Tesis sobre el tema "Existence of equilibrium"

COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>Propomos um modelo de equilíbrio geral com securitização de ativos pro- tegidos por colateral físico e discutimos o papel do colateral para evitar, em equilíbrio, expectativas pessimistas em relação as taxas futuras de default. As promessas dos ativos garantidos por colateral são securitizadas em famílias de derivativos Pass-through, que dividem o default de forma prorata, ou de Collateralized Loans Obligations (CLO), que permitem diferentes níveis de prioridade entre os ativos dentro de uma mesma família. No caso de default, os tomadores de empréstimo podem sofrer penalidades extra-econômicas, que são internalizadas nas suas preferências não ordenadas. Nesse contexto, provamos a existência de um equilíbrio no qual os investidores não são muito pessimistas sobre os pagamentos dos derivativos.<br>We propose a specification of a general equilibrium model with securitization of collateral-backed promises and discuss the role of physical collateral to avoid, in equilibrium, pessimistic beliefs about the future rates of default. Promises are pooled in either pass-through securities or collateralized loans obligations (CLO), allowing the existence of different seniority levels among tranches in the same CLO. In case of default, borrowers may also suffer extra-economic penalties, which are internalized into a structure of nonordered preferences. In this context, we prove the existence of an equilibrium in that investors are not over-pessimistic about the payments of derivatives.

In this work the problematic issue of price determination in economic theory is re-examined. In the first chapter a state-of-the-art survey regarding the existence of equilibrium and the structure of the equilibrium set in price-setting games is provided. In chapter two a new core concept, the Bertrand core, is introduced and characterized. In chapter three a revealed preference perspective upon the Nash equilibria in price-setting games is provided. In chapter four, the issue of Bayesian equilibrium existence is addressed when traders have incomplete information regarding each others' types. Finally, a summary of possible future avenues for research in this area is provided.

Cette thèse est centrée sur l’étude d’équations issues de la théorie cinétique des gaz. Dans tous les problèmes qui y sont explorés, une analyse des problèmes linéaires ou linéarisés associés est réalisée d’un point de vue spectral et du point de vue des semi-groupes. A cela s’ajoute une analyse de la stabilité non linéaire lorsque le modèle est non linéaire. Plus précisément, dans une première partie, nous nous intéressons aux équations de Fokker-Planck fractionnaire et Boltzmann sans cut-off homogène en espace et nous prouvons un retour vers l’équilibre des solutions de ces équations avec un taux exponentiel dans des espaces de type L1 à poids polynomial. Concernant l’équation de Landau inhomogène en espace, nous développons une théorie de Cauchy de solutions perturbatives dans des espaces de type L2 avec différents poids (polynomiaux ou exponentiels) et nous prouvons également la stabilité exponentielle de ces solutions.Nous démontrons ensuite pour l’équation de Boltzmann inélastique inhomogène avec terme diffusif le même type de résultat dans des espaces L1 à poids polynomial dans un régime de faible inélasticité. Pour finir, nous étudions dans un cadre général et uniforme des modèles qui convergent vers l’équation de Fokker-Planck du point de vue de l’analyse spectrale et des semi-groupes<br>The topic of this thesis is the study of models coming from kinetic theory. In all the problems that are addressed, the associated linear or linearized problem is analyzed from a spectral point of view and from the point of view of semigroups. Tothat, we add the study of the nonlinear stability when the equation is nonlinear. More precisely, to begin with, we treat the problem of trend to equilibrium for the fractional Fokker-Planck and Boltzmann without cut-off equations, proving an exponential decay to equilibrium in spaces of type L1 with polynomial weights. Concerning the inhomogeneous Landau equation, we develop a Cauchy theory of perturbative solutions in spaces of type L2 with various weights such as polynomial and exponential weights and we also prove the exponential stability of these solutions. Then, we prove similar results for the inhomogeneous inelastic diffusively driven Boltzmann equation in a small inelasticity regime in L1 spaces with polynomial weights. Finally, we study in the same and uniform framework from the spectral analysis point of view with a semigroup approach several Fokker-Planck equations which converge towards the classical one

Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2015.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 63-65).<br>Competitive equilibrium provides a natural steady state for iterative combinatorial auctions that maximize social welfare, and therefore the first step in auction design is to establish its existence. Recent work by Baldwin and Klemperer (2012) has proved that the "demand type" of valuations being "unimodular" is a necessary and sufficient condition for the existence of a competitive equilibrium, but under the general setting where both buyers and sellers as well as multiple copies of items may exist, and the supply could be any combination of items available. In this work, we investigate the same condition under the more restrictive but standard setting for combinatorial auctions, where only buyers and a single copy of each distinct item are allowed and the supply is fixed to be the set of all available items. First, we provide an alternative proof of the sufficiency result for unimodular "complements" demand type, which defines a subclass of valuations for which a competitive equilibrium exists according to Baldwin and Klemperer (2012). While their original proof and analysis use tools from tropical geometry, our approach is based on linear programming. Relying on a result from Bikhchandani and Mamer (1999) that a competitive equilibrium exists if and only if a related linear program has an integral optimal solution, we provide a direct proof that the linear program has an integral optimal solution. Our analysis provides a fundamental understanding of the structure of the linear program and leads to various properties which may be helpful in auction design. Second, we provide an algorithm to determine the demand types of sign-consistent tree graphical valuations, for which competitive equilibria are known to exist due to Candogan et al. (2013). We then analyze the relationship between the set of the demand types of sign-consistent tree graphical valuations and the set of unimodular demand types. Our analysis implies that the unimodularity of demand type is not necessary for the existence of a competitive equilibrium in combinatorial auctions.<br>by Ji Young Lee.<br>S.M.

Ce mémoire de thèse est consacré à l’équation de Fokker-Planckpartial_ f=∆f+div(Ef).Il est subdivisé en deux parties :une partie linéaire et une partie non linéaire. Dans la partie linéaire on considère un champ de vecteur E(x) dépendant seulement de x. Cette partie est constituée des chapitres 3, 4 et 5. Dans le chapitre 3 on montre que l’opérateur linéaire Lf :=∆ f + div(E f ) est le générateur d’un semi-groupe fortement continu (SL(t))_{t≥0} dans tous les espaces L^p. On y établit également que le semi-groupe (SL(t))_{t≥0} est positif et ultracontractif. Dans le chapitre 4 nous montrons comment est qu’une décomposition adéquate de l’opérateur L permet d’établir certaines propriétés du semi-groupe (SL(t))_{t≥0} notamment sa bornitude. Le chapitre 5 est consacré à l’existence d’un état d’équilibre. De plus on y montre que cet état d’équi- libre est asymptotiquement stable. Dans la partie non linéaire on considère un champ de vecteur de la forme E(x,f) := x+nabla (a*f) ou a et f sont des fonctions assez régulières et * est l’opérateur de convolution. Cette parties est contituée des chapitre 6 et 7. Dans le chapitre 6 nous établissons que poura appartenant à W^{2,infini}_locl’équation de Fokker-Planck non linéaire admet une unique solution locale dans l’espace L^2_{K_alpha} (R^d). Dans le dernier chapitre nous montrons que le problème non linéaire admet une solution globale. De plus cette solution dépend continument des données<br>This thesis is devoted to the Fokker-Planck équation partial_t f =∆f + div(E f).It is divided into two parts. The rst part deals with the linear problem. In this part we consider a vector E(x) depending only on x. It is composed of chapters 3, 4 and 5. In chapter 3 we prove that the linear operator Lf :=∆f + div(Ef ) is an in nitesimal generator of a strong continuous semigroup (SL(t))_{t≥0}. We establish also that (SL(t))_{t≥0} is positive and ultracontractive. In chapter 4 we show how an adequate decomposition of the linear operator L allows us to deduce interesting properties for the semigroup (SL(t))_{t≥0}. Indeed using this decomposition we prove that (SL(t))_{t≥0} is a bounded semigroup. In the last chapter of this part we establish that the linear Fokker-Planck admits a unique steady state. Moreover this stationary solution is asymptotically stable.In the nonlinear part we consider a vector eld of the form E(x, f ) := x +nabla (a *f ), where a and f are regular functions. It is composed of two chapters. In chapter 6 we establish that fora in W^{2,infini}_locthe nonlinear problem has a unique local solution in L^2_{K_alpha}(R^d); . To end this part we prove in chapter 7 that the nonlinear problem has a unique global solution in L^2_k(R^d). This solution depends continuously on the data

Cette thèse se focalise sur les jeux de potentiel et une généralisation d'un jeu d'ordonnancement dans un graphe que nous avons appelé jeu de placement.Dans ce jeu, le coût d'un joueur est impacté par son voisinage.Nous pouvons illustrer cela avec un exemple : le placement de joueurs dans un train, pour lesquels la présence de voisins directs influe sur le bien-être.Les résultats de cette thèse se divisent en deux parties.Tout d'abord, nous étudions ces jeux en considérant l'existence et les propriétés de structure des équilibres.Nous nous posons la question fondamentale de savoir s'il existe des équilibres de Nash dans le jeu de placement.Si tel est le cas, nous tachons de déterminer si ces équilibres sont facilement calculables.Dans le cas où il n'existe pas d'équilibre nous prouvons la NP-complétude du problème.Dans un second temps nous nous intéressons à la notion de calcul distribué d'équilibre de Nash dans des jeux de placement.En particulier nous considérons un jeu basé sur le problème de Max-Cut, qui a été plus étudié en théorie des graphes.Cela nous a permis d'étendre nos travaux à une application aux réseaux mobiles pour la gestion d'interférences dans les réseaux sans fils.Nous avons pu, pour les différents jeux, mettre en place des algorithmes distribués de calcul d'équilibres et étudier leur convergence.Parallèlement, nous avons étendu les travaux de Max-Cut à un problème de sélection d'offre de qualité de service parmi divers fournisseurs d'accès.Nous comparons les performances d'algorithmes de calcul distribué d'équilibres et de minimisation de regret<br>This thesis focuses on potential games and a generalized load balancing game in a graph we called placement game.In this game, the cost of a player is affected by its neighbors.We can illustrate this with an example: the placement of players on a train, where the presence of direct neighbors affects their well-being.The results of this thesis are divided into two parts.First, we study these games considering the existence and structural properties of equilibria.We ask ourselves the fundamental question of whether there are Nash equilibria in the placement game.If this is the case we aim to determine if they are easily calculable, if there is no such equilibria we prove the NP-completeness of the problem.Secondly we focus on the concept of distributed algorithms to compute Nash equilibria in placement games.In particular we consider a game based on the Max-Cut problem, which has been more frequently studied.This allowed us to expand our work to a mobile network application for managing interference in wireless networks.We were able, for those different games, to implement distributed algorithms to compute equilibria and study their convergence.Meanwhile, we have expanded the Max-Cut works with a selection of QoS offers problem from various network providers.We compare the performance of distributed algorithms and regret minimization

Cette thèse est consacrée à l’étude du transport réactif dans les réserves en eaux. Elle est structurée en deux parties distinctes : la première porte sur l’élaboration de solveurs chimiques et la seconde sur l’étude mathématique d’une classe de modèles décrivant des écoulements en eaux peu profondes en interaction avec les eaux de surface.Dans la première partie du travail, on s’intéresse à la résolution numérique des équilibres thermodynamiques qui conduisent à des systèmes non linéaires complexes et très mal conditionnés. Dans ce travail, on combine une formulation particulière du système d’équilibre chimique, appelée la méthode des fractions continues positives, avec deux méthodes numériques itératives, la méthode d’Accélération d’Anderson et des méthodes d’extrapolation vectorielle, à savoir les méthodes MPE (minimal polynomial extrapolation) et RRE (reduced rank extrapolation). Le principal avantage de ces approches est d’éviter de former la matrice jacobienne et donc d’éviter les problèmes liés aux mauvais conditionnements de la matrice. Des tests numériques sont faits, notamment sur le cas test de l’acide gallique et sur le cas test 1D de référence du benchmark MoMas. Ces essais illustrent la grande efficacité de cette approche par rapport aux résolutions classiques résultant de la méthode de Newton-Raphson. Dans la seconde partie de la thèse, on introduit et étudie des modèles de type Richards-Dupuit pour décrire les écoulements dans des aquifères peu profonds. L’idée est de coupler les deux types d’écoulements principaux présents dans l’aquifère : celui de la partie insaturée avec celui de la partie saturée. Le premier est décrit par le problème classique de Richards dans la frange capillaire supérieure.Le second résulte de l’approximation de Dupuit après intégration verticale des lois de conservation entre le fond de l’aquifère et l’interface de saturation. Le modèle final consiste en un système fortement couplé d’edp de type parabolique qui sont définies sur un domaine dépendant du temps. Nous montrons comment la prise en compte de la faible compressibilité du fluide permet d’éliminer la dégénérescence présente dans la dérivée temporelle de l’équation de Richards. Puis nous utilisons le cadre général des équations paraboliques dans des domaines non cylindriques introduit par Lions pour donner un résultat d’existence global en temps. Nous présentons l’analyse mathématique du premier modèle qui correspond au cas isotrope et non conservatif. Puis nous généralisons l’étude au cas anisotrope et conservatif<br>This thesis is devoted to the study of reactive transport in water reserves. It is structured in two distinct parts : the first deals with the development of chemical solvers and the second with the mathematical study of a class of models describing flows in shallow water interacting with the surface water. In the first part of the work, we focus on the numerical resolution of thermodynamic equilibria which lead to complex and very badly conditioned nonlinear systems. In this work, we combine aparticular formulation of the chemical equilibrium system, called the method of positive continuous fractions, with two iterative numerical methods, the Anderson Acceleration method and vector extrapolation methods, namely the MPE (minimal polynomial extrapolation) and RRE (reduced rank extrapolation) methods.The main advantage of these approaches is to avoid forming the Jacobian matrix and thus to avoid problems linked to bad conditioning of the matrix. Numerical tests are performed, especially on the test case of gallic acid and on the reference 1D case of the MoMas benchmark. These tests illustrate the great efficiency of this approach compared to classical solutions resulting from the Newton-Raphson method. In the second part of the thesis, we introduce and study Richards-Dupuit type models to describe flows in shallow aquifers. The idea is to couple the two main types of flows in the aquifer : that of the unsaturated part with that of the saturated part. The first is described by the classic Richardsproblem in the upper capillary fringe. The second results from Dupuit’s approximation after vertical integration of the conservation laws between the bottom of the aquifer and the saturation interface. The final model consists of a strongly coupled system of parabolic type pde which are defined on a time dependent domain. We show how taking into account the low compressibility of the fluid makes it possible to eliminate the degeneration in the time derivative term of the Richards equation.Then we use the general framework of parabolic equations in non-cylindrical domains introduced by Lions to give a global existence result in time. We present the mathematical analysis of the first model which corresponds to the isotropic and non-conservative case. Then we generalize the study to the anisotropic and conservative case

Cette thèse est essentiellement composée de deux parties traitant des problèmes de Cauchy ou des problèmes périodiques. Dans la première partie, on étudie la stabilité de solutions régulières au voisinage d'états d'équilibre non constants pour un système d'Euler-Maxwell isentropique compressible bipolaire. Par des estimations d'énergie classiques et un argument de récurrence sur l'ordre des dérivées des solutions, on montre l'existence globale et l'unicité des solutions régulières du système lorsque les données initiales sont proches des états d'équilibre. On obtient aussi le comportement asymptotique des solutions quand le temps tend vers l'infini. Dans la deuxième partie, on considère la stabilité en temps long des solutions régulières de systèmes d'Euler-Maxwell et de Navier-Stokes-Maxwell compressibles dans le cas non isentropique lorsque les états d'équilibre sont constants. Grâce à des choix convenables de symétriseurs des systèmes et à des estimations d'énergie, on montre l'existence globale et l'unicité des solutions régulières des systèmes avec données initiales petites. De plus, par le principe de Duhamel et l'outil d'analyse de Fourier, on obtient des taux de décroissance des solutions quand le temps tend vers l'infini<br>This thesis is essentially composed of two parts dealing with Cauchy problems and periodic problems. In the first part, we study the stability of smooth solutions near non constant equilibrium states for a two-fluid isentropic compressible Euler-Maxwell system.By classical energy estimates together with an induction argument on the order of the derivatives of solutions, we prove the existence and uniqueness of global solutions to the system when the given initial data are near the equilibrium states. We also obtain the asymptotic behavior of solutions when the time goes to infinity. In the second part, we consider the long time stability of the global smooth solutions for compressible Euler-Maxwell and Navier-Stokes-Maxwell systems in non isentropic case when the equilibrium solutions are constants. With the help of suitable choices of symmetrizers and energy estimates, we prove the existence and uniqueness of global solutions to the systems with given small initial data. Furthermore, using the Duhamel principle and the Fourier analysis tool, we obtain the decay rates of smooth solutions as the time goes to infinity

La croissante demande d'énergie électrique a conduit à la conception de systèmes électriques de grande complexité où les combustibles fossiles constituent la principale source d'énergie. Néanmoins, les préoccupations environnementales poussent à un changement majeur dans les pratiques de production d'électricité, avec un passage marqué des énergies fossiles aux énergies renouvelables et des architectures centralisées à distribuées. Les problèmes de stabilité dus à la présence de ce qu'on appelle les Charges à Puissance Constante (CPLs) constituent l’un des principaux défis auxquels sont confrontés les systèmes électriques distribués. On sait que ces charges, que l’on trouve couramment dans les installations de technologie de l’information et de la communication, réduisent l’amortissement effectif des circuits qui les alimentent, ce qui peut provoquer des oscillations de tension, voire une chute. Dans cette thèse, les principales contributions sont centrées sur la compréhension et la résolution de divers problèmes rencontrés dans l'analyse et le contrôle de systèmes électriques contenant des CPLs. Les contributions sont énumérées comme suit. (i) Des conditions simplement vérifiables sont proposées pour certifier la non existence d'états en régime permanent pour des réseaux multi-ports, à courant alternatif avec une distribution de CPLs. Ces conditions, qui reposent sur les inégalités matricielles linéaires, permettent d’écarter les valeurs des puissances des charges qui produiraient certainement un effondrement de la tension sur l’ensemble du réseau. (ii) Pour des modèles généraux de certains systèmes électriques modernes, y compris les réseaux de transmission à courant continu haute tension et les microréseaux, il est montré que, si des équilibres existent, il existe un équilibre caractéristique à haute tension qui domine tous les autres. En outre, dans le cas des systèmes d'alimentation en courant alternatif sous l'hypothèse de découplage standard, cet équilibre caractéristique s'avère stable à long terme. (iii) Une classe de systèmes port-Hamiltoniens, dans laquelle les variables de contrôle agissent directement sur l'équation du balance de puissance, est explorée. Il est démontré que ces systèmes sont décalés de manière passive lorsque leurs trajectoires sont contraintes à des ensembles facilement définissables. Ces dernières propriétés sont exploitées pour analyser la stabilité de leurs équilibres intrinsèquement non nuls. Il a également été montré que la stabilité des réseaux électriques à courant continu multiports et des générateurs synchrones, tous deux connectés à des CPLs, peuvent naturellement être étudiée avec le cadre proposé. (iv) Le problème de la régulation de la tension de sortie du convertisseur buck-boost alimentant une CPL non connu est résolu. L'un des principaux obstacles à la conception de commandes linéaires classiques provient du fait que le modèle du système est de phase non minimale par rapport à chacune de ses variables d'état. Cette thèse rapporte un contrôleur adaptatif non linéaire capable de rendre un équilibre souhaité asymptotiquement stable; de plus, une estimation de la région d'attraction peut être calculée. (v) La dernière contribution concerne l'amortissement actif d'un système d'alimentation de petite taille à courant continu avec une CPL. Au lieu de connecter des éléments passifs peu pratiques et énergétiquement inefficaces au réseau existant, l’ajout d’un convertisseur de puissance contrôlé est exploré. La contribution principale rapportée ici est la conception d'une loi de contrôle non linéaire basée sur l'observateur pour le convertisseur. La nouveauté de la proposition réside dans le fait qu'il n'est pas nécessaire de mesurer le courant électrique du réseau ni la valeur de la CPL, soulignant ainsi son applicabilité pratique. L'efficacité du schéma de contrôle est ensuite validée par des expériences sur un réseau à courant continu réel<br>The continuously increasing demand of electrical energy has led to the conception of power systems of great complexity that may extend even through entire countries. In the vast majority of large-scale power systems the main primary source of energy are fossil fuels. Nonetheless, environmental concerns are pushing a major change in electric energy production practices, with a marked shift from fossil fuels to renewables and from centralized architectures to more distributed ones. One of the main challenges that distributed power systems face are the stability problems arising from the presence of the so-called Constant Power Loads (CPLs). These loads, which are commonly found in information and communication technology facilities, are known to reduce the effective damping of the circuits that energize them, which can cause voltage oscillations or even voltage collapse. In this thesis, the main contributions are focused in understanding and solving diverse problems found in the analysis and control of electrical power systems containing CPLs. The contributions are listed as follows. (i) Simply verifiable conditions are proposed to certify the non existence of steady states in general, multi-port, alternating current (AC) networks with a distributed array of CPLs. These conditions, which are based on Linear Matrix Inequalities, allow to discard the values of the loads' powers that would certainly produce a voltage collapse in the whole network. (ii) For general models of some modern power systems, including High-Voltage Direct Current transmission networks and microgrids, it is shown that if equilibria exist, then there is a characteristic high-voltage equilibrium that dominates, entry-wise, all the other ones. Furthermore, for the case of AC power systems under the standard decoupling assumption, this characteristic equilibrium is shown to be long-term stable. (iii) A class of port-Hamiltonian systems, in which the control variables act directly on the power balance equation, is explored. These systems are shown to be shifted passive when their trajectories are constrained to easily definable sets. The latter properties are exploited to analyze the stability of their---intrinsically non zero---equilibria. It is also shown that the stability of multi-port DC electrical networks and synchronous generators, both with CPLs, can be naturally studied with the proposed framework. (iv) The problem of regulating the output voltage of the versatile DC buck-boost converter feeding an {em unknown} CPL is addressed. One of the main obstacles for conventional linear control design stems from the fact that the system's model is non-minimum phase with respect to each of its state variables. As a possible solution to this problem, this thesis reports a nonlinear, adaptive controller that is able to render a desired equilibrium asymptotically stable; furthermore an estimate of the region of attraction can be computed. (v) The last contribution concerns the active damping of a DC small-scale power system with a CPL. Instead of connecting impractical, energetically inefficient passive elements to the existing network, the addition of a controlled DC-DC power converter is explored. The main contribution reported here is the design of a nonlinear, observer-based control law for the converter. The novelty of the proposal lies in the non necessity of measuring the network's electrical current nor the value of the CPL, highlighting its practical applicability. The effectiveness of the control scheme is further validated through experiments on a real DC network

This thesis consists of three parts: The first and second parts focus on long-time asymptotics of macroscopic and kinetic models respectively, while in the third part we connect these regimes using different scaling approaches. (1) Keller–Segel-type aggregation-diffusion equations: We study a Keller–Segel-type model with non-linear power-law diffusion and non-local particle interaction: Does the system admit equilibria? If yes, are they unique? Which solutions converge to them? Can we determine an explicit rate of convergence? To answer these questions, we make use of the special gradient flow structure of the equation and its associated free energy functional for which the overall convexity properties are not known. Special cases of this family of models have been investigated in previous works, and this part of the thesis represents a contribution towards a complete characterisation of the asymptotic behaviour of solutions. (2) Hypocoercivity techniques for a fibre lay-down model: We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modelling the fibre lay-down process in non-woven textile production. Further, we prove convergence to equilibrium with an explicit rate. This part of the thesis is an extension of previous work which considered the case of a stationary conveyor belt. Adding the movement of the belt, the global equilibrium state is not known explicitly and a more general hypocoercivity estimate is needed. Although we focus here on a particular application, this approach can be used for any equation with a similar structure as long as it can be understood as a certain perturbation of a system for which the global Gibbs state is known. (3) Scaling approaches for collective animal behaviour models: We study the multi-scale aspects of self-organised biological aggregations using various scaling techniques. Not many previous studies investigate how the dynamics of the initial models are preserved via these scalings. Firstly, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local kinetic 1D and 2D models to simpler models existing in the literature. Secondly, we investigate how some of the kinetic spatio-temporal patterns are preserved via these scalings using asymptotic preserving numerical methods.

碩士<br>東吳大學<br>數學系<br>94<br>In this paper, we assume that the individually rational utility set is compact and use demand approach to prove an equilibrium existence theorem for finite dimensional economies. This extends the result of Dana, Le Van, and Magnien(1999)from exchange economies to production economies.

The problem of necessary and sufficient conditions for the existence of a unique Cournot equilibrium has been rarely dealt with in the literature, To date there exists virtually only the formulation of the problem given by Kolstad and Mathiesen (1987). Yet several aspects of their approach are debatable. Here, a new formulation is proposed which considers sets of oligopolies rather than single oligopolies. This allows to obtain a new result which brings out the necessity of some important conditions assumed in Kolstad and Mathiesen and often employed to prove the existence of a Cournot equilibrium.

博士<br>中原大學<br>數學系<br>88<br>This thesis contains the following three main theorems, the first theorem settles the Shih-Ho''s conjecture. Theorem. Let F : {0, 1}^n --> {0, 1}^n. If ρ(F''(x)) = 0 for all x in {0, 1}^n, then F has a unique fixed point. Theorem. Let F : {0, 1}^n --> {0, 1}^n. If ρ(F''(x)) = 0 for all x in {0, 1}^n, then F has a unique fixed point ξ. If, in addition, F(V_1(x)) included in V_1(F(x)) for all x in V_{n-2}(ξ), then ξ is a strongly global attractor. The V_{n-2}(ξ) is optimal. Theorem. Let F : {0, 1}^n --> {0, 1}^n. Suppose ρ(F''(x)) =0 for all x in {0, 1}^n, and F(V_1(x)) included in V_1(F(x)) for all x in {0, 1}n. Then for each x in {0, 1}^n there exist a unique point α in x[{1, ..., n-1}|n] _ and a unique point β in x[{1, ..., n-1}\,|\,n] such that f_j(α) = α_j and f_j(β) = β_j for all j = 1, ..., n-1. Further, if ρ_H(α,β ) = n, then there exist two positive integers p, q less or equal to n such that F^p(x) =α and G^q(x) = α for all x in {0, 1}^n or F^p(x) =β and G^q(x) =β for all x in {0, 1}^n. Consider the set of all points in n-dimensional space with each coordinate equal to zero or one. These points may be thought of as the corners of a n-dimensional cube. We let these points correspond to processors, and we consider a communication link for every two points differing in a single coordinate. The resulting network is called a n-cube {0, 1}^n. The n-cube architecture has many attractive features (see Bertsekas and Tsitsiklis [2]). Theory of Automata Network was introduced by S. Ulam [16], W. McCulloch [9] and J. von Neumann ([17],[18]). Automata Networks are discrete dynamical systems, in time and space. Roughly speaking , they are defined by a graph, where each vertex takes states in a finite set. Thus a discrete dynamical system is modelled by F : {0,1}^n --> {0, 1}^n. A basic theory of discrete dynamical system modelled by F : {0, 1}^n --> {0, 1}^n was given in Robert [11] and [12] (see also [13],[14]). A recent work by Shih and Ho [15] deepened the theory of Robert. Among the results of Shih and Ho, Shih and Ho made the following conjecture. Conjecture. Let F : {0, 1}^n --> {0, 1}^n. If ρ(F''(x)) = 0 for all x in { 0, 1}^n, then F has a unique fixed point. This conjecture was motivated by the outstanding Jacobian conjecture in Algebraic Geometry formulated by O. H. Keller in 1939 (see Bass, Connell and Wright [1]). In 1995, Cima , Gasull and Ma$\tilde{n}$osas proved that the Jacobian Conjecture is equivalent to the following : If F : C^n --> C^n is a polynomial mapping (i.e., each f_i is a polynomial) and ρ(F''(x)) < 1 for all x in C^n, then F has a unique fixed point. The purpose of this thesis has three folds. The first fold of this thesis is to prove Shih-Ho''s conjecture in Sect. 3, Theorem 3.1. The second fold of this thesis is to determine the optimal Hamming sphere such that the fixed point is strongly attractive in {0, 1}^n. Our answer of the optimal Hamming sphere is V_{n-2}(ξ). The main result is given in Theorem 4.1. The third fold of this thesis is to give a partial answer of the conjecture raised by Shih concerning the dynamical behavior of the Gauss-Seidel operator for F : {0, 1}^n --> {0, 1}^n. The main result is given in Theorem 5.1.The content of this thesis is organized as follows. Section 2 is a development of the tools and the spectra of Boolean matrices, needed to prove the main theorem. Section 3, the main theorem of this thesis, is to prove Shih-Ho''s conjecture. Section 4 gives a study of dynamical behavior of hypercube mappings. Section 5 is a study of Gauss-Seidel iteration. It should be mentioned here that the motivation of this thesis is motivated by the study of the Associated Memory in Automata Network, see for example, Fogelman, Robert and Tchuente [5], Goles and Martinez [6] , Kohonen [8], and Robert [13] and [14].

碩士<br>國立彰化師範大學<br>數學系<br>91<br>In this paper, we first establish the existence theorems of generalized abstract economy with a lower semicontinuous constraint correspondence and a fuzzy constraint correspondence defined on non-compact and non-paracompact strategy sets. As consequence of our results, we get existence theorems for a solution to the system of generalized vector quasi-equilibrium problems. As applications, we derive some existence results for the system of mixed vector quasi-variational-like inequality problems, Debreu type vector equilibrium problems and Nash type equilibrium problem for vector-valued mappings.

In a transferable utility context, Choo and Siow (2006) introduced a competitive model of the marriage market with gumbel distributed stochastic part, and derived its equilibrium output, a marriage match- ing function. The marriage matching function defines the gains generated by a marriage between agents of prescribed types in terms of the observed frequency of such marriages within the population, relative to the number of unmarried individuals of the same types. Left open in their work is the issue of existence and uniqueness of equilibrium. We resolve this question in the affirmative, assuming the norm of the gains matrix (viewed as an operator) to be less than two. Our method adapts a strategy called the continuity method,more commonly used to solve elliptic partial differen- tial equations, to the new setting of isolating positive roots of polynomial systems. Finally, the data estimated in [4] falls within the scope of our results.

碩士<br>國立彰化師範大學<br>數學系<br>91<br>In this paper, we first establish two existence theorems of equilibrium for a non-compact generalized game with a fuzzy constraint. Next, as applications, we shall prove system of generalized quasi-minimax inequalities by using equilibrium existence theorems.

碩士<br>國立彰化師範大學<br>數學系所<br>93<br>In this paper, we establish existence theorems of quasi-variational inclusion, from which we establish existence theorems of mathematical programs with quasi-variational inclusion constraint, bilevel problems, mathematical programs with equilibrium constraint and semi-infinite problems.

This thesis provides new equilibrium existence results for markets with information asymmetries and relates them in a novel way to the role of economic intermediation. Under atomless uncertainty, single market linear price equilibria are known not to exist prevalently even when agents are risk averse expected utility maximizers. The notion of prevalence involves essentially picking an economy at random. Bypassing the nonexistence problem is one of the achievements of the nonlinear price decentralization theory. This thesis contributes by reconciling the nonlinear price decentralization theory to a large extent with certain competitive market structures. We do this in Chapter 1 by defining linear price equilibrium with multiple markets and establishing its existence. Each market has its own price vector (linear functional), and agents’ involvement in various markets is heterogeneous. As a result, price differences across markets may prevail in equilibrium. We present an example in which single market linear price equilibrium does not exist but certain corresponding equilibrium with two markets does. Our equilibrium with multiple markets has a more standard economic interpretation than equilibrium with nonlinear prices used in nonlinear decentralization theory. Our framework can potentially accommodate even more nonlinearities if economic intermediaries are explicitly introduced into the model. Despite the nonexistence problem, single market linear price equilibrium with infinitely many states is still known to exist under restrictive assumptions on the information structure. In Chapter 2, we introduce two new results on the existence of single market linear price (Radner) equilibrium with infinitely many states under economically meaningful conditions. Our first result requires that agents have independent information, while the second assumes that the total endowment of the economy is common knowledge. In Chapter 3, we explore how economic agents can test the scope of their knowledge and, in particular, the informational content of equilibrium prices under asymmetric information. We show that one can go far in arguing that equilibrium prices tend to be fully revealing.

碩士<br>國立彰化師範大學<br>數學系所<br>93<br>In this paper, we introduce systems of vector quasi equilibrium problems and prove the existence of their solutions. As applications of our results, we derive the theorem for solution of system of vector quasi-saddle point problem and then prove the existences of a solution of system of generalized quasi-minimax inequalities. Further application of our results is also given to the mathematical program with equilibrium constraint, semi-infinite and bilevel problems.

碩士<br>國立臺灣大學<br>數學研究所<br>101<br>Brouwer’s fixed point theorem is critical to the existence of general equilibria in a complete market. However, [4] pointed out that in an incomplete market, a general equilibrium does not always exist. In this paper, following [2] and [6], we use the invariant property of mod 2 Euler number of Grassmannian manifold to derive a fixed-point-like theorem, and therefore, prove the existence of a weaker “pseudo-equilibrium”. At last, we discuss the genericity of general equilibria, that is, the economy in which the equilibria exist is an open and dense set.

碩士<br>國立彰化師範大學<br>數學系<br>90<br>In this pager, we deal with KKM-type theorems. With these theorems, we obtain some coincidence theorems and show their equivalences to some KKM-type theorems. We apply these results to establish the existence of the solution to generalized vector equilibrium problems where a bimap f is involved with considering by some sufficient conditions on f .

碩士<br>國立彰化師範大學<br>數學系<br>90<br>In this paper, we first estabilish the existence results of the Stampacchia type equilibrium problems. Then we apply the maximal element theorem to establish the exisence theorems of Minty type equilibrium problems. We also establish another existing results of the Minty type without constrainted correspondence by an equilibrium theorem.