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Some Concrete Aspects Of Hilbert's 17th Problem
 In Contemporary Mathematics
, 1996
"... This paper is dedicated to the memory of Raphael M. Robinson and Olga TausskyTodd. 1. Introduction ..."
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Cited by 156 (7 self)
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This paper is dedicated to the memory of Raphael M. Robinson and Olga TausskyTodd. 1. Introduction
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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Cited by 101 (22 self)
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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
Decomposition of quantics in sums of powers of linear forms
 Signal Processing
, 1996
"... Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent complementary use of HighOrder Statistics (HOS). However, very few special purpose tools are at disposal for manipulating such objects in engineering prob ..."
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Cited by 91 (23 self)
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Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent complementary use of HighOrder Statistics (HOS). However, very few special purpose tools are at disposal for manipulating such objects in engineering problems. In this paper, the decomposition of a symmetric tensor into a sum of simpler ones is focused on, and links with the theory of homogeneous polynomials in several variables (i.e. quantics) are pointed out. This decomposition may be seen as a formal extension of the Eigen Value Decomposition (EVD), known for symmetric matrices. By reviewing the state of the art, quite surprising statements are emphasized, that explain why the problem is much more complicated in the tensor case than in the matrix case. Very few theoretical results can be applied in practice, even for cubics or quartics, because proofs are not constructive. Nevertheless in the binary case, we have more freedom to devise numerical algorithms. Keywords. Tensors, Polynomials, Diagonalization, EVD, HighOrder Statistics, Cumulants. 1
Tensor decompositions, state of the art and applications
 MATHEMATICS IN SIGNAL PROCESSING V
"... ..."
Canonical Tensor Decompositions
 ARCC WORKSHOP ON TENSOR DECOMPOSITION
, 2004
"... The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short pap ..."
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Cited by 42 (16 self)
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The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short paper, such as the maximal achievable rank of a tensor of given dimensions, or the computation of a CanD. Some questions are answered, and it turns out that the answers depend on the choice of the underlying field, and on tensor symmetry structure, which outlines a major difference compared to matrices.
McLaren’s improved snub cube and other new spherical designs in three dimensions
 Discrete and Computational Geometry
, 1996
"... Evidence is presented to suggest that, in three dimensions, spherical 6designs with N ..."
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Cited by 40 (1 self)
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Evidence is presented to suggest that, in three dimensions, spherical 6designs with N
Truncated Kmoment problems in several variables
 J. OPERATOR THEORY
, 2005
"... Let β ≡ β (2n) be an Ndimensional real multisequence of degree 2n, with associated moment matrix M(n) ≡ M(n)(β), and let r: = rank M(n). We prove that if M(n) is positive semidefinite and admits a rankpreserving moment matrix extension M(n+1), then M(n+1) has a unique representing measure µ, w ..."
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Cited by 40 (11 self)
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Let β ≡ β (2n) be an Ndimensional real multisequence of degree 2n, with associated moment matrix M(n) ≡ M(n)(β), and let r: = rank M(n). We prove that if M(n) is positive semidefinite and admits a rankpreserving moment matrix extension M(n+1), then M(n+1) has a unique representing measure µ, which is ratomic, with suppµ equal to V(M(n + 1)), the algebraic variety of M(n + 1). Further, β has an ratomic (minimal) representing measure supported in a semialgebraic set KQ subordinate to a family Q ≡ {qi} m i=1 ⊆ R[t1,..., tN] if and only if M(n) is positive semidefinite and admits a rankpreserving extension M(n + 1) for which the associated localizing matrices Mqi (n + [ 1+deg qi]) are positive semidefinite (1 ≤ i ≤ m); in this case, µ (as 2 above) satisfies supp µ ⊆ KQ, and µ has precisely rank M(n) − rank Mqi (n + [ 1+deg qi]) atoms in 2 Z(qi) ≡ { t ∈ R N: qi(t) = 0} , 1 ≤ i ≤ m.
The proof of Tchakaloff’s Theorem
, 2005
"... Abstract. We provide a simple proof of Tchakaloff’s Theorem on the existence of cubature formulas of degree m for Borel measures with moments up to order m. The result improves known results for noncompact support, since we do not need conditions on (m + 1)st moments. In fact we reduce the classica ..."
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Cited by 30 (2 self)
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Abstract. We provide a simple proof of Tchakaloff’s Theorem on the existence of cubature formulas of degree m for Borel measures with moments up to order m. The result improves known results for noncompact support, since we do not need conditions on (m + 1)st moments. In fact we reduce the classical assertion of Tchakaloff’s Theorem to a wellknown statement going back to F. Riesz. We consider the question of existence of cubature formulas of degree m for Borel measures µ, i. e. a measure defined on the Borel σalgebra, where moments up to degree m exist: Definition 1. Let µ be a positive Borel measure on RN and m ≥ 1 such that ‖x ‖ k µ(dx) < ∞
Symmetries of Polynomials
 J. Symb. Comp
"... New algorithms for determining discrete and continuous symmetries of polynomials  also known as binary forms in classical invariant theory  are presented. Implementations in Mathematica and Maple are discussed and compared. The results are based on a new, comprehensive theory of moving frames ..."
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Cited by 24 (17 self)
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New algorithms for determining discrete and continuous symmetries of polynomials  also known as binary forms in classical invariant theory  are presented. Implementations in Mathematica and Maple are discussed and compared. The results are based on a new, comprehensive theory of moving frames that completely characterizes the equivalence and symmetry properties of submanifolds under general Lie group actions. This work was partially supported by NSF Grant DMS 9803154. 1 Introduction. The purpose of this paper is to explain the detailed implementation of a new algorithm for determining the symmetries of polynomials (binary forms). The method was first described in the second author's new book [24], and the present paper adds details and refinements. We shall demonstrate that the symmetry group of both real and complex binary forms can be completely determined by solving two simultaneous bivariate polynomial equations, which are based on two fundamental covariants of the for...