The primary purpose of this thesis is to carefully develop and prove some of the fundamental, classical theorems of the differential calculus for functions of two real variables.

by Chow Wai Chuen. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Bibliography: leaves 66-67.

by Ho Wing Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 85). / Introduction --- p.1 / Chapter 1 --- Preliminaries --- p.3 / Chapter 1.1 --- Gateaux and Frechet Differentiability --- p.4 / Chapter 1.2 --- β-Differentiability --- p.9 / Chapter 1.3 --- Monotone Operators and Usco Maps --- p.14 / Chapter 2 --- Variational Principle --- p.25 / Chapter 2.1 --- A Generalized Variational Principle --- p.27 / Chapter 2.2 --- A Smooth Variational Principle --- p.37 / Chapter 3 --- Differentiability of Convex Functions --- p.47 / Chapter 3.1 --- On Banach Spaces with β-Smooth Bump Functions --- p.48 / Chapter 3.2 --- A Characterization of Asplund Spaces --- p.64 / Chapter 4 --- More on Differentiability --- p.70 / Chapter 4.1 --- Introduction --- p.70 / Chapter 4.2 --- Differentiability Theorems --- p.75

We consider a portfolio selection problem in the presence of transaction costs. Transaction costs on each asset are assumed to be a convex function of the amount sold or bought. This function can be nondifferentiable in a finite number of points. The objective function of this problem is a sum of a convex twice differentiable function and a separable convex nondifferentiable function. We first consider the problem in the presence of linear constraints and later generalize the results to the case when the constraints are given by the convex piece-wise linear functions. <br /><br /> Due to the special structure, this problem can be replaced by an equivalent differentiable problem in a higher dimension. It's main drawback is efficiency since the higher dimensional problem is computationally expensive to solve. <br /><br /> We propose several alternative ways to solve this problem which do not require introducing new variables or constraints. We derive the optimality conditions for this problem using subdifferentials. First, we generalize an active set method to this class of problems. We solve the problem by considering a sequence of equality constrained subproblems, each subproblem having a twice differentiable objective function. Information gathered at each step is used to construct the subproblem for the next step. We also show how the nonsmoothness can be handled efficiently by using spline approximations. The problem is then solved using a primal-dual interior-point method. <br /><br /> If a higher accuracy is needed, we do a crossover to an active set method. Our numerical tests show that we can solve large scale problems efficiently and accurately.

We consider a portfolio selection problem in the presence of transaction costs. Transaction costs on each asset are assumed to be a convex function of the amount sold or bought. This function can be nondifferentiable in a finite number of points. The objective function of this problem is a sum of a convex twice differentiable function and a separable convex nondifferentiable function. We first consider the problem in the presence of linear constraints and later generalize the results to the case when the constraints are given by the convex piece-wise linear functions. <br /><br /> Due to the special structure, this problem can be replaced by an equivalent differentiable problem in a higher dimension. It's main drawback is efficiency since the higher dimensional problem is computationally expensive to solve. <br /><br /> We propose several alternative ways to solve this problem which do not require introducing new variables or constraints. We derive the optimality conditions for this problem using subdifferentials. First, we generalize an active set method to this class of problems. We solve the problem by considering a sequence of equality constrained subproblems, each subproblem having a twice differentiable objective function. Information gathered at each step is used to construct the subproblem for the next step. We also show how the nonsmoothness can be handled efficiently by using spline approximations. The problem is then solved using a primal-dual interior-point method. <br /><br /> If a higher accuracy is needed, we do a crossover to an active set method. Our numerical tests show that we can solve large scale problems efficiently and accurately.

Submitted by ANTONIO DOS SANTOS DE FREITAS null (toninhosantosfreitas123@gmail.com) on 2017-06-03T18:53:33Z No. of bitstreams: 1 Dissertaçao de Mestrado Antonio dos Santos de Freitas.pdf: 3789052 bytes, checksum: 877acec1e0f42977ad82b4aada999922 (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-06-06T14:47:57Z (GMT) No. of bitstreams: 1 freitas_as_me_rcla.pdf: 3789052 bytes, checksum: 877acec1e0f42977ad82b4aada999922 (MD5) / Made available in DSpace on 2017-06-06T14:47:57Z (GMT). No. of bitstreams: 1 freitas_as_me_rcla.pdf: 3789052 bytes, checksum: 877acec1e0f42977ad82b4aada999922 (MD5) Previous issue date: 2017-05-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho aborda em especial a análise dos pontos críticos de uma função diferenciável. Fazemos inicialmente uma abordagem sobre funções diferenciáveis com duas variáveis e outros temas necessários para a compreensão de algumas demonstrações e conceitos que serão abordados neste trabalho e em seguida apresentamos uma abordagem sobre curvas e superfícies. Depois, apresentamos um estudo sobre pontos críticos e as funções de Morse, que estão relacionadas ao estudo dos pontos críticos não degenerados de uma função diferenciável f: X → IR em uma superfície, e mostramos ainda que toda função diferenciável em torno de um ponto crítico não degenerado pode ser escrita como um polinômio quadrático. Para finalizar o trabalho, fazemos uma proposta de abordagem dos pontos críticos de uma função diferenciável destinada à 3ª série do ensino médio usando o conceito de derivada com uma variável. / This work deals in particular with the analysis of the critical points of a differentiable function. We make an initial approach on differentiable functions with two variables and other topics necessary for the understanding of the sampled concepts and concepts that will be approached in this work and next we present an approach on curves and surfaces.Then, we present a study on critical points and Morse functions, which are related to the study of the nondegenerate critical points of a differentiable f: X → IR function on a surface, and we show that any differentiable function around a nondegenerated critical point can be written as a quadratic polynomial.To finalize the work, we make a proposal to approach the critical points of a differentiable function destined for the 3rd grade of high school using the concept of derivative with one variable.

Orientador: Jorge Tulio Mujica Ascui / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T12:22:20Z (GMT). No. of bitstreams: 1 Araujo_MariaAngelica_M.pdf: 648625 bytes, checksum: 6d3a2ed1bef26a3213e505351e408696 (MD5) Previous issue date: 2014 / Resumo: O objetivo desta dissertação é apresentar e demonstrar alguns teoremas da Análise matemática, são eles, O Teorema de Aproximação de Weierstrass, o Teorema de Kakutani-Stone, os Teoremas de Stone-Weierstrass e o Teorema de Nachbin. Para demonstrá-los relembraremos algumas definições e resultados básicos da teoria de Análise e Topologia e abordaremos as demais ferramentas necessárias para suas respectivas demonstrações / Abstract: The aim of this dissertation is to present and prove some theorems of mathematical analysis, that are, the Weierstrass Approximation Theorem, the Kakutani-Stone Theorem, the Stone-Weierstrass Theorems and the Nachbin Theorem. To prove them we recall some basic definitions and results of analysis and topology and we discuss other tools that are necessary for their respective proofs / Mestrado / Matematica / Mestra em Matemática

In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function 𝑓 which is the uniform limit of a sequence of sawtooth functions 𝑓ₙ : [0, 1] → [0, 1] with increasingly sharp teeth. Let 𝑋 = [0, 1] x [0, 1] and 𝐹(𝑋) be the Hausdorff metric space determined by 𝑋. We define contraction maps 𝑤₁ , 𝑤₂ , 𝑤₃ on 𝑋. These maps define a contraction map 𝑤 on 𝐹(𝑋) via 𝑤(𝐴) = 𝑤₁(𝐴) ⋃ 𝑤₂(𝐴) ⋃ 𝑤₃(𝐴). The iteration under 𝑤 of the diagonal in 𝑋 defines a sequence of graphs of continuous functions 𝑓ₙ. Since 𝑤 is a contraction map in the compact metric space 𝐹(𝑋), 𝑤 has a unique fixed point. Hence, these iterations converge to the fixed point-which turns out to be the graph of our continuous, nowhere-differentiable function 𝑓. Chapter 2 contains the background we will need to engage our task. Chapter 3 includes two results from the Baire Category Theorem. The first is the well known fact that the set of continuous, nowhere-differentiable functions on [0,1] is a residual set in 𝐶[0,1]. The second fact is that the set of continuous functions on [0,1] which have a dense set of proper local extrema is residual in 𝐶[0,1]. In the fourth and last chapter we actually construct our function and prove it is continuous, nowhere-differentiable and has a dense set of proper local extrema. Lastly we iterate the set {(0,0), (1,1)} under 𝑤 and plot its points. Any terms not defined in Chapters 2 through 4 may be found in [2,4]. The same applies to the basic properties of metric spaces which have not been explicitly stated. Throughout, we will let 𝒩 and 𝕽 denote the natural numbers and the real numbers, respectively.

of the dissertation thesis Title: Sigma-porous sets and the differentiation theory Author: Martin Koc Department: Department of mathematical analysis Supervisor: Prof. RNDr. Luděk Zajíček, DrSc., Department of mathematical analysis Abstract: The thesis consists of five research articles. In the first one, it is shown that there exists a closed upper porous (in a strong sense) subset of a nonempty, topolo- gically complete metric space without isolated points that is not σ-lower porous (in a weak sense). In the second article, a new notion of porosity with respect to a measure, that generalizes the upper porosity of a measure, is introduced. Several natural definitions of this notion are investigated. The main result of this chapter is a decomposition theorem for sets that are σ-porous with respect to a measure. The third article deals with sets of points at which arbitrary real functions are Lipschitz from one side and not Lipschitz from another side. A full characterization of the system generated by sets of this type is proved. In the fourth article, several results on relations among metric derived numbers for functions with values in metric spaces are shown. The last chapter deals with existence of differentiable extensions for functions defined on closed subsets of Rn . Its main result...