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Extensions And Generalizations Of Graphs
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 * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Extensionality
The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. The axiom defines what a Set (mathematics), set is. Informally, the axiom means that the two set (mathematics), sets ''A'' and ''B'' are equal if and only if ''A'' and ''B'' have the same members. Etymology The term ''extensionality'', as used in '''Axiom of Extensionality has its roots in logic. An intensional definition describes the necessary and sufficient conditions for a term to apply to an object. For example: "An even number is an integer which is divisible by 2." An extensional definition instead lists all objects where the term applies. For example: "An even number is any one of the following integers: 0, 2, 4, 6, 8..., -2, -4, -6, -8..." In logic, the Extension (logic), extension of a Predicate (mathematical logic), predicate is the set of all things for which the predicate is true. The logical term was introduce ... [...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] |
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] |
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] |
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] |
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] |
Extension (George Braith Album)
''Extension'' is the third album by American saxophonist George Braith recorded in 1964 and released on the Blue Note label.Blue Note Records discography accessed November 17, 2010 Reception The review by Stephen Thomas Erlewine awarded the album 4½ stars and stated "George Braith turned in his strongest record with ''Extension''. Largely freed from the restraints of the dueling horns, Braith is able to explore the outer reaches of his music. He still remains grounded in soul-jazz -- any guitar-organ combo is bound to have soul-jazz roots -- but he pushes the music toward adventurous hard bop, often with rewarding results".[...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] |
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] |
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] |
