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# Topic: Tensor product

 Tensor product - Wikipedia, the free encyclopedia It means that if a pair of tensors are juxtaposed (placed side-by-side) then they combine by mere aggregation to form a new tensor which can be subsequently called the tensor product of the pair of juxtaposed tensors. The tensor product inherits all the indices of its factors. In J the tensor product is the dyadic form of */ (for example a */ b or a */ b */ c). en.wikipedia.org /wiki/Tensor_product   (1132 words)

 PlanetMath: tensor product (vector spaces) Historically, the tensor product was called the outer product, and has its origins in the absolute differential calculus (the theory of manifolds). This relationship is the source of the terminology contravariant tensor'' and contravariant index'', and I surmise that it is this very medieval pit of darkness and confusion that spawned the present-day notion of contravariant functor''. This is version 7 of tensor product (vector spaces), born on 2002-02-18, modified 2005-08-02. planetmath.org /encyclopedia/TensorProductClassical.html   (594 words)

 Tensor product - Encyclopedia, History, Geography and Biography In mathematics, the tensor product, denoted by $\otimes$, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. The tensor product $V \otimes W$ of two vector spaces V and W has a formal definition by the method of generators and relations. The resulting tensor product $V \otimes W$ is itself a vector space, which can be verified by directly checking the vector space axioms.Given bases $\\left\{v_i\\right\}$ and $\\left\{w_i\\right\}$ for V and W respectively, the tensors of the form $v_i \otimes w_j$forms a basis for $V \otimes W$. www.arikah.com /encyclopedia/Tensor_product   (1457 words)

 Tensor product of fields - Wikipedia, the free encyclopedia In mathematics, the theory of fields in abstract algebra lacks a direct product: the direct product of two fields, considered as ring is never itself a field. One can prove this by calculating the dimension of the tensor product over Q as 9, and observing that the splitting field does contain two (indeed three) copies of K, and is the compositum of two of them. In algebraic number theory, tensor products of fields are (implicitly, often) a basic tool. en.wikipedia.org /wiki/Tensor_product_of_fields   (680 words)

 Encyclopedia: Tensor product The tensor product in fact satifies the universal property of being a fibered coproduct. Because the tensor product is left exact, not exact, this may not be a minimal generating set, even if the original generating sets are minimal. When defining the tensor product, we need to be careful about the ring: most modules can be considered as modules over several different rings or over the same ring with a different actions of the ring on the module elements. www.nationmaster.com /encyclopedia/Tensor-product   (445 words)

 A   (Site not responding. Last check: 2007-10-06) the vector product of pseudo-vectors is a pseudo-vector. the vector product of a pseudo-tensor and a psuedo-vector is a pseudo-tensor. product of a vector and its inverse yields the identity vector. www.cchem.berkeley.edu /jsngrp/Che250/HW1.htm   (958 words)

 TENSOR PRODUCT The dimension of the tensor product therefore is the product of dimensions. Universal property of tensor product: The space of all multilinear maps from V xW to R is naturally isomorphic to the space of all linear maps from V ⊗W to R. Tensor spaces allow us to use the theory of linear operators to study multi-linear operators, and this says the bilinear case is the main hurdle. www.websters-online-dictionary.org /definition/TENSOR+PRODUCT   (1467 words)

 THE MEANING OF "TENSOR" These are more general than the tensor product which is very often reserved on certain occasions to be direct products of linear (vector) spaces. Roughly, according to a classical theorem (Levi's theorem) on the subject, any Lie group is a semidirect product of a solvable algebra by a semisimple algebra, meaning that the structure problem is broken into determining the structure problems for solvable and semisimple algebras. Tensor Product The tensor product grows from the idea of the "Cartesian Product" of the one dimensional x-axis as a vector space, and the one dimensional y-axis in elementary algebra to form the two dimensional Cartesian plane. graham.main.nc.us /~bhammel/MATH/tensor.html   (1952 words)

 BME 332: Mathematical Preliminaries This means that the product of 1 cross 2 is a positive 3, the product of 2 cross 3 is a positive 1 and the product of 3 cross 1 is a positive 2 going clockwise. The product of 2 cross 1 is a negative 3, the product of 1 cross 3 is a negative 2 and the product of 3 cross 2 is a negative 1. Tensors may be thought of mathematically as linear operators that act on either a vector or tensor to generate another vector or tensor. www.engin.umich.edu /class/bme332/ch1mathprelim/bme332mathprelim.htm   (5215 words)

 Tensor product of algebras - Wikipedia, the free encyclopedia In mathematics, the tensor product of two R-algebras is also an R-algebra in a natural way. There the coproduct is given by a more general free product of algebras). The tensor product of algebras is of constant use in algebraic geometry: working in the opposite category to that of commutative R-algebras, it provides pullbacks of affine schemes, otherwise known as fiber products. en.wikipedia.org /wiki/Tensor_product_of_R-algebras   (264 words)

 PlanetMath: tensor product The tensor product is a formal bilinear multiplication of two modules or vector spaces. In essence, it permits us to replace bilinear maps from two such objects by an equivalent linear map from the tensor product of the two objects. This is version 5 of tensor product, born on 2002-02-17, modified 2005-08-11. planetmath.org /encyclopedia/TensorProduct.html   (256 words)

 Tensor product - explanation-Guide.info - for information, definition, meaning, reference - free encyclopedia, glossary ... In abstract algebra, the subject of linear algebra is upgraded to multilinear algebra by introducing the tensor product of two vector spaces. This functor is representable and the representing abelian group is the tensor product. Their tensor product over R is an abelian group P together with an R-bilinear operator T: M × N → P such that for every R-bilinear operator B: M × N → O there is a unique group homomorphism L: P → O such that L o T = B. explanation-guide.info /meaning/Tensor-product.html   (1763 words)

 PlanetMath: graded tensor product to be the ordinary tensor product as graded modules, but with multiplication - called the super product - defined by is itself a graded algebra, as we grade the super tensor product of This is version 5 of graded tensor product, born on 2002-06-07, modified 2004-04-30. planetmath.org /encyclopedia/SuperTensorProduct.html   (89 words)

 R: Define tensor product smooths in GAM formulae   (Site not responding. Last check: 2007-10-06) Smooths of several covariates can be constructed from tensor products of the bases used to represent smooths of one (or sometimes more) of the covariates. The basis dimension of the whole smooth is the product of the basis dimensions of the marginal smooths. Tensor product smooths are especially useful for representing functions of covariates measured in different units, although they are typically not quite as nicely behaved as t.p.r.s. www.oulu.fi /atkk/tkpalv/unix/R/library/mgcv/html/te.html   (524 words)

 Projects   (Site not responding. Last check: 2007-10-06) Tensor product bases having as best N term approximation spaces anisotropic Besov spaces are so far only known for cube-like domains. The use of tensor product wavelet schemes is a promising approach, since it allows for approximation rates which are up to logarithmic terms independent of the dimension. We show that tensor product wavelet bases are suitable for the approximation of elliptic singularities and give the maximal rates valid for smooth parts of the solution to be approximated. www.sam.math.ethz.ch /~andrej/projects.shtml   (480 words)

 Glossary of tensor theory - InfoSearchPoint.com   (Site not responding. Last check: 2007-10-06) Einstein summation convention This states that in a product of two indexed arrays, if an index letter in the first is repeated in the second, then the (default) interpretation is that the product is summed over all values of the index. This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol. The wedge product is the anti-symmetic form of the $\otimes$ operation. www.infosearchpoint.com /display/Glossary_of_tensor_theory   (552 words)

 Tensor Corporation, Provisional Acceptance of a Settlement Agreement and Order   (Site not responding. Last check: 2007-10-06) A halogen lamp is a consumer product'' and Tensor is a manufacturer'' of a consumer product'', which is distributed in commerce'' as those terms are defined in Sections 3 (a)(1),(4), (11) and (12) of the CPSA, 15 U.S.C. 2052 (a)(1),(4), (11) and (12). Tensor shall pay sixty-two thousand five hundred dollars ($62,500.00) within 10 calendar days of receiving service of the final Settlement Agreement and Order, and sixty-two thousand five hundred dollars ($62,500.00) to be paid no later than one year from the date on which this Settlement Agreement and Order became final. Tensor agrees to the entry of the attached Order, which is incorporated herein by reference, and agrees to be bound by its terms. www.cpsc.gov /businfo/frnotices/fr01/tensor.html   (1183 words)

 Tensor product space ANOVA models in multivariate function estimation To deal with the curse of dimensionality in high dimensional nonparametric problems, we consider using tensor product space ANOVA models, which extend the popular additive models and are able to capture interactions of any order. This means that the optimal rate for the tensor product space ANOVA models is very close to the optimal rate for one dimensional models. Thus in a sense, curse of dimensionality is overcome by the tensor product space ANOVA models. repository.upenn.edu /dissertations/AAI9829937   (289 words)

 [No title]   (Site not responding. Last check: 2007-10-06) This, however, does not mean that every vector in E is a product vector, only that it can be written as a linear combination of product vectors. Tensor products occur whenever a system has two or more independent degrees of freedom. For further discussion of the tensor product of state see Cohen-Tannoudji, chapter II F and Complement D III. electron6.phys.utk.edu /qm1/modules/m8/tensorpoduct.htm   (483 words)

 The Tensor of Two Functions Tensor a and b with a module m, and tensor a → b with the identity map from m onto m. Therefore, tensoring a → b with a direct sum m is the direct sum of the individual tensor products with m As a corollary, tensor with a freee module and obtain the direct sum of multiple instances of a → b. www.mathreference.com /mod-pit,tensfunc.html   (561 words)

 Tensor products Tensor products obey a number of nice rules. The tensor product of several matrices is unitary if and only if each one of the matrices is unitary up to a constant. This in turn implies (reminds us?) that the tensor product space is actually the equivalence classes in a quotient space. astarte.csustan.edu /~tom/MISC/qc-article/node3.html   (160 words)

 Tensor Product Networks Tensor product networks were applied early to the task of dealing with symbolic structures in connectionist systems, by Paul Smolensky. Notice that the outer product of two vectors is actually the Kronecker product of those vectors regarded as an n×1 matrix (column vector) and a 1×n matrix (row vector). This information can be represented, and stored in the tensor in the usual way, (putting the animal' in the side we have been calling variable', and the food' in the side we have been calling filler') and then we can try retrieving it. www.cse.unsw.edu.au /~billw/cs9444/tensor-stuff/tensor-intro-04.html   (3435 words)

 [No title] For tensor products of three and more vector spaces: another "abstract nonsense" fact can be proved, namely that (V ox W) ox Z is (explicitly, "canonically") isomorphic to V ox (W ox Z) so that we can drop the parentheses if we are not finicky about isomorphisms of this kind. Such a tensor can be regarded as an element of the tensor product of the duals of the corresponding vector spaces as above. A tensor field may *also* be regarded as a multilinear map on the vector space of differential functions and it's dual. www.math.niu.edu /~rusin/known-math/00_incoming/tensor   (1480 words)

 Levi-Civita symbol and cross product vector/tensor   (Site not responding. Last check: 2007-10-06) The cross product in 3 dimensions is actually a tensor of rank 2 with 3 independent coordinates. is the dimension of the vector, the cross product tensor The cross product is connected to rotations and has a structure which also looks like rotations, called a simplectic structure. folk.uio.no /patricg/teaching/a112/levi-civita   (293 words)

 [No title]   (Site not responding. Last check: 2007-10-06) For a simple Lie algebra g there is a ring homomorphism from the representation ring of its finite dimensional representations, to the ring of "level-restricted" representations that takes the usual tensor product to the fusion product. We are interested in a $q$-analogue of the tensor product that is compatible with the level restriction map; here by $q$-analogue we mean a grading or filtration. Affine crystal theory yields a natural grading of both the ordinary and fusion tensor product multiplicity, but at the cost of replacing the irreducible tensor factors with suitable representations (which outside of type A, no longer be irreducible). www-math.mit.edu /~combin/abstracts/mar99/shimozono.html   (179 words)

 A Tensor Product For Gray-Categories - Crans (ResearchIndex)   (Site not responding. Last check: 2007-10-06) A Tensor Product For Gray-Categories - Crans (ResearchIndex) I show that this tensor product of Gray-categories satisfies a universal property with respect to quasi-functors of two variables, which are defined in terms of lax-natural transformations... For Gray categories C and D, their tensor product C Omega D has as generators expressions c Omega d of dimension p q for c 2C p... citeseer.ist.psu.edu /59320.html   (659 words)

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