No description available.

No description available.

We classify the smooth complete intersections with positive-definite intersection form on their middle cohomology. There are two families. The first family are quadric hypersurfaces in P(4k+1) with k a positive integer. The middle cohomology is always of rank two and the intersection lattice corresponds to the identity matrix. The second family are complete intersections of two quadrics in P(4k+2) (k a positive integer). Here the intersection lattices are the Gamma(4(k+1)) lattices; in particular, the intersection lattice of a smooth complete intersection of two quadrics in P(6) is the famous E8 lattice. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2012-10-15 13:19:42.654

This thesis is concerned with the solution of large systems of linear algebraic equations in which the matrix of coefficients is sparse. Such systems occur in the numerical solution of elliptic partial differential equations by finite-difference methods. By applying some well-known iterative methods, usually used to solve linear PDE systems, the thesis investigates their applicability to solve a set of four mildly nonlinear test problems. In Chapter 4 we study the basic iterative methods and semiiterative methods for linear systems. In particular, we derive and apply the CS, SOR, SSOR methods and the SSOR method extrapolated by the Chebyshev acceleration strategy. In Chapter 5, three ways of accelerating the SOR method are described together with the applications to the test problems. Also the Newton-SOR method and the SOR-Newton method are derived and applied to the same problems. In Chapter 6, the Alternating Directions Implicit methods are described. Two versions are studied in detail, namely, the Peaceman-Rachford and the Douglas-Rachford methods. They have been applied to the test problems for cycles of 1, 2 and 3 parameters. In Chapter 7, the conjugate gradients method and the conjugate gradient acceleration procedure are described together with some preconditioning techniques. Also an approximate LU-decomposition algorithm (ALUBOT algorithm) is given and then applied in conjunction with the Picard and Newton methods. Chapter 8 contains the final conclusions.

No description available.

<p>This thesis analyzes the Chow motives of 3 types of smooth projective varieties: the desingularized elliptic self fiber product, the Fano surface of lines on a cubic threefold and an ample hypersurface of an Abelian variety. For the desingularized elliptic self fiber product, we use an isotypic decomposition of the motive to deduce the Murre conjectures. We also prove a result about the intersection product. For the Fano surface of lines, we prove the finite-dimensionality of the Chow motive. Finally, we prove that an ample hypersurface on an Abelian variety possesses a Chow-Kunneth decomposition for which a motivic version of the Lefschetz hyperplane theorem holds.</p> / Dissertation

A thesis submitted to the School of Computer Science, Faculty of Science, University of the Witwatersrand, Johannesburg in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 15 May 2013 / In this thesis a number of different constructions on ordered algebraic structures are studied. In particular, two types of constructions are considered: completions and finite embeddability property constructions. A main theme of this thesis is to determine, for each construction under consideration, whether or not a class of ordered algebraic structures is closed under the construction. Another main focus of this thesis is, for a particular construction, to give a syntactical description of properties preserved by the construction. A property is said to be preserved by a construction if, whenever an ordered algebraic structure satisfies it, then the structure obtained through the construction also satisfies the property. The first four constructions investigated in this thesis are types of completions. A completion of an ordered algebraic structure consists of a completely lattice ordered algebraic structure and an embedding that embeds the former into the latter. Firstly, different types of filters (dually, ideals) of partially ordered sets are investigated. These are then used to form the filter (dually, ideal) completions of partially ordered sets. The other completions of ordered algebraic structures studied here include the MacNeille completion, the canonical extension (also called the completion with respect to a polarization) and finally a prime filter completion. A class of algebras has the finite embeddability property if every finite partial subalgebra of some algebra in the class can be embedded into some finite algebra in the class. Firstly, two constructions that establish the finite embeddability property for residuated ordered structures are investigated. Both of these constructions are based on completion constructions: the first on the Mac- Neille completion and the second on the canonical extension. Finally, algebraic filtrations on modal algebras are considered and a duality between algebraic and relational versions of filtrations is established.

We give an overview of the numerical solution of the initial value differential-algebraic equation (DAB) from the underlying theory, through both the development of numerical techniques and software and a survey of the major areas of application, to the implementation of available software codes in the solution of DABs arising in applications. The experimental part serves to demonstrate the need for specific DAB, rather than simply ordinary differential equation (ODE), methods and the special considerations requited for the successful numerical solution of DAEs, as well.as verifying predictions made in the theory. It is hoped that '.his dissertation can be used as a reference by those working in areas of application, or at least as a pointer to the relevant literature. / GR 2016

A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg, 2014. / This study investigates to what extent and how teaching informed by variation theory could improve the Grade 11 learners’ interpretation of algebraic functions. The study adopted a learning study approach, where learner difficulties are elicited in a pre-test, and on the basis of the results of the pre-test, a lesson is planned, informed by variation theory, to make it possible for learners to discern what they found difficult. In this study, a pre-test on functions was given to three groups of Grade 11 learners (85 learners in all) in the researcher’s school that enabled the identification of aspects of functions learners found to be most problematic. The lesson was then taught successively to each group. A post-test at the end of each lesson, together with reflection on the lesson led to refinements for the next lesson to the next group. The study describes the changes made to the lessons, and the results of the pre and post tests for each of the three groups. The results showed that while each group improved in the post test, the third group outperformed the others, confirming that an intervention, informed by variation theory, did improved learning. All learners were afforded the opportunity to discern the object of learning and their interpretation of functions improved, with the third group improving the most. Keywords: object of learning, patterns of variation theory, interpretation of functions.

Nesse trabalho combinamos a teoria de Nielsen de pontos fixos com a teoria dos grupos de transformações para dar condições necessárias e sucientes para realizar um subconjunto A localmente contrátil de X G - como o conjunto de pontos xos de uma apli- h : X X em uma classe de homotopia equivariante dada, onde G é X é uma G -variedade suave e compacta. Além disso, se X é o espaço total de um G -brado localmente trivial demos condições cação equivariante um grupo de Lie compacto e necessárias e sucientes para o correspondente problema de realização para aplicações G -equivariantes que preservam bra, onde G é um grupo nito. / In this work, we combine the Nielsen fixed point theory with the transformation group theory to present necessary and sucient conditions for the realization of a locally contrac- G -subset A of X as the xed point set of a map h : X X in a given G -homotopy class. Here, G is a compact Lie group and X is a compact smooth G -manifold. In adition, if X is the total space of a G -ber bundle we present necessary and sucient conditions for the corresponding realization problem for G -ber-preserving maps when G tible is a nite group.