Groupoid normalizers of tensor products

We consider an inclusion B⊆M of finite von Neumann algebras satisfying B^′∩M⊆B. A partial isometry v∈M is called a groupoid normalizer if vBv^∗,v^∗Bv⊆B. Given two such inclusions Bi⊆Mi, i=1,2, we find approximations to the groupoid normalizers of in , from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis , i=1,2. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries v∈M satisfying vBv^∗⊆B and v^∗v,vv^∗∈B.     

Atomic decompositions for tensor products and polynomial spaces

We study the existence of atomic decompositions for tensor products of Banach spaces and spaces of homogeneous polynomials. If a Banach space X admits an atomic decomposition of a certain kind, we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product , for any symmetric tensor norm μ. In addition, the reciprocal statement is investigated and analogous consequences for the full tensor product are obtained. Finally we apply the previous results to establish the existence of monomial atomic decompositions for certain ideals of polynomials on X.     

Hamilton cycle decompositions of the tensor products of complete bipartite graphs and complete multipartite graphs

In this paper, it is shown that the tensor product of the complete bipartite graph, K_r,r,r≥2, and the regular complete multipartite graph, , is Hamilton cycle decomposable.     

Note on the wave equation and tensor products

In this study we apply convolution and tensor products of distribution to solve the non-homogenous wave equation with initial condition and discuss the uniqueness and continuity of solution. We also show that the tensor product can be applied to compute the some singular integrals.     

Weyl's theorem, tensor products and multiplication operators

The transmission of “Weyl's theorem” from operators on Banach spaces to their tensor products, and also to their associated multiplication operators, is deconstructed.     

Pointwise products of some Banach function spaces and factorization

The well-known factorization theorem of Lozanovski? may be written in the form L¹≡E⊙E^′$L^1\equiv E\odot E^{\prime }$, where ⊙ means the pointwise product of Banach ideal spaces. A natural generalization of this problem would be the question when one can factorize F through E  , i.e., when F≡E⊙M(E,F)$F\equiv E\odot M(E,F)$, where M(E,F)$M(E,F)$ is the space of pointwise multipliers from E to F  . Properties of M(E,F)$M(E,F)$ were investigated in our earlier paper [41] and here we collect and prove some properties of the construction E⊙F$E\odot F$. The formulas for pointwise product of Calderón–Lozanovski? _Eφ$E_{\phi }$-spaces, Lorentz spaces and Marcinkiewicz spaces are proved. These results are then used to prove factorization theorems for such spaces. Finally, it is proved in Theorem 11 that under some natural assumptions, a rearrangement invariant Banach function space may be factorized through a Marcinkiewicz space.     

Fractional chromatic numbers of tensor products of three graphs

The tensor product $(G_1,G_2,G_3)$ of graphs $G_1$, $G_2$ and $G_3$ is defined by $V(G_1,G_2,G_3)=V\left(G_1\right)\times V\left(G_2\right)\times V\left(G_3\right)$and $E(G_1,G_2,G_3)=\left\{\left((u_1,u_2,u_3),(v_1,v_2,v_3)\right):\left|\{i:(u_i,v_i)\in E\left(G_i\right)\}\right|\ge 2\right\}.$Let ${\chi }_f\left(G\right)$ be the fractional chromatic number of a graph $G$. In this paper, we prove that if one of the three graphs $G_1$, $G_2$ and $G_3$ is a circular clique, ${\chi }_f(G_1,G_2,G_3)=min\{{\chi }_f\left(G_1\right){\chi }_f\left(G_2\right),{\chi }_f\left(G_1\right){\chi }_f\left(G_3\right),{\chi }_f\left(G_2\right){\chi }_f\left(G_3\right)\}.$     

Interpolation of Banach algebras and tensor products of Banach couples

Using tensor products of Banach couples we study a class of interpolation functors with the property that to every Banach couple of Banach algebras they give an interpolation space which is a Banach algebra. For the real θ,1-method we give a complete answer to the question of when the interpolation space is unital.     

Cn中单位球上两个全纯函数空间之间的点乘子

在Cn中单位球上讨论了加权Bergman空间Aαp和Bloch型空间βq之间的点乘子.根据α,p,q的不同情况,得到了Aαp空间到βq空间的所有点乘子,并研究了βq空间到Apα空间的点乘子的性质.     

Weyl's theorem and tensor products: A counterexample

Approximately fifty percent of Weyl's theorem fails to transfer from Hilbert space operators to their tensor product. As a biproduct we find that the product of circles in the complex plane is a limaçon.     

Second-order conditions for an exact penalty function

In this paper we give first- and second-order conditions to characterize a local minimizer of an exact penalty function. The form of this characterization gives support to the claim that the exact penalty function and the nonlinear programming problem are closely related.In addition, we demonstrate that there exist arguments for the penalty function from which there are no descent directions even though these points are not minimizers.This research is partially supported by the Natural Science and Engineering Research Council Grant No. A8639 and the U.S. Department of Energy.     

Nonlinear tensor product approximation of functions

We are interested in approximation of a multivariate function $f(x_1,\dots ,x_d)$ by linear combinations of products $u^1\left(x_1\right)\cdots u^d\left(x_d\right)$ of univariate functions $u^i\left(x_i\right)$, $i=1,\dots ,d$. In the case $d=2$ it is the classical problem of bilinear approximation. In the case of approximation in the $L_2$ space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel $f(x_1,x_2)$. There are known results on the rate of decay of errors of best bilinear approximation in $L_p$ under different smoothness assumptions on $f$. The problem of multilinear approximation (nonlinear tensor product approximation) in the case $d\ge 3$ is more difficult and much less studied than the bilinear approximation problem. We will present results on best multilinear approximation in $L_p$ under mixed smoothness assumption on $f$.     

Optimal cubature formulas for tensor products of certain classes of functions

We study the problem of constructing an optimal formula of approximate integration along a d-dimensional parallelepiped. Our construction utilizes mean values along intersections of the integration domain with n hyperplanes of dimension (d−1), each of which is perpendicular to some coordinate axis. We find an optimal cubature formula of this type for two classes of functions. The first class controls the moduli of continuity with respect to all variables, whereas the second class is the intersection of certain periodic multivariate Sobolev classes. We prove that all node hyperplanes of the optimal formula in each case are perpendicular to a certain coordinate axis and are equally spaced and the weights are equal. For specific moduli of continuity and for sufficiently large n, the formula remains optimal for the first class among cubature formulas with arbitrary positions of hyperplanes.     

Resolvable even cycle decompositions of the tensor product of complete graphs

In this paper, we consider resolvable k-cycle decompositions (for short, k-RCD) of K_m×K_n, where × denotes the tensor product of graphs. It has been proved that the standard necessary conditions for the existence of a k-RCD of K_m×K_n are sufficient when k is even.     

Hamilton cycle decompositions of the tensor product of complete multipartite graphs

In this paper, tensor product of two regular complete multipartite graphs is shown to be Hamilton cycle decomposable. Using this result, it is immediate that the tensor product of two complete graphs with at least three vertices is Hamilton cycle decomposable thereby providing an alternate proof of this fact.     

On the tensor algebra of a non-abelian group and applications

We introduce the tensor algebra J() of a crossed complex and give its relations with the James construction for a filtered space. For a group G, we define via multi-derivations the derived algebra I _* G which is isomorphic to the algebra C(JG) of chains on JG We derive from a certain exact sequence for the homology H*JG applications to the homotopy of the suspended classifying space BG.     

Pointwise multipliers for localized morrey-campanato spaces on RD-spaces

In this article, the authors characterize pointwise multipliers for localized Morrey-Campanato spaces, associated with some admissible functions on RD-spaces, which include localized BMO spaces as a special case. The results obtained are applied to Schrödinger operators and some Laguerre operators.