Our aim in this research was to study monotone operators in Banach spaces. In particular, the most important concept in this theory, the maximal monotone operators. Here we make an attempt to describe most of the important results and concepts on maximal monotone operators and how they all tie together. We will take a brief look at subdifferentials, which generalize the notion of a derivative. The subdifferential is a maximal monotone operator and it has proved to be of fundamental importance for the study of maximal monotone operators. The theory of maximal monotone operators is somewhat complete in reflexive Banach spaces. However, in nonreflexive Banach spaces it is still to be developed fully. As such, here we will describe most of the important results about maximal monotone operators in Banach spaces and we will distinguish between the reflexive Banach spaces and nonreflexive Banach spaces when a property is known to hold only in reflexive Banach spaces. In the latter case, we will state what the corresponding situation is in nonreflexive Banach spaces and we will give counter examples whenever such a result is known to fail in nonreflexive Banach spaces. The representations of monotone operators by convex functions have found to be extremely useful for the study of maximal monotone operators and it has generated a lot of interest of late. We will discuss some of those key representations and their properties. We will also demonstrate how these representations could be utilized to obtain results about maximal monotone operators. We have included a discussion about the very important Rockafellar sum theorem and some its generalizations. This key result and its generalizations have only been proved in reflexive Banach spaces. We will also discuss several special cases where the Rockafellar sum theorem is known to be true in nonreflexive Banach spaces. The subclasses which provide a basis for the study of monotone operators in nonreflexive Banach spaces are also discussed here

No description available.

Includes bibliographical references (p. 55-57).

Ce travail de thèse présente le développement de modèles numériques pour la modélisation de l’endommagement des matériaux composites à fibres continues et matrice polymère. L’objectif est de fournir des outils numériques efficaces pour prédire l’endommagement sous chargement monotone et cyclique. Deux modèles ont été développés, un basé sur une approche multi-échelle d’homogénéisation et l’autre défini dans le cadre de la mécanique de l’endommagement. Ce dernier est utilisé dans une nouvelle approche pour prédire la durée de vie des matériaux composites. La première approche est une méthode multi-échelle afin d’étudier l’influence de l’endommagement à l’échelle microscopique sur le comportement macroscopique du composite. Le comportement macroscopique est déterminé par homogénéisation d’une cellule unitaire. L’approche est appliquée au cas d’un composite unidirectionnel afin d’étudier l’influence de l’endommagement de la matrice. Pour contourner les difficultés d’une approche multi-échelle, un modèle mésoscopique basé sur le couplage entre la plasticité et l’endommagement est proposé. Le modèle est validé et confronté à d’autres modèles ainsi qu’à des résultats expérimentaux. La dernière contribution est le développement d’une nouvelle approche pour la prédiction de la durée de vie des matériaux composites s’appuyant sur l’hypothèse que le matériau atteint un état stabilisé d’endommagement. Cette hypothèse permet d’utiliser l’analyse simplifiée pour prédire les états stabilisés. Le modèle de fatigue proposé est une loi puissance entre le nombre de cycles et les forces thermodynamiques associées à l’endommagement. L’approche est validée à partir de résultats expérimentaux. / A numerical framework for the modeling of the damage in fibre reinforced polymer composite materials has been developped. The objectives were to provide efficient numerical tools to predict the damage under static and cyclic loading. Two different models were proposed, one based on a fully computational multi-scale homogenization technique and a second one under the hypothesis of the meso-mechanics associated with a coupled damage-plastic constitutive model. The latter has been used in a new approach developped to predict the life of composite structures. As a first approach, a multi-scale application has been developed to better understand the influence of the damages occuring at lower scales on the macroscopic response. The macroscopic is defined by homogenization of a unit cell. The scheme is used to analyze the effect of the matrix damage on the material response in the case of unidirectional composites. To overcome the difficulties of the multi-scale approach, a meso-scale phenomenological model based on the coupled plasticity with continuous damage mechanics is proposed. All comparisons of the simulation with experiments and other models have shown good agreements. The third contribution is the development of a new approach to predict the life of composite materials based on the assumption that the material reaches a damage stabilized state during his life. This assumption makes it possible to use the simplified analysis. The life of the material is considered depending on the level of the thermodynamical forces associated with damage at the stabilized state by the use of a power law. Good agreements are obtained with experimental results.

本文旨在討論非線性拋物型積分微分方程式(組)的解之存在性．首先藉由與上解及下解相關的若干假設，我們得到一個比較性的結果．然後我們利用單調法建構出兩個單調收歛到方程式解的序列，從而驗證了方程式解的存在性． / In this paper, the existence of the solutions for nonlinear integro-differential equations and systems is discussed. First, by the assumption of weak upper and weak lower solutions for the given problem, we obtain the comparison result. Next, we provide the method of monotony and construct two sequences which converge monotonically to the solution.

Awareness of biological invasions is becoming widespread and several mathematical tools have been used to study this problem. Interacting particle systems, specifically the contact process, have been used to study systems with invasion/infection type dynamics. The Propp-Wilson algorithm is a method for exact sampling from the stationary distribution of an ergodic monotone Markov chain using a method called coupling from the past. The contact process is monotone so we can sample exactly from the stationary distribution of a modified finite grid version using the Propp-Wilson algorithm. In order to study an invasion, we would like to include at least 2 species; however, monotonicity is not well defined for contact processes with more than 2 particle types. Here we develop a general theory of monotonicity for interaction map particle systems, which are interacting particle systems with contact process type dynamics. This allows us to create monotone models with any number of particles and to use the Propp-Wilson algorithm for not only sampling from the stationary distribution, but analyzing the path of invasion leading to equilibrium. Virtual particle invasion models that fall into this new theoretical framework, which we develop here, present a wide range of biological dynamics. Computer simulation of the stochastic system and mean field analysis are two powerful tools that we use for analyzing these types of models. Statistics gathered along the path to invasion help us understand the spatial dynamics of this ecological process and what the stationary behavior looks like. This allows us to understand when the invasion is successful or if coexistence occurs and how these depend on the transition rates and interactions within the process.

In this paper, we report existing convergence results on monotone variational inequalities where the governing monotone operators are either strongly monotone or inverse strongly monotone. We reformulate the variational inequality problem as an equivalent fixed point problem and then use fixed point iteration method to solve the original variational inequality problem. In the case of strong monotonicity case we use the Banach¡¦s contraction principle to define out iteration sequence; while in the case of inverse strong monotonicity we use the technique of averaged mappings to define our iteration sequence. In both cases we prove strong convergence for our iteration methods. An application to a minimization problem is also included.

In many practical situations when analyzing a dependence of one or more explanatory variables on a response variable it is essential to assume that the relationship of interest obeys certain shape constraints, such as monotonicity or monotonicity and convexity/concavity. In this thesis a new approach to shape preserving smoothing within generalized additive models has been developed. In contrast with previous quadratic programming based methods, the project develops intermediate rank penalized smoothers with shape constrained restrictions based on re-parameterized B-splines and penalties based on the P-spline ideas of Eilers and Marx (1996). Smoothing under monotonicity constraints and monotonicity together with convexity/concavity for univariate smooths; and smoothing of bivariate functions with monotonicity restrictions on both covariates and on only one of them are considered. The proposed shape constrained smoothing has been incorporated into generalized additive models with a mixture of unconstrained and shape restricted smooth terms (mono-GAM). A fitting procedure for mono-GAM is developed. Since a major challenge of any flexible regression method is its implementation in a computationally efficient and stable manner, issues such as convergence, rank deficiency of the working model matrix, initialization, and others have been thoroughly dealt with. A question about the limiting posterior distribution of the model parameters is solved, which allows us to construct Bayesian confidence intervals of the mono-GAM smooth terms by means of the delta method. The performance of these confidence intervals is examined by assessing realized coverage probabilities using simulation studies. The proposed modelling approach has been implemented in an R package monogam. The model setup is the same as in mgcv(gam) with the addition of shape constrained smooths. In order to be consistent with the unconstrained GAM, the package provides key functions similar to those associated with mgcv(gam). Performance and timing comparisons of mono-GAM with other alternative methods has been undertaken. The simulation studies show that the new method has practical advantages over the alternatives considered. Applications of mono-GAM to various data sets are presented which demonstrate its ability to model many practical situations.

Alon 和Krivelevich (SIAM J. Discrete Math. 15(2): 211-227 (2002)) 證明了如果一個圖是ε -非二部圖，那麼階數為Ỡ(1/ε) 的隨機導出于圖以很大概率是非二部圖。我們進一步猜想，這個導出子圖以很大概率是Ω(ε)-非二部圖。Gonen 和Ron (RANDOM 2007) 證明了當圖的最大度不超過O (εn )時猜想成立。我們將對更一般的情形給出證明，對於任意d，所有d 正則(或幾乎d 正則)的圖猜想成立。 / 假設猜想成立的情況下，我們證明二分屬性是可以被單側誤差在O(1/ε^c ) 時間內檢驗的，其中c 是一個嚴格小於2 的常數，而這個結果也改進了Alon 和Krivelevich 的檢驗算法。由於己知對二分屬性的非適應性的檢驗算法需要Ω(1 /ε²) 的複雜性(Bogdanov 和Trevisan ， CCC 2004) ，我們的結果也得出，假設猜想成立，適應性對檢驗二分屬性是有幫助的。 / 另外一個有很多屬性檢驗問題被廣泛研究的領域是布爾函數。對布爾函數單調性的檢驗也是屬性檢驗的經典問題。給定對布爾函數f: {0,1}{U+207F} → {0,1} 的訪問，在[18]中，證明了檢驗算法複雜性的下界是Ω(√n) 。另一方面，在[21]中，作者們構造了一個複雜性為O(n) 的算法。在本文中，我們刻畫一些單調布爾函數的本質，設計算法并分析其對於一些困難例子的效果。最近，在[14] 中， 一個類似的算法被證明是非適應性，單側誤差，複雜性為Ỡ (n⁵/⁶ ε⁻⁵/³) 的算法。 / Alon and Krivelevich (SIAM J. Discrete Math. 15(2): 211-227 (2002)) show that if a graph is ε-far from bipartite, then the subgraph induced by a random subset of Ỡ (1/ε) vertices is not bipartite with high probability. We conjecture that the induced subgraph is Ω(ε)-far from bipartite with high probability. Gonen and Ron (RANDOM 2007) proved this conjecture in the case when the degrees of all vertices are at most O(εn). We give a more general proof that works for any d-regular (or almost d-regular) graph for arbitrary degree d. / Assuming this conjecture, we prove that bipartiteness is testable with one-sided error in time O(1=ε{U+1D9C}), where c is a constant strictly smaller than two, improving upon the tester of Alon and Krivelevich. As it is known that non-adaptive testers for bipartiteness require Ω(1/ε²) queries (Bogdanov and Trevisan, CCC2004), our result shows, assuming the conjecture, that adaptivity helps in testing bipartiteness. / The other area in which various properties have been well studied is boolean function. Testing monotonicity of Boolean functions is a classical question in property testing. Given oracle access to a Boolean function f : {0, 1}{U+207F} →{0, 1}, in [18], it is shown a lower bound of testing is Ω(√n). On the other hand, in [21], the authors introduced an algorithm to test monotonicity using O(n) queries. In this paper, we characterize some nature of monotone functions, design a tester and analyze the performance on some generalizations of the hard case. Recently, in [14], a similar tester is shown to be a non-adaptive, one-sided error tester making Ỡ (n⁵/⁶ ε⁻⁵/³) queries. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Li, Fan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 76-79). / Abstracts also in Chinese. / Abstract --- p.i / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Property Testing --- p.1 / Chapter 1.2 --- Testing Bipartiteness --- p.4 / Chapter 1.3 --- Testing Monotonicity --- p.7 / Chapter 2 --- Testing Bipartiteness --- p.11 / Chapter 2.1 --- Background --- p.11 / Chapter 2.1.1 --- Our result --- p.11 / Chapter 2.1.2 --- The algorithms of Gonen and Ron --- p.13 / Chapter 2.1.3 --- Our proof --- p.16 / Chapter 2.1.4 --- Notation --- p.19 / Chapter 2.2 --- Splitting the vertices by degree --- p.19 / Chapter 2.3 --- The algorithm for high degree vertices --- p.20 / Chapter 2.4 --- Eliminating the high degree vertices --- p.22 / Chapter 2.5 --- From an XOR game to a bipartite graph --- p.33 / Chapter 2.6 --- Proof of the main theorem --- p.35 / Chapter 2.7 --- Proof of the conjecture for regular graphs --- p.37 / Chapter 3 --- Testing Monotonicity --- p.40 / Chapter 3.1 --- Towards an improved tester --- p.40 / Chapter 3.1.1 --- Properties of Distribution D --- p.42 / Chapter 3.1.2 --- An Alternative Representation of D --- p.46 / Chapter 3.1.3 --- Performance of D on Decreasing Functions --- p.51 / Chapter 3.1.4 --- Functions Containing Ω(2{U+207F}) Disjoint Violating Edges --- p.54 / Chapter 3.2 --- A o(n) Monotonicity Tester [14] and Some Improvements --- p.62 / Chapter 3.2.1 --- A o(n) Monotonicity Tester [14] --- p.62 / Chapter 3.2.2 --- An Alternative Proof of Theorem 3.2.2 --- p.65 / Chapter 4 --- Other Related Results --- p.67 / Chapter 4.1 --- Short Odd Cycles in Graphs that are Far From Bipartiteness --- p.67 / Chapter 4.2 --- Fourier Analysis on Boolean Functions --- p.69 / Bibliography --- p.76

Mean-field games (MFGs) model the behavior of large populations of rational agents. Each agent seeks to minimize an individual cost that depends on the statistical distribution of the population. Roughly speaking, MFGs are given by the limit of differential games with N agents as N goes to infinity. This limit describes an average effect of the population’s behavior. Instead of modeling large systems for all agents, we consider two coupled equations: the Hamilton–Jacobi equation and the Fokker–Planck equation. A solution to MFGs is given by two functions: a value function and a population density. From the point of view of mathematics, monotonicity conditions for MFGs are a natural way to obtain the uniqueness of solutions and the stability of systems. In this thesis, we develop a new framework to establish the existence of solutions to MFGs through monotonicity. First, we study first-order stationary monotone MFGs with Dirichlet boundary conditions. In MFGs, boundary conditions arise when agents can leave the domain. There are exit costs for agents given by Dirichlet boundary conditions. Here, we establish the existence of solutions to MFGs that fulfill those boundary conditions in the trace sense. In particular, our solution is continuous up to the boundary in the one-dimensional case. Second, we consider time-dependent monotone MFGs with space-periodic boundary conditions. To solve the time-dependent monotone MFG, we introduce a mono- tone high-order regularized elliptic problem in Rn+1, although the original MFG is a parabolic type. To preserve monotonicity, we need to determine the specific boundary conditions for the time variable. Then, we can apply our method of stationary MFGs to this regularization. In particular, we prove that a solution to the problem exists for any terminal time. Third, we investigate stationary MFGs with hypoelliptic operators that are degenerate differential operators. Those models arise from stochastic control problems with the Stratonovich integration. We study a hypoelliptic MFG with the standard quadratic Hamiltonian. Under standard assumptions, although there is no uniform elliptic condition in hypoelliptic operators, we verify that there is a unique solution to our hypoelliptic MFG.