Word Equations and Related Topics. Independence, Decidability and Characterizations

The three main topics of this work are independent systems and chains of word equations, parametric solutions of word equations on three unknowns, and unique decipherability in the monoid of regular languages. The most important result about independent systems is a new method giving an upper bound for their sizes in the case of three unknowns. The bound depends on the length of the shortest equation. This result has generalizations for decreasing chains and for more than three unknowns. The method also leads to shorter proofs and generalizations of some old results. Hmelevksii’s theorem states that every word equation on three unknowns has a parametric solution. We give a significantly simplified proof for this theorem. As a new result we estimate the lengths of parametric solutions and get a bound for the length of the minimal nontrivial solution and for the complexity of deciding whether such a solution exists. The unique decipherability problem asks whether given elements of some monoid form a code, that is, whether they satisfy a nontrivial equation. We give characterizations for when a collection of unary regular languages is a code. We also prove that it is undecidable whether a collection of binary regular languages is a code.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes

In this thesis we examine four well-known and traditional concepts of combinatorics on words. However the contexts in which these topics are treated are not the traditional ones. More precisely, the question of avoidability is asked, for example, in terms of k-abelian squares. Two words are said to be k-abelian equivalent if they have the same number of occurrences of each factor up to length k. Consequently, k-abelian equivalence can be seen as a sharpening of abelian equivalence. This fairly new concept is discussed broader than the other topics of this thesis. The second main subject concerns the defect property. The defect theorem is a well-known result for words. We will analyze the property, for example, among the sets of 2-dimensional words, i.e., polyominoes composed of labelled unit squares. From the defect effect we move to equations. We will use a special way to define a product operation for words and then solve a few basic equations over constructed partial semigroup. We will also consider the satisfiability question and the compactness property with respect to this kind of equations. The final topic of the thesis deals with palindromes. Some finite words, including all binary words, are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. The famous Thue-Morse word has the property that for each positive integer n, there exists a factor which cannot be generated by fewer than n palindromes. We prove that in general, every non ultimately periodic word contains a factor which cannot be generated by fewer than 3 palindromes, and we obtain a classification of those binary words each of whose factors are generated by at most 3 palindromes. Surprisingly these words are related to another much studied set of words, Sturmian words.