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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 al... | https://en.wikipedia.org/wiki/Homogeneous_equation |
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. Ea... | https://en.wikipedia.org/wiki/Underdetermined_system |
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 whe... | https://en.wikipedia.org/wiki/Underdetermined_system |
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, ..., x... | https://en.wikipedia.org/wiki/System_of_parameters |
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... | https://en.wikipedia.org/wiki/T-norm |
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. | https://en.wikipedia.org/wiki/Tall_cardinal |
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 se... | https://en.wikipedia.org/wiki/Moderated_Topology |
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. | https://en.wikipedia.org/wiki/Tangent_Lie_group |
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... | https://en.wikipedia.org/wiki/Tangent_vector |
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 arc... | https://en.wikipedia.org/wiki/Algebraic_tangle |
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 instea... | https://en.wikipedia.org/wiki/Algebraic_tangle |
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. | https://en.wikipedia.org/wiki/Tempered_representation |
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... | https://en.wikipedia.org/wiki/Tensor_order |
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 susceptibi... | https://en.wikipedia.org/wiki/Tensor_order |
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 abs... | https://en.wikipedia.org/wiki/Tensor_order |
In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables. | https://en.wikipedia.org/wiki/Ternary_cubic_form |
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 equiva... | https://en.wikipedia.org/wiki/Ternary_equivalence_relation |
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. | https://en.wikipedia.org/wiki/Ternary_operation |
In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables. | https://en.wikipedia.org/wiki/Ternary_quartic |
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 pro... | https://en.wikipedia.org/wiki/Triadic_relation |
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 gene... | https://en.wikipedia.org/wiki/Tertiary_ideal |
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{ }}\ex... | https://en.wikipedia.org/wiki/Tertiary_ideal |
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... | https://en.wikipedia.org/wiki/Formal_theorem |
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 bec... | https://en.wikipedia.org/wiki/Formal_theorem |
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 theorem... | https://en.wikipedia.org/wiki/Formal_theorem |
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 bun... | https://en.wikipedia.org/wiki/Theta_characteristic |
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 for... | https://en.wikipedia.org/wiki/Theta_constant |
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... | https://en.wikipedia.org/wiki/Thick_set |
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;... | https://en.wikipedia.org/wiki/Thin_set_(Serre) |
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. | https://en.wikipedia.org/wiki/Univariate_time_series |
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, ma... | https://en.wikipedia.org/wiki/Univariate_time_series |
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 ... | https://en.wikipedia.org/wiki/Univariate_time_series |
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... | https://en.wikipedia.org/wiki/Univariate_time_series |
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 d... | https://en.wikipedia.org/wiki/Univariate_time_series |
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 ... | https://en.wikipedia.org/wiki/Abelian_topological_group |
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. | https://en.wikipedia.org/wiki/Stereotype_algebra |
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 exten... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Topological_game |
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. | https://en.wikipedia.org/wiki/Topological_game |
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 alwa... | https://en.wikipedia.org/wiki/Topological_game |
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 rep... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Topological_graph |
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://en.wikipedia.org/wiki/Discrete_group |
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'. | https://en.wikipedia.org/wiki/Discrete_group |
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 g... | https://en.wikipedia.org/wiki/Discrete_group |
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. | https://en.wikipedia.org/wiki/Direct_sum_of_topological_groups |
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... | https://en.wikipedia.org/wiki/Topological_half-exact_functor |
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous. | https://en.wikipedia.org/wiki/Topological_module |
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 multiplic... | https://en.wikipedia.org/wiki/Topological_ring |
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. | https://en.wikipedia.org/wiki/Topological_semigroup |
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 al... | https://en.wikipedia.org/wiki/Completely_uniformizable_space |
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. | https://en.wikipedia.org/wiki/Contractible_space |
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 ... | https://en.wikipedia.org/wiki/Sequentially_compact |
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 uniformizab... | https://en.wikipedia.org/wiki/Fine_uniformity |
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... | https://en.wikipedia.org/wiki/Fine_uniformity |
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 {\displaysty... | https://en.wikipedia.org/wiki/D-space |
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 {\di... | https://en.wikipedia.org/wiki/Collectionwise_normal_space |
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 preci... | https://en.wikipedia.org/wiki/Countably_generated_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://en.wikipedia.org/wiki/Baire_space |
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://en.wikipedia.org/wiki/Limit_point_compact |
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://en.wikipedia.org/wiki/Separable_space |
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 pseudoco... | https://en.wikipedia.org/wiki/Feebly_compact_space |
For a completely regular Hausdorff space the properties of being feebly compact and pseudocompact are equivalent. Any maximal feebly compact space is submaximal. == References == | https://en.wikipedia.org/wiki/Feebly_compact_space |
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. | https://en.wikipedia.org/wiki/Ultraconnected_space |
In mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial. | https://en.wikipedia.org/wiki/Weakly_contractible |
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 count... | https://en.wikipedia.org/wiki/Σ-compact_space |
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 defin... | https://en.wikipedia.org/wiki/Topological_structure |
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... | https://en.wikipedia.org/wiki/Topological_structure |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Vector_topology |
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://en.wikipedia.org/wiki/Logical_functor |
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://en.wikipedia.org/wiki/Toral_Lie_algebra |
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://en.wikipedia.org/wiki/Toric_manifold |
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 fig... | https://en.wikipedia.org/wiki/Toroid |
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. | https://en.wikipedia.org/wiki/Toroid |
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. | https://en.wikipedia.org/wiki/Toroid |
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}}} ... | https://en.wikipedia.org/wiki/Torsion_sheaf |
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}:... | https://en.wikipedia.org/wiki/Linearly_ordered_set |
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. | https://en.wikipedia.org/wiki/Linearly_ordered_set |
), 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 c... | https://en.wikipedia.org/wiki/Linearly_ordered_set |
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). | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Totally_disconnected_group |
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://en.wikipedia.org/wiki/Totally_positive_matrix |
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. | https://en.wikipedia.org/wiki/Infinite_tower_of_fields |
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://en.wikipedia.org/wiki/Toy_theorem |
In mathematics, a trace identity is any equation involving the trace of a matrix. | https://en.wikipedia.org/wiki/Trace_identity |
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 {\displaysty... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Transcendence_degree |
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://en.wikipedia.org/wiki/Transcendental_functions |
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 ... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Transcendental_number |
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://en.wikipedia.org/wiki/Mathematical_transformations |
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://en.wikipedia.org/wiki/Generating_function_transformation |
+ f 1 1 ! z + f 2 2 ! z 2 + ⋯ . | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Generating_function_transformation |
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