The classical spherical and Euclidean geometries are easy to visualize and correspond to spaces with constant curvature 0 and +1 respectively. The geometry with constant curvature −1, hyperbolic geometry, is much more complex. A powerful theorem of Mostow and Prasad states that in all dimensions at least 3, the geometry of a finite-volume hyperbolic manifold (a space with local d-dimensional hyperbolic geometry) is determined by the manifold's fundamental group (a topological invariant of the manifold). A cusp is a part of a finite-volume hyperbolic manifold that is infinite but has finite volume (cf. the surface of revolution of a tractrix has finite area but is infinite). All non-compact hyperbolic manifolds have cusps, but only finitely many of them. In the fundamental group of such a manifold, each cusp corresponds to a cusp subgroup, and each cusp subgroup is associated to a point on the boundary of H^d, which can be identified with the (d − 1)-sphere. It is known that there are many one-cusped two- and three-dimensional hyperbolic manifolds. This thesis studies restrictions on the existence of 1-cusped hyperbolic d-dimensional manifolds for d ≥ 3. Congruence subgroups belong to a special class of hyperbolic manifolds called arithmetic manifolds. Much is known about arithmetic hyperbolic 3- manifolds, but less is known about arithmetic hyperbolic manifolds of higher dimensions. An important infinite class of arithmetic d-manifolds is obtained using SO(n, 1; Z), a subset of the integer matrices with determinant 1. This is known to produce 1-cusped examples for small d. Taking special congruence conditions modulo a fixed number, we obtain congruence subgroups of SO(n, 1; Z) which also have cusps but possibly more than one. We ask what congruence subgroups with one cusp exist in SO(n, 1; Z). We consider the prime congruence level case, then generalize to arbitrary levels. Covering space theory implies a relation between the number of cusps and the image of a cusp in the mod p reduced group SO(d+ 1, p), an analogue of the classical rotation Lie group. We use the sizes of maximal subgroups of groups SO(d + 1, p), and the maximal subgroups' geometric actions on finite vector spaces, to bound the number of cusps from below. Let Ω(d, 1; Z) be the index 2 subgroup in SO(d, 1; Z) that consists of all elements of SO(d, 1; Z) with spinor norm +1. We show that for d = 5 and d ≥ 7 and all q not a power of 2, there is no 1-cusped level-q congruence subgroup of Ω(d, 1; Z). For d = 4, 6 and all q not of the form 2^a3^b, there is no 1-cusped level-q congruence subgroup of Ω(d, 1; Z). / Mathematics

A logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a group G of order Qk j=1 pj mj , the length of any LS for G satises `() Pk j=1mjpj . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS). The MLS conjecture states that every finite simple group has an MLS. If the conjecture is true then every finite group will have an MLS. The conjecture was shown to be true by a number of researchers for a few classes of finite simple groups. However, the problem is still wide open. This dissertation addresses the MLS conjecture for the classical simple groups. In particular, it is shown that MLS's exist for the symplectic groups Sp2n(q), the orthogonal groups O 2n(q0) and the corresponding simple groups PSp2n(q) and 2n(q0) for all n 2 N, prime power q and even prime power q0. The existence of an MLS is also shown for all unitary groups GUn(q) for all odd n and q = 2s under the assumption that an MLS exists for GUn 1(q). The methods used are very general and algorithmic in nature and may be useful for studying all nite simple groups of Lie type and possibly also the sporadic groups. The blocks of logarithmic signatures constructed in this dissertation have cyclic structure and provide a sort of cyclic decomposition for these classical groups. / by Nikhil Singhi. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.

The minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups, An, PSLn(q) when gcd(n; q 1) is 1, 4 or a prime and several sporadic groups [10, 9, 12, 14, 18]. This dissertation is devoted to proving that the conjecture is true for a large class of simple groups of Lie type called classical groups. The methods developed use the structure of these groups as isometry groups of bilinear or quadratic forms. A large part of the construction is also based on the Bruhat and Levi decompositions of parabolic subgroups of these groups. In this dissertation the conjecture is shown to be true for the following families of simple groups: the projective special linear groups PSLn(q), the projective symplectic groups PSp2n(q) for all n and q a prime power, and the projective orthogonal groups of positive type + 2n(q) for all n and q an even prime power. During the process, the existence of minimal logarithmic signatures (MLS's) is also proven for the linear groups: GLn(q), PGLn(q), SLn(q), the symplectic groups: Sp2n(q) for all n and q a prime power, and for the orthogonal groups of plus type O+ 2n(q) for all n and q an even prime power. The constructions in most of these cases provide cyclic MLS's. Using the relationship between nite groups of Lie type and groups with a split BN-pair, it is also shown that every nite group of Lie type can be expressed as a disjoint union of sets, each of which has an MLS. / by NIdhi Singhi. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.

In this dissertation, we will investigate aspects of Auslander-Reiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute Auslander-Reiten quivers of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the dissertation is an adaptation to this situation of the Auslander and Ringel-Tachikawa Theorem which states that for an artinian ring R of finite representation type, each R-module is a direct sum of finite-length indecomposable R-modules. In cases where this applies, the indecomposable objects obtained in the Auslander-Reiten quiver give the building blocks for the objects in the category. We also briefly discuss in which cases systems of submodule embeddings form a Frobenius category, and for a few examples explore pointwise Calabi-Yau dimension of such a category. / by Audrey Moore. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

by Lok Tsan-ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 74-75). / Introduction --- p.2 / Chapter 1 --- "Rings, Modules and Categories" --- p.4 / Chapter 1.1 --- "Rings, Subrings and Ideals" --- p.5 / Chapter 1.2 --- Modules and Categories --- p.8 / Chapter 1.3 --- Module Theory --- p.13 / Chapter 2 --- Isomorphisms between Endomorphism rings of Quasiprogener- ators --- p.24 / Chapter 2.1 --- Preliminaries --- p.24 / Chapter 2.2 --- The Fundamental Theorem --- p.31 / Chapter 2.3 --- Isomorphisms Induced by Semilinear Maps --- p.41 / Chapter 2.4 --- Isomorphisms of General linear groups --- p.46 / Chapter 3 --- Endomorphism ring of projective module --- p.54 / Chapter 3.1 --- Preliminaries --- p.54 / Chapter 3.2 --- Main Theorem --- p.60 / Bibliography --- p.74

In this thesis, we compute minimal presentations in terms of generators and relations for the oriented cohomology rings of several semisimple linear algebraic groups of ranks 1 and 2 over algebraically closed fields of characteristic 0. The main tools we use in this thesis are the combinatorics of Coxeter groups and formal group laws, and recent results of Calm\`es, Gille, Petrov, Zainoulline, and Zhong, which relate the oriented cohomology rings of flag varieties and semisimple linear algebraic groups to the dual of the formal affine Demazure algebra.

Digital signal processing (DSP) is widely used in many applications spanning the spectrum from audio processing to image and video processing to radar and sonar processing. At the core of digital signal processing applications is the digital filter which are implemented in two ways, using either finite impulse response (FIR) filters or infinite impulse response (IIR) filters. The primary difference between FIR and IIR is that for FIR filters, the output is dependent only on the inputs, while for IIR filters the output is dependent on the inputs and the previous outputs. FIR filters also do not sur from stability issues stemming from the feedback of the output to the input that aect IIR filters. In this thesis, an architecture for FIR filtering based on distributed arithmetic is presented. The proposed architecture has the ability to implement large FIR filters using minimal hardware and at the same time is able to complete the FIR filtering operation in minimal amount of time and delay when compared to typical FIR filter implementations. The proposed architecture is then used to implement the fast affine projection adaptive algorithm, an algorithm that is typically used with large filter sizes. The fast affine projection algorithm has a high computational burden that limits the throughput, which in turn restricts the number of applications. However, using the proposed FIR filtering architecture, the limitations on throughput are removed. The implementation of the fast affine projection adaptive algorithm using distributed arithmetic is unique to this thesis. The constructed adaptive filter shares all the benefits of the proposed FIR filter: low hardware requirements, high speed, and minimal delay.

Cette thèse est consacrée aux propriétés géométriques des opérations de tores algébriques dans les variétés affines. Elle est issue de trois prépublications qui correspondent aux points (1), (2), (3) ci-après. Soit X une variété affine munie d’une opération d’un tore algébrique T. Nous appelons complexité la codimension de l’orbite générale de T dans X. Sous l’hypothèse de normalité et lorsque le corps de base est algébriquement clos de caractéristique 0, la variété X admet une description combinatoire en termes de géométrie convexe. Cette description, obtenue en 2006 par Altmann et Hausen, généralise celle classique des variétes toriques. Notre but consiste à étudier des problèmes nouveaux concernant les propriétés algébriques et géométriques de X lorsque l’operation de T dans X est de complexité 1. (1) Dans la première partie, un résultat donne une manière explicite de déterminer la clôture intégrale de toute variété affine définie sur un corps algébriquement clos de caractérisque 0 munie d’une opération de T de complexité 1 en termes de la description combinatoire d’Altmann-Hausen. Comme application, nous donnons une classification complète des idéaux intégralement clos homogènes de l’algèbre des fonctions régulières de X et généralisons un théorème de Reid-Roberts-Vitulli sur la description de certains idéaux normaux de l’algèbre des polynômes à plusieurs variables. (2) Les calculs de la première partie suggèrent une démonstration de la validité de la présentation d’Altmann-Hausen sur un corps quelconque dans le cas de complexité 1. Ce qui est fait dans la deuxième partie. Dans la situation non déployée, la descente galoisienne d’une variété affine normale munie d’une opération d’un tore algébrique de complexité 1 est décrite par un nouvel objet combinatoire que nous appelons diviseur polyédral Galois stable. (3) Dans la troisième partie, lorsque que le corps de base est parfait, nous classifions toutes les opérations du groupe additif dans X normalisées par l’action de T de complexité 1. Cette classification généralise des travaux classiques de Flenner et Zaidenberg dans le cas des surfaces et de Liendo dans le cas où le corps ambiant est algébriquement clos de caractéristique 0. / This thesis is devoted to the study of geometric properties of affine algebraic varieties endowed with an action of an algebraic torus. It comes from three preprints which correspond to the indicated points (1), (2), (3). Let X be an affine variety equipped with an action of the algebraic torus T. The complexity of the T-action on X is the codimension of the general T-orbit. Under the assumption of normality and when the ground field is algebraically closed of characteristic 0, the variety X admits a combinatorial description in terms of convex geometry. This description obtained by Altmann and Hausen in the year 2006 generalizes the classical one for toric varieties. Our purpose is to investigate new problems on the algebraic and geometric properties of the variety X when the T-action on X is of complexity 1. (1) In the first part, a result gives an effective method to determine the integral closure of any affine variety defined over an algebraically field of characteristic 0 with a T-action of complexity 1 in terms of the combinatorial description of Altmann-Hausen. As an application, we provide an entire classification of the homogeneous integrally closed ideals of the algebra of regular functions on X and generalize the Reid-Roberts-Vitulli's theorem on the description of certain normal ideals of the polynomial algebra. (2) The calculations of the first part suggest a proof of the validity of the presentation of Altmann-Hausen in the case of complexity 1 over an arbitrary ground field. This is done in the second part of this thesis. In the non-split situation, the Galois descent of normal affine varieties with a T-action of complexity 1 is described by a new combinatorial object which we call a Galois invariant polyhedral divisor. (3) In the third part, when the base field is perfect, we classify all the actions of the additive group on X normalized by the T-action of complexity 1. This classification generalizes classical works of Flenner and Zaidenberg in the surface case and of Liendo when the base field is algebraically closed of characteristic 0.

Les graphes définis à partir de structures algébriques possèdent d’excellentes propriétés de symétries particulièrement intéressantes. L’exemple le plus flagrant est la notion de graphe de Cayley qui s’est révélée très riche non seulement du point de vue théorique mais aussi pratique par ses applications à de nombreux domaines incluant l’architecture des réseaux ou les machines parallèles. Néanmoins, la régularité des graphes de Cayley se révèle parfois être une limite étant donné qu’ils sont toujours sommet-transitifs et donc en particulier non pertinents pour générer des réseaux semiréguliers.Cette observation a motivé, en 2005, la définition d’une nouvelle classe de graphes définis à partir d’un groupe, appelés G-graphes. Ils possèdent aussi de nombreuses propriétés de régularité mais de manière moins restrictive.Cette thèse propose un nouveau regard sur cette classe de graphes par une approche plutôt orientée recherche opérationnelle alors que la grande majorité des études précédentes est dominée par des approches essentiellement algébriques. Nous-nous sommes alors intéressés à plusieurs questions :— La caractérisation des G-graphes : nous proposons des améliorations par rapport aux précédents résultats.— Identifier des classes de graphes comme des G-graphes grâce à des isomorphismes ou en utilisant le théorème de caractérisation.— Etudier la structure et les propriétés de ces graphes, en particulier pour de possibles applications aux réseaux : colorations semi-régulières, symétries et robustesse.— Une approche algorithmique pour la reconnaissance de cette classe avec notamment un premier exemple de cas polynomial lorsque le groupe est abélien. / Interactions between graph theory and group theory have already led to interesting results for both domains. Graphs defined from algebraic groups have highly symmetrical structure giving birth to interesting properties. The most famous example is Cayley graphs, which revealed to be particularly interesting both from a theoretical and a practical point of view due to their applications in several domains including network architecture or parallel machines. Nevertheless, the regularity of Cayley graphs is also a limit as they are always vertex-transitive and therefore not relevant to generate semi-regular networks. This observation motivated the definition, in 2005, of a new family of graphs defined from a group, called G-graphs. They also have many regular properties but are less restrictive. These graphs are in particular semi-regular k-partite, with a chromatic number k directly given in the group representation and they can be either transitive or not.This thesis proposes a new insight into this class of graphs using an approach based on operational research while most of previous studies have been so far dominated by algebraic approaches. Then, the thesis addresses different kind of questions:— Characterizing G-graphs: we propose improvements of previous results.— Identifying some classes of graphs as G-graphs through isomorphism or using the characterization theorem.— Studying the structure and properties of these graphs, in particular for possible applications to networks: semi-regular coloring, symmetries and robustness.— Algorithmic approach for recognizing this class with a first example of polynomial case when the group is abelian.

Working over an arbitrary field F of characteristic not 2, we consider linear algebraic groups over F. We view these as functors, represented by finitely generated F-Hopf algebras, from the category of commutative, associative, F-algebras Alg_F, to the category of groups. Classical examples of these groups, such as the special linear group SL_n are split, however there are also linear algebraic groups arising from central simple F-algebras which are non-split. For example, associated to a non-split central simple F-algebra A of degree n is a non-split special linear group SL(A). It is well known that central simple algebras are twisted forms of matrix algebras. This means that over the separable closure of F, denoted F_sep, we have A⊗_F F_sep ∼= M_n(F_sep) and that there is a twisted Gal(F_sep/F)-action on M_n(F_sep) whose fixed points are A. We show that a similar method of twisted Galois descent can be used to obtain all non-split semisimple linear algebraic groups associated to central simple algebras as fixed points within their split counterparts. In particular, these techniques can be used to construct the spin and half-spin groups Spin(A, τ ) and HSpin(A, τ ) associated to a central simple F-algebra of degree 4n with orthogonal involution. Furthermore, we develop a theory of twisted Galois descent for Hopf algebras and show how the fixed points obtained this way are the representing Hopf algebras of our non-split groups. Returning to the view of group schemes as functors, we discuss how the group schemes we consider are sheaves on the étale site of Alg_F whose stalks are Chevalley groups over local, strictly Henselian F-algebras. This allows us to use the generators and relations presentation of Chevalley groups to explicitly describe group scheme morphisms. After showing how the Kronecker tensor product of matrices induces maps between simply connected groups, we give an explicit description of these maps in terms of Chevalley generators. This allows us to compute the kernel of these new maps composed with standard isogenies and thereby construct new tensor product maps between non-simply connected split groups. These new maps are Gal(F_sep/F)-morphisms and so we apply our techniques of twisted Galois descent to also obtain new tensor product morphisms between non-split groups schemes. Finally, we use one of our new split tensor product maps to compute the degree three cohomological invariants of HSpin_4n for all n.