Call a set of 2n + k elements Kneser colored when its n-subsets are put into classes such that disjoint n-subsets are in different classes. Kneser showed that k + 2 classes are sufficient to Kneser-color the n-subsets of a 2n + k element set. There are several proofs that this same number is necessary which rely on fixed-point theorems related to the Lusternik-Schnirelmann- Borsuk (LSB) theorem. By employing generalizations of these theorems we expand the proofs mentioned to obtain proofs of an original result we call the Subcoloring theorem. The Subcoloring theorem asserts the existence of a partition of a Kneser-colored set that halves its classes in a special way. We demonstrate both a topological proof and a combinatorial proof of this main result. We present an original corollary that extends the Subcoloring theorem by providing bounds on the size of the pieces of the asserted partition. Throughout, we formulate our results both in combinatorial and graph theoretic terminology.

Thèse de doctorat : Mathématiques : Lille 1 : 2002. / N° d'ordre (Lille) : 3131. Résumé en français. Bibliogr. f. 79-82.

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

In 1971, Ganea put forth a conjecture that the LS category of the Cartesian product of a topological space X with a sphere Sn is always exactly 1 higher than the LS category of X by itself. Several special cases of this conjecture were proven in the years following, however the question remained open until 1998 when Iwase produced not just one, but infinitely many counterexamples. In this thesis, we study the methods implemented by Iwase, culminating in the construction of his counterexample.

Neste trabalho estudamos a existência e a multiplicidade de soluções clássicas positivas para uma classe de problemas de quarta-ordem sob a condição de fronteira de Navier, relacionando o número de soluções com a topologia do domínio, mais precisamente, com sua categoria de Lusternik-Schnirelman. Introduzimos também uma noção de regiões crítica e não-crítica associadas a um de nossos problemas, a fim de garantir condições para existência de solução / In this work we study the existence and multiplicity of positive classical solutions for a class of fourth-order problems under Navier boundary condition, relating the number of solutions to the domain topology, more specifically, to its Lusternik-Schnirelman category. We also introduce the notion of critical and noncritical regions related to one of our problems, in order to ensure conditions to existence of solutions

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-15T12:49:10Z No. of bitstreams: 1 arquivototal.pdf: 1323834 bytes, checksum: 530efbd6b56f11c5cc1b4369c8c44888 (MD5) / Made available in DSpace on 2017-08-15T12:49:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1323834 bytes, checksum: 530efbd6b56f11c5cc1b4369c8c44888 (MD5) Previous issue date: 2016-02-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we establish existence, multiplicity and concentration of positive solutions for the following class of problem 8<: 􀀀div􀀀 2 ( jruj)ru + V (x) (juj)u = f(u); in RN; u 2 W1; (RN); u > 0 in RN; where N 2, is a positive parameter, ; V; f are functions satisfying technical conditions that will be presented throughout the thesis and (t) = Rjtj 0 (s)sds. The main tools used are Variational methods, Lusternik-Schnirelman of category, Penalization methods and properties of Orlicz-Sobolev spaces. / Neste trabalho estabelecemos resultados de existência, multiplicidade e concentração de soluções positivas para a seguinte classe de problemas quasilineares 8<: 􀀀div􀀀 2 ( jruj)ru + V (x) (juj)u = f(u); em RN; u 2 W1; (RN); u > 0 em RN; onde N 2, é um parâmetro positivo, ; V; f são funções satisfazendo condições técnicas que serão apresentadas ao longo da tese e (t) = Rjtj 0 (s)sds. As principais ferramentas utilizadas são os Métodos Variacionais, Categoria de Lusternik-Schnirelman, Método de Penalização e propriedades dos espaços de Orlicz-Sobolev.

Made available in DSpace on 2015-05-15T11:46:26Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 549935 bytes, checksum: f7562c326b5af177cb80a71a184aa0c9 (MD5) Previous issue date: 2010-08-04 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this dissertation, we study the multiplicity of solutions for the following class of semilinear elliptic problems involving the critical Sobolev exponent, ---u = - juj2---2 u + f (x; u) ; x 2 e u = 0; x 2 @ ; where N - 3, - RN is a smooth and bounded domain, - is a positive real parameter and 2- = 2N= (N - 2) is the critical Sobolev exponent. In obtaining our result, we use variational methods, such as, minimax theorems, Lusternik-Schnirelman theorems, as well as, concentration-compactness lemma. / Nesta dissertação, estudamos a multiplicidade de soluções para a seguinte classe de problemas elípticos semilineares envolvendo o expoente crítico de Sobolev, --u = - juj2---2 u + f (x; u) ; x 2 e u (x) = 0; x 2 @ ; onde N - 3, - RN é um dominio suave e limitado, - é um parâmetro real positivo e 2* = 2N= (N - 2) é o expoente crítico de Sobolev. Na prova dos resultados, usamos métodos variacionais, tais como, teoremas do tipo minimax, teoremas do tipo Lusternik-Schnirelman, bem como, lemas de concentração-compacidade.

Neste trabalho estudamos a existência e a multiplicidade de soluções clássicas positivas para uma classe de problemas de quarta-ordem sob a condição de fronteira de Navier, relacionando o número de soluções com a topologia do domínio, mais precisamente, com sua categoria de Lusternik-Schnirelman. Introduzimos também uma noção de regiões crítica e não-crítica associadas a um de nossos problemas, a fim de garantir condições para existência de solução / In this work we study the existence and multiplicity of positive classical solutions for a class of fourth-order problems under Navier boundary condition, relating the number of solutions to the domain topology, more specifically, to its Lusternik-Schnirelman category. We also introduce the notion of critical and noncritical regions related to one of our problems, in order to ensure conditions to existence of solutions

If X is a simply connected CW complex, then it has a unique (up to isomorphism) minimal Sullivan model. There is an important rational homotopy invariant, called the rational Lusternik–Schnirelmann of X, denoted cat0(X), which has an algebraic formulation in terms of the minimal Sullivan model of X. We study another such numerical invariant called the rational retraction index of X, denoted r0(X), which is defined in terms of the minimal Sullivan model of X and satisfies 0 ≤ r0(X) ≤ cat0(X). It was introduced by Cuvilliez et al. as a tool to estimate the rational Lusternik–Schnirelmann category of the total space of a fibration. In this thesis we compute the rational retraction index on a range of rationally elliptic spaces, including for example spheres, complex projective space, the biquotient Sp(1) \ Sp(3) / Sp(1) × Sp(1), the homogeneous space Sp(3)/U(3) and products of these. In particular, we focus on formal spaces and formulate a conjecture to answer a question posed in the original article of Cuvilliez et al., “If X is formal, what invariant of the algebra H∗(X;Q) is r0(X)?”

We studied some semilinear elliptic equations on the entire space R^N. Our approach was variational, and the major obstacle was the breakdown in compactness due to the unboundedness of the domain. First, we considered an asymptotically linear Scltrodinger equation under the presence of a steep potential well. Using Lusternik-Schnirelmann theory, we obtained multiple solutions depending on the interplay between the linear, and nonlinear parts. We also exploited the nodal structure of the solutions. For periodic potentials, we constructed infinitely many homoclinic-type multibump solutions. This recovers the analogues result for the superlinear case. Finally, we introduced weights on the linear and nonlinear parts, and studied how their interact ion affects the local and global compactness of the problem. Our approach is based on the Caffarelli-Kohn-Nirenberg inequalities.