The concept of LR-nets was introduced and investigated firstly by H.P. Lotz in [27] and by F. Raebiger in [30]. Therefore we call such nets Lotz-Raebiger nets, shortly LR-nets. In this thesis we treat two problems on asymptotic behavior of these operator nets. First problem is to generalize well known theorems for Ces`aro averages of a single operator to LR-nets, namely to generalize the Eberlein and Sine theorems. The second problem is related to the strong convergence of Markov LR-nets on L1-spaces. We prove that the existence of a lower-bound functions is necessary and sufficient for asymptotic stability of LR-nets of Markov operators.

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study.

The dissertation consists of two parts. In the first part approximate methods for multidimensional weakly singular integral operators with operator-valued kernels are investigated. Convergence results and error estimates are given. There is considered an application of these methods to solving radiation transfer problems. Numerical results are presented, too. In the second part we consider a polynomial collocation method for the numerical solution of a singular integral equation over the interval. More precisely, the operator of our integral equation is supposed to be of the form \ $aI + b \mu^{-1} S \mu I $\ with \ $S$\ the Cauchy singular integral operator, with piecewise continuous coefficients \ $a$\ and \ $b,$\ and with a Jacobi weight \ $\mu.$\ To the equation we apply a collocation method, where the collocation points are the Chebyshev nodes of the first kind and where the trial space is the space of polynomials multiplied by another Jacobi weight. For the stability and convergence of this collocation method in weighted \ $L^2$\ spaces, we derive necessary and sufficient conditions. Moreover, the extension of these results to an algebra generated by the sequences of the collocation method applied to the mentioned singular integral operators is discussed and the behaviour of the singular values of the discretized operators is investigated. / Die Dissertation beschäftigt sich insgesamt mit der numerischen Analysis singulärer Integralgleichungen, besteht aber aus zwei voneinander unabhängigen Teilen. Der este Teil behandelt Diskretisierungsverfahren für mehrdimensionale schwach singuläre Integralgleichungen mit operatorwertigen Kernen. Darüber hinaus wird hier die Anwendung dieser allgemeinen Resultate auf ein Strahlungstransportproblem diskutiert, und numerische Ergebnisse werden präsentiert. Im zweiten Teil betrachten wir ein Kollokationsverfahren zur numerischen Lösung Cauchyscher singulärer Integralgleichungen auf Intervallen. Der Operator der Integralgleichung hat die Form \ $aI + b \mu^{-1} S \mu I $\ mit dem Cauchyschen singulären Integraloperator \ $S,$\ mit stückweise stetigen Koeffizienten \ $a$\ und \ $b,$\ und mit einem klassischen Jacobigewicht \ $\mu.$\ Als Kollokationspunkte dienen die Nullstellen des n-ten Tschebyscheff-Polynoms erster Art und Ansatzfunktionen sind ein in einem geeigneten Hilbertraum orthonormales System gewichteter Tschebyscheff-Polynome zweiter Art. Wir erhalten notwendige und hinreichende Bedingungen für die Stabilität und Konvergenz dieses Kollokationsverfahrens. Außerdem wird das Stabilitätskriterium auf alle Folgen aus der durch die Folgen des Kollokationsverfahrens erzeugten Algebra erweitert. Diese Resultate liefern uns Aussagen über das asymptotische Verhalten der Singulärwerte der Folge der diskreten Operatoren.

No description available.

The authors study parameter choice strategies for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depend on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskii principle. For the convenience of the reader the authors review in an appendix a few instances where the validity of a variational inequality can be established.

This thesis consists of four research papers. In the first paper we construct nonmetrizable compact convex sets with pathological sets of simpliciality, show- ing that the properties of the set of simpliciality known in the metrizable case do not hold without the assumption of metrizability. In the second paper we construct an example concerning remotal sets, answering thus a question of Martín and Rao, and present a new proof of the fact that in every infinite- dimensional Banach space there exists a closed convex bounded set which is not remotal. The third paper is a study of the relations between polynomials on Banach spaces and linear identities. We investigate under which conditions a linear identity is satisfied only by polynomials, and describe the space of poly- nomials satisfying such linear identity. In the last paper we study the coarse and uniform embeddability between Orlicz sequence spaces. We show that the embeddability between two Orlicz sequence spaces is in most cases determined only by the values of their upper Matuszewska-Orlicz indices. 1

Title: Exceptional Sets in Mathematical Analysis Author: Martin Rmoutil Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Ondřej Kalenda, Ph.D., DSc., Department of Mathematical Analysis Abstract: The present thesis consists of four research articles. In the first paper we study the notion of σ-lower porous set; our main result is the existence of two closed sets A, B ⊂ R which are not σ-lower porous, but their product in R2 is lower porous. In the second and third article we use a set-theoretical method of el- ementary submodels involving the Lwenheim-Skolem theorem to prove that certain σ-ideals of sets in Banach spaces are separably determined. In the second article we do so for σ-porous sets and σ-lower porous sets. In the next article we refine these methods obtaining separable determination of a wide class of σ-ideals. In both cases we derive interesting corollaries which extend known theorems in separable spaces to the nonseparable setting; for example, we obtain the following theorem. Any continuous convex function on an Asplund space is Frchet differentiable outside a cone small set. In the fourth article we introduce the following notion. A closed set A ⊂ Rd is said to be c-removable if the following is true: Every real function on Rd is convex whenever it is continuous on Rd...

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-06T14:21:44Z No. of bitstreams: 1 MOISÉS DANTAS DOS SANTOS - DISSERTAÇÃO PPGMAT 2005..pdf: 524230 bytes, checksum: a8d6e23eaf6da89e369eb29e888e7a1a (MD5) / Made available in DSpace on 2018-07-06T14:21:44Z (GMT). No. of bitstreams: 1 MOISÉS DANTAS DOS SANTOS - DISSERTAÇÃO PPGMAT 2005..pdf: 524230 bytes, checksum: a8d6e23eaf6da89e369eb29e888e7a1a (MD5) Previous issue date: 2005-12 / Neste trabalho usaremos métodos variacionais para mostrar a existência de solução fraca para dois tipos de problema. O primeiro consiste num problema não-linear, O segundo, trata-se de uma Equação Diferencial Ordinária / In this work we use variational methods to show the existence of weak solutions for two types problems. The first of them is a nonlinear problem, The second, is related with a following Ordinary Differential Equations.

This thesis is divided into four chapters. In the first one, all the pre-requisite theory of semigroups and groupoids is introduced, as well as a few new results - such as a short study of ∨-ideals and quotients in distributive semigroups and a non-commutative Loomis-Sikorski Theorem. In the second chapter, we motivate and describe the sofic property for probability measure-preserving groupoids and prove several permanence properties for the class of sofic groupoids. This provides a common ground for similar results in the particular cases of groups and equivalence relations. In particular, we prove that soficity is preserved under finite index extensions of groupoids. We also prove that soficity can be determined in terms of the full group alone, answering a question by Conley, Kechris and Tucker-Drob. In the third chapter we turn to the classical problem of reconstructing a topological space from a suitable structure on the space of continuous functions. We prove that a locally compact Hausdorff space can be recovered from classes of functions with values on a Hausdorff space together with an appropriate notion of disjointness, as long as some natural regularity hypotheses are satisfied. This allows us to recover (and even generalize) classical theorem by Kaplansky, Milgram, Banach-Stone, among others, as well as recent results of the similar nature, and obtain new consequences as well. Furthermore, we extend the techniques used here to obtain structural theorems related to topological groupoids. In the fourth and final chapter, we study dynamical aspects of partial actions of inverse semigroups, and in particular how to construct groupoids of germs and (partial) crossed products and how do they relate to each other. This chapter is based on joint work with Viviane Beuter.

The main topic of this thesis is separation of points and w∗ -derived sets in dual Banach spaces. We show, that in duals of reflexive spaces w∗ -derived set of a convex subset coincides with its w∗ -closure. We also show, that subspace of a dual reflexive space is norming, if and only if it is total. Later we show, that in the dual of every non-reflexive space we can find a convex subset whose w∗ -derived set is not w∗ -closed. Hence, this statement is a characterisation of reflexive spaces. Next we show, that subspaces in duals of quasi-reflexive spaces are norming, if and only if they are total. Later we show, that in the dual of every non-quasi-reflexive space we can find a subspace which is total but not norming; thus, the previous statement is a characterisation of quasi-reflexive spaces. We also show, that for absolutely convex subsets of duals of quasi-reflexive spaces w∗ -derived set coincides with w∗ -closure. In the last section we define w∗ -derived sets of higher orders and show, that in the dual of every non-quasi-reflexive separable Banach space there exist subspaces of order of each countable non-limit ordinal and no other. 1