A Mealy machine with polynomial growth of irrational degree

We consider a very simple Mealy machine (three states over a two-symbol alphabet), and derive some properties of the semigroup it generates. In particular, this is an infinite, finitely generated semigroup; we show that the growth function of its balls behaves asymptotically like n^2.4401..., where this constant is 1 + log(2)/log((1+sqrt(5))/2); that the semigroup satisfies the identity g^6=g^4; and that its lattice of two-sided ideals is a chain.Comment: 20 pages, 1 diagra

Раціональність функцій росту ініціальних автоматів Мілі

Функція росту gA(n) ініціального автомата Мілі A обчислює кількість станів у композиції автоматів A^n = Ao…o A (n разів) після мінімізації, які досягаються з ініціального стану. Досліджено, коли генератриса функції росту є раціональною для таких класів ініціальних автоматів: стискуючих з нільпотентною автоматною групою, біреверсивних, поліноміальних.Функция роста gA(n) инициального автомата Мили A подcчитывает количество состояний в композиции автоматов A^n = Ao…o A (n раз) после минимизации, достижимых с инициального состояния. Исследовано, когда генератриса функции роста является рациональной для следующих классов автоматов: стягивающих с нильпотентной автоматной группой, биреверсивных, полиномиальных.The growth function γA(n) of an initial Mealy automaton A counts the number of states in a composition of automata A^n = Ao…o A (n times) after the minimization that are reachable from the initial state. We study the question when the generating function of the growth function is rational for the following automata classes: contracting with a nilpotent automaton group, bireversible, and polynomial ones

From self-similar groups to self-similar sets and spectra

The survey presents developments in the theory of self-similar groups leading to applications to the study of fractal sets and graphs, and their associated spectra

Integrable and Chaotic Systems Associated with Fractal Groups

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 80-s of the last century with the purpose to solve some famous problems in mathematics, including the question raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor's question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schr\"odinger operators. One of important developments is the relation of them to the multi-dimensional dynamics, theory of joint spectrum of pencil of operators, and spectral theory of Laplace operator on graphs. The paper gives a quick access to these topics, provide calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains discussion of the dichotomy "integrable-chaotic" in the considered model, and suggests a possible probabilistic approach to the study of discussed problems.Comment: 48 pages, 15 figure

MFCS\u2798 Satellite Workshop on Cellular Automata

For the 1998 conference on Mathematical Foundations of Computer Science (MFCS\u2798) four papers on Cellular Automata were accepted as regular MFCS\u2798 contributions. Furthermore an MFCS\u2798 satellite workshop on Cellular Automata was organized with ten additional talks. The embedding of the workshop into the conference with its participants coming from a broad spectrum of fields of work lead to interesting discussions and a fruitful exchange of ideas. The contributions which had been accepted for MFCS\u2798 itself may be found in the conference proceedings, edited by L. Brim, J. Gruska and J. Zlatuska, Springer LNCS 1450. All other (invited and regular) papers of the workshop are contained in this technical report. (One paper, for which no postscript file of the full paper is available, is only included in the printed version of the report). Contents: F. Blanchard, E. Formenti, P. Kurka: Cellular automata in the Cantor, Besicovitch and Weyl Spaces K. Kobayashi: On Time Optimal Solutions of the Two-Dimensional Firing Squad Synchronization Problem L. Margara: Topological Mixing and Denseness of Periodic Orbits for Linear Cellular Automata over Z_m B. Martin: A Geometrical Hierarchy of Graph via Cellular Automata K. Morita, K. Imai: Number-Conserving Reversible Cellular Automata and Their Computation-Universality C. Nichitiu, E. Remila: Simulations of graph automata K. Svozil: Is the world a machine? H. Umeo: Cellular Algorithms with 1-bit Inter-Cell Communications F. Reischle, Th. Worsch: Simulations between alternating CA, alternating TM and circuit families K. Sutner: Computation Theory of Cellular Automat

Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators

We show that the lamplighter group L has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the spectrum of its Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family of convolution operators on L. Our results show that the spectrum is a pure point spectrum for each value of the parameter, the eigenvalues are solutions of algebraic equations involving Chebyshev polynomials of the second kind, and the topological structure of the spectrum makes a bifurcation when the parameter passes the points 1 and -1

Modeling of systems

The handbook contains the fundamentals of modeling of complex systems. The classification of mathematical models is represented and the methods of their construction are given. The analytical modeling of the basic types of processes in the complex systems is considered. The principles of simulation, statistical and business processes modeling are described. The handbook is oriented on students of higher education establishments that obtain a degree in directions of “Software engineering” and “Computer science” as well as on lecturers and specialists in the domain of computer modeling