13 research outputs found
Todd-Coxeter methods for inverse monoids
Let P be the inverse monoid presentation (X|U) for the inverse monoid M, let π be the set of generators for a right congruence on M and let u Є M. Using the work of J. Stephen [15], the current work demonstrates a coset enumeration technique for the R-class Rᵤ similar to the coset enumeration algorithm developed by J. A. Todd and H. S. M. Coxeter for groups. Furthermore it is demonstrated how to test whether Rᵤ = Rᵥ, for u, v Є M and so a technique for enumerating inverse monoids is described. This technique is generalised to enumerate the H-classes of M. The algorithms have been implemented in GAP 3.4.4 [25], and have been used to analyse some examples given in Chapter 6. The thesis concludes by a related discussion of normal forms and automaticity of free inverse semigroups
Identities and bases in plactic, hypoplactic, sylvester, and related monoids
The ubiquitous plactic monoid, also known as the monoid of Young tableaux, has deep
connections to several areas of mathematics, in particular, to the theory of symmetric
functions. An active research topic is the identities satisfied by the plactic monoids of
finite rank. It is known that there is no “global" identity satisfied by the plactic monoid
of every rank. In contrast, monoids related to the plactic monoid, such as the hypoplactic
monoid (the monoid of quasi-ribbon tableaux), sylvester monoid (the monoid of binary
search trees) and Baxter monoid (the monoid of pairs of twin binary search trees), satisfy
global identities, and the shortest identities have been characterized.
In this thesis, we present new results on the identities satisfied by the hypoplactic,
sylvester, #-sylvester and Baxter monoids. We show how to embed these monoids, of any
rank strictly greater than 2, into a direct product of copies of the corresponding monoid
of rank 2. This confirms that all monoids of the same family, of rank greater than or equal
to 2, satisfy exactly the same identities. We then give a complete characterization of those
identities, thus showing that the identity checking problems of these monoids are in the
complexity class P, and prove that the varieties generated by these monoids have finite
axiomatic rank, by giving a finite basis for them. We also give a subdirect representation
ofmultihomogeneous monoids by finite subdirectly irreducible Rees factor monoids, thus
showing that they are residually finite.O ubíquo monóide plático, também conhecido como o monóide dos diagramas de Young,
tem ligações profundas a várias áreas de Matemática, em particular à teoria das funções
simétricas. Um tópico de pesquisa ativo é o das identidades satisfeitas pelos monóides
pláticos de característica finita. Sabe-se que não existe nenhuma identidade “global” satisfeita
pelos monóides pláticos de cada característica. Em contraste, sabe-se que monóides
ligados ao monóide plático, como o monóide hipoplático (o monóide dos diagramas quasifita),
o monóide silvestre (o monóide de árvores de busca binárias) e o monóide de Baxter
(o monóide de pares de árvores de busca binária gémeas), satisfazem identidades globais,
e as identidades mais curtas já foram caracterizadas.
Nesta tese, apresentamos novos resultados acerca das identidades satisfeitas pelos monóides
hipopláticos, silvestres, silvestres-# e de Baxter. Mostramos como mergulhar estes
monóides, de característica estritamente maior que 2, num produto direto de cópias do
monóide correspondente de característica 2. Confirmamos assim que todos os monóides
da mesma família, de característica maior ou igual a 2, satisfazem exatamente as mesmas
identidades. A seguir, damos uma caracterização completa dessas identidades, mostrando
assim que os problemas de verificação de identidades destes monóides estão na classe de
complexidade P, e provamos que as variedades geradas por estes monóides têm característica
axiomática finita, ao apresentar uma base finita para elas. Também damos uma
representação subdireta de monóides multihomogéneos por monóides fatores de Rees
finitos e subdiretamente irredutíveis, mostrando assim que são residualmente finitos
Algebraic hierarchical decomposition of finite state automata : a computational approach
The theory of algebraic hierarchical decomposition of finite state automata
is an important and well developed branch of theoretical computer science
(Krohn-Rhodes Theory). Beyond this it gives a general model for some
important aspects of our cognitive capabilities and also provides possible
means for constructing artificial cognitive systems: a Krohn-Rhodes decomposition
may serve as a formal model of understanding since we comprehend
the world around us in terms of hierarchical representations. In order to
investigate formal models of understanding using this approach, we need
efficient tools but despite the significance of the theory there has been no
computational implementation until this work.
Here the main aim was to open up the vast space of these decompositions
by developing a computational toolkit and to make the initial steps of the
exploration. Two different decomposition methods were implemented: the
VuT and the holonomy decomposition. Since the holonomy method, unlike
the VUT method, gives decompositions of reasonable lengths, it was chosen
for a more detailed study.
In studying the holonomy decomposition our main focus is to develop
techniques which enable us to calculate the decompositions efficiently, since
eventually we would like to apply the decompositions for real-world problems.
As the most crucial part is finding the the group components we
present several different ways for solving this problem. Then we investigate
actual decompositions generated by the holonomy method: automata with
some spatial structure illustrating the core structure of the holonomy decomposition,
cases for showing interesting properties of the decomposition
(length of the decomposition, number of states of a component), and the
decomposition of finite residue class rings of integers modulo n.
Finally we analyse the applicability of the holonomy decompositions as
formal theories of understanding, and delineate the directions for further
research
Discrete Clifford analysis
Doutoramento em MatemáticaEsta tese estuda os fundamentos de uma teoria discreta de funções em dimensões superiores usando a linguagem das Álgebras de Clifford. Esta abordagem combina as ideias do Cálculo Umbral e Formas Diferenciais. O potencial desta abordagem assenta essencialmente da osmose entre ambas as linguagens. Isto permitiu a construção de operadores de entrelaçamento entre estruturas contínuas e discretas, transferindo resultados conhecidos do contínuo para o discreto. Adicionalmente, isto resultou numa transcrição mimética de bases de polinómios, funções geradoras, Decomposição de Fischer, Lema de Poincaré, Teorema de Stokes, fórmula de Cauchy e fórmula de Borel-Pompeiu. Esta teoria também inclui a descrição dos homólogos discretos de formas diferenciais, campos vectores e integração discreta. De facto, a construção resultante de formas diferenciais, campos vectores e integração discreta em termos de coordenadas baricêntricas conduz à correspondência entre a teoria de Diferenças Finitas e a teoria de Elementos Finitos, dando um núcleo de aplicações desta abordagem promissora em análise numérica. Algumas ideias preliminares deste ponto de vista foram apresentadas nesta tese. Também foram apresentados resultados preliminares na teoria discreta de funções em complexos envolvendo simplexes. Algumas ligações com Combinatória e Mecânica Quântica foram também apresentadas ao longo desta tese.This thesis studies the fundamentals of a higher dimensional discrete function theory using the Clifford Algebra setting. This approach combines the ideas of Umbral Calculus and Differential Forms. Its powerful rests mostly on the interplay between both languages. This allowed the construction of intertwining operators between continuous and discrete structures, lifting the well known results from continuum to discrete. Furthermore, this resulted in a mimetic transcription of basis polynomial, generating functions, Fischer Decomposition, Poincaré and dual-Poincaré lemmata, Stokes theorem and Cauchy’s formula. This theory also includes the description discrete counterparts of differential forms, vector-fields and discrete integration. Indeed the resulted construction of discrete differential forms, discrete vector-fields and discrete integration in terms of barycentric coordinates leads to the correspondence between the theory of Finite Differences and the theory of Finite Elements, which gives a core of promising applications of this approach in numerical analysis. Some preliminary ideas on this point of view were presented in this thesis. We also developed some preliminary results in the theory of discrete monogenic functions on simplicial complexes. Some connections with Combinatorics and Quantum Mechanics were also presented along this thesis
Wavelets on Lie groups and homogeneous spaces
Within the past decades, wavelets and associated wavelet transforms have been intensively investigated in both applied and pure mathematics. They and the related multi-scale analysis provide essential tools to describe, analyse and modify signals, images or, in rather abstract concepts, functions, function spaces and associated operators. We introduce the concept of diffusive wavelets where the dilation operator is provided by an evolution like process that comes from an approximate identity. The translation operator is naturally defined by a regular representation of the Lie group where we want to construct wavelets. For compact Lie groups the theory can be formulated in a very elegant way and also for homogeneous spaces of those groups we formulate the theory in the theory of non-commutative harmonic analysis. Explicit realisation are given for the Rotation group SO(3), the k-Torus, the Spin group and the n-sphere as homogeneous space. As non compact example we discuss diffusive wavelets on the Heisenberg group, where the construction succeeds thanks to existence of the Plancherel measure for this group. The last chapter is devoted to the Radon transform on SO(3), where the application on diffusive wavelets can be used for its inversion. The discussion of a variational spline approach provides criteria for the choice of points for measurements in concrete applications
Wavelets on Lie groups and homogeneous spaces
Within the past decades, wavelets and associated wavelet transforms have been intensively investigated in both applied and pure mathematics. They and the related multi-scale analysis provide essential tools to describe, analyse and modify signals, images or, in rather abstract concepts, functions, function spaces and associated operators. We introduce the concept of diffusive wavelets where the dilation operator is provided by an evolution like process that comes from an approximate identity. The translation operator is naturally defined by a regular representation of the Lie group where we want to construct wavelets. For compact Lie groups the theory can be formulated in a very elegant way and also for homogeneous spaces of those groups we formulate the theory in the theory of non-commutative harmonic analysis. Explicit realisation are given for the Rotation group SO(3), the k-Torus, the Spin group and the n-sphere as homogeneous space. As non compact example we discuss diffusive wavelets on the Heisenberg group, where the construction succeeds thanks to existence of the Plancherel measure for this group. The last chapter is devoted to the Radon transform on SO(3), where the application on diffusive wavelets can be used for its inversion. The discussion of a variational spline approach provides criteria for the choice of points for measurements in concrete applications
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Relating Thompson's group V to graphs of groups and Hecke algebras
This thesis is in two main sections, both of which feature Thompson's group , relating it to classical constructions involving automorphism groups on trees or to representations of symmetric groups. In the first section, we take to be a graph of groups, which acts on its universal cover, the Bass-Serre tree, by tree automorphisms. Brownlowe, Mundey, Pask, Spielberg and Thomas constructed a -algebra for a graph of groups, writtten , which bears many similarities to the -algebra of a directed graph . Inspired by the fact that directed graph -algebras have algebraic analogues in Leavitt path algebras , we define a Leavitt graph-of-groups algebra for . We extend Leavitt path algebra results to , including uniqueness theorems describing homomorphisms out of , and establish a wider context for the algebras by showing they are Steinberg algebras of a particular \'{e}tale groupoid. Finally we show that certain unitaries in form a group we can understand as a variant of Thompson's , combining features of both Nekrashevych-R\"{o}ver groups and Matui's topological full groups of one-sided shifts. We prove finiteness and simplicity results for these Thompson variants. The latter section of this thesis turns to representation theory. We briefly state some results about representations of (due to Dudko and Grigorchuk) which we generalize to the new family of Thompson groups, including a discussion of representations of finite factor type and Koopman representations. Then, we describe how one would try to construct a Hecke algebra for , built from copies of the Iwahori-Hecke algebra of in a way inspired by how can be constructed from copies of the symmetric group. We survey attempts to construct this and demonstrate what we believe to be the closest possible analogue to the theory. We discuss how this construction could prove useful for understanding further representation theory.I was sponsored by the research council EPSRC for the first 3.5 years of my thesis