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Very often, and in this article, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.
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https://en.wikipedia.org/wiki/Homogeneous_equation
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In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns (in contrast to an overdetermined system, where there are more equations than unknowns). The terminology can be explained using the concept of constraint counting. Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom.
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https://en.wikipedia.org/wiki/Underdetermined_system
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Therefore, the critical case (between overdetermined and underdetermined) occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint removing a degree of freedom. The underdetermined case, by contrast, occurs when the system has been underconstrained—that is, when the unknowns outnumber the equations.
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https://en.wikipedia.org/wiki/Underdetermined_system
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In mathematics, a system of parameters for a local Noetherian ring of Krull dimension d with maximal ideal m is a set of elements x1, ..., xd that satisfies any of the following equivalent conditions: m is a minimal prime over (x1, ..., xd). The radical of (x1, ..., xd) is m. Some power of m is contained in (x1, ..., xd). (x1, ..., xd) is m-primary.Every local Noetherian ring admits a system of parameters.It is not possible for fewer than d elements to generate an ideal whose radical is m because then the dimension of R would be less than d. If M is a k-dimensional module over a local ring, then x1, ..., xk is a system of parameters for M if the length of M / (x1, ..., xk) M is finite.
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https://en.wikipedia.org/wiki/System_of_parameters
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In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name triangular norm refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize the triangle inequality of ordinary metric spaces.
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https://en.wikipedia.org/wiki/T-norm
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In mathematics, a tall cardinal is a large cardinal κ that is θ-tall for all ordinals θ, where a cardinal is called θ-tall if there is an elementary embedding j: V → M with critical point κ such that j(κ) > θ and Mκ ⊆ M. Tall cardinals are equiconsistent with strong cardinals.
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https://en.wikipedia.org/wiki/Tall_cardinal
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In mathematics, a tame topology is a hypothetical topology proposed by Alexander Grothendieck in his research program Esquisse d’un programme under the French name topologie modérée (moderate topology). It is a topology in which the theory of dévissage can be applied to stratified structures such as semialgebraic or semianalytic sets.Some authors consider an o-minimal structure to be a candidate for realizing tame topology in the real case. There are also some other suggestions.
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https://en.wikipedia.org/wiki/Moderated_Topology
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In mathematics, a tangent Lie group is a Lie group whose underlying space is the tangent bundle TG of a Lie group G. As a Lie group, the tangent bundle is a semidirect product of a normal abelian subgroup with underlying space the Lie algebra of G, and G itself.
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https://en.wikipedia.org/wiki/Tangent_Lie_group
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In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point x {\displaystyle x} is a linear derivation of the algebra defined by the set of germs at x {\displaystyle x} .
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https://en.wikipedia.org/wiki/Tangent_vector
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In mathematics, a tangle is generally one of two related concepts: In John Conway's definition, an n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2n marked points on the ball's boundary. In link theory, a tangle is an embedding of n arcs and m circles into R 2 × {\displaystyle \mathbf {R} ^{2}\times } – the difference from the previous definition is that it includes circles as well as arcs, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance. (A quite different use of 'tangle' appears in Graph minors X. Obstructions to tree-decomposition by N. Robertson and P. D. Seymour, Journal of Combinatorial Theory B 52 (1991) 153–190, who used it to describe separation in graphs. This usage has been extended to matroids.)
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https://en.wikipedia.org/wiki/Algebraic_tangle
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The balance of this article discusses Conway's sense of tangles; for the link theory sense, see that article. Two n-tangles are considered equivalent if there is an ambient isotopy of one tangle to the other keeping the boundary of the 3-ball fixed. Tangle theory can be considered analogous to knot theory except instead of closed loops, strings whose ends are nailed down are used. See also braid theory.
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https://en.wikipedia.org/wiki/Algebraic_tangle
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In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp space L2+ε(G)for any ε > 0.
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https://en.wikipedia.org/wiki/Tempered_representation
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In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.
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https://en.wikipedia.org/wiki/Tensor_order
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Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another.
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https://en.wikipedia.org/wiki/Tensor_order
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This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors". Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing the earlier work of Bernhard Riemann, Elwin Bruno Christoffel, and others – as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.
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https://en.wikipedia.org/wiki/Tensor_order
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In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables.
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https://en.wikipedia.org/wiki/Ternary_cubic_form
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In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation. A ternary equivalence relation is symmetric, reflexive, and transitive. The classic example is the relation of collinearity among three points in Euclidean space. In an abstract set, a ternary equivalence relation determines a collection of equivalence classes or pencils that form a linear space in the sense of incidence geometry. In the same way, a binary equivalence relation on a set determines a partition.
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https://en.wikipedia.org/wiki/Ternary_equivalence_relation
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In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A. In computer science, a ternary operator is an operator that takes three arguments as input and returns one output.
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https://en.wikipedia.org/wiki/Ternary_operation
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In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables.
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https://en.wikipedia.org/wiki/Ternary_quartic
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In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian product A × B of some sets A and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product A × B × C of three sets A, B and C. An example of a ternary relation in elementary geometry can be given on triples of points, where a triple is in the relation if the three points are collinear. Another geometric example can be obtained by considering triples consisting of two points and a line, where a triple is in the ternary relation if the two points determine (are incident with) the line.
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https://en.wikipedia.org/wiki/Triadic_relation
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In mathematics, a tertiary ideal is a two-sided ideal in a perhaps noncommutative ring that cannot be expressed as a nontrivial intersection of a right fractional ideal with another ideal. Tertiary ideals generalize primary ideals to the case of noncommutative rings. Although primary decompositions do not exist in general for ideals in noncommutative rings, tertiary decompositions do, at least if the ring is Noetherian. Every primary ideal is tertiary.
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https://en.wikipedia.org/wiki/Tertiary_ideal
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Tertiary ideals and primary ideals coincide for commutative rings. To any (two-sided) ideal, a tertiary ideal can be associated called the tertiary radical, defined as t ( I ) = { r ∈ R | ∀ s ∉ I , ∃ x ∈ ( s ) x ∉ I and ( x ) ( r ) ⊂ I } . {\displaystyle t(I)=\{r\in R{\mbox{ }}|{\mbox{ }}\forall s\notin I,{\mbox{ }}\exists x\in (s){\mbox{ }}x\notin I{\text{ and }}(x)(r)\subset I\}.} Then t(I) always contains I. If R is a (not necessarily commutative) Noetherian ring and I a right ideal in R, then I has a unique irredundant decomposition into tertiary ideals I = T 1 ∩ ⋯ ∩ T n {\displaystyle I=T_{1}\cap \dots \cap T_{n}} .
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https://en.wikipedia.org/wiki/Tertiary_ideal
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In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems.
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https://en.wikipedia.org/wiki/Formal_theorem
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Moreover, many authors qualify as theorems only the most important results, and use the terms lemma, proposition and corollary for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements become well-formed formulas of some formal language.
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https://en.wikipedia.org/wiki/Formal_theorem
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A theory consists of some basis statements called axioms, and some deducing rules (sometimes included in the axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules. This formalization led to proof theory, which allows proving general theorems about theorems and proofs. In particular, Gödel's incompleteness theorems show that every consistent theory containing the natural numbers has true statements on natural numbers that are not theorems of the theory (that is they cannot be proved inside the theory). As the axioms are often abstractions of properties of the physical world, theorems may be considered as expressing some truth, but in contrast to the notion of a scientific law, which is experimental, the justification of the truth of a theorem is purely deductive.
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https://en.wikipedia.org/wiki/Formal_theorem
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In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle. In terms of algebraic geometry, the equivalent definition is as an invertible sheaf, which squares to the sheaf of differentials of the first kind. Theta characteristics were introduced by Rosenhain (1851)
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https://en.wikipedia.org/wiki/Theta_characteristic
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In mathematics, a theta constant or Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction θm(τ) = θm(τ,0) of a theta function θm(τ,z) with rational characteristic m to z = 0. The variable τ may be a complex number in the upper half-plane in which case the theta constants are modular forms, or more generally may be an element of a Siegel upper half plane in which case the theta constants are Siegel modular forms. The theta function of a lattice is essentially a special case of a theta constant.
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https://en.wikipedia.org/wiki/Theta_constant
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In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set T {\displaystyle T} , for every p ∈ N {\displaystyle p\in \mathbb {N} } , there is some n ∈ N {\displaystyle n\in \mathbb {N} } such that { n , n + 1 , n + 2 , . . . , n + p } ⊂ T {\displaystyle \{n,n+1,n+2,...,n+p\}\subset T} .
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https://en.wikipedia.org/wiki/Thick_set
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In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or may not be the case; solving within K a polynomial that does not always factorise. One is also allowed to take finite unions.
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https://en.wikipedia.org/wiki/Thin_set_(Serre)
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In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data.
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https://en.wikipedia.org/wiki/Univariate_time_series
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Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.
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https://en.wikipedia.org/wiki/Univariate_time_series
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Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series.
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https://en.wikipedia.org/wiki/Univariate_time_series
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Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses).
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https://en.wikipedia.org/wiki/Univariate_time_series
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A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility). Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language).
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https://en.wikipedia.org/wiki/Univariate_time_series
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In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group. That is, a TAG is both a group and a topological space, the group operations are continuous, and the group's binary operation is commutative. The theory of topological groups applies also to TAGs, but more can be done with TAGs. Locally compact TAGs, in particular, are used heavily in harmonic analysis.
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https://en.wikipedia.org/wiki/Abelian_topological_group
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In mathematics, a topological algebra A {\displaystyle A} is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
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https://en.wikipedia.org/wiki/Stereotype_algebra
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In mathematics, a topological game is an infinite game of perfect information played between two players on a topological space. Players choose objects with topological properties such as points, open sets, closed sets and open coverings. Time is generally discrete, but the plays may have transfinite lengths, and extensions to continuum time have been put forth. The conditions for a player to win can involve notions like topological closure and convergence.
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https://en.wikipedia.org/wiki/Topological_game
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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 same time, some topological properties that arise naturally in topological games can be generalized beyond a game-theoretic context: by virtue of this duality, topological games have been widely used to describe new properties of topological spaces, and to put known properties under a different light. There are also close links with selection principles.
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https://en.wikipedia.org/wiki/Topological_game
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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 by topological games"; this approach is based on analogies with matrix games, differential games and statistical games, and defines and studies topological games within topology. After more than 35 years, the term “topological game” became widespread, and appeared in several hundreds of publications.
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https://en.wikipedia.org/wiki/Topological_game
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The survey paper of Telgársky emphasizes the origin of topological games from the Banach–Mazur game. There are two other meanings of topological games, but these are used less frequently. The term topological game introduced by Leon Petrosjan in the study of antagonistic pursuit–evasion games.
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https://en.wikipedia.org/wiki/Topological_game
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The trajectories in these topological games are continuous in time. The games of Nash (the Hex games), the Milnor games (Y games), the Shapley games (projective plane games), and Gale's games (Bridg-It games) were called topological games by David Gale in his invited address . The number of moves in these games is always finite. The discovery or rediscovery of these topological games goes back to years 1948–49.
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https://en.wikipedia.org/wiki/Topological_game
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In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and the edges by Jordan arcs (connected pieces of Jordan curves) joining the corresponding pairs of points. The points representing the vertices of a graph and the arcs representing its edges are called the vertices and the edges of the topological graph. It is usually assumed that any two edges of a topological graph cross a finite number of times, no edge passes through a vertex different from its endpoints, and no two edges touch each other (without crossing). A topological graph is also called a drawing of a graph.
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https://en.wikipedia.org/wiki/Topological_graph
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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 properties of topological graphs, in particular, with the crossing patterns of their edges. It is closely related to graph drawing, a field which is more application oriented, and topological graph theory, which focuses on embeddings of graphs in surfaces (that is, drawings without crossings).
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https://en.wikipedia.org/wiki/Topological_graph
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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 subgroup if H is discrete when endowed with the subspace topology from G. In other words there is a neighbourhood of the identity in G containing no other element of H. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not. Any group can be endowed with the discrete topology, making it a discrete topological group. Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups.
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https://en.wikipedia.org/wiki/Discrete_group
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Hence, there is an isomorphism between the category of groups and the category of discrete groups. Discrete groups can therefore be identified with their underlying (non-topological) groups. There are some occasions when a topological group or Lie group is usefully endowed with the discrete topology, 'against nature'.
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https://en.wikipedia.org/wiki/Discrete_group
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This happens for example in the theory of the Bohr compactification, and in group cohomology theory of Lie groups. A discrete isometry group is an isometry group such that for every point of the metric space the set of images of the point under the isometries is a discrete set. A discrete symmetry group is a symmetry group that is a discrete isometry group.
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https://en.wikipedia.org/wiki/Discrete_group
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In mathematics, a topological group G {\displaystyle G} is called the topological direct sum of two subgroups H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} if the map is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.
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https://en.wikipedia.org/wiki/Direct_sum_of_topological_groups
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In mathematics, a topological half-exact functor F is a functor from a fixed topological category (for example CW complexes or pointed spaces) to an abelian category (most frequently in applications, category of abelian groups or category of modules over a fixed ring) that has a following property: for each sequence of spaces, of the form: X → Y → C(f)where C(f) denotes a mapping cone, the sequence: F(X) → F(Y) → F(C(f))is exact. If F is a contravariant functor, it is half-exact if for each sequence of spaces as above, the sequence F(C(f)) → F(Y) → F(X) is exact. Homology is an example of a half-exact functor, and cohomology (and generalized cohomology theories) are examples of contravariant half-exact functors. If B is any fibrant topological space, the (representable) functor F(X)= is half-exact.
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https://en.wikipedia.org/wiki/Topological_half-exact_functor
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In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
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https://en.wikipedia.org/wiki/Topological_module
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In mathematics, a topological ring is a ring R {\displaystyle R} that is also a topological space such that both the addition and the multiplication are continuous as maps: where R × R {\displaystyle R\times R} carries the product topology. That means R {\displaystyle R} is an additive topological group and a multiplicative topological semigroup. Topological rings are fundamentally related to topological fields and arise naturally while studying them, since for example completion of a topological field may be a topological ring which is not a field.
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https://en.wikipedia.org/wiki/Topological_ring
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In mathematics, a topological semigroup is a semigroup that is simultaneously a topological space, and whose semigroup operation is continuous.Every topological group is a topological semigroup.
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https://en.wikipedia.org/wiki/Topological_semigroup
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In mathematics, a topological space (X, T) is called completely uniformizable (or Dieudonné complete) if there exists at least one complete uniformity that induces the topology T. Some authors additionally require X to be Hausdorff. Some authors have called these spaces topologically complete, although that term has also been used in other meanings like completely metrizable, which is a stronger property than completely uniformizable.
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https://en.wikipedia.org/wiki/Completely_uniformizable_space
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In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.
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https://en.wikipedia.org/wiki/Contractible_space
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In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in X {\displaystyle X} . Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.
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https://en.wikipedia.org/wiki/Sequentially_compact
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In mathematics, a topological space X is uniformizable if there exists a uniform structure on X that induces the topology of X. Equivalently, X is uniformizable if and only if it is homeomorphic to a uniform space (equipped with the topology induced by the uniform structure). Any (pseudo)metrizable space is uniformizable since the (pseudo)metric uniformity induces the (pseudo)metric topology. The converse fails: There are uniformizable spaces that are not (pseudo)metrizable.
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https://en.wikipedia.org/wiki/Fine_uniformity
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However, it is true that the topology of a uniformizable space can always be induced by a family of pseudometrics; indeed, this is because any uniformity on a set X can be defined by a family of pseudometrics. Showing that a space is uniformizable is much simpler than showing it is metrizable. In fact, uniformizability is equivalent to a common separation axiom: A topological space is uniformizable if and only if it is completely regular.
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https://en.wikipedia.org/wiki/Fine_uniformity
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In mathematics, a topological space X {\displaystyle X} is a D-space if for any family { U x: x ∈ X } {\displaystyle \{U_{x}:x\in X\}} of open sets such that x ∈ U x {\displaystyle x\in U_{x}} for all points x ∈ X {\displaystyle x\in X} , there is a closed discrete subset D {\displaystyle D} of the space X {\displaystyle X} such that ⋃ x ∈ D U x = X {\displaystyle \bigcup _{x\in D}U_{x}=X} .
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https://en.wikipedia.org/wiki/D-space
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In mathematics, a topological space X {\displaystyle X} is called collectionwise normal if for every discrete family Fi (i ∈ I) of closed subsets of X {\displaystyle X} there exists a pairwise disjoint family of open sets Ui (i ∈ I), such that Fi ⊆ Ui. Here a family F {\displaystyle {\mathcal {F}}} of subsets of X {\displaystyle X} is called discrete when every point of X {\displaystyle X} has a neighbourhood that intersects at most one of the sets from F {\displaystyle {\mathcal {F}}} . An equivalent definition of collectionwise normal demands that the above Ui (i ∈ I) themselves form a discrete family, which is stronger than pairwise disjoint. Some authors assume that X {\displaystyle X} is also a T1 space as part of the definition. The property is intermediate in strength between paracompactness and normality, and occurs in metrization theorems.
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https://en.wikipedia.org/wiki/Collectionwise_normal_space
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In mathematics, a topological space X {\displaystyle X} is called countably generated if the topology of X {\displaystyle X} is determined by the countable sets in a similar way as the topology of a sequential space (or a Fréchet space) is determined by the convergent sequences. The countably generated spaces are precisely the spaces having countable tightness—therefore the name countably tight is used as well.
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https://en.wikipedia.org/wiki/Countably_generated_space
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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 combined with the properties of Baire spaces has numerous applications in topology, geometry, analysis, in particular functional analysis. For more motivation and applications, see the article Baire category theorem. The current article focuses more on characterizations and basic properties of Baire spaces per se. Bourbaki introduced the term "Baire space" in honor of René Baire, who investigated the Baire category theorem in the context of Euclidean space R n {\displaystyle \mathbb {R} ^{n}} in his 1899 thesis.
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https://en.wikipedia.org/wiki/Baire_space
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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, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.
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https://en.wikipedia.org/wiki/Limit_point_compact
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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 other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces.
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https://en.wikipedia.org/wiki/Separable_space
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In mathematics, a topological space is feebly compact if every locally finite cover by nonempty open sets is finite. The concept was introduced by S. Mardeĉić and P. Papić in 1955.Some facts: Every compact space is feebly compact. Every feebly compact paracompact space is compact. Every feebly compact space is pseudocompact but the converse is not necessarily true.
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https://en.wikipedia.org/wiki/Feebly_compact_space
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For a completely regular Hausdorff space the properties of being feebly compact and pseudocompact are equivalent. Any maximal feebly compact space is submaximal. == References ==
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https://en.wikipedia.org/wiki/Feebly_compact_space
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In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint. Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no T1 space with more than one point is ultraconnected.
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https://en.wikipedia.org/wiki/Ultraconnected_space
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In mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial.
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https://en.wikipedia.org/wiki/Weakly_contractible
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In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.A space is said to be σ-locally compact if it is both σ-compact and (weakly) locally compact. That terminology can be somewhat confusing as it does not fit the usual pattern of σ-(property) meaning a countable union of spaces satisfying (property); that's why such spaces are more commonly referred to explicitly as σ-compact (weakly) locally compact, which is also equivalent to being exhaustible by compact sets.
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https://en.wikipedia.org/wiki/Σ-compact_space
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In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness.
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https://en.wikipedia.org/wiki/Topological_structure
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Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called point-set topology or general topology.
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https://en.wikipedia.org/wiki/Topological_structure
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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 (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness.
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https://en.wikipedia.org/wiki/Vector_topology
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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.
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https://en.wikipedia.org/wiki/Vector_topology
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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 this article, the scalar field of a topological vector space will be assumed to be either the complex numbers C {\displaystyle \mathbb {C} } or the real numbers R , {\displaystyle \mathbb {R} ,} unless clearly stated otherwise.
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https://en.wikipedia.org/wiki/Vector_topology
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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 topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory.
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https://en.wikipedia.org/wiki/Logical_functor
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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 algebra is abelian; thus, its elements are simultaneously diagonalizable.
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https://en.wikipedia.org/wiki/Toral_Lie_algebra
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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 combinatorics on the quotient polytope and obtain information on the manifold above. For example, the Euler characteristic and the cohomology ring of the manifold can be described in terms of the polytope.
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https://en.wikipedia.org/wiki/Toric_manifold
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In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus.
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https://en.wikipedia.org/wiki/Toroid
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The term toroid is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A g-holed toroid can be seen as approximating the surface of a torus having a topological genus, g, of 1 or greater.
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https://en.wikipedia.org/wiki/Toroid
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The Euler characteristic χ of a g holed toroid is 2(1-g).The torus is an example of a toroid, which is the surface of a doughnut. Doughnuts are an example of a solid torus created by rotating a disk, and should not be confused with toroids. Toroidal structures occur in both natural and synthetic materials.
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https://en.wikipedia.org/wiki/Toroid
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In mathematics, a torsion sheaf is a sheaf of abelian groups F {\displaystyle {\mathcal {F}}} on a site for which, for every object U, the space of sections Γ ( U , F ) {\displaystyle \Gamma (U,{\mathcal {F}})} is a torsion abelian group. Similarly, for a prime number p, we say a sheaf F {\displaystyle {\mathcal {F}}} is p-torsion if every section over any object is killed by a power of p. A torsion sheaf on an étale site is the union of its constructible subsheaves.
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https://en.wikipedia.org/wiki/Torsion_sheaf
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In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X}: a ≤ a {\displaystyle a\leq a} (reflexive). If a ≤ b {\displaystyle a\leq b} and b ≤ c {\displaystyle b\leq c} then a ≤ c {\displaystyle a\leq c} (transitive). If a ≤ b {\displaystyle a\leq b} and b ≤ a {\displaystyle b\leq a} then a = b {\displaystyle a=b} (antisymmetric).
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https://en.wikipedia.org/wiki/Linearly_ordered_set
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a ≤ b {\displaystyle a\leq b} or b ≤ a {\displaystyle b\leq a} (strongly connected, formerly called total).Reflexivity (1.) already follows from connectedness (4.
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https://en.wikipedia.org/wiki/Linearly_ordered_set
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), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders.A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term chain is sometimes defined as a synonym of totally ordered set, but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order.
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https://en.wikipedia.org/wiki/Linearly_ordered_set
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In mathematics, a totally disconnected group is a topological group that is totally disconnected. Such topological groups are necessarily Hausdorff. Interest centres on locally compact totally disconnected groups (variously referred to as groups of td-type, locally profinite groups, or t.d. groups).
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https://en.wikipedia.org/wiki/Totally_disconnected_group
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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, opened up the field by showing that every locally compact totally disconnected group contains a so-called tidy subgroup and a special function on its automorphisms, the scale function, giving a quantifiable parameter for the local structure. Advances on the global structure of totally disconnected groups were obtained in 2011 by Caprace and Monod, with notably a classification of characteristically simple groups and of Noetherian groups.
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https://en.wikipedia.org/wiki/Totally_disconnected_group
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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 eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly, except that all the minors must be non-negative (positive or zero). Some authors use "totally positive" to include all totally non-negative matrices.
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https://en.wikipedia.org/wiki/Totally_positive_matrix
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In mathematics, a tower of fields is a sequence of field extensions F0 ⊆ F1 ⊆ ... ⊆ Fn ⊆ ...The name comes from such sequences often being written in the form ⋮ | F 2 | F 1 | F 0 . {\displaystyle {\begin{array}{c}\vdots \\|\\F_{2}\\|\\F_{1}\\|\\\ F_{0}.\end{array}}} A tower of fields may be finite or infinite.
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https://en.wikipedia.org/wiki/Infinite_tower_of_fields
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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. In many cases, a toy theorem is used to illustrate the claim of a theorem, while in other cases, studying the proofs of a toy theorem (derived from a non-trivial theorem) can provide insight that would be hard to obtain otherwise. Toy theorems can also have educational value as well. For example, after presenting a theorem (with, say, a highly non-trivial proof), one can sometimes give some assurance that the theorem really holds, by proving a toy version of the theorem.
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https://en.wikipedia.org/wiki/Toy_theorem
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In mathematics, a trace identity is any equation involving the trace of a matrix.
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https://en.wikipedia.org/wiki/Trace_identity
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In mathematics, a transcendental extension L / K {\displaystyle L/K} is a field extension such that there exists an element in the field L {\displaystyle L} that is transcendental over the field K {\displaystyle K} ; that is, an element that is not a root of any univariate polynomial with coefficients in K {\displaystyle K} . In other words, a transcendental extension is a field extension that is not algebraic. For example, C , R {\displaystyle \mathbb {C} ,\mathbb {R} } are both transcendental extensions of Q . {\displaystyle \mathbb {Q} .}
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https://en.wikipedia.org/wiki/Transcendence_degree
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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.
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https://en.wikipedia.org/wiki/Transcendence_degree
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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, the dimension of an algebraic variety is the transcendence degree of its function field. Also, global function fields are transcendental extensions of degree one of a finite field, and play in number theory in positive characteristic a role that is very similar to the role of algebraic number fields in characteristic zero.
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https://en.wikipedia.org/wiki/Transcendence_degree
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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 functions include the exponential function, the logarithm, and the trigonometric functions.
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https://en.wikipedia.org/wiki/Transcendental_functions
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In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are π and e.Though only a few classes of transcendental numbers are known – partly because it can be extremely difficult to show that a given number is transcendental – transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. The converse is not true: Not all irrational numbers are transcendental.
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https://en.wikipedia.org/wiki/Transcendental_number
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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 \varphi } or ϕ {\displaystyle \phi } ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x2 − x − 1 = 0. The quality of a number being transcendental is called transcendence.
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https://en.wikipedia.org/wiki/Transcendental_number
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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 transformations, such as rotations, reflections and translations.
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https://en.wikipedia.org/wiki/Mathematical_transformations
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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) or weighted sums over the higher-order derivatives of these functions (see derivative transformations). Given a sequence, { f n } n = 0 ∞ {\displaystyle \{f_{n}\}_{n=0}^{\infty }} , the ordinary generating function (OGF) of the sequence, denoted F ( z ) {\displaystyle F(z)} , and the exponential generating function (EGF) of the sequence, denoted F ^ ( z ) {\displaystyle {\widehat {F}}(z)} , are defined by the formal power series F ( z ) = ∑ n = 0 ∞ f n z n = f 0 + f 1 z + f 2 z 2 + ⋯ {\displaystyle F(z)=\sum _{n=0}^{\infty }f_{n}z^{n}=f_{0}+f_{1}z+f_{2}z^{2}+\cdots } F ^ ( z ) = ∑ n = 0 ∞ f n n ! z n = f 0 0 !
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https://en.wikipedia.org/wiki/Generating_function_transformation
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+ f 1 1 ! z + f 2 2 ! z 2 + ⋯ .
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https://en.wikipedia.org/wiki/Generating_function_transformation
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{\displaystyle {\widehat {F}}(z)=\sum _{n=0}^{\infty }{\frac {f_{n}}{n! }}z^{n}={\frac {f_{0}}{0! }}+{\frac {f_{1}}{1!
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https://en.wikipedia.org/wiki/Generating_function_transformation
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}}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 {\displaystyle z} when the context of this notation is clear.
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https://en.wikipedia.org/wiki/Generating_function_transformation
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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 generating functions (DGFs), Lambert series, and Newton series. In this article we focus on transformations of generating functions in mathematics and keep a running list of useful transformations and transformation formulas.
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https://en.wikipedia.org/wiki/Generating_function_transformation
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