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Extended
Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (proof theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Extension (semantics), the set of things to which a property applies * Extension (simplicial set) * Extension by definitions * Extensional definition, a definition that enumerates every individual a term applies to * Extensionality Other uses * Extension of a function, defined on a larger domain * Extension of a polyhedron, in geometry * Extension of a line segment (finite) into an infinite line (e.g., extended base) * Exterior algebra, Grassmann's theory of extension, in geometry * Field extension, in Galois theory * Group extension, in abstract algebra and homological algebra * Homotopy extension property, in topology * Kolmogorov extension theorem, in probability theory * Linear extension, in order theory * Sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector v in V. The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol \wedge and the fact that the product of two elements of V is "outside" V. The wedge product of k vectors v_1 \wedge v_2 \wedge \dots \wedge v_k is called a ''blade (geometry), blade of degree k'' or ''k-blade''. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude (mathematics), magnitude of a bivector, -blade v\wedge w is the area of the parallelogram defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the Parallelepiped#Parallelotope, parallelotope defined by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tietze Extension Theorem
In topology, the Tietze extension theorem (also known as the Tietze– Urysohn– Brouwer extension theorem or Urysohn-Brouwer lemma) states that any real-valued, continuous function on a closed subset of a normal topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ... can be extended to the entire space, preserving boundedness if necessary. Formal statement If X is a normal space and f : A \to \R is a continuous map from a closed subset A of X into the real numbers \R carrying the standard topology, then there exists a of f to X; that is, there exists a map F : X \to \R continuous on all of X with F(a) = f(a) for all a \in A. Moreover, F may be chosen such that \sup \ ~=~ \sup \, that is, if f is bounded then F may be chosen to be bounded (with the same bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Base
In geometry, a base is a Edge (geometry), side of a polygon or a face (geometry), face of a polyhedron, particularly one oriented perpendicular to the direction in which Height#In mathematics, height is measured, or on what is considered to be the "bottom" of the figure. This term is commonly applied in plane geometry to triangles, parallelograms, trapezoids, and in solid geometry to Cylinder (geometry), cylinders, Cone (geometry), cones, Pyramid (geometry), pyramids, parallelepipeds, Prism (geometry), prisms, and frustums. The side or point opposite the base is often called the ''apex (geometry), apex'' or ''summit'' of the shape. Of a triangle In a triangle, any arbitrary side can be considered the ''base''. The two endpoints of the base are called ''base vertices'' and the corresponding angles are called ''base angles''. The third vertex opposite the base is called the ''apex''. The extended base of a triangle (a particular case of an extended side) is the line (geometry), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension Of A Polyhedron
In convex geometry and polyhedral combinatorics, the extension complexity of a convex polytope P is the smallest number of facets among convex polytopes Q that have P as a projection. In this context, Q is called an extended formulation of P; it may have much higher dimension than P. The extension complexity depends on the precise shape of P, not just on its combinatorial structure. For instance, regular polygons with n sides have extension complexity O(\log n) (expressed using big O notation), but some other convex n-gons have extension complexity at least proportional to \sqrt. If a polytope describing the feasible solutions to a combinatorial optimization problem has low extension complexity, this could potentially be used to devise efficient algorithms for the problem, using linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Extension
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Definitions Linear extension of a partial order A partial order is a reflexive, transitive and antisymmetric relation. Given any partial orders \,\leq\, and \,\leq^*\, on a set X, \,\leq^*\, is a linear extension of \,\leq\, exactly when # \,\leq^*\, is a total order, and # For every x, y \in X, if x \leq y, then x \leq^* y. It is that second property that leads mathematicians to describe \,\leq^*\, as extending \,\leq. Alternatively, a linear extension may be viewed as an order-preserving bijection from a partially ordered set P to a chain C on the same ground set. Linear extension of a preorder A preorder is a reflexive and transitive relation. The difference between a preorder and a partial- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Extension Property
In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations. Definition Let X\,\! be a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ..., and let A \subset X. We say that the pair (X,A)\,\! has the homotopy extension property if, given a homotopy f_\bullet\colon A \rightarrow Y^I and a map \tilde_0\colon X \rightarrow Y such that \tilde_0\circ \iota = \left.\tilde_0\_A = f_0 = \pi_0 \circ f_\bullet, then there exists an ''extension'' of f_\bullet to a homotopy \tilde_\bullet\colon X \rightarrow Y^I such that \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Extension Theorem
In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if ''A'' is a closed subset of a Euclidean space, then it is possible to extend a given function of ''A'' in such a way as to have prescribed derivatives at the points of ''A''. It is a result of Hassler Whitney. Statement A precise statement of the theorem requires careful consideration of what it means to prescribe the derivative of a function on a closed set. One difficulty, for instance, is that closed subsets of Euclidean space in general lack a differentiable structure. The starting point, then, is an examination of the statement of Taylor's theorem. Given a real-valued ''C''''m'' function ''f''(x) on R''n'', Taylor's theorem asserts that for each a, x, y ∈ R''n'', there is a function ''R''''α''(x,y) approaching 0 uniformly as x,y → a such that where the sum is over multi-indices ''α' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov Extension Theorem
In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to the English mathematician Percy John Daniell and the Russia, Russian mathematician Andrey Kolmogorov, Andrey Nikolaevich Kolmogorov. Statement of the theorem Let T denote some Interval (mathematics), interval (thought of as "time"), and let n \in \mathbb. For each k \in \mathbb and finite sequence of distinct times t_, \dots, t_ \in T, let \nu_ be a probability measure on (\mathbb^)^. Suppose that these measures satisfy two consistency conditions: 1. for all permutations \pi of \ and measurable sets F_ \subseteq \mathbb^, :\nu_ \left( F_ \times \dots \times F_ \right) = \nu_ \left( F_ \times \dots \times F_ \right); 2. for all measurable sets F_ \subseteq \mathbb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special case of an ''arc (geometry), arc'', with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum (symbol), vinculum) above the symbols for the two endpoints, such as in . Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension (music)
In music, an extension is a set of musical notes that lie outside the standard range or tessitura. Staff A note that lies outside the lines of a musical staff is an extension of the staff. The note will lie on a ledger line. Middle C, for example, is an extension note on both treble and bass clefs, however is not outside the grand staff. Soprano C and Deep C lie two ledger lines above treble and below bass respectively (as well as the grand staff). Instruments An instrumental extension is a range of playable notes outside the normal range of the instrument. A baritone horn, if played by a skillful player, can be played an octave above the normal range. Since this is not standard, these notes would be an extension. (See also: Crook (music)). With the bowed string instruments, lower pitches than the standard range are sometimes used through scordatura in which the lowest string is tuned down a note or two. The double bass sometimes uses a C extension extending th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension (Clare Fischer Album)
''Extension'' is the third album by composer/arranger/keyboardist Clare Fischer, and his first for big band, recorded and released in 1963 on the Pacific Jazz label, reissued on CD (together with the 1967 LP, ''Songs for Rainy Day Lovers'') in 2002 as ''America the Beautiful'', and, under its original name, in 2012. WorldCat. Retrieved 2013-03-18 According to Ed Beach, disk jockey of the WRVR 106.7 FM New York radio program “Just Jazz,” on this album, all of the solos, except for those of Clare Fischer and tenor saxophonist Jerry Coker, were written out (by Fischer), not improvised. Reception Reviewing the 2012 CD reissue for All About Jazz, Troy Collins calls ''Extension'' Fischer's "mas ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended (Solar Fields Album)
Magnus Birgersson, better known by his stage name Solar Fields, is a Swedish electronic music artist. As of 2014, he has released fifteen albums, and has also scored all interactive in-game music for the Electronic Arts game '' Mirror's Edge'' as well as its reboot, '' Mirror's Edge Catalyst''. His latest album, ''Formations'', was released in November 2022. Biography Gothenburg-based Swedish composer, sound designer, and multi-instrumentalist Magnus Birgersson created Solar Fields in the late 1990s. Birgersson was raised in a musical family and began playing piano and synthesizers in the 1970s. In the mid-1980s he began combining synthesizers with computers. In addition to his ambient work, he has also been a guitar player in rock bands, a pianist in jazz funk bands, and keyboard player in drum and bass bands. He has collaborated with Vincent Villuis, a.k.a. AES Dana, on H.U.V.A. Network and T.S.R. in the company of Daniel Segerstad and Johannes Hedberg from Carbon B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
