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Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
General existence principles for nonlocal boundary value problems with φLaplacian and their applications
"... The paper presents general existence principles which can be used for a large class of nonlocal boundary value problems of the form (φ(x′)) ′ = f1(t,x,x′) + f2(t,x,x′)F1x + f3(t,x,x′)F2x,α(x) = 0, β(x) = 0, where f j satisfy local Carathéodory conditions on some [0,T]× j ⊂ R2, f j are either regu ..."
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The paper presents general existence principles which can be used for a large class of nonlocal boundary value problems of the form (φ(x′)) ′ = f1(t,x,x′) + f2(t,x,x′)F1x + f3(t,x,x′)F2x,α(x) = 0, β(x) = 0, where f j satisfy local Carathéodory conditions on some [0,T]× j ⊂ R2, f j are either regular or have singularities in their phase variables ( j = 1,2,3), Fi: C1[0,T] → C0[0,T] (i = 1,2), and α,β: C1[0,T] → R are continuous. The proofs are based on the LeraySchauder degree theory and use regularization and sequential techniques. Applications of general existence principles to singular BVPs are given. Copyright © 2006 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.