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Extensions
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 theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extensions (Dave Holland Album)
''Extensions'' is an album by the Dave Holland Quartet, recorded in September 1989 and released on ECM the following year—Holland's eight album for the label. The quartet features Holland Quintet saxophonist Steve Coleman and drummer Marvin "Smitty" Smith alongside guitarist Kevin Eubanks, in his first appearance on a Holland record. Reception The AllMusic AllMusic (previously known as All-Music Guide and AMG) is an American online database, online music database. It catalogs more than three million album entries and 30 million tracks, as well as information on Musical artist, musicians and Mus ... review by Brian Olewnick called the album a "tight and enjoyable quartet date," writing, "One of his better albums from this period, ''Extensions'' should please any Holland fan, and is an agreeable and non-threatening jumping in point for the curious."Olewnick, BAllMusic Reviewaccessed August 19, 2011. ''Extensions'' was voted Album of the Year (1989) by '' DownBeat''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extensions (McCoy Tyner Album)
''Extensions'' is the eleventh album by jazz pianist McCoy Tyner released on the Blue Note label. It was recorded on February 9, 1970, but not released until January 1973. It has performances by Tyner with alto saxophonist Gary Bartz, tenor saxophonist Wayne Shorter, bassist Ron Carter, drummer Elvin Jones, and features Alice Coltrane playing harp on three of the four tracks. Reception In his AllMusic review Scott Yanow says, "The all-star sextet stretches out on lengthy renditions of four of Tyner's modal originals, and there is strong solo space for the leader and the two saxophonists... Stimulating music". Reviewing the album for jazz.com, Jared Pauley says, "McCoy Tyner finds himself among elite company on ''Extensions.'' Recorded as jazz was entering the fusion period, this is a great example of just how good straight-ahead swing can sound... This performance matches the superb quality of previous Shorter and Tyner albums where members of the Davis and Coltran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extensions (The Manhattan Transfer Album)
''Extensions'' is the fifth studio album by the Manhattan Transfer, released on October 31, 1979, by Atlantic Records. Marking a new era for the group, the album was the first one with Cheryl Bentyne, who replaced Laurel Massé in early 1979. It was also their first album with Jay Graydon in the producer's chair and their first to contain songs that were hits in both the jazz and pop categories. The song "Twilight Zone/Twilight Tone" reached No. 4 on the ''Billboard'' Disco chart and No. 30 on the Hot 100. "Trickle, Trickle" reached No. 73 on the Hot 100. The album reached No. 55 on the ''Billboard'' Top LPs chart. The most widely known song from this album, " Birdland" by Weather Report, won the Grammy Award for Best Jazz Fusion. Best Jazz Fusion Performance in 1981. Jon Hendricks wrote lyrics for the vocalese version on the album and Janis Siegel received a Grammy for her vocal arrangement of "Birdland". Critical reception ''The New York Times'' wrote that the album " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extensions (Ahmad Jamal Album)
''Extensions'' is an album by American jazz pianist Ahmad Jamal featuring performances recorded in 1965 and released on the Argo label.Ahmad Jamal discography accessed May 18, 2012 Critical reception awarded the album 3 stars.Allmusic Reviewaccessed May 18, 20121 Track listing # "Extensions" (Ahmad Jamal) – 13:10 # "Dance to the Lady" ( John Handy) – 5:50 # "This Terrible Planet" (Bob Williams ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overset\;G\;\overset\;Q \to 1. If G is an extension of Q by N, then G is a group, \iota(N) is a normal subgroup of G and the quotient group G/\iota(N) is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is also used by some. Since any finite group G possesses a maximal normal subgroup N with simple factor group G/\iota(N), all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N lies in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ''L''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains the multiplicative identity 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension (proof Theory)
In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the original. More formally stated, a theory T_2 is a ( proof theoretic) conservative extension of a theory T_1 if every theorem of T_1 is a theorem of T_2, and any theorem of T_2 in the language of T_1 is already a theorem of T_1. More generally, if \Gamma is a set of formulas in the common language of T_1 and T_2, then T_2 is \Gamma-conservative over T_1 if every formula from \Gamma provable in T_2 is also provable in T_1. Note that a conservative extension of a consistent theory is consistent. If it were not, then by the principle of explosion, every formula in the language of T_2 would be a theorem of T_2, so every formula in the language of T_1 would be a theorem of T_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension By Definitions
In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol \emptyset for the set that has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant \emptyset and the new axiom \forall x(x\notin\emptyset), meaning "for all ''x'', ''x'' is not a member of \emptyset". It can then be proved that doing so adds essentially nothing to the old theory, as should be expected from a definition. More precisely, the new theory is a conservative extension of the old one. Definition of relation symbols ''Let'' T be a first-order theory and \phi(x_1,\dots,x_n) a formula of T such that x_1, ..., x_n are distinct and include the variables free in \phi(x_1,\dots,x_n). Form a new first-order theory T' from T by adding a new n-ary relation symbol R, the logica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension Of A Function
In mathematics, the restriction of a function f is a new function, denoted f\vert_A or f , obtained by choosing a smaller domain A for the original function f. The function f is then said to extend f\vert_A. Formal definition Let f : E \to F be a function from a set E to a set F. If a set A is a subset of E, then the restriction of f to A is the function _A : A \to F given by _A(x) = f(x) for x \in A. Informally, the restriction of f to A is the same function as f, but is only defined on A. If the function f is thought of as a relation (x,f(x)) on the Cartesian product E \times F, then the restriction of f to A can be represented by its graph, :G(_A) = \ = G(f)\cap (A\times F), where the pairs (x,f(x)) represent ordered pairs in the graph G. Extensions A function F is said to be an ' of another function f if whenever x is in the domain of f then x is also in the domain of F and f(x) = F(x). That is, if \operatorname f \subseteq \operatorname F and F\big\vert_ = f. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mystic Moods Orchestra
The Mystic Moods Orchestra was a group known for mixing orchestral pop, environmental sounds, and pioneering recording techniques. It was created by audiophile Brad Miller. The first Mystic Moods Orchestra album, ''One Stormy Night'', was released in 1966 through the label Philips. Throughout the rest of the 1960s and 1970s, the group continued to release similar styled recordings and their recordings continued to be reissued throughout the 1980s and 1990s. History Brad Miller was born in Burbank, California, and had developed an interest in railroading in his teens. After a few years of hanging around railyards and learning all the lore of steam and diesel engines, he decided to record the sounds of some of the last steam locomotives operating on a major rail line. Eventually, around 1958, he and his friend, Jim Connella, formed a company called Mobile Fidelity Records and started cutting records from these field recordings, which they released through railroading magazines and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extensible Cardinal
In mathematics, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe of sets start to look similar, in the sense that each is elementarily embeddable into a later one. Definition For every ordinal ''η'', a cardinal κ is called η-extendible if for some ordinal ''λ'' there is a nontrivial elementary embedding ''j'' of ''V''κ+η into ''V''λ, where ''κ'' is the critical point of ''j'', and as usual ''Vα'' denotes the ''α''th level of the von Neumann hierarchy. A cardinal ''κ'' is called an extendible cardinal if it is ''η''-extendible for every nonzero ordinal ''η'' (Kanamori 2003). Properties For a cardinal \kappa, say that a logic L is \kappa-compact if for every set A of L-sentences, if every subset of A or cardinality <\kappa has a model, then has a model. (The usual |
Extension (semantics)
In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension (logic), comprehension or intension, which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question. In philosophical semantics or the philosophy of language, the 'extension' of a concept or expression is the set of things it extends to, or applies to, if it is the sort of concept or expression that a single object by itself can satisfy. Concepts and expressions of this sort are monad (Greek philosophy), monadic or "one-place" concepts and expressions. So the extension of the word "dog" is the set of all (past, present and future) dogs in the world: the set includes Fido, Rover, Lassie, Rex, and so on. The extension of the ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
