text stringlengths 42 3.65k | source stringclasses 1
value |
|---|---|
Akivis algebra Summary Akivis_algebra {\displaystyle \left,z\right]+\left,x\right]+\left,y\right]=\left+\left+\left-\left-\left-\left.} An Akivis algebra with = 0 {\displaystyle \left=0} is a Lie algebra, for the Akivis identity reduces to the Jacobi identity. Note that the terms on the right hand side have positive s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stable manifold Summary Stable_space In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graph continuous function Summary Graph_continuous_function In mathematics, and in particular the study of game theory, a function is graph continuous if it exhibits the following properties. The concept was originally defined by Partha Dasgupta and Eric Maskin in 1986 and is a version of continuity that finds applicat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular module Summary Left_regular_representation In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation. One distinguishes the left regular representation λ given by left tra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
N-ary group Summary N-ary_group In mathematics, and in particular universal algebra, the concept of an n-ary group (also called n-group or multiary group) is a generalization of the concept of a group to a set G with an n-ary operation instead of a binary operation. By an n-ary operation is meant any map f: Gn → G from... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudo inverse Summary Generalized_inverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudo inverse Summary Generalized_inverse A matrix A g ∈ R n × m {\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{n\times m}} is a generalized inverse of a matrix A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} if A A g A = A . {\displaystyle AA^{\mathrm {g} }A=A.} A generalized inverse exists for an arbitra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Perron formula Summary Perron's_formula In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Primorial Summary Primorial In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Eberhard's theorem Summary Eberhard's_theorem In mathematics, and more particularly in polyhedral combinatorics, Eberhard's theorem partially characterizes the multisets of polygons that can form the faces of simple convex polyhedra. It states that, for given numbers of triangles, quadrilaterals, pentagons, heptagons, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Refinement (topology) Summary Finite_cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X {\displaystyle X} is a family of subsets of X {\displaystyle X} whose union is all of X {\displaystyle X} . More formally, if C = { U α: α ∈ A } {\displaystyle C=\lbrace U_{\alpha }:\alpha \i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Restricted partial quotients Summary Restricted_partial_quotients In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction x is said to be restricted, or composed of restricted partial quotients, if the sequence of denominators of its partial qu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Wallis' integrals Summary Wallis'_integrals In mathematics, and more precisely in analysis, the Wallis integrals constitute a family of integrals introduced by John Wallis. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nilpotent semigroup Summary Nilpotent_semigroup In mathematics, and more precisely in semigroup theory, a nilsemigroup or nilpotent semigroup is a semigroup whose every element is nilpotent. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Variety of finite semigroups Summary Variety_of_finite_semigroups In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups having some nice algebraic properties. Those classes can be defined in two distinct ways, using either algebraic notions or topological notion... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dehn-Nielsen theorem Summary Dehn–Nielsen_theorem In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Involutive algebra Summary Involutive_algebra In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras genera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudo-ring Summary Pseudo-ring In mathematics, and more specifically in abstract algebra, a pseudo-ring is one of the following variants of a ring: A rng, i.e., a structure satisfying all the axioms of a ring except for the existence of a multiplicative identity. A set R with two binary operations + and ⋅ such that (R... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rng (algebra) Summary Rng_(algebra) In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term rng (IPA: ) is meant to suggest that it i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self adjoint Summary Self_adjoint In mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if x ∗ = x {\displaystyle x^{*}=x} . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self adjoint Summary Self_adjoint For example, if x ∗ = y {\displaystyle x^{*}=y} then since y ∗ = x ∗ ∗ = x {\displaystyle y^{*}=x^{**}=x} in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements. In functional analysis, a linear operator A: H → H {\displaystyle A:H\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Self adjoint Summary Self_adjoint If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is self-adjoint if and only if the matrix describing A with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also calle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Euler–Poincaré characteristic Summary Euler's_polyhedral_formula In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or struc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Euler–Poincaré characteristic Summary Euler's_polyhedral_formula It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In mode... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Holonomic function Summary Holonomic_function In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Holonomic function Summary Holonomic_function When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called holonomic. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivaria... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zero-divisor graph Summary Zero-divisor_graph In mathematics, and more specifically in combinatorial commutative algebra, a zero-divisor graph is an undirected graph representing the zero divisors of a commutative ring. It has elements of the ring as its vertices, and pairs of elements whose product is zero as its edge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular chain Summary Regular_chain In mathematics, and more specifically in computer algebra and elimination theory, a regular chain is a particular kind of triangular set of multivariate polynomials over a field, where a triangular set is a finite sequence of polynomials such that each one contains at least one more ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multivariate division algorithm Summary Saturation_(commutative_algebra) In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K over a field K. A Gröbner... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multivariate division algorithm Summary Saturation_(commutative_algebra) He named them after his advisor Wolfgang Gröbner. In 2007, Buchberger received the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work. However, the Russian mathematician Nikolai Günther had introduced a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multivariate division algorithm Summary Saturation_(commutative_algebra) These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. An analogous concept for multivariate power series was developed independently by Heisuke Hironaka in 1964, who named them sta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hermitian manifold Summary Hermitian_structure In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hermitian manifold Summary Hermitian_structure By dropping this condition, we get an almost Hermitian manifold. On any almost Hermitian manifold, we can introduce a fundamental 2-form (or cosymplectic structure) that depends only on the chosen metric and the almost complex structure. This form is always non-degenerate.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Parametrization invariance Summary Parametrization_invariance In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Parametrization invariance Summary Parametrization_invariance The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system. F... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Parametrization invariance Summary Parametrization_invariance If x, y, z are the coordinates of the point, the movement is thus described by a parametric equation x = f ( t ) y = g ( t ) z = h ( t ) , {\displaystyle {\begin{aligned}x&=f(t)\\y&=g(t)\\z&=h(t),\end{aligned}}} where t is the parameter and denotes the time.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weighted digraph Summary Simple_directed_graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multigraph Summary Directed_multigraph In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edges), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge. There are 2 distinct notion... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multigraph Summary Directed_multigraph Edges with own identity: Edges are primitive entities just like nodes. When multiple edges connect two nodes, these are different edges.A multigraph is different from a hypergraph, which is a graph in which an edge can connect any number of nodes, not just two. For some authors, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Oriented tree Summary Oriented_tree In mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree or singly connected network) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Oriented tree Summary Oriented_tree In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic. A polytree is an example of an oriented graph. The term polytree was coined in 1987 by Rebane and Pearl. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Minimal resolution (algebra) Summary Resolution_(algebra) In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Minimal resolution (algebra) Summary Resolution_(algebra) Thus one speaks of a P resolution. In particular, every module has free resolutions, projective resolutions and flat resolutions, which are left resolutions consisting, respectively of free modules, projective modules or flat modules. Similarly every module has ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Splitting lemma Summary Splitting_lemma In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence 0 ⟶ A ⟶ q B ⟶ r C ⟶ 0. {\displaystyle 0\longrightarrow A\mathrel {\overset {q}{\longrightarrow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Linear endomorphism Summary Linear_isomorphism In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V → W {\displaystyle V\to W} between two vector spaces that preserves th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Linear endomorphism Summary Linear_isomorphism In the case where V = W {\displaystyle V=W} , a linear map is called a linear endomorphism. Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasiz... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Vector subspace Summary Vector_subspace In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Householder's method Summary Householder's_method In mathematics, and more specifically in numerical analysis, Householder's methods are a class of root-finding algorithms that are used for functions of one real variable with continuous derivatives up to some order d + 1. Each of these methods is characterized by the n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of order structures in mathematics Summary List_of_order_structures_in_mathematics In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise Lattices, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Duhamel's principle Summary Duhamel's_principle In mathematics, and more specifically in partial differential equations, Duhamel's principle is a general method for obtaining solutions to inhomogeneous linear evolution equations like the heat equation, wave equation, and vibrating plate equation. It is named after Jean... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Duhamel's principle Summary Duhamel's_principle The philosophy underlying Duhamel's principle is that it is possible to go from solutions of the Cauchy problem (or initial value problem) to solutions of the inhomogeneous problem. Consider, for instance, the example of the heat equation modeling the distribution of heat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Duhamel's principle Summary Duhamel's_principle By contrast, the inhomogeneous problem for the heat equation, corresponds to adding an external heat energy f (x, t) dt at each point. Intuitively, one can think of the inhomogeneous problem as a set of homogeneous problems each starting afresh at a different time slice t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Goldberg polyhedron Summary Goldberg_polyhedra In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three properties: each face is either a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Goldberg polyhedron Summary Goldberg_polyhedra A Goldberg polyhedron is a dual polyhedron of a geodesic sphere. A consequence of Euler's polyhedron formula is that a Goldberg polyhedron always has exactly twelve pentagonal faces. Icosahedral symmetry ensures that the pentagons are always regular and that there are alwa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Goldberg polyhedron Summary Goldberg_polyhedra If the vertices are not constrained to a sphere, the polyhedron can be constructed with planar equilateral (but not in general equiangular) faces. Simple examples of Goldberg polyhedra include the dodecahedron and truncated icosahedron. Other forms can be described by taki... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Goldberg polyhedron Summary Goldberg_polyhedra Such a polyhedron is denoted GP(m,n). A dodecahedron is GP(1,0) and a truncated icosahedron is GP(1,1). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Goldberg polyhedron Summary Goldberg_polyhedra A similar technique can be applied to construct polyhedra with tetrahedral symmetry and octahedral symmetry. These polyhedra will have triangles or squares rather than pentagons. These variations are given Roman numeral subscripts denoting the number of sides on the non-he... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective frame Summary Projective_frame In mathematics, and more specifically in projective geometry, a projective frame or projective basis is a tuple of points in a projective space that can be used for defining homogeneous coordinates in this space. More precisely, in a projective space of dimension n, a projectiv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective frame Summary Projective_frame Let P(V) be a projective space of dimension n, where V is a K-vector space of dimension n + 1. Let p: V ∖ { 0 } → P ( V ) {\displaystyle p:V\setminus \{0\}\to \mathbf {P} (V)} be the canonical projection that maps a nonzero vector v to the corresponding point of P(V), which is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective frame Summary Projective_frame In particular, the only homography fixing the points of a frame is the identity map. This result is much more difficult in synthetic geometry (where projective spaces are defined through axioms). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective frame Summary Projective_frame It is sometimes called the first fundamental theorem of projective geometry.Every frame can be written as ( p ( e 0 ) , … , p ( e n ) , p ( e 0 + ⋯ + e n ) ) , {\displaystyle (p(e_{0}),\ldots ,p(e_{n}),p(e_{0}+\cdots +e_{n})),} where ( e 0 , … , e n ) {\displaystyle (e_{0},\dot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective frame Summary Projective_frame Commonly, the projective space Pn(K) = P(Kn+1) is considered. It has a canonical frame consisting of the image by p of the canonical basis of Kn+1 (consisting of the elements having only one nonzero entry, which is equal to 1), and (1, 1, ..., 1). On this basis, the homogeneous... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective frame Summary Projective_frame The projective coordinates of a point a on the frame F are the homogeneous coordinates of h(a) on the canonical frame of Pn(K). In the case of a projective line, a frame consists of three distinct points. If P1(K) is identified with K with a point at infinity ∞ added, then its ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Krull's theorem Summary Krull's_theorem In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic ideal Summary Extension_and_contraction_of_ideals In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic ideal Summary Extension_and_contraction_of_ideals Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic ideal Summary Extension_and_contraction_of_ideals There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory). The related, but distinct, concept of an ideal in order theory is derived from the notion of ideal in ring theory. A fractional id... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Noncommutative torus Summary Noncommutative_torus In mathematics, and more specifically in the theory of C*-algebras, the noncommutative tori Aθ, also known as irrational rotation algebras for irrational values of θ, form a family of noncommutative C*-algebras which generalize the algebra of continuous functions on the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Crossed product Summary Crossed_product In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. It is related to the semidirect product construction for groups. (Roughly spe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spread of a matrix Summary Spread_of_a_matrix In mathematics, and more specifically matrix theory, the spread of a matrix is the largest distance in the complex plane between any two eigenvalues of the matrix. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hyperfactorial Summary Hyperfactorial In mathematics, and more specifically number theory, the hyperfactorial of a positive integer n {\displaystyle n} is the product of the numbers of the form x x {\displaystyle x^{x}} from 1 1 {\displaystyle 1^{1}} to n n {\displaystyle n^{n}} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superfactorial Summary Superfactorial In mathematics, and more specifically number theory, the superfactorial of a positive integer n {\displaystyle n} is the product of the first n {\displaystyle n} factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factoria... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frostman lemma Summary Frostman's_lemma In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets. Lemma: Let A be a Borel subset of Rn, and let s > 0. Then the following are equivalent: Hs(A) > 0, where Hs deno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frostman lemma Summary Frostman's_lemma The generalization to Borel sets is more involved, and requires the theory of Suslin sets. A useful corollary of Frostman's lemma requires the notions of the s-capacity of a Borel set A ⊂ Rn, which is defined by C s ( A ) := sup { ( ∫ A × A d μ ( x ) d μ ( y ) | x − y | s ) − 1: ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frostman lemma Summary Frostman's_lemma (Here, we take inf ∅ = ∞ and 1⁄∞ = 0. As before, the measure μ {\displaystyle \mu } is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rn d i m H ( A ) = sup { s ≥ 0: C s ( A ) > 0 } . {\displaystyle \mathrm {dim} _{H}(A)=\sup\{s\geq 0:C_{s}(A)>0\}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coherence axiom Summary Coherence_axiom In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Escaping set Summary Escaping_set In mathematics, and particularly complex dynamics, the escaping set of an entire function ƒ consists of all points that tend to infinity under the repeated application of ƒ. That is, a complex number z 0 ∈ C {\displaystyle z_{0}\in \mathbb {C} } belongs to the escaping set if and only ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Helly space Summary Helly_space In mathematics, and particularly functional analysis, the Helly space, named after Eduard Helly, consists of all monotonically increasing functions ƒ: → , where denotes the closed interval given by the set of all x such that 0 ≤ x ≤ 1. In other words, for all 0 ≤ x ≤ 1 we have 0 ≤ ƒ(x)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Half-disk topology Summary Half-disk_topology In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set X {\displaystyle X} , given by all points ( x , y ) {\displaystyle (x,y)} in the plane such that y ≥ 0 {\displaystyle y\geq 0} . The set X {\displaystyle X... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Steenrod problem Summary Steenrod_problem In mathematics, and particularly homology theory, Steenrod's Problem (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Diamond principle Summary Diamond_principle In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe (L) and that implies the continuum hypothesis. Jensen extracted the diamond pri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Clubsuit Summary Clubsuit In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principles that are a weaker version of the corresponding ◊S; it was introduced in 1975 by Adam Ostaszewski. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polygraph (mathematics) Summary Polygraph_(mathematics) In mathematics, and particularly in category theory, a polygraph is a generalisation of a directed graph. It is also known as a computad. They were introduced as "polygraphs" by Albert Burroni and as "computads" by Ross Street.In the same way that a directed multi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fichera's existence principle Summary Fichera's_existence_principle In mathematics, and particularly in functional analysis, Fichera's existence principle is an existence and uniqueness theorem for solution of functional equations, proved by Gaetano Fichera in 1954. More precisely, given a general vector space V and tw... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dimension (graph theory) Summary Dimension_(graph_theory) In mathematics, and particularly in graph theory, the dimension of a graph is the least integer n such that there exists a "classical representation" of the graph in the Euclidean space of dimension n with all the edges having unit length. In a classical represe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Primary pseudoperfect number Summary Primary_pseudoperfect_number In mathematics, and particularly in number theory, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation 1 N + ∑ p | N 1 p = 1 , {\displaystyle {\frac {1}{N}}+\sum _{p\,|\;\!N}{\frac {1}{p}}=1,} where the sum is over only the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dirichlet principle Summary Dirichlet_principle In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Universe (mathematics) Summary Universe_(mathematics) In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. In set theory, universes are often classes that c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Universe (mathematics) Summary Universe_(mathematics) Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations. For instance, the canonical motivating example of a category is Set, the category of all sets, which cannot be formalized in a set theory without some... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hadamard factorization theorem Summary Hadamard_factorization_theorem In mathematics, and particularly in the field of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving its zeroes and an exponential of a polynomial. It is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weierstrass factorization theorem Summary Weierstrass_factorization_theorem In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Shortlex order Summary Shortlex_order In mathematics, and particularly in the theory of formal languages, shortlex is a total ordering for finite sequences of objects that can themselves be totally ordered. In the shortlex ordering, sequences are primarily sorted by cardinality (length) with the shortest sequences firs... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Monodromy operator Summary Monodromy_matrix In mathematics, and particularly ordinary differential equations (ODEs), a monodromy matrix is the fundamental matrix of a system of ODEs evaluated at the period of the coefficients of the system. It is used for the analysis of periodic solutions of ODEs in Floquet theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characteristic multiplier Summary Characteristic_multiplier In mathematics, and particularly ordinary differential equations, a characteristic multiplier is an eigenvalue of a monodromy matrix. The logarithm of a characteristic multiplier is also known as characteristic exponent. They appear in Floquet theory of period... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Milnor number Summary Milnor_number In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If f is a complex-valued holomorphic function germ then the Milnor number of f, denoted μ(f), is either a nonnegative integer, or is infinite. It can b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trivialization (mathematics) Summary Fiber_bundles In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space E {\displaystyle E} a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trivialization (mathematics) Summary Fiber_bundles The space E {\displaystyle E} is known as the total space of the fiber bundle, B {\displaystyle B} as the base space, and F {\displaystyle F} the fiber. In the trivial case, E {\displaystyle E} is just B × F , {\displaystyle B\times F,} and the map π {\displaystyle \pi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trivialization (mathematics) Summary Fiber_bundles Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trivialization (mathematics) Summary Fiber_bundles Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, and the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself (with the identity mapping as project... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Connection one-form Summary Connection_1-form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.