Nesta dissertação foram discutidas abordagens do problema isoperimétrico que podem ser aplicadas no ensino médio e para alunos de Licenciatura plena em Matemática. Foi realizada inicialmente uma abordagem histórica e posteriormente a discussão de casos particulares e gerais de desigualdade isoperimétrica tanto no plano como no espaço. A abordagem principal deste texto é no plano, no qual foram analisadas as áreas dos triângulos, quadriláteros e polígonos regulares dado um perímetro fixo. / In this dissertation isoperimetric problem approaches were discussed that can be applied in high school and full degree students in mathematics. It was initially performed a historical approach and then the discussion of individual and general cases of isoperimetric inequality both in the plane and in space . The main approach of this text is in the plan, in which the areas of the triangles were analyzed , quadrilaterals and regular polygons given a fixed perimeter.

Nesta dissertação foram discutidas abordagens do problema isoperimétrico que podem ser aplicadas no ensino médio e para alunos de Licenciatura plena em Matemática. Foi realizada inicialmente uma abordagem histórica e posteriormente a discussão de casos particulares e gerais de desigualdade isoperimétrica tanto no plano como no espaço. A abordagem principal deste texto é no plano, no qual foram analisadas as áreas dos triângulos, quadriláteros e polígonos regulares dado um perímetro fixo. / In this dissertation isoperimetric problem approaches were discussed that can be applied in high school and full degree students in mathematics. It was initially performed a historical approach and then the discussion of individual and general cases of isoperimetric inequality both in the plane and in space . The main approach of this text is in the plan, in which the areas of the triangles were analyzed , quadrilaterals and regular polygons given a fixed perimeter.

Inspired by a planar partitioning problem involving multiple unbounded chambers, this thesis investigates using classical techniques what can be said of the existence, uniqueness, and regularity of minimizers in a certain free-endpoint isoperimetric problem. In two cases, a full existence-uniqueness-regularity result is proved using a convexity technique inspired by work of Talenti. The problem studied here can be interpreted physically as the identification of the equilibrium shape of a sessile liquid drop in half-space (in the absence of gravity). This is a well-studied variational problem whose full resolution requires the use of geometric measure theory, in particular the theory of sets of finite perimeter. A crash course on the theory required for the modern statement of the equilibrium shape theorem is presented in an appendix. / Thesis / Master of Science (MSc)

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2014-08-28T15:33:45Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertacao Marcos Antonio da Costa.pdf: 1237372 bytes, checksum: e4392b806f26f0293114e12b4ed829c9 (MD5) / Made available in DSpace on 2014-08-28T15:33:45Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertacao Marcos Antonio da Costa.pdf: 1237372 bytes, checksum: e4392b806f26f0293114e12b4ed829c9 (MD5) Previous issue date: 2013-04-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We deal with extremum values problems. Our focus is the high school students. We present simple ideas and techniques on solving classical optimization problems. Among other problems we cite the classical isoperimetric ploblem and the Heron0s problem. We are based on the book Stories About Maxima and Minima by Tikhomirov which lead with these classical problems using only elementary mathematical subjects. / Estudamos problemas envolvendo valores extremos, com foco nos estudantes do Ensino Médio. Apresentamos de forma simples e resumida, algumas ideias e teorias para a solução de tais problemas. Dentre os quais citamos o Problema de Dido e o de Heron. O principal referencial teórico para confecção deste trabalho foi o livro de Tikhomirov intitulado Stories About Maxima and Minima. Baseados em tal livro, aplicamos métodos e teorias elementares para solucionarmos problemas clássicos de máximos e mínimos.

In this thesis we study the isoperimetric problem on trees and graphs with bounded treewidth. Let G = (V,E) be a finite, simple and undirected graph. For let δ(S,G)= {(u,v) ε E : u ε S and v ε V – S }be the edge boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the edge isoperimetric value of G at I be defined as be(i,G)= mins v;|s|= i | δ(S,G)|. For S V, let φ(S,G) = {u ε V – S : ,such that be the vertex boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the vertex isoperimetric value of G at I be defined as bv(i,G)= The edge isoperimetric peak of G is defined as be(G) =. Similarly the vertex isoperimetric peak of G is defined as bv(G)= .The problem of determining a lower bound for the vertex isoperimetric peak in complete k-ary trees of depth d,Tdkwas recently considered in[32]. In the first part of this thesis we provide lower bounds for the edge and vertex isoperimetric peaks in complete k-ary trees which improve those in[32]. Our results are then generalized to arbitrary (rooted)trees. Let i be an integer where . For each i define the connected edge isoperimetric value and the connected vertex isoperimetric value of G at i as follows: is connected and is connected A meta-Fibonacci sequence is given by the reccurence a(n)= a(x1(n)+ a1′(n-1))+ a(x2(n)+ a2′(n -2)), where xi: Z+ → Z+ , i =1,2, is a linear function of n and ai′(j)= a(j) or ai′(j)= -a(j), for i=1,2. Sequences belonging to this class have been well studied but in general their properties remain intriguing. In the second part of the thesis we show an interesting connection between the problem of determining and certain meta-Fibonacci sequences. In the third part of the thesis we study the problem of determining be and bv algorithmically for certain special classes of graphs. Definition 0.1. A tree decomposition of a graph G = (V,E) is a pair where I is an index set, is a collection of subsets of V and T is a tree whose node set is I such that the following conditions are satisfied: (For mathematical equations pl see the pdf file)