This work is devoted to investigating the behaviour of invariant algebraic curves for the two dimensional Lotka-Volterra systems and examining almost a geometrical approach for finding invariant algebraic surfaces in three dimensional Lotka-Volterra systems. We consider the twenty three cases of invariant algebraic curves found in Ollagnier (2001) of the two dimensional Lotka-Volterra system in the complex plane and then we explain the geometric nature of each curve, especially at the critical points of the mentioned system. We also investigate the local integrability of two dimensional Lotka-Volterra systems at its critical points using the monodromy method which we extend to use the behaviour of some of the invariant algebraic curves mentioned above. Finally, we investigate invariant algebraic surfaces in three dimensional Lotka- Volterra systems by a geometrical method related with the intersection multiplicity of algebraic surfaces with the axes including the lines at infinity. We will classify both linear and quadratic invariant algebraic surfaces under some assumptions and commence a study of the cubic surfaces.

No description available.

No description available.

In this work we study the space of group homomorphisms Hom(Γ,G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.

The aim of this dissertation at the outset is to give a survey of obstruction theories after Steenrod and to describe the various techniques employed by different researchers, the intricate perhaps subtle relation from one technique to another. Owing to the difficulty in computing higher co-homology operations, one is led naturally to K-theory and the Eilenberg-Moore spectral sequence. However, these and other recent developments especially those in the study of stable Postnikov systems go beyond the intention of this modest survey. / Science, Faculty of / Mathematics, Department of / Graduate

The purpose of this thesis is to determine the ordinary and p-modular irreducible characters and the characters of the principal indecomposable modules of the group SL(2,q), q=pⁿ, for odd p. The decomposition matrix and the Cartan matrix for SL(2,q) are also given. / Science, Faculty of / Mathematics, Department of / Graduate

The graded cohomology groups of a cartesian product of two cellular spaces are expressible in terms of the cohomology groups of the factors. This relationship is given by the (split) short exact Runneth sequence. However the multiplicative structure on the cohomology of a cartesian product can in general not be derived by solely referring to the Runneth formula. In this thesis we explicitly exhibit the cup product structure on a cartesian product of two (standard) lens spaces. This result is obtained by analyzing the Runneth sequence and by making use of the particular geometry of the spaces involved. / Science, Faculty of / Mathematics, Department of / Graduate

This paper is devoted to finding integral solutions of algebraic equations. Only algebraic equations with integral coefficients are considered. The elementary properties of integers are assumed.

Given a set A a (internal) law of composition on A is a function on AxA into A. A set endowed with one or more such laws of composition which in turn may be subject to certain axioms (e. g. associativity, commutativity) is termed an algebraic structure. The most fundamental algebraic structures are the group (which has a single law of composition) and the ring (which has two laws of composition). The structures of the "field" and "domain of integrity" are merely particular cases of the ring. This paper is devoted to an exposition of the basic algebraic theory of ring structure with some attention being devoted to the above cited particular cases of rings as well as to certain rather special substructures (ideals) of rings. In this development use is made throughout of the notion of the "algebraic image" of a structure, that is of homomorphism and isomorphism theory. This paper lays the groundwork for, but does not embark upon, the study of the structure theory of rings; which theory has been so beautifully developed in the last three decades by Artin, Noether and Jacobson. / Advisor: Robert N. Tompson, Professor Directing Paper. / Typescript. / "August, 1954." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Includes bibliographical references (leaf 39).