Roughly speaking, a separating algebra is a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this thesis, we introduce the notion of a geometric separating algebra, a more geometric notion of a separating algebra. We find two geometric formulations for the notion of separating algebra which allow us to prove, for geometric separating algebras, the results found in the literature for separating algebras, generally removing the hypothesis that the base field be algebraically closed. Using results from algebraic geometry allows us to prove that, for finite groups, when a polynomial separating algebra exists, the group is generated by reflections, and when a complete intersection separating algebra exists, the group is generated by bireflections. We also consider geometric separating algebras having a small number of generators, giving an upper bound on the number of generators required for a geometric separating algebra. We end with a discussion of two methods for obtaining new separating sets from old. Interesting, and relevant examples are presented throughout the text. Some of these examples provide answers to questions which previously appeared in print. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-08-28 14:14:04.138

Thesis--Wisconsin. / Vita. Includes bibliographical references (leaf 111).

First published in the Bulletin of the American Mathematical Society in Vol.68 1962, published by the American Mathematical Society

Notre travail s’inscrit dans un domaine initié en 1934 par Lusternik et Schnirelmann qui associent a une variété un invariant appelé catégorie qui permet de minorer le nombre des points critiques d’une fonction différentiable sur cette variété. Nous nous intéressons à une généralisation au cas des applications continues entre espaces topologiques, auxquelles nous associons un invariant appelé sigma-i-catégorie. Nous obtenons plusieurs caractérisations de la sigma-i-catégorie d’une application. Nous examinons ensuite l’effet sur la sigma-i-catégorie d’un attachement d’une cellule à la source d’une application. Cette étude est faite au moyen d’un nouvel invariant, appelé invariant de Hopf relatif. Enfin nous examinons les relations entre les catégories de produit et de produit smash. / Our work is registered in a field initiated in 1934 by Lusternik and Schnirelmann, which associate with a variety an invariant called category, which allows to undervalue the number of the critical points of a differentiable function on this variety. We are interested in a generalization in the case of the continuous applications between topological spaces in wich we associate an invariant called sigma-i-category. We obtain several characterizations of the sigma-i-category on an application We examine then the effect on the sigma-i-category of a cell attachment on an application source. This study is made with a new invariant, called invariant of relative Hopf. Finally we examine the relations between the categories of product and product smach.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2009. / Title from title screen (site viewed January 5, 2010). PDF text: viii, 65 p. : ill. ; 507 K. UMI publication number: AAT 3360158. Includes bibliographical references. Also available in microfilm and microfiche formats.

In the first chapter; we consider the invariant measure mu generated by an integral self-affine IFS. We prove that any integral self-affine measure with a common contracting matrix can be expressed as a vector-valued self-affine measure with an IFS satisfying the open set condition (OSC). The same idea can also be applied to scaling functions of refinement equations, we extended a well known necessary and sufficient condition for the existence of L1-solutions of lattice refinement equations. We then apply this vector-valued form to study the integral self-affine sets, we obtain an algorithm for the Lebesgue measure of integral self-affine region and an algorithm for the Hausdorff dimension of a class of self-affine sets. The vector-value setup also provides an easy way to consider the L q-spectrum and the multifractal formulism for self-similar measures. As an application we can conclude the differentiability of the Lq spectrum (for q > 0) of any integral self-similar measure with a common contracting matrix. / In this thesis, we study the invariant measures and sets generated by iterated function systems (IFS). The systems have been extensively studied in the frame work of Hutchinson [Hut]. For the iteration, it is often assumed that the IFS satisfies the open set condition (OSC), a non-overlap condition in the iteration. One of the advantage of the OSC is that the point in K can be uniquely represented in a symbolic space except for a mu-zero set and many important results have been obtained. Our special interest in this thesis is to transform an invariant measure with overlaps to a vector-valued form with non overlaps. The advantage of this vector-valued form is that locally the measure can be expressed as a product of matrices. / The problem considered in the third chapter is on the choice of the invariant open set in the finite type condition (FTC). From definition, the FTC depends on the choice of the invariant open set. We show that, in one dimensional case, if the IFS satisfies the FTC for some invariant open interval then it satisfies the FTC with all invariant open sets. To our surprising, we find a counter-example to show that, in high dimensional case, the invariant open set can not be chosen arbitrarily even if the IFS satisfies the OSC and generates a tile. / The second chapter is devoted to the absolute continuity of self-affine (real-valued or vector-valued) measures and some properties of the boundary of the invariant set. For self-similar IFS with a common contracting ratio, there is a necessary and sufficient condition for the self-similar measure to be absolutely continuous with respect to the Lebesgue measure (under the weak separation condition (WSC)). In our consideration we first extend the definition of WSC to self-affine IFS. Then we generalize the previous condition to obtain a necessary and sufficient condition for the self-affine vector-valued measures to be absolutely continuous with respect to the Lebesgue measure. As an application, we prove that the boundary of all integral self-affine set has zero Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding invariant (real-valued or vector-valued) measure is supported either in V or in ∂ V. / by Deng Qirong. / "March 2005." / Adviser: Ka-Sing Lau. / Source: Dissertation Abstracts International, Volume: 67-01, Section: B, page: 0301. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 87-91). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract in English and Chinese. / School code: 1307.

by Wang Xiang-Yang. / "May 2002." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (p. 95-99). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Problems involving partial or ordinary differential equations arise in various fields of science. Therefore, the task of obtaining exact solutions of differential equations is of primary importance, and attracts high attention. The main purpose of the current thesis is the development of a Maple-based, symbolic software package for symmetry reduction of differential equations and computation of symmetry-invariant solutions. The package developed in the current thesis is compatible with and can be viewed as an extension of the package GeM for symbolic symmetry analysis, developed by Prof. Alexei Cheviakov. The reduction procedure is based on the Lie's classical symmetry reduction method involving canonical coordinates. The developed package is applicable for obtaining solutions arising from extension of Lie's method, in particular, nonlocal and approximate symmetries. The developed software is applied to a number of PDE problems to obtain exact invariant solutions. The considered equations include the one-dimensional nonlinear heat equation, the potential Burgers' equation, as well as equations arising in nonlinear elastostatics and elastodynamics.

Problems involving partial or ordinary differential equations arise in various fields of science. Therefore, the task of obtaining exact solutions of differential equations is of primary importance, and attracts high attention. The main purpose of the current thesis is the development of a Maple-based, symbolic software package for symmetry reduction of differential equations and computation of symmetry-invariant solutions. The package developed in the current thesis is compatible with and can be viewed as an extension of the package GeM for symbolic symmetry analysis, developed by Prof. Alexei Cheviakov. The reduction procedure is based on the Lie's classical symmetry reduction method involving canonical coordinates. The developed package is applicable for obtaining solutions arising from extension of Lie's method, in particular, nonlocal and approximate symmetries. The developed software is applied to a number of PDE problems to obtain exact invariant solutions. The considered equations include the one-dimensional nonlinear heat equation, the potential Burgers' equation, as well as equations arising in nonlinear elastostatics and elastodynamics.

Consider a simple model of an age-structured population with two age-classes and stochastically varying survival rate of young. Let $m_{1,y},m_{2,t}$ be birth rates per capital and $P_{1,t}$ be a survival rate. egin{eqnarray} left( egin{array}{clr} N_{1,t+1}N_{2,t+1} end{array} ight) = left( egin{array}{clr} m_{1,t+1} & m_{2,t+1} P_{1,t+1} & 0 end{array} ight) left( egin{array}{clr} N_{1,t}N_{2,t} end{array} ight) end{eqnarray} we want to study the large term behavior of $(N_{1,t},N_{2,t})$ the age-structured population through the theory of random matrix product.