text stringlengths 42 3.65k | source stringclasses 1
value |
|---|---|
Projective spaces Summary Projective_space_over_a_division_algebra In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective spaces Summary Projective_space_over_a_division_algebra There are two classes of definitions. In synthetic geometry, point and line are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective spaces Summary Projective_space_over_a_division_algebra Using linear algebra, a projective space of dimension n is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space V of dimension n + 1. Equivalently, it is the quotient set of V \ {0} by the equivalence rel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective spaces Summary Projective_space_over_a_division_algebra A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane. Projective spaces are widely used in geometry, as allowing simpler statements and simpler proofs. For example, in affine geometry, two d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relatively hyperbolic group Summary Relatively_hyperbolic_group In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residuated mapping Summary Residuated_mapping In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function. If A, B are posets, a function f: A → B is defined to be monotone if it is order-preserving: that is, if x ≤ y implies f(x) ≤ f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residuated mapping Summary Residuated_mapping In general the preimage under f of a principal down-set need not be a principal down-set. If all of them are, f is called residuated. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residuated mapping Summary Residuated_mapping The notion of residuated map can be generalized to a binary operator (or any higher arity) via component-wise residuation. This approach gives rise to notions of left and right division in a partially ordered magma, additionally endowing it with a quasigroup structure. (One... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equations defining abelian varieties Summary Equations_defining_abelian_varieties In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimensi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left inverse element Summary I-semigroup In mathematics, the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers. Given an operation denoted here ∗, and an identity element denoted e, if x ∗ y = e, one says that x is a left inverse of y, and that y is a right inverse ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left inverse element Summary I-semigroup Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. In a ring, an invertible element, also called a u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left inverse element Summary I-semigroup Inverses are commonly used in groups—where every element is invertible, and rings—where invertible elements are also called units. They are also commonly used for operations that are not defined for all possible operands, such as inverse matrices and inverse functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left inverse element Summary I-semigroup This has been generalized to category theory, where, by definition, an isomorphism is an invertible morphism. The word 'inverse' is derived from Latin: inversus that means 'turned upside down', 'overturned'. This may take its origin from the case of fractions, where the (multipl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graph dynamical system Summary Graph_dynamical_system In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of GDSs is to relate their structural properties (e.g. the networ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graph dynamical system Summary Graph_dynamical_system As such, the research typically involves techniques from, e.g., graph theory, combinatorics, algebra, and dynamical systems rather than differential geometry. In principle, one could define and study GDSs over an infinite graph (e.g. cellular automata or probabilist... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Groupoid algebra Summary Groupoid_algebra In mathematics, the concept of groupoid algebra generalizes the notion of group algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Irreducible (mathematics) Summary Irreducibility_(mathematics) In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Irreducible (mathematics) Summary Irreducibility_(mathematics) Similarly, an irreducible module is another name for a simple module. Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Irreducible (mathematics) Summary Irreducibility_(mathematics) In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space. A matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Irreducible (mathematics) Summary Irreducibility_(mathematics) A detailed definition is given here. Also, a Markov chain is irreducible if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state. In the theory of manifolds, an n-manifold is irreducible if any e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Irreducible (mathematics) Summary Irreducibility_(mathematics) Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds. The notions of irreducibility in algebra and manifold theory are related. An n-manifold is called ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Irreducible (mathematics) Summary Irreducibility_(mathematics) An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the 3-manifold topologist) finds the definition above more u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Irreducible (mathematics) Summary Irreducibility_(mathematics) See, for example, Prime decomposition (3-manifold). A topological space is irreducible if it is not the union of two proper closed subsets. This notion is used in algebraic geometry, where spaces are equipped with the Zariski topology; it is not of much sig... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Irreducible (mathematics) Summary Irreducibility_(mathematics) See also irreducible component, algebraic variety. In universal algebra, irreducible can refer to the inability to represent an algebraic structure as a composition of simpler structures using a product construction; for example subdirectly irreducible. A 3... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical quantity Background Mathematical_quantity > Background In mathematics, the concept of quantity is an ancient one extending back to the time of Aristotle and earlier. Aristotle regarded quantity as a fundamental ontological and scientific category. In Aristotle's ontology, quantity or quantum was classified... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical quantity Background Mathematical_quantity > Background Plurality means that which is divisible potentially into non-continuous parts, magnitude that which is divisible into continuous parts; of magnitude, that which is continuous in one dimension is length; in two breadth, in three depth. Of these, limited... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical quantity Background Mathematical_quantity > Background For Aristotle and Euclid, relations were conceived as whole numbers (Michell, 1993). John Wallis later conceived of ratios of magnitudes as real numbers: When a comparison in terms of ratio is made, the resultant ratio often leaves the genus of quanti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular solid Symmetry groups Platonic_polyhedron > Symmetry > Symmetry groups In mathematics, the concept of symmetry is studied with the notion of a mathematical group. Every polyhedron has an associated symmetry group, which is the set of all transformations (Euclidean isometries) which leave the polyhedron invarian... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular solid Symmetry groups Platonic_polyhedron > Symmetry > Symmetry groups The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral groups. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the ver... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular solid Symmetry groups Platonic_polyhedron > Symmetry > Symmetry groups One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is regular if and only if it is vertex-uniform, edge-uniform, and fa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular solid Symmetry groups Platonic_polyhedron > Symmetry > Symmetry groups This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice versa. The three polyhedral groups are: the tetrahedral group T, the octahedral group O (which is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular solid Symmetry groups Platonic_polyhedron > Symmetry > Symmetry groups The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular solid Symmetry groups Platonic_polyhedron > Symmetry > Symmetry groups All Platonic solids except the tetrahedron are centrally symmetric, meaning they are preserved under reflection through the origin. The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ess sup Summary Essential_infimum In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost ever... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conductor of an elliptic curve Summary Conductor_of_an_elliptic_curve In mathematics, the conductor of an elliptic curve over the field of rational numbers, or more generally a local or global field, is an integral ideal analogous to the Artin conductor of a Galois representation. It is given as a product of prime idea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conductor-discriminant formula Summary Conductor-discriminant_formula In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by Hasse (1926, 1930) for abelian extensions and by Artin (1931) for Galois extensions, is a formula calculating the relative discriminant of a finite... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cone condition Summary Cone_condition In mathematics, the cone condition is a property which may be satisfied by a subset of a Euclidean space. Informally, it requires that for each point in the subset a cone with vertex in that point must be contained in the subset itself, and so the subset is "non-flat". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cone of curves Summary Cone_theorem In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X {\displaystyle X} is a combinatorial invariant of importance to the birational geometry of X {\displaystyle X} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conformal dimension Summary Conformal_dimension In mathematics, the conformal dimension of a metric space X is the infimum of the Hausdorff dimension over the conformal gauge of X, that is, the class of all metric spaces quasisymmetric to X. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conformal group of spacetime Summary Conformal_group In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conforma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conformal group of spacetime Summary Conformal_group If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V Q ( T x ) = λ 2 Q ( x ) {\displaystyle Q(Tx)=\lambda ^{2}Q(x)} For a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conformal radius Summary Conformal_radius In mathematics, the conformal radius is a way to measure the size of a simply connected planar domain D viewed from a point z in it. As opposed to notions using Euclidean distance (say, the radius of the largest inscribed disk with center z), this notion is well-suited to use i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Huhn's theorem Summary Huhn's_theorem In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open probl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Preconditioned conjugate gradient method Summary Conjugate_gradient In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Preconditioned conjugate gradient method Summary Conjugate_gradient The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, who programmed it on the Z4, and extensively researched it.The bic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conjugate (square roots) Summary Conjugate_(square_roots) In mathematics, the conjugate of an expression of the form a + b d {\displaystyle a+b{\sqrt {d}}} is a − b d , {\displaystyle a-b{\sqrt {d}},} provided that d {\displaystyle {\sqrt {d}}} does not appear in a and b. One says also that the two expressions are conj... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Adjoint matrix Summary Hermitian_Transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m × n {\displaystyle m\times n} complex matrix A {\displaystyle {\boldsymbol {A}}} is an n × m {\displaystyle n\times m} matrix obtained by transposing A {\displaystyle {\boldsymbol {A}}} and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Connective constant Summary Connective_constant In mathematics, the connective constant is a numerical quantity associated with self-avoiding walks on a lattice. It is studied in connection with the notion of universality in two-dimensional statistical physics models. While the connective constant depends on the choice... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constant problem Summary Constant_problem In mathematics, the constant problem is the problem of deciding whether a given expression is equal to zero. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constant sheaf Summary Constant_sheaf In mathematics, the constant sheaf on a topological space X {\displaystyle X} associated to a set A {\displaystyle A} is a sheaf of sets on X {\displaystyle X} whose stalks are all equal to A {\displaystyle A} . It is denoted by A _ {\displaystyle {\underline {A}}} or A X {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constant sheaf Summary Constant_sheaf The constant sheaf associated to A {\displaystyle A} is the sheafification of the constant presheaf associated to A {\displaystyle A} . This sheaf identifies with the sheaf of locally constant A {\displaystyle A} -valued functions on X {\displaystyle X} .In certain cases, the set A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous Hahn polynomials Summary Continuous_Hahn_polynomials In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by p n ( x ; a , b , c , d ) = i n ( a + ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous Hahn polynomials Summary Continuous_Hahn_polynomials }}{}_{3}F_{2}\left({\begin{array}{c}-n,n+a+b+c+d-1,a+ix\\a+c,a+d\end{array}};1\right)} Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. Closely related polynomials include the dual Hahn polynomials ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous big q-Hermite polynomials Summary Continuous_big_q-Hermite_polynomials In mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous dual Hahn polynomials Summary Continuous_dual_Hahn_polynomials In mathematics, the continuous dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by S n ( x 2 ; a , b , c )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous dual q-Hahn polynomials Summary Continuous_dual_q-Hahn_polynomials In mathematics, the continuous dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their prope... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous q-Hahn polynomials Summary Continuous_q-Hahn_polynomials In mathematics, the continuous q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous q-Hermite polynomials Summary Continuous_q-Hermite_polynomials In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous q-Jacobi polynomials Summary Continuous_q-Jacobi_polynomials In mathematics, the continuous q-Jacobi polynomials P(α,β)n(x|q), introduced by Askey & Wilson (1985), are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (201... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous q-Laguerre polynomials Summary Continuous_q-Laguerre_polynomials In mathematics, the continuous q-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous q-Legendre polynomials Summary Continuous_q-Legendre_polynomials In mathematics, the continuous q-Legendre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme.Koekoek, Lesky & Swarttouw (2010) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous wavelet transform Summary Continuous_wavelet_transform In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. The continuous wavele... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous wavelet transform Summary Continuous_wavelet_transform x ( t ) = C ψ − 1 ∫ 0 ∞ ∫ − ∞ ∞ X w ( a , b ) 1 | a | 1 / 2 ψ ~ ( t − b a ) d b d a a 2 {\displaystyle x(t)=C_{\psi }^{-1}\int _{0}^{\infty }\int _{-\infty }^{\infty }X_{w}(a,b){\frac {1}{|a|^{1/2}}}{\tilde {\psi }}\left({\frac {t-b}{a}}\right)\,db\ {\fr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuum function Summary Continuum_function In mathematics, the continuum function is κ ↦ 2 κ {\displaystyle \kappa \mapsto 2^{\kappa }} , i.e. raising 2 to the power of κ using cardinal exponentiation. Given a cardinal number, it is the cardinality of the power set of a set of the given cardinality. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Courant–Friedrichs–Lewy condition Summary Courant–Friedrichs–Lewy_condition In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of exp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Converse (logic) Converse of a theorem Converse_(logic) > Implicational converse > Converse of a theorem In mathematics, the converse of a theorem of the form P → Q will be Q → P. The converse may or may not be true, and even if true, the proof may be difficult. For example, the Four-vertex theorem was proved in 1912, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Converse relation Summary Converse_relation In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if X {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Converse relation Summary Converse_relation In set-builder notation, L T = { ( y , x ) ∈ Y × X: ( x , y ) ∈ L } . {\displaystyle L^{\operatorname {T} }=\{(y,x)\in Y\times X:(x,y)\in L\}.} The notation is analogous with that for an inverse function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Converse relation Summary Converse_relation Although many functions do not have an inverse, every relation does have a unique converse. The unary operation that maps a relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, mor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Converse relation Summary Converse_relation Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation is also called the transpose relation. It has also been called the opposite or dual of the original relation, or th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convolution power Summary Convolution_power In mathematics, the convolution power is the n-fold iteration of the convolution with itself. Thus if x {\displaystyle x} is a function on Euclidean space Rd and n {\displaystyle n} is a natural number, then the convolution power is defined by x ∗ n = x ∗ x ∗ x ∗ ⋯ ∗ x ∗ x ⏟ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convolution power Summary Convolution_power Equivalently, x ∗ n / σ n {\displaystyle x^{*n}/\sigma {\sqrt {n}}} tends weakly to the standard normal distribution. In some cases, it is possible to define powers x*t for arbitrary real t > 0. If μ is a probability measure, then μ is infinitely divisible provided there exis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convolution power Summary Convolution_power {\displaystyle \mu _{1/n}^{*n}=\mu .} That is, a measure is infinitely divisible if it is possible to define all nth roots. Not every probability measure is infinitely divisible, and a characterization of infinitely divisible measures is of central importance in the abstract ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convolution power Summary Convolution_power Intuitively, a measure should be infinitely divisible provided it has a well-defined "convolution logarithm." The natural candidate for measures having such a logarithm are those of (generalized) Poisson type, given in the form π α , μ = e − α ∑ n = 0 ∞ α n n ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convolution power Summary Convolution_power μ ∗ n . {\displaystyle \pi _{\alpha ,\mu }=e^{-\alpha }\sum _{n=0}^{\infty }{\frac {\alpha ^{n}}{n! }}\mu ^{*n}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convolution power Summary Convolution_power In fact, the Lévy–Khinchin theorem states that a necessary and sufficient condition for a measure to be infinitely divisible is that it must lie in the closure, with respect to the vague topology, of the class of Poisson measures (Stroock 1993, §3.2). Many applications of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convolution power Summary Convolution_power {\displaystyle F^{*}(x)=a_{0}\delta _{0}+\sum _{n=1}^{\infty }a_{n}x^{*n}.} If x ∈ L1(Rd) or more generally is a finite Borel measure on Rd, then the latter series converges absolutely in norm provided that the norm of x is less than the radius of convergence of the original ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convolution power Summary Convolution_power . {\displaystyle \exp ^{*}(x)=\delta _{0}+\sum _{n=1}^{\infty }{\frac {x^{*n}}{n!}}.} It is not generally possible to extend this definition to arbitrary distributions, although a class of distributions on which this series still converges in an appropriate weak sense is iden... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convolution theorem Summary Convolution_theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corona set Summary Corona_set In mathematics, the corona or corona set of a topological space X is the complement βX\X of the space in its Stone–Čech compactification βX. A topological space is said to be σ-compact if it is the union of countably many compact subspaces, and locally compact if every point has a neighbou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corona theorem Summary Corona_theorem In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by Kakutani (1941) and proved by Lennart Carleson (1962). The commutative Banach algebra and Hardy space H∞ consists of the bounded holomorphic ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corona theorem Summary Corona_theorem In 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in (Koosis 1980) and (Gamelin 1980). Cole later showed that this result cannot be extended to all open Riemann surfaces (Gamelin 1978). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corona theorem Summary Corona_theorem As a by-product, of Carleson's work, the Carleson measure was invented which itself is a very useful tool in modern function theory. It remains an open question whether there are versions of the corona theorem for every planar domain or for higher-dimensional domains. Note that if ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Correlation immunity Summary Correlation_immunity In mathematics, the correlation immunity of a Boolean function is a measure of the degree to which its outputs are uncorrelated with some subset of its inputs. Specifically, a Boolean function is said to be correlation-immune of order m if every subset of m or fewer var... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coset construction Summary Coset_construction In mathematics, the coset construction (or GKO construction) is a method of constructing unitary highest weight representations of the Virasoro algebra, introduced by Peter Goddard, Adrian Kent and David Olive (1986). The construction produces the complete discrete series o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coshc function Summary Coshc_function In mathematics, the coshc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For z ≠ 0 {\displaystyle z\neq 0} , it is defined as It is a solution of the following differential equation: | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cotangent complex Summary Cotangent_complex In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X → Y {\displaystyle f:X\to Y} is a morphism of geometric or algebraic objects, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cotangent complex Summary Cotangent_complex Restricted versions of cotangent complexes were first defined in various cases by a number of authors in the early 1960s. In the late 1960s, Michel André and Daniel Quillen independently came up with the correct definition for a morphism of commutative rings, using simplicial... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Covariant differential Summary Covariant_differential In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Covariant differential Summary Covariant_differential The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation.This article presents an introduct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Crank conjecture Summary Crank_conjecture In mathematics, the crank conjecture was a conjecture about the existence of the crank of a partition that separates partitions of a number congruent to 6 mod 11 into 11 equal classes. The conjecture was introduced by Dyson (1944) and proved by Andrews and Garvan (1987). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Crenel function Summary Crenel_function In mathematics, the crenel function is a periodic discontinuous function P(x) defined as 1 for x belonging to a given interval and 0 outside of it. It can be presented as a difference between two Heaviside step functions of amplitude 1. It is used in crystallography to account fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Crenel function Summary Crenel_function {\displaystyle P_{k}(\Delta ,x)={\frac {\exp(2\pi i\,kx)\sin(\pi k\Delta )}{\pi k}}=\Delta \cdot \mathrm {sinc} (\pi k\Delta )\cdot \mathrm {e} ^{2\pi i\,kx}.} with the Sinc function. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Xyzzy (mnemonic) Summary Cross_Product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E {\displaystyle E} ), and is denoted by the sy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Xyzzy (mnemonic) Summary Cross_Product It should not be confused with the dot product (projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are not linearly independent), or if either one has zero length, then their cross product is zero. More g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Xyzzy (mnemonic) Summary Cross_Product The cross product is anticommutative (that is, a × b = − b × a) and is distributive over addition (that is, a × (b + c) = a × b + a × c). The space E {\displaystyle E} together with the cross product is an algebra over the real numbers, which is neither commutative nor associative... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Xyzzy (mnemonic) Summary Cross_Product In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions (with a bivector or 2-form result) and is independent of the orientation of the space. The product can be generalized in various ways, using the orientation and metric structu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Curvature of a measure Summary Curvature_of_a_measure In mathematics, the curvature of a measure defined on the Euclidean plane R2 is a quantification of how much the measure's "distribution of mass" is "curved". It is related to notions of curvature in geometry. In the form presented below, the concept was introduced ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Curve complex Summary Curve_complex In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed curves on S. The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Curve-shortening flow Summary Gage–Hamilton–Grayson_theorem In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a geometric flow, ... | 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.