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## Nanowhiskers politípicos - uma abordagem teórica baseada em teoria de grupos e no método k.p; Polytypical nanowhiskers - a theoretical approach based on group theory and k.p method

Faria Júnior, Paulo Eduardo de
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
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Nanowhiskers semicondutores de compostos III-V apresentam grande potencial para aplicações tecnológicas. Controlando as condições de crescimento, tais como temperatura e diâmetro, é possível alternar entre as fases cristalinas zincblend e wurtzita, dando origem ao politipismo. Esse efeito tem grande influência nas propriedades eletrônicas e óticas do sistema, gerando novas formas de confinamento para os portadores. Um modelo teórico capaz de descrever com exatidão as propriedades eletrônicas e óticas presentes nessas nanoestruturas politípicas pode ser utilizado para o estudo e desenvolvimento de novos tipos de nanodispositivos. Neste trabalho, apresento a construção do Hamiltoniano k.p no ponto Γ para as estruturas cristalinas zincblend e wurtzita baseada no formalismo da teoria de grupos. Utilizando o grupo de simetria do ponto Γ, é possível obter as representações irredutíveis das bandas de energia, partindo de orbitais atômicos e do número de átomos na célula primitiva unitária. Além disso, as operações de simetria do grupo são utilizadas para calcular os elementos de matriz não nulos e independentes do Hamiltoniano k.p. O estudo da simetria dos estados de base pertencentes às representações irredutíveis das bandas de energia...

## Aplicações de metodos de topologia algebrica em teoria de grupos; Aplications of methods of algebraic topology in group theory

Patricia Massae Kitani
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Dissertação de Mestrado Formato: application/pdf
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Este trabalho consistiu no estudo das aplicações de topologia algébrica (recobrimentos, teorema de Van Kampen) em teoria de grupos e também, no estudo detalhado do resultado de R. Bieri, R. Strebel [Proc. London Math. Soc. (3) 41 (1980), no. 3, 439?464], que para um grupo G do tipo FP2, ou G contém subgrupo livre não cíclico ou para qualquer subgrupo normal N C G tal que Q = G/N é abeliano, N/[N,N] é um ZQ-módulo manso via conjugação. A definição de módulo manso usa o invariante de Bieri-Strebel §A(Q), nesse caso A = N/[N,N]; This work consisted of the study of the applications of algebraic topology (covering maps, Van Kampen theorem) in group theory and also, in the detailed study of a result of R. Bieri, R. Strebel [Proc. London Math. Soc. (3) 41 (1980), no. 3, 439?464], that for a group G of type FP2, either G has a free non-cyclic subgroup or for any normal subgroup N C G such that Q = G/N is abelian, N/[N,N] is a tame ZQ-module where Q acts via conjugation. The definition of tame module uses the Bieri-Strebel invariant §A(Q), in this case A = N/[N,N]

## Some group theory problems

Sapir, M. V.
Tipo: Artigo de Revista Científica
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This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For each problem, I try to mention the author. If the author is not given, the problem, to the best of my knowledge, was formulated by me first.; Comment: 25 pages

## Group theory in cryptography

Blackburn, Simon R.; Cid, Carlos; Mullan, Ciaran
Tipo: Artigo de Revista Científica
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This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.; Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, UK

## Symmetric-group decomposition of SU(N) group-theory constraints on four-, five-, and six-point color-ordered amplitudes

Edison, Alexander C.; Naculich, Stephen G.
Tipo: Artigo de Revista Científica
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Color-ordered amplitudes for the scattering of n particles in the adjoint representation of SU(N) gauge theory satisfy constraints that arise from group theory alone. These constraints break into subsets associated with irreducible representations of the symmetric group S_n, which allows them to be presented in a compact and natural way. Using an iterative approach, we derive the constraints for six-point amplitudes at all loop orders, extending earlier results for n=4 and n=5. We then decompose the four-, five-, and six-point group-theory constraints into their irreducible S_n subspaces. We comment briefly on higher-point two-loop amplitudes.; Comment: 35 pages; v2: typos in eqs. 4.14 and 4.20 corrected; v3: typo in eq. 7.3 corrected

## Group Operads and Homotopy Theory

Zhang, Wenbin
Tipo: Artigo de Revista Científica
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We introduce the classical theory of the interplay between group theory and topology into the context of operads and explore some applications to homotopy theory. We first propose a notion of a group operad and then develop a theory of group operads, extending the classical theories of groups, spaces with actions of groups, covering spaces and classifying spaces of groups. In particular, the fundamental groups of a topological operad is naturally a group operad and its higher homotopy groups are naturally operads with actions of its fundamental groups operad, and a topological $K(\pi,1)$ operad is characterized by and can be reconstructed from its fundamental groups operad. Two most important examples of group operads are the symmetric groups operad and the braid groups operad which provide group models for $\Omega^{\infty} \Sigma^{\infty} X$ (due to Barratt and Eccles) and $\Omega^2 \Sigma^2 X$ (due to Fiedorowicz) respectively. We combine the two models together to produce a free group model for the canonical stabilization $\Omega^2 \Sigma^2 X \hookrightarrow \Omega^{\infty} \Sigma^{\infty} X$, in particular a free group model for its homotopy fibre.; Comment: submitted; 39 pages; part of the author's Ph.D. thesis; Abstract and Introduction rewritten; Remarks 2.14 and 2.32 added concerning extending any group and G-space to a group operad and G-operad; Acknowledgements added; numerous minor corrections and changes made

## Representation Theory of Finite Groups

Singh, Anupam
Tipo: Artigo de Revista Científica
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The point of view of these notes on the topic is to bring out the flavor that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of base field. These notes cover completely the theory over complex numbers which is Character Theory. A large number of worked out examples are the main feature of these notes. The prerequisite for this note is basic group theory and linear algebra.; Comment: These notes grew out of a course on Representation Theory of finite groups given to undergraduate students at IISER Pune. This is first draft and hence readers are advised to use their own judgment

## The Classification of Partition Homogeneous Groups with Applications to Semigroup Theory

André, Jorge; Araújo, João; Cameron, Peter J.
Tipo: Artigo de Revista Científica
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Let $\lambda=(\lambda_1,\lambda_2,...)$ be a \emph{partition} of $n$, a sequence of positive integers in non-increasing order with sum $n$. Let $\Omega:=\{1,...,n\}$. An ordered partition $P=(A_1,A_2,...)$ of $\Omega$ has \emph{type} $\lambda$ if $|A_i|=\lambda_i$. Following Martin and Sagan, we say that $G$ is \emph{$\lambda$-transitive} if, for any two ordered partitions $P=(A_1,A_2,...)$ and $Q=(B_1,B_2,...)$ of $\Omega$ of type $\lambda$, there exists $g\in G$ with $A_ig=B_i$ for all $i$. A group $G$ is said to be \emph{$\lambda$-homogeneous} if, given two ordered partitions $P$ and $Q$ as above, inducing the sets $P'=\{A_1,A_2,...\}$ and $Q'=\{B_1,B_2,...\}$, there exists $g\in G$ such that $P'g=Q'$. Clearly a $\lambda$-transitive group is $\lambda$-homogeneous. The first goal of this paper is to classify the $\lambda$-homogeneous groups. The second goal is to apply this classification to a problem in semigroup theory. Let $\trans$ and $\sym$ denote the transformation monoid and the symmetric group on $\Omega$, respectively. Fix a group $H\leq \sym$. Given a non-invertible transformation $a\in \trans\setminus \sym$ and a group $G\leq \sym$, we say that $(a,G)$ is an \emph{$H$-pair} if the semigroups generated by $\{a\}\cup H$ and $\{a\}\cup G$ contain the same non-units...

## Orbit Equivalence and Measured Group Theory

Gaboriau, Damien
Tipo: Artigo de Revista Científica
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We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions.; Comment: 2010 Hyderabad ICM proceeding; Dans Proceedings of the International Congress of Mathematicians, Hyderabad, India - International Congress of Mathematicians (ICM), Hyderabad : India (2010)

## Combinatorial group theory and public key cryptography

Tipo: Artigo de Revista Científica
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After some excitement generated by recently suggested public key exchange protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al., it is a prevalent opinion now that the conjugacy search problem is unlikely to provide sufficient level of security if a braid group is used as the platform. In this paper we address the following questions: (1) whether choosing a different group, or a class of groups, can remedy the situation; (2) whether some other "hard" problem from combinatorial group theory can be used, instead of the conjugacy search problem, in a public key exchange protocol. Another question that we address here, although somewhat vague, is likely to become a focus of the future research in public key cryptography based on symbolic computation: (3) whether one can efficiently disguise an element of a given group (or a semigroup) by using defining relations.; Comment: 12 pages

## Aspects of Nonabelian Group Based Cryptography: A Survey and Open Problems

Fine, Benjamin; Habeeb, Maggie; Kahrobaei, Delaram; Rosenberger, Gerhard
Tipo: Artigo de Revista Científica
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Most common public key cryptosystems and public key exchange protocols presently in use, such as the RSA algorithm, Diffie-Hellman, and elliptic curve methods are number theory based and hence depend on the structure of abelian groups. The strength of computing machinery has made these techniques theoretically susceptible to attack and hence recently there has been an active line of research to develop cryptosystems and key exchange protocols using noncommutative cryptographic platforms. This line of investigation has been given the broad title of noncommutative algebraic cryptography. This was initiated by two public key protocols that used the braid groups, one by Ko, Lee et.al.and one by Anshel, Anshel and Goldfeld. The study of these protocols and the group theory surrounding them has had a large effect on research in infinite group theory. In this paper we survey these noncommutative group based methods and discuss several ideas in abstract infinite group theory that have arisen from them. We then present a set of open problems.

## Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version)

Mazurov, V. D.; Khukhro, E. I.
Tipo: Artigo de Revista Científica
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This is a collection of open problems in Group Theory proposed by more than 300 mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965, now also in English. This is the 18th edition, which contains 120 new problems and a number of comments on about 1000 problems from the previous editions.; Comment: several new solutions and references have been added, as well as a few corrections; in particular, the corrected version of 2.2 in Archive

## Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; Yan, C. H.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)$\times$(center) is an integer vector. This series of papers explain such properties. A {\em Descartes configuration} is a set of four mutually tangent circles with disjoint interiors. We describe the space of all Descartes configurations using a coordinate system $\sM_\DD$ consisting of those $4 \times 4$ real matrices $\bW$ with $\bW^T \bQ_{D} \bW = \bQ_{W}$ where $\bQ_D$ is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 -{1/2}(x_1 +x_2 +x_3 + x_4)^2$ and $\bQ_W$ of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. There are natural group actions on the parameter space $\sM_\DD$. We observe that the Descartes configurations in each Apollonian packing form an orbit under a certain finitely generated discrete group, the {\em Apollonian group}. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups...

## Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings

Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; Yan, C. H.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.69%
Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. It observed there exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group, the super-Apollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object we call a super-packing. The circles in a super-packing never cross each other but are nested to an arbitrary depth. Certain Apollonian packings and super-packings are strongly integral in the sense that the curvatures of all circles are integral and the curvature$\times$centers of all circles are integral. We show that (up to scale) there are exactly 8 different (geometric) strongly integral super-packings, and that each contains a copy of every integral Apollonian circle packing (also up to scale). We show that the super-Apollonian group has finite volume in the group of all automorphisms of the parameter space of Descartes configurations...

## Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; Yan, C. H.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.69%
This paper gives $n$-dimensional analogues of the Apollonian circle packings in parts I and II. We work in the space $\sM_{\dd}^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, ACC-coordinates, as those $(n+2) \times (n+2)$ real matrices $\bW$ with $\bW^T \bQ_{D,n} \bW = \bQ_{W,n}$ where $Q_{D,n} = x_1^2 +... + x_{n+2}^2 - \frac{1}{n}(x_1 +... + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + ... + 2x_{n+2}^2$, and $\bQ_{D,n}$ and $\bQ_{W,n}$ are their corresponding symmetric matrices. There are natural actions on the parameter space $\sM_{\dd}^n$. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set $S$ depending on the dimension. We show that the the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group...

## Search and witness problems in group theory

Tipo: Artigo de Revista Científica
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Decision problems are problems of the following nature: given a property P and an object O, find out whether or not the object O has the property P. On the other hand, witness problems are: given a property P and an object O with the property P, find a proof of the fact that O indeed has the property P. On the third hand(?!), search problems are of the following nature: given a property P and an object O with the property P, find something "material" establishing the property P; for example, given two conjugate elements of a group, find a conjugator. In this survey our focus is on various search problems in group theory, including the word search problem, the subgroup membership search problem, the conjugacy search problem, and others.

## Torsion-free crystallographic groups with indecomposable holonomy group

Bovdi, V. A.; Gudivok, P. M.; Rudko, V. P.
Tipo: Artigo de Revista Científica
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Let K be a principal ideal domain, G a finite group, and M a KG-module which as K-module is free of finite rank, and on which $G$ acts faithfully. A generalized crystallographic group (introduced by the authors in volume 5 of Journal of Group Theory) is a group $\frak C$ which has a normal subgroup isomorphic to M with quotient G, such that conjugation in $\frak C$ gives the same action of G on M that we started with. (When $K=\Bbb Z$, these are just the classical crystallographic groups.) The K-free rank of M is said to be the dimension of $\frak C$, the holonomy group of $\frak C$ is G, and $\frak C$ is called indecomposable if M is an indecomposable KG-module. Let K be either $\Bbb Z$, or its localization $\Bbb Z_{(p)}$ at the prime p, or the ring $\Bbb Z_p$ of p-adic integers, and consider indecomposable torsionfree generalized crystallographic groups whose holonomy group is noncyclic of order p^2. In Theorem 2, we prove that (for any given p) the dimensions of these groups are not bounded. For $K=\Bbb Z$, we show in Theorem 3 that there are infinitely many non-isomorphic indecomposable torsionfree crystallographic groups with holonomy group the alternating group of degree 4. In Theorem 1, we look at a cyclic G whose order |G| satisfies the following condition: for all prime divisors p of |G|...

## Survey on geometric group theory

Lueck, Wolfgang
Tipo: Artigo de Revista Científica
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This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research. The sections are: classical examples, basics about quasiisometry,properties and invariants of groups invariant under quasiisometry, rigidity, hyperbolic spaces and CAT(k)-spaces, the boundary of a hyperbolic space, hyperbolic groups, CAT(0)-groups, classifying spaces for proper actions, measurable group theory, some open problems.; Comment: 28 pages. Following the two detailed referee reports we have improved the exposition and corrected typos. The paper will appear in the Muenster Journal for Mathematics

## The classification of p-compact groups and homotopical group theory

Grodal, Jesper
Tipo: Artigo de Revista Científica
We give a precise definition of generic-case complexity'' and show that for a very large class of finitely generated groups the classical decision problems of group theory - the word, conjugacy and membership problems - all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs.; Comment: Revised version