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The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. By Mincs conjecture, there exists a reachable upper bound on the permanent of 2-dimensional (0,1)-matrices. In this paper we obtain some generalizations of Mincs conjecture to the multidimensional case. For this purpose we prove and compare several bounds on the permanent of multidimensional (0,1)-matrices. Most estimates can be used for matrices with nonnegative bounded entries.
The trace norm of a matrix is the sum of its singular values. This paper presents results on the minimum trace norm $psi_{n}left( mright) $ of $left( 0,1right) $-matrices of size $ntimes n$ with exactly $m$ ones. It is shown that: (1) if $ngeq2$ an
A $d$-dimensional matrix is called emph{$1$-polystochastic} if it is non-negative and the sum over each line equals~$1$. Such a matrix that has a single $1$ in each line and zeros elsewhere is called a emph{$1$-permutation} matrix. A emph{diagonal} o
We give upper bounds on the order of the automorphism group of a simple graph
We design a deterministic polynomial time $c^n$ approximation algorithm for the permanent of positive semidefinite matrices where $c=e^{gamma+1}simeq 4.84$. We write a natural convex relaxation and show that its optimum solution gives a $c^n$ approxi
Let ${mathcal D}(n)$ be the maximal determinant for $n times n$ ${pm 1}$-matrices, and $mathcal R(n) = {mathcal D}(n)/n^{n/2}$ be the ratio of ${mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on