In English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics. Mathematics


Volume 54 No. 1 March 2005


On informational properties of covers; 3–25

Tõnu Lausmaa

Abstract. Any partition on a finite set gives us partial information about an object represented by this set. This algebraic approach to information has proved very useful for solving various problems concerning structural complexity of finite objects. Like a partition, a cover on a finite set can be interpreted as information about structural complexity of an object represented by this cover. But unlike partitions, the quantitative measure of the information content of covers is not yet established. In this paper an effort is made to provide covers with a quantitative measure of their information content from the axiomatic point of view, by finding an extropy for a special class of covers which reflect the information difference between two partitions. This result breaks ground in providing all relevant for practice covers with a quantative measure, which would make it possible to evaluate the structural complexity for a wide range of finite objects.

Key words: entropy, extropy, partition, cover.

Searching: models and methods; 26–45

Volodymyr G. Skobelev

Abstract. Searching in a partially ordered structure under the supposition that the set of elementary operators consists only of monotone ones is developed on the basis of schemes determined via some predicate. Models for the design of preset and adaptive solutions, namely an -model and an -model, are determined and investigated. Interrelations between these models are established in the general case. Schemes for the design of solutions of all basic (minimal, irreducible, cooperative, and adaptive) types are developed. It is shown that the suggested approach can be applied to resolve some basic problems of discrete mathematics, such as the design of irreducible sets of representatives for a family of sets, design of experiments with finite automata, design of minterms and disjunctive normal forms consisting of minterms only.

Key words: searching schemes and models, preset and adaptive solutions, experiments with finite automata, disjunctive normal forms.

Languages recognized by two-sided automata of graphs; 46–54

Andrei V. Kelarev, Mirka Miller, and Olga V. Sokratova

Abstract. We introduce two-sided automata defined by directed graphs and describe all languages recognized by these automata.

Key words: labelled graphs, finite state automata, formal languages.

Controller and controllability canonical forms for discrete-time nonlinear systems; 55–62

Ülle Kotta

Abstract. The paper studies the controller and controllability canonical forms for single-input discrete-time nonlinear systems applying the algebraic formalism of one-forms. Necessary and sufficient conditions are given under which the system can be transformed locally into the controller or controllability form by means of a coordinate transformation. These conditions are formulated in terms of integrability of certain subspaces of one-forms, classified according to their relative degree, and in terms of factorizability of a given function in a certain way.

Key words: nonlinear systems, discrete-time systems, algebraic methods, canonical forms, accessibility.

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