Considering n × n × n stochastic tensors (a ijk ) (i.e., nonnegative hypermatrices in which every sum over one index i, j, or k, is 1), we study the polytope (Ω n ) of all these tensors, the convex set (L n ) of all tensors in Ω n with some positive diagonals, and the polytope (∆ n ) generated by the permutation tensors. We show that L n is almost the same as Ω n except for some boundary points. We also present an upper bound for the number of vertices of Ω n .
By an embedding approach and through tensor products, some inequalities for generalized matrix functions (of positive semidefinite matrices) associated with any subgroup of the permutation group and any irreducible character of the subgroup are obtainned.
We present an inequality for tensor product of positive operators on Hilbert spaces by considering the tensor product of operators as words on certain alphabets (i.e., a set of letters). As applications of the operator inequality and by a multilinear approach, we show some matrix inequalities concerning induced operators and generalized matrix functions (including determinants and permanents as special cases).
We present inequalities concerning the entries of correlation matrices, density matrices, and partial isometries through the positivity of 3 × 3 matrices. We extend our discussions to the inequalities concerning the triangle triplets with metric-preserving and subadditive functions.
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