RAMOS, Igor Rochaid Oliveira. Estudo de um sistema clássico de dipolos magnéticos carregados em estruturas de bicamadas. 2010. 75 f. Dissertação (Mestrado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2010. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2014-10-31T20:38:15Z No. of bitstreams: 1 2010_dis_iroramos.pdf: 3512055 bytes, checksum: ba5eb29d22412048c4f8a2972b874c7e (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2014-10-31T21:08:54Z (GMT) No. of bitstreams: 1 2010_dis_iroramos.pdf: 3512055 bytes, checksum: ba5eb29d22412048c4f8a2972b874c7e (MD5) / Made available in DSpace on 2014-10-31T21:08:54Z (GMT). No. of bitstreams: 1 2010_dis_iroramos.pdf: 3512055 bytes, checksum: ba5eb29d22412048c4f8a2972b874c7e (MD5) Previous issue date: 2010 / We study the structural and dynamical properties of a two-dimensional (2D) classical bi-layer crystal of charged magnetic dipolar particles in a setup where the dipoles are oriented perpendicular to the layers and equal density of charged dipolar particles in each layer. The energy of the system is due to the charge - charge interaction (Coulomb interaction) and the dipole - dipole interaction. Due to the long-range nature of the interactions, we use the Ewald summation method to obtain an expression for the energy involving rapidly convergent sums. By comparing the energies of a number of possible crystal geometries, we determine the phase diagram of the system as a function of the parameter η (which is related to the separation between the layers of charged magnetic dipoles and the particle density) and the relative intensity of the magnetic and electrical interactions. By changing the relative intensity of the dipole - dipole interaction with respect to electrical one, we are able to find six diferent stable crystalline structures as a function of η. An interesting feature of the present model system is the possibility to tune between the matched and staggered arrangements by varying the magnetic interaction between the dipoles, e.g. through an external magnetic field. The phase boundaries of the crystalline structures consist of both continuous and discontinuous transitions. In order to investigate the stability of the minimum energy arrangements we also calculate the phonon spectra of the system within the harmonic approximation. In this case, we resort again on the Ewald technique to obtain the rapidly convergent sums. The analysis of the phonon spectra reveals interesting features which are useful in the study of melting. / Estudamos as estruturas e as propriedades dinâmicas de um cristal clássico bidimensional (2D) em bicamadas de partículas dipolares magnéticas carregadas em um arranjo no qual os dipolos são perpendiculares às camadas e com mesma densidade de partículas em cada camada. A energia do sistema é devido à interação carga - carga (interação coulom- biana) e a interação dipolo - dipolo. Devido ao fato dessas interações serem de longo alcance, usamos o método da soma de Ewald para obter uma expressão para a energia envolvendo somas que convergem rapidamente. Comparando as energias de possíveis geometrias do cristal, determinamos o diagrama de fase do sistema em função do parâmetro η (que está relacionado com a distância entre as camadas de dipolos magnéticos carregados e a densidade de partículas) e da intensidade relativa das interações elétrica e magnética. Mudando a intensidade relativa da interacão dipolo - dipolo com respeito à interação elétrica, podemos encontrar seis diferentes estruturas cristalinas estáveis em função de η. Uma característica interessante desse sistema é a possibilidade de permanecer em arranjos nos quais as camadas são ou não deslocadas uma em relação a outra, apenas variando a interação magnética entre os dipolos, por exemplo, através de um campo magnético externo. As transições entre as estruturas cristalinas podem ser contínuas e descontínuas. No intuito de investigar a estabilidade das configurações de mínima energia, calculamos o espectro dos fônons do sistema usando a aproximação harmônica. Para isto, recorremos novamente a técnica de Ewald para obter somas que convergem rapidamente. A análise da relação de dispersão (fônons) revela características do sistema que são de grande utilidade no estudo da transição sólido-líquido (fusão).

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2018-10-04T12:33:28Z No. of bitstreams: 2 Dissertação - Raul Rodrigues dos Santos - 2018.pdf: 1879658 bytes, checksum: 96c8a59837af9f467513e7904dbac69e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-10-05T14:11:09Z (GMT) No. of bitstreams: 2 Dissertação - Raul Rodrigues dos Santos - 2018.pdf: 1879658 bytes, checksum: 96c8a59837af9f467513e7904dbac69e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-10-05T14:11:09Z (GMT). No. of bitstreams: 2 Dissertação - Raul Rodrigues dos Santos - 2018.pdf: 1879658 bytes, checksum: 96c8a59837af9f467513e7904dbac69e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-07-27 / This dissertation will present a proposal of Arithmetic teaching, starting from the initial years of Education to Higher Education. The reader will find in this work the following contents: Topics of the History of Mathematics, Arithmetic Progressions, Divisibility, Congruence and Summaries. In the sections, we will have examples solved and proposed activities to be solved by the reader. We have two objectives in this research, the first one is to present a proposal that interrelates mathematical contents of Arithmetic, knowing some historical curiosities and demonstrating Theorems, Propositions, Corollaries, solving examples and questions of the Brazilian Games of Mathematics of the Public Schools (OBMEP). And the second is to propose two Theorems and two Corollas of divisibility and congruence. The theorems, definitions, corollaries, demonstrations and etc. of this bibliographical research were, to a large extent, based on established authors as [12], [14], [16], [17], [18], [19], [20], [22] e [23]. / Esta dissertação apresentará uma proposta de ensino de Aritmética, partindo dos anos iniciais da Educação Básica até o Ensino Superior. O leitor encontrará neste trabalho os seguintes conteúdos: Tópicos da História da Matemática, Progressões Aritméticas, Divisibilidade, Congruência e Somatórios. Nas seções, teremos exemplos resolvidos e atividades propostas a serem solucionados pelo leitor. Temos dois objetivos nesta pesquisa, o primeiro é apresentar uma proposta que inter-relaciona conteúdos matemáticos da Aritmética, conhecendo algumas curiosidades históricas e demonstrando Teoremas, Proposições, Corolários, resolvendo exemplos e questões das Olimpíadas Brasileiras de Matemática das Escolas Públicas (OBMEP). E o segundo é, propor dois Teoremas e dois Corolários de divisibilidade e congruência. Os teoremas, definições, corolários, demonstrações e etc., desta pesquisa bibliográfica, foram, em grande parte, baseados em autores consagrados como [12], [14], [16], [17], [18], [19], [20], [22] e [23].

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<p> Jakeman's random walk model with step number fluctuations describes the amplitude scattered from a rough medium in terms as the coherent summation of (independent) individual scatterers' contributions. For a population following a birthdeath- immigration (BDI) model, the resulting statistics are k-distributed and the multiplicative representation of the amplitude as a Gaussian speckle modulated by a Gamma radar cross-section (RCS) is recovered. The main objective of the present thesis is to discuss techniques for the inference of the RCS in local time in order to facilitate anomaly detection. We first show how the Pearson class of diffusions, which we derive on the basis of a discrete population model analogous to the BDI, encompasses this Gamma texture as well as other texture models studied in the literature. Next we recall how Field & Tough derived, in an Ito calculus framework, the dynamics and the auto-correlation function of the scattered amplitude from the random walk model. In particular, they showed how the RCS was observable through the intensity-weighted squared fluctuations of the phase. Thanks to a discussion of the sources of discrepancy arising during this process, we derive an analytical expression for the inference error based on its asymptotic behaviours, together with a condition to minimize it. Our results are then extended to the Pearson class of diffusions whose importance for radar clutters is described. Next, we consider an experimental caveat, namely the presence of an additional white noise. The finite impulse response Wiener filter enables the design of the optimal filter to retrieve the scattered amplitude when it lies in superposition with thermal noise, thus enabling the usage of our inference technique. Finally, we consider weak scattering when a coherent signal lies in superposition with the aforementioned (strongly) scattered amplitude. Strong and weak scattering patterns differ regarding the correlation structure of their radial and angular fluctuations. Investigating these geometric characteristics yields two distinct procedures to infer the scattering cross-section from the phase and intensity fluctuations of the weakly scattered amplitude, thus generalizing the results obtained in the strong scattering case. </p> / Thesis / Doctor of Philosophy (PhD)

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Secure multiparty computation refers to the problem where a number of users wish to securely compute a function on their inputs without revealing any unnecessary information. This dissertation focuses on the fundamental limits of secure summation under different constraints. We first focus on the minimal model of secure computation, in which two users each hold an input and wish to securely compute a function of their inputs at the server. We propose a novel scheme base on the algebraic structure of finite field and modulo ring of integers. Then we extend the minimal model of secure computation, in which K users wish to securely compute the sum of their inputs at the server. We prove a folklore result on the limits of communication cost and randomness cost. Then we characterized the optimal communication cost with user dropouts constraint, when some users may lose connection to the server and the server wishes to compute the sum of remaining inputs. Next, we characterize the optimal communication and randomness cost for symmetric groupwise keys and find the feasibility condition for arbitrary groupwise keys. Last, we study the secure summation with user selection, such that the server may select any subset of users to compute the sum of their inputs. This leads us to the MDS variable generation problem. We characterize the optimal individual key rate and the result is interestingly the harmonic number.

The computation of the Coulomb potentials and forces in charged particle systems under 3d-periodic boundary conditions is possible in an efficient way by utilizing the Ewald summation formulas and applying the fast Fourier transform (FFT). In this paper we consider the particle-particle NFFT (P2NFFT) approach, which is based on the fast Fourier transform for nonequispaced data (NFFT) and compare the error behaviors regarding different window functions, which are used in order to approximate the given continuous charge distribution by a mesh based charge density. While typically B-splines are applied in the scope of particle mesh methods, we consider for the first time also an approximation by Bessel functions. We show how the resulting root mean square errors in the forces can be predicted precisely and efficiently. The results show that if the parameters are tuned appropriately the Bessel window function can keep up with the B-spline window and is in many cases even the better choice with respect to computational costs.

The present thesis provides a modularized view on the structure of fast numerical methods for computing Coulomb interactions between charged particles in three-dimensional space. Thereby, the common structure is given in terms of three self-contained algorithmic frameworks that are built on top of each other, namely fast Fourier transform (FFT), nonequispaced fast Fourier transform (NFFT) and NFFT based particle-mesh methods (P²NFFT). For each of these frameworks algorithmic enhancement and parallel implementations are presented with special emphasis on scalability up to hundreds of thousands of parallel processes. In the context of FFT massively parallel algorithms are composed from hardware adaptive low level modules provided by the FFTW software library. The new algorithmic NFFT concepts include pruned NFFT, interlacing, analytic differentiation, and optimized deconvolution in Fourier space with respect to a mean square aliasing error. Enabled by these generalized concepts it is shown that NFFT provides a unified access to particle-mesh methods. Especially, mixed-periodic boundary conditions are handled in a consistent way and interlacing can be incorporated more efficiently. Heuristic approaches for parameter tuning are presented on the basis of thorough error estimates. / Die vorliegende Dissertation beschreibt einen modularisierten Blick auf die Struktur schneller numerischer Methoden für die Berechnung der Coulomb-Wechselwirkungen zwischen Ladungen im dreidimensionalen Raum. Die gemeinsame Struktur ist geprägt durch drei selbstständige und auf einander aufbauenden Algorithmen, nämlich der schnellen Fourier-Transformation (FFT), der nicht äquidistanten schnellen Fourier-Transformation (NFFT) und der NFFT-basierten Teilchen-Gitter-Methode (P²NFFT). Für jeden dieser Algorithmen werden Verbesserungen und parallele Implementierungen vorgestellt mit besonderem Augenmerk auf massiv paralleler Skalierbarkeit. Im Kontext der FFT werden parallele Algorithmen aus den Hardware adaptiven Modulen der FFTW Softwarebibliothek zusammengesetzt. Die neuen NFFT-Konzepte beinhalten abgeschnittene NFFT, Versatz, analytische Differentiation und optimierte Entfaltung im Fourier-Raum bezüglich des mittleren quadratischen Aliasfehlers. Mit Hilfe dieser Verallgemeinerungen bietet die NFFT einen vereinheitlichten Zugang zu Teilchen-Gitter-Methoden. Insbesondere gemischt periodische Randbedingungen werden einheitlich behandelt und Versatz wird effizienter umgesetzt. Heuristiken für die Parameterwahl werden auf Basis sorgfältiger Fehlerabschätzungen angegeben.

This paper presents a theory of Fourier transforms of complex-valued functions on a finite abelian group and investigates two applications of this theory. Chapter I is an introduction with remarks on notation. Basic theory, including Pontrvagin duality and the Poisson Summation formula, is the subject of Chapter II. In Chapter III the Fourier transform is viewed as an intertwining operator for certain unitary group representations. The solution of the eigenvalue problem of the Fourier transform of functions on the group Z/n of integers module n leads to a proof of the quadratic reciprocity law in Chapter IV. Chapter V addresses the, use of the Fourier transform in computing.