Sobolev mappings and Cantor type counterexamples Author: Martin Fiala Supervisor: doc. RNDr. Stanislav Hencl, Ph.D. Abstract: The aim of this work is to show one of the general con- structions of the mappings, which can be used to create different coun- terexamples in the theory of Sobolev mappings. The construction is described in detail and then it is used for a number of examples. The last chapter is devoted to a slight generalization of this construction. 1

This thesis presents a study on possible genetic consequences of the exposure to the space environment during space missions The present study was undertaken in co-operation with the Canadian Space Agency, and involved the analysis of the lymphocyte samples taken from experienced cosmonauts and trainees. For the analysis of genotoxicity of the space environment, a T-lymphocyte hprt clonal assay has been employed. In order to distinguish between artefacts associated with this method and the spaceflight-related effects, we have conducted a series of in vitro reconstruction experiments. In these experiments we have analysed interactions between plating efficiency (PE) of T-lymphocytes and efficiency of mutant recovery. Using 12 pairs of independent wild type (WT) and mutant clones, we have demonstrated an inverse correlation between initial viability of the WT cells and survival of mutant cells (r = 0.3496, p < 0.05). Our data suggest that the presence of WT cells in the selection plates does suppress the recovery of mutants in HPRT assay. This effect is stronger in samples with high PE, and may be a source of large error in estimation of mutant frequencies (approx. 3-fold in the range of PEs from 10% to 60%), which is especially relevant when samples with different PEs are compared. Analysis of samples from cosmonauts was conducted in two experiments. The first experiment involved 5 samples taken in 1992 from cosmonauts who have completed spaceflights ranging in duration from 7 to 365 days. Hprt mutant frequencies (MF) in these samples were 2.5–5 times higher than the age-corrected values for healthy, unexposed subjects in Western countries (Tates et al., 1991; Branda et al., 1993), and 2-3-fold higher than those determined for unexposed individuals residing in Russia (Jones et al., 1995). The cosmonaut mutational spectrum differed from that of unexposed healthy subjects (p = 0.042), and showed a higher incidence of splicing errors, frameshifts, and complex mutations. Distribution of base substitutions was remarkably similar to that observed in Russian twins sampled at the same period (Curry et al., 1998), thus suggestive of possible environmental, diet, or life-style related exposures. The second study was conducted on samples taken 5 years later and involved trainees and a group of cosmonauts with more uniform (at least 6 months) and recent flight experience. Hprt MFs in both cosmonaut and trainee groups were virtually identical (17.2 ± 0.6 and 17.6 ± 4.7 × 10⁻⁶ respectively), and approximately 2-fold higher than in matching Western controls, although considerably lower than in our previous observations. Mutational spectra in both datasets were very similar to that observed in our earlier study, and were significantly different from spontaneous data (p = 0.031–0.038). Distribution of base substitutions, however, did not show any differences. Our data indicate that the space environment is not genotoxic at the hprt locus. At the same time, uniformly high MFs observed in all studied groups suggest that the level of the mutagenic burden in at least megalopolis areas of Russia may be considerably larger than in the West. Also, there are some indications of a possible restructuring of mutagenic burden in post-transitional Russia. / Graduate

NDLTD Innovative ETD Award 2007. No abstract available / Dissertation (MArch(Prof))--University of Pretoria, 2007. / Architecture / unrestricted

A performance evaluation of CSTC (Cooperative Space Time Coding) with spatially cor-related fading and imperfect channel estimation in Gaussian as well as impulsive noise is presented. Closed form expressions for the pairwise error probability conditioned on the estimated channel gains are derived by assuming the components of the received vector are independent given the estimated channel gains. An expurgated union bound using the limiting before averaging technique given the estimated channel gains is then obtained. Although this assumption is not strictly valid, simulation results show that the bound is accurate in estimating the diversity order as long the channel estimation is not very poor. It is found that CSTC with block fading channels can reduce the frame error rate (FER) relative to SUSTC (Single User Space Time Coding) with quasi-static fading channels, even when the channel gains for each user are strongly correlated and when the channel estimations are very poor. A decision metric for CSTC with spatially correlated fading, imperfect channel estimation, and impulsive mixture Gaussian noise is derived which yields lower FERs than the Gaussian noise decision metric. Simulation results show that the FER performance of CSTC with mixture Gaussian noise outperforms CSTC with Gaussian noise at low SNR. At high SNR, the FER performance of CSTC with Gaussian noise is better than the FER performance of CSTC with mixture Gaussian noise due to the heavy tail of the mixture Gaussian noise. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

Certain properties of an unknown element u in a Hilbert space are investigated. For u satisfying certain linear constraints, it is shown that approximations to u and error bounds for the approximations may be obtained in terms of functional representers. The general approximation method is applied to homogeneous systems of ordinary linear differential equations and various formulae are derived. An Alwac III-E digital computer was used to compute optimal approximations and error bounds with the aid of these formulae. Numerous applications to particular systems are mentioned. On the basis of the numerical results, certain remarks are given as a guide for the numerical application of the method, at least in the framework of ordinary differential equations. From the cases studied it is seen that this can be a practicable method for the numerical solution of differential equations. / Science, Faculty of / Mathematics, Department of / Graduate

Pour le présent mémoire de maîtrise, nous avons retenu le roman de J.M.G. Le Clézio, Désert, dans lequel nous avons essayé d'analyser le temps et l'espace. Plus brièvement nous avons examiné la fonction des personnages dans ce même roman. Au cours de notre recherche, nous avons appris que l'écriture est le point de départ de la pensée de Le Clézio. La première dimension temporelle du roman est celle de l'histoire. Désert est un roman binaire. Il est divisé en deux grandes parties : l'histoire des hommes bleus et l'histoire de Lalla, jeune immigrée de dix-sept ans. La première est racontée au passé, la deuxième au présent. Le roman n'est pas construit chronologiquement, nous avons choisi le terme de mosaïque pour définir le découpage passë-présent-passé. La composition et le mode narratif permettent d'exprimer le temps ; deux niveaux temporels : le temps de la fiction et le temps de la narration. La multiplicité temporelle, c'est-à-dire temps de l'histoire et temps du récit a constitué une partie de notre étude. Le temps nous a conduit à l'espace qui s'est avéré un point essentiel de notre étude. Il n'est pas seulement le lieu où se joue l'histoire. Espace et description sont indissociables. Nous remarquons que la description chez Le Clézio est étroitement liée à la conscience et à l'état d'âme des personnages. L'espace contribue à l'élaboration de la narration. Sous des formes diverses, la description suggère la découverte de certains mythes dont nous avons analysé la valeur symbolique à travers les images présentées. La fonction des personnages n'est pas dissociable du reste de notre analyse. Lieux et descriptions expliquent les personnages. A travers leur caractère nous retrouvons uneanalyse psychologique et sociale. / Arts, Faculty of / French, Hispanic, and Italian Studies, Department of / Graduate

In the study of the spaces (formula omitted) of functions for which the pth powers of all the derivatives up to order ℓ are summable in the domain Ω⊂R, it has been found that there are mutual relations between various spaces. These relations were developed under the name "embedding theorems". The first embedding theorem (for spaces (formula omitted) were proved by Sobolev [3]*. Subsequently these spaces became known as Sobolev Spaces. However, in the study of existence of solutions for well-posed boundary value problems, there arose the necessity to consider spaces of distributions: an example is the space dual to (formula omitted). For a thorough development of distributions see L. Schwarz's texts [4]. Furthermore, the classes of Sobolev spaces had to be widened to fractional values of ℓ, the latter spaces being particularly useful in the study of non-linear problems. This thesis follows the development of generalized Sobolev spaces as in Volevich and Panayakh [1]. In section I we prove the basic theorems in this formulation. In section II, the existence of a function is proved using the formulation of section II. The proof of the proposition in which a modification has been made was given by Agranovich and Vishik [2]. The proposition is essential to the applications of Sobolev spaces to differential operators. The result states that ll u ll µ ≤ constant ll u,Ω ll µ for (formula omitted) for the particular case when the weighting function is (formula omitted) and Ω is a half-line. (For definitions see section I). Section III is devoted to a brief comparison of this formulation of Sobolev spaces to other approaches. / Science, Faculty of / Mathematics, Department of / Graduate

The purpose of this exploratory study was 1) to identify and illustrate a number of components of a scheme of interpretation used by children to comprehend and act according to social rules and conventions governing the appropriation of space, and 2) to develop and test a model for examining qualitative differences in such a scheme of interpretation, within and between various age groups. The model was based on two propositions: first that knowledge of the social structure and the social organization of space are two components of a scheme of interpretation, and second that such a scheme of interpretation will become more abstract, differentiated, and integrated with age. In the final study fifteen children were interviewed, including ten six year olds and five twelve and thirteen year olds. The interview focussed on the children's awareness of and justifications for social rules and conventions governing access to and use of private, semi-private, and public spaces in the neighborhood, school, and home. The model was successfully used in the analysis of the children's responses. Various aspects of their conceptions of the social structure and the social organization of space were explored. Their responses were also classified according to four levels posited in the model. It was found that most of the six year olds had an undifferentiated and concrete scheme of interpretation. They were aware of context-specific rules and conventions. When asked to explain or justify the rules they typically referred to the physical or social characteristics of the setting. By twelve or thirteen years of age the children had developed a more abstract, differentiated, and integrated scheme of interpretation. They often referred to concepts such as power or ownership to explain or justify the ability of various individuals to appropriate space. Thus, the results of this study suggest that an understanding of spatial appropriation involves not simply internalizing a 'catalogue' of social rules and conventions, but rather constructing a more complex scheme of interpretation consisting of knowledge of 1) the social structure, 2) the social organization of space, and 3) the relations between them. / Applied Science, Faculty of / Community and Regional Planning (SCARP), School of / Graduate

The purpose of this paper is to make a detailed study of vector spaces and a certain vector-like system.

This thesis explores how familiar objects such as the homely recipe book hold our affection and shape our personal worlds. It takes its inspiration from a body of literature that only recently has explored in detail our relationship to mundane objects, subjecting these objects - and our feelings about them - to a serious scrutiny. The thesis is concerned with a material culture that takes us into domestic space, and to the objects within it to which we attach importance. Specifically, the inquiry explores the cultural mores surrounding the practice of cooking and writing food. It considers the interplay between public and private, male and female, self and other and the significance of the domestic space in each case. It asks how the culture of the recipe book helps shape female domestic identity, that is, the personae of women within the home, and as wives and mothers, as opposed to their public personae. This thesis studies the (until-recently) under-researched yet broad field - previously regarded as both too trivial and too formulaic to merit study - of homely recipe books. It considers the large collection of historic manuscripts of this genre available at the National Library of South Africa, in particular the collection of Louis C Leipoldt, and it regards these as part of a continuum with my own mother's recipe book. An important leitmotif of the study is the evolution of the recipe book from manuscript to printed, and from single copy to mass-produced text. On the one hand using recipe books as historical sources for the study of food and material culture, this study is also concerned with the affective impact of these texts, and more specifically what they say about the individuals and societies that made them. A central theme of the study is the role played in women's lives by the collecting and archiving of recipes through hand-written texts. My purpose is twofold: first, to bring these hidden histories to light, opening the kitchen door to the lives of ordinary women through their private writings; and second, to explore why the practice of writing food continues to be relevant into the present. I trace how homely recipe books are both exercises in personal authority as well as material traces of women's internal worlds and archives of the communities in which they exist. This study ultimately sees the return of the personalised recipe book as a route back to a positive and affirming female domestic identity, through a practice which is both therapeutic and self-actualising and which, through the act of archiving, brings together both past and present.