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wiki_3500_chunk_0 | Topological game | It turns out that some fundamental topological constructions have a natural counterpart in topological games; examples of these are the Baire property, Baire spaces, completeness and convergence properties, separation properties, covering and base properties, continuous images, Suslin sets, and singular spaces. At the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3501_chunk_0 | Topological game | The term topological game was first introduced by Claude Berge, who defined the basic ideas and formalism in analogy with topological groups. A different meaning for topological game, the concept of “topological properties defined by games”, was introduced in the paper of Rastislav Telgársky, and later "spaces defined ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3502_chunk_0 | Topological graph | An important special class of topological graphs is the class of geometric graphs, where the edges are represented by line segments. (The term geometric graph is sometimes used in a broader, somewhat vague sense.) The theory of topological graphs is an area of graph theory, mainly concerned with combinatorial propertie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3503_chunk_0 | Discrete group | In mathematics, a topological group G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood which only contains that element). Equivalently, the group G is discrete if and only if its identity is isolated.A subgroup H of a topological group G is a discrete sub... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3504_chunk_0 | Baire space | In mathematics, a topological space X {\displaystyle X} is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces. The Baire category theorem combi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3505_chunk_0 | Limit point compact | In mathematics, a topological space X {\displaystyle X} is said to be limit point compact or weakly countably compact if every infinite subset of X {\displaystyle X} has a limit point in X . {\displaystyle X.} This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactnes... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3506_chunk_0 | Separable space | In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence { x n } n = 1 ∞ {\displaystyle \{x_{n}\}_{n=1}^{\infty }} of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3507_chunk_0 | Vector topology | In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3508_chunk_0 | Vector topology | Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3509_chunk_0 | Vector topology | Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vector spaces are spaces of functions, or linear operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions. In thi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3510_chunk_0 | Logical functor | In mathematics, a topos (US: , UK: ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topolo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3511_chunk_0 | Toral Lie algebra | In mathematics, a toral subalgebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are semisimple (or diagonalizable over an algebraically closed field). Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every toral Lie algeb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3512_chunk_0 | Toric manifold | In mathematics, a toric manifold is a topological analogue of toric variety in algebraic geometry. It is an even-dimensional manifold with an effective smooth action of an n {\displaystyle n} -dimensional compact torus which is locally standard with the orbit space a simple convex polytope.The aim is to do combinatoric... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3513_chunk_0 | Totally disconnected group | The compact case has been heavily studied – these are the profinite groups – but for a long time not much was known about the general case. A theorem of van Dantzig from the 1930s, stating that every such group contains a compact open subgroup, was all that was known. Then groundbreaking work by George Willis in 1994, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3514_chunk_0 | Totally positive matrix | In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number. A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenval... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3515_chunk_0 | Toy theorem | In mathematics, a toy theorem is a simplified instance (special case) of a more general theorem, which can be useful in providing a handy representation of the general theorem, or a framework for proving the general theorem. One way of obtaining a toy theorem is by introducing some simplifying assumptions in a theorem.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3516_chunk_0 | Trace identity | In mathematics, a trace identity is any equation involving the trace of a matrix. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3517_chunk_0 | Transcendence degree | A transcendence basis of a field extension L / K {\displaystyle L/K} (or a transcendence basis of L {\displaystyle L} over K {\displaystyle K} ) is a maximal algebraically independent subset of L {\displaystyle L} over K . {\displaystyle K.} Transcendence bases share many properties with bases of vector spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3518_chunk_0 | Transcendence degree | In particular, all transcendence bases of a field extension have the same cardinality, called the transcendence degree of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is positive. Transcendental extensions are widely used in algebraic geometry. For example... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3519_chunk_0 | Transcendental functions | In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed algebraically using a finite amount of terms. Examples of transcendental func... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3520_chunk_0 | Transcendental number | Hence, the set of real numbers consists of non-overlapping rational, algebraic non-rational and transcendental real numbers. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x2 − 2 = 0. The golden ratio (denoted φ {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3521_chunk_0 | Mathematical transformations | In mathematics, a transformation is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine transfor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3522_chunk_0 | Generating function transformation | In mathematics, a transformation of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas applied to a sequence generating function (see integral transformations)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3523_chunk_0 | Generating function transformation | + f 1 1 ! z + f 2 2 ! z 2 + ⋯ . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3524_chunk_0 | Generating function transformation | {\displaystyle {\widehat {F}}(z)=\sum _{n=0}^{\infty }{\frac {f_{n}}{n! }}z^{n}={\frac {f_{0}}{0! }}+{\frac {f_{1}}{1! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3525_chunk_0 | Generating function transformation | }}z+{\frac {f_{2}}{2! }}z^{2}+\cdots .} In this article, we use the convention that the ordinary (exponential) generating function for a sequence { f n } {\displaystyle \{f_{n}\}} is denoted by the uppercase function F ( z ) {\displaystyle F(z)} / F ^ ( z ) {\displaystyle {\widehat {F}}(z)} for some fixed or formal z {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3526_chunk_0 | Generating function transformation | Additionally, we use the bracket notation for coefficient extraction from the Concrete Mathematics reference which is given by F ( z ) := f n {\displaystyle F(z):=f_{n}} . The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet genera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3527_chunk_0 | Translation plane | A line l in a projective plane Π is a translation line if the group of all elations with axis l acts transitively on the points of the affine plane obtained by removing l from the plane Π, Πl (the affine derivative of Π). A projective plane with a translation line is called a translation plane. The affine plane obtaine... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3528_chunk_0 | Transverse knot | In mathematics, a transverse knot is a smooth embedding of a circle into a three-dimensional contact manifold such that the tangent vector at every point of the knot is transverse to the contact plane at that point. Any Legendrian knot can be C0-perturbed in a direction transverse to the contact planes to obtain a tran... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3529_chunk_0 | XML tree | In mathematics, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. Any connected graph without simple cycles is a tree. A tree data structure simulates a hierarchical tree structure with a set of linked nodes. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3530_chunk_0 | XML tree | A hierarchy consists of an order defined on a set. The term hierarchy is used to stress a hierarchical relation among the elements. The XML specification defines an XML document as a well-formed text if it satisfies a list of syntax rules defined in the specification. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3531_chunk_0 | Tree of primitive Pythagorean triples | In mathematics, a tree of primitive Pythagorean triples is a data tree in which each node branches to three subsequent nodes with the infinite set of all nodes giving all (and only) primitive Pythagorean triples without duplication. A Pythagorean triple is a set of three positive integers a, b, and c having the propert... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3532_chunk_0 | Tree of primitive Pythagorean triples | This was first discovered by B. Berggren in 1934.F. J. M. Barning showed that when any of the three matrices A = B = C = {\displaystyle {\begin{array}{lcr}A={\begin{bmatrix}1&-2&2\\2&-1&2\\2&-2&3\end{bmatrix}}&B={\begin{bmatrix}1&2&2\\2&1&2\\2&2&3\end{bmatrix}}&C={\begin{bmatrix}-1&2&2\\-2&1&2\\-2&2&3\end{bmatrix}}\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3533_chunk_0 | Triangular form | In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero. Because matrix equations with triangular matr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3534_chunk_0 | Exact triangle | In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3535_chunk_0 | Trident curve | In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula: x y + a x 3 + b x 2 + c x = d {\displaystyle xy+ax^{3}+bx^{2}+cx=d} Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x = 0, y ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3536_chunk_0 | Trinomial expansion | k ! . {\displaystyle {n \choose i,j,k}={\frac {n!}{i!\,j!\,k!}}\,.} This formula is a special case of the multinomial formula for m = 3. The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3537_chunk_0 | Tube domain | In mathematics, a tube domain is a generalization of the notion of a vertical strip (or half-plane) in the complex plane to several complex variables. A strip can be thought of as the collection of complex numbers whose real part lie in a given subset of the real line and whose imaginary part is unconstrained; likewise... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3538_chunk_0 | Tube domain | In mathematical physics, the future tube is the tube domain associated to the interior of the past null cone in Minkowski space, and has applications in relativity theory and quantum gravity. Certain tubes over cones support a Bergman metric in terms of which they become bounded symmetric domains. One of these is the S... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3539_chunk_0 | Empty tuple | Braces "{ }" are used to specify arrays in some programming languages but not in mathematical expressions, as they are the standard notation for sets. The term tuple can often occur when discussing other mathematical objects, such as vectors. In computer science, tuples come in many forms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3540_chunk_0 | Empty tuple | Most typed functional programming languages implement tuples directly as product types, tightly associated with algebraic data types, pattern matching, and destructuring assignment. Many programming languages offer an alternative to tuples, known as record types, featuring unordered elements accessed by label. A few pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3541_chunk_0 | Twisted cubic | In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the twisted cubic, therefore). In algebraic geometry, the twisted cubic is a simple example of a projective vari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3542_chunk_0 | Noncommutative polynomials | In mathematics, a twisted polynomial is a polynomial over a field of characteristic p {\displaystyle p} in the variable τ {\displaystyle \tau } representing the Frobenius map x ↦ x p {\displaystyle x\mapsto x^{p}} . In contrast to normal polynomials, multiplication of these polynomials is not commutative, but satisfies... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3543_chunk_0 | Two-graph | In mathematics, a two-graph is a set of (unordered) triples chosen from a finite vertex set X, such that every (unordered) quadruple from X contains an even number of triples of the two-graph. A regular two-graph has the property that every pair of vertices lies in the same number of triples of the two-graph. Two-graph... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3544_chunk_0 | Unary function | In mathematics, a unary function is a function that takes one argument. A unary operator belongs to a subset of unary functions, in that its range coincides with its domain. In contrast, a unary function's domain may or may not coincide with its range. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3545_chunk_0 | Unicoherent space | In mathematics, a unicoherent space is a topological space X {\displaystyle X} that is connected and in which the following property holds: For any closed, connected A , B ⊂ X {\displaystyle A,B\subset X} with X = A ∪ B {\displaystyle X=A\cup B} , the intersection A ∩ B {\displaystyle A\cap B} is connected. For example... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3546_chunk_0 | Uniformly bounded representation | The result on unitarizability of uniformly bounded representations was extended in 1950 by Dixmier, Day and Nakamura-Takeda to all locally compact amenable groups, following essentially the method of proof of Sz-Nagy. The result is known to fail for non-amenable groups such as SL(2,R) and the free group on two generato... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3547_chunk_0 | Uniformly smooth space | In mathematics, a uniformly smooth space is a normed vector space X {\displaystyle X} satisfying the property that for every ϵ > 0 {\displaystyle \epsilon >0} there exists δ > 0 {\displaystyle \delta >0} such that if x , y ∈ X {\displaystyle x,y\in X} with ‖ x ‖ = 1 {\displaystyle \|x\|=1} and ‖ y ‖ ≤ δ {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3548_chunk_0 | Totally unimodular | In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N that is its inverse (these are equivalent under Cramer's rule). Thus every equation Mx = b, where M and b both have integ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3549_chunk_0 | Unimodular polynomial matrix | In mathematics, a unimodular polynomial matrix is a square polynomial matrix whose inverse exists and is itself a polynomial matrix. Equivalently, a polynomial matrix A is unimodular if its determinant det(A) is a nonzero constant. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3550_chunk_0 | Unipotent group | In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. In particular, a square matrix M is a unipotent matrix if and only if its characteristic polynomial P(t) is a power of t − 1. Thus all the eigenvalues of a unipotent matrix are 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3551_chunk_0 | Unipotent representation | In mathematics, a unipotent representation of a reductive group is a representation that has some similarities with unipotent conjugacy classes of groups. Informally, Langlands philosophy suggests that there should be a correspondence between representations of a reductive group and conjugacy classes of a Langlands dua... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3552_chunk_0 | Unique factorisation | In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3553_chunk_0 | Unique sink orientation | In mathematics, a unique sink orientation is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope as one of the faces), there is exactly one vertex for which all adjoining edges are oriented inward (i.e. towards that vertex). If a polytope is given together wi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3554_chunk_0 | Uniqueness theorem | In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems include: Alexandrov's uniqueness theorem of three-dimensional polyhedra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3555_chunk_0 | Uniqueness theorem | Fundamental theorem of arithmetic, the uniqueness of prime factorization. Holmgren's uniqueness theorem for linear partial differential equations with real analytic coefficients. Picard–Lindelöf theorem, the uniqueness of solutions to first-order differential equations. Thompson uniqueness theorem in finite group theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3556_chunk_0 | Unistochastic matrix | In mathematics, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix. A square matrix B of size n is doubly stochastic (or bistochastic) if all its entries are non-negative real numbers and each o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3557_chunk_0 | Unistochastic matrix | Since all orthogonal matrices are necessarily unitary matrices, all orthostochastic matrices are also unistochastic. The converse, however, is not true. First, all 2-by-2 doubly stochastic matrices are both unistochastic and orthostochastic, but for larger n this is not the case. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3558_chunk_0 | Unistochastic matrix | For example, take n = 3 {\displaystyle n=3} and consider the following doubly stochastic matrix: B = 1 2 . {\displaystyle B={\frac {1}{2}}{\begin{bmatrix}1&1&0\\0&1&1\\1&0&1\end{bmatrix}}.} This matrix is not unistochastic, since any two vectors with moduli equal to the square root of the entries of two columns (or ro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3559_chunk_0 | Unit circle | Thus, by the Pythagorean theorem, x and y satisfy the equation Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant. The interior of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3560_chunk_0 | Hat operator | In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in v ^ {\displaystyle {\hat {\mathbf {v} }}} (pronounced "v-hat"). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3561_chunk_0 | Unit vector | In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in v ^ {\displaystyle {\hat {\mathbf {v} }}} (pronounced "v-hat"). The term direction vector, commonly denoted as d, is used to de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3562_chunk_0 | Unit vector | 2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., u ^ = u ‖ u ‖ {\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3563_chunk_0 | Unitary representation | In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly conti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3564_chunk_0 | Univariate | In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate cases is fundamental; for example, the fundamental theorem of algebra and E... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3565_chunk_0 | Univariate | In statistics, a univariate distribution characterizes one variable, although it can be applied in other ways as well. For example, univariate data are composed of a single scalar component. In time series analysis, the whole time series is the "variable": a univariate time series is the series of values over time of a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3566_chunk_0 | Univariate | Correspondingly, a "multivariate time series" characterizes the changing values over time of several quantities. In some cases, the terminology is ambiguous, since the values within a univariate time series may be treated using certain types of multivariate statistical analyses and may be represented using multivariate... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3567_chunk_0 | Properties of polynomial roots | In mathematics, a univariate polynomial of degree n with real or complex coefficients has n complex roots, if counted with their multiplicities. They form a multiset of n points in the complex plane. This article concerns the geometry of these points, that is the information about their localization in the complex plan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3568_chunk_0 | Properties of polynomial roots | Such bounds are widely used for root-finding algorithms for polynomials, either for tuning them, or for computing their computational complexity. Some other properties are probabilistic, such as the expected number of real roots of a random polynomial of degree n with real coefficients, which is less than 1 + 2 π ln ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3569_chunk_0 | Universal C*-algebra | In mathematics, a universal C*-algebra is a C*-algebra described in terms of generators and relations. In contrast to rings or algebras, where one can consider quotients by free rings to construct universal objects, C*-algebras must be realizable as algebras of bounded operators on a Hilbert space by the Gelfand-Naimar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3570_chunk_0 | Universal graph | However it is not the smallest such graph: it is known that there is a universal graph for n-vertex trees, with only n vertices and O(n log n) edges, and that this is optimal. A construction based on the planar separator theorem can be used to show that n-vertex planar graphs have universal graphs with O(n3/2) edges, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3571_chunk_0 | Universal graph | It is also possible to construct universal graphs for planar graphs that have n1+o(1) vertices.Sumner's conjecture states that tournaments are universal for polytrees, in the sense that every tournament with 2n − 2 vertices contains every polytree with n vertices as a subgraph.A family F of graphs has a universal graph... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3572_chunk_0 | Wonders of Numbers | Similarly, 136,948 is a vampire because 136,948 = 146 × 938. Vampire numbers first appeared in a 1994 post by Clifford A. Pickover to the Usenet group sci.math, and the article he later wrote was published in chapter 30 of his book Keys to Infinity.In addition to "Vampire numbers", a term Pickover actually coined, he h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3573_chunk_0 | Variable (logics) | In mathematics, a variable (from Latin variabilis, "changeable") is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set.Algebraic computations with variables as if they were explicit numbers solve a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3574_chunk_0 | Variational inequalities | In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initially developed to deal with equilibrium problems, precisely the Signorini ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3575_chunk_0 | Vector bundle morphism | In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X {\displaystyle X} (for example X {\displaystyle X} could be a topological space, a manifold, or an algebraic variety): to every point x {\displaystyle x} of the space X... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3576_chunk_0 | Vector bundle morphism | Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3577_chunk_0 | Vector bundle morphism | Vector bundles are almost always required to be locally trivial, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3578_chunk_0 | Flat vector bundle | In mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3579_chunk_0 | Lyapunov's theorem | In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3580_chunk_0 | Vector-valued differential form | In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-value... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3581_chunk_0 | Unit quaternion | In mathematics, a versor is a quaternion of norm one (a unit quaternion). Each versor has the form q = exp ( a r ) = cos a + r sin a , r 2 = − 1 , a ∈ , {\displaystyle q=\exp(a\mathbf {r} )=\cos a+\mathbf {r} \sin a,\quad \mathbf {r} ^{2}=-1,\quad a\in ,} where the r2 = −1 condition means that r is a unit-length... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3582_chunk_0 | Vertex cycle cover | Similar definitions exist for digraphs, in terms of directed cycles. Finding a vertex-disjoint cycle cover of a directed graph can also be performed in polynomial time by a similar reduction to perfect matching. However, adding the condition that each cycle should have length at least 3 makes the problem NP-hard. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3583_chunk_0 | Vertex algebra | In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geomet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3584_chunk_0 | Vertex algebra | Borcherds formulated the notion of vertex algebra by axiomatizing the relations between the lattice vertex operators, producing an algebraic structure that allows one to construct new Lie algebras by following Frenkel's method. The notion of vertex operator algebra was introduced as a modification of the notion of vert... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3585_chunk_0 | Vertex algebra | Motivated by this observation, they added the Virasoro action and bounded-below property as axioms. We now have post-hoc motivation for these notions from physics, together with several interpretations of the axioms that were not initially known. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3586_chunk_0 | Vertex algebra | Physically, the vertex operators arising from holomorphic field insertions at points in two-dimensional conformal field theory admit operator product expansions when insertions collide, and these satisfy precisely the relations specified in the definition of vertex operator algebra. Indeed, the axioms of a vertex opera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3587_chunk_0 | Volume element | In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form d V = ρ ( u 1 , u 2 , u 3 ) d u 1 d u 2 d u 3 {\displaystyle \mathrm {d} V=\rho ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3588_chunk_0 | Volume element | The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transform... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3589_chunk_0 | Riemannian volume form | An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. A volume form provides a means to define the integral of a function on a di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3590_chunk_0 | Riemannian volume form | In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a twisted volume form or pseudo-volume form. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3591_chunk_0 | Von Neumann algebras | In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3592_chunk_0 | Von Neumann algebras | Two basic examples of von Neumann algebras are as follows: The ring L ∞ ( R ) {\displaystyle L^{\infty }(\mathbb {R} )} of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L 2 (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3593_chunk_0 | Von Neumann regular element | In mathematics, a von Neumann regular ring is a ring R (associative, with 1, not necessarily commutative) such that for every element a in R there exists an x in R with a = axa. One may think of x as a "weak inverse" of the element a; in general x is not uniquely determined by a. Von Neumann regular rings are also call... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3594_chunk_0 | Von Neumann regular element | Von Neumann regular rings should not be confused with the unrelated regular rings and regular local rings of commutative algebra. An element a of a ring is called a von Neumann regular element if there exists an x such that a = axa. An ideal i {\displaystyle {\mathfrak {i}}} is called a (von Neumann) regular ideal if f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3595_chunk_0 | Wavelet transforms | In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3596_chunk_0 | Lie algebra bundle | In mathematics, a weak Lie algebra bundle ξ = ( ξ , p , X , θ ) {\displaystyle \xi =(\xi ,p,X,\theta )\,} is a vector bundle ξ {\displaystyle \xi \,} over a base space X together with a morphism θ: ξ ⊗ ξ → ξ {\displaystyle \theta :\xi \otimes \xi \rightarrow \xi } which induces a Lie algebra structure on each fibre ξ x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3597_chunk_0 | Lie algebra bundle | Let denote the Lie bracket of s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} and deform it by the real parameter as: x = x ⋅ {\displaystyle _{x}=x\cdot } for X , Y ∈ s o ( 3 ) {\displaystyle X,Y\in {\mathfrak {so}}(3)} and x ∈ R {\displaystyle x\in \mathbb {R} } . Lie's third theorem states that every bundle of Lie a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3598_chunk_0 | Lie algebra bundle | In general globally the total space might fail to be Hausdorff. But if all fibres of a real Lie algebra bundle over a topological space are mutually isomorphic as Lie algebras, then it is a locally trivial Lie algebra bundle. This result was proved by proving that the real orbit of a real point under an algebraic group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3599_chunk_0 | Lie algebra bundle | Suppose the base space is Hausdorff and fibers of total space are isomorphic as Lie algebras then there exists a Hausdorff Lie group bundle over the same base space whose Lie algebra bundle is isomorphic to the given Lie algebra bundle. Every semi simple Lie algebra bundle is locally trivial. Hence there exist a Hausdo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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