Call number: LD2668 .T4 1963 C54 / Master of Science

This thesis explores the geometry at infinity for certain hermitian locally symmetric spaces. Let Gamma < SU(r + 1, r) be a maximal nonuniform arithmetic lattice determined by automorphisms of a hermitian form on k^{2 r + 1}, where k is an imaginary quadratic field. We give a formula for the number of cusps of X / Gamma, where X is the hermitian symmetric space on which SU(r + 1, r) acts. If r > 1 and 2 r + 1 is prime, this completely determines the number of cusps for minimal finite volume orbifolds with X-geometry, and there are only finitely many commensurability classes of noncompact finite volume quotients of X containing a one-cusped orbifold. In the case r = 1, which corresponds to the complex hyperbolic plane, we show that this holds for any N: there are only finitely many commensurability classes of arithmetic lattices in SU(2, 1) which contain an N-cusped orbifold. / text

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Let L be a lattice. For x ∈ L, we say x is a consistent join-irreducible if x V y is a join-irreducible of the lattice [y,1] for all y in L. We say L is consistent if every join-irreducible of L is consistent. In this dissertation, we study the notion of consistent elements in semimodular lattices.

A criptografia de chave pública é uma área do conhecimento sujeita que é tema de intensa atividade contemporânea de pesquisa. Novos protocolos, primitivas e ataques são propostos com frequência, com semelhanças e diferenças mútuas que podem ser mais ou menos evidentes. Algumas primitivas criptográficas de chave pública mostram-se extremamente férteis em termos de flexibilidade, eficiência e segurança. Duas vertentes que se enquadram nesta categoria são os emparelhamentos e os reticulados. Por possuírem semelhanças em suas funcionalidades a despeito de possuírem naturezas completamente díspares, além de exibirem uma versatilidade rara em toda a área de criptografia de chave pública, alguns autores propuseram chamar os reticulados de os novos emparelhamentos, conforme a ordem cronológica em que essas primitivas passaram a atrair interesse mais vívido de pesquisa. Neste cenário, um estudo comparativo entre elas é de razoável interesse, em particular sobre vantagens e desvantagens que o estado da arte revela sobre a eficiência de cada uma delas. A pesquisa aqui relatada contempla esse estudo, e contribui técnicas de implementação eficiente de emparelhamentos (com ênfase no uso de coordenadas afins, pouco exploradas na literatura), novos parâmetros para a construção de reticulados compactos (na forma das chamadas álgebras discretas de Rojo) e uma técnica inovadora para instanciar reticulados na prática (especificamente, um algoritmo simples e natural para amostrar vetores normalmente distribuídos nos reticulados comumente adotados em sistemas criptográficos). / Public key cryptography is an area of knowledge undergoing intense research at present. New protocols, primitives and attacks are often proposed, with mutual similarities and differences that may be more or less evident. Some public key cryptographic primitives tend to be extremely prolific in terms of flexibility, efficiency and security. Two trends that fit this category are pairings and lattices. Because of their similar functionalities despite their completely disparate nature, and because of their rare versatility within the whole area of public key cryptography, some authors proposed to call lattices \"the new pairings,\" according to the chronological order by which these primitives began to attract more vivid research interest. In this scenario, a comparative study between them is of reasonable interest, in particular on the advantages and disadvantages that the state of the art reveals about the efficiency of each one. The research reported herein addresses this study, and also contributes efficient pairing implementation techniques (focusing on affine coordinates, which are scarcely explored in literature), new parameters for building compact lattices (in the form of the so-called discrete Rojo algebras) and an innovative technique to instantiate lattices in practical (specifically, a simple and natural algorithm for sampling normally distributed vectors in the lattices that are commonly adopted in cryptographic systems).

Digitized by Kansas Correctional Industries

A criptografia de chave pública é uma área do conhecimento sujeita que é tema de intensa atividade contemporânea de pesquisa. Novos protocolos, primitivas e ataques são propostos com frequência, com semelhanças e diferenças mútuas que podem ser mais ou menos evidentes. Algumas primitivas criptográficas de chave pública mostram-se extremamente férteis em termos de flexibilidade, eficiência e segurança. Duas vertentes que se enquadram nesta categoria são os emparelhamentos e os reticulados. Por possuírem semelhanças em suas funcionalidades a despeito de possuírem naturezas completamente díspares, além de exibirem uma versatilidade rara em toda a área de criptografia de chave pública, alguns autores propuseram chamar os reticulados de os novos emparelhamentos, conforme a ordem cronológica em que essas primitivas passaram a atrair interesse mais vívido de pesquisa. Neste cenário, um estudo comparativo entre elas é de razoável interesse, em particular sobre vantagens e desvantagens que o estado da arte revela sobre a eficiência de cada uma delas. A pesquisa aqui relatada contempla esse estudo, e contribui técnicas de implementação eficiente de emparelhamentos (com ênfase no uso de coordenadas afins, pouco exploradas na literatura), novos parâmetros para a construção de reticulados compactos (na forma das chamadas álgebras discretas de Rojo) e uma técnica inovadora para instanciar reticulados na prática (especificamente, um algoritmo simples e natural para amostrar vetores normalmente distribuídos nos reticulados comumente adotados em sistemas criptográficos). / Public key cryptography is an area of knowledge undergoing intense research at present. New protocols, primitives and attacks are often proposed, with mutual similarities and differences that may be more or less evident. Some public key cryptographic primitives tend to be extremely prolific in terms of flexibility, efficiency and security. Two trends that fit this category are pairings and lattices. Because of their similar functionalities despite their completely disparate nature, and because of their rare versatility within the whole area of public key cryptography, some authors proposed to call lattices \"the new pairings,\" according to the chronological order by which these primitives began to attract more vivid research interest. In this scenario, a comparative study between them is of reasonable interest, in particular on the advantages and disadvantages that the state of the art reveals about the efficiency of each one. The research reported herein addresses this study, and also contributes efficient pairing implementation techniques (focusing on affine coordinates, which are scarcely explored in literature), new parameters for building compact lattices (in the form of the so-called discrete Rojo algebras) and an innovative technique to instantiate lattices in practical (specifically, a simple and natural algorithm for sampling normally distributed vectors in the lattices that are commonly adopted in cryptographic systems).

Chan, Yun San = 光子晶格系統中相干行為的研究 / 陳潤燊. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 109-112). / Abstracts in English and Chinese. / Chan, Yun San = Guang zi jing ge xi tong zhong xiang gan xing wei de yan jiu / Chen Runshen. / Abstract --- p.i / 摘要 --- p.iii / Acknowledgements --- p.V / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Periodic system and photonic crystal --- p.1 / Chapter 1.1.1 --- Properties and Applications --- p.1 / Chapter 1.1.2 --- Coherent Phenomena --- p.2 / Chapter 1.1.3 --- Quantum Optical Analogue --- p.9 / Chapter 1.2 --- Coupled Optical Waveguides --- p.11 / Chapter 1.2.1 --- Coupled-mode Theory --- p.11 / Chapter 1.2.2 --- Field Evolution Analysis (FEA) --- p.14 / Chapter 1.2.3 --- Hamiltonian Optics (HO) --- p.15 / Chapter 1.3 --- Experimental Realization --- p.17 / Chapter 1.4 --- Objectives --- p.17 / Chapter 2 --- Parabolic Optical Waveguide Array --- p.19 / Chapter 2.1 --- Introduction --- p.19 / Chapter 2.1.1 --- Generalized Bloch Oscillation --- p.19 / Chapter 2.1.2 --- DO-BO Transition --- p.20 / Chapter 2.2 --- Model and Formalism --- p.20 / Chapter 2.3 --- Results --- p.25 / Chapter 2.3.1 --- Dipole Oscillation --- p.29 / Chapter 2.3.2 --- Bloch Oscillation --- p.29 / Chapter 2.3.3 --- Right Reflection --- p.31 / Chapter 2.3.4 --- Mechanical Analogue --- p.32 / Chapter 2.3.5 --- Lift-n-Shift Process --- p.33 / Chapter 2.4 --- Summary --- p.39 / Chapter 3 --- Binary POWA --- p.40 / Chapter 3.1 --- Introduction --- p.40 / Chapter 3.2 --- Model and Formalism --- p.41 / Chapter 3.3 --- Results --- p.45 / Chapter 3.3.1 --- Dipole Oscillation --- p.48 / Chapter 3.3.2 --- Bloch-dipole-Zener Oscillation --- p.51 / Chapter 3.3.3 --- Bloch-Zener oscillation --- p.54 / Chapter 3.4 --- Viable Experimental Realization --- p.57 / Chapter 3.5 --- Summary --- p.58 / Chapter 4 --- Parabolically Graded Square Lattice --- p.60 / Chapter 4.1 --- Introduction --- p.60 / Chapter 4.2 --- Model and Formalism --- p.61 / Chapter 4.3 --- Results --- p.65 / Chapter 4.3.1 --- Orthogonal Coupling --- p.65 / Chapter 4.3.2 --- Weak Diagonal Coupling --- p.76 / Chapter 4.4 --- Summary --- p.81 / Chapter 5 --- Elliptical Optical Waveguide Array --- p.82 / Chapter 5.1 --- Introduction --- p.82 / Chapter 5.2 --- Model and Formalism --- p.83 / Chapter 5.2.1 --- Kac Matrix --- p.83 / Chapter 5.2.2 --- Kac Matrix and Spin --- p.85 / Chapter 5.2.3 --- System Configuration --- p.86 / Chapter 5.3 --- Results --- p.91 / Chapter 5.3.1 --- Upper Dipole Oscillation --- p.92 / Chapter 5.3.2 --- Lower Dipole Oscillation --- p.94 / Chapter 5.3.3 --- Bloch Oscillation --- p.95 / Chapter 5.3.4 --- Upper Reflection --- p.96 / Chapter 5.3.5 --- Lower Reflection --- p.98 / Chapter 5.3.6 --- Harmonic Oscillations --- p.98 / Chapter 5.3.7 --- Lift-n-Shift Process --- p.101 / Chapter 5.4 --- Summary --- p.102 / Chapter 6 --- Conclusion --- p.104 / Chapter 6.1 --- Suggestion of Future Works --- p.106 / Chapter 6.1.1 --- POWA --- p.106 / Chapter 6.1.2 --- BPOWA --- p.106 / Chapter 6.1.3 --- PGSL --- p.107 / Chapter 6.1.4 --- EOWA --- p.107 / Chapter A --- List of abbreviations --- p.108 / Bibliography --- p.109

In this thesis, we provide an algorithm for constructing strongly eutactic lattices from vertex transitive graphs. We show that such construction produces infinitely many strongly eutactic lattices in arbitrarily large dimensions. We demonstrate our algorithm on the example of the famous Petersen graph using Maple computer algebra system. We also discuss some additional examples of strongly eutactic lattices obtained from notable vertex transitive graphs. Further, we study the properties of the lattices generated by product graphs, complement graphs, and line graphs of vertex transitive graphs. This thesis is based on the research paper written by the author jointly with L. Fukshansky, D. Needell and J. Park.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references. Also available on the Internet.