Tensor Product of Massey Products

In this paper, we interpret Massey products in terms of realizations （twitsting cochains） of certain differential graded coalgebras with values in differential graded algebras. In the case where the target algebra is the cobar construction of a differential graded commutative Hopf algebra, we construct the tensor product of realizations and show that the tensor product is strictly associative, and commutative up to homotopy.     

Regularity of Tensor Product Approximations to?Square Integrable Functions

We investigate first-order conditions for canonical and optimal subspace (Tucker format) tensor product approximations to square integrable functions. They reveal that the best approximation and all of its factors have the same smoothness as the approximated function itself. This is not obvious, since the approximation is performed in L ².     

Tensor Product Weight Representations of the Neveu–Schwarz Algebra

We study the tensor product of a highest weight module with an intermediate series module over the Neveu–Schwarz algebra. If the highest weight module is nontrivial, the weight spaces of such a tensor product are infinite dimensional. We show that such a tensor product is indecomposable. Using a “shifting technique” developed by H. Chen, X. Guo, and K. Zhao for the Virasoro algebra case, we give necessary and sufficient conditions for such a tensor product to be irreducible. Furthermore, we give necessary and sufficient conditions for two such tensor products to be isomorphic.     

Graded Limits of Simple Tensor Product of Kirillov–Reshetikhin Modules for

We study graded limits of simple -modules which are isomorphic to tensor products of Kirillov–Reshetikhin modules associated to a fixed fundamental weight. We prove that every such module admits a graded limit which is isomorphic to the fusion product of the graded limits of its tensor factors. Moreover, using recent results of Naoi, we exhibit a set of defining relations for these graded limits.     

The Characterizations of Tensor Product Functors and Their Derived Functors of Left Modules

In 1979, Zhou Boxun introduced concepts of the tensor product of left modules, so that there exist tensor products of left modules which is an extension of the tensor products of modules over commutative rings. In [2], it was proved that - M preserves direct limits and (-M, Horn(M, -)) is an adjoint pair.In this paper, we characterize functor -M and its derived functors. These results can be consider as extensions of Watts′theorems     

Some Invariant Properties of Semi-symmetric Metric Recurrent Connections and Curvature Tensor Expressions

The projective transformation of the special semi-symmetric metric recurrent connection is studied in this paper. First of all, an invariant under this transformation is granted; Secondly, by inducing of the invariant and making use of the properties that the corresponding covariant derivative keeps being fixed under the distinctness connection, the curvature tensor expression of the Riemannian manifold is posed at the same time.     

Generalized Tractability for Multivariate Problems Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complex. 23:262–295, 2007. We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approximate an operator S _d given as the d-fold tensor product of a compact linear operator S ₁ for d=1,2,…, with ‖S ₁‖=1 and S ₁ having at least two positive singular values. Let n(ε,S _d ) be the minimal number of information evaluations needed to approximate S _d to within ε∈[0,1]. We study generalized tractability by verifying when n(ε,S _d ) can be bounded by a multiple of a power of T(ε ⁻¹,d) for all (ε ⁻¹,d)∈Ω⊆[1,∞)×ℕ. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T(ε ⁻¹,d) whose multiple bounds n(ε,S _d ). We also study weak tractability, i.e., when . In our previous paper, we studied generalized tractability for proper subsets Ω of [1,∞)×ℕ, whereas in this paper we take the unrestricted domain Ω ^unr=[1,∞)×ℕ. We consider the three cases for which we have only finitely many positive singular values of S ₁, or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S ₁ decay slightly faster than logarithmically. We provide necessary and sufficient conditions on the function T such that generalized tractability holds. These conditions are obtained in terms of the singular values of S ₁ and mostly asymptotic properties of T. The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T(x,y)=x ^1+ln y . We also study tractability functions T of product form, T(x,y)=f ₁(x)f ₂(x). Assume that a _i =lim inf  _x→∞(ln ln f _i (x))/(ln ln x) is finite for i=1,2. Then generalized tractability takes place iff

The Killing–Yano Tensor

We obtain a general solution of the equations determining the Killing–Yano tensor of rank p on an n-dimensional (1 p n – 1) pseudo-Riemannian manifold of constant curvature and discuss possible applications of the obtained result.     

Über die Tensorübertragung

Zusammenfassung Eine Zusammenstellung der bisherigen Ergebnisse in der Theorie der Tensorübertragugen samt einigen eigenen Resultaten die haupts?chlich in den Paragraphen 5 und 6 enthalten sind. Enrico Bompiani zu seinem wissenschaftlichen Jubil?um     

Product Ł ukasiewicz Logic

u logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of u logic by adding a new connective which expresses multiplication. The resulting logic, P, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of P logic is introduced and developed too.The work of the first author was supported by the Grant Agency of the Czech Republic under project GACR 201/02/1540, by the Grant Agency of the Czech Technical University in Prague under project CTU 0208613, and by Net CEEPUS SK-042.The work of the second author was supported by grant IAA1030004 of the Grant Agency of the Academy of Sciences of the Czech Republic.     

Branching Rules, Kostka–Foulkes Polynomials and q-multiplicities in Tensor Product for the Root Systems B n , C n and D n

The Kostka–Foulkes polynomials related to a root system can be defined as alternating sums running over the Weyl group associated to . By restricting these sums over the elements of the symmetric group when is of type or , we obtain again a class of Kostka–Foulkes polynomials. When is of type or there exists a duality between these polynomials and some natural -multiplicities and in tensor products [11]. In this paper we first establish identities for the which implies in particular that they can be decomposed as sums of Kostka–Foulkes polynomials with nonnegative integer coefficients. Moreover these coefficients are branching coefficients This allows us to clarify the connection between the -multiplicities and the polynomials defined by Shimozono and Zabrocki. Finally we show that and coincide up to a power of with the one dimension sum introduced by Hatayama and co-workers when all the parts of are equal to , which partially proves some conjectures of Lecouvey and Shimozono and Zabrocki.Presented by P. Littelmann.     

Tensor Products of Approximately Cohen–Macaulay Rings

Our aim in this article is to study a problem originally raised by Grothendieck. We show that the approximately Cohen–Macaulay property is preserved for the tensor product of algebras over a field k. We also discuss the converse problem.     

Kronecker Product of Two-dimensional Arrays

Kronecker sequences constructed from short sequences are good sequences for spread spectrum communication systems. In this paper we study a similar problem for two-dimensional arrays, and we determine the linear complexity of the Kronecker product of two arrays, Our result shows that similar good property on linear complexity holds for Kronecker product of arrays.     

Smash Product的循环同调

本文讨论了群分次代数A与其SmashProductA#G的循环同调群之间的关系,并且在A分别是有限分次,强分次,以及非负分次的情形下得出一些刻划两者内在联系的结论.     

Smash Product的循环同调

本文讨论了群分次代数A与其SmashProductA＃G的循环同调群之间的关系，并且在A分别是有限分次，强分次，以及非负分次的情形下得出一些刻划两者内在联系的结论。     

Product Representations in ΠΣ-Fields

-fields are a very general class of difference fields that enable one to discover and prove multisum identities arising in combinatorics and special functions. In this article we focus on the problem how such multisums can be represented in terms of -fields. In particular we consider product representations and their simplifications in -fields.Received July 09, 2003     

Product of matrical discs ∗

We determine the matrical disc with minimal radius for the set of products XY when X and Y run over a (possibly distinct) matrical disc, respectively. Also we determine the union of the numerical ranges W(XY).     

Tensor products, multiplications and Weyl’s theorem

Tensor productsZ=T ₁⊗T ₂ and multiplicationsZ=L _{T 1} R _{T 2} do not inherit Weyl’s theorem from Weyl’s theorem forT ₁ andT ₂. Also, Weyl’s theorem does not transfer fromZ toZ*. We prove that ifT _i,i=1, 2, has SVEP (=the single-valued extension property) at points in the complement of the Weyl spectrumσ _w(T_i) ofT _i, and if the operatorsT _i are Kato type at the isolated points ofσ(T_i), thenZ andZ* satisfy Weyl’s theorem.     

On Direct Product of HX Groups

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