Let K be a closed convex cone with dual K∗ in a finite-dimensional real Hilbert space V . A positive operator on K is a linear operator L on V such that L (K) ⊆ K. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. We say that L is a Z-operator on K if 〈L (x), s〉 ≤ 0 for all (x, s) ∈ K ×K such that 〈x, s〉 = 0. The Z-operators are generalizat...

The entropy of a closure operator has been recently proposed for the study of network coding and secret sharing. In this paper, we study closure operators in relation to their entropy. We first introduce four different kinds of rank functions for a given closure operator, which determine bounds on the entropy of that operator. This yields new axioms for matroids based on their closure operators...

The entropy of a closure operator has been recently proposed for the study of network coding and secret sharing. In this paper, we study closure operators in relation to their entropy. We first introduce four different kinds of rank functions for a given closure operator, which determine bounds on the entropy of that operator. This yields new axioms for matroids based on their closure operators...

The notion of the shell of a Hilbert space operator, which is a useful generalization (proposed by Wielandt) of the numerical range, is extended to operators in spaces with an indefinite inner product. For the most part, finite dimensional spaces are considered. Geometric properties of shells (convexity, boundedness, being a subset of a line, etc.) are described, as well as shells of operators ...

In a recent preprint by Amaral & Letchford (2006) convex hulls of sets of matrices corresponding to permutations and path-metrics are studied. A symmetric n × n-matrix is a path metric, if there exist points x1, . . . , xn ∈ R such that the matrix entries are just the pairwise distances |xk − xl| between the points and if these distances are at least one whenever k 6= l. The convex hull of the ...

In a recent preprint by Amaral & Letchford (2006) convex hulls of sets of matrices corresponding to permutations and certain metrics are studied. For each n-tupel of points x1, . . . , xn ∈ R with |xk − xl| ≥ 1 for k 6= l, we define a metric in form of a symmetric n× n-matrix whose entries are the pairwise distances |xk − xl| between the points. The convex hull of these metrics is denoted by Qn...

In a recent preprint by Amaral & Letchford (2006) convex hulls of sets of matrices corresponding to permutations and path-metrics are studied. A symmetric n × n-matrix is a path metric, if there exist points x1, . . . , xn ∈ R such that the matrix entries are just the pairwise distances |xk − xl| between the points and if these distances are at least one whenever k 6= l. The convex hull of the ...

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on 5 separates the points of S. If X is a Banach space and T: X -* F is a continuous linear operator, then T is quasi-convex if T(U) is quasiconvex, where U is the unit ball of X. In the case when T is compact, T(U) is quasi-convex if and only if it is af...

Suppose L is a differential operator of order k. Moreover, let u be a function defined and k times continuously differentiable in the closure of a domain Ω of R. Provided the adjoint differential operator possesses a fundamental solution, we shall see that u can be recovered from Lu and the boundary values of u. Strictly speaking, we shall get an integral representation of u in form of the sum ...

Many applications in the area of production and statistical estimation are problems of convex optimization subject to ranking constraints that represent a given partial order. This problem – which we call the convex cost closure problem, or (CCC) – is a generalization of the known maximum (or minimum) closure problem and the isotonic regression problem. For a (CCC) problem on n variables and m ...