### Article

## Closed Classes of Three-Valued Logic Generated by Systems Containing Symmetric Functions

Closed classes of functions of three-valued logic whose generating systems include nonmonotone symmetric functions taking values in the set {0,1} are studied. It is shown that in some cases the problems of existence of a basis and existence of a finite basis can be reduced to a similar problem for reduced generated systems.

Closed classes of functions of three-valued logic whose generating systems include nonmonotone symmetric functions taking values in the set {0,1} and taking value 1 on restricted number of layers are studied. Cryteria of existence of basis and existence of finite basis has been obtained.

Closed classes are considered of three-valued logic functions generated by symmetric functions taking values in the set {0, 1}. Criteria for existence of bases and for existence of finite generating systems are obtained for some classes generated by elementary periodic symmetric functions.

The complexity of realization of *k*-valued logic functions by circuits in a special infinite basis is under study. This basis consists of Post negation (i.e. function *x*+1(mod *k*)) and all monotone functions. The complexity of the circuit is the total number of elements of this circuit. For an arbitrary function *f*, we find the lower and upper bounds of complexity, which differ from one another at most by 1. The complexity has the form 3log_3 (*d*(*f*)+1)+*O*(1), here *d*(*f*) is the maximum number of the value decrease of the value of *f* taken over all increasing chains of tuples of variable values. We find the exact value of the corresponding Shannon function which characterizes the complexity of the most complex function of a given number of variables.

This paper reviews the theory of Credit Default Swaps (CDS), the main characteristics of the CDS market, and how to estimate the non-default component of the yield spreads as the basis between the actual CDS premium and the hypothetical CDS premium implied by bond yields. We then analyze the most liquid CDS on Russian companies and compute the relative CDS-Bond basis from 2005 till 2010, paying particular attention to the period when a short selling ban was into effect in Russian financial markets from September 18, 2008 till June 15, 2009. We found that, while the basis was mainly negative before the ban, it then became largely positive during the period the ban was enforced. After the ban was lifted, the basis has started to decrease but still remains positive for all companies examined. This evidence therefore seems to support the hypothesis that a positive basis can be justified by the difficulty of arbitrage caused by short selling costs

The collection represents proceedings of the 5th school-seminar "Syntax and Semantics of Logic Systems" (Ulan-Ude, 08.08.2017 - 12.08.2017). The conference subject area includes: theory of models and universal algebra; theory of boolean and finite-valued functions; formal languages and logic calculus; mathematical logic in education.

Jan Lukasiewicz (1878-1956) was one of the most important members of the Lwow-Warsaw school of logic. The thirteen translated articles in this volume demonstrate the protean form of Lukasiewiczs work, from his texts on Aristotle and the principle of non-contradiction and syllogistics to modal logic, intuitionism, and multivalent logics. The articles show in particular his preoccupations with logical precision and the problem of human liberty.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.