In this work, an application of the algebraic reconstruction technique to a borehole reconstruction problem is considered. The formulation of the borehole problem gives the attendant electromagnetic wave equations in matrix form. The algebraic reconstruction technique is used to reconstruct a solution. Three sources of errors are identified in the reconstruction process. Suggestions are made which will help minimize or predict the effects of these errors. General limitations of the algebraic reconstruction technique are discussed. The limitations in terms of the borehole problem are explained. Practical limitations for the borehole problem are thus obtained and quantified mathematically. It is found that even in some practical situations, the borehole reconstruction process is impossible.

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No description available.

In this thesis we determine the submodule structure of a number of Weyl modules for algebraic groups with root systems B2 and G2. We use the Jantzen sum formula to find the composition factors of Weyl modules and go on to use homomorphisms between Weyl modules, given by H.H. Andersen, and the comparison of two filtrations of tensor products of Weyl modules to establish submodule structure. A computer program in the Prolog language is given which calculates the Jantzen sum formula. In addition we find one 2-dimensional Ext group for simple modules for type G2 in characteristic greater than or equal to 7.

The ways that physics students’ conceptualize – the way they experience – the use of algebraic signs in vector-kinematics has not been extensively studied. The most comprehensive of these few studies was carried out in South Africa 15 years ago. This study found that the variation in the ways that students experience the use of algebraic signs could be characterized by five qualitatively different categories. The consistency of the nature of this experience across either the same or different educational settings has never given further consideration. This project sets out to do this using two educational settings; one similar to the original South African one, and one at the natural science preparatory programme known as basåret at Uppsala University in Sweden. The study was carried out under the auspices of the Division of Physics Education Research at the Department of Physics and Astronomy at Uppsala University in collaboration with Nadaraj Govender, University of KwaZulu-Natal, who performed the original study while completing his PhD at the University of the Western Cape, South Africa. This study is situated in the kinematics section of introductory physics with participants drawn from the natural science preparatory programme at Uppsala University and physical science preservice teachers’ programme at the University of KwaZulu-Natal, South Africa. The participating students completed a specially designed questionnaire on the use of signs in kinematics problem solving. A sub-group of these students was also purposefully selected to take part in semi-structured interviews that aimed at further exploring their experiences of algebraic signs. The students’ descriptions and answers were categorized using Nadaraj Govender’s set of categories, which had been constructed using the phenomenographic research approach. This approach is designed to enable finding the variation of ways people experience a phenomenon. The process of sorting the data was grounded in this phenomenographic perspective. From this categorization it was possible to identify four of the original five categories amongst the participating students. The results suggest that these four categories remain educationally relevant today even if the context is not the same as the one for the original findings. Although one of the original five categories was not found, the analysis cannot be taken to definitely eliminate this from the original outcome space of results. A more extensive study would be needed for this and thus a proposal is made that further studies be undertaken around this issue. The study ends by suggesting that physics teachers at the introductory level need to obtain a broader understanding of their students’ difficulties and develop their teaching to better deal with the challenges that become more visible in this broader understanding. / På vilka sätt fysikstudenter föreställer sig och förstår användandet av algebraiska tecken i vektorkinematik har endast studerats i mindre utsträckning. Den mest omfattande av dessa få studier genomfördes i Sydafrika för 15 år sedan. Denna studie upptäckte att variationen av de sätt studenter upplever användandet av algebraiska tecken på kunde karaktäriseras genom fem kvalitativt olika kategorier. Hur solida dessa upplevelser är i en liknande eller helt annan utbildningsmiljö har däremot inte studerats vidare. Detta projekt ämnar till att göra detta genom att använda två olika studentgrupper; en liknande den ursprungliga gruppen i Sydafrika, samt det tekniskt-naturvetenskapliga basåret vid Uppsala universitet, Sverige. Studien har genomförts med stöd från avdelningen för fysikens didaktik vid institutionen för fysik och astronomi vid Uppsala universitet i samarbete med Nadaraj Govender, University of KwaZulu-Natal, Sydafrika, som genomförde den ursprungliga studien under sin doktorandutbildning vid University of the Westen Cape, Sydafrika. Denna studie är begränsad till den del av den grundläggande fysiken som behandlar kinematik och innefattade deltagare från det tekniskt-naturvetenskapliga basåret vid Uppsala universitet samt tredje års studenter vid physical science preservice teachers’ programme, University of KwaZulu-Natal, Sydafrika. De deltagande studenterna genomförde ett specialdesignat frågeformulär kring användandet av algebraiska tecken för att lösa kinematiska problem. En del av dessa studenter valdes sedan ut för att delta i semi-strukturerade intervjuer som syftade till att vidare utforska deras upplevelser kring algebraiska tecken. Studenternas beskrivningar och svar kategoriserades med hjälp av Nadaraj Govenders fem kategorier som tagits fram genom ett fenomenografiskt tillvägagångssätt. Detta tillvägagångssätt är framtaget för att kunna hitta variationen av hur människor upplever ett fenomen. Sorteringsprocessen grundades i detta fenomenografiska perspektiv. Från denna kategorisering var det möjligt att identifiera fyra av de fem ursprungliga kategorierna bland de deltagande studenterna. Fyra av de fem ursprungliga kategorierna som föreslagits av Govender återfanns genom denna studie varför dessa kategorier föreslås förbli relevanta idag även om utbildningsmiljön skiljer sig från den ursprungliga. Trots att den femte kategorin inte hittades kan denna inte definitivt exkluderas från det outcome space som beskriver studenters upplevelser för algebraiska tecken. Det föreslås att vidare studier undersöker förekomsten av denna kategori. Studien avslutas med att föreslå att fysik lärare på grundnivå behöver få en bättre förståelse för sina studenters svårigheter samt att de behöver utveckla sin undervisning för att bättre kunna hantera dessa svårigheter och på så sätt göra undervisningen mer anpassad för mångfalden av studenterna.

We compare two methods for software specification: <em>algebraic specifications</em> and automata. While algebraic specifications have been around since the 1970s and have been studied extensively, specification by automata is relatively new. Its origins are in another veteran method called <em>trace assertions</em>, which considers a software module as a set of traces, that is, a sequences of function executions. A module is specified by a set of canonical traces and an equivalence relation matching one of the canonical traces to each non-canonical trace. It has been recently shown that trace assertions is an equivalent method to specification by automata. In continuation of this work on trace assertions and automata, we study how automata compare with algebraic specifications. We prove that every specification using an automaton can be converted into an algebraic specification describing the same abstract data type. This conversion utilises a set of canonical words, representing states in the automaton. We next consider varieties of monoids as a heuristic for obtaining more concise algebraic specifications from automata. Finally, we discuss the opposite conversion of algebraic specifications into automata. We show that, while an automaton always exists for every abstract data type described by an algebraic specification, this automaton may not be finitely describable and therefore may not be considered as a viable method for software specification.

This thesis investigates two problems that are discrete analogues of two harmonic analytic problems which lie in the heart of research in the field. More specifically, we consider discrete analogues of the maximal Kakeya operator conjecture and of the recently solved endpoint multilinear Kakeya problem, by effectively shrinking the tubes involved in these problems to lines, thus giving rise to the problems of counting joints and multijoints with multiplicities. In fact, we effectively show that, in R3, what we expect to hold due to the maximal Kakeya operator conjecture, as well as what we know in the continuous case due to the endpoint multilinear Kakeya theorem by Guth, still hold in the discrete case. In particular, let L be a collection of L lines in R3 and J the set of joints formed by L, that is, the set of points each of which lies in at least three non-coplanar lines of L. It is known that |J| = O(L3/2) ( first proved by Guth and Katz). For each joint x ∈ J, let the multiplicity N(x) of x be the number of triples of non-coplanar lines through x. We prove here that X x2J N(x)1=2 = O(L3=2); while we also extend this result to real algebraic curves in R3 of uniformly bounded degree, as well as to curves in R3 parametrized by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, involving three finite collections of lines in R3; a multijoint formed by them is a point that lies in (at least) three non-coplanar lines, one from each collection. We finally present some results regarding the joints problem in different field settings and higher dimensions.

In this thesis we make several contributions to the theory of moduli spaces of smooth manifolds, especially in dimension two. In Chapter 2 (joint with Soren Galatius) we give a new geometric proof of a generalisation of the Madsen-Weiss theorem, which does not rely on the tangential structure under investigation having homological stability. This allows us to compute the stable homology of moduli spaces of surfaces equipped with many different tangential structures. In Chapter 3 we give a general approach to homological stability problems, especially focused on stability for moduli spaces of surfaces with tangential structure. We give a sufficient condition for a structure to exhibit homological stability, and thus obtain stability ranges for many tangential structures of current interest (orientations, maps to a simply-connected background space, etc.), which match or improve the previously known ranges in all cases. In Chapter 4 we define and study the cobordism category of submanifolds of a fixed background manifold, and extend the work of Galatius-Madsen-Tillmann-Weiss to identify the homotopy type of these categories. We describe several applications of this theory. In Chapter 5 we compute the stable (co)homology of the non-orientable mapping class group, and find a family of geometrically-defined torsion cohomology classes. This is in contrast to the oriented mapping class group, where few are known. In Chapter 6 (joint with Johannes Ebert) we study the divisibility of certain characteristic classes of bundles of unoriented surfaces introduced by Wahl, analogues of the Miller-Morita-Mumford classes for unoriented surfaces. We show them to be indivisible in the free quotient of cohomology.

A thesis submitted to the Faculty of Science, University of the Witwatersrand and to the University Cheikh Anta Diop of Dakar(UCAD) in fulfillment of the requirements for a Dual-degree for Doctor in Philosophy in Mathematics. November 6th, 2017. / This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form '(jaunj) = jbvmj; where ' is the Euler totient function, fungn 0 and fvmgm 0 are two non-degenerate binary recurrence sequences and a; b some positive integers. More precisely, we study problems involving members of the recurrent sequences being rep-digits, Lehmer numbers, whose Euler’s function remain in the same sequence. We prove that there is no Lehmer number neither in the Lucas sequence fLngn 0 nor in the Pell sequence fPngn 0. The main tools used in this thesis are lower bounds for linear forms in logarithms of algebraic numbers, the so-called Baker-Davenport reduction method, continued fractions, elementary estimates from the theory of prime numbers and sieve methods. / LG2018

It is well known that the n-th eigenfunction to one-dimensional Sturm-Liouville eigenvalue problems has exactly n-1 nodes, i.e. non-degenerate zeros. For higher dimensions, it is much more complicated to obtain general statements on the zeros of eigenfunctions. The author states a new conjecture on the number of nodal domains of spherical harmonics, i.e. of connected components of S^2 \ N(u) with the nodal set N(u) = (x in S^2 : u(x) = 0) of the eigenfunction u, and proves it for the first six eigenvalues. It is a sharp upper bound, thus improving known bounds as the Courant nodal domain theorem, see S. Y. Cheng, Comment. Math. Helv. 51, 43-55 (1976; Zbl 334.35022). The proof uses facts on real projective plane algebraic curves (see D. A. Gudkov, Usp. Mat. Nauk 29(4), 3-79, Russian Math. Surveys 29(4), 1-79 (1979; Zbl 316.14018)), because they are the zero sets of homogeneous polynomials, and the spherical harmonics are the restrictions of spherical harmonic homogeneous polynomials in the space to the plane. / Series: Preprint Series / Department of Applied Statistics and Data Processing