貨幣需求在總體經濟學中一直扮演著重要的角色。同時，貨幣與貨幣性資 產間替代之研究亦是一般貨幣經濟學者關心的課題之一。因為瞭解了這層 關係後，對於如何定義貨幣、貨幣總計數之衡量及貨幣政策的制定等，將 會有所助益。傳統的貨幣需求分析，實質貨幣餘額需求為實質所得( 產 出 )，預期通貨膨脹率與名目利率的函數。但實證結果顯示，使用這些變 數對於貨幣需求的預測或是制定、評估貨幣政策時，並不十分有用。近來 ，許多學者嘗試以符合個體基礎的方式來估計貨幣需求。然而，大部份的 實證結果亦是令人沮喪。本論文將回顧估計貨幣需求的一般化個體─經濟 計量方法，並嘗試使用較新的個體─經濟計量模型─漸近理想化模型( The Asymptotically Id eal Model)來估計台灣地區的貨幣需求。同時也 討論下列問題：ぇ貨幣性資產間的替代性╱互補性。えAIM 與Translog貨 幣需求系統之比較。ぉ效用最大化條件之比較。お一階AIM貨幣需求系統 之動態分析。實證研究的主要結果如下：LTL、HTL貨幣需求系統，不但違 反滿足效用函數彎曲性的必要條件且與需求法則相違背。由一階 AIM貨幣 需求系統估計之彈性值發現，活期儲蓄存款加郵局存簿儲金與活期存款、 郵局劃撥儲金呈現淨互補的關係，印證了交易性存款與儲蓄性存款彼此替 代性不高的現象。就滿足效用最大化條件而言，一階AIM貨幣需求系統滿 足Regular ity條件的情況仍優於LTL、HTL貨幣需求系統。就一階AIM貨幣 性資產成長率之模擬而言，通貨淨額加支票存款之實際與模擬成長率配適 的最佳，其餘次之。

Neste projeto apresentamos algumas direções de pesquisa desenvolvidas no estudo da geometria/topologia da singularidade, no ambiente real e complexo, para funções e aplicações polinomiais. Para isso, utilizaremos as ferramentas da teoria de estratificação, técnicas de decomposição Open book, condições de regularidade no sentido Malgrange, t-regularidade, \'rho\'E-regularidade e trivialidade topológica no infinito / On this project we present some research lines developed in the study of the geometry/ topology of singularity, on the real and complex settings, for functions and polynomial mappings. For this, we use tools from stratification theory, techniques of Open Book decomposition, Malgrange regularity condition, t-regularity condition, \'rho\'E-regularity and topological triviality at infinity

Let f : \'K POT. \' be a \'C POT. 2\' semi-algebraic mapping for K = R and a polynomial mapping for K = C. It is well-known that f is a locally trivial topological fibration over the complement of the bifurcation set B(f), also called atypical set. In this work, we consider the notion of t-regularity and \'ho E\'-regularity to study the bifurcation set of semi-algebraic mappings f : \'R POT. n\' \'ARROW\' \'R POT. p\' and polynomial mappings f : \'C POT. n\' \'ARROW\' \'C POT. p\'. We show that t-regularity is equivalent to regularity conditions at infinity which have been used by Rabier (1997), Gaffney (1999), Kurdyka, Orro and Simon (2000) and Jelonek (2003) in order to control the asymptotic behaviour of mappings. In addition, we prove that t-regularity implies \'ho E\'-regularity. The \'ho E\'-regularity enables one to define the set of asymptotic non \'ho E\'-regular values S(f) \'This contained\' \' K POT. p\', and the set \'A IND. \'ho E\'\' := f(Singf) U S(f). For \'C POT. 2\' semi-algebraic mappings f : \'R POT. n\' ARROW \' \'R POT. p\' and polynomial mappings f : \'C POT. n\' \'ARROW\' \'C POT. p\', based on a partial Thom stratification at infinity, we rove that S(f) and \'A IND. ho E\' are closed real semi-algebraic sets of dimension at most p - 1 (real dimension at most 2p - 2, for f : \'C POT. n\' \'ARROW\' \'C POT. p\'). Moreover, based on a new fibration theorem at infinity, i.e. holding in the complement of a sufficiently large ball, we obtain B(f) \'this contained\' \'A IND. ho E\'. We study two special classes of polynomial mappings f : \'R POT. n\' \"ARROW\' \'R POT. p\', the class of fair polynomial mappings and the class of Newton non-degenerate polynomial mappings. For fair polynomial mappings, we give an interpretation of t-regularity in terms of integral closure of modules, which is a real counterpart of Gaffney\'s result (1999). For non-degenerate polynomial mappings, we obtain an approximation for B(f) through a set which depends on the Newton polyhedron of f (results like this have been obtained by Némethi and Zaharia (1990) for polynomial functions f : \'C POT. n\' \'ARROW\' C and recently for mixed polynomial functions by Chen and Tibar (2012)). To finish, we discuss some simple consequences of our work: the equivalence t regularity Rabier (equivalently Gaffney, Kuo-KOS, Jelonek) condition for mappings f : X \'ARROW\' \'K POT. p\', where X \'this contained\' \'K POT. n\' is a smooth ane variety; the problem of bijectivity of semi-algebraic mappings; and a formula to compute the Euler characteristic of regular fibres of polynomial mappings f : \'R POT. n\' \'AROOW\' \'R POT. n-1\'. The above results are also extensions of some results obtained, for polynomial functions f : \'K POT. n\' \'ARROW K, by Némethi and Zaharia (1990), Siersma and Tibar (1995), Paunescu and Zaharia (1997), Parusinski (1995) and Tibar (1998). Title: Regularity at infinity and global fibrations of real algebraic maps / Considere f : \'K POT. n\' \"SETA\' \'K POT. p\' uma aplicação semi-algébrica de classe \'C POT. 2\' para K = R e uma aplicação polinomial para K = C. Por resultados clássicos, sabe-se que f é uma fibração topologicamente trivial sobre o complementar dos valores de bifurcação B(f), também chamado de valores atípicos. Neste trabalho, consideramos a t-regularidade e a \'ho E\'-regularidade no estudo dos valores de bifurcação de aplicações semi-algébricas f : \'R POT. n\' \'SETA\' \'R POT. p\' de classe \'C POT. 2\' e aplicações polinomiais f : \'C POT. n\' \'SETA\' \'C POT. p\'. Mostramos que t-regularidade é equivalente às condições de regularidade no infinito usadas por Rabier (1997), Gaffney (1999), Kurdyka, Orro e Simon (2000) e Jelonek (2003) no controle do comportamento assintótico de aplicações. Também mostramos que t-regularidade implica \'ho E\'-regularidade. Através da \'ho E\'-regularidade, definimos o conjunto dos valores assintóticos não \'ho E\'- regulares S(f) \'K POT. p\', e o conjunto \'A IND. ho E\' : = f(Singf) U S(f). Para aplicações semialgébricas f : \'R POT. n\' \'SETA\' \'R POT. p\' de classe \'C POT. 2\' e aplicações polinomiais f : \'C POT. \' \'SETA\' \'C POT. p\', baseados na existência de uma estraticação parcial de Thom no infinito, provamos que S(f) e \'A IND. ho E\' são conjuntos semi-algébricos reais de dimensão no máximo p - 1 (dimensão real no máximo 2p 2, para f : \'C POT. \' \'SETA\' \' C POT. p\'). Além disso, baseados em um novo teorema de fibração no infinito, ou seja na existência de fibração no complementar de uma bola de raio suficientemente grande, obtemos que o conjunto de bifurcação B(f) está contido no conjunto \'A IND. ho E\'. Estudamos também duas classes de aplicações polinomiais f : \'R POT. n\' \'SETA\' \'R POT. p\', a classe de aplicações polinomiais fair e a classe de aplicações Newton não degeneradas. Para aplicações polinomiais fair, obtemos uma interpretação da t-regularidade em termos da teoria de fecho integral de módulos, estendendo para o caso real os resultados de Gaffney (1999). Para aplicações não degeneradas, obtemos uma aproximação de B(f) através de um conjunto que depende do poliedro de Newton de f (resultados deste tipo foram obtidos por Némethi e Zaharia (1990) para funções polinomiais f : \'C POT. \' \'SETA\' C e recentemente para funções polinomiais mistas por Chen e Tibar (2012)). No final, discutimos algumas consequências simples do nosso trabalho: a equivalência t-regularidade condição de Rabier (equivalentemente Gaffney, Kuo-KOS, Jelonek) para aplicações f : X \'SETA\' \'K POT. p\', onde X \'está contido\' \'K POT. n\' é uma variedade suave afim; o problema de bijetividade de aplicações semi-algébricas; e uma fórmula para o cálculo da característica de Euler de fibras regulares de aplicações polinomiais f : \'R POT. n\' \'SETA\' \'R POT. n-1\'. Os resultados acima também são extensões de alguns resultados obtidos para funções polinomiais f : \'K POT. n\' \'SETA\' K, por Némethi e Zaharia (1990), Siersma e Tibar (1995), Paunescu e Zaharia (1997), Parusinski (1995) e Tibar (1998). Título: Regularidade no infinito e fibrações globais de aplicações algébricas reais

Neste projeto apresentamos algumas direções de pesquisa desenvolvidas no estudo da geometria/topologia da singularidade, no ambiente real e complexo, para funções e aplicações polinomiais. Para isso, utilizaremos as ferramentas da teoria de estratificação, técnicas de decomposição Open book, condições de regularidade no sentido Malgrange, t-regularidade, \'rho\'E-regularidade e trivialidade topológica no infinito / On this project we present some research lines developed in the study of the geometry/ topology of singularity, on the real and complex settings, for functions and polynomial mappings. For this, we use tools from stratification theory, techniques of Open Book decomposition, Malgrange regularity condition, t-regularity condition, \'rho\'E-regularity and topological triviality at infinity

Let f : \'K POT. \' be a \'C POT. 2\' semi-algebraic mapping for K = R and a polynomial mapping for K = C. It is well-known that f is a locally trivial topological fibration over the complement of the bifurcation set B(f), also called atypical set. In this work, we consider the notion of t-regularity and \'ho E\'-regularity to study the bifurcation set of semi-algebraic mappings f : \'R POT. n\' \'ARROW\' \'R POT. p\' and polynomial mappings f : \'C POT. n\' \'ARROW\' \'C POT. p\'. We show that t-regularity is equivalent to regularity conditions at infinity which have been used by Rabier (1997), Gaffney (1999), Kurdyka, Orro and Simon (2000) and Jelonek (2003) in order to control the asymptotic behaviour of mappings. In addition, we prove that t-regularity implies \'ho E\'-regularity. The \'ho E\'-regularity enables one to define the set of asymptotic non \'ho E\'-regular values S(f) \'This contained\' \' K POT. p\', and the set \'A IND. \'ho E\'\' := f(Singf) U S(f). For \'C POT. 2\' semi-algebraic mappings f : \'R POT. n\' ARROW \' \'R POT. p\' and polynomial mappings f : \'C POT. n\' \'ARROW\' \'C POT. p\', based on a partial Thom stratification at infinity, we rove that S(f) and \'A IND. ho E\' are closed real semi-algebraic sets of dimension at most p - 1 (real dimension at most 2p - 2, for f : \'C POT. n\' \'ARROW\' \'C POT. p\'). Moreover, based on a new fibration theorem at infinity, i.e. holding in the complement of a sufficiently large ball, we obtain B(f) \'this contained\' \'A IND. ho E\'. We study two special classes of polynomial mappings f : \'R POT. n\' \"ARROW\' \'R POT. p\', the class of fair polynomial mappings and the class of Newton non-degenerate polynomial mappings. For fair polynomial mappings, we give an interpretation of t-regularity in terms of integral closure of modules, which is a real counterpart of Gaffney\'s result (1999). For non-degenerate polynomial mappings, we obtain an approximation for B(f) through a set which depends on the Newton polyhedron of f (results like this have been obtained by Némethi and Zaharia (1990) for polynomial functions f : \'C POT. n\' \'ARROW\' C and recently for mixed polynomial functions by Chen and Tibar (2012)). To finish, we discuss some simple consequences of our work: the equivalence t regularity Rabier (equivalently Gaffney, Kuo-KOS, Jelonek) condition for mappings f : X \'ARROW\' \'K POT. p\', where X \'this contained\' \'K POT. n\' is a smooth ane variety; the problem of bijectivity of semi-algebraic mappings; and a formula to compute the Euler characteristic of regular fibres of polynomial mappings f : \'R POT. n\' \'AROOW\' \'R POT. n-1\'. The above results are also extensions of some results obtained, for polynomial functions f : \'K POT. n\' \'ARROW K, by Némethi and Zaharia (1990), Siersma and Tibar (1995), Paunescu and Zaharia (1997), Parusinski (1995) and Tibar (1998). Title: Regularity at infinity and global fibrations of real algebraic maps / Considere f : \'K POT. n\' \"SETA\' \'K POT. p\' uma aplicação semi-algébrica de classe \'C POT. 2\' para K = R e uma aplicação polinomial para K = C. Por resultados clássicos, sabe-se que f é uma fibração topologicamente trivial sobre o complementar dos valores de bifurcação B(f), também chamado de valores atípicos. Neste trabalho, consideramos a t-regularidade e a \'ho E\'-regularidade no estudo dos valores de bifurcação de aplicações semi-algébricas f : \'R POT. n\' \'SETA\' \'R POT. p\' de classe \'C POT. 2\' e aplicações polinomiais f : \'C POT. n\' \'SETA\' \'C POT. p\'. Mostramos que t-regularidade é equivalente às condições de regularidade no infinito usadas por Rabier (1997), Gaffney (1999), Kurdyka, Orro e Simon (2000) e Jelonek (2003) no controle do comportamento assintótico de aplicações. Também mostramos que t-regularidade implica \'ho E\'-regularidade. Através da \'ho E\'-regularidade, definimos o conjunto dos valores assintóticos não \'ho E\'- regulares S(f) \'K POT. p\', e o conjunto \'A IND. ho E\' : = f(Singf) U S(f). Para aplicações semialgébricas f : \'R POT. n\' \'SETA\' \'R POT. p\' de classe \'C POT. 2\' e aplicações polinomiais f : \'C POT. \' \'SETA\' \'C POT. p\', baseados na existência de uma estraticação parcial de Thom no infinito, provamos que S(f) e \'A IND. ho E\' são conjuntos semi-algébricos reais de dimensão no máximo p - 1 (dimensão real no máximo 2p 2, para f : \'C POT. \' \'SETA\' \' C POT. p\'). Além disso, baseados em um novo teorema de fibração no infinito, ou seja na existência de fibração no complementar de uma bola de raio suficientemente grande, obtemos que o conjunto de bifurcação B(f) está contido no conjunto \'A IND. ho E\'. Estudamos também duas classes de aplicações polinomiais f : \'R POT. n\' \'SETA\' \'R POT. p\', a classe de aplicações polinomiais fair e a classe de aplicações Newton não degeneradas. Para aplicações polinomiais fair, obtemos uma interpretação da t-regularidade em termos da teoria de fecho integral de módulos, estendendo para o caso real os resultados de Gaffney (1999). Para aplicações não degeneradas, obtemos uma aproximação de B(f) através de um conjunto que depende do poliedro de Newton de f (resultados deste tipo foram obtidos por Némethi e Zaharia (1990) para funções polinomiais f : \'C POT. \' \'SETA\' C e recentemente para funções polinomiais mistas por Chen e Tibar (2012)). No final, discutimos algumas consequências simples do nosso trabalho: a equivalência t-regularidade condição de Rabier (equivalentemente Gaffney, Kuo-KOS, Jelonek) para aplicações f : X \'SETA\' \'K POT. p\', onde X \'está contido\' \'K POT. n\' é uma variedade suave afim; o problema de bijetividade de aplicações semi-algébricas; e uma fórmula para o cálculo da característica de Euler de fibras regulares de aplicações polinomiais f : \'R POT. n\' \'SETA\' \'R POT. n-1\'. Os resultados acima também são extensões de alguns resultados obtidos para funções polinomiais f : \'K POT. n\' \'SETA\' K, por Némethi e Zaharia (1990), Siersma e Tibar (1995), Paunescu e Zaharia (1997), Parusinski (1995) e Tibar (1998). Título: Regularidade no infinito e fibrações globais de aplicações algébricas reais

The goal of this thesis is two-fold. On the one hand, it pursues to provide a contribution to the conjugate duality by proposing a new duality concept, which can be understood as an umbrella for different meaningful perturbation methods. On the other hand, this thesis aims to investigate minimax location problems by means of the duality concept introduced in the first part of this work, followed by a numerical approach using epigraphical splitting methods. After summarizing some elements of the convex analysis as well as introducing important results needed later, we consider an optimization problem with geometric and cone constraints, whose objective function is a composition of n+1 functions. For this problem we propose a conjugate dual problem, where the functions involved in the objective function of the primal problem are decomposed. Furthermore, we formulate generalized interior point regularity conditions for strong duality and give necessary and sufficient optimality conditions. As applications of this approach we determine the formulae of the conjugate as well as the biconjugate of the objective function of the primal problem and analyze an optimization problem having as objective function the sum of reciprocals of concave functions. In the second part of this thesis we discuss in the sense of the introduced duality concept three classes of minimax location problems. The first one consists of nonlinear and linear single minimax location problems with geometric constraints, where the maximum of nonlinear or linear functions composed with gauges between pairs of a new and existing points will be minimized. The version of the nonlinear location problem is additionally considered with set-up costs. The second class of minimax location problems deals with multifacility location problems as suggested by Drezner (1991), where for each given point the sum of weighted distances to all facilities plus set-up costs is determined and the maximal value of these sums is to be minimized. As the last and third class the classical multifacility location problem with geometrical constraints is considered in a generalized form where the maximum of gauges between pairs of new facilities and the maximum of gauges between pairs of new and existing facilities will be minimized. To each of these location problems associated dual problems will be formulated as well as corresponding duality statements and necessary and sufficient optimality conditions. To illustrate the results of the duality approach and to give a more detailed characterization of the relations between the location problems and their corresponding duals, we consider examples in the Euclidean space. This thesis ends with a numerical approach for solving minimax location problems by epigraphical splitting methods. In this framework, we give formulae for the projections onto the epigraphs of several sums of powers of weighted norms as well as formulae for the projection onto the epigraphs of gauges. Numerical experiments document the usefulness of our approach for the discussed location problems.

In this work we present the most recent developments in the direction of local fibrations structures of analytic singularities. Using techniques and tools from stratification theory we prove structural theorems in the stratified sense, which will be called singular Milnor tube fibration and Milnor-Hamm sphere fibration. In addition, we present algorithms with the purpose of creating a large number of examples in this new setting and compare our results obtained with the current ones found in the literature. Our results generalize all previous result in both cases: in the classical and in the stratified ones. / Neste trabalho apresentamos os mais recentes desenvolvimentos na direção de estruturas de fibrações locais de singularidades analíticas. Usando técnicas e ferramentas da teoria de estratificação, provamos alguns teoremas estruturais no sentido estratificado, os quais serão chamados fibração singular de Milnor sobre o tubo e fibração de Milnor-Hamm sobre a esfera. Além disso, apresentamos algoritmos com o intuito de criar uma ampla variedade de exemplos e comparamos nossos resultados com os atuais encontrados na literatura. Nossos resultados generalizam todos os previamente existentes tanto no caso clássico, quanto no sentido estratificado.

In this work we present the most recent developments in the direction of local fibrations structures of analytic singularities. Using techniques and tools from stratification theory we prove structural theorems in the stratified sense, which will be called singular Milnor tube fibration and Milnor-Hamm sphere fibration. In addition, we present algorithms with the purpose of creating a large number of examples in this new setting and compare our results obtained with the current ones found in the literature. Our results generalize all previous result in both cases: in the classical and in the stratified ones. / Neste trabalho apresentamos os mais recentes desenvolvimentos na direção de estruturas de fibrações locais de singularidades analíticas. Usando técnicas e ferramentas da teoria de estratificação, provamos alguns teoremas estruturais no sentido estratificado, os quais serão chamados fibração singular de Milnor sobre o tubo e fibração de Milnor-Hamm sobre a esfera. Além disso, apresentamos algoritmos com o intuito de criar uma ampla variedade de exemplos e comparamos nossos resultados com os atuais encontrados na literatura. Nossos resultados generalizam todos os previamente existentes tanto no caso clássico, quanto no sentido estratificado.

The goal of this thesis is two-fold. On the one hand, it pursues to provide a contribution to the conjugate duality by proposing a new duality concept, which can be understood as an umbrella for different meaningful perturbation methods. On the other hand, this thesis aims to investigate minimax location problems by means of the duality concept introduced in the first part of this work, followed by a numerical approach using epigraphical splitting methods. After summarizing some elements of the convex analysis as well as introducing important results needed later, we consider an optimization problem with geometric and cone constraints, whose objective function is a composition of n+1 functions. For this problem we propose a conjugate dual problem, where the functions involved in the objective function of the primal problem are decomposed. Furthermore, we formulate generalized interior point regularity conditions for strong duality and give necessary and sufficient optimality conditions. As applications of this approach we determine the formulae of the conjugate as well as the biconjugate of the objective function of the primal problem and analyze an optimization problem having as objective function the sum of reciprocals of concave functions. In the second part of this thesis we discuss in the sense of the introduced duality concept three classes of minimax location problems. The first one consists of nonlinear and linear single minimax location problems with geometric constraints, where the maximum of nonlinear or linear functions composed with gauges between pairs of a new and existing points will be minimized. The version of the nonlinear location problem is additionally considered with set-up costs. The second class of minimax location problems deals with multifacility location problems as suggested by Drezner (1991), where for each given point the sum of weighted distances to all facilities plus set-up costs is determined and the maximal value of these sums is to be minimized. As the last and third class the classical multifacility location problem with geometrical constraints is considered in a generalized form where the maximum of gauges between pairs of new facilities and the maximum of gauges between pairs of new and existing facilities will be minimized. To each of these location problems associated dual problems will be formulated as well as corresponding duality statements and necessary and sufficient optimality conditions. To illustrate the results of the duality approach and to give a more detailed characterization of the relations between the location problems and their corresponding duals, we consider examples in the Euclidean space. This thesis ends with a numerical approach for solving minimax location problems by epigraphical splitting methods. In this framework, we give formulae for the projections onto the epigraphs of several sums of powers of weighted norms as well as formulae for the projection onto the epigraphs of gauges. Numerical experiments document the usefulness of our approach for the discussed location problems.

The aim of this thesis is to present new results concerning duality in scalar optimization. We show how the theory can be applied to optimization problems arising in the theory of risk measures, portfolio optimization and machine learning. First we give some notations and preliminaries we need within the thesis. After that we recall how the well-known Lagrange dual problem can be derived by using the general perturbation theory and give some generalized interior point regularity conditions used in the literature. Using these facts we consider some special scalar optimization problems having a composed objective function and geometric (and cone) constraints. We derive their duals, give strong duality results and optimality condition using some regularity conditions. Thus we complete and/or extend some results in the literature especially by using the mentioned regularity conditions, which are weaker than the classical ones. We further consider a scalar optimization problem having single chance constraints and a convex objective function. We also derive its dual, give a strong duality result and further consider a special case of this problem. Thus we show how the conjugate duality theory can be used for stochastic programming problems and extend some results given in the literature. In the third chapter of this thesis we consider convex risk and deviation measures. We present some more general measures than the ones given in the literature and derive formulas for their conjugate functions. Using these we calculate some dual representation formulas for the risk and deviation measures and correct some formulas in the literature. Finally we proof some subdifferential formulas for measures and risk functions by using the facts above. The generalized deviation measures we introduced in the previous chapter can be used to formulate some portfolio optimization problems we consider in the fourth chapter. Their duals, strong duality results and optimality conditions are derived by using the general theory and the conjugate functions, respectively, given in the second and third chapter. Analogous calculations are done for a portfolio optimization problem having single chance constraints using the general theory given in the second chapter. Thus we give an application of the duality theory in the well-developed field of portfolio optimization. We close this thesis by considering a general Support Vector Machines problem and derive its dual using the conjugate duality theory. We give a strong duality result and necessary as well as sufficient optimality conditions. By considering different cost functions we get problems for Support Vector Regression and Support Vector Classification. We extend the results given in the literature by dropping the assumption of invertibility of the kernel matrix. We use a cost function that generalizes the well-known Vapnik's ε-insensitive loss and consider the optimization problems that arise by using this. We show how the general theory can be applied for a real data set, especially we predict the concrete compressive strength by using a special Support Vector Regression problem.