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Subspace Lattice
Subspace may refer to: In mathematics * A space inheriting all characteristics of a parent space * A subset of a topological space endowed with the subspace topology * Linear subspace, in linear algebra, a subset of a vector space that is closed under addition and scalar multiplication * Flat (geometry), a Euclidean subspace * Affine subspace, a geometric structure that generalizes the affine properties of a flat * Projective subspace, a geometric structure that generalizes a linear subspace of a vector space * Multilinear subspace in multilinear algebra, a subset of a tensor space that is closed under addition and scalar multiplication In science fiction * a concept similar to that of hyperspace In science fiction, hyperspace (also known as nulspace, subspace, overspace, jumpspace and similar terms) is a concept relating to higher dimensions as well as parallel universes and a faster-than-light (FTL) method of interstellar travel. ... * Subspace (''Star Trek''), a fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space (mathematics)
In mathematics, a space is a set (sometimes called a universe) with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can be elements of a set, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships between the points of a three-dimensional Euclidean space are uniquely determined by Euclid's axioms, and all three-dimensional Euclidean spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subspace Topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced topology, or the trace topology). Definition Given a topological space (X, \tau) and a subset S of X, the subspace topology on S is defined by :\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace. That is, a subset of S is open in the subspace topology if and only if it is the intersection of S with an open set in (X, \tau). If S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of (X, \tau). Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated. Alternatively we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map :\iota: S \hookrightarrow X is continuous. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, linear subspaces, flats, and affine subspaces are also called ''linear manifolds'' for emphasizing that there are also manifolds. is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a ''subspace'' when the context serves to distinguish it from other types of subspaces. Definition If ''V'' is a vector space over a field ''K'' and if ''W'' is a subset of ''V'', then ''W'' is a linear subspace of ''V'' if under the operations of ''V'', ''W'' is a vector space over ''K''. Equivalently, a nonempty subset ''W'' is a subspace of ''V'' if, whenever are elements of ''W'' and are elements of ''K'', it follows that is in ''W''. As a corollary, all vector spaces are equipped with at least two ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat (geometry)
In geometry, a flat or Euclidean subspace is a subset of a Euclidean space that is itself a Euclidean space (of lower dimension). The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes. In a -dimensional space, there are flats of every dimension from 0 to ; flats of dimension are called ''hyperplanes''. Flats are the affine subspaces of Euclidean spaces, which means that they are similar to linear subspaces, except that they need not pass through the origin. Flats occur in linear algebra, as geometric realizations of solution sets of systems of linear equations. A flat is a manifold and an algebraic variety, and is sometimes called a ''linear manifold'' or ''linear variety'' to distinguish it from other manifolds or varieties. Descriptions By equations A flat can be described by a system of linear equations. For example, a line in two-dimensional space can be described by a single linear equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Subspace
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead '' displacement vectors'', also called ''translation'' vectors or simply ''translations'', between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be viewed as an affine spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multilinear Subspace
Multilinear subspace learning is an approach to dimensionality reduction.M. A. O. Vasilescu, D. Terzopoulos (2003"Multilinear Subspace Analysis of Image Ensembles" "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI, June, 2003"M. A. O. Vasilescu, D. Terzopoulos (2002"Multilinear Analysis of Image Ensembles: TensorFaces" Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002M. A. O. Vasilescu,(2002"Human Motion Signatures: Analysis, Synthesis, Recognition" "Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460." Dimensionality reduction can be performed on a data tensor that contains a collection of observations have been vectorized, or observations that are treated as matrices and concatenated into a data tensor.X. He, D. Cai, P. NiyogiTensor subspace analysis in: Advances in Neural Information Processing Systemsc 18 (N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperspace
In science fiction, hyperspace (also known as nulspace, subspace, overspace, jumpspace and similar terms) is a concept relating to dimension#Additional dimensions, higher dimensions as well as parallel universes in fiction, parallel universes and a faster-than-light (FTL) method of interstellar travel. Its use in science fiction originated in the magazine ''Amazing Stories Quarterly'' in 1931 and within several decades it became one of the most popular Trope (literature), tropes of science fiction, popularized by its use in the works of authors such as Isaac Asimov and Edwin Charles Tubb, E. C. Tubb, and media franchises such as ''Star Wars''. One of the main reasons for the popularity of the concept is the prohibition against faster-than-light travel in ordinary space, which hyperspace allows writers to bypass. In most works, hyperspace is described as a higher dimension through which the shape of our three-dimensional space can be distorted to bring distant points close to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subspace (Star Trek)
The technology in ''Star Trek'' has borrowed many ideas from the scientific world. Episodes often contain technologies named after real-world scientific phenomena, such as tachyon beams, baryon sweeps, quantum slipstream drives, and photon torpedoes. Some of the technologies created for the ''Star Trek'' universe were done so out of financial necessity. For instance, the transporter was created because the limited budget of ''Star Trek: The Original Series'' (''TOS'') in the 1960s did not allow expensive shots of spaceships landing on planets. ''Discovery Channel Magazine'' stated that cloaking devices, faster-than-light travel, and dematerialized transport were only dreams at the time ''TOS'' was made, but physicist Michio Kaku believes all these things are possible. William Shatner, who portrayed James T. Kirk in ''TOS'', believes this as well, and went on to co-write the book ''I'm Working on That'', in which he investigates how ''Star Trek'' technology was becoming feasibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SubSpace (video Game)
''SubSpace'' is a 2D computer graphics, 2D space shooter video game created in 1995 and released in 1997 by Virgin Interactive which was a finalist for the Academy of Interactive Arts & Sciences Online Game of the Year Award in 1998. ''SubSpace'' incorporates quasi-realistic zero-friction physics into a massively multiplayer online game. It is no longer operated by VIE; instead, fans and players of the game provide servers and technical updates. The action is viewed from above, which presents challenges very different from those of a 3D computer graphics, three-dimensional game. The game has no built-in story or set of goals; players may enter a variety of server (computing), servers, each of which have differing objectives, maps, sounds, and graphics. ''SubSpace'' is considered an early entry in the massively multiplayer online game, massively multiplayer online genre due to its unprecedented player counts. History ''SubSpace'' evolved from a game originally called ''Sniper'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Subspace Emissary
''Super Smash Bros. Brawl'' is a 2008 crossover fighting video game developed by Sora Ltd. and published by Nintendo for the Wii. The third installment in the ''Super Smash Bros.'' series, it was announced at a pre-E3 2005 press conference by Nintendo president Satoru Iwata. Masahiro Sakurai, director of the previous two games in the series, assumed the role of director at Iwata's request. Game development began in October 2005 with a creative team that included members from several Nintendo and third-party development teams. After delays due to development problems, the game was released worldwide in 2008. The number of playable characters in ''Brawl'' has grown from that in ''Super Smash Bros. Melee'', although some characters from ''Melee'' were cut in ''Brawl''. ''Brawl'' is the first game in the series to have playable third-party characters. Like that of its predecessors, the objective of ''Brawl'' is to knock opponents off the screen. It is a departure from traditiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subspace (song)
Subspace may refer to: In mathematics * A space inheriting all characteristics of a parent space * A subset of a topological space endowed with the subspace topology * Linear subspace, in linear algebra, a subset of a vector space that is closed under addition and scalar multiplication * Flat (geometry), a Euclidean subspace * Affine subspace, a geometric structure that generalizes the affine properties of a flat * Projective subspace, a geometric structure that generalizes a linear subspace of a vector space * Multilinear subspace Multilinear subspace learning is an approach to dimensionality reduction.M. A. O. Vasilescu, D. Terzopoulos (2003"Multilinear Subspace Analysis of Image Ensembles" "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVP ... in multilinear algebra, a subset of a tensor space that is closed under addition and scalar multiplication In science fiction * a concept similar to that of hyperspace * Subspace (''Star Trek''), a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
