Motzkin Decomposable Sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. We have provided several characterizations of the larger class of closed convex sets, Motzkin decomposable, in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed. Another result establishes that a closed convex set is Motzkin decomposable if and only if the set of extreme points of its intersection with the linear subspace orthogonal to its lineality is bounded. We characterize the class of the extended functions whose epigraphs are Motzkin decomposable sets showing, in particular, that these functions attain their global minima when they are bounded from below. Calculus of Motzkin decomposable sets and functions is provided.

Recognizing Decomposition of a Partial Boolean Function

A hard combinatorial problem is investigated which has useful application in design of discrete devices: the two-block decomposition of a partial Boolean function. The key task is regarded: finding such a weak partition on the set of arguments, at which the considered function can be decomposed. Solving that task is essentially speeded up by the way of preliminary discovering traces of the sought-for partition. Efficient combinatorial operations are used by that, based on parallel execution of operations above adjacent units in the Boolean space.

Decomposition of Banach Space into a Direct Sum of Separable and Reflexive Subspaces and Borel Maps

* This paper was supported in part by the Bulgarian Ministry of Education, Science and Technologies under contract MM-506/95.

Decomposition of Boolean Functions – Recognizing a Good Solution by Traces

The problem of sequent two-block decomposition of a Boolean function is regarded in case when a good solution does exist. The problem consists mainly in finding an appropriate weak partition on the set of arguments of the considered Boolean function, which should be decomposable at that partition. A new fast heuristic combinatorial algorithm is offered for solving this task. At first the randomized search for traces of such a partition is fulfilled. The recognized traces are represented by some "triads" - the simplest weak partitions corresponding to non-trivial decompositions. After that the whole sought-for partition is restored from the discovered trace by building a track initialized by the trace and leading to the solution. The results of computer experiments testify the high practical efficiency of the algorithm.

Implementation of a Heuristic Method of Decomposition of Partial Boolean Functions

An original heuristic algorithm of sequential two-block decomposition of partial Boolean functions is researched. The key combinatorial task is considered: finding of suitable partition on the set of arguments, i. e. such one, on which the function is separable. The search for suitable partition is essentially accelerated by preliminary detection of its traces. Within the framework of the experimental system the efficiency of the algorithm is evaluated, the boundaries of its practical application are determined.

Asymmetric Cipher Protocol Using Decomposition Problem

The asymmetric cipher protocol based on decomposition problem in matrix semiring M over semiring of natural numbers N is presented. The security parameters are defined and preliminary security analysis is presented.

Application of Wavelet Decomposition to Document Line Segmentation

ACM Computing Classification System (1998): I.7, I.7.5.

Sobolev Type Decomposition of Paley-Wiener-Schwartz Space with Application to Sampling Theory

2000 Mathematics Subject Classification: 94A12, 94A20, 30D20, 41A05.