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.

The role of metal ions in selective and sustainable processes

Sustainability and recycling are core values in today’s industrial operations. New materials, products and processes need to be designed in such a way as to consume fewer of the diminishing resources we have available and to put as little strain on the environment as possible. An integral part of this is cleaning and recycling. New processes are to be designed to improve the efficiency in this aspect. Wastewater, including municipal wastewaters, is treated in several steps including chemical and mechanical cleaning of waters. Well-cleaned water can be recycled and reused. Clean water for everyone is one of the greatest challenges we are facing today. Ferric sulphate, made by oxidation from ferrous sulphate, is used in water purification. The oxidation of ferrous sulphate, FeSO4, to ferric sulphate in acidic aqueous solutions of H2SO4 over finely dispersed active carbon particles was studied in a vigorously stirred batch reactor. Molecular oxygen was used as the oxidation agent and several catalysts were screened: active carbon, active carbon impregnated with Pt, Rh, Pd and Ru. Both active carbon and noble metal-active carbon catalysts enhanced the oxidation rate considerably. The order of the noble metals according to the effect was: Pt >> Rh > Pd, Ru. By the use of catalysts, the production capacities of existing oxidation units can be considerably increased. Good coagulants have a high charge on a long polymer chain effectively capturing dirty particles of the opposite charge. Analysis of the reaction product indicated that it is possible to obtain polymeric iron-based products with good coagulation properties. Systematic kinetic experiments were carried out at the temperature and pressure ranges of 60B100°C and 4B10 bar, respectively. The results revealed that both non-catalytic and catalytic oxidation of Fe2+ to Fe3+ take place simultaneously. The experimental data were fitted to rate equations, which were based on a plausible reaction mechanism: adsorption of dissolved oxygen on active carbon, electron transfer from Fe2+ ions to adsorbed oxygen and formation of surface hydroxyls. A comparison of the Fe2+ concentrations predicted by the kinetic model with the experimentally observed concentrations indicated that the mechanistic rate equations were able to describe the intrinsic oxidation kinetics of Fe2+ over active carbon and active carbon-noble metal catalysts. Engineering aspects were closely considered and effort was directed to utilizing existing equipment in the production of the new coagulant. Ferrous sulphate can be catalytically oxidized to produce a novel long-chained polymeric iron-based flocculent in an easy and affordable way in existing facilities. The results can be used for modelling the reactors and for scale-up. Ferric iron (Fe3+) was successfully applied for the dissolution of sphalerite. Sphalerite contains indium, gallium and germanium, among others, and the application can promote their recovery. The understanding of the reduction process of ferric to ferrous iron can be used to develop further the understanding of the dissolution mechanisms and oxidation of ferrous sulphate. Indium, gallium and germanium face an ever-increasing demand in the electronics industry, among others. The supply is, however, very limited. The fact that most part of the material is obtained through secondary production means that real production quota depends on the primary material production. This also sets the pricing. The primary production material is in most cases zinc and aluminium. Recycling of scrap material and the utilization of industrial waste, containing indium, gallium and geranium, is a necessity without real options. As a part of this study plausible methods for the recovery of indium, gallium and germanium have been studied. The results were encouraging and provided information about the precipitation of these valuables from highly acidic solutions. Indium and gallium were separated from acidic sulphuric acid solutions by precipitation with basic sulphates such as alunite or they were precipitated as basic sulphates of their own as galliunite and indiunite. Germanium may precipitate as a basic sulphate of a mixed composition. The precipitation is rapid and the selectivity is good. When the solutions contain both indium and gallium then the results show that gallium should be separated before indium to achieve a better selectivity. Germanium was separated from highly acidic sulphuric acid solutions containing other metals as well by precipitating with tannic acid. This is a highly selective method. According to the study other commonly found metals in the solution do not affect germanium precipitation. The reduction of ferric iron to ferrous, the precipitation of indium, gallium and germanium, and the dissolution of the raw materials are strongly depending on temperature and pH. The temperature and pH effect were studied and which contributed to the understanding and design of the different process steps. Increased temperature and reduced pH improve the reduction rate. Finally, the gained understanding in the studied areas can be employed to develop better industrial processes not only on a large scale but also increasingly on a smaller scale. The small amounts of indium, gallium and germanium may favour smaller and more locally bound recovery.