The study of orderable groups is a topic that is all too often overlooked as a topic in algebra. The subject of orderable groups is a field of study which is directly associated with algebraic group theory, algebraic topology, and set theory. This paper will act as a guide into the world of orderable groups. It begins by enlightening the reader to the fundamental axioms of orderable groups, as well as, noting various important groups on which orders are often considered. We will then consider more interesting groups, on which the placement of orders is considered less often. After considering the orderings placed on various groups, we wish to consider in further detail the topologies of the sets of these orders. In particular, it is important to consider whether the set of orders placed on a particular group is finite or uncountable. We prove the latter by creating a homeomorphism from the group to the Cantor set, a set which is known for its uncountability. / Master of Science

This thesis presents studies of the low energy properties of nseveral frustrated spin-1/2 Heisenberg antiferromagnets using various analytic and computational methods. The models studied include the union jack model, the alternating Heisenberg chain, the Heisenberg bilayer model, and the spin-Peierls model. The union jack model is a Heisenberg antiferromagnetic spin model with frustration, and is analyzed using spin-wave theory. For small values of the frustrating coupling $\alpha$, the system is N{\' e}el ordered, while for large $\alpha$ the frustration is found to induce a canted phase. Spin wave theory with second order corrections finds the critical coupling at $\alpha \simeq 0.645$,which agrees quantitatively with series expansion results. No intermediate spin-liquid phase is found to exist between the two phases. The alternating Heisenberg chain is studied using an alternative triplet-wave expansion formalism for dimerized spin systems, modification of the ??bond operator?? formalism of Sachdev and Bhatt. Projection operators are used to confine the system to the physical subspace, rather than constraint equations. Comparisons are made with the results of dimer series expansions and exact diagonalization. The S=1/2 Heisenberg bilayer spin model at zero temperature is studied in the dimerized phase using analytic triplet-wave expansions and dimer series expansions. The occurrence of two-triplon bound states in the S=0 and S=1 channels, and antibound states in the S=2 channel, is predicted with triplet-wave theory and confirmed by series expansions. All bound states are found to vanish at or before the critical coupling separating the dimerized phase from the N{\' e}el phase. The critical behavior of the total and single-particle static transverse structure factors is also studied by series expansion methods and found to conform with theoretical expectations. The Heisenberg spin-Peierls model with dispersive, gapless phonons is studied with Density Matrix Renormalization Group methods. We investigate the zero temperature properties of the model using the crossover method. The calculations were found to converge poorly and no conclusive results could be found using this method. An analysis of the convergence problems and the discovery of an anomalous triplet ground state is presented in this chapter.

Real space renormalisation group scaling techniques are used to investigate the static critical behaviour of the pure and dilute, classical, anisotropic Heisenberg model. Transfer matrix methods are employed to obtain asymptotically exact expressions for the correlation lengths and susceptibilities of the one-dimensional system. The resulting scaling relationships are combined with an approximate bond moving scheme to treat pure and dilute models in higher dimensionalities. Detailed discussions are given for the dependence of correlation lengths and susceptibilities on temperature, anisotropy and concentration, and fcr the critical temperature on anisotropy and concentration. Particular emphasis is given to the weakly anisotropic system near percolation threshold and comparisons are made between the results of the present analysis and those of neutron-scattering experiments on dilute quasi-two- and three-dimensional systems.

This thesis presents studies of the low energy properties of nseveral frustrated spin-1/2 Heisenberg antiferromagnets using various analytic and computational methods. The models studied include the union jack model, the alternating Heisenberg chain, the Heisenberg bilayer model, and the spin-Peierls model. The union jack model is a Heisenberg antiferromagnetic spin model with frustration, and is analyzed using spin-wave theory. For small values of the frustrating coupling $\alpha$, the system is N{\' e}el ordered, while for large $\alpha$ the frustration is found to induce a canted phase. Spin wave theory with second order corrections finds the critical coupling at $\alpha \simeq 0.645$,which agrees quantitatively with series expansion results. No intermediate spin-liquid phase is found to exist between the two phases. The alternating Heisenberg chain is studied using an alternative triplet-wave expansion formalism for dimerized spin systems, modification of the ??bond operator?? formalism of Sachdev and Bhatt. Projection operators are used to confine the system to the physical subspace, rather than constraint equations. Comparisons are made with the results of dimer series expansions and exact diagonalization. The S=1/2 Heisenberg bilayer spin model at zero temperature is studied in the dimerized phase using analytic triplet-wave expansions and dimer series expansions. The occurrence of two-triplon bound states in the S=0 and S=1 channels, and antibound states in the S=2 channel, is predicted with triplet-wave theory and confirmed by series expansions. All bound states are found to vanish at or before the critical coupling separating the dimerized phase from the N{\' e}el phase. The critical behavior of the total and single-particle static transverse structure factors is also studied by series expansion methods and found to conform with theoretical expectations. The Heisenberg spin-Peierls model with dispersive, gapless phonons is studied with Density Matrix Renormalization Group methods. We investigate the zero temperature properties of the model using the crossover method. The calculations were found to converge poorly and no conclusive results could be found using this method. An analysis of the convergence problems and the discovery of an anomalous triplet ground state is presented in this chapter.

Stuttgart, Univ., Diss., 2000.

Augsburg, Univ., Diss., 2009.

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2018. / Submitted by Raquel Viana (raquelviana@bce.unb.br) on 2018-10-30T21:20:30Z No. of bitstreams: 1 2018_BrunoMarinoXavier.pdf: 1783146 bytes, checksum: 58567e91e514f4dced7a3ca18f00354c (MD5) / Approved for entry into archive by Fabiana Santos (fabianacamargo@bce.unb.br) on 2018-11-12T18:03:20Z (GMT) No. of bitstreams: 1 2018_BrunoMarinoXavier.pdf: 1783146 bytes, checksum: 58567e91e514f4dced7a3ca18f00354c (MD5) / Made available in DSpace on 2018-11-12T18:03:20Z (GMT). No. of bitstreams: 1 2018_BrunoMarinoXavier.pdf: 1783146 bytes, checksum: 58567e91e514f4dced7a3ca18f00354c (MD5) Previous issue date: 2018-11-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) e Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). / Obtemos uma transformação de Bäcklund entre superfı́cies mı́nimas em Nil3 aplicando uma correspondência de Calabi entre uma superfı́cie CMC-1/2 em L3 e sua superfı́cie associada em Nil3 e fazendo uma transformação de Ribaucour na superfı́cie original relacionamos a geometria dessas duas superfı em L3. Em seguida, ́cies usando a segunda forma de Abresch- Rosenberg. Adiante, estendemos a definição de superfı́cies mı́nimas de Laguerre a formas espaciais enquanto relacionamos estas às superfı́cies mı́nimas em L3 e mı́nimas em outros espaços produto M2(k) × R e M2 (k) × R1 , com k = ±1. / We obtain a Bäcklund transformation between minimal surfaces in Nil3 by performing a Calabi correspondence between a CMC-1/2 surface in L3 and its associated minimal surface in Nil3 and sau Rrfiabcaeusc ouusrin tgra tnhsef oArmbr eosnc hth-eR oosreigninbaelr gs usrefaccoen din foLr3m .. NFeuxrtth, ewrem orerela,t ew eth ee xgteenodm ethtrey doef fbinoittiho nth oefs ea Laguerre minimal surface to space forms whilst relating these to the minimal immersions on L3 and minimal surfaces on other product spaces M2(k) × R and M2 (k) × R1 , with k = ±1.

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013. / Submitted by Alaíde Gonçalves dos Santos (alaide@unb.br) on 2014-03-25T11:46:09Z No. of bitstreams: 1 2013_DhiegoLoioladeAraujo.pdf: 300749 bytes, checksum: 88e56177211a24bab917702ee10a7a0a (MD5) / Approved for entry into archive by Guimaraes Jacqueline(jacqueline.guimaraes@bce.unb.br) on 2014-06-06T14:11:39Z (GMT) No. of bitstreams: 1 2013_DhiegoLoioladeAraujo.pdf: 300749 bytes, checksum: 88e56177211a24bab917702ee10a7a0a (MD5) / Made available in DSpace on 2014-06-06T14:11:39Z (GMT). No. of bitstreams: 1 2013_DhiegoLoioladeAraujo.pdf: 300749 bytes, checksum: 88e56177211a24bab917702ee10a7a0a (MD5) / Neste trabalho, estudamos superfícies tipo-espaço imersas no espaço de Lorentz-Heisenberg tridimensional que possuem curvatura média constante nula, denominadas superfícies máximas. Mostramos que a aplicação de Gauss de tais superfícies é uma aplicação harmônica na esfera de Riemann trivial, C∪{∞}, munida com uma métrica conforme. Resolvemos o problema de Calabi-Bernstein mostrando a não existência de gráficos máximos inteiros no espaço de Lorentz-Heisenberg e perturbando a diferencial de Hopf, obtemos diferenciais quadráticas holomorfas em superfícies máximas neste espaço. Por fim, construímos uma correspondência entre superfícies de curvatura média constante não nula em R3 e superfícies máximas no espaço de Lorentz-Heisenberg. ______________________________________________________________________________________ ABSTRACT / In this paper, we study the spacelike surfaces with zero mean curvature immersed in the Lorentz-Heisenberg space, called maximal surfaces. We prove that de Gauss map of maximal surfaces are harmonic maps into the trivial Riemann sphere, C∪{∞}, endowed with a conformal metric. We solve the Calabi-Bernstein problem showing the nonexistence of entire maximal graphs in the Lorentz-Heisenberg space, and disturbing the Hopf differential, we obtain holomorphic quadratic differentials on the maximal surfaces. We build a correspondence between non-zero constant mean curvature surfaces in R3 and maximal surfaces in the Lorentz-Heisenberg space.

Made available in DSpace on 2018-08-01T22:30:00Z (GMT). No. of bitstreams: 1 tese_6812_Tese final Thiago Oakes20141114-121507.pdf: 378469 bytes, checksum: 13e58c69ea83014550c9b1bd4e49cfd3 (MD5) Previous issue date: 2013-08-23 / Neste trabalho calculamos a energia do estado fundamental do átomo de hidrogênio levando em conta as contribuições advindas da presença de um comprimento mínimo. O cenário de comprimento mínimo é introduzido pela modificação da equação de Dirac através da álgebra de Heisenberg modificada (álgebra de Kempf). Com a introdução do potencial Coulombiano no novo operador de energia de Dirac, calculamos a mudança na energia do estado fundamental do átomo de hidrogênio em primeira ordem do parâmetro relacionado ao comprimento mínimo via teoria de perturbação.

Submitted by Isaac Francisco de Souza Dias (isaac.souzadias@ufpe.br) on 2016-04-22T17:28:54Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) DISSERTAÇÃO Wellington Moreira da Silva.pdf: 1184862 bytes, checksum: 9174a55ce0d6f3d7204587f06752d247 (MD5) / Made available in DSpace on 2016-04-22T17:28:54Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) DISSERTAÇÃO Wellington Moreira da Silva.pdf: 1184862 bytes, checksum: 9174a55ce0d6f3d7204587f06752d247 (MD5) Previous issue date: 2015-03-11 / FACEPE / O trabalho consiste no estudo de cadeias quânticas ferrimagnéticas associadas a compostos ferrimagnéticos quase-unidimensionais de spins mistos s − S, com s = 1/2 e S = 1. Estudamos as fases induzidas pela aplicação de um campo magnético, B, e efeitos de temperatura, T; em particular as excitações de baixa energia e as curvas de magnetização em função de B. A modelagem teórica e realizada através do hamiltoniano de Heisenberg com acoplamentos de superexchange antiferromagnéticos, J, entre sítios vizinhos e uniformes ao longo da cadeia. O estado fundamental e as excitações de baixa energia de sistemas finitos são obtidos através do m´método de ondas de spin não interagente (SW, do inglês Spin Wave) e interagente (ISW, do inglês Interacting Spin Wave), além do m´método de Lanczos de diagonalização exata; enquanto que as propriedades termodinâmicas são obtidas pelo M´método de Lanczos de Temperatura Finita (FTLM, do inglês Finite Temperature Lanczos Method) e analiticamente pela teoria ISW com a restrição de Takahashi adaptada para ferrimagnetos. Discutimos um modelo analítico efetivo para estudar o efeito de B nas propriedades t´térmicas da cadeia, além de analisar seu limite de validade comparando seus resultados com aqueles obtidos por FTLM. Esboçamos o diagrama de fases T −B do sistema, identi ficando os pontos críıtico-quânticos em T = 0 e as linhas de crossover em temperatura finita, em particular aquelas que limitam a fase Líquida de Luttinger do diagrama. / The work consists in the study of the spin 1=2 1 ferrimagnetic chains which are associated with quasi-one-dimensional ferrimagnetic compounds. We study the phases induced by a magnetic eld, B, and the e ect of temperature, T; in particular the magnetization curves as a function of B. The theoretical modeling is performed through the Heisenberg Hamiltonian with antiferromagnetic superexchange couplings ,J, between neighboring sites and uniform along the chain. The ground state and the low energy excitations of nite systems are calculated through not interaction Spin Wave (SW) and Interacting Spin Wave (ISW) methods, beyond the Lanczos exact diagonalization method, while the thermodynamic properties are obtained throguh the Finite Temperature Lanczos Method (FTLM ) and analitycally by ISW theory with the Takahashi restriction adaptated for ferrimagnets. We discuss a analitycal effetive model to study the e ect of B in the termical properties of the chain, beyond to analyse your limit of validity comparing theirs resuts with those obtained by FTLM. We sketh the phase diagram T B of the system, identifying the quantum critical points in T = 0 and the crossover lines in nite temperature, in particular those that limit the phase Luttinger liquid of the diagram