Let K be a compact subset of R " with K=~nt K. Necessary conditions on an n-dimensional subspace U, of C(K) are given so that for each f~ C(K) there exists a unique best L'(w)-approximation from U,,, for every fixed positive weight function w. Let K be a compact subset of R ". For convenience we assume that K= int K. W will denote the set of bounded, integrable functions on K for which inf(w(x)...

In this work we prove the necessity of the existence of a density function for a globally asymptotically stable dynamical system.

We study time-frequency localization operators of the form A12 a , where a is the symbol of the operator and φ1, φ2 are the analysis and synthesis windows, respectively. It is shown in [3] that a sufficient condition for A12 a ∈ Sp(L(R)), the Schatten class of order p, is that a belongs to the modulation space Mp,∞(R2d) and the window functions to the modulation space M1. Here we prove a partia...

Hoffman proved that for a simple graph G, the chromatic number χ(G) obeys χ(G) ≥ 1 − λ1 λn where λ1 and λn are the maximal and minimal eigenvalues of the adjacency matrix of G respectively. Lovász later showed that χ(G) ≥ 1− λ1 λn for any (perhaps negatively) weighted adjacency matrix. In this paper, we give a probabilistic proof of Lovász’s theorem, then extend the technique to derive generali...

Molecular dynamics is employed to study the nature and magnitude of beam cooling that is required in order to achieve a crystalline beam. Analysis is presented of a number of cooling systems now in use, or whose use is contemplated, none of which has been shown to be able to achieve the crystalline state. However, for an adequately strong cooling system that produces on average a constant angul...

The multiobjective bilevel program is a sequence of two optimization problems, with the upper-level problem being multiobjective and the constraint region of the upper level problem being determined implicitly by the solution set to the lower-level problem. In the case where the Karush-Kuhn-Tucker (KKT) condition is necessary and sufficient for global optimality of all lower-level problems near...

Necessary conditions of optimality, in the form of a maximum principle, are derived for a class of optimal control problems, certain of whose controls are represented by measures and whose state trajectories are functions of bounded variation. State trajectories are interpreted as robust solutions of the dynamic equations, a concept of solutions which takes account of the interaction between th...

In this paper, it is proved that a symmetric tensor is (strictly) copositive if and only if each of its principal subtensors has no (non-positive) negative H++-eigenvalue. Necessary and sufficient conditions for (strict) copositivity of a symmetric tensor are also given in terms of Z++-eigenvalues of the principal sub-tensors of that tensor. This presents a method for testing (strict) copositiv...

Necessary and sufficient conditions are given for a quantummechanical system to possess a coordinate system with respect to which its behaviour at discrete times is that of a universal digital computer . The form of the diagonal representation for the unitary time evolution operator for quantum universal computers is derived ; aspects of the transformation between the diagonal representation an...

What assumptions are required to achieve an unconditionally secure distributed circuit evaluation in a fully connected network? This question was addressed with respect to the allowed number of malicious players [BGW, CCD, RB], given that every channel is unconditionally secure. In this paper we investigate whether the security of all channels is also a necessary condition. [BGW, CCD] showed ho...