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Tensor Product Of Algebras
In mathematics, the tensor product of two algebras over a commutative ring ''R'' is also an ''R''-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the product of algebra representations. Definition Let ''R'' be a commutative ring and let ''A'' and ''B'' be ''R''-algebras. Since ''A'' and ''B'' may both be regarded as ''R''-modules, their tensor product :A \otimes_R B is also an ''R''-module. The tensor product can be given the structure of a ring by defining the product on elements of the form by :(a_1\otimes b_1)(a_2\otimes b_2) = a_1 a_2\otimes b_1b_2 and then extending by linearity to all of . This ring is an ''R''-algebra, associative and unital with the identity element given by . where 1''A'' and 1''B'' are the identity elements of ''A'' and ''B''. If ''A'' and ''B'' are commutative, then the tensor product is commutative as well. The tensor product turns the category of ''R''-alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multilinear Subspace Learning
Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing 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." The Dimensionality reduction can be performed on a data tensor that contains a collection of observations that have been vectorized, or observations that are treated as matrices and concatenated into a data tensor.X. He, D. Cai, P. NiyogiTensor s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearly Disjoint
In mathematics, algebras ''A'', ''B'' over a field ''k'' inside some field extension \Omega of ''k'' are said to be linearly disjoint over ''k'' if the following equivalent conditions are met: *(i) The map A \otimes_k B \to AB induced by (x, y) \mapsto xy is injective. *(ii) Any ''k''- basis of ''A'' remains linearly independent over ''B''. *(iii) If u_i, v_j are ''k''-bases for ''A'', ''B'', then the products u_i v_j are linearly independent over ''k''. Note that, since every subalgebra of \Omega is a domain, (i) implies A \otimes_k B is a domain (in particular reduced). Conversely if ''A'' and ''B'' are fields and either ''A'' or ''B'' is an algebraic extension of ''k'' and A \otimes_k B is a domain then it is a field and ''A'' and ''B'' are linearly disjoint. However, there are examples where A \otimes_k B is a domain but ''A'' and ''B'' are not linearly disjoint: for example, ''A'' = ''B'' = ''k''(''t''), the field of rational functions over ''k''. One also has: ''A'', ''B'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Fields
In mathematics, the tensor product of two field (mathematics), fields is their tensor product of algebras, tensor product as algebra over a field, algebras over a common subfield (mathematics), subfield. If no subfield is explicitly specified, the two fields must have the same characteristic (algebra), characteristic and the common subfield is their prime field, prime subfield. The tensor product of two fields is sometimes a field, and often a product ring, direct product of fields; In some cases, it can contain non-zero nilpotent elements. The tensor product of two fields expresses in a single structure the different way to embed the two fields in a common extension field. Compositum of fields First, one defines the notion of the compositum of fields. This construction occurs frequently in field theory (mathematics), field theory. The idea behind the compositum is to make the smallest field containing two other fields. In order to formally define the compositum, one must firs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension Of Scalars
In algebra, a change of rings is an operation of changing a coefficient ring to another. Constructions Given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a right ''R''-module ''M'' and a right ''S''-module ''N'', one can form *f_! M = M\otimes_R S, the induced module, formed by extension of scalars, *f_* M = \operatorname_R(S, M), the coinduced module, formed by co-extension of scalars, and *f^* N = N_R, formed by restriction of scalars. They are related as adjoint functors: :f_! : \text_R \leftrightarrows \text_S : f^* and :f^* : \text_S \leftrightarrows \text_R : f_*. This is related to Shapiro's lemma. Operations Restriction of scalars Throughout this section, let R and S be two rings (they may or may not be commutative, or contain an identity), and let f:R \to S be a homomorphism. Restriction of scalars changes ''S''-modules into ''R''-modules. In algebraic geometry, the term "restriction of scalars" is ofte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded-commutative Ring
In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy :xy = (-1)^ yx, where , ''x'', and , ''y'', denote the degrees of ''x'' and ''y''. A commutative (non-graded) ring, with trivial grading, is a basic example. For a nontrivial example, an exterior algebra is generally not a commutative ring but is a ''graded''-commutative ring. A cup product on cohomology satisfies the skew-commutative relation; hence, a cohomology ring is graded-commutative. In fact, many examples of graded-commutative rings come from algebraic topology and homological algebra. References * David Eisenbud, ''Commutative Algebra. With a view toward algebraic geometry'', Graduate Texts in Mathematics, vol 150, Springer-Verlag, New York, 1995. * See also * DG algebra * graded-symmetric algebra *alternating algebra In mathematics, an alternating algebra is a -graded algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Product
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is written :. Usually the morphisms and are omitted from the notation, and then the pullback is written :. The pullback comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square. The dual concept of the pullback is the '' pushout''. Universal property Explicitly, a pullback of the morphisms f and g consists of an object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenization of a polynomial, homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse function, inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. If the defining polynomial of a plane algebraic curve is irreducible polynomial, irreducible, then one has an ''irreducible plane algebraic curve''. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its ''Irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise '' Éléments de géométrie algébrique'' (EGA); one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Schemes elaborate the fundamental idea that an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
