The starting point of the research developed in this thesis was the work done by Arek Goetz in his PhD thesis [Dynamics of Piecewise Isometrics, University of Illinois, 1996). Following his dissertation, we have considered the simple example of a piecewise rotation in two convex atoms defined in the whole plane (now commonly known as the Goetz map) as our main source of motivation. The first main achievement of our work was the construction of a new family of polygonal sets which are, in fact, global attractors. These examples are very similar in nature to the Sierpinski-gasket triangle presented in Goetz [1998c]. The next natural step was to argue that the definition of a piecewise isometric attractor is not entirely suitable in this context due to the lack of uniqueness. Under a new definition of attractor, some results are proved involving the properties of quasi-invariance and regularity existing in all examples available in the literature on the Goetz map. Following that, we attempt to generalise the results of Goetz regarding periodic cells and periodic points to unbounded spaces. We prove that there is a fundamental discrepancy between piecewise rotations in odd and even dimensions. In the odd-dimensional case the existence of periodic points is rare; hence those that exist must be unstable under almost all perturbations, whereas in even dimensions periodic points are stable for a prevalent set of piecewise rotations. Furthermore, if a piecewise rotation is such that the free monoid generated by the linear parts of the induced rotations does not contain the identity map then it follows trivially that all diverging orbits must be irrationally coded. This implies, together with a result on the coding of the open connected components in the complement of the closure of the exceptional set, that there exist examples of piecewise isometries possessing irrational cells with positive measure. This scenario was not considered previously in the results proved by Goetz. Using a common recurrence argument and Goetz's characterisation of the closure of the exceptional set (see for instance, Goetz [2001]) we prove that every recurrent point (i.e., such that w(x) ? ?) must be rationally coded. Given an invertible piecewise isometry in a compact space we also show that the closure of the exceptional set equals that of its inverse. This sustains the common idea that forward and backward iterates of the discontinuity yield similar graphics. In the context of the Goetz map we have also investigated the appearance of symmetric patterns when plotting the closure of the exceptional set. Although the Goetz map is by its nature discontinuous, the existence of symmetry is still possible under the broader framework of almost-everywhere-symmetry. Finally, we briefly note that the local symmetry properties of symmetric patterns arising in the invertible Goetz map are in part due to the existence of a reversing-symmetry, which generates piecewise continuous reversing symmetries under iteration of the original Goetz map.

Our aim is to investigate perfect isometry groups, which are invariants for blocks of finite groups. There are two subgoals. First is to study some properties of perfect isometry groups in general. We found that every perfect isometry has essentially a unique sign. This allowed us to show that, in many cases, a perfect isometry group contains a direct factor generated by -id. The second subgoal is to calculate perfect isometry groups for various blocks. Notable results include the perfect isometry groups for blocks with defect 1, abelian p-groups, extra special p-groups, and the principal 2-block of the Suzuki group Sz(q). In the case of blocks with defect 1, we also showed that every perfect isometry can be induced by a derived equivalence. With the help of a computer, we also calculated perfect isometry groups for some blocks of sporadic simple groups.Apart from perfect isometries, we also investigated self-isotypies in the special case where C_G(x) is a p-group whenever x is a p-element. We applied our result to calculate isotypies in cyclic p-groups and the principal 2-blocks of the Suzuki group Sz(q).

In their paper [2] Cuntz, Deninger, and Laca introduced a C*-algebra \mathfrak{T}[R] associated to a number ring R and showed that it was functorial for injective ring homomorphisms and had an interesting KMS-state structure, which they computed directly. Although isomorphic to the Toeplitz algebra of the ax+b-semigroup R⋊R^× of R, their C*-algebra \mathfrak{T}[R] was defined in terms of relations on a generating set of isometries and projections. They showed that a homomorphism φ:\mathfrak{T}[R]→ A is injective if and only if φ is injective on a certain commutative *-subalgebra of \mathfrak{T}[R]. In this thesis we give a direct proof of this result, and go on to show that there is a countable collection of projections which detects injectivity, which allows us to simplify their characterization of faithful representations of \mathfrak{T}[R]. / Graduate / 0405 / jaspar.wiart@gmail.com

Fuchsian groups are discrete subgroups of isometries of the hyperbolic plane. This thesis will primarily work with the upper half-plane model, though we will provide an example in the disk model. We will define Fuchsian groups and examine their properties geometrically and algebraically. We will also discuss the relationships between fundamental regions, Dirichlet regions and Ford regions. The goal is to see how a Ford region can be constructed with isometric circles.

No description available.

Apresentamos conceitos da teoria de Grupos de Lie e Ações Próprias e descrevemos a demonstração da Conjectura de Palais-Terng efetuada por Alexandrino. Tal conjectura garante que uma folheação riemanniana singular com distribuição normal é uma folheação riemanniana singular com seções. Adaptamos para o caso particular das ações isométricas. / We present some aspects of the theory of Lie Groups and Proper Actions, and we review the proof of the Palais-Terng Conjecture given by Alexandrino. This theorem assures that a singular Riemannian foliation with integrable normal distribution is a singular Riemannian foliation with section. We adapt the proof for isometric actions.

In classical and quantum information theory there are different types of error-correcting codes being used. We study the equivalence of codes via a classification of their isometries. The isometries of various codes over Frobenius alphabets endowed with various weights typically have a rich and predictable structure. On the other hand, when the alphabet is not Frobenius the isometry group behaves unpredictably. We use character theory to develop a duality theory of partitions over Frobenius bimodules, which is then used to study the equivalence of codes. We also consider instances of codes over non-Frobenius alphabets and establish their isometry groups. Secondly, we focus on quantum stabilizer codes over local Frobenius rings. We estimate their minimum distance and conjecture that they do not underperform quantum stabilizer codes over fields. We introduce symplectic isometries. Isometry groups of binary quantum stabilizer codes are established and then applied to the LU-LC conjecture.

Let M Φ and Mψ be elements of PSL(2,C) representing orientation preserving isometries on the upper half-space model of hyperbolic 3-space Φ and ψ respectively. The parameters β = tr2(M Φ) - 4, β1 = tr2(Mψ) - 4, γ = tr[M Φ,Mψ] - 2, determine the discrete group (Φ ,ψ) uniquely up to conjugacy whenever γ ≠ 0. This thesis is concerned with explicitly lifting this parameterisation of (Φ , ψ) to PSO(1, 3) realised as a discrete 2 generator subgroup of orientation preserving isometries on the hyperboloid model of hyperbolic 3-space. We particularly focus on the case where both Φ and ψ are elliptic.

Cette these presente quelques aspects de la theorie des blocs generalises pour les groupes finis. Apres une breve description des theories classique et generalisee, on y etudie les proprietes des blocs generalises de certains groupes. On montre l'existence d'isometries parfaites generalisees dans trois familles de groupes de Lie de rang 1, generalisant ainsi une conjecture de M. Broue. On etudie ensuite le concept de groupe de Cartan generalise, et une formule est donnee pour l'ordre dans le groupe de Cartan des caracteres du groupe symetrique. Enfin, on definit des blocs generalises dans les groupes lineaires finis, et on montre que certaines unions de blocs de caracteres unipotents satisfont un analogue de la Conjecture de Nakayama ainsi qu'un analogue du Deuxieme Theoreme de Brauer.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-10-26T19:45:34Z No. of bitstreams: 1 DissLFGP.pdf: 5615571 bytes, checksum: f5636cf5ae2059f71859a66b025ed4cf (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-11-08T18:31:26Z (GMT) No. of bitstreams: 1 DissLFGP.pdf: 5615571 bytes, checksum: f5636cf5ae2059f71859a66b025ed4cf (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-11-08T18:31:35Z (GMT) No. of bitstreams: 1 DissLFGP.pdf: 5615571 bytes, checksum: f5636cf5ae2059f71859a66b025ed4cf (MD5) / Made available in DSpace on 2016-11-08T18:31:41Z (GMT). No. of bitstreams: 1 DissLFGP.pdf: 5615571 bytes, checksum: f5636cf5ae2059f71859a66b025ed4cf (MD5) Previous issue date: 2016-09-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / This actual study is an investigation about the importance of using computers and other information and communication technologies (ICTs) in the teaching of Mathematics. We believe in the thesis that ICT can contribute as an educational means for the improvement of teaching and learning situations, and consequently in solving problems such as school failure and school indiscipline. Therefore, we designed and applied a didactic sequence using the Geogebra for the development of Isometries topic in a class of 6th grade of elementary school. The didactic sequence was structured within the perspectives of Didactic Engineering and adopted a qualitative methodology. The results indicate that ICTs contribute significantly in the process of teaching and learning, but some external factors are still obstacles to the effective implementation of this resource in schools. / O presente estudo é uma investigação acerca da importância da utilização de computadores e outras tecnologias de informação e comunicação (as TIC’s) no ensino de Matemática. Acreditamos na tese de que as TIC’s podem contribuir como meio educativo para a melhora das situações de ensino aprendizagem e, consequentemente, na solução de problemas como o fracasso escolar e a indisciplina escolar. Nesse intuito, idealizamos e aplicamos uma sequência didática com o uso do Geogebra para o desenvolvimento do tópico Transformações Geométricas em uma turma do sexto ano do Ensino Fundamental. A sequência didática foi estruturada dentro das perspectivas da Engenharia Didática e seguiu uma metodologia qualitativa. Os resultados indicam que as TIC’s contribuem de forma significativa no processo de ensino e aprendizagem, contudo alguns fatores externos ainda são entraves para a efetiva implementação deste recurso nas escolas.