Dissertations / Theses on the topic 'Vector valued functions'

This thesis is concerned with operators on certain vector-valued function spaces. Namely, Bergman spaces of \mathbb{C}^n$-valued functions and L^2(\mathbb{R},\mathbb{C}^n,V)$, where $V$ is a matrix weight. We will study products of Toeplitz operators on the vector Bergman space $L^2_a(\mathbb{C}^n)$. We also study various operators, including the dyadic shift and the Hilbert transform, between $L^2(\mathbb{R},\mathbb{C}^n,V)$ and $L^2(\mathbb{R},\mathbb{C}^n,U)$. These function spaces are generalizations of normed vector spaces of functions which take values in $\mathbb{C}$. The thesis is split into two distinct areas of function space theory: analytic function spaces and harmonic analysis. There is, however, a common theme of matrix weights, particularly the reverse Hölder condition on matrix weights and a generalization of the $A_p$ conditions on matrix weights for $p=2$ and $p=\infty$.

This thesis treats different aspects of time-frequency analysis and pseudodifferential operators, with particular emphasis on techniques involving vector-valued functions and operator-valued symbols. The vector (Banach) space is either motivated by an application as in Paper I, where it is a space of stochastic variables, or is part of a general problem as in Paper II, or arises naturally from problems for scalar-valued operators and function spaces, as in Paper V. Paper III and IV fall outside this framework and treats algebraic aspects of time-frequency analysis and pseudodifferential operators for scalar-valued symbols and functions that are members of modulation spaces. Paper IV builds upon Paper III and applies the results to a filtering problem for second-order stochastic processes. Paper I treats the Wigner distribution of a Gaussian weakly harmonizable stochastic process defned on Rd. Paper II extends recent continuity results for pseudodifferential and localization operators, with symbols in modulation spaces, to the vector/operator-framework, where the vector space is a Hilbert or a Banach space. In Paper III we give algebraic results for the Weyl product acting on modulation spaces. We give suffcient conditions for a weighted modulation space to be an algebra under theWeyl product, and we also give necessary conditions for unweighted modulation spaces. In Paper IV we discretize the results of Paper III by means of a Gabor frame delined by a Gaussian function. Finally, Paper V deals with pseudodifferential operators with symbols that are almost periodic in the first variable. We show that such operators may be transformed to Fourier multiplier operators with operator- valued symbols such that the transformation preserves positivity and operator composition.

The main goal of this work is to study vector-valued Bergman spaces and to obtain the weak factorization of these spaces. In order to do that we need to study small Hankel operators with operator-valued holomorphic symbols. We also study the big Hankel operator acting on vector-valued Bergman spaces. In Chapter 1 we collect all the previous results and notations needed to follow the rest of the manuscript. More concretely, some of the topics covered in this chapter are the Bochner integral, the integral for vector-valued functions appearing first in Bochner; the Bergman metric, results of the metric used in Bn; harmonic and subharmonic function; basic notions of differentiation, where the differential operators R(a, t) are presented which is important in the next chapters and in the final section we recall some topics on Banach spaces, as the Rademacher type and cotype of a Banach space and some other related results. Having all that in mind, in Chapter 2, the vector-valued Bergman spaces are presented. The vector-valued Bloch type spaces play a similar role and therefore we dedícate one full chapter to these spaces. Chapter 3 is devoted to present and characterize the vector-valued Bloch type spaces. Since we mention Hankel operators, in Chapter 4 we prove the characterization of the boundedness of the small Hankel operator with analytic operator-valued symbols between vector-valued Bergman spaces (of different type). We explain what this means in the following. Another very important consequence of the boundedness of the small Hankel operator between vector-valued Bergman spaces is shown in Chapter 5. We establish the weak factorization of the vector-valued Bergman spaces. Factorization of analytic functions is a very big topic and many people worked on it during many years and it is known to have many applications. Therefore, in Chapter 6 we fully characterize the boundedness of the big Hankel operator on vector-valued Bergman spaces in terms of its operator-valued holomorphic symbol for all cases of p > 1 and q > 1, and so we solve and generalize the previous problem. Finally, in Chapter 7 we discuss some open problems we have not been able to solve, as well as some other interesting problems in the same line as this work in order to look on the future.

En el año 1960 Komatsu introdujo ciertas clases de funciones infinitamente derivables definidas mediante estimaciones del crecimiento de los sucesivos iterados de un operador en derivadas parciales cuando estudiaba propiedades de regularidad de las soluciones de ciertas ecuaciones en derivadas parciales. Esta línea de investigación ha sido muy activa hasta la actualidad a través de los trabajos de muchos autores. Destacamos, entre otros, Bolley, Camus, Kotake, Langenbruch, Métivier, Narasimhan, Newberger, Rodino, Zanghirati y Zielezny. Toda esta bibliografía involucra el llamado problema de los iterados que consiste, grosso modo, en caracterizar las funciones de una cierta clase en términos del comportamiento de los iterados de un operador previamente fijado. En la primera parte de esta tesis seguimos con la investigación mencionada antes en un contexto más general: clases no casi analíticas de funciones ultradiferenciables en el sentido de Braun, Meise y Taylor. El estudio de estas clases no casi analíticas es una área de investigación muy activa debido a sus aplicaciones a la teoría de operadores en derivadas parciales: destacamos entre otros el trabajo de Bonet, Braun, Domanski, Fernández, Frerick, Galbis, Taylor y Vogt. En el Capítulo 1 introducimos estas clases y enunciamos las propiedados que utilizaremos a lo largo de esta tesis. En el Capítulo 2 definimos las clases no casi analíticas con respecto a los iterados de un operador en derivadas parciales P(D) y estudiamos sus propiedades topológicas como la completitud y la nuclearidad. En particular, demostramos que estas clases son un espacio localmente convexo completo si y sólo si el operador P(D) es hipoelíptico y vemos que en tal caso son además un espacio nuclear. A continuación, demostramos que estas clases verifican un teorema de tipo Paley-Wiener. En el Capítulo 3 tenemos como objetivo obtener resultados sobre el problema de los iterados en clases no casi analíticas. Generalizamos varios resultados de Newberger, Zielezny, Métivier y Komatsu y damos caracterizaciones de cuándo una clase no casi analítica definida en términos de los iterados de un operador coincide con una clase no casi analítica según Braun, Meise y Taylor. Toda la investigación que se había hecho sobre espacios de funciones definidos por iterados de operadores se había centrado en clases de tipo Roumieu. Sin embargo, demostramos que los resultados dados en los Capítulos 2 y 3 también son válidos para clases de tipo Beurling. En el año 1990, Langenbruch y Voigt demostraron que todo espacio de Fréchet formado por distribuciones que sea invariante bajo la acción de un operador hipoelíptico está continuamente incluido en C¥. En el capítulo 4 introducimos los operadores ultradiferenciales e investigamos extensiones del resultado de Langenbruch y Voigt al contexto ultradiferenciable. El nuevo concepto de espacio de Fréchet (w, P(D))-estable involucra a los iterados de P(D) mediante una condición de equicontinuidad y nos permite mostrar la relación de este tipo de resultados con el problema de los iterados. La segunda parte de esta tesis se centra en el estudio de funciones con valores vectoriales en un espacio localmente convexo.
Juan Huguet, J. (2011). Iterates of differential operators and vector valued functions on non quasi analytic classes [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/9401
Palancia

Thesis (Ph.D.)--Kent State University, 2008.
Title from PDF t.p. (viewed Jan. 26, 2010). Advisor: Joseph Diestel. Keywords: absolute continiuty, range of a vector measure. Includes bibliographical references (p. 40-41).

Para um espaço localmente compacto de Hausdorff K e um espaço de Banach X, denotamos por C0(K,X) o espaço de todas as funções a valores em X contínuas sobre K que se anulam no infinito, munido da norma do supremo. No espírito do clássico teorema de Banach-Stone 1937, estabelecemos que se C0(K1,X) é isomorfo a C0(K2,X), onde X é um espaço de Banach de cotipo finito e tal que X é separável ou X* tem a propriedade de Radon-Nikodým, então ou K1 e K2 são ambos finitos ou K1 e K2 tem a mesma cardinalidade. Trata-se de uma extensão vetorial de um resultado de Cengiz 1978, o caso escalar X = R ou X = C. Demonstramos também que se K1 e K2 são intervalos compactos de ordinais e X é um espaço de Banach de cotipo finito, então a existência de um isomorfismo T de C(K1,X) em C(K2,X) com ||T||||T-1|| < 3 implica que uma certa soma topológica finita de K1 é homeomorfa a alguma soma topológica finita de K2. Mais ainda, se Xn não contém subespaço isomorfo a Xn+1 para todo n ∈ N, então K1 é homeomorfo a K2. Em outras palavras, obtemos um teorema tipo Banach-Stone vetorial que é uma extensão de um teorema de Gordon de 1970 e ao mesmo tempo uma extensão de um teorema de Behrends e Cambern de 1988. Mostramos que se existe um isomorfismo T de C(K1) em um subespaço de C(K2,X) com ||T||||T-1|| < 3, então a cardinalidade do α-ésimo derivado de K2 ou é finita ou é maior do que a cardinalidade do α-ésimo derivado de K1, para todo ordinal α. Em seguida, seja n um inteiro positivo, Γ um conjunto infinito munido da topologia discreta e X um espaço de Banach de cotipo finito. Estabelecemos que se o n-ésimo derivado de K for não vazio, então a distância de Banach-Mazur entre C0(K,X) e C0(Γ,X) é maior ou igual a 2n + 1. Também demonstramos que para quaisquer inteiros positivos n e k, a distância de Banach-Mazur entre C([1,ωnk],X) e C0(N,X) é exatamente 2n+1. Estes resultados fornecem extensões vetoriais para alguns teoremas de Cambern de 1970. Para um ordinal enumerável α, denotando por C(α) o espaço de Banach das funções contínuas no intervalo de ordinal [1, α], obtemos cotas superiores H(n, k) e cotas inferiores G(n, k) para as distâncias de Banach-Mazur entre os espaços C(ω) e C(ωnk), 1 < n, k < ω, verificando H(n, k) - G(n, k) < 2. Estas estimativas fornecem uma resposta para uma questão de Bessaga e Peczynski de 1960 sobre as distâncias de Banach-Mazur entre C(ω) e cada um dos espaços C(α), ω<α<ωω.
For a locally compact Hausdorff space K and a Banach space X, we denote by C0(K,X) the space of X-valued continuous functions on K which vanish at infinity, endowed with the supremum norm. In the spirit of the classical 1937 Banach-Stone theorem, we prove that if C0(K1,X) is isomorphic to C0(K2,X), where X is a Banach space having finite cotype and such that X is separable or X* has the Radon-Nikodým property, then either K1 and K2 are finite or K1 and K2 have the same cardinality. It is a vector-valued extension of a 1978 Cengiz result, the scalar case X = R or X = C. We also prove that if K1 and K2 are compact ordinal spaces and X is Banach space having finite cotype, then the existence of an isomorphism T from C(K1,X) onto C(K2,X) with ||T||||T-1|| < 3 implies that some finite topological sum of K1 is homeomorphic to some finite topological sum of K2. Moreover, if Xn contains no subspace isomorphic to Xn+1 for every n ∈ N, then K1 is homeomorphic to K2. In other words, we obtain a vector-valued Banach-Stone theorem which is an extension of a 1970 Gordon theorem and at same time an improvement of a 1988 Behrends and Cambern theorem. We show that if there is an embedding T of a C(K1) into C(K2,X) with ||T||||T-1|| < 3, then the cardinality of the α-th derivative of K2 is either finite or greater than the cardinality of the α-th derivative of K1, for every ordinal α. Next, let n be a positive integer, Γ an infinite set with the discrete topology and X is a Banach space having finite cotype. We prove that if the n-th derivative of K is not empty, then the Banach Mazur distance between C0(K,X) and C0(Γ,X) is greater than or equal to 2n + 1. Thus, we also show that for every positive integers n and k, the Banach Mazur distance between C([1,ωnk],X) and C0(N,X) is exactly 2n+1. These results provide vector-valued versions of some 1970 Cambern theorems. For a countable ordinal α, writing C(α) for the Banach space of continuous functions on the interval of ordinal [1, α], we give lower bounds H(n, k) and upper bounds G(n, k) on the Banach- Mazur distances between C(ω) and C(ωnk), 1 < n, k < ω, such that H(n, k) - G(n, k) < 2. These estimates provide an answer to a 1960 Bessaga and Peczynski question on the Banach-Mazur distances between C(ω) and each of the C(α) spaces, ω<α<ωω.

We present the new framework based on BFKL and DGLAP evolution equations in which the leading in(Q(_2)) and in(l/x) terms are treated on equal footing. We introduce a pair of coupled integro-difFerential equations for the quark singlet and the unintegrated gluon distribution. The observable structure functions are calculated using high energy factorisation approach. We also include the sub-leading in (l/x) effects via consistency constraint. We argue that the use of this constraint leads to more stable solution to the Pomeron intercept than that based on the NLO calculation of the BFKL equation alone and generates resummation to all orders of the major part of the subleading in (l/x) effects. The global fit to all available deep inelastic data is performed using a simple parametrisation of the non-perturbative region. We also present the results for the longitudinal structure function and the charm component of the F(_2) structure function. Next; we extend this approach to the low Q(^2) domain. At small distances we use the perturbative approach based on the unified BFKL/DGLAP equations and for large distances we use Vector Meson Dominance Model and, for the higher mass qq states, the additive quark approach. We show the results for the total cross section and for the ratio of the longitudinal and transverse structure functions. Finally, we calculate the dijet production and consider the decorrelation effects in the azimuthal distributions caused by the diffusion in the transverse momentum k(_r) of the exchanged gluon. Using the gluon distribution which is fixed by the fit to the DIS data we are able to make absolute predictions. We show the results for the dF(_r)/dɸ, the total cross section and also the distributions in Q(^2) as well as in the longitudinal momentum fraction of the gluon. Our theoretical predictions are confronted with the measurements made using ZEUS detector at HERA.

Neurological diseases in newborns are usually first revealed by seizures, which are characterised by a synchronous discharge of a large number of neurons. Failure to control seizures may lead to brain damage or even death. The importance of this problem prompted many researchers to look for accurate automatic methods for seizure detection. Nonstationarity and multicomponent behaviour of newborn EEG signals made this task very challenging. The significant overlap in the characteristic of background and seizure activities in newborn EEG signals added to the difficulty of seizure detection. This research uses time-frequency based methods for automatic seizure detection. Since time-frequency signal analysis methods use joint representation in both time and frequency domains, they proved to be very suitable for analysis and processing of nonstationary and multicomponent signals such as newborn EEG. Before using any seizure detector, the EEG data is pre-processed in order to reduce the noise effects using a time-frequency based technique. The proposed method is based on the singular value decomposition (SVD) technique applied to the matrix representing the time-frequency distribution (TFD) of the EEG signal. It has been shown that by appropriately filtering the singular vectors associated with the TFD, one can effectively enhance the desired information embedded in the signal. Neonatal EEG seizures can have signatures in both low frequency (lower than 10 Hz) and high frequency (higher than 70 Hz) areas. The seizure detection techniques proposed in the literature concentrated on using either low frequency or high frequency signatures but not both simultaneously. These methods tend to miss the seizures that reveal themselves only in one of the two frequency areas. In this research, we propose a detection method that uses seizure features in both low and high frequency areas. To detect EEG seizures using the low frequency signatures, an SVD-based technique is employed. The technique uses the estimated distribution function of the singular vectors associated with the time-frequency distribution of EEG epochs to discriminate between seizure and nonseizure patterns. The high frequency signatures of seizures are mostly the result of spike events in the EEG signals. To detect these spike events, the signal is mapped into the TF domain. The high instantaneous energy of spikes is reflected as a localised energy in the high frequency area of the TF domain. Consequently, a spike can be seen as a ridge in this area of the TF domain. It has been shown that during seizure activity there is regularity in the distribution of the interspike intervals. This feature has been used as the basis for discriminating between seizure and nonseizure patterns. The performance results obtained by applying the proposed methods on EEG signals extracted from a number of newborns show the superiority of these methods over the existing ones.

Neurological diseases in newborns are usually first revealed by seizures, which are characterised by a synchronous discharge of a large number of neurons. Failure to control seizures may lead to brain damage or even death. The importance of this problem prompted many researchers to look for accurate automatic methods for seizure detection. Nonstationarity and multicomponent behaviour of newborn EEG signals made this task very challenging. The significant overlap in the characteristic of background and seizure activities in newborn EEG signals added to the difficulty of seizure detection. This research uses time-frequency based methods for automatic seizure detection. Since time-frequency signal analysis methods use joint representation in both time and frequency domains, they proved to be very suitable for analysis and processing of nonstationary and multicomponent signals such as newborn EEG. Before using any seizure detector, the EEG data is pre-processed in order to reduce the noise effects using a time-frequency based technique. The proposed method is based on the singular value decomposition (SVD) technique applied to the matrix representing the time-frequency distribution (TFD) of the EEG signal. It has been shown that by appropriately filtering the singular vectors associated with the TFD, one can effectively enhance the desired information embedded in the signal. Neonatal EEG seizures can have signatures in both low frequency (lower than 10 Hz) and high frequency (higher than 70 Hz) areas. The seizure detection techniques proposed in the literature concentrated on using either low frequency or high frequency signatures but not both simultaneously. These methods tend to miss the seizures that reveal themselves only in one of the two frequency areas. In this research, we propose a detection method that uses seizure features in both low and high frequency areas. To detect EEG seizures using the low frequency signatures, an SVD-based technique is employed. The technique uses the estimated distribution function of the singular vectors associated with the time-frequency distribution of EEG epochs to discriminate between seizure and nonseizure patterns. The high frequency signatures of seizures are mostly the result of spike events in the EEG signals. To detect these spike events, the signal is mapped into the TF domain. The high instantaneous energy of spikes is reflected as a localised energy in the high frequency area of the TF domain. Consequently, a spike can be seen as a ridge in this area of the TF domain. It has been shown that during seizure activity there is regularity in the distribution of the interspike intervals. This feature has been used as the basis for discriminating between seizure and nonseizure patterns. The performance results obtained by applying the proposed methods on EEG signals extracted from a number of newborns show the superiority of these methods over the existing ones.

Dans cette thèse, nous présentons une généralisation de l'approche polyédrale classique du problème de satisfiabilité d'un ensemble de fonctions booléennes {fl,…, fm} à n variables. Nous proposons d'élargir la représentation spatiale au graphe d'une application booléenne f = (fl,…, fm). Autrement dit, nous construisons le sous-ensemble des points de l'hypercube de dimension n+m dont les coordonnées sont les lignes de la table de vérité de fl,…, fm. Nous considérons ensuite l'enveloppe convexe de ces points. L'objet de ce travail est de donner un large ensemble d'applications booléennes dont le polytope associé admet par projection, une caractérisation polynomiale, c'est-à-dire une formulation compacte. La satisfiabilité des fonctions fl,…, fm pourra alors être déterminée en temps polynomial par des méthodes de programmation linéaire. La thèse est organisée comme suit: premièrement nous présentons le principe d'approche utilisée ici, que nous appelons projection par famille génératrice, et les résultats théoriques sur lesquels elle s'appuie. Deuxièmement, nous étudions trois classes d'applications booléennes. La première est bien connue, c'est celle des applications symétriques. Les deux autres qui la contiennent, sont nouvelles et sont les fonctions k-quasi―symétriques à n variables, 1≤k≤n, et leur concaténation. Finalement, nous appliquons la méthode de projection à ces applications pour obtenir des formulations compactes.

Esta tesis doctoral consta de dos partes claramente diferenciadas. En la primera de ellas se utiliza una herramienta matemática llamada current para caracterizar cuerpos geométricos y poder trabajar con ellos en un espacio de Hilbert con propiedades prácticas: un vector-valued Reproducing Kernel Hilbert Space. Al representar los cuerpos geométricos mediante funciones en este espacio, podemos utilizar la teoría de Análisis de Datos Funcionales para adaptar técnicas estadísticas (como algoritmos de clasificación o métodos de regresión) a un conjunto de cuerpos geométricos. Finalmente, se aplican los modelos teóricos desarrollados a una base de datos formada por escáneres de cuerpos de niños y niñas, para resolver problemas relacionados con el tallaje infantil. La segunda parte del trabajo se desarrolla dentro del ámbito de la Estereología. En él obtenemos fórmulas rotacionales para el área y para las integrales de curvatura media de la superficie frontera de un dominio compacto en un espacio de curvatura constante λ.
This doctoral thesis consists of two clearly differentiated parts. In the first one, a mathematical tool called current is used to characterize geometric bodies as functions in a vector-valued Reproducing Kernel Hilbert Space, which is a Hilbert spaces with practical properties. Using Functional Data Analysis Theory we can apply statistical techniques, such us classification algorithms or regression methods, to a set of functions representing geometric bodies. Later, the theoretical models that have been developed are applied to a database consisting of scans of bodies of children to solve problems related to sizing children. The second part of the work is developed within the scope of Stereology. In this part we obtain rotational formulas for the area and for the average curvature integrals of the boundary surface of a compact domain in a space of constant curvature λ.

A distribuição bivariada de Marshall-Olkin é estendida, relaxando-se a hipótese de choques exponencialmente distribuídos e assumindo-se dependência entre os choques individuais. Abordagem semelhante é considerada para sua versão dual. Representação por meio de cópula, propriedades probabilísticas e de confiabilidade assim como resultados em valores extremos são então obtidos. A propriedade de falta de memória bivariada é estendida assumindo-se uma função de dependência sem memória. Uma nova classe de distribuições caracterizada por essa propriedade estendida é introduzida. Correspondentes interpretações geométricas, procedimentos de construção, representação estocástica, relação com cópula de sobrevivência e propriedades de confiabilidade são derivadas.
Bivariate Marshall-Olkin model, Dual model, Exponential representation, Dependence function, Bivariate aging, Copula, Survival copula, Stochastic order, Bivariate extreme value distribution, Pickands measure, Pickands dependence function, Failure rate, Bivariate hazard gradient, Bivariate lack-of-memory, Residual lifetime vector, Characterization.

碩士
國立彰化師範大學
數學系
90
In this paper, we first prove some existence theorems of generalized vector saddle point for multivalued map or vector valued function. As a consequence, we establish the existence theoremsof generalized vector minimax theorem. We also establish the existencetheorem of generalized vector quasi-variational-like inequality problem for vector valued function and the relationship between the solutions of the generalized vector saddle point problem and the generalized vectorquasi-variational-like inequality problem for vector valued functions.From which we establish another existence theorem of generalized vector saddle point problems.

The results of this thesis are motivated by the investigation of abstract Cauchy problems. Our primary contribution is encapsulated in two new theorems. The first main theorem is a generalisation of a result of E. M. Stein. In particular, we show that every symmetric diffusion semigroup acting on a complex-valued Lebesgue space has a tensor product extension to a UMD-valued Lebesgue space that can be continued analytically to sectors of the complex plane. Moreover, this analytic continuation exhibits pointwise convergence almost everywhere. Both conclusions hold provided that the UMD space satisfies a geometric condition that is weak enough to include many classical spaces. The theorem is proved by showing that every symmetric diffusion semigroup is dominated by a positive symmetric diffusion semigoup. This allows us to obtain (a) the existence of the semigroup's tensor extension, (b) a vector-valued version of the Hopf--Dunford--Schwartz ergodic theorem and (c) an holomorphic functional calculus for the extension's generator. The ergodic theorem is used to prove a vector-valued version of a maximal theorem by Stein, which, when combined with the functional calculus, proves the pointwise convergence theorem. The second part of the thesis proves the existence of abstract Strichartz estimates for any evolution family of operators that satisfies an abstract energy and dispersive estimate. Some of these Strichartz estimates were already announced, without proof, by M. Keel and T. Tao. Those estimates which are not included in their result are new, and are an abstract extension of inhomogeneous estimates recently obtained by D. Foschi. When applied to physical problems, our abstract estimates give new inhomogeneous Strichartz estimates for the wave equation, extend the range of inhomogeneous estimates obtained by M. Nakamura and T. Ozawa for a class of Klein--Gordon equations, and recover the inhomogeneous estimates for the Schr??dinger equation obtained independently by Foschi and M. Vilela. These abstract estimates are applicable to a range of other problems, such as the Schr??dinger equation with a certain class of potentials.

by Yan Shing.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1999.
Includes bibliographical references (leaves 78-80).
Abstract also in Chinese.
Chapter 1 --- Admissible Points of Convex Sets --- p.7
Chapter 1.1 --- Introduction and Notations --- p.7
Chapter 1.2 --- The Main Result --- p.7
Chapter 1.2.1 --- The Proof of Theoreml.2.1 --- p.8
Chapter 1.3 --- An Application --- p.10
Chapter 2 --- A Generalization on The Theorems of Admissible Points --- p.12
Chapter 2.1 --- Introduction and Notations --- p.12
Chapter 2.2 --- Fundamental Lemmas --- p.14
Chapter 2.3 --- The Main Result --- p.16
Chapter 3 --- Introduction to Variational Inequalities --- p.21
Chapter 3.1 --- Variational Inequalities in Finite Dimensional Space --- p.21
Chapter 3.2 --- Problems Which Relate to Variational Inequalities --- p.25
Chapter 3.3 --- Some Variations on Variational Inequality --- p.28
Chapter 3.4 --- The Vector Variational Inequality Problem and Its Relation with The Vector Optimization Problem --- p.29
Chapter 3.5 --- Variational Inequalities in Hilbert Space --- p.31
Chapter 4 --- Vector Variational Inequalities --- p.36
Chapter 4.1 --- Preliminaries --- p.36
Chapter 4.2 --- Notations --- p.37
Chapter 4.3 --- Existence Results of Vector Variational Inequality --- p.38
Chapter 5 --- The Generalized Quasi-Variational Inequalities --- p.44
Chapter 5.1 --- Introduction --- p.44
Chapter 5.2 --- Properties of The Class F0 --- p.46
Chapter 5.3 --- Main Theorem --- p.53
Chapter 5.4 --- Remarks --- p.58
Chapter 6 --- A set-valued open mapping theorem and related re- sults --- p.61
Chapter 6.1 --- Introduction and Notations --- p.61
Chapter 6.2 --- An Open Mapping Theorem --- p.62
Chapter 6.3 --- Main Result --- p.63
Chapter 6.4 --- An Application on Ordered Normed Spaces --- p.66
Chapter 6.5 --- An Application on Open Decomposition --- p.70
Chapter 6.6 --- An Application on Continuous Mappings from Order- infrabarreled Spaces --- p.72
Bibliography

碩士
國立中央大學
數學研究所
89
The purpose of this thesis is to study, by means of some r-dimensional linear operators, the approximation of vector-valued functions defined on a bounded subset. We use an approximation theorem of Korovkin type to prove that these operators converge uniformly, and then use another Korovkin-type theorem with rate to estimate their pointwise convergence rates. Finally, we apply some of these concrete approximation processes to derive some representation formulas for r-parameter semigroups of bounded linear operators.