Includes bibliographical references (p. 69-77).

A biological reaction diusion model has gained much attention recently. This model is formulated as a system of nonlinear partial dierential equations that contains an unknown function of one dependent variable. How to determine this unknown function is complicated but also useful. This model is considered in this master thesis. The generators of the equivalence groups and invariant solutions are calculated.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 195-199).

First published in the Bulletin of the American Mathematical Society in 1963, published by the American Mathematical Society

First published in the Bulletin of the American Mathematical Society in Vol.83, 1977, published by the American Mathematical Society

This thesis is a study of the solutions of the equations of radiative transfer by an invariant imbedding approach.

Commutative algebra finds its roots in invariant theory and the connection is drawn from a modern standpoint. The Hilbert Basis Theorem and the Nullstellenstatz were considered lemmas for classical invariant theory. The Groebner basis is a modern tool used and is implemented with the computer algebra system Mathematica. Number 14 of Hilbert\'s 23 problems is discussed along with the notion of invariance under a group action of GLn(C). Computational difficulties are also discussed in reference to Groebner bases and Invariant theory.The straitening law is presented from a Groebner basis point of view and is motivated as being a key piece of machinery in proving First Fundamental Theorem of Invariant Theory. / Master of Science

Ce mémoire revient sur l'invariant de Gromov-Witten dans le contexte de topologie symplectique. D'abord, on présente un survol des notions nécessaires de la topologie symplectique, qui inclut les espaces vectoriels symplectiques, les variétés symplectiques, les structures presque complexes et la première classe de Chern. Ensuite, on présente une définition de l'invariant de Gromov-Witten, qui utilise les courbes pseudoholomorphes, les espaces de modules ainsi que les applications d'évaluation. Finalement, on donne quelques exemples de calcul d'invariant à la fin de ce mémoire. / The present work reviews the Gromov-Witten invariant in the context of symplectic topology. First, we showcase the basic concepts required for the understanding of the matter, which includes symplectic vector spaces, symplectic manifolds, almost complex structures and the first Chern class. Then, we provide a definition of the Gromov-Witten invariant, after studying pseudoholomorphic curves, moduli spaces and evaluation maps. In the end, we present some examples of Gromov-Witten invariant calculations.

This thesis is concerned with numerical techniques for resolving and continuing closed, compact invariant manifolds in parameter-dependent dynamical systems with specific emphasis on invariant tori under flows. In the first part, we review several numerical methods of continuing invariant tori and concentrate on one choice called the ``orthogonality condition'. We show that the orthogonality condition is equivalent to another condition on the smooth level and show that they both descend from the same geometrical relationship. Then we show that for hyperbolic, periodic orbits in the plane, the linearization of the orthogonality condition yields a scalar system whose characteristic multiplier is the same as the non-unity multiplier of the orbit. In the second part, we demonstrate that one class of discretizations of the orthogonality condition for periodic orbits represents a natural extension of collocation. Using this viewpoint, we give sufficient conditions for convergence of a periodic orbit. The stability argument does not extend to higher-dimensional tori, however, and we prove that the method is unconditionally unstable for some common types of two-tori embedded in R^3 with even numbers of points in both angular directions. In the third part, we develop several numerical examples and demonstrate that the convergence properties of the method and discretization can be quite complicated. In the fourth and final part, we extend the method to the general case of p-tori in R^n in a different way from previous implementations and solve the continuation problem for a three-torus embedded in R^8.

Abstract not available. / text