph The maximum number of edges for an undirected graph is ( | V | 2 ) = | V | ( | V | − 1 ) 2 {\displaystyle {\binom {|V|}{2}}={\frac {|V|(|V|-1)}{2}}} , so the maximal density is 1 (for complete graphs) and the minimal density is 0 (Coleman & Moré 1983). For families of graphs of increasi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dependence relation Summary Dependence_relation In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence. Let X {\displaystyle X} be a set. A (binary) relation ◃ {\displaystyle \triangleleft } between an element a {\displaystyle a} of X {\displaystyle X} and a subse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dependence relation Summary Dependence_relation If S ⊆ T {\displaystyle S\subseteq T} , then S {\displaystyle S} is said to span T {\displaystyle T} if t ◃ S {\displaystyle t\triangleleft S} for every t ∈ T . {\displaystyle t\in T.} S {\displaystyle S} is said to be a basis of X {\displaystyle X} if S {\displaystyle S}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dependence relation Summary Dependence_relation {\displaystyle X.} Remark. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dependence relation Summary Dependence_relation If X {\displaystyle X} is a non-empty set with a dependence relation ◃ {\displaystyle \triangleleft } , then X {\displaystyle X} always has a basis with respect to ◃ . {\displaystyle \triangleleft .} Furthermore, any two bases of X {\displaystyle X} have the same cardinal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homogeneous derivation Summary Derivation_(abstract_algebra) In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D: A → A that satisfies Leibniz's law: D ( a b ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homogeneous derivation Summary Derivation_(abstract_algebra) The collection of all K-derivations of A to itself is denoted by DerK(A). The collection of K-derivations of A into an A-module M is denoted by DerK(A, M). Derivations occur in many different contexts in diverse areas of mathematics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homogeneous derivation Summary Derivation_(abstract_algebra) The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homogeneous derivation Summary Derivation_(abstract_algebra) The Pincherle derivative is an example of a derivation in abstract algebra. If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. That is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Locally nilpotent derivation Summary Locally_nilpotent_derivation In mathematics, a derivation ∂ {\displaystyle \partial } of a commutative ring A {\displaystyle A} is called a locally nilpotent derivation (LND) if every element of A {\displaystyle A} is annihilated by some power of ∂ {\displaystyle \partial } . One mo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dessin d'enfant Summary Dessin_d'enfant In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for a "child's drawing"; its plural is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dessin d'enfant Summary Dessin_d'enfant The faces of the embedding are required be topological disks. The surface and the embedding may be described combinatorially using a rotation system, a cyclic order of the edges surrounding each vertex of the graph that describes the order in which the edges would be crossed by a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dessin d'enfant Summary Dessin_d'enfant It is natural to ask which Riemann surfaces arise in this way. The answer is provided by Belyi's theorem, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as algebraic curves over the field of algebraic numbers. The a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Determinantal point process Summary Determinantal_point_process In mathematics, a determinantal point process is a stochastic point process, the probability distribution of which is characterized as a determinant of some function. Such processes arise as important tools in random matrix theory, combinatorics, physics, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Developable surface Summary Developable_surface In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). Conversely, it is a surface ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fermat hypersurface Summary Fermat_hypersurface In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is ∑ i = 1 n a i x i m {\displaystyle \sum _{i=1}^{n}a_{i}{x_{i}}^{m}\ } for some given degree m. Such forms F, and the hypers... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Directed cut Summary Dicut In mathematics, a dicut is a partition of the vertices of a directed graph into two subsets, so that each edge that has an endpoint in both subsets is directed from the first subset to the second. Each strongly connected component of the graph must be entirely contained in one of the two subs... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Directed cut Summary Dicut The dual graph of a directed graph, embedded in the plane, is a graph with a vertex for each face of the given graph, and a dual edge between two dual vertices when the corresponding two faces are separated by an edge. Each dual edge crosses one of the original graph edges, turned by 90° cloc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Directed cut Summary Dicut Woodall's conjecture, an unsolved problem in this area, states that in any directed graph the minimum number of edges in a dicut (the unweighted minimum closure) equals the maximum number of disjoint dijoins that can be found in the graph (a packing of dijoins). A fractional weighted version ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Diffeology Summary Diffeology In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are. The concept was first introduced by Jean-Marie Souriau in the 1980s under the name Espace différentiel and later develop... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Diffeomorphism group Summary Diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuously differentiable Summary Continuously_differentiable_function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuously differentiable Summary Continuously_differentiable_function In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). f is said to be differentiable on U if it is differentiable at every point of U. f is said to be continuously differentiable if its derivative is also a conti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Smooth manifold Summary Smooth_manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Smooth manifold Summary Smooth_manifold In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Smooth manifold Summary Smooth_manifold In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Smooth manifold Summary Smooth_manifold A locally differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields. Differentiable manifolds are very important in physics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Smooth manifold Summary Smooth_manifold Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, and Yang–Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Parallelizable manifold Summary Parallelizable_manifold In mathematics, a differentiable manifold M {\displaystyle M} of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at every point p {\displaystyle p} of M {\displaystyle M} the tangent vectors provide a basis of th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differential algebraic group Summary Differential_algebraic_group In mathematics, a differential algebraic group is a differential algebraic variety with a compatible group structure. Differential algebraic groups were introduced by Cassidy (1972). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Order (differential equation) Summary Separable_ordinary_differential_equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Order (differential equation) Summary Separable_ordinary_differential_equation The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are soluble by explicit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differentially closed field Summary Differentially_closed_field In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by Robinson (1959). Differentia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differential invariant Summary Differential_invariant In mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves the derivatives of graphs of functions in the space. Differential invariants are fundamental in projective differential geometry, and the curvature is oft... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differential invariant Summary Differential_invariant Differential invariants are contrasted with geometric invariants. Whereas differential invariants can involve a distinguished choice of independent variables (or a parameterization), geometric invariants do not. Élie Cartan's method of moving frames is a refinement ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ring of differential operators Summary Ring_of_differential_operators In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns anoth... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differential poset Summary Differential_poset In mathematics, a differential poset is a partially ordered set (or poset for short) satisfying certain local properties. (The formal definition is given below.) This family of posets was introduced by Stanley (1988) as a generalization of Young's lattice (the poset of inte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differential variational inequality Summary Differential_variational_inequality In mathematics, a differential variational inequality (DVI) is a dynamical system that incorporates ordinary differential equations and variational inequalities or complementarity problems. DVIs are useful for representing models involving ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differential variational inequality Summary Differential_variational_inequality Differential variational inequalities were first formally introduced by Pang and Stewart, whose definition should not be confused with the differential variational inequality used in Aubin and Cellina (1984). Differential variational inequa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Diffiety Summary Vinogradov_sequence In mathematics, a diffiety () is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way. The term was coined in 19... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Digital manifold Summary Digital_manifold In mathematics, a digital manifold is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space. A combinatorial manifold is a kind of manifold which is a discretization of a manifold. It usually means a piecewise linear manifold made by si... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dihedral symmetry Summary Dihedral_Group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dihedral symmetry Summary Dihedral_Group In geometry, Dn or Dihn refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D2n refers to this same dihedral group. This article uses the geometric convention, Dn. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dijoin Summary Dijoin In mathematics, a dijoin is a subset of the edges of a directed graph, with the property that contracting every edge in the dijoin produces a strongly connected graph. Equivalently, a dijoin is a subset of the edges that, for every dicut, includes at least one edge crossing the dicut. Here, a dicu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dijoin Summary Dijoin A fractional weighted version of the conjecture, posed by Jack Edmonds and Rick Giles, was refuted by Alexander Schrijver.The Lucchesi–Younger theorem states that the minimum size of a dijoin, in any given directed graph, equals the maximum number of disjoint dicuts that can be found in the graph.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dijoin Summary Dijoin Each dual edge crosses one of the original graph edges, turned by 90° clockwise. A feedback arc set is a subset of the edges that includes at least one edge from every directed cycle. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dijoin Summary Dijoin For a dijoin in the given graph, the corresponding set of edges forms a directed cut in the dual graph, and vice versa. This relationship between these two problems allows the feedback arc set problem to be solved efficiently for planar graphs, even though it is NP-hard for other types of graphs. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dilation theory Summary Dilation_(metric_space) In mathematics, a dilation is a function f {\displaystyle f} from a metric space M {\displaystyle M} into itself that satisfies the identity d ( f ( x ) , f ( y ) ) = r d ( x , y ) {\displaystyle d(f(x),f(y))=rd(x,y)} for all points x , y ∈ M {\displaystyle x,y\in M} , wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Direct limit Summary Direct_system_(mathematics) In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Direct limit Summary Direct_system_(mathematics) (This is a slight abuse of notation as it suppresses the system of homomorphisms that is crucial for the structure of the limit.) Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are also a spe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Direct limit of groups Summary Direct_limit_of_groups In mathematics, a direct limit of groups is the direct limit of a direct system of groups. These are central objects of study in algebraic topology, especially stable homotopy theory and homological algebra. They are sometimes called stable groups, though this term ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Directed set Summary Directed_subset In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A {\displaystyle A} together with a reflexive and transitive binary relation ≤ {\displaystyle \,\leq \,} (that is, a preorder), with the additional property that every pair of elements has an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Directed set Summary Directed_subset The notion defined above is sometimes called an upward directed set. A downward directed set is defined analogously, meaning that every pair of elements is bounded below. Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Directed set Summary Directed_subset Other authors call a set directed if and only if it is directed both upward and downward.Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast partially ordered sets, which need not be directed). Join-semil... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Directed set Summary Directed_subset Likewise, lattices are directed sets both upward and downward. In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generall... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete series representation Summary Limit_of_discrete_series_representation In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²(G). In the Plancherel measure, such ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete valuation field Summary Discrete_valuation In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function: ν: K → Z ∪ { ∞ } {\displaystyle \nu :K\to \mathbb {Z} \cup \{\infty \}} satisfying the conditions: ν ( x ⋅ y ) = ν ( x ) + ν ( y ) {\displaystyle \nu (x\cdot y)=\nu (x)+\nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Disjoint unions Summary Disjoint_unions In mathematics, a disjoint union (or discriminated union) of a family of sets ( A i: i ∈ I ) {\displaystyle (A_{i}:i\in I)} is a set A , {\displaystyle A,} often denoted by ⨆ i ∈ I A i , {\textstyle \bigsqcup _{i\in I}A_{i},} with an injection of each A i {\displaystyle A_{i}} in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Disjoint unions Summary Disjoint_unions The disjoint union of two sets A {\displaystyle A} and B {\displaystyle B} is written with infix notation as A ⊔ B {\displaystyle A\sqcup B} . Some authors use the alternative notation A ⊎ B {\displaystyle A\uplus B} or A ∪ ⋅ B {\displaystyle A\operatorname {{\cup }\!\!\! {\cdo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Disjoint unions Summary Disjoint_unions {\cdot }\,} _{i\in I}A_{i}} ). A standard way for building the disjoint union is to define A {\displaystyle A} as the set of ordered pairs ( x , i ) {\displaystyle (x,i)} such that x ∈ A i , {\displaystyle x\in A_{i},} and the injection A i → A {\displaystyle A_{i}\to A} as x ↦ (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dispersive PDE Summary Dispersive_PDE In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dissipative operator Summary Accretive_operator In mathematics, a dissipative operator is a linear operator A defined on a linear subspace D(A) of Banach space X, taking values in X such that for all λ > 0 and all x ∈ D(A) ‖ ( λ I − A ) x ‖ ≥ λ ‖ x ‖ . {\displaystyle \|(\lambda I-A)x\|\geq \lambda \|x\|.} A couple of e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ample line bundle Summary Ample_line_bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ample line bundle Summary Ample_line_bundle In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ample line bundle Summary Ample_line_bundle More strongly, a line bundle on a complete variety X is very ample if it has enough sections to give a closed immersion (or "embedding") of X into projective space. A line bundle is ample if some positive power is very ample. An ample line bundle on a projective variety X has... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Distinguished limit Summary Distinguished_limit In mathematics, a distinguished limit is an appropriately chosen scale factor used in the method of matched asymptotic expansions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free distributive lattice Summary Free_distributive_lattice In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and interse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Summation method Summary Summability_theory In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Summation method Summary Summability_theory A counterexample is the harmonic series 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯ = ∑ n = 1 ∞ 1 n . {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n}}.} The divergence of the harmonic series was proven by the medieval ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Summation method Summary Summability_theory In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A summability method or summation method is a partial function from the set of series ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Summation method Summary Summability_theory Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety of summability methods; these are discussed in greater detai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Diversity (mathematics) Summary Diversity_(mathematics) In mathematics, a diversity is a generalization of the concept of metric space. The concept was introduced in 2012 by Bryant and Tupper, who call diversities "a form of multi-way metric". The concept finds application in nonlinear analysis.Given a set X {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Diversity (mathematics) Summary Diversity_(mathematics) Bryant and Tupper observe that these axioms imply monotonicity; that is, if A ⊆ B {\displaystyle A\subseteq B} , then δ ( A ) ≤ δ ( B ) {\displaystyle \delta (A)\leq \delta (B)} . They state that the term "diversity" comes from the appearance of a special case of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Divisibility sequence Summary Divisibility_sequence In mathematics, a divisibility sequence is an integer sequence ( a n ) {\displaystyle (a_{n})} indexed by positive integers n such that if m ∣ n then a m ∣ a n {\displaystyle {\text{if }}m\mid n{\text{ then }}a_{m}\mid a_{n}} for all m, n. That is, whenever one index ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Divisor (number theory) Summary Proper_divisor In mathematics, a divisor of an integer n {\displaystyle n} , also called a factor of n {\displaystyle n} , is an integer m {\displaystyle m} that may be multiplied by some integer to produce n {\displaystyle n} . In this case, one also says that n {\displaystyle n} is a m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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