This thesis presents two applications of the incompleteness and one application of the imperfection of the market economy. The first application, Chapter 2, studies the decision making problem of an individual seeking to accumulate an optimal amount of human capital realizing that the wage income derived from the accumulated human capital is subject to incompletely insured uncertainty. In other words, the financial market that insures against wage income risk is not fully functional. We find that the individual's inability to diversify wage income risk tends to increase the need to accumulate more human capital in order to elevate wage path and compensate for the burden of its associated risk. This is particularly true when (i) the wage income risk is positively correlated with the rate-of-return risk in the financial market, resulting in an even greater risk burden to the individual, and (ii) the individual is more risk averse. There are two possibilities that no human capital is needed. The first possibility occurs when it is optimal to work as an unskilled worker because both the burden from wage income risk and the rate of return from education are low. The second possibility is the case where the risk burden is so high that the optimal time spent in school to acquire sufficient human capital to cover the risk is so long that the discounted rate of return from education is negative. In this case, the best strategy is to invest in financial assets alone and forfeit the opportunity to earn wage income - either as an educated or as an unskilled worker - to avoid its associated risk. Chapter 3 applies equilibrium unemployment theory with a frictional labor market to study the impact of immigration on the local labor market. Markets are imperfect in the sense that job matching takes time and recruitment is costly. We find that labor market outcomes of both the natives and existing immigrants depend crucially on how the economic surplus from successful matching is divided between the firms and the workers or, in other words, on the bargaining power of the workers. An arrival of immigrants with low bargaining power tends to benefit both the natives and the existing immigrants. A disparity between the two worker types in the matching efficiency also plays a major role. An inferior matching technology among the immigrants, interpreted here as reflecting their less established social network, lowers their wage rate and increases their unemployment rate. The natives are more likely to benefit from additional immigration than the existing immigrants and, when they do, the overall benefit can be decomposed into "job creation spillover" effect resulting from the immigrants' low bargaining power, and "job stealing" effect resulting from the immigrants' less efficient matching. The implications on the pattern of international migration flows are also discussed. In Chapter 4, a simple macroeconomic model is constructed and applied quantitatively to OECD countries, to analyze the effect of incomplete insurance on saving, growth and welfare in a closed economy. In this economy, precautionary saving motivated by uninsured idiosyncratic shocks raises growth rates but lowers risk-free returns. Welfare is measured by the sum of growth rates and risk-free rates of return, not growth rates alone. This welfare measure takes the negative impact of precautionary saving into consideration. Applied to the OECD data, three major results emerge: (i) the heterogeneous performance of growth and saving across the countries reflects different degrees of insurance incompleteness, (ii) since the externality of growth on productivity was very strong in the 1960's, the heavily constrained insurance market itself improves productivity by promoting growth, thereby enhancing welfare, (iii) while the externality of growth became weaker in the 1980's, the development of insurance markets lowered growth, but still contributed to a raise in welfare.

The research object of this thesis is the sum of a random number of summands of independent identically distributed random variables with positive weights. Such sums appear as models, for example, in insurance, finance mathematics. Throughout the thesis, it is assumed that the random number of summands is independent of the summands, the summands satisfy S. N. Bernstein's condition, and the random number of summands together with weights satisfy some compatibility conditions. The aim of this dissertation is a normal approximation to a distribution of the sum of a random number of summands of independent identically distributed random variables with positive weights that takes into consideration large deviations in both the Cramer and the power Linnik zones. / Disertacinio darbo tyrimo objektas yra atsitiktinio dėmenų skaičiaus nepriklausomų vienodai pasiskirsčiusių atsitiktinių dydžių su teigiamais svoriniais koeficientais sumos, kurios kaip modelis sutinkamos, pavyzdžiui, finansų, draudos matematikose. Daromos prielaidos, kad atsitiktinis dėmenų skaičius yra nepriklausomas nuo sumos dėmenų, atsitiktiniai dėmenys tenkina apibendrintą S. N. Bernšteino sąlygą, o atsitiktinis dėmenų skaičius kartu su svoriais tenkina tam tikras suderinamumo sąlygas. Disertacijos tikslas yra standartizuotos (centruotos ir normuotos) minėtos atsitiktinės sumos skirstinio aproksimacija standartiniu normaliuoju dėsniu didžiųjų nuokrypių tiek Kramero, tiek ir laipsninėse Liniko zonose.

Disertacinio darbo tyrimo objektas yra atsitiktinio dėmenų skaičiaus nepriklausomų vienodai pasiskirsčiusių atsitiktinių dydžių su teigiamais svoriniais koeficientais sumos, kurios kaip modelis sutinkamos, pavyzdžiui, finansų, draudos matematikose. Daromos prielaidos, kad atsitiktinis dėmenų skaičius yra nepriklausomas nuo sumos dėmenų, atsitiktiniai dėmenys tenkina apibendrintą S. N. Bernšteino sąlygą, o atsitiktinis dėmenų skaičius kartu su svoriais tenkina tam tikras suderinamumo sąlygas. Disertacijos tikslas yra standartizuotos (centruotos ir normuotos) minėtos atsitiktinės sumos skirstinio aproksimacija standartiniu normaliuoju dėsniu didžiųjų nuokrypių tiek Kramero, tiek ir laipsninėse Liniko zonose. / The research object of this thesis is the sum of a random number of summands of independent identically distributed random variables with positive weights. Such sums appear as models, for example, in insurance, finance mathematics. Throughout the thesis, it is assumed that the random number of summands is independent of the summands, the summands satisfy S. N. Bernstein's condition, and the random number of summands together with weights satisfy some compatibility conditions. The aim of this dissertation is a normal approximation to a distribution of the sum of a random number of summands of independent identically distributed random variables with positive weights that takes into consideration large deviations in both the Cramer and the power Linnik zones.

In this thesis we shall examine the role of measurability in the theory of Lebesgue Integration. This shall be done in the context of the real line where we define the notion of an integral of a bouuded real-valued function over a set of bounded outer measure without a prior assumption of measurability concerning the function and the domain of integration

A domain-specific embedded language (DSEL) is a domain-specific programming language with no concrete syntax of its own. Defined as a set of combinators encapsulated in a module, it borrows the syntax and tools (such as type-checkers and compilers) of its host language; hence it is economical to design, introduce, and maintain. Unfortunately, this economy is counterbalanced by a lack of room for growth. DSELs cannot match sophisticated domain-specific languages that offer tools for domainspecific error-checking and optimisation. These tools are usually based on syntactic analyses, so they do not work on DSELs. Abstract interpretation is a technique ideally suited to the analysis of DSELs, due to its semantic, rather than syntactic, approach. It is based upon the observation that analysing a program is equivalent to evaluating it over an abstract semantic domain. The mathematical properties of the abstract domain are such that evaluation reduces to solving a mutually recursive set of equations. This dissertation shows how abstract interpretation can be applied to a DSEL by replacing it with an abstract implementation of the same interface; evaluating a program with the abstract implementation yields an analysis result, rather than an executable. The abstract interpretation of DSELs provides a foundation upon which to build sophisticated error-checking and optimisation tools. This is illustrated with three examples: an alphabet analyser for CSP, an ambiguity test for parser combinators, and a definedness test for attribute grammars. Of these, the ambiguity test for parser combinators is probably the most important example, due to the prominence of parser combinators and their rather conspicuous lack of support for the well-known LL(k) test. In this dissertation, DSELs and their signatures are encoded using the polymorphic lambda calculus. This allows the correctness of the abstract interpretation of DSELs to be proved using the parametricity theorem: safety is derived for free from the polymorphic type of a program. Crucially, parametricity also solves a problem commonly encountered by other analysis methods: it ensures the correctness of the approach in the presence of higher-order functions.

It is well known that mathematics students have to be able to understand and prove theorems. From our experience we know that engineering students should also be able to do the same, since a good theoretical knowledge of mathematics is essential for solving practical problems and constructing models. Proving theorems gives students a much better understanding of the subject, and helps them to develop mathematical thinking. The proof of a theorem consists of a logical chain of steps. Students should understand the need and the legitimacy of every step. Moreover, they have to comprehend the reasoning behind the order of the chain’s steps. For our research students were provided with proofs whose steps were either written in a random order or had missing parts. Students were asked to solve the \"puzzle\" – find the correct logical chain or complete the proof. These \"puzzles\" were meant to discourage students from simply memorizing the proof of a theorem. By using our examples students were encouraged to think independently and came to improve their understanding of the subject.

We will survey some of the major directions of research in arithmetic combinatorics and their connections to other fields. We will then discuss three new results. The first result will generalize a structural theorem from Balog and Szemerédi. The second result will establish a new tool in incidence geometry, which should prove useful in attacking combinatorial estimates. The third result evolved from the famous sum-product problem, by providing a partial categorization of bivariate polynomial set functions which induce exponential expansion on all finite sets of real numbers.

Neste trabalho demonstram-se os teoremas de ponto fixo de Brouwer e Kakutani com o objetivo de provar a existência do equilíbrio de Nash em jogos finitos em forma normal. No primeiro capítulo apresentam-se as definições de teoria dos jogos, começando com jogos finitos em forma normal e terminando com o conceito de equilíbrio de Nash. Na primeira seção do capítulo dois desenvolve-se a teoria de simplexes, em Rn, e se demonstra o teorema de Brouwer. Na seção seguinte, são relacionadas as propriedades de semi-continuidade superior e gráfico fechado em set functions, para então provar os teoremas de Celina e von Neumann que, em conjunto com o teorema de Brouwer, resultam no teorema de Kakutani no fim da seção. Como último resultado é demonstrado o teorema de existência do equilíbrio de Nash em jogos finitos em forma normal através do teorema de Kakutani, mostrando que o equilíbrio de Nash é um ponto fixo de uma set function. / In this work, the fixed-point theorems of Kakutani and Brouwer are proved with the intention of showing the existence of Nash equilibrium in finite normal-form games. In the first chapter the needed definitions of game theory are shown, starting with finite normal-form games and ending with the concept of Nash equilibrium. In the first section of chapter two, simplex theory in Rn is developed and then the Brouwer fixer point theorem is proved. In the next section, some relations of upper hemi-continuity and closed graph in set functions are shown, then proving the theorems of Celina and von Neumann that, along with Brouwer theorem, result in Kakutani fixed-point theorem in the end of the section. As the last result, the existence of Nash equilibrium in finite normal-form games is proved through Kakutani’s theorem, relating the Nash equilibrium to the fixed-point of a set function.

Many concrete problems, ranging from Portfolio selection to Water resource management, may be cast into a multiobjective programming framework. The simplistic way of superseding blindly conflictual goals by one objective function let no chance to the model but to churn out meaningless outcomes. Hence interest of discussing ways for tackling Multiobjective Programming Problems. More than this, in many real-life situations, uncertainty and imprecision are in the state of affairs. In this dissertation we discuss ways for solving Multiobjective Programming Problems with fixed and fuzzy coefficients. No preference, a priori, a posteriori, interactive and metaheuristic methods are discussed for the deterministic case. As far as the fuzzy case is concerned, two approaches based respectively on possibility measures and on Embedding Theorem for fuzzy numbers are described. A case study is also carried out for the sake of illustration. We end up with some concluding remarks along with lines for further development, in this field. / Operations Research / M. Sc. (Operations Research)

This study focuses on the discussion regarding mathematical knowledge involved in the work of rural farmers, the possible relations existing within the farmers‟ mathematical concepts, the social-cultural experience of students and the pedagogical practice of teachers of the rural Municipal Schools in the region of Irecê, BA, with respect to area calculation. The main objective was to investigate the practical mathematical knowledge of agricultural workers, specifically, area calculation, in order to establish possible conceptual differences between the informal conduct of rural workers and the formal conduct used in the school. The methodology used in this study was qualitative, as it evaluates the attitudes of the individuals in their social-cultural environment. The research was based on two distinct categories: one belonging to the school, comprising of students from Eighth and Ninth grades, and Math teachers; the other not being part of the school, represented by rural workers, all the residents of the city of Gameleira dos Crentes and the municipality of João Dourado, BA, situated in the micro-region of Irecê, Bahia. The theoretical references were taken from Ethnomathematics, as a possibility of interlocution between formal mathematical knowledge and knowledge stemming from everyday activities, as well as from the discussion of contributions of the social-cultural theory based on Vigotsky, on the anlaysis of concept development proposed by Vergnaud, by which the use of area calculation with fathoms, squares, acres and hectares can be considered in the context of this research, as theorems-in-action. / Este estudo centra-se na discussão sobre os conhecimentos matemáticos envolvidos na prática dos trabalhadores rurais, as possíveis relações existentes entre os conceitos matemáticos dos agricultores, a vivência sócio cultural dos alunos e a prática pedagógica dos professores das Escolas Municipais rurais na região de Irecê/BA, no que diz respeito ao cálculo de área. O objetivo principal foi investigar o conhecimento matemático prático de trabalhadores rurais, em especial, o cálculo de área, a fim de estabelecer possíveis diferenças conceituais entre os procedimentos não formais dos trabalhadores e os procedimentos formais usados na escola. A metodologia empregada configura-se como qualitativa, na medida em que avalia as atitudes dos indivíduos em seu ambiente sócio-cultural. A referida pesquisa conta com duas categorias distintas; uma pertencente à escola, constituída por alunos dos 8º e 9º anos do Ensino Fundamental, professores de matemática; e outra não pertencente à escola, representada por trabalhadores rurais, todos residentes no distrito de Gameleira dos Crentes e do município de João Dourado/BA, situados na micro-região rural de Irecê, na Bahia. Tomamos como referencial teórico as abordagens da Etnomatemática, como uma possibilidade de interlocução entre os saberes matemáticos formais e os saberes do cotidiano, bem como a discussão das contribuições da teoria sócio-cultural embasadas em Vigotsky, na análise do desenvolvimento dos conceitos, proposta por Vergnaud, pela qual a operacionalização de cálculos de área com braças, quadros, tarefas e aceros poderá ser considerada no contexto desta pesquisa, como teoremas-em-ação.