Minimal rank of abelian group matrices

The minimal rank of abelian group matrices with positive integral entries is determined.The corresponding problem for circulant matrices have been investigated by Ingleton and more recently by Shiu-Ma-Fang. Our work can be viewed as a generalization of their results, since a group matrix becomes circulant when the group is cyclic.     

Minimal rank completions of partial banded matrices

It is proven that the minimal rank of a partial banded matrix equals the maximum of the minimal ranks of all triangular subpatterns. This proves partly the minimal rank conjecture in a paper by N. Cohen, C. R. Johnson, L. Rodman and H. J. Woerdeman (Operator Theory: Advances and Applications 40 (1989), 165-185).

The results are applied to the problem of simultaneously completing a matrix and its inverse.     

Minimal condition number for positive definite Hankel matrices using semidefinite programming

We present a semidefinite programming approach for computing optimally conditioned positive definite Hankel matrices of order n. Unlike previous approaches, our method is guaranteed to find an optimally conditioned positive definite Hankel matrix within any desired tolerance. Since the condition number of such matrices grows exponentially with n, this is a very good test problem for checking the numerical accuracy of semidefinite programming solvers. Our tests show that semidefinite programming solvers using fixed double precision arithmetic are not able to solve problems with n>30. Moreover, the accuracy of the results for 24?n?30 is questionable. In order to accurately compute minimal condition number positive definite Hankel matrices of higher order, we use a Mathematica 6.0 implementation of the SDPHA solver that performs the numerical calculations in arbitrary precision arithmetic. By using this code, we have validated the results obtained by standard codes for n?24, and we have found optimally conditioned positive definite Hankel matrices up to n=100.     

The Product of Minimal Functions is Minimal

Using Doob's h-processes, a simple proof is given of a resultof Freire: every product of minimal eigenfunctions for Brownianmotions on each one of a product of complete Riemannian manifoldsis itself minimal for the Brownian motion on the product manifold.     

Minimal Frame Operator Norms Via Minimal Theta Functions

We investigate sharp frame bounds of Gabor frames with chirped Gaussians and rectangular lattices or, equivalently, the case of the standard Gaussian and general lattices. We prove that for even redundancy and standard Gaussian window the hexagonal lattice minimizes the upper frame bound using a result by Montgomery on minimal theta functions.     

Clean matrices and unit-regular matrices

An element in a ring R is said to be clean (respectively unit-regular) if it is the sum (respectively product) of an idempotent element and an invertible element. If all elements in R are unit-regular, it is known that all elements in R are clean. In this note, we show that a single unit-regular element in a ring need not be clean. More generally, a criterion is given for a matrix to be clean in a matrix ring M₂(K) over any commutative ring K. For K=Z, this criterion shows, for instance, that the unit-regular matrix is not clean. Also, this turns out to be the “smallest” such example.     

Minimal inequalities

We provide a characterization of minimal inequalities for bounded mixed integer programs, in terms of subadditive functions. The condition on the columns of the integer-constrained variables is analogous to that obtained earlier for the group problem, and we also determine the condition on the columns of the continuous variables.The research reported here was done at Carnegie-Mellon University.The report was prepared as part of the activities of the Management Sciences Research Group, Carnegie-Mellon University, under Contract N00014-67-A-0314-0007 NR 047-048 with U.S. Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

Minimal Submanifolds

This paper provides a survey of both old and new results aboutminimal surfaces and submanifolds. 2000 Mathematics SubjectClassification 54C40, 14E20 (primary), 46E25, 20C20 (secondary).     

Embedding matrices in Hadamard matrices

It is shown that if A is any n×n matrix of zeros and ones, and if k is the smallest number not less than n which is the order of an Hadamard matrix, then A is a submatrix of an Hadamard matrix of order k².     

Minimal lattice-subspaces

In this paper the existence of minimal lattice-subspaces of a vector lattice containing a subset of is studied (a lattice-subspace of is a subspace of which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology on and is -closed (especially if is a Banach lattice with order continuous norm), then minimal lattice-subspaces with -closed positive cone exist (Theorem 2.5).

In the sequel it is supposed that is a finite subset of , where is a compact, Hausdorff topological space, the functions are linearly independent and the existence of finite-dimensional minimal lattice-subspaces is studied. To this end we define the function where . If is the range of and the convex hull of the closure of , it is proved:

(i)

There exists an -dimensional minimal lattice-subspace containing if and only if is a polytope of with vertices (Theorem 3.20).

(ii)

The sublattice generated by is an -dimensional subspace if and only if the set contains exactly points (Theorem 3.7).

This study defines an algorithm which determines whether a finite-dimensional minimal lattice-subspace (sublattice) exists and also determines these subspaces.

     

Minimal graphs

Elementary properties of harmonic maps between Riemannian manifolds are interpreted via their graphs, viewed as nonparametric minimal submanifolds (Proposition 1). Then examples, are given of nonparametric submanifolds of compact Riemannian manifolds which cannot be deformed-through nonparametric submanifolds-to nonparametric minimal submanifolds (Propositions 2 and 4).     

Minimal measures

For a finite composition of exact, area preserving, positive twist diffeomorphisms of the infinite cylinder, we will show that an invariant measure whose support consists of minimal orbits strictly minimizes the average action.     

Approximating Runge-Kutta matrices by triangular matrices

The implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers [5] substitute the Runge-Kutta matrixA in the Newton process for a triangular matrixT that approximatesA, hereby making the method suitable for parallel implementation. The matrixT is constructed according to a simple procedure, such that the stiff error components in the numerical solution are strongly damped. In this paper we prove for a large class of Runge-Kutta methods that this procedure can be carried out and that the diagnoal entries ofT are positive. This means that the linear systems that are to be solved have a non-singular matrix. The research reported in this paper was supported by STW (Dutch Foundation for Technical Sciences).     

On double-one matrices and double-zero matrices

We investigate two special classes of matrices over GF(2) with certain interesting properties. These properties can be applied to construct nonsingular matrix pairs efficiently and thus provide a solution to the long-key problems of McEIiece's public-key cryptosystem.     

On double-one matrices and double-zero matrices

We investigate two special classes of matrices over GF(2) with certain interesting properties. These properties can be applied to construct nonsingular matrix pairs efficiently and thus provide a solution to the long-key problems of McEIiece's public-key cryptosystem.     

The Product of Minimal Functions is Minimal: Erratum

Corollary 2.3 is false as stated, but is correct if one assumesthat both generators satisfy the separation property.