let a; b; k 2 k and u ; v 2 u(k). we show for any idempotent e 2 k, ( a 0 b 0 ) is e-clean i ( a 0 u(vb + ka) 0 ) is e-clean and if ( a 0 b 0 ) is 0-clean, ( ua 0 u(vb + ka) 0 ) is too.

7 We first characterize submatrices of a unimodular integral matrix. We then prove that if n entries of an 8 n× n partial integral matrix are prescribed and these n entries do not constitute a row or a column, then this 9 matrix can be completed to a unimodular matrix. Consequently an n× n partial integral matrix with n− 1 10 prescribed entries can always be completed to a unimodular matrix. 11...

The idea behind the SizeReduce(B) subroutine is, in the Gram-Schmidt decomposition B = B̃ ·U, to shift the entries in the upper triangle of U by integers (via unimodular transformations), so that they lie in [−12 , 1 2). Because changing an entry of U may affect the ones above it (but not below it) in the same column, we must make the changes upward in each column. Formally, the algorithm works ...

An integral square matrix A is called principally unimodular (PU if every nonsingular principal submatrix is unimodular (that is, has determinant \1). Principal unimodularity was originally studied with regard to skew-symmetric matrices; see [2, 4, 5]; here we consider symmetric matrices. Our main theorem is a generalization of Tutte's excluded minor characterization of totally unimodular matri...

We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular row and column operations to the origina...

We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodu...

We provide preliminary results regarding the existence of a polynomial time approximation scheme (PTAS) for minimizing a linear function over a 0/1 covering polytope which is integral, with one additional covering constraint. Our algorithm is based on extending the methods of Goemans and Ravi for the constrained minimum spanning tree problem and, in particular, implies the existence of a PTAS f...

A Las Vegas randomized algorithm is given to compute the Smith multipliers for a nonsingular integer matrix A, that is, unimodular matrices U and V such AV=US, with S normal form of A. The expected running time about same as required multiply together two dimension size entries Explicit bounds are in both multipliers. main tool used by massager, relaxed version V, specifying column operations c...

It is shown that if there is an extremal even unimodular lattice in dimension 72, then there is an optimal odd unimodular lattice in that dimension. Hence, the first example of an optimal odd unimodular lattice in dimension 72 is constructed from the extremal even unimodular lattice which has been recently found by G. Nebe.

We use the property of unimodular functions to perform approximate inference with dual decomposition in binary labeled graphs. Exact inference is possible for a subclass of binary labeled graphs that have unimodular functions. We call such graphs unimodular graphs. These are graphs where the submodular and nonsubmodular edges follow a specific pattern– essentially that an isomorphism or ”flippi...