2024-03-29T07:22:02Zhttps://research-repository.st-andrews.ac.uk/oai/requestoai:research-repository.st-andrews.ac.uk:10023/28432019-12-05T03:03:15Zcom_10023_94com_10023_28col_10023_99
Presentations and efficiency of semigroups
Ayik, Hayrullah
Campbell, C. M. (Colin Matthew)
O'Connor, John J. (John Joseph)
QA171.S3A9
Group theory
Representations of groups
In this thesis we consider in detail the following two problems for semigroups:
(i) When are semigroups finitely generated and presented?
(ii) Which families of semigroups can be efficiently presented?
We also consider some other finiteness conditions for semigroups, homology of
semigroups and wreath product of groups.
In Chapter 2 we investigate finite presentability and some other finiteness conditions
for the O-direct union of semigroups with zero. In Chapter 3 we investigate
finite generation and presentability of Rees matrix semigroups over semigroups.
We find necessary and sufficient conditions for finite generation and presentability.
In Chapter 4 we investigate some other finiteness conditions for Rees matrix
semigroups.
In Chapter 5 we consider groups as semigroups and investigate their semigroup
efficiency. In Chapter 6 we look at "proper" semigroups, that is semigroups
that are not groups. We first give examples of efficient and inefficient "proper"
semigroups by computing their homology and finding their minimal presentations.
In Chapter 7 we compute the second homology of finite simple semigroups and
find a "small" presentation for them. If that "small" presentation has a special
relation, we prove that finite simple semigroups are efficient. Finally, in Chapter
8, we investigate the efficiency of wreath products of finite groups as groups and
as semigroups. We give more examples of efficient groups and inefficient groups.
1998
2012-06-22T15:36:12Z
2012-06-22T15:36:12Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/2843
en
application/pdf
201
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/33582019-07-01T10:16:26Zcom_10023_94com_10023_28col_10023_99
Endomorphisms of Fraïssé limits and automorphism groups of algebraically closed relational structures
McPhee, Jillian Dawn
Quick, M. R. (Martyn R.)
Mitchell, James David
Fraïssé limit
Relational structure
Algebraically closed
Green's relations
H-class
D-class
J-class
Semigroup
Maximal subgroups
Random graph
Random bipartite graph
Random tournament
Henson's graphs
Rationals
Endomorphism semigroup
Endomorphism monoid
Retracts
QA182.M7
Semigroups of endomorphisms
Semigroups
Maximal subgroups
Let Ω be the Fraïssé limit of a class of relational structures. We seek to
answer the following semigroup theoretic question about Ω. What are the group H-classes, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for which we answer this question include the random graph
R, the random directed graph D, the random tournament T, the random bipartite graph B, Henson's graphs G[subscript n] (for n greater or equal to 3) and the total order Q. The maximal subgroups of End(Ω) are closely connected to the automorphism groups of the relational structures induced by the images of idempotents from End(Ω). It has been shown that the relational structure induced by the image of an idempotent from End(Ω) is algebraically closed. Accordingly, we investigate which groups can be realised as the automorphism group of an algebraically closed relational structure in order to
determine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph and Ω = R,D,B,
then there exist 2[superscript aleph-naught] maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsets
of Q which are the image of an idempotent from End(Q). We call these subsets retracts of Q and show that if Ω is a total order and f is an embedding of Ω into Q such that im f is a retract of Q, then there exist 2[superscript aleph-naught] maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Zⁿ for some natural number n. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2[superscript aleph-naught] regular D-classes of End(Ω) and when Ω = R,D,B there exist 2[superscript aleph-naught] J-classes of End(Ω). Additionally we show
that if Ω = R,D then all regular D-classes contain 2[superscript aleph-naught] group H-classes. On the other hand, we show that when
Ω = B,Q there exist regular D-classes
which contain countably many group H-classes.
2012-11-30
2013-02-18T11:22:16Z
2013-02-18T11:22:16Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/3358
en
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
http://creativecommons.org/licenses/by-nc-nd/3.0/
application/pdf
162
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/17102019-07-01T10:14:46Zcom_10023_94com_10023_28col_10023_99
A commutative noncommutative fractal geometry
Samuel, Anthony
Falconer, K. J.
Stratmann, Bernd
Fractal geometry
Multifractal analysis
Symbolic dynamics
Ergodic theory
Thermodynamic formalism
Renewal theory
Noncommutative geometry
Spectral triples
QA614.86S2
Fractals
Symbolic dynamics
Ergodic theory
In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained.
Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S.
Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}}^{{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.
2010
2011-03-24T14:49:12Z
2011-03-24T14:49:12Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/1710
en
Creative Commons Attribution 3.0 Unported
http://creativecommons.org/licenses/by/3.0/
application/pdf
118
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/38692021-03-10T13:08:06Zcom_10023_94com_10023_28col_10023_99
Dimension theory and fractal constructions based on self-affine carpets
Fraser, Jonathan M.
Falconer, K. J.
Olsen, Lars
Engineering and Physical Sciences Research Council (EPSRC)
Fractal
Iterated function system
Self-affine
Dimension theory
Hausdorff dimension
Box dimension
Random fractal
Inhomogeneous attractor
QA614.86F82
Fractals
Dimension theory (Topology)
Hausdorff measures
Attractors (Mathematics)
The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student.
The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012.
In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'.
In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions.
In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation.
Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.
2013-11-29
2013-07-24T14:32:08Z
2013-07-24T14:32:08Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/3869
en
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
http://creativecommons.org/licenses/by-nc-nd/3.0/
application/pdf
vi, 115 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/76292019-04-01T08:34:51Zcom_10023_94com_10023_28col_10023_99
On generators, relations and D-simplicity of direct products, Byleen extensions, and other semigroup constructions
Baynes, Samuel
Ruškuc, Nik
Semigroups
D-simple
Bisimple
Byleen extension
Relative rank
Embedding
QA182.B2
Semigroups
Group theory--Generators
Group theory--Relations
Embeddings (Mathematics)
In this thesis we study two different topics, both in the context of semigroup constructions. The first is the investigation of an embedding problem, specifically the problem of whether it is possible to embed any given finitely presentable semigroup into a D-simple finitely presentable semigroup. We consider some well-known semigroup constructions, investigating their properties to determine whether they might prove useful for finding a solution to our problem. We carry out a more detailed study into a more complicated semigroup construction, the Byleen extension, which has been used to solve several other embedding problems. We prove several results regarding the structure of this extension, finding necessary and sufficient conditions for an extension to be D-simple and a very strong necessary condition for an extension to be finitely presentable.
The second topic covered in this thesis is relative rank, specifically the sequence obtained by taking the rank of incremental direct powers of a given semigroup modulo the diagonal subsemigroup. We investigate the relative rank sequences of infinite Cartesian products of groups and of semigroups. We characterise all semigroups for which the relative rank sequence of an infinite Cartesian product is finite, and show that if the sequence is finite then it is bounded above by a logarithmic function. We will find sufficient conditions for the relative rank sequence of an infinite Cartesian product to be logarithmic, and sufficient conditions for it to be constant. Chapter 4 ends with the introduction of a new topic, relative presentability, which follows naturally from the topic of relative rank.
2015-11-30
2015-10-08T09:19:34Z
2015-10-08T09:19:34Z
Thesis
Doctoral
PhD Doctor of Philosophy
uk.bl.ethos.667526
http://hdl.handle.net/10023/7629
en
application/pdf
vi, 146 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/4312019-07-01T10:18:51Zcom_10023_94com_10023_28col_10023_99
Simplicity in relational structures and its application to permutation classes
Brignall, Robert
Ruškuc, Nik
Engineering and Physical Sciences Research Council (EPSRC)
Combinatorics
Finite mathematics
QA165.B8
Combinatorial analysis
The simple relational structures form the units, or atoms, upon which all other relational structures are constructed by means of the substitution decomposition. This decomposition appears to have first been introduced in 1953 in a talk by Fraïssé, though it did not appear in an article until a paper by Gallai in 1967. It has subsequently been frequently rediscovered from a wide variety of perspectives, ranging from game theory to combinatorial optimization.
Of all the relational structures - a set which also includes graphs, tournaments and posets - permutations are receiving ever increasing amounts of attention. A simple permutation is one that maps every nontrivial contiguous set of indices to a set of indices that is never contiguous. Simple permutations and intervals of permutations are important in biomathematics, while permutation classes - downsets under the pattern containment order - arise naturally in settings ranging from sorting to algebraic geometry.
We begin by studying simple permutations themselves, though always aim to establish this theory within the broader context of relational structures. We first develop the technology of "pin sequences", and prove that every sufficiently long simple permutation must contain either a long horizontal or parallel alternation, or a long pin sequence. This gives rise to a simpler unavoidable substructures result, namely that every sufficiently long simple permutation contains a long alternation or oscillation.
ErdÅ s, Fried, Hajnal and Milner showed in 1972 that every tournament could be extended to a simple tournament by adding at most two additional points. We prove analogous results for permutations, graphs, and posets, noting that in these three cases we may need to extend a structure by adding (n+1)/2 points in the case of permutations and posets, and logâ (n+1) points in the graph case.
The importance of simple permutations in permutation classes has been well established in recent years. We extend this knowledge in a variety of ways, first by showing that, in a permutation class containing only finitely many simple permutations, every subset defined by properties belonging to a finite "query-complete set" is enumerated by an algebraic generating function. Such properties include being an even or alternating permutation, or avoiding generalised (blocked or barred) permutations. We further indicate that membership of a permutation class containing only finitely many simple permutations can be computed in linear time.
Using the decomposition of simple permutations, we establish, by representing pin sequences as a language over an eight-letter alphabet, that it is decidable if a permutation class given by a finite basis contains only finitely many simple permutations. We also discuss possible approaches to the same question for other relational structures, in particular the difficulties that arise for graphs. The pin sequence technology provides a further result relating to the wreath product of two permutation classes, namely that C â D is finitely based whenever D does not admit arbitrarily long pin sequences. As a partial converse, we also exhibit a number of explicit examples of wreath products that are not finitely based.
2007-11-30
2008-03-04T10:09:26Z
2008-03-04T10:09:26Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/431
en
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
http://creativecommons.org/licenses/by-nc-nd/3.0/
1259089 bytes
application/pdf
application/pdf
204
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/159642019-04-01T08:34:52Zcom_10023_94com_10023_28col_10023_99
Some group presentations with few defining relations
Gill, David Michael
Campbell, C. M. (Colin Matthew)
QA171.G5
Presentations of groups (Mathematics)
We consider two classes of groups with two generators and three relations. One class has a similar presentation to groups considered in the paper by C.M. Campbell and R.M. Thomas, ‘On (2,n)-Groups related to Fibonacci Groups’, (Israel J. Math., 58), with one generator of order three instead of order two . We attempt to find the order of these groups and in one case find groups which have the alternating group A₅ as a subgroup of index equal to the order of the second generator of the group. Questions remain as to the order of some of the other groups.
The second class has already been considered in the paper 'Some families of finite groups having two generators and two relations' by C.M. Campbell , H.S.M. Coxeter and E.F. Robertson, (Proc. R. Soc. London A. 357, 423-438 (1977)), in which a formula for the orders of these groups was found. We attempt to find simpler formulae based on recurrence relations for subclasses and write Maple programs to enable us to do this. We also find a formula, again based on recurrence relations, for an upper bound for the orders of the groups.
1990
2018-09-04T13:53:51Z
2018-09-04T13:53:51Z
Thesis
Masters
MSc Master of Science
http://hdl.handle.net/10023/15964
en
application/pdf
81 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/36712019-07-01T10:15:44Zcom_10023_94com_10023_28col_10023_99
Adventures in applying iteration lemmas
Pfeiffer, Markus Johannes
Ruškuc, Nik
Neunhöffer, Max
University of St Andrews. School of Mathematics and Statistics
Engineering and Physical Sciences Research Council (EPSRC)
Automata
Computation
Decidability
Semigroups
QA182.P44
Semigroups
Word problems (Mathematics)
Iterative methods (Mathematics)
The word problem of a finitely generated group is commonly defined to be a formal language over a finite generating set.
The class of finite groups has been characterised as the class of finitely generated groups that have word problem decidable by a finite state automaton.
We give a natural generalisation of the notion of word problem from finitely generated groups to finitely generated semigroups by considering relations of strings. We characterise the class of finite semigroups by the class of finitely generated semigroups whose word problem is decidable by finite state automata.
We then examine the class of semigroups with word problem decidable by asynchronous two tape finite state automata. Algebraic properties of semigroups in this class are considered, towards an algebraic characterisation.
We take the next natural step to further extend the classes of semigroups under consideration to semigroups that have word problem decidable by a finite collection of asynchronous automata working independently.
A central tool used in the derivation of structural results are so-called iteration lemmas.
We define a hierarchy of the considered classes of semigroups and connect our original results with previous research.
2013-06-28
2013-06-12T14:33:42Z
2013-06-12T14:33:42Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/3671
en
Creative Commons Attribution 3.0 Unported
http://creativecommons.org/licenses/by/3.0/
application/pdf
221
University of St Andrews
The University of St Andrews
School of Mathematics and Statistics
oai:research-repository.st-andrews.ac.uk:10023/86452019-04-01T08:34:53Zcom_10023_94com_10023_28col_10023_99
Graph automatic semigroups
Carey, Rachael Marie
Ruškuc, Nik
Engineering and Physical Sciences Research Council (EPSRC)
Semigroups
Regular languages
QA182.C7
Semigroups
Cayley graphs
In this thesis we examine properties and constructions of graph automatic semigroups, a generalisation of both automatic semigroups and finitely generated FA-presentable semigroups.
We consider the properties of graph automatic semigroups, showing that they are independent of the choice of generating set, have decidable word problem, and that if we have a graph automatic structure for a semigroup then we can find one with uniqueness.
Semigroup constructions and their effect on graph automaticity are considered. We show that finitely generated direct products, free products, finitely generated Rees matrix semigroup constructions, zero unions, and ordinal sums all preserve unary graph automaticity, and examine when the converse also holds. We also demonstrate situations where semidirect products, Bruck-Reilly extensions, and semilattice constructions preserve graph automaticity, and consider the conditions we may impose on such constructions in order to ensure that graph automaticity is preserved.
Unary graph automatic semigroups, that is semigroups which have graph automatic structures over a single letter alphabet, are also examined. We consider the form of an automaton recognising multiplication by generators in such a semigroup, and use this to demonstrate various properties of unary graph automatic semigroups. We show that infinite periodic semigroups are not unary graph automatic, and show that we may always find a uniform set of normal forms for a unary graph automatic semigroup. We also determine some necessary conditions for a semigroup to be unary graph automatic, and use this to provide examples of semigroups which are not unary graph automatic.
Finally we consider semigroup constructions for unary graph automatic semigroups. We show that the free product of two semigroups is unary graph automatic if and only if both semigroups are trivial; that direct products do not always preserve unary graph automaticity; and that Bruck-Reilly extensions are never unary graph automatic.
2016-06-24
2016-04-20T09:00:38Z
2016-04-20T09:00:38Z
Thesis
Doctoral
PhD Doctor of Philosophy
uk.bl.ethos.685059
http://hdl.handle.net/10023/8645
en
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
application/pdf
xiv, 158 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/7652019-07-01T10:12:09Zcom_10023_94com_10023_28col_10023_99
Intersection problems in combinatorics
Brunk, Fiona
Ruškuc, Nik
Huczynska, Sophie
t-intersecting
Erdős-Ko-Rado
Injections
Posets
Intersecting families
Fixing
Saturation
Extremal combinatorics
QA164.B88
Combinatorial analysis
Intersection theory
With the publication of the famous Erdős-Ko-Rado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is t-intersecting if any two of its elements mutually t-intersect, where the latter concept needs to be specified separately in each instance. This thesis is split into two parts; the first is concerned with intersecting injections while the second investigates intersecting posets.
We classify maximum 1-intersecting families of injections from {1, ..., k} to {1, ..., n}, a generalisation of the corresponding result on permutations from the early 2000s. Moreover, we obtain classifications in the general t>1 case for different parameter limits:
if n is large in terms of k and t, then the so-called fix-families, consisting of all injections which map some fixed set of t points to the same image points, are the only t-intersecting injection families of maximal size. By way of contrast, fixing the differences k-t and n-k while increasing k leads to optimal families which are equivalent to one of the so-called saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=|_ (k-t)/2 _|. Furthermore we demonstrate that, among injection families with t-intersecting and left-compressed fixed point sets, for some value of r the saturation family has maximal size .
The concept that two posets intersect if they share a comparison is new. We begin by classifying maximum intersecting families in several isomorphism classes of posets which are linear, or almost linear. Then we study the union of the almost linear classes, and derive a bound for an intersecting family by adapting Katona's elegant cycle method to posets. The thesis ends with an investigation of the intersection structure of poset classes whose elements are close to the antichain.
The overarching theme of this thesis is fixing versus saturation: we compare the sizes and structures of intersecting families obtained from these two distinct principles in the context of various classes of combinatorial objects.
2009-11-30
2009-11-02T15:55:10Z
2009-11-02T15:55:10Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/765
en
Creative Commons Attribution 3.0 Unported
http://creativecommons.org/licenses/by/3.0/
application/pdf
187
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/6822019-07-01T10:06:39Zcom_10023_94com_10023_28col_10023_99
Inhomogeneous self-similar sets and measures
Snigireva, Nina
Olsen, Lars
Inhomogeneous self-similar sets
Inhomogeneous self-similar measures
Hausdorff dimension
Box dimension
Packing dimension
L^q-spectra
Renyi dimensions
Multifractal spectra
Fourier transforms
QA248.S65
Set theory
Hausdorff measures
The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and
measures. In particular, we show that these sets and measures
are natural generalizations of the well known self-similar sets and
measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal
dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary)
self-similar sets. In the third chapter we investigate the L^{q}
spectra and the Renyi dimensions of inhomogeneous self-similar
measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case.
Namely, we show that inhomogeneous self-similar measures may
have phase transitions which is in sharp contrast to the
behaviour of the
L^{q} spectra
of (ordinary) self-similar
measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra
of inhomogeneous self-similar measures. We show that
the multifractal spectra
of
inhomogeneous self-similar
measures
may be non-concave which is again in sharp contrast to the
behaviour of the
multifractal spectra
of (ordinary) self-similar
measures satisfying the Open Set Condition. Then we present a number of
applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of
inhomogeneous self-similar measures and again we present a
number of applications of our results, in particular to non-linear
self-similar measures.
2008
2009-05-01T13:17:19Z
2009-05-01T13:17:19Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/682
en
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
http://creativecommons.org/licenses/by-nc-nd/3.0/
1056016 bytes
application/pdf
application/pdf
111
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/99132019-04-01T08:34:54Zcom_10023_94com_10023_28col_10023_99
Generating "large" subgroups and subsemigroups
Jonušas, Julius
Mitchell, James David
QA174.2J76
Group theory
Semigroups
In this thesis we will be exclusively considering uncountable groups and semigroups.
Roughly speaking the underlying problem is to find “large” subgroups
(or subsemigroups) of the object in question, where we consider three different
notions of “largeness”:
(i) We classify all the subsemigroups of the set of all mapping from a countable
set back to itself which contains a specific uncountable subsemigroup;
(ii) We investigate topological “largeness”, in particular subgroups which are
finitely generated and dense;
(iii) We investigate if it is possible to find an integer r such that any countable
collection of elements belongs to some r-generated subsemigroup, and more
precisely can these elements be obtained by multiplying the generators in a
prescribed fashion.
2016
2016-12-01T14:11:19Z
2016-12-01T14:11:19Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/9913
en
application/pdf
vi, 168 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/35782019-07-01T10:13:17Zcom_10023_94com_10023_28col_10023_99
Random generation and chief length of finite groups
Menezes, Nina E.
Quick, M. R. (Martyn R.)
Roney-Dougal, Colva Mary
Finite group theory
Random generation
Chief length
Probabilistic group theory
Almost simple groups
Permutation groups
Matrix groups
Computational group theory
Group theory
QA177.M4
Finite groups
Combinatorial probabilities
Group theory--Generators
Part I of this thesis studies P[subscript(G)](d), the probability of generating a nonabelian
simple group G with d randomly chosen elements, and extends this
idea to consider the conditional probability P[subscript(G,Soc(G))](d), the probability
of generating an almost simple group G by d randomly chosen elements,
given that they project onto a generating set of G/Soc(G). In particular
we show that for a 2-generated almost simple group, P[subscript(G,Soc(G))](2) 53≥90,
with equality if and only if G = A₆ or S₆. Furthermore P[subscript(G,Soc(G))](2) 9≥10
except for 30 almost simple groups G, and we specify this list and provide
exact values for P[subscript(G,Soc(G))](2) in these cases. We conclude Part I by showing
that for all almost simple groups P[subscript(G,Soc(G))](3)≥139/150.
In Part II we consider a related notion. Given a probability ε, we wish
to determine d[superscript(ε)] (G), the number of random elements needed to generate a finite group G with failure probabilty at most ε. A generalisation of a result
of Lubotzky bounds d[superscript(ε)](G) in terms of l(G), the chief length of G, and d(G),
the minimal number of generators needed to generate G. We obtain bounds
on the chief length of permutation groups in terms of the degree n, and
bounds on the chief length of completely reducible matrix groups in terms
of the dimension and field size. Combining these with existing bounds on
d(G), we obtain bounds on d[superscript(ε)] (G) for permutation groups and completely
reducible matrix groups.
2013
2013-05-30T15:34:00Z
2013-05-30T15:34:00Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/3578
en
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
http://creativecommons.org/licenses/by-nc-nd/3.0/
application/pdf
iv, 204
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/292902024-02-20T03:01:30Zcom_10023_94com_10023_28col_10023_99
Modern computational methods for finitely presented monoids
Tsalakou, Maria
Mitchell, James David
University of St Andrews. School of Mathematics and Statistics
Cyprus State Scholarship Foundation
Semigroup theory
Semigroups
Monoids
Computational algebra
Computational semigroup theory
In this thesis we are mainly interested in the development of practical algorithms for semigroups and monoids defined by finite presentations. Although in general nearly every problem about finitely presented semigroups is undecidable, many finitely presented semigroups and monoids of interest are more tractable. Semigroup and monoid presentations have been widely studied in the literature more or less since the inception of the field of semigroup theory. The aim of computational semigroup theory, of which
this thesis forms a part, is to develop algorithms and software tools for computing with semigroups, and on the applications of these tools to research problems.
In this thesis we develop the concept of words graphs, which form the basis for the work presented in the first half of the thesis. We describe an algorithm that computes one-sided congruences of finitely presented semigroups. This is the semigroup theoretic
analogue of an algorithm described by Sims for computing subgroups of small index in finitely presented groups. Furthermore, we focus on the Todd-Coxeter Algorithm, one of the most widely studied algorithms in computational semigroup theory. We describe
a more general version of the Todd-Coxeter Algorithm than the versions available in the literature for computing congruences of finitely presented semigroups.
The remaining part of this thesis is focused on a class of finitely presented monoids, called small overlap monoids. These are, in some sense, the generic finitely presented monoids. They have decidable word problem that can be solved in linear time. We
present the results related to the word problem and the combinatorial theory for small overlap monoids developed by Kambites. In addition, we discuss methods appearing in the literature for normal forms in small overlap monoids and we present a new practical
algorithm for computing normal forms.
2024-06-11
2024-02-19T15:08:08Z
2024-02-19T15:08:08Z
Thesis
Doctoral
PhD Doctor of Philosophy
https://hdl.handle.net/10023/29290
https://doi.org/10.17630/sta/780
en
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International
http://creativecommons.org/licenses/by-nc-sa/4.0/
application/pdf
application/zip
222
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/25292019-04-01T08:34:54Zcom_10023_94com_10023_28col_10023_99
Generation problems for finite groups
McDougall-Bagnall, Jonathan M.
Quick, M. R. (Martyn R.)
Group theory
Generating sets
Finite groups
Basis property
Frattini subgroup
QA177.M3
Finite groups
Frattini subgroups
It can be deduced from the Burnside Basis Theorem that if G is a finite p-group with d(G)=r then given any generating set A for G there exists a subset of A of size r that generates G. We have denoted this property B. A group is said to have the basis property if all subgroups have property B. This thesis is a study into the nature of these two properties. Note all groups are finite unless stated otherwise.
We begin this thesis by providing examples of groups with and without property B and several results on the structure of groups with property B, showing that under certain conditions property B is inherited by quotients. This culminates with a result which shows that groups with property B that can be expressed as direct products are exactly those arising from the Burnside Basis Theorem.
We also seek to create a class of groups which have property B. We provide a method for constructing groups with property B and trivial Frattini subgroup using finite fields. We then classify all groups G where the quotient of G by the Frattini subgroup is isomorphic to this construction. We finally note that groups arising from this construction do not in general have the basis property.
Finally we look at groups with the basis property. We prove that groups with the basis property are soluble and consist only of elements of prime-power order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.
2011-11-30
2012-03-29T11:48:28Z
2012-03-29T11:48:28Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/2529
en
application/pdf
101
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/268952023-08-18T20:52:14Zcom_10023_94com_10023_28col_10023_99
Diameters of graphs related to groups and base sizes of primitive groups
Freedman, Saul Daniel
Roney-Dougal, Colva Mary
Cameron, Peter J. (Peter Jephson)
EPSRC
University of St Andrews. School of Mathematics and Statistics
University of St Andrews. St Leonard's College
Intersection graph
Non-commuting, non-generating graph
Base size
Primitive groups
Subspace actions
Finite simple groups
Graphs defined on groups
QA166.185F8
Intersection graph theory
Finite simple groups
In this thesis, we study three problems. First, we determine new bounds for base sizes b(G,Ω) of primitive subspace actions of finite almost simple classical groups G. Such base sizes are useful statistics in computational group theory. We show that if the underlying set Ω consists of k-dimensional subspaces of the natural module V = F_q^n for G, then b(G,Ω) ≥ ⌈n/k⌉ + c, where c ∈ {-2,-1,0,1} depends on n, q, k and the type of G. If instead Ω consists of pairs {X,Y} of subspaces of V with k:=dim(X) < dim(Y), and G is generated by PGL(n,q) and the graph automorphism of PSL(n,q), then b(G,Ω) ≤ max{⌈n/k⌉,4}.
The second part of the thesis concerns the intersection graph Δ_G of a finite simple group G. This graph has vertices the nontrivial proper subgroups of G, and its edges are the pairs of subgroups that intersect nontrivially. We prove that Δ_G has diameter at most 5, and that a diameter of 5 is achieved only by the graphs of the baby monster group and certain unitary groups of odd prime dimension. This answers a question posed by Shen.
Finally, we study the non-commuting, non-generating graph Ξ(G) of a group G, where G/Z(G) is either finite or non-simple. This graph is closely related to the hierarchy of graphs introduced by Cameron. The graph's vertices are the non-central elements of G, and its edges are the pairs {x,y} such that ⟨x, y⟩ ≠ G and xy ≠ yx. We show that if Ξ(G) has an edge, then either the graph is connected with diameter at most 5; the graph has exactly two connected components, each of diameter 2; or the graph consists of isolated vertices and a component of diameter at most 4. In this last case, either the nontrivial component has diameter 2, or G/Z(G) is a non-simple insoluble primitive group with every proper quotient cyclic.
2022-11-29
2023-02-03T15:58:06Z
2023-02-03T15:58:06Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/26895
https://doi.org/10.17630/sta/254
EP/R014604/1
en
Diameters of graphs related to groups and base sizes of primitive groups - GAP and Magma code (thesis data)
Freedman, S. D., University of St Andrews, 16 Aug 2022.
DOI: https://doi.org/10.17630/56ceed97-0a86-4684-b0a9-e454c1a7440b
https://doi.org/10.17630/56ceed97-0a86-4684-b0a9-e454c1a7440b
Creative Commons Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
application/pdf
application/zip
218
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/17002019-04-01T08:34:55Zcom_10023_94com_10023_28col_10023_99
The lesser names : the teachers of the Edinburgh Mathematical Society and other aspects of Scottish mathematics, 1867–1946
Hartveit, Marit
Robertson, E. F.
O'Connor, John J. (John Joseph)
Edinburgh Mathematical Society
Mathematical publication
A. C. Aitken
Sir D'Arcy Thompson
G. T. Bennett
Edinburgh Association for the University Education of Women
John Watt Butters
Scottish mathematics
Mathematical teachers
QA1.E4H2
Edinburgh Mathematical Society--History
Mathematics teachers--Scotland--History
Mathematics--Research--Scotland--History
Mathematics--Scotland--History
The Edinburgh Mathematical Society started out in 1883 as a society with a large proportion of teachers. Today, the member base is mainly academical and there are only a few teachers left. This thesis explores how and when this change came about, and discusses what this meant for the Society.
It argues that the exit of the teachers is related to the rising standard of mathematics, but even more to a change in the Society’s printing policy in the 1920s, that turned the Society’s Proceedings into a pure research publication and led to the death of the ‘teacher journal’, the Mathematical Notes. The thesis also argues that this change, drastic as it may seem, does not represent a change in the Society’s nature.
For this aim, the role of the teachers within the Society has been studied and compared to that of the academics, from 1883 to 1946. The mathematical contribution of the teachers to the Proceedings is studied in some detail, in particular the papers by John Watt Butters.
A paper in the Mathematical Notes by A. C. Aitken on the Bell numbers is considered in connection with a series of letters on the same topic from 1938–39. These letters, written by Aitken, Sir D’Arcy Thompson, another EMS member, and the Cambridge mathematician G. T. Bennett, explores the relation between the three and gives valuable insight into the status of the Notes.
Finally, the role of the first women in the Society is studied. The first woman joined without any official university education, but had received the necessary mathematical background from her studies under the Edinburgh Association for the University Education of Women. The final chapter is largely an assessment of this Association’s mathematical classes.
2011-06-22
2011-03-18T08:21:33Z
2011-03-18T08:21:33Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/1700
en
application/pdf
243
University of St Andrews
The University of St Andrews
School of Mathematics and Statistics
oai:research-repository.st-andrews.ac.uk:10023/21092019-04-01T08:34:55Zcom_10023_94com_10023_28col_10023_99
Directed graph iterated function systems
Boore, Graeme C.
Falconer, K. J.
Olsen, Lars
Engineering and Physical Sciences Research Council (EPSRC)
Fractal geometry
Iterated function systems
Exact Hausdorff measure of attractors
Directed graphs
Multifractal analysis
Qth packing moment
QA614.86B7
Fractals
Hausdorff measures
Directed graphs
Iterative methods (Mathematics)
This thesis concerns an active research area within fractal geometry.
In the first part, in Chapters 2 and 3, for directed graph iterated function systems
(IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors
that cannot be the attractors of standard (1-vertex directed graph) IFSs, with
or without separation conditions. We also calculate their exact Hausdorff measure.
Thus we are able to identify a new class of attractors for which the exact Hausdorff
measure is known.
We give a constructive algorithm for calculating the set of gap lengths of any
attractor as a finite union of cosets of finitely generated semigroups of positive real
numbers. The generators of these semigroups are contracting similarity ratios of
simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ
with no limit on the number of vertices in the directed graph, provided a separation
condition holds.
The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We
obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth
packing moment of μᵤ, the self-similar measure at a vertex u, for the non-lattice case,
with a corresponding limit for the lattice case. We do this
(i) for any q ∈ ℝ if the strong separation condition holds,
(ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative
matrix associated with the system is irreducible.
In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum
of μᵤ to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper
multifractal q box-dimension with respect to μᵤ, of the set consisting of all the intersections
of the components of Fᵤ, is strictly less than the multifractal q Hausdorff
dimension with respect to μᵤ of Fᵤ.
2011-11-30
2011-12-12T16:38:24Z
2011-12-12T16:38:24Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/2109
en
application/pdf
160
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/113622019-04-01T08:34:55Zcom_10023_94com_10023_28col_10023_99
Constructing 2-generated subgroups of the group of homeomorphisms of Cantor space
Hyde, James Thomas
Bleak, Collin Patrick
Ruškuc, Nik
QA174.2H8
Group theory
Cantor sets
Homeomorphisms
We study finite generation, 2-generation and simplicity of subgroups of H[sub]c, the
group of homeomorphisms of Cantor space.
In Chapter 1 and Chapter 2 we run through foundational concepts and notation. In Chapter 3 we study vigorous subgroups of H[sub]c. A subgroup G of H[sub]c is vigorous if for any non-empty clopen set A with proper non-empty clopen subsets B and C there exists g ∈ G with supp(g) ⊑ A and Bg ⊆ C. It is a corollary of the main theorem of Chapter 3 that all finitely generated simple vigorous subgroups of H[sub]c are in fact 2-generated. We show the family of finitely generated, simple, vigorous subgroups of H[sub]c is closed under several natural constructions.
In Chapter 4 we use a generalised notion of word and the tight completion construction from [13] to construct a family of subgroups of H[sub]c which generalise Thompson's group V . We give necessary conditions for these groups to be finitely generated and simple. Of these we show which are vigorous. Finally we give some examples.
2017
2017-08-03T12:13:20Z
2017-08-03T12:13:20Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/11362
en
application/pdf
vi, 141 leaves
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/264702024-03-08T12:15:34Zcom_10023_94com_10023_28col_10023_99
Synchronising and separating permutation groups through graphs
Aljohani, Mohammed
Cameron, Peter J. (Peter Jephson)
Michell, James David
Saudi Arabia. Wizārat al-Taʻlīm al-ʻĀlī
Saudi Arabia. Safārah (London, England)
Finite permutation groups
Synchronisation
Separation
Association schemes
Graphs
QA175.A6
QA175.A6
About 15 years ago, Araújo, Arnold and Steinberg introduced the notion of synchronisation to the theory of finite permutation groups. Synchronisation property is closely related to another property which is called separation, but they are not the same. Interestingly, the study of the two properties for finite groups involves many combinatorial problems. In this thesis, we tried to extend the current knowledge about synchronising and separating groups and suggest some questions. The introduction and the background are represented in Chapter 1 and Chapter 2, respectively. The main work is divided into three chapters.
In Chapter 3, we started by extending the notions of synchronisation and separation to association schemes. Then, we considered two important families of almost simple permutation groups. Firstly, the group 𝐺 induced by the action of the symmetric group Sym(𝑛) on the set Ω of 𝑘-element subsets of an 𝑛-set, say {1, ..., 𝑛} (we call this the first group). Secondly, the group 𝐺 induced by the action of the symmetric group Sym(𝑛) on the set Ω of uniform 𝑙-partitions of an 𝑛-set, {1, ..., 𝑛}, into subsets of size 𝑘 where 𝑛 = 𝑘𝑙 (we call this the second group).
For first group, when 𝑘 = 2, 3, 4 and 5, we showed that for large enough 𝑛 the group is non-separating (resp. non-synchronizing) if and only if there is a Steiner system S(𝑡, 𝑘, 𝑛) (resp. large set) for some 𝑡 < 𝑘. In general, we stated a conjecture that is if true would be a crucial extension of the remarkable result by Peter Keevash that considers the existence of Steiner systems. For the second group, we gave similar results to the first group when 𝐾 = 2, 3, 4, 5, 6 and 𝑙 = 2. We stated conjecture for 𝑘 > 6 and 𝑙 = 2. Also, we showed that the group is non-synchronising when 𝑙 > 2.
In Chapter 4, the synchronisation property of a ne distance transitive permutation groups is considered. We showed that the separation and the synchronising properties are equivalent for a ne groups. We determined when some groups are synchronising, for example, automorphism groups of Hamming graphs, halved graphs, folded halved graphs, bilinear form graphs, some alternating form graphs and cosets graphs of some Golay codes. In addition, we stated a conjecture for distance regular graphs which connects this chapter and the previous one.
In Chapter 5, we started by defining the diagonal factorisation of finite groups and proved some related basic results. Then, we showed that the diagonal group 𝐷(𝑇, 2) is non-separating if and only if 𝑇 admits a diagonal factorisation. Also, we showed that the group 𝐷(𝑇, 2) is non-separating when 𝑇 = 𝐴ₙ. We proved that the diagonal group 𝐷(𝑇, 𝑑) for 𝑑 ≥ 3, is non-synchronising. In the last section, we showed the equivalence between the separation and the synchronisation properties for groups of diagonal types.
"This work was supported by the Ministry of Higher Education in Saudi Arabia
(Taibah University) and the Saudi Cultural Bureau in London." -- Acknowledgements
2022-06-14
2022-11-23T10:56:09Z
2022-11-23T10:56:09Z
Thesis
Doctoral
PhD Doctor of Philosophy
https://hdl.handle.net/10023/26470
https://doi.org/10.17630/sta/227
en
application/pdf
application/octet-stream
227
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/25612019-04-01T08:34:55Zcom_10023_94com_10023_28col_10023_99
Topics in computational group theory : primitive permutation groups and matrix group normalisers
Coutts, Hannah Jane
Roney-Dougal, Colva Mary
QA174.7D36C7
Group theory--Data processing
Permutation groups
Matrix groups
Part I of this thesis presents methods for finding the primitive permutation
groups of degree d, where 2500 ≤ d < 4096, using the O'Nan-Scott Theorem
and Aschbacher's theorem. Tables of the groups G are given for each O'Nan-Scott class. For the non-affine groups, additional information is given: the
degree d of G, the shape of a stabiliser in G of the primitive action, the
shape of the normaliser N in S[subscript(d)] of G and the rank of N.
Part II presents a new algorithm NormaliserGL for computing the normaliser
in GL[subscript(n)](q) of a group G ≤ GL[subscript(n)](q). The algorithm is implemented in
the computational algebra system MAGMA and employs Aschbacher's theorem
to break the problem into several cases. The attached CD contains the
code for the algorithm as well as several test cases which demonstrate the
improvement over MAGMA's existing algorithm.
2011-11
2012-04-17T09:37:55Z
2012-04-17T09:37:55Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/2561
en
application/zip
application/pdf
vi, 170
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/233812021-08-17T09:41:47Zcom_10023_94com_10023_28col_10023_99
Coincidence and disparity of fractal dimensions
Burrell, Stuart Andrew
Fraser, Jonathan M.
Falconer, K. J.
Carnegie Trust for the Universities of Scotland
Fractal geometry
Intermediate dimensions
Assouad spectrum
Inhomogeneous attractors
Fractional Brownian motion
Projection
Elliptical polynomial spirals
QA614.86B88
Fractals
Attractors (Mathematics)
Brownian motion processes
We investigate the dimension and structure of four fractal families: inhomogeneous attractors, fractal projections, fractional Brownian images, and elliptical polynomial spirals. For each family, particular attention is given to the relationships between different notions of dimension. This may take the form of determining conditions for them to coincide, or, in the case they differ, calculating the spectrum of dimensions interpolating between them. Material for this thesis is drawn from the papers [6,7,8,9,10].
First, we develop the dimension theory of inhomogeneous attractors for non-linear and affine iterated function systems. In both cases, we find natural quantities that bound the upper box-counting dimension from above and identify sufficient conditions for these bounds to be obtained. Our work improves and unifies previous theorems on inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer's seminal results on homogeneous self-affine sets.
Second, we prove that the intermediate dimensions of the orthogonal projection of a Borel set 𝐸 ⸦ ℝⁿ onto a linear subspace 𝑉 are almost surely independent of the choice of subspace. Similar methods identify the almost sure value of the dimension of Borel sets under index-α fractional Brownian motion. Various applications are given, including a surprising result that relates the box dimension of the Hölder images of a set to the Hausdorff dimension of the preimages.
Finally, we investigate fractal aspects of elliptical polynomial spirals; that is, planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis, computing explicitly their intermediate, box-counting and Assouad-type dimensions. Relying on this, we bound the Hölder regularity of maps that deform one spiral into another, generalising the `winding problem’ of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents.
"This work was supported by a PhD scholarship from The Carnegie Trust for the Universities
of Scotland [grant number EP/L1234567/8]." -- Funding
2021-06-29
2021-06-17T16:01:37Z
2021-06-17T16:01:37Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/23381
EP/L1234567/8
en
S. A. Burrell. On the dimension and measure of inhomogeneous attractors. Real Anal. Exchange, 44 (1), 199–216 (2019)
S. A. Burrell and J. M. Fraser. The dimensions of inhomogeneous self-affine sets. Ann. Acad. Sci. Fenn. Math., 45, 313–324 (2020)
S. A. Burrell. Dimensions of fractional Brownian images. Preprint (2020)
S. A. Burrell, K. J. Falconer and J. M. Fraser. The fractal structure of elliptical polynomial spirals. Preprint (2020)
S. A. Burrell, K. J. Falconer and J. M. Fraser. Projection theorems for intermediate dimensions. J. Fractal Geom., 8 (2), 95–116 (2021)
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
application/pdf
viii, 116 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/126002019-04-01T08:34:56Zcom_10023_94com_10023_28col_10023_99
The construction of finite soluble factor groups of finitely presented groups and its application
Wegner, Alexander
Robertson, E. F.
QA171.W3
Group theory--Data processing
Finite groups
Presentation of groups (Mathematics)
Computational group theory deals with the design, analysis and computer implementation of algorithms for solving computational problems involving groups, and with the applications of the programs produced to interesting questions in group theory, in other branches of mathematics, and in other areas of science. This thesis describes an implementation of a proposal for a Soluble Quotient Algorithm, i.e. a description of the algorithms used and a report on the findings of an empirical study of the behaviour of the programs, and gives an account of an application of the programs. The programs were used for the construction of soluble groups with interesting properties, e.g. for the construction of soluble groups of large derived length which seem to be candidates for groups having efficient presentations. New finite soluble groups of derived length six with trivial Schur multiplier and efficient presentations are described. The methods for finding efficient presentations proved to be only practicable for groups of moderate order. Therefore, for a given derived length soluble groups of small order are of interest. The minimal soluble groups of derived length less than or equal to six are classified.
1992
2018-01-24T13:14:38Z
2018-01-24T13:14:38Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/12600
en
application/pdf
130, 11 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/120242019-04-01T08:34:56Zcom_10023_94com_10023_28col_10023_99
Counting subwords and other results related to the generalised star-height problem for regular languages
Bourne, Thomas
Ruškuc, Nik
Roney-Dougal, Colva Mary
Engineering and Physical Sciences Research Council (EPSRC)
Regular language
Generalised star-height
Semigroup
Monoid
Subwords
QA267.3B78
Formal languages
The Generalised Star-Height Problem is an open question in the field of formal language theory that concerns a measure of complexity on the class of regular languages; specifically, it asks whether or not there exists an algorithm to determine the generalised star-height of a given regular language. Rather surprisingly, it is not yet known whether there exists a regular language of generalised star-height greater than one.
Motivated by a theorem of Thérien, we first take a combinatorial approach to the problem and consider the languages in which every word features a fixed contiguous subword an exact number of times. We show that these languages are all of generalised star-height zero. Similarly, we consider the languages in which every word features a fixed contiguous subword a prescribed number of times modulo a fixed number and show that these languages are all of generalised star-height at most one.
Using these combinatorial results, we initiate work on identifying the generalised star-height of the languages that are recognised by finite semigroups. To do this, we establish the generalised star-height of languages recognised by Rees zero-matrix semigroups over nilpotent groups of classes zero and one before considering Rees zero-matrix semigroups over monogenic semigroups.
Finally, we explore the generalised star-height of languages recognised by finite groups of a given order. We do this through the use of finite state automata and 'count arrows' to examine semidirect products of the form 𝐴 ⋊ ℤ[sub]𝑟, where 𝐴 is an abelian group and ℤ[sub]𝑟 is the cyclic group of order 𝑟.
2017-12-07
2017-11-07T14:16:13Z
2017-11-07T14:16:13Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/12024
en
application/pdf
vii, 93 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/249452022-02-24T03:03:35Zcom_10023_94com_10023_28col_10023_99
On singular pencils of matrices
Ledermann, Walter
Turnbull, Herbert Westren
"This thesis is a study of Singular Matrix Pencils under various aspects. In part (I) a new derivation of the Canonical Form of matrix pencils is given. This suggests investigation of the transformations of a pencil into itself (part (II)). Finally, part (III) deals with the canonical form of singular pencils of special types, namely those whose members are induced (or invariant) matrices." -- Preface
1936
2022-02-23T15:31:20Z
2022-02-23T15:31:20Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/24945
https://doi.org/10.17630/sta/158
en
application/pdf
97 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/28212019-07-01T10:13:29Zcom_10023_94com_10023_28col_10023_99
Semigroup presentations
Ruškuc, Nik
Howie, John M. (John Mackintosh)
Robertson, E. F.
QA171.S3R9
Group theory
Representations of groups
Semigroups
In this thesis we consider in detail the following two fundamental problems for
semigroup presentations:
1. Given a semigroup find a presentation defining it.
2. Given a presentation describe the semigroup defined by it.
We also establish two links between these two approaches: semigroup constructions
and computational methods.
After an introduction to semigroup presentations in Chapter 3, in Chapters 4
and 5 we consider the first of the two approaches. The semigroups we examine in
these two chapters include completely O-simple semigroups, transformation semigroups,
matrix semigroups and various endomorphism semigroups. In Chapter 6
we find presentations for the following semi group constructions: wreath product,
Bruck-Reilly extension, Schiitzenberger product, strong semilattices of monoids,
Rees matrix semigroups, ideal extensions and subsemigroups. We investigate in
more detail presentations for subsemigroups in Chapters 7 and 10, where we prove
a number of Reidemeister-Schreier type results for semigroups. In Chapter 9
we examine the connection between the semi group and the group defined by the
same presentation. The general results from Chapters 6, 7, 9 and 10 are applied
in Chapters 8, 11, 12 and 13 to subsemigroups of free semigroups, Fibonacci
semigroups, semigroups defined by Coxeter type presentations and one relator
products of cyclic groups. Finally, in Chapter 14 we describe the Todd-Coxeter
enumeration procedure and introduce three modifications of this procedure.
1995
2012-06-20T11:09:54Z
2012-06-20T11:09:54Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/2821
en
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
http://creativecommons.org/licenses/by-nc-nd/3.0/
application/pdf
271
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/248962022-04-21T08:05:09Zcom_10023_94com_10023_28col_10023_99
Studies in form-theory : 1. Mixed determinants - 2. The pedal correspondence
Vaidyanathaswamy, R.
1924
2022-02-16T15:06:32Z
2022-02-16T15:06:32Z
Thesis
Doctoral
DSc Doctor of Science
http://hdl.handle.net/10023/24896
en
application/pdf
122 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/113702019-04-01T08:34:57Zcom_10023_94com_10023_28col_10023_99
Fractal, group theoretic, and relational structures on Cantor space
Donoven, Casey Ryall
Falconer, K. J.
Bleak, Collin Patrick
QA174.2D7
Cantor sets
Fractals
Group theory
Topological spaces
Cantor space, the set of infinite words over a finite alphabet, is a type of metric space
with a `self-similar' structure. This thesis explores three areas concerning Cantor space
with regard to fractal geometry, group theory, and topology.
We find first results on the dimension of intersections of fractal sets within the Cantor
space. More specifically, we examine the intersection of a subset E of the n-ary Cantor
space, C[sub]n with the image of another subset Funder a random isometry. We obtain
almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the
intersection, and a lower bound for the essential supremum of the Hausdorff dimension.
We then consider a class of groups, denoted by V[sub]n(G), of homeomorphisms of the
Cantor space built from transducers. These groups can be seen as homeomorphisms
that respect the self-similar and symmetric structure of C[sub]n, and are supergroups of the
Higman-Thompson groups V[sub]n. We explore their isomorphism classes with our primary
result being that V[sub]n(G) is isomorphic to (and conjugate to) V[sub]n if and only if G is a
semiregular subgroup of the symmetric group on n points.
Lastly, we explore invariant relations on Cantor space, which have quotients homeomorphic to fractals in many different classes. We generalize a method of describing these
quotients by invariant relations as an inverse limit, before characterizing a specific class
of fractals known as Sierpiński relatives as invariant factors. We then compare relations
arising through edge replacement systems to invariant relations, detailing the conditions
under which they are the same.
2016
2017-08-03T15:38:28Z
2017-08-03T15:38:28Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/11370
en
application/pdf
vi, 121 leaves
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/106372019-04-01T08:34:57Zcom_10023_94com_10023_28col_10023_99
Multifractal zeta functions
Mijović, Vuksan
Olsen, Lars
Todd, Michael John
University of St Andrews
Multifractal zeta functions
Fine multifractal spectra
Coarse multifractal spectra
Renyi dimension
Geometric zeta function
Dynamical zeta function
Multifractal pressure
QA614.86M5
Multifractals
Functions, Zeta
Multifractals have during the past 20 − 25 years been the focus of enormous attention in the mathematical literature. Loosely speaking there are two main ingredients in multifractal analysis: the multifractal spectra and the Renyi dimensions. One of the main goals in multifractal analysis is to understand these two ingredients and their relationship with each other. Motivated by the powerful techniques provided by the use of the Artin-Mazur zeta-functions in number theory and the use of the Ruelle zeta-functions in dynamical systems, Lapidus and collaborators (see books by Lapidus & van Frankenhuysen [32, 33] and the references therein) have introduced and pioneered use of zeta-functions in fractal geometry. Inspired by this development, within the past 7−8 years several authors have paralleled this development by introducing zeta-functions into multifractal geometry. Our result inspired by this work will be given in section 2.2.2. There we introduce geometric multifractal zeta-functions providing precise information of very general classes of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. Results in that section are based on paper [37].
Dynamical zeta-functions have been introduced and developed by Ruelle [63, 64] and others, (see, for example, the surveys and books [3, 54, 55] and the references therein). It has been a major challenge to introduce and develop a natural and meaningful theory of dynamical multifractal zeta-functions paralleling existing theory of dynamical zeta functions. In particular, in the setting of self-conformal constructions, Olsen [49] introduced a family of dynamical multifractal zeta-functions designed to provide precise information of very general classes of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. However, recently it has been recognised that while self-conformal constructions provide a useful and important framework for studying fractal and multifractal geometry, the more general notion of graph-directed self-conformal constructions provide a substantially more flexible and useful framework, see, for example, [36] for an elaboration of this. In recognition of this viewpoint, in section 2.3.11 we provide main definitions of the multifractal pressure and the multifractal dynamical zeta-functions and we state our main results. This section is based on paper [38].
Setting we are working unifies various different multifractal spectra including fine multifractal spectra of self-conformal measures or Birkhoff averages of continuous function. It was introduced by Olsen in [43]. In section 2.1 we propose answer to problem of defining Renyi spectra in more general settings and provide slight improvement of result regrading multifractal spectra in the case of Subshift of finite type.
2017-06-23
2017-04-21T09:08:29Z
2017-04-21T09:08:29Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/10637
en
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
application/pdf
[8], 86 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/78542019-04-01T08:34:57Zcom_10023_94com_10023_28col_10023_99
Dimension and measure theory of self-similar structures with no separation condition
Farkas, Ábel
Falconer, K. J.
Engineering and Physical Sciences Research Council (EPSRC)
Self-similar set
Projection theorem
Projections of self-similar sets
Graph-directed attractors
Subshifts of finite type
Separation condition
Fractal
Hausdorff dimension
Hausdorff measure
QA248.F28
Set theory
Fractals
Hausdorff measures
Attractors (Mathematics)
We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets.
We investigate how the Hausdorff dimension and measure of a self-similar set
K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0.
We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.
We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0.
2015-11-30
2015-11-26T10:41:57Z
2015-11-26T10:41:57Z
Thesis
Doctoral
PhD Doctor of Philosophy
uk.bl.ethos.675217
http://hdl.handle.net/10023/7854
en
application/pdf
v, 86 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/275492023-06-07T11:55:23Zcom_10023_94com_10023_28col_10023_99
Solving decision problems in finitely presented groups via generalized small cancellation theory
Jurina, Simon
Roney-Dougal, Colva Mary
Cameron, Peter J. (Peter Jephson)
Engineering and Physical Sciences Research Council (EPSRC)
University of St Andrews. School of Mathematics and Statistics
Hyperbolic groups
Geometric/Computational group theory
Decision problems
MAGMA
Pregroups
Van Kampen diagrams
QA171.J88
Hyperbolic groups
Geometric group theory
Statistical decision
Pregroups
This thesis studies two decision problems for finitely presented groups. Using a standard RAM model of computation, in which the basic arithmetical operations on integers are assumed to take constant time, in Part I we develop an algorithm IsConjugate, which on input a (finite) presentation defining a hyperbolic group 𝐺, correctly decides whether 𝑤₁ ϵ 𝑋* and 𝑤₂ ϵ 𝑋* are conjugate in 𝐺, and if so, then for each 𝑖 ϵ {1,2}, returns a cyclically reduced word 𝑟ᵢ that is conjugate in 𝐺 to 𝑤ᵢ, and an 𝑥 ϵ 𝑋* such that r₂= G 𝑥^{-1} r_1 x (hence if 𝑤₁ and 𝑤₂ are already cyclically reduced, then it returns an 𝑥 ϵ 𝑋* such that 𝑤₂=_G x^{-1} w_1 x). Moreover, IsConjugate can be constructed in polynomial-time in the input presentation < 𝑋 | 𝑅 >, and IsConjugate runs in time O((|𝑤₁| + |𝑤₂| min{|𝑤₁|, |𝑤₂|}).
IsConjugate has been implemented in the MAGMA software, and our experiments show that the run times agree with the worst-case time complexities. Thus, IsConjugate is the most efficient general practically implementable conjugacy problem solver for hyperbolic groups.
It is undecidable in general whether a given finitely presented group is hyperbolic. In Part II of this thesis, we present a polynomial-time procedure VerifyHypVertex which on input a finite presentation for a group G, returns true only if G is hyperbolic. VerifyHypVertex generalizes the methods from [34], and in particular succeeds on all presentations on which the implementation from [34] succeeds, and many additional presentations as well. The algorithms have been implemented in MAGMA, and the experiments show that they return a positive answer on many examples on which other comparable publicly available methods fail, such as KBMAG.
2023-06-13
2023-05-10T08:26:42Z
2023-05-10T08:26:42Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/27549
https://doi.org/10.17630/sta/429
EP/R513337/1
en
Solving decision problems in finitely presented groups via generalized small cancellation theory (thesis data)
Jurina, S., University of St Andrews, 21 Mar 2023. DOI: https://doi.org/10.17630/0f7a7c71-26f5-4346-9bf1-3da33e354721
https://doi.org/10.17630/0f7a7c71-26f5-4346-9bf1-3da33e354721
application/pdf
application/x-tex
application/zip
195
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/235582021-07-15T02:03:51Zcom_10023_94com_10023_28col_10023_99
Enumerating 0-simple semigroups
Russell, Christopher
Mitchell, James David
Ruškuc, Nik
Engineering and Physical Sciences Research Council (EPSRC)
Semigroups
Computational semigroup theory
Combinatorics
Matrices
Binary matrices
0-simple semigroups
Congruence free semigroups
E-unitary inverse semigroups
GAP
E-unitary inverse covers
Rees 0-matrix semigroups
Computational semigroup theory involves the study and implementation of algorithms to compute with semigroups. Efficiency is of central concern and often follows from the insight of semigroup theoretic results. In turn, computational methods allow for analysis of semigroups which can provide intuition leading to theoretical breakthroughs. More efficient algorithms allow for more cases to be computed and increases the potential for insight. In this way, research into computational semigroup theory and abstract semigroup theory forms a feedback loop with each benefiting the other.
In this thesis the primary focus will be on counting isomorphism classes of finite 0-simple semigroups. These semigroups are in some sense the building blocks of finite semigroups due to their correspondence with the Greens 𝒟-classes of a semigroup. The theory of Rees 0-matrix semigroups links these semigroups to matrices with entries from 0-groups. Special consideration will be given to the enumeration of certain sub-cases, most prominently the case of congruence free semigroups. The author has implemented these enumeration techniques and applied them to count isomorphism classes of 0-simple semigroups and congruence free semigroups by order. Included in this thesis are tables of the number of 0-simple semigroups of orders less than or equal to 130, up to isomorphism. Also included are tables of the numbers of congruence free semigroups, up to isomorphism, with m Green’s ℒ-classes and n Green’s ℛ-classes for all mn less than or equal to 100, as well as for various other values of m,n. Furthermore a database of finite 0-simple semigroups has been created and we detail how this was done. The implementation of these enumeration methods and the database are publicly available as GAP code. In order to achieve these results pertaining to finite 0-simple semigroups we invoke the theory of group actions and prove novel combinatorial results. Most notably, we have deduced formulae for enumerating the number of binary matrices with distinct rows and columns up to row and column permutations.
There are also two sections dedicated to covers of E-unitary inverse semigroups, and presentations of factorisable orthodox monoids, respectively. In the first, we explore the concept of a minimal E-unitary inverse cover, up to isomorphism, by defining various sensible orderings. We provide examples of Clifford semigroups showing that, in general, these orderings do not have a unique minimal element. Finally, we pose conjectures about the existence of unique minimal E-unitary inverse covers of Clifford semigroups, when considered up to an equivalence weaker than isomorphism. In the latter section, we generalise the theory of presentations of factorisable inverse monoids to the more general setting of factorisable orthodox monoids. These topics were explored early in the authors doctoral studies but ultimately in less depth than the research on 0-simple semigroups.
2021-06-29
2021-07-14T13:53:38Z
2021-07-14T13:53:38Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/23558
https://doi.org/10.17630/sta/109
EP/N509759/1
en
application/pdf
xxi, 196 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/287692023-11-28T03:03:22Zcom_10023_94com_10023_28col_10023_99
On constructing topology from algebra
Elliott, Luke
Bleak, Collin Patrick
Mitchell, James David
University of St Andrews. School of Mathematics and Statistics
In this thesis we explore natural procedures through which topological structure can be constructed from specific semigroups. We will do this in two ways: 1) we equip the semigroup object itself with a topological structure, and 2) we find a topological space for the semigroup to act on continuously. We discuss various minimum/maximum topologies which one can define on an arbitrary semigroup (given some topological restrictions). We give explicit descriptions of each these topologies for the monoids of binary relations, partial transformations, transformations, and partial bijections on the set N. Using similar methods we determine whether or not each of these semigroups admits a unique Polish semigroup topology. We also do this for the following semigroups: the monoid of all injective functions on N, the monoid of continuous transformations of the Hilbert cube [0, 1]N, the monoid of continuous transformations of the Cantor space 2N, and the monoid of endomorphisms of the countably infinite atomless boolean algebra. With the exception of the continuous transformation monoid of the Hilbert cube, we also show that all of the above semigroups admit a second countable semigroup topology such that every semigroup homomorphism from the semigroup to a second countable topological semigroup is continuous. In a recent paper, Bleak, Cameron, Maissel, Navas, and Olukoya use a theorem of Rubin to describe the automorphism groups of the Higman-Thompson groups Gₙ,ᵣ via their canonical Rubin action on the Cantor space. In particular they embed these automorphism groups into the rational group R of transducers introduced by Grigorchuk, Nekrashevich, and Sushchanskii. We generalise these transducers to be more suitable to higher dimensional Cantor spaces and give a similar description of the automorphism groups of the Brin-Thompson groups 𝑑Vₙ (although we do not give an embedding into R). Using our description, we show that the outer automorphism group Out(𝑑V₂) of 𝑑V₂ is isomorphic to the wreath product of Out(1V₂) with the symmetric group on 𝑑 points.
2022-06-14
2023-11-27T16:37:01Z
2023-11-27T16:37:01Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/28769
https://doi.org/10.17630/sta/671
en
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
application/pdf
136
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/9482019-04-01T08:34:57Zcom_10023_94com_10023_28col_10023_99
Stable and multistable processes and localisability
Liu, Lining
Falconer, K. J.
Multistable
Localisability
QA274.L58
Stochastic processes
Characteristic functions
We first review recent work on stable and multistable random processes and their
localisability. Then most of the thesis concerns a new approach to these topics
based on characteristic functions.
Our aim is to construct processes on R, which are α(x)-multistable, where the
stability index α(x) varies with x. To do this we first use characteristic functions
to define α(x)-multistable random integrals and measures and examine their properties.
We show that an α(x)-multistable random measure may be obtained as the
limit of a sequence of measures made up of α-stable random measures restricted
to small intervals with α constant on each interval.
We then use the multistable random integrals to define multistable random
processes on R and study the localisability of these processes. Thus we find conditions
that ensure that a process locally ‘looks like’ a given stochastic process
under enlargement and appropriate scaling. We give many examples of multistable
random processes and examine their local forms.
Finally, we examine the dimensions of graphs of α-stable random functions
defined by series with α-stable random variables as coefficients.
2010-06-23
2010-06-30T13:34:12Z
2010-06-30T13:34:12Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/948
en
application/pdf
91
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/285912023-11-01T03:03:35Zcom_10023_94com_10023_28col_10023_99
Interpolating between Hausdorff and box dimension
Banaji, Amlan
Fraser, Jonathan M.
Falconer, K. J.
Leverhulme Trust
Fractal geometry
Hausdorff dimension
Box dimension
Intermediate dimensions
Dimension interpolation
Moran set
Iterated function system
Self-conformal
Self-affine
Bedford–McMullen carpet
QA614.86B262
Fractals
Dimension theory (Topology)
Mathematical analysis
Hausdorff and box dimension are two familiar notions of fractal dimension. Box dimension can be larger than Hausdorff dimension, because in the definition of box dimension, all sets in the cover have the same diameter, but for Hausdorff dimension there is no such restriction. This thesis focuses on a family of dimensions parameterised by θ ∈ (0,1), called the intermediate dimensions, which are defined by requiring that diam(U) ⩽ (diam(V))ᶿ for all sets U, V in the cover.
We begin by generalising the intermediate dimensions to allow for greater refinement in how the relative sizes of the covering sets are restricted. These new dimensions can recover the interpolation between Hausdorff and box dimension for compact sets whose intermediate dimensions do not tend to the Hausdorff dimension as θ → 0. We also use a Moran set construction to prove a necessary and sufficient condition, in terms of Dini derivatives, for a given function to be realised as the intermediate dimensions of a set.
We proceed to prove that the intermediate dimensions of limit sets of infinite conformal iterated function systems are given by the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This applies to sets defined using continued fraction expansions, and has applications to dimensions of projections, fractional Brownian images, and general Hölder images.
Finally, we determine a formula for the intermediate dimensions of all self-affine Bedford–McMullen carpets. The functions display features not witnessed in previous examples, such as having countably many phase transitions. We deduce that two carpets have equal intermediate dimensions if and only if the multifractal spectra of the corresponding uniform Bernoulli measures coincide. This shows that if two carpets are bi-Lipschitz equivalent then the multifractal spectra are equal.
"This work was supported by a Leverhulme Trust Research Project Grant (RPG-2019-034)." -- Funding
2023-11-28
2023-10-30T15:21:21Z
2023-10-30T15:21:21Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/28591
https://doi.org/10.17630/sta/642
RPG-2019-034
en
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
application/pdf
application/zip
vii, 174 p.
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/111832019-04-01T08:34:58Zcom_10023_94com_10023_28col_10023_99
Generalized Bernstein polynomials and total positivity
Oruç, Halil
Phillips, G. M. (George McArtney)
QA404.5O8
Polynomials
"This thesis submitted for Ph.D. degree deals mainly with geometric properties of generalized Bernstein polynomials which replace the single Bernstein polynomial by a one-parameter family of polynomials. It also provides a triangular decomposition and 1-banded factorization of the Vandermonde matrix.
We first establish the generalized Bernstein polynomials for monomials, which leads to a definition of Stirling polynomials of the second kind. These are q-analogues of Stirling numbers of the second kind. Some of the properties of the Stirling numbers are generalized to their q-analogues.
We show that the generalized Bernstein polynomials are monotonic in degree n when the function ƒ is convex...
Shape preserving properties of the generalized Bernstein polynomials are studied by making use of the concept of total positivity. It is proved that monotonic and convex functions produce monotonic and convex generalized Bernstein polynomials. It is also shown that the generalized Bernstein polynomials are monotonic in the parameter q
for the class of convex functions.
Finally, we look into the degree elevation and degree reduction processes on the generalized Bernstein polynomials." -- from the Abstract.
1999
2017-07-11T12:44:36Z
2017-07-11T12:44:36Z
Thesis
Doctoral
PhD Doctor of Philosophy
uk.bl.ethos.576989
http://hdl.handle.net/10023/11183
en
application/pdf
106 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/110232019-04-01T08:34:58Zcom_10023_94com_10023_28col_10023_99
Restricted permutations, antichains, atomic classes and stack sorting
Murphy, Maximilian M.
Ruskuc, Nik
QA171.C2M9
Group theory
Permutations
Involvement is a partial order on all finite permutations, of infinite dimension and having subsets isomorphic to every countable partial order with finite descending chains. It has attracted the attention of some celebrated mathematicians including Paul Erdős and, due to its close links with sorting devices, Donald Knuth.
We compare and contrast two presentations of closed classes that depend on the partial order of involvement: Basis or Avoidance Set, and Union of Atomic Classes. We examine how the basis is affected by a comprehensive list of closed class constructions and decompositions.
The partial order of involvement contains infinite antichains. We develop the concept of a fundamental antichain. We compare the concept of 'fundamental' with other definitions of minimality for antichains, and compare fundamental permutation antichains with fundamental antichains in graph theory. The justification for investigating fundamental antichains is the nice patterns they produce. We forward the case for classifying the fundamental permutation antichains.
Sorting devices have close links with closed classes. We consider two sorting devices, constructed from stacks in series, in detail.
We give a comment on an enumerative conjecture by Ira Gessel.
We demonstrate, with a remarkable example, that there exist two closed classes, equinumerous, one of which has a single basis element, the other infinitely many basis elements.
We present this paper as a comprehensive analysis of the partial order of permutation involvement. We regard the main research contributions offered here to be the examples that demonstrate what is, and what is not, possible; although there are numerous structure results that do not fall under this category. We propose the classification of fundamental permutation antichains as one of the principal problems for closed classes today, and consider this as a problem whose solution will have wide significance for the study of partial orders, and mathematics as a whole.
2003
2017-06-19T13:10:42Z
2017-06-19T13:10:42Z
Thesis
Doctoral
PhD Doctor of Philosophy
uk.bl.ethos.527125
http://hdl.handle.net/10023/11023
en
application/pdf
360 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/288742024-01-29T16:01:04Zcom_10023_94com_10023_28col_10023_99
Finiteness conditions on semigroups relating to their actions and one-sided congruences
Miller, Craig
Ruškuc, Nik
Engineering and Physical Sciences Research Council (EPSRC)
QA182.M5
Semigroups
Congruences and residues
Monoids
The purpose of this thesis is threefold: firstly, to develop a systematic theory of presentations
of monoid acts; secondly, to study finiteness conditions on semigroups
relating to finite generation of one-sided congruences; and thirdly, to establish
connections between each of these finiteness conditions, restricted to the class of
monoids, with finite presentability of acts.
We find general presentations for various monoid act constructions/components,
leading to a number of finite presentability results. In particular, we consider
subacts, Rees quotients, unions of subacts, direct products and wreath products.
A semigroup 𝑆 is called right noetherian if every right congruence on 𝑆 is finitely
generated. We present some fundamental properties of right noetherian semigroups,
discuss how semigroups relate to their substructures with regard to the
property of being right noetherian, and investigate whether this property is preserved
under various semigroup constructions.
Finally, we introduce and study the condition that every right congruence of finite index on a semigroup is finitely generated. We call semigroups satisfying this condition f-noetherian. It turns out that every finitely generated semigroup is
f-noetherian. We investigate, for various semigroup classes, whether the property of being f-noetherian coincides with finite generation.
2020-12-01
2023-12-14T12:58:12Z
2023-12-14T12:58:12Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/28874
https://doi.org/10.17630/sta/678
en
[iii], vi, 140 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/146522019-04-01T08:34:58Zcom_10023_94com_10023_28col_10023_99
Commutativity and free products in Thompson's group V
Bieniecka, Ewa
Bleak, Collin Patrick
Quick, M. R. (Martyn R.)
Gibson-Sykora Trust
Engineering and Physical Sciences Research Council (EPSRC)
Thompson's groups
V
Dynamics
Revealing pair
Balanced pair
Chains graph
Flow graph
Detecting torsion
Centralisers
Free products
Ping-Pong
Thompson
QA178.B5
Group theory
Infinite groups
We broaden the theory of dynamical interpretation, investigate the property of commutativity and explore the subject of subgroups forming free products in Thompson's group V.
We expand Brin's terminology for a revealing pair to an any tree pair. We use it to analyse the dynamical behaviour of an arbitrary tree pair which cannot occur in a revealing pair. Hence, we design a series of algorithms generating Brin's revealing pair from any tree pair, by successively eliminating the undesirable structures. To detect patterns and transitioning between tree pairs, we introduce a new combinatorial object called the chains graph. A newly defined, unique and symmetrical type of a tree pair, called a balanced tree pair, stems from the use of the chains graphs.
The main theorem of Bleak et al. in "Centralizers in the R. Thompson's Group V_n" states the necessary structure of the centraliser of an element of V. We provide a converse to this theorem, by proving that each of the predicted structures is realisable. Hence we obtain a complete classification of centralisers in V. We give an explicit construction of an element of V with prescribed centraliser. The underlying concept is to embed a Cayley graph of a finite group into the flow graph (introduced in Bleak et al.) of the desired element. To reflect the symmetry, we present the resulting element in terms of a balanced tree pair.
The group V is conjectured to be a universal coCF group, which generates interest in studying its subgroups. We develop a better understanding of embeddings into V by providing a necessary and sufficient dynamical condition for two subgroups (not both torsion) to form a free product in V. For this, we use the properties, explored in Bleak and Salazar-Díaz "Free Products in Thompson's Group V", of sets of so--called important points, and the Ping-Pong action induced on them.
2018-06-26
2018-06-27T08:39:44Z
2018-06-27T08:39:44Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/14652
en
'Centralizers in the R. Thompson's group V_n' (2013) / Collin Bleak, Hannah Bowman, Alison Gordon Lynch, Garrett Graham, Jacob Hughes, Francesco Matucci, Eugenia Sapir, Groups Geom. Dyn. 7, no. 4, 821-865
'Free products in R. Thompson's group V' (2013) / Collin Bleak and Olga Salazar-Díaz, trans. Amer. Math. Soc. 365, no. 11, 5967-5997
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
application/pdf
v, 203 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/156312019-04-01T08:34:59Zcom_10023_94com_10023_28col_10023_99
On plausible counterexamples to Lehnert's conjecture
Bennett, Daniel
Bleak, Collin Patrick
Quick, M. R. (Martyn R.)
Engineering and Physical Sciences Research Council (EPSRC)
QA174.2B4
Formal languages
Group theory
Semigroups
A group whose co-word problem is a context free language is called co𝐶𝐹 . Lehnert's conjecture states that a group 𝐺 is co𝐶𝐹 if and only if 𝐺 embeds as a finitely generated subgroup of R. Thompson's group V . In this thesis we explore a class of groups, Faug, proposed by Berns-Zieze, Fry, Gillings, Hoganson, and Mathews to contain potential counterexamples to Lehnert's conjecture. We create infinite and finite presentations for such groups and go on to prove that a certain subclass of 𝓕𝑎𝑢𝑔 consists of groups that do embed into 𝑉.
By Anisimov a group has regular word problem if and only if it is finite. It is also known
that a group 𝐺 is finite if and only if there exists an embedding of 𝐺 into 𝑉 such that
its natural action on 𝕮₂:= {0, 1}[super]𝜔 is free on the whole space. We show that the class of
groups with a context free word problem, the class of 𝐶𝐹 groups, is precisely the class of finitely generated demonstrable groups for 𝑉 . A demonstrable group for V is a group 𝐺 which is isomorphic to a subgroup in 𝑉 whose natural action on 𝕮₂ acts freely on an open subset. Thus our result extends the correspondence between language theoretic properties of groups and dynamical properties of subgroups of V . Additionally, our result also shows that the final condition of the four known closure properties of the class of co𝐶𝐹 groups also holds for the set of finitely generated subgroups of 𝑉.
2018
2018-07-23T09:29:21Z
2018-07-23T09:29:21Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/15631
en
application/pdf
v, 141 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/35902019-04-01T08:34:59Zcom_10023_94com_10023_28col_10023_99
Ends of semigroups
Craik, Simon
Mitchell, James David
Ruškuc, Nik
QA182.C8
Semigroups
Directed graphs
The aim of this thesis is to understand the algebraic structure of a semigroup
by studying the geometric properties of its Cayley graph. We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups
and extensions of finite Rees index. We prove an analogue of Hopfs Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellative
semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.
We classify all semigroups with one end and make use of this classification
to prove various finiteness properties for semigroups with one end.
We also consider the ends of digraphs with certain algebraic properties.
We prove that two quasi-isometric digraphs have isomorphic end sets. We also prove that vertex transitive digraphs have 1, 2 or infinitely many ends and construct a topology that reflects the properties of the ends of a digraph.
2013
2013-06-03T11:52:12Z
2013-06-03T11:52:12Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/3590
en
application/pdf
iii, 198
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/110712019-04-01T08:34:59Zcom_10023_94com_10023_28col_10023_99
Flatness, extension and amalgamation in monoids, semigroups and rings
Renshaw, James Henry
Howie, John M. (John Mackintosh)
Science and Engineering Research Council (SERC)
QA355.R6
Amalgams (Group theory)
Semigroups
We begin our study of amalgamations by examining some ideas which are well-known for the category of R-modules. In particular we look at such notions as direct limits, pushouts, pullbacks, tensor products and flatness in the category of S-sets.
Chapter II introduces the important concept of free extensions and uses this to describe the amalgamated free product.
In Chapter III we define the extension property and the notion of purity. We show that many of the important notions in semigroup amalgams are intimately connected to these. In Section 2 we deduce that 'the extension property implies amalgamation' and more
surprisingly that a semigroup U is an amalgamation base if and only if it has the extension property in every containing semigroup.
Chapter IV revisits the idea of flatness and after some technical results we prove a result, similar to one for rings, on flat amalgams.
In Chapter V we show that the results of Hall and Howie on perfect amalgams can be proved using the same techniques as those used in Chapters III and IV.
We conclude the thesis with a look at the case of rings. We show that almost all of the results for semi group amalgams examined in the previous chapters, also hold for ring amalgams.
1986
2017-06-23T12:46:08Z
2017-06-23T12:46:08Z
Thesis
Doctoral
PhD Doctor of Philosophy
uk.bl.ethos.373042
http://hdl.handle.net/10023/11071
en
application/pdf
150 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/133622019-04-01T08:35:00Zcom_10023_94com_10023_28col_10023_99
Proof search issues in some non-classical logics
Howe, Jacob M.
Dyckhoff, Roy
QA9.54H7
Proof theory
This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cut- free LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1-1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called 'permutation-free' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics; Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a 'permutation-free' calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to be sound and complete with respect to a proof-theoretic semantics and (weak) cut- elimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter 4.
1999
2018-05-17T09:17:58Z
2018-05-17T09:17:58Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/13362
en
application/pdf
216 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/99122019-04-01T08:35:00Zcom_10023_94com_10023_28col_10023_99
Synchronizing permutation groups and graph endomorphisms
Schaefer, Artur
Ruškuc, Nik
Cameron, Peter J. (Peter Jephson)
Engineering and Physical Sciences Research Council (EPSRC)
University of St Andrews. School of Mathematics and Statistics
QA175.S3
Permutation groups
Endomorphisms (Group theory)
The current thesis is focused on synchronizing permutation groups and on graph endo-
morphisms. Applying the implicit classification of rank 3 groups, we provide a bound
on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph
endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of
class r, establish their relation to mixed MDS codes, investigate G-decompositions of
(non)-synchronizing semigroups, and analyse the kernel graph construction used in the
theorem of Cameron and Kazanidis which identifies non-synchronizing transformations
with graph endomorphisms [20].
The contribution lies in the following points:
1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a
complete list of small non-synchronizing groups of permutation rank 3 is provided
(see Chapter 3).
2. The singular endomorphisms of the Hamming graph and some related graphs are
characterised (see Chapter 5).
3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper-
cuboids for small values are counted, and their correspondence to mixed MDS
codes is unveiled (see Chapter 6).
4. The research on normalizing groups from [3] is extended to semigroups of the
form <G, T>, and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling
problems (see Chapter 7).
5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are
hulls and it is conjectured that a hull on n vertices has minimal generating set of at
most n generators (see Chapter 8).
2016
2016-12-01T12:52:24Z
2016-12-01T12:52:24Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/9912
en
application/pdf
248 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/158852019-04-01T08:35:00Zcom_10023_94com_10023_28col_10023_99
Decision problems in groups of homeomorphisms of Cantor space
Olukoya, Feyisayo
Bleak, Collin Patrick
Carnegie Trust for the Universities of Scotland
Group theory
Decision problems
Automorphisms
Thompson groups
QA178.O6
Group theory
Automorphisms
Infinite groups
The Thompson groups $F, T$ and $V$ are important groups in geometric group theory: $T$ and $V$ being the first discovered examples of finitely presented infinite simple groups. There are many generalisations of these groups including, for $n$ and $r$ natural numbers and $1 < r < n$, the groups $F_{n}$, $T_{n,r}$ and $G_{n,r}$ ($T ≅ T_{2,1}$ and $V ≅ G_{2,1}$). Automorphisms of $F$ and $T$ were characterised in the seminal paper of Brin ([16]) and, later on, Brin and Guzman ([17]) investigate automorphisms of $T_{n, n-1}$ and $F_{n}$ for $n>2$. However, their techniques give no information about automorphisms of $G_{n,r}$.
The second chapter of this thesis is dedicated to characterising the automorphisms of $G_{n,r}$. Presenting results of the author's article [10], we show that automorphisms of $G_{n,r}$ are homeomorphisms of Cantor space induced by transducers (finite state machines) which satisfy a strong synchronizing condition.
In the rest of Chapter 2 and early sections of Chapter 3 we investigate the group $\out{G_{n,r}}$ of outer automorphisms of $G_{n,r}$. Presenting results of the forthcoming article [6] of the author's, we show that there is a subgroup $\hn{n}$ of $\out{G_{n,r}}$, independent of $r$, which is isomorphic to the group of automorphisms of the one-sided shift dynamical system. Most of Chapter 3 is devoted to the order problem in $\hn{n}$ and is based on [44]. We give necessary and sufficient conditions for an element of $\hn{n}$ to have finite order, although these do not yield a decision procedure.
Given an automorphism $\phi$ of a group $G$, two elements $f, g ∈ G$ are said to be $\phi$-twisted conjugate to one another if for some $h ∈ G$, $g = h⁻¹ f (h)\phi$. This defines an equivalence relation on $G$ and $G$ is said to have the $\rfty$ property if it has infinitely many $\phi$-twisted conjugacy classes for all automorphisms $\phi ∈ \aut{G}$. In the final chapter we show, using the description of $\aut{G_{n,r}}$, that for certain automorphisms, $G_{n,r}$ has infinitely many twisted conjugacy classes. We also show that for certain $\phi ∈ \aut{G_{2,1}}$ the problem of deciding when two elements of $G_{2,1}$ are $\phi$-twisted conjugate to one another is soluble.
2018-12-06
2018-08-27T14:32:38Z
2018-08-27T14:32:38Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/15885
en
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
application/pdf
169 p.
University of St Andrews
The University of St Andrews
Centere for Interdisciplinary Research in Computational Algebra
oai:research-repository.st-andrews.ac.uk:10023/8672019-04-01T08:35:01Zcom_10023_94com_10023_28col_10023_99
Generating uncountable transformation semigroups
Péresse, Yann
Quick, M. R. (Martyn R.)
Mitchell, James David
QA182.P48
Semigroups
Transformations (Mathematics)
We consider naturally occurring, uncountable transformation semigroups S and investigate the following three questions.
(i) Is every countable subset F of S also a subset of a finitely generated subsemigroup of S? If so, what is the least number n such that for every countable
subset F of S there exist n elements of S that generate a subsemigroup of S
containing F as a subset.
(ii) Given a subset U of S, what is the least cardinality of a subset A of S such
that the union of A and U is a generating set for S?
(iii) Define a preorder relation ≤ on the subsets of S as follows. For subsets V and
W of S write V ≤ W if there exists a countable subset C of S such that V
is contained in the semigroup generated by the union of W and C. Given a
subset U of S, where does U lie in the preorder ≤ on subsets of S?
Semigroups S for which we answer question (i) include: the semigroups of the injec-
tive functions and the surjective functions on a countably infinite set; the semigroups
of the increasing functions, the Lebesgue measurable functions, and the differentiable
functions on the closed unit interval [0, 1]; and the endomorphism semigroup of the
random graph.
We investigate questions (ii) and (iii) in the case where S is the semigroup Ω[superscript Ω] of all functions on a countably infinite set Ω. Subsets U of Ω[superscript Ω] under consideration
are semigroups of Lipschitz functions on Ω with respect to discrete metrics on Ω and
semigroups of endomorphisms of binary relations on Ω such as graphs or preorders.
2009
2010-04-09T10:30:54Z
2010-04-09T10:30:54Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/867
en
application/pdf
184
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/213332021-07-19T15:35:52Zcom_10023_94com_10023_28col_10023_99
Subdirect products of free semigroups and monoids
Clayton, Ashley
Ruškuc, Nik
Engineering and Physical Sciences Research Council (EPSRC)
University of St Andrews. School of Mathematics and Statistics
Semigroups
Subdirect products
Fiber products
Algebra
Free
QA182.C6
Semigroups
Group theory
Subdirect products are special types of subalgebras of direct products. The purpose of this thesis is to initiate a study of combinatorial properties of subdirect products and fiber products of semigroups and monoids, motivated by the previous work on free groups, and some recent advances in general algebra.
In Chapter 1, we outline the necessary preliminary definitions and results, including elements of algebraic semigroup theory, formal language theory, automata theory and universal algebra.
In Chapter 2, we consider the number of subsemigroups and subdirect products of ℕ𝗑ℕ up to isomorphism. We obtain uncountably many such objects, and characterise the finite semigroups 𝘚 for which ℕ𝗑𝘚 has uncountable many subsemigroups and subdirect products up to isomorphism.
In Chapter 3, we consider particular finite generating sets for subdirect products of free semigroups introduced as "sets of letter pairs". We classify and count these sets which generate subdirect and fiber products, and discuss their abundance.
In Chapter 4, we consider finite generation and presentation for fiber products of free semigroups and monoids over finite fibers. We give a characterisation for finite generation of the fiber product of two free monoids over a finite fiber, and show that this implies finite presentation. We show that the fiber product of two free semigroups over a finite fiber is never finitely generated, and obtain necessary conditions on an infinite fiber for finite generation.
In Chapter 5, we consider the problem of finite generation for fiber products of free semigroups and monoids over a free fiber. We construct two-tape automata which we use to determine the language of indecomposable elements of the fiber product, which algorithmically decides when they are finitely generated.
Finally in Chapter 6, we summarise our findings, providing some further questions based on the results of the thesis.
"I would like to thank the EPSRC (grant number EP/N509759/1) for their generous
funding and skills training. I would also like to thank the entire School
of Mathematics & Statistics of the University of St Andrews for their nancial
support..." -- Acknowledgements
2020-12-01
2021-01-27T13:46:35Z
2021-01-27T13:46:35Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/21333
https://doi.org/10.17630/sta/19
EP/N509759/1
en
application/pdf
vii, 171 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/69232019-04-01T08:35:01Zcom_10023_94com_10023_28col_10023_99
Dots and lines : geometric semigroup theory and finite presentability
Awang, Jennifer S.
Ruškuc, Nik
Quick, M. R. (Martyn R.)
Semigroups
Cayley graph
Finite presentability
QA183.A8
Geometric group theory
Representations of semigroups
Cayley graphs
Geometric semigroup theory means different things to different people, but it is agreed that it involves associating a geometric structure to a semigroup and deducing properties of the semigroup based on that structure.
One such property is finite presentability. In geometric group theory, the geometric structure of choice is the Cayley graph of the group. It is known that in group theory finite presentability is an invariant under quasi-isometry of Cayley graphs.
We choose to associate a metric space to a semigroup based on a Cayley graph of that semigroup. This metric space is constructed by removing directions, multiple edges and loops from the Cayley graph. We call this a skeleton of the semigroup.
We show that finite presentability of certain types of direct products, completely (0-)simple, and Clifford semigroups is preserved under isomorphism of skeletons. A major tool employed in this is the Švarc-Milnor Lemma.
We present an example that shows that in general, finite presentability is not an invariant property under isomorphism of skeletons of semigroups, and in fact is not an invariant property under quasi-isometry of Cayley graphs for semigroups.
We give several skeletons and describe fully the semigroups that can be associated to these.
2015-06-26
2015-07-07T13:47:15Z
2015-07-07T13:47:15Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/6923
en
application/pdf
vi, 206 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/110332019-04-01T08:35:01Zcom_10023_94com_10023_28col_10023_99
Dimension theory of random self-similar and self-affine constructions
Troscheit, Sascha
Falconer, K. J.
Todd, Michael John
Engineering and Physical Sciences Research Council (EPSRC)
Institute for Computational and Experimental Research in Mathematics (ICERM)
Instituto de Matemática Pura e Aplicada (Brazil)
Brazilian-European partnership in Dynamical Systems (BREUDS)
Fractal geometry
Random attractors
Iterated function systems
Dimension theory
QA614.86T8
Fractals
Attractors (Mathematics)
Dimension theory (Topology)
This thesis is structured as follows.
Chapter 1 introduces fractal sets before recalling basic mathematical concepts from dynamical systems, measure theory, dimension theory and probability theory.
In Chapter 2 we give an overview of both deterministic and stochastic sets obtained from iterated function systems.
We summarise classical results and set most of the basic notation.
This is followed by the introduction of random graph directed systems in Chapter 3, based on the single authored paper [T1] to be published in Journal of Fractal Geometry. We prove that these attractors have equal Hausdorff and upper box-counting dimension irrespective of overlaps. It follows that the same holds for the classical models introduced in Chapter 2. This chapter also contains results about the Assouad dimensions for these random sets.
Chapter 4 is based on the single authored paper [T2] and establishes the box-counting dimension for random box-like self-affine sets using some of the results and the notation developed in Chapter 3. We give some examples to illustrate the results.
In Chapter 5 we consider the Hausdorff and packing measure of random attractors and show that for reasonable random systems the Hausdorff measure is zero almost surely. We further establish bounds on the gauge functions necessary to obtain positive or finite Hausdorff measure for random homogeneous systems.
Chapter 6 is based on a joint article with J. M. Fraser and J.-J. Miao [FMT] to appear in Ergodic Theory and Dynamical Systems. It is chronologically the first and contains results that were extended in the paper on which Chapter 3 is based.
However, we will give some simpler, alternative proofs in this section and crucially also find the Assouad dimension of some random self-affine carpets and show that the Assouad dimension is always `maximal' in both measure theoretic and topological meanings.
2017-06-23
2017-06-20T15:18:18Z
2017-06-20T15:18:18Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/11033
en
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
application/pdf
xiii, 124 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/28342019-07-01T10:10:15Zcom_10023_94com_10023_28col_10023_99
Semigroups of order-decreasing transformations
Umar, Abdullahi
Howie, John M. (John Mackintosh)
QA171.S3U6
Group theory
Representations of groups
Group theory--Problems, exercises, etc.
Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial one-to-one) transformations of X. In this
Thesis the study of order-increasing full (partial, partial one-to-one) transformations
has been reduced to that of order-decreasing full (partial, partial one-to-one)
transformations and the study of order-decreasing partial transformations to that of
order-decreasing full transformations for both the finite and infinite cases.
For the finite order-decreasing full (partial one-to-one) transformation
semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden
(1990) concerning products of idempotents (quasi-idempotents), and concerning
combinatorial and rank properties. By contrast with the semigroups of order-preserving
transformations and the full transformation semigroup, the semigroups of orderdecreasing
full (partial one-to-one) transformations and their Rees quotient semigroups
are not regular. They are, however, abundant (type A) semigroups in the sense of
Fountain (1982,1979). An explicit characterisation of the minimum semilattice
congruence on the finite semigroups of order-decreasing transformations and their Rees
quotient semigroups is obtained.
If X is an infinite chain then the semigroup S of order-decreasing full
transformations need not be abundant. A necessary and sufficient condition on X is
obtained for S to be abundant. By contrast, for every chain X the semigroup of
order-decreasing partial one-to-one transformations is type A.
The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing
full (partial one-to-one) transformations have been investigated.
1992
2012-06-21T13:40:02Z
2012-06-21T13:40:02Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/2834
en
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
http://creativecommons.org/licenses/by-nc-nd/3.0/
application/pdf
115
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/7192019-07-01T10:09:33Zcom_10023_94com_10023_28col_10023_99
Resonances for graph directed Markov systems, and geometry of infinitely generated dynamical systems
Hille, Martial R.
Stratmann, Bernd
Resonances
Graph directed Markov systems
Hausdorff dimension
Zeta function
Limit set
Discrepancy type
In the first part of this thesis we transfer a result of Guillopé et al. concerning the
number of zeros of the Selberg zeta function for convex
cocompact Schottky groups to the setting of certain types of graph directed Markov systems (GDMS).
For these systems the zeta function will be a type of Ruelle zeta function.
We show that for a finitely generated primitive conformal GDMS S, which satisfies the strong separation
condition (SSC) and the nestedness condition (NC), we have for each
c>0 that the following holds, for each w \in\$C$ with Re(w)>-c, |\Im(w)|>1 and for all k \in\$N$ sufficiently large:
log | zeta(w) | <<e^{delta(S).log(Im|w|)} and card{w \in\ Q(k) | zeta(w)=0} << k^{delta(S)}.
Here, Q(k)\subset\%C$ denotes a certain box of height k, and
delta(S) refers to the Hausdorff dimension of the limit set of S.
In the second part of this thesis we show that in any dimension
m \in\$N$ there are GDMSs for which the Hausdorff dimension of the uniformly radial limit set
is equal to a given arbitrary number d \in\(0,m) and the Hausdorff dimension of the Jørgensen limit set
is equal to a given arbitrary number j \in\ [0,m).
Furthermore, we derive various relations between the exponents of
convergence and the Hausdorff dimensions of certain different types of limit sets for iterated function systems (IFS), GDMSs, pseudo GDMSs and normal subsystems
of finitely generated GDMSs.
Finally, we apply our results to Kleinian groups and generalise
a result of Patterson by showing that in any dimension m \in\$N$ there
are Kleinian groups for which the Hausdorff dimension of their uniformly
radial limit set is less than a given arbitrary number d \in\ (0,m) and the Hausdorff dimension
of their Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m).
2009-06-24
2009-06-25T15:15:17Z
2009-06-25T15:15:17Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/719
en
Creative Commons Attribution-NoDerivs 3.0 Unported
http://creativecommons.org/licenses/by-nd/3.0/
687271 bytes
application/pdf
application/pdf
98
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/184652019-09-12T08:13:49Zcom_10023_94com_10023_28col_10023_99
Orderings on words and permutations
McDevitt, Matthew
Ruškuc, Nik
Engineering and Physical Sciences Research Council (EPSRC)
QA171.48M3
Ordered sets
Combinatorial analysis
Permutations
Substructure orderings are ubiquitous throughout combinatorics and all of mathematics.
In this thesis we consider various orderings on words, as well as the consecutive
involvement ordering on permutations. Throughout there will be a focus
on deciding certain order-theoretic properties, primarily the properties of being well-quasi-ordered
(WQO) and of being atomic.
In Chapter 1, we establish the background material required for the remainder of
the thesis. This will include concepts from order theory, formal language theory, automata
theory, and the theory of permutations. We also introduce various orderings
on words, and the consecutive involvement ordering on permutations.
In Chapter 2, we consider the prefix, suffix and factor orderings on words. For the
prefix and suffix orderings, we give a characterisation of the regular languages which
are WQO, and of those which are atomic. We then consider the factor ordering and
show that the atomicity is decidable for finitely-based sets. We also give a new proof
that WQO is decidable for finitely-based sets, which is a special case of a result of
Atminas et al.
In Chapters 3 and 4, we consider some general families of orderings on words. In
Chapter 3 we consider orderings on words which are rational, meaning that they can
be generated by transducers. We discuss the class of insertion relations introduced
in a paper by the author, and introduce a generalisation. In Chapter 4, we
consider three other variations of orderings on words. Throughout these chapters we
prove various decidability results.
In Chapter 5, we consider the consecutive involvement on permutations. We generalise
our results for the factor ordering on words to show that WQO and atomicity
are decidable. Through this investigation we answer some questions which have been
asked (and remain open) for the involvement on permutations.
2019-12-03
2019-09-11T15:45:33Z
2019-09-11T15:45:33Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/18465
en
application/pdf
xii, 157 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/2372019-07-01T10:16:59Zcom_10023_94com_10023_28col_10023_99
On permutation classes defined by token passing networks, gridding matrices and pictures : three flavours of involvement
Waton, Stephen D.
Ruškuc, Nik
Engineering and Physical Sciences Research Council (EPSRC)
Combinatorics
Finite mathematics
QA165.W2
Permutations
The study of pattern classes is the study of the involvement order on finite
permutations. This order can be traced back to the work of Knuth. In recent
years the area has attracted the attention of many combinatoralists and there
have been many structural and enumerative developments. We consider permutations
classes defined in three different ways and demonstrate that asking the same
fixed questions in each case
motivates a different view of involvement. Token passing networks encourage us
to consider permutations as sequences of integers; grid classes encourage us to
consider them as point sets; picture classes, which are developed for the first
time in this thesis, encourage a purely geometrical approach. As we journey
through each area we present several new results.
We begin by
studying the basic definitions of a permutation. This is followed by a discussion
of the questions one would wish to ask of permutation classes. We concentrate on
four particular areas: partial well order, finite basis, atomicity and
enumeration. Our third chapter asks these questions of token passing networks;
we also develop the concept of completeness
and show that it is decidable whether or not a particular network is
complete. Next we move onto grid classes, our analysis using generic sets yields
an algorithm for determining when a grid class is atomic; we also present a new
and elegant proof which demonstrates that certain grid classes are partially
well ordered.
The final chapter
comprises the development and analysis of picture classes. We completely classify
and enumerate those permutations which can be drawn from a circle, those which can be drawn from an X and
those which can be drawn from some convex polygon. We exhibit the first
uncountable set of closed classes to be found in a natural setting; each class
is drawn from three parallel lines. We present a permutation version
of the famous `happy ending' problem of Erdös and Szekeres. We conclude with a
discussion of permutation classes in higher dimensional space.
2007-06-19
2007-03-26T09:11:54Z
2007-03-26T09:11:54Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/237
en
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
http://creativecommons.org/licenses/by-nc-nd/3.0/
1248459 bytes
application/pdf
application/pdf
197
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/294772024-03-14T03:07:22Zcom_10023_94com_10023_28col_10023_99
Rearrangement groups of connected spaces
Khalid, Nayab
Bleak, Collin Patrick
Quick, M. R. (Martyn R.)
Commonwealth Scholarship Commission in the United Kingdom
QA614.86K5
Fractals
We develop a combinatorial framework that assists in finding natural infinite “geometric” presentations for a large subclass of rearrangement groups of fractals – defined by Belk and Forrest, namely rearrangement groups acting on F-type topological spaces. In this framework, for a given fractal set with its group of “rearrangements”, the group generators have a natural one-to-one correspondence with the standard basis of the fractal set, and the relations are all conjugacy relations.
We use this framework to produce a presentation for Richard Thompson’s group F. This presentation has been mentioned before by Dehornoy, but a combinatorial method to find the length of an element in terms of the generating set of this presentation has been hitherto unknown. We provide algorithms that express an element of F in terms of our generating set and reduce a word representing the identity in F to the trivial word.
We conjecture that this framework can be used to find infinite presentations for all groups in the subclass of rearrangement groups acting on F-type topological spaces.
"I have been supported in this PhD by a Commonwealth Scholarship (PKCS-2015-496), funded by the UK Government."--Funding
2020-07-28
2024-03-13T11:07:11Z
2024-03-13T11:07:11Z
Thesis
Doctoral
PhD Doctor of Philosophy
https://hdl.handle.net/10023/29477
https://doi.org/10.17630/sta/816
en
application/pdf
219
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/36872019-07-01T10:19:12Zcom_10023_94com_10023_28col_10023_99
Finiteness conditions for unions of semigroups
Abu-Ghazalh, Nabilah Hani
Ruškuc, Nik
Princess Nora University
Free monogenic semigroup
Disjoint unions
Copies
Clifford
Rees matrix
Finiteness conditions
Finitely presented
Residually finite
Word problem
Membership problem
QA182.A3
Semigroups
In this thesis we prove the following:
The semigroup which is a disjoint union of two or three copies of a group is a Clifford semigroup, Rees matrix semigroup or a combination between a Rees matrix semigroup and a group. Furthermore, the semigroup which is a disjoint union of finitely many copies of a finitely presented (residually finite) group is finitely presented (residually finite) semigroup.
The constructions of the semigroup which is a disjoint union of two copies of the free monogenic semigroup are parallel to the constructions of the semigroup which is a disjoint union of two copies of a group, i.e. such a semigroup is Clifford (strong semilattice of groups) or Rees matrix semigroup. However, the semigroup which is a disjoint union of three copies of the free monogenic semigroup is not just a strong semillatice of semigroups, Rees matrix semigroup or combination between a Rees matrix semigroup and a semigroup, but there are two more semigroups which do not arise from the constructions of the semigroup which is a disjoint union of three copies of a group. We also classify semigroups which are disjoint unions of two or three copies of the free monogenic semigroup. There are three types of semigroups which are unions of two copies of the free monogenic semigroup and nine types of semigroups which are unions of three copies of the free monogenic semigroup. For each type of such semigroups we exhibit a presentation defining semigroups of this type.
The semigroup which is a disjoint union of finitely many copies of the free monogenic semigroup is finitely presented, residually finite, hopfian, has soluble word problem and has soluble subsemigroup membership problem.
2013-06-28
2013-06-13T10:23:41Z
2013-06-13T10:23:41Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/3687
en
Creative Commons Attribution 3.0 Unported
http://creativecommons.org/licenses/by/3.0/
application/pdf
152
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/239562021-09-15T02:07:17Zcom_10023_94com_10023_28col_10023_99
Contributions to the theory of apolarity
Vaidyanathaswamy, R.
Turnbull, Herbert Westren
University of Madras
Madras (India : Presidency)
Great Britain. Department of Scientific and Industrial Research
QA201.V2
Invariants
Geometry, Algebraic
“I am the holder of a Scholarship from the University of Madras and one from the Madras Government. Also of a grant from the British Department of Scientific Research.” -- Career
1924
2021-09-14T09:52:10Z
2021-09-14T09:52:10Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/23956
https://doi.org/10.17630/sta/133
en
application/pdf
44 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/32262019-04-01T08:35:03Zcom_10023_94com_10023_28col_10023_99
Topics in combinatorial semigroup theory
Maltcev, Victor
Ruškuc, Nik
Mitchell, James David
QA182.M2
Semigroups
Combinatorial group theory
In this thesis we discuss various topics from Combinatorial Semigroup Theory: automaton semigroups; finiteness conditions and their preservation under certain semigroup theoretic notions of index; Markov semigroups; word-hyperbolic semigroups; decision problems for finitely presented
and one-relator monoids. First, in order to show that general ideas from Combinatorial Semigroup Theory can apply to uncountable semigroups, at the beginning of the thesis we discuss semigroups with Bergman’s property. We prove that an automaton semigroup generated by a Cayley machine
of a finite semigroup S is itself finite if and only if S is aperiodic, which yields a new characterisation of finite aperiodic monoids. Using this, we derive some further results about Cayley automaton semigroups.
We investigate how various semigroup finiteness conditions, linked to
the notion of ideal, are preserved under finite Rees and Green indices. We
obtain a surprising result that J = D is preserved by supersemigroups of finite Green index, but it is not preserved by subsemigroups of finite Rees index even in the finitely generated case. We also consider the question of preservation of hopficity for finite Rees index. We prove that in general hopficity is preserved neither by finite Rees index subsemigroups, nor by finite Rees index extensions. However, under finite generation assumption,
hopficity is preserved by finite Rees index extensions. Still, there is
an example of a finitely generated hopfian semigroup with a non-hopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent context-free monadic rewriting systems are word-hyperbolic, and provide an example of such a monoid, which does not admit a word-hyperbolic structure with uniqueness.
This answers in the negative a question of Duncan & Gilman. We initiate in this thesis a study of Markov semigroups. We investigate
how the property of being Markov is preserved under finite Rees and
Green indices. For various semigroup properties P we examine whether P , ¬P are Markov properties, and whether P is decidable for finitely presented and
one-relator monoids.
2012-11-30
2012-10-25T13:25:11Z
2012-10-25T13:25:11Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/3226
en
application/pdf
157
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/181572021-03-24T16:41:11Zcom_10023_94com_10023_28col_10023_99
Assouad type dimensions and dimension spectra
Yu, Han
Fraser, Jonathan M.
University of St Andrews. School of Mathematics and Statistics
University of Manchester
Assouad type spectra
Assouad dimension
QA611.3Y8
Dimension theory (Topology)
Metric spaces
Fractals
In the first part of this thesis we introduce a new dimension spectrum motivated by the Assouad dimension; a familiar notion of dimension which, for a given metric space, returns the minimal exponent α ≥ 0 such that for any pair of scales 0 < r < R, any ball of radius R may be covered by a constant times (R/r)ᵅ balls of radius r. To each 𝛩 ∈ (0,1), we associate the appropriate analogue of the Assouad dimension with the restriction that the two scales r and R used in the definition satisfy log R/log r = 𝛩. The resulting 'dimension spectrum' (as a function of 𝛩) thus gives finer geometric information regarding the scaling structure of the space and, in some precise sense, interpolates between the upper box dimension and the Assouad dimension. This latter point is particularly useful because the spectrum is generally better behaved than the Assouad dimension. We also consider the corresponding 'lower spectrum', motivated by the lower dimension, which acts as a dual to the Assouad spectrum. We conduct a detailed study of these dimension spectra; including analytic and geometric properties. We also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with sub-exponential and monotonic winding. We also give several applications of our results, including: dimension distortion estimates under bi-Hölder maps for Assouad dimension. We compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of 'tail densities' defined for subsets of the integers.
In the second part of this thesis, we study the Assouad dimension of sets of integers and deduce a weak solution to the Erdős-Turán conjecture. Let 𝐹 ⊂ ℕ. If $\sum_{n\in F}n^{-1}=\infty$ then 𝐹 "asymptotically" contains arbitrarily long arithmetic progressions.
2019-12-03
2019-07-24T15:04:04Z
2019-07-24T15:04:04Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/18157
https://doi.org/10.17630/10023-18157
en
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
application/pdf
vii, 101 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/65582019-04-01T08:35:03Zcom_10023_94com_10023_28col_10023_99
Cayley automaton semigroups
McLeman, Alexander Lewis Andrew
Ruškuc, Nik
Bleak, Collin Patrick
Semigroup
Automaton
QA182.M6
Semigroups
Cayley graphs
Let S be a semigroup, C(S) the automaton constructed from the right Cayley
graph of S with respect to all of S as the generating set and ∑(C(S)) the
automaton semigroup constructed from C(S). Such semigroups are termed
Cayley automaton semigroups. For a given semigroup S we aim to establish
connections between S and ∑(C(S)).
For a finite monogenic semigroup S with a non-trivial cyclic subgroup C[sub]n we
show that ∑(C(S)) is a small extension of a free semigroup of rank n, and
that in the case of a trivial subgroup ∑(C(S)) is finite.
The notion of invariance is considered and we examine those semigroups S
satisfying S ≅ ∑(C(S)). We classify which bands satisfy this, showing that
they are those bands with faithful left-regular representations, but exhibit
examples outwith this classification. In doing so we answer an open problem
of Cain.
Following this, we consider iterations of the construction and show that for
any n there exists a semigroup where we can iterate the construction n times
before reaching a semigroup satisfying S ≅ ∑(C(S)). We also give an example of a semigroup where repeated iteration never produces a semigroup
satisfying S ≅ ∑(C(S)).
Cayley automaton semigroups of infinite semigroups are also considered and
we generalise and extend a result of Silva and Steinberg to cancellative semigroups. We also construct the Cayley automaton semigroup of the bicyclic
monoid, showing in particular that it is not finitely generated.
2015-06-26
2015-04-23T14:46:35Z
2015-04-23T14:46:35Z
Thesis
Doctoral
PhD Doctor of Philosophy
uk.bl.ethos.644842
http://hdl.handle.net/10023/6558
en
application/pdf
167
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/202182021-02-04T14:44:33Zcom_10023_94com_10023_28col_10023_99
On the regularity dimensions of measures
Howroyd, Douglas Charles
Fraser, Jonathan M.
Falconer, K. J.
Engineering and Physical Sciences Research Council (EPSRC)
Fractal
Dimension theory
Assouad dimension
Upper regularity dimension
Lower regularity dimension
Brownian motion
Self-similar
Self-affine
QA611.3H7
Dimension theory (Topology)
Measure theory
Fractals
Brownian motion processes
This body of work is based upon the following three papers that the author wrote during his PhD with Jonathan Fraser and Han Yu: [FH20, HY17, How19].
Chapter 1 starts by introducing many of the common tools and notation that will be used throughout this thesis. This will cover the main notions of dimensions discussed from both the set and the measure perspectives. An emphasis will be placed on their relationships where possible. This will provide a solid base upon which to expand. Many of the standard results in this part can be found in fractal geometry textbooks such as [Fal03, Mat95] if further reading was desired.
The first results discussed in Chapter 2 will cover some of the regularity dimensions’ properties such as general bounds in relation to the Assouad and lower dimensions, local dimensions and the Lq-spectrum. The Assoaud and lower dimensions are known to interact pleasantly with weak tangents and these ideas are discussed in the regularity dimension setting. We then calculate the regularity dimensions for several specific example measures such as self-similar and self-affine measures which provides an opportunity to discuss the sharpness of the previously obtained bounds. This work originates in [FH20] where the upper regularity dimension was studied, with many of the lower regularity dimension results being natural extensions.
In Chapter 3 we continue the study of the upper and lower regularity dimensions with an emphasis on how they can be used to quantify doubling and uniform perfectness of measures. This starts with an explicit relation between the upper regularity dimension and the doubling constants along with a similar link between the lower regularity dimension and the constants of uniform perfectness. We then turn our attention to a technical result which can be made more quantitative thanks to the regularity dimensions. It is interesting to study how properties, such as doubling, change under pushforwards by different types of maps, here we study the regularity dimensions of pushforward measures with respect to quasisymmetric homeomorphisms. We round this chapter out with an interesting application of the lower regularity to Diophantine approximation by noting the equivalence between uniform perfectness and weakly absolutely α-decaying measures. The original material for this part can be found in [How19] with part of the first section integrating a result of [FH20].
Finally, in Chapter 4, we will consider graphs of Brownian motion, and more generally, graphs of Levy processes. This will involve the calculation of the lower and Assouad dimensions for such sets and then the regularity dimensions of measures pushed onto these graphs from the real line. These graphs are the only examples in this thesis for which the Assouad and lower dimensions had not been previously calculated so we delve deeper into the area, studying graphs of functions defined as stochastic integrals as well. This chapter is based on the paper [HY17] for the set theoretic half, with the regularity dimension results coming from [How19].
"I would aso like to thank the EPSRC for funding me with the Doctoral training Grant (EP/N509759/1)" -- Acknowledgements
2020
2020-07-08T11:26:03Z
2020-07-08T11:26:03Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/20218
https://doi.org/10.17630/10023-20218
en
Fraser, Jonathan & Howroyd, Douglas. (2017). On the upper regularity dimensions of measures. Indiana University Mathematics Journal. 69.
Howroyd, Douglas & Yu, Han. (2017). Assouad Dimension of Random Processes. Proceedings of the Edinburgh Mathematical Society. 10.
Howroyd, Douglas. (2019). Regularity dimensions: quantifying doubling and uniform perfectness.
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
application/pdf
110 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/8382019-04-01T08:35:04Zcom_10023_94com_10023_28col_10023_99
The geometry of self-affine fractals
Miao, Jun Jie
QA614.86M52
Fractals
In this thesis we study the dimension theory of self-affine sets. We begin by
introducing a number of notions from fractal geometry, in particular, dimensions,
measure properties and iterated functions systems. We give a review of existing
work on self-affine sets. We then develop a variety of new results on self-affine
sets and their dimensional properties.
This work falls into three parts:
Firstly, we look at the dimension formulae for a class of self-affine sets generated
by upper triangular matrices. In this case, we simplify the affine dimension
formula into equations only involving the diagonal elements of the matrices.
Secondly, since the Hausdorff dimensions of self-affine sets depend not only
on the linear parts of the contractions but also on the translation parameters, we
obtain an upper bound for the dimensions of exceptional sets, that is, the set of
parameters such that the Hausdorff dimension of the attractor is smaller than the
affine dimension.
Thirdly, we investigate dimensions of a class of random self-affine sets, aiming
to extend the ‘almost sure’ formula for random self-similar sets to random self-affine
sets.
2008
2009-12-16T15:34:06Z
2009-12-16T15:34:06Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/838
en
application/pdf
86 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/258262022-08-12T08:47:42Zcom_10023_94com_10023_28col_10023_99
Base size and generating graphs of primitive permutation groups
Kelsey, Veronica
Roney-Dougal, Colva Mary
Quick, M. R. (Martyn R.)
University of St Andrews. School of Mathematics and Statistics
Base size
Generating graphs
Simple groups
Permutation group
Relational complexity
Computational complexity
Alternating groups
Symmetric groups
QA175.K4
Permutation groups
In this thesis we consider base size and properties of the generating graph for finite groups.
Let Ω = {1,...,n}, let Sₙ = Sym({1,...,n}) and let G ≤ Sₙ. A base for G is a sequence Λ = (ω₁, . . . , ωₖ) of points in Ω such that the pointwise stabilizer, G_{ω₁,...,ωₖ} , is the identity. The base size of G, denoted by b(G, Ω) or b(G), is the length of the shortest base. We say that Λ is an irredundant base if
G > G_{ω₁} > G_{ω₁,ω₂} > ··· > G_{ω₁,ω₂,...,ωₖ} = 1.
If no irredundant base is longer than Λ, then we say that Λ is a maximal irredundant base for G and denote its length by I(G). A group is called large base if it is either a product action or almost simple group, and its socle is one or more copies of the alternating group Aᵣ acting on k-sets.
Let G be a primitive subgroup of Sₙ that is not large base. We prove that any irredundant base for G has size at most 5log₂n. This bound is best possible up to a small multiplicative constant and is the first logarithmic bound on the size of an irredundant base for such groups. We show that for any constant c, there are infinitely many primitive groups with maximal irredundant base size at least c times the minimal base size. As a corollary of the first result, the relational complexity of G, denoted RC(G) (see Definition 2.2.10), is at most 5log₂n + 1. In addition the maximal size of a minimal base and the height, denoted B(G) and H(G) (see Definitions 2.2.1 and 2.2.5), are both at most 5log₂n. Furthermore, we deduce that a base for G of size at most 5log₂n can be computed in polynomial time.
The generating graph Γ(G) of a finite group G has vertex set the non-identity elements of G, with two elements connected exactly when they generate G. A coclique in a graph is an empty induced subgraph, so a coclique in Γ(G) is a subset of G such that no pair of elements generate G. A coclique is maximal if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of G form a coclique in Γ(G), but this coclique need not be maximal.
Let G = Sₙ or Aₙ. We first determine when the intransitive maximal subgroups of G are maximal cocliques in Γ(G), and when they are not we find the unique maximal coclique in which they are contained. We then show that for sufficiently large n, the imprimitive maximal subgroups of G are all maximal cocliques in Γ(G).
In addition, using the result on intransitive maximal subgroups we prove that a conjecture of Cameron, Lucchini, and Roney-Dougal holds for G under certain restrictions on n. Namely we prove that two elements of G have identical sets of neighbours in Γ(G) if and only if they belong to exactly the same maximal subgroups. Finally under another set of restrictions on n we then determine precisely which maximal subgroups are maximal cocliques in Γ(G).
2022-06-14
2022-08-11T14:41:22Z
2022-08-11T14:41:22Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/25826
https://doi.org/10.17630/sta/192
en
application/pdf
xii, 204 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/70672019-04-01T08:35:04Zcom_10023_94com_10023_28col_10023_99
The maximal subgroups of the classical groups in dimension 13, 14 and 15
Schröder, Anna Katharina
Roney-Dougal, Colva Mary
Group theory
Classical groups
Classification of finite simple groups
Maximal subgroups
QA177.S3
Finite simple groups
Maximal subgroups
One might easily argue that the Classification of Finite Simple Groups is
one of the most important theorems of group theory. Given that any finite
group can be deconstructed into its simple composition factors, it is of great
importance to have a detailed knowledge of the structure of finite simple
groups.
One of the classes of finite groups that appear in the classification theorem
are the simple classical groups, which are matrix groups preserving
some form. This thesis will shed some new light on almost simple classical
groups in dimension 13, 14 and 15. In particular we will determine their
maximal subgroups.
We will build on the results by Bray, Holt, and Roney-Dougal who
calculated the maximal subgroups of all almost simple finite classical groups
in dimension less than 12. Furthermore, Aschbacher proved that the maximal
subgroups of almost simple classical groups lie in nine classes. The maximal
subgroups in the first eight classes, i.e. the subgroups of geometric type,
were determined by Kleidman and Liebeck for
dimension greater than 13.
Therefore this thesis concentrates on the ninth class of Aschbacher’s
Theorem. This class roughly consists of subgroups which are almost simple
modulo scalars and do not preserve a geometric structure. As our final
result we will give tables containing all maximal subgroups of almost simple
classical groups in dimension 13, 14 and 15.
2015-11-30
2015-07-28T15:24:59Z
2015-07-28T15:24:59Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/7067
en
J.N. Bray, D.F. Holt and C.M. Roney-Dougal. The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. London Mathematical Society Lecture Note Series 407. Cambridge University Press, Cambridge, 2013.
application/pdf
application/zip
198
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/9452019-07-01T10:16:43Zcom_10023_94com_10023_28col_10023_99
Classification and enumeration of finite semigroups
Distler, Andreas
Ruškuc, Nik
Linton, Stephen
University of St Andrews
Nilpotent semigroups
Enumerative combinatorics
Data library
Power group enumeration
3-nilpotent
Computer search
Constraint satisfaction
Computer algebra
QA182.D5
Semigroups
Combinatorial analysis
The classification of finite semigroups is difficult even for small
orders because of their large number. Most finite semigroups are
nilpotent of nilpotency rank 3. Formulae for their number up to
isomorphism, and up to isomorphism and anti-isomorphism of any order
are the main results in the theoretical part of this thesis. Further
studies concern the classification of nilpotent semigroups by rank,
leading to a full classification for large ranks.
In the computational part, a method to find and
enumerate multiplication tables of semigroups and subclasses is
presented. The approach combines the advantages of computer algebra
and constraint satisfaction, to allow for an efficient and fast
search. The problem of avoiding isomorphic and anti-isomorphic
semigroups is dealt with by supporting standard methods from
constraint satisfaction with structural knowledge about the semigroups
under consideration. The approach is adapted to various
problems, and realised using the computer algebra system GAP and the
constraint solver Minion. New results include the numbers of
semigroups of order 9, and of monoids and bands of order 10. Up to
isomorphism and anti-isomorphism there are 52,989,400,714,478 semigroups
with 9 elements, 52,991,253,973,742 monoids with 10 elements, and
7,033,090 bands with 10 elements. That constraint satisfaction can also
be utilised for the analysis of algebraic objects is demonstrated by
determining the automorphism groups of all semigroups with 9 elements.
A classification of the semigroups of orders 1 to 8 is made available
as a data library in form of the GAP package Smallsemi. Beyond the
semigroups themselves a large amount of precomputed properties is
contained in the library. The package as well as the code used to
obtain the enumeration results are available on the attached DVD.
2010-06-23
2010-06-29T08:43:31Z
2010-06-29T08:43:31Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/945
en
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported
http://creativecommons.org/licenses/by-nc-sa/3.0/
application/pdf
application/zip
application/zip
application/zip
xviii+154
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/173502021-02-19T13:26:26Zcom_10023_94com_10023_28col_10023_99
Semigroup congruences : computational techniques and theoretical applications
Torpey, Michael
Mitchell, James David
Engineering and Physical Sciences Research Council (EPSRC)
Semigroups
Computational algebra
Computational mathematics
Congruences
Algorithms
Diagram semigroups
Bipartitions
Bipartition semigroups
Partition semigroups
Motzkin monoid
Finite semigroups
Finitely presented semigroups
Simple semigroups
0-simple semigroups
Inverse semigroups
QA182.T7
Semigroups--Data processing
Algorithms
Computational semigroup theory is an area of research that is subject to growing interest. The development of semigroup algorithms allows for new theoretical results to be discovered, which in turn informs the creation of yet more algorithms. Groups have benefitted from this cycle since before the invention of electronic computers, and the popularity of computational group theory has resulted in a rich and detailed literature. Computational semigroup theory is a less developed field, but recent work has resulted in a variety of algorithms, and some important pieces of software such as the Semigroups package for GAP.
Congruences are an important part of semigroup theory. A semigroup’s congruences determine its homomorphic images in a manner analogous to a group’s normal subgroups. Prior to the work described here, there existed few practical algorithms for computing with semigroup congruences. However, a number of results about alternative representations for congruences, as well as existing algorithms that can be borrowed from group theory, make congruences a fertile area for improvement. In this thesis, we first consider computational techniques that can be applied to the study of congruences, and then present some results that have been produced or precipitated by applying these techniques to interesting examples.
After some preliminary theory, we present a new parallel approach to computing with congruences specified by generating pairs. We then consider alternative ways of representing a congruence, using intermediate objects such as linked triples. We also present an algorithm for computing the entire congruence lattice of a finite semigroup. In the second part of the thesis, we classify the congruences of several monoids of bipartitions, as well as the principal factors of several monoids of partial transformations. Finally, we consider how many congruences a finite semigroup can have, and examine those on semigroups with up to seven elements.
2019-06-25
2019-03-22T15:02:32Z
2019-03-22T15:02:32Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/17350
https://doi.org/10.17630/10023-17350
en
application/pdf
194 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/249302022-02-24T03:10:11Zcom_10023_94com_10023_28col_10023_99
The invariant theory of linear complexes associated with a quaternary quadric
DasGupta, Pramathanath
Turnbull, Herbert Westren
"The invariants and covariants of a quaternary system of a linear complex and a quadric have been discussed by Weitzenbock, but he excludes the mixed concomitants. In the present paper the concomitants (which will include mixed forms) of a quadric and two linear complexes are discussed. The Reduced Prepared System is given in §3, the Complete System in §8 and some invariants of the covariant forms in §9. The concomitants of a quadric and 𝑛 linear complexes have next been considered and the corresponding reduced system of typical forms, is given at the end of the paper." -- From the Introduction.
1929
2022-02-23T10:34:02Z
2022-02-23T10:34:02Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/24930
https://doi.org/10.17630/sta/157
en
application/pdf
71 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/165212019-04-01T08:35:09Zcom_10023_94com_10023_28col_10023_99
Computational techniques in finite semigroup theory
Wilson, Wilf A.
Mitchell, James David
Carnegie Trust for the Universities of Scotland
University of St Andrews. School of Mathematics and Statistics
Semigroup theory
Computational algebra
Maximal subsemigroups
Semigroups
Computational semigroup theory
Rees matrix semigroups
Rees 0-matrix semigroups
Direct products
Algorithms
Transformation semigroups
Diagram monoids
Partition monoids
Monoids
Generating sets
Green's relations
QA182.W5
Semigroups--Data processing
Algebra--Data processing
Group theory
A semigroup is simply a set with an associative binary operation; computational semigroup theory is the branch of mathematics concerned with developing techniques for computing with semigroups, as well as investigating semigroups with the help of computers. This thesis explores both sides of computational semigroup theory, across several topics, especially in the finite case.
The central focus of this thesis is computing and describing maximal subsemigroups of finite semigroups. A maximal subsemigroup of a semigroup is a proper subsemigroup that is contained in no other proper subsemigroup. We present novel and useful algorithms for computing the maximal subsemigroups of an arbitrary finite semigroup, building on the paper of Graham, Graham, and Rhodes from 1968. In certain cases, the algorithms reduce to computing maximal subgroups of finite groups, and analysing graphs that capture information about the regular ℐ-classes of a semigroup. We use the framework underpinning these algorithms to describe the maximal subsemigroups of many families of finite transformation and diagram monoids. This reproduces and greatly extends a large amount of existing work in the literature, and allows us to easily see the common features between these maximal subsemigroups.
This thesis is also concerned with direct products of semigroups, and with a special class of semigroups known as Rees 0-matrix semigroups. We extend known results concerning the generating sets of direct products of semigroups; in doing so, we propose techniques for computing relatively small generating sets for certain kinds of direct products. Additionally, we characterise several features of Rees 0-matrix semigroups in terms of their underlying semigroups and matrices, such as their Green's relations and generating sets, and whether they are inverse. In doing so, we suggest new methods for computing Rees 0-matrix semigroups.
2019-06-25
2018-11-21T16:52:04Z
2018-11-21T16:52:04Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/16521
en
application/pdf
197 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/89262019-04-01T08:35:09Zcom_10023_94com_10023_28col_10023_99
Aspects of order and congruence relations on regular semigroups
Gomes, Gracinda Maria dos Santos
Howie, John M. (John Mackintosh)
Instituto Nacional de Investigação Científica (Portugal)
Universidade de Lisboa
Overseas Research Students Fees Support Scheme
University of St Andrews
QA171.G7
Group theory
On a regular semigroup S natural order relations have been defined
by Nambooripad and by Lallement. Different characterisations and
relationships between the Nambooripad order J, Lallement's order λ and
a certain relation k are considered in Chapter I. It is shown that on
a regular semigroup S the partial order J is left compatible if and
only if S is locally R-unipotent. This condition in the case where S
is orthodox is equivalent to saying that E(S) is a left seminormal
band. It is also proved that λ is the least compatible partial order
contained in J and that k = λ if and only if k is compatible and k
if and only if J is compatible. A description of λ and J in the
semigroups T(X) and PT(X) is presented.
In Chapter II, it is proved that in an orthodox semigroup S the
band of idempotents E(S) is left quasinormal if and only if there
exists a local isomorphism from S onto an R-unipotent semigroup. It is
shown that there exists a least R-unipotent congruence on any orthodox
semigroup, generated by a certain left compatible equivalence R. This
equivalence is a congruence if and only if E(S) is a right semiregular
band.
The last Chapter is particularly concerned with the description of
R-unipotent congruences on a regular semigroup S by means of their
kernels and traces. The lattice RC(S) of all R-unipotent congruences
on a regular semigroup S is studied. A congruence≡ on the lattice
RC(S) is considered and the greatest and the least element of each
≡-class are described.
1983
2016-06-03T10:13:08Z
2016-06-03T10:13:08Z
Thesis
Doctoral
PhD Doctor of Philosophy
uk.bl.ethos.687032
http://hdl.handle.net/10023/8926
en
application/pdf
iv, 119 p.
University of St Andrews
The University of St Andrews
oai:research-repository.st-andrews.ac.uk:10023/237862021-08-18T02:08:51Zcom_10023_94com_10023_28col_10023_99
Multifractal measures : from self-affine to nonlinear
Lee, Lawrence David
Falconer, K. J.
Fraser, Jonathan M.
Engineering and Physical Sciences Research Council (EPSRC)
Fractal geometry
Fractals
Multifractals
Self-affine
Nonlinear
𝐿𝑞-spectra
𝐿[sup]𝑞-spectra
Ledrappier-Young formulae
Dimension theory
Box dimension
Exact dimensionality
Measure theory
QA614.86L44
Multifractals
Dimension theory (Topology)
Measure theory
This thesis is based on three papers the author wrote during his time as a PhD student
[28, 17, 33].
In Chapter 2 we study 𝐿[sup]𝑞-spectra of planar self-affine measures generated by diagonal
matrices. We introduce a new technique for constructing and understanding examples
based on combinatorial estimates for the exponential growth of certain split binomial
sums. Using this approach we find counterexamples to a statement of Falconer and Miao
from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for
the generalised dimensions of generic self-affine measures.
We also answer a question of Fraser from 2016 in the negative by proving that a certain natural closed form expression does not generally give the 𝐿[sup]𝑞-spectrum. As a further
application we provide examples of self-affine measures whose 𝐿[sup]𝑞-spectra exhibit new
types of phase transitions. Finally, we provide new non-trivial closed form bounds for
the 𝐿[sup]𝑞-spectra, which in certain cases yield sharp results.
In Chapter 3 we study 𝐿[sup]𝑞-spectra of measures in the plane generated by certain nonlinear maps. In particular we study attractors of iterated function systems consisting
of maps whose components are 𝐶[sup](1+α) and for which the Jacobian is a lower triangular
matrix at every point subject to a natural domination condition on the entries. We
calculate the 𝐿[sup]𝑞-spectrum of Bernoulli measures supported on such sets using an appropriately defined analogue of the singular value function and an appropriate pressure function.
In Chapter 4 we study a more general class of invariant measures supported on the attractors introduced in Chapter 3. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier-Young formula.
"The work in this thesis was supported by an Engineering and Physical Sciences Research
Council (EPSRC) Doctoral Training Grant (EP/N509759/1)." -- Funding
2021-12-01
2021-08-17T10:00:52Z
2021-08-17T10:00:52Z
Thesis
Doctoral
PhD Doctor of Philosophy
http://hdl.handle.net/10023/23786
https://doi.org/10.17630/sta/123
EP/N509759/1
en
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
application/pdf
viii, 119 p.
University of St Andrews
The University of St Andrews