Given a rectangular matrix with more columns than rows, nd a base of linear combinations of the row vectors such that these contain as many zero entries as possible. This process is called \sparsiication" (of the matrix). A combinatorial search method to solve sparsiication is presented which needs exponentially many arithmetic operations (in terms of the size of the matrix). However, various p...

We show that the Nakayama automorphism of a Frobenius algebra R over a field k is independent of the field (Theorem 4). Consequently, the k-dual functor on left R-modules and the bimodule isomorphism type of the k-dual of R, and hence the question of whether R is a symmetric k-algebra, are independent of k. We give a purely ring-theoretic condition that is necessary and sufficient for a finite-...

Let Zm be the ring of integers modulo m. The m-rank of an integer matrix is the largest order of a square submatrix whose determinant is not divisible by m. We determine the probability that a random rectangular matrix over Zm has a specified m-rank and, if it is square, a specified determinant. These results were previously known only for prime m. CommentsOnly the Abstract is given here. T...

Let X1,…,Xn be operators in a finite von Neumann algebra and consider their division closure the affiliated unbounded operators. We address question when this is skew field (aka ring) it free field. show that first property equivalent to strong Atiyah second can characterized terms of noncommutative distribution X1,…,Xn. More precisely, generate if only there exist no non-zero rank T1,…,Tn such...