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Jos M.F. Ten Berge Alwin Stegeman Mohammed Bennani Dosse 《Linear algebra and its applications》2009,430(2-3):818-829
The Candecomp/Parafac algorithm approximates a set of matrices by products of the form , with diagonal, . Carroll and Chang have conjectured that, when the matrices are symmetric, the resulting and will be column wise proportional. For cases of perfect fit, Ten Berge et al. have shown that the conjecture holds true in a variety of cases, but may fail when there is no unique solution. In such cases, obtaining proportionality by changing (part of) the solution seems possible. The present paper extends and further clarifies their results. In particular, where Ten Berge et al. solved all cases, now all cases, and also the cases for , and 9 are clarified. In a number of cases, and necessarily have column wise proportionality when Candecomp/Parafac is run to convergence. In other cases, proportionality can be obtained by using specific methods. No cases were found that seem to resist proportionality. 相似文献
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Let be any ordinal. We consider the class of relativized diagonal free set algebras of dimension α. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class corresponds to first order logic, without equality symbol, with α-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory. 相似文献
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Let be an arbitrary integral domain, let be a multiset of elements of , let be a permutation of let be positive integers such that , and for let . We are interested in the problem of finding a block matrix with spectrum and such that for . Cravo and Silva completely characterized the existence of such a matrix when is a field. In this work we construct a solution matrix that solves the problem when is an integral domain with two exceptions: (i) ; (ii) , and for some .What makes this work quite unique in this area is that we consider the problem over the more general algebraic structure of integral domains, which includes the important case of integers. Furthermore, we provide an explicit and easy to implement finite step algorithm that constructs an specific solution matrix (we point out that Cravo and Silva’s proof is not constructive). 相似文献
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We exploit the structure of the critical orbital sets of symmetry classes of tensors associated to sign uniform partitions and we establish new connections between symmetry classes of tensors, matchings on bipartite graphs and coding theory. In particular, we prove that the orthogonal dimension of the critical orbital sets associated to single hook partitions equals the value of the coding theoretic function . When we reobtain this number as the independence number of the Dynkin diagram . 相似文献
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