Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-09-05T12:56:54Z No. of bitstreams: 1 arquivototal.pdf: 2044930 bytes, checksum: 05bad0799c40d5bf256cf504f0a8b5ab (MD5) / Approved for entry into archive by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-09-05T15:29:04Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 2044930 bytes, checksum: 05bad0799c40d5bf256cf504f0a8b5ab (MD5) / Made available in DSpace on 2017-09-05T15:29:04Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2044930 bytes, checksum: 05bad0799c40d5bf256cf504f0a8b5ab (MD5) Previous issue date: 2016-08-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We will do here a theoretical study of the Discrete Fourier Transform on the finite circle ℤ/nℤ. Our main objective is to see if we can get properties analogous to those found in the Fourier transform for the continuous case. In this work we show that ℤ/nℤ has a ring structure, providing conditions for the development of extensively discussed topics in arithmetic, for example, The Chinese Remainder Theorem, Euler’s Phi Function and primitive roots, themes these to be dealt with in first chapter. The main subject of this study is developed in the second chapter, which define the space L2(ℤ/nℤ) and prove that this is a finite-dimensional inner product vector space, with an orthonormal basis. This fact is of utmost importance when we are determining the matrix and demonstrating the properties of the discrete Fourier transform. We will also make geometric interpretations of the Chinese Remainder Theorem and the finite circle ℤ/nℤ as well as give a graphical representation of the DFT of some functions that calculate. During the development of this study we will make recurrent use of definitions and results treated in Arithmetic, Algebra and Linear Algebra. / Faremos, aqui, um estudo teórico sobre a Transformada Discreta de Fourier no círculo finito ℤ/nℤ. Nosso principal objetivo é verificar se podemos obter propriedades análogas às encontradas nas transformadas de Fourier para o caso contínuo. Nesse trabalho mostraremos que ℤ/nℤ tem uma estrutura de anel, dando condições para o desenvolvimento de temas bastante discutidos na Aritmética como, por exemplo, o Teorema Chinês do Resto, função Phi de Euler e raízes primitivas, temas estes que serão tratados no primeiro capítulo. O assunto principal desse estudo é desenvolvido no segundo capítulo, onde definiremos o espaço L2(ℤ/nℤ) e provaremos que este é um espaço vetorial com produto interno, dimensão finita e uma base ortonormal. Tal fato será de extrema importância quando estivermos determinando a matriz e demonstrando as propriedades da transformada discreta de Fourier. Também faremos interpretações geométricas do Teorema Chinês do Resto e do círculo finito ℤ/nℤ assim como daremos a representação gráfica da DFT de algumas funções que calcularemos. Durante o desenvolvimento desse estudo faremos uso recorrente de definições e resultados tratados na Aritmética, Álgebra e Álgebra Linear.

Quasi-hom-Lie algebras (qhl-algebras) were introduced by Larsson and Silvestrov (2004) as a generalisation of hom-Lie algebras, which are a deformation of Lie algebras. Lie algebras are defined by an operation called bracket, [·,·], and a three-term Jacobi identity. By the theorem from Hartwig, Larsson, and Silvestrov (2003), this bracket and the three-term Jacobi identity are deformed into a new bracket operation, <·,·>, and a six-term Jacobi identity, making it a quasi-hom-Lie algebra. Throughout this thesis we deform the Lie algebra sl2(F), where F is a field of characteristic 0. We examine the quasi-deformed relations and six-term Jacobi identities of the following polynomial algebras: F[t], F[t]/(t2), F[t]/(t3), F[t]/(t4), F[t]/(t5), F[t]/(tn), where n is a positive integer ≥2, and F[t]/((t-t0)3). Larsson and Silvestrov (2005) and Larsson, Sigurdsson, and Silvestrov (2008) have already examined some of these cases, which we repeat for the reader's convenience. We further investigate the following σ-twisted derivations, and how they act in the different cases of mentioned polynomial algebras: the ordinary differential operator, the shifted difference operator, the Jackson q-derivation operator, the continuous q-difference operator, the Eulerian operator, the divided difference operator, and the nilpotent imaginary derivative operator. We also introduce a new, general, σ-twisted derivation operator, which is σ(t) as a polynomial of degree k.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / A cyclic code is a speci c type of linear code. Its relevance consists in the fact that all its main information is intrinsic to the structure of the ideals in the quotient ring K[x]=(xn - 1) via an isomorphism. In this work, we characterize the cyclic codes in biunivocal correspondence with the ideals of this quotient ring. We will also present its generating matrix, the parity matrix and we will discuss its codi cation and decoding. / Um código cíclico é um tipo específico de código linear. Sua relevância consiste no fato de que todas suas principais informações são intrinsecas à estrutura dos ideais no anel quociente K[x]=(xn 1) via um isomorfismo. Neste trabalho, caracterizamos os códigos cíclicos em correspondência biunívoca com os ideais deste anel quociente. Apresentaremos também sua matriz geradora, a matriz de paridade e abordaremos sua codificação e decodificação.