Arithmetic fuchsian groups identified in quaternion orders for the construction of signal constellations

Orientadores: Reginaldo Palazzo Jr., Mercio Botelho FariaTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de ComputaçãoResumo: Dentro do contexto de projetar sistema de comunicação digital em espaços homogêneos, em particular, em espaços hiperbólicos, é necessário estabelecer um procedimento sistemático para construção de reticulados O, como elemento base para construção de constelações de sinais geometricamente uniformes. E através desse procedimento que identificamos as estruturas algébrica e geométrica além de construir códigos geometricamente uniformes em espaços homogêneos. Propomos, a partir desses reticulados, a construção de grupos fuchsianos aritméticos Tp obtidos de tesselações hiperbólicas {p; q}, derivados de álgebras de divisão dos quaternios A sobre corpos de números K. Generalizamos o processo de identificação desses grupos em ordens dos quatérnios (reticulados hiperbólicos), associadas às constelações de sinais geometricamente uniformes, provenientes de grupos discretos. Esse procedimento permite rotular os sinais das constelações construídas por elementos de uma estrutura algébricaAbstract: Within the context of digital communications system in homogeneous space in particular, in hyperbolic spaces, it is necessary to establish systematic procedure for the construction of lattices O ; as the basic entity for construction of eometrically uniforms signal constellations. By this procedure we identify the algebraic and geometric structures to construct geometrically uniforms codes in homogeneous spaces. We propose, from lattices, the construction of arithmetic fuchsian groups ¡p obtained by hyperbolic tessellations {p; q}, derived from division quaternion algebras A over numbers fields K. We generalize the process of identification of these groups in quaternion orders (hyperbolic lattices), which are associated with geometrically uniforms signal constellations, proceeding from discrete groups. This procedure allows us to realize the labelling of the signals belonging to such constellations by elements of an algebraic structureDoutoradoTelecomunicações e TelemáticaDoutor em Engenharia Elétric

Irracionalidade rec\'iproca

Prime numbers play a key role in number theory and have applications beyond Mathematics. In particular, in the Theory of Codes and also in Cryptography, the properties of prime numbers are relevant, because, from them, it is possible to guarantee the storage of data and the sending of messages in a secure way. And this is evident in e-commerce when personal data must be kept confidential. The proof that $\sqrt{p}$ is an irrational number, for every positive prime $p$, is known, if not by everyone, at least by the majority of Mathematics students, and such a proof is, in general, given by means of a basic property of numbers primes: if $p$ divides the product of two integers, then it divides at least one of them. This result forms the basis of other equally important results, such as, for example, what is given by the Fundamental Theorem of Arithmetic, which is the basic result of the Theory of Numbers. In this article, we present a proof of the irrationality of $\sqrt[2n]{p}$ using results from Quadratic Residue Theory, especially, by Gauss's Law of Quadratic Reciprocity.Comment: in Portuguese languag

Algebraic and Geometric Characterizations Related to the Quantization Problem of the C2,8C_{2,8} Channel

In this paper, we consider the steps to be followed in the analysis and interpretation of the quantization problem related to the $C_{2,8}$ channel, where the Fuchsian differential equations, the generators of the Fuchsian groups, and the tessellations associated with the cases $g=2$ and $g=3$, related to the hyperbolic case, are determined. In order to obtain these results, it is necessary to determine the genus $g$ of each surface on which this channel may be embedded. After that, the procedure is to determine the algebraic structure (Fuchsian group generators) associated with the fundamental region of each surface. To achieve this goal, an associated linear second-order Fuchsian differential equation whose linearly independent solutions provide the generators of this Fuchsian group is devised. In addition, the tessellations associated with each analyzed case are identified. These structures are identified in four situations, divided into two cases $(g=2$ and $g=3)$, obtaining, therefore, both algebraic and geometric characterizations associated with quantizing the $C_{2,8}$ channel.Comment: 31 pages, 9 figure

Generalized Edge-pairings For The Family Of Hyperbolic Tessellations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)In this paper we present generalized edge-pairings for the family of hyperbolic tessellations , with the purpose to obtain the corresponding discrete group of isometries. These tessellations have greater density packing than the self-dual tessellations implying that the associated codes achieve the least error probability, or equivalently, that these codes are optimum codes.3512943PROPESQ/UEPB [02/2010]FAPESP [04/15328-2, 2007/56052-8]CNPq [505258/2008-0, 303059/2010-9]CNPq under grant UFVFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq

Generalized edge-pairings for the family of hyperbolic tessellations {10λ,2λ}

In this paper we present generalized edge-pairings for the family of hyperbolic tessellations {10λ,2λ}{10λ,2λ} , with the purpose to obtain the corresponding discrete group of isometries. These tessellations have greater density packing than the self-dual tessellations {4λ,4λ}{4λ,4λ} implying that the associated codes achieve the least error probability, or equivalently, that these codes are optimum codes

Generalized edge-pairings for the family of hyperbolic tessellations {10λ,2λ}

In this paper we present generalized edge-pairings for the family of hyperbolic tessellations {10λ,2λ}, with the purpose to obtain the corresponding discrete group of isometries. These tessellations have greater density packing than the self-dual tessellations {4λ,4λ} implying that the associated codes achieve the least error probability, or equivalently, that these codes are optimum codes352943CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP505258/2008-0; 303059/2010-904/15328-2; 2007/56052-