Thesis--Bonn. / Extra t.p. with thesis statement inserted. Includes bibliographical references (p. 55).

Thesis--Bonn. / Extra t.p. with thesis statement inserted. Includes bibliographical references (p. 55).

Thesis (Ph.D.)--Ohio University, June, 2005. / Includes bibliographical references (p. 96-102)

Thesis (Ph.D.)--Ohio University, June, 2005. / Title from PDF t.p. Includes bibliographical references (p. 96-102)

Under skolgången ska elever utveckla flera grundläggande kunskaper inom matematiken. En sådan kunskap handlar om de fyra räknesättens olika egenskaper. Kommutativitet är en egenskap hos addition och multiplikation som innebär att termer eller faktorer kan byta plats utan att summan eller produkten förändras, exempelvis att 2+4 är lika mycket som 4+2 och att 3‧2 är lika mycket som 2‧3. Egenskaper räknesätt har sammanfattas i olika räknelagar, de vanligaste är kommutativa-, associativa- och distributiva lagen. Syftet med denna studie är att skapa ny kunskap om hur kommutativa lagen presenteras för elever i matematikläromedel. Studien bygger på en läromedelsgranskning som genomförs med en kvalitativ innehållsanalys och inspireras av variationsteorin. Tre läromedelsserier avsedda för årskurs 1–6 har granskats. I analysen har det framkommit att kommutativitet sällan beskrivs som en fristående egenskap utan ofta presenteras tillsammans med något annat. Främst är det tillsammans med talkamrater, tillsammans med sambandet mellan addition och subtraktion och när elever ska överföra något bildligt till matematiska uttryck som kommutativitet beskrivs. Det går också att urskilja en viss progression av kommutativitetbegreppen i läromedel. / Students should develop several basic skills in mathematics during their school years. For instance, they should develop solid understanding of the properties of the four rules of arithmetic. Commutativity is a property of addition and multiplication that means that terms or factors can change place without changing the sum or the product, for example, that 2 + 4 is equal to 4 + 2 and 3‧2 is equal to 2‧3. The purpose of this study is to expand our understanding about how the commutative law is presented to students in mathematics textbooks. This work is a textbook review which is a qualitative content analysis and is inspired by the theory of variation. Three series of mathematics textbooks for school years 1 through 6 in Sweden have been reviewed. The analysis of the textbooks show that commutativity is rarely described on its own,but is often presented together with something else: Primarily with part-whole relations, along with the connection between addition and subtraction and when students are going to transfer something figuratively to math expressions. A progression of the commutativity concepts in the textbooks can be distinguished.

This thesis is concerned with the existence of pushouts in two different settings of algebraic geometry. At first, we study the pushouts in the cat- egory of affine algebraic sets over an infinite field. We show that this can be regarded as an instance of much general problem whether the pullback of finitely generated algebras over a commutative Noetherian ring is finitely generated. We give a partial solution to this problem and study some ex- amples. Secondly, we examine the existence of pushouts in the category of schemes with an emphasis on diagrams of affine schemes. We use the methods of Ferrand [2003] and Schwede [2004] and generalise some of their results. We conclude by giving some examples and suggest another approach to the problem.

Doctor of Philosophy / Department of Mathematics / Yan Soibelman / Given a quiver Q with/without potential, one can construct an algebra structure on the cohomology of the moduli stacks of representations of Q. The algebra is called Cohomological Hall algebra (COHA for short). One can also add a framed structure to quiver Q, and discuss the moduli space of the stable framed representations of Q. Through these geometric constructions, one can construct two representations of Cohomological Hall algebra of Q over the cohomology of moduli spaces of stable framed representations. One would get the double of the representations of Cohomological Hall algebras by putting these two representations together. This double construction implies that there are some relations between Cohomological Hall algebras and some other algebras. In this dissertation, we focus on the quiver without potential case. We first define Cohomological Hall algebras, and then the above construction is stated under some assumptions. We computed two examples in detail: A₁-quiver and Jordan quiver. It turns out that A₁-COHA and its double representations are related to the half infinite Clifford algebra, and Jordan-COHA and its double representations are related to the infinite Heisenberg algebra. Then by the fact that the underlying vector spaces of these two COHAs are isomorphic to each other, we get a COHA version of Boson-Fermion correspondence.

Orientador: Trajano Pires da Nóbrega Neto / Banca: Antonio Aparecido de Andrade / Banca: Jos'e Valter Lopes Nunes / Resumo: O propósito deste trabalho é estudar alguns corpos abelianos, mais especificamente, as extensões reais maximais contidas nos corpos ciclotômicos de grau 8 e, os subcorpos dos corpos ciclotômicos Q(ζ_7) e Q(ζ_17). Em tais corpos, determinamos base integral, discriminante, grupo de Galois e construimos submódulos de posto máximo do anel dos inteiros algébricos com sua respectiva representação geométrica. Além disso, calculamos a densidade de centro destes reticulados / Abstract: The purpose of this work is to investigate some Abelian Number Fields, especifically the maximal extension contained in the cyclotomic fields of degree 8, and the subfields of the cyclotomic fields Q(ζ7) and Q(ζ17). In such fields, we compute: integral bases, discriminant, Galois group and submoduli with maximal rank in the ring of algebraic integers, its geometrical realization with the respective center density / Mestre

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-11T13:46:47Z No. of bitstreams: 1 arquivototal.pdf: 4626287 bytes, checksum: 422f70b41a38fd74eb6520e392f6d65b (MD5) / Made available in DSpace on 2017-09-11T13:46:47Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 4626287 bytes, checksum: 422f70b41a38fd74eb6520e392f6d65b (MD5) Previous issue date: 2017-03-07 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we developed a complete study on the relativistic Landau model and non-commutative geometry, the latter was derived from level projection, in order to describe curved graphene systems. In developing of the theory, we address the problem of the eigenvalues from the relativistic Dirac-Landau operador on the sphere with a magnetic monopole in its center. The relativistic fuzzy spheres are introduced using the eigenstates of the relativistic Landau levels and we compare it with non-relativistic cases. Under mass deformation, the fuzzy spheres relative to the relativistic symmetric Landau levels change their sizes, however zero-modes there are no variation of size for the corresponding fuzzy sphere. Consecutively we verify that the relativistic Landau model and non-relativistic system of Pauli-Schr odinger are related by gauge transformation SU(2). And nally, the application of the whole theoretical graphene's framework show a simmetric spectrum with respect to its zero energy, and it maintains itself under mass deformation. On the other hand, the in uence of the mass parameters M 6= 0 on the four fuzzy spheres (two for each valley) is such that, if n 6= 0, two of them are enlarged and the other two are diminished, but for n = 0 the fuzzy spheres (at M) do not change their sizes. / Neste trabalho, desenvolvemos um estudo completo do modelo de Landau relativ stico e da geometria n~ao-comutativa, esta ultima derivada a partir da proje c~ao de n vel, a m de descrever sistemas curvos de grafeno. No desenvolvimento da teoria, abordamos o problema de autovalores do operador Dirac-Landau relativ stico sobre uma esfera com um monopolo magn etico em seu centro. As esferas fuzzy relativ sticas s~ao introduzidas usando-se os autoestados dos n veis de Landau relativ sticos e uma compara c~ao e feita com os casos n~ao-relativ sticos. Sob deforma c~ao da massa, as esferas fuzzy correspondentes a n veis de Landau relativ sticos sim etricos modi cam seus tamanhos, mas para modos-zero n~ao h a varia c~ao do tamanho para a esfera fuzzy correspondente. Em sequ^encia veri ca-se que o modelo de Landau relativ stico e o sistema n~ao-relativ stico de Pauli-Schr odinger est~ao relacionados por uma transforma c~ao de gauge SU(2). Finalmente, a aplica c~ao de todo o esse arcabou co te orico no grafeno, mostra que seu espectro e sim etrico com respeito a energia zero, e mant em-se mesmo sob deforma c~ao da massa. Por outro lado, a in u^encia do par^ametro de massa M 6= 0 nas quatro esferas fuzzy (2 para cada vale) e tal que, se n 6= 0, duas delas aumentam e as outras duas diminuem, mas para n = 0 as esferas fuzzy (em M) n~ao mudam seus tamanhos

Neste trabalho estudamos os modelos de Gross-Neveu e Sigma Não-Linear empregando a expansão 1/N no espaço não-comutativo. Consideramos três formalismos distintos para implementar a não-comutatividade. Dois deles tratam a não-comutatividade através da deformação do produto entre funções, porém, um dos formalismos é invariante de Lorentz e o outro não. O terceiro método trata a não-comutatividade através dos estados coerentes. Nesses diferentes contextos, calculamos o propagador do campo auxiliar de ambos os modelos, além da correção radiativa para o propagador do campo básico no modelo Sigma Não-Linear. / ln this work we studied the Gross-Neveu and nonlinear sigma models employing the 1/N expansion in the non-commutative espace. We describe three different manner of introduce non-commutativity. Two of them treat the non-commutativity through the deformation of the product of functions, however one of the method is Lorentz invariant and the other not. The third manner treat the non-commutativity through coherent states. We calculated the propagator of the auxiliary field in both models in these different settings and, besides that, we compute the leading radiative correction to propagator of the basic field of the nonlinear sigma model.