DSpace Community:
http://hdl.handle.net/10023/94
Sat, 08 Mar 2014 00:21:54 GMT
20140308T00:21:54Z

Generalized dimensions of images of measures under Gaussian processes
http://hdl.handle.net/10023/4319
Abstract: We show that for certain Gaussian random processes and fields X:RN→Rd, Dq(μx) = min {d, 1/α Dq (μ)} a.s., for an index α which depends on Hölder properties and strong local nondeterminism of X, where q>1, where Dq denotes generalized qdimension μX is the image of the measure μ under X. In particular this holds for indexα fractional Brownian motion, for fractional Riesz–Bessel motions and for certain infinity scale fractional Brownian motions.
Description: 26 pages
Sat, 15 Feb 2014 00:00:00 GMT
http://hdl.handle.net/10023/4319
20140215T00:00:00Z
Falconer, Kenneth
Xiao, Yimin
We show that for certain Gaussian random processes and fields X:RN→Rd, Dq(μx) = min {d, 1/α Dq (μ)} a.s., for an index α which depends on Hölder properties and strong local nondeterminism of X, where q>1, where Dq denotes generalized qdimension μX is the image of the measure μ under X. In particular this holds for indexα fractional Brownian motion, for fractional Riesz–Bessel motions and for certain infinity scale fractional Brownian motions.

Inhomogeneous parabolic equations on unbounded metric measure spaces
http://hdl.handle.net/10023/4061
Abstract: We study the inhomogeneous semilinear parabolic equation ut = Δu + up + f(x), with source term f independent of time and subject to f(x) ≥ 0 and with u(0, x) = φ(x) ≥ 0, for the very general setting of a metric measure space. By establishing Harnacktype inequalities in time t and some powerful estimates, we give sufficient conditions for nonexistence, local existence and global existence of weak solutions, depending on the value of p relative to a critical exponent.
Mon, 01 Oct 2012 00:00:00 GMT
http://hdl.handle.net/10023/4061
20121001T00:00:00Z
Falconer, Kenneth John
Hu, Jiaxin
Sun, Yuhua
We study the inhomogeneous semilinear parabolic equation ut = Δu + up + f(x), with source term f independent of time and subject to f(x) ≥ 0 and with u(0, x) = φ(x) ≥ 0, for the very general setting of a metric measure space. By establishing Harnacktype inequalities in time t and some powerful estimates, we give sufficient conditions for nonexistence, local existence and global existence of weak solutions, depending on the value of p relative to a critical exponent.

Strong renewal theorems and Lyapunov spectra for alphaFarey and alphaLuroth systems
http://hdl.handle.net/10023/3933
Abstract: In this paper, we introduce and study the alphaFarey map and its associated jump transformation, the alphaLuroth map, for an arbitrary countable partition alpha of the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called alphasumlevel sets for the alphaLuroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the alphaFarey map and the alphaLuroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partition alpha.
Fri, 01 Jun 2012 00:00:00 GMT
http://hdl.handle.net/10023/3933
20120601T00:00:00Z
Kesseboehmer, Marc
Munday, Sara
Stratmann, Bernd O.
In this paper, we introduce and study the alphaFarey map and its associated jump transformation, the alphaLuroth map, for an arbitrary countable partition alpha of the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called alphasumlevel sets for the alphaLuroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the alphaFarey map and the alphaLuroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partition alpha.

Dimension and measure for generic continuous images
http://hdl.handle.net/10023/3902
Abstract: We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, X, into Rn. The key question is 'what is the generic dimension of f(X)?' and we consider two different approaches to answering it: Baire category and prevalence. In the Baire category setting we prove that typically the packing and upper box dimensions are as large as possible, n, but find that the behaviour of the Hausdorff, lower box and topological dimensions is considerably more subtle. In fact, they are typically equal to the minimum of n and the topological dimension of X. We also study, the typical Hausdorff and packing measures of f (X) and, in particular, give necessary and sufficient conditions for them to be zero, positive and finite, or infinite. It is interesting to compare the Baire category results with results in the prevalence setting. As such we also discuss a result of Dougherty on the prevalent topological dimension of f (X) and give some simple applications concerning the prevalent dimensions of graphs of realvalued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux.
Tue, 01 Jan 2013 00:00:00 GMT
http://hdl.handle.net/10023/3902
20130101T00:00:00Z
Balka, Richard
Farkas, Abel
Fraser, Jonathan M.
Hyde, James T.
We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, X, into Rn. The key question is 'what is the generic dimension of f(X)?' and we consider two different approaches to answering it: Baire category and prevalence. In the Baire category setting we prove that typically the packing and upper box dimensions are as large as possible, n, but find that the behaviour of the Hausdorff, lower box and topological dimensions is considerably more subtle. In fact, they are typically equal to the minimum of n and the topological dimension of X. We also study, the typical Hausdorff and packing measures of f (X) and, in particular, give necessary and sufficient conditions for them to be zero, positive and finite, or infinite. It is interesting to compare the Baire category results with results in the prevalence setting. As such we also discuss a result of Dougherty on the prevalent topological dimension of f (X) and give some simple applications concerning the prevalent dimensions of graphs of realvalued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux.

Dimension theory and fractal constructions based on selfaffine carpets
http://hdl.handle.net/10023/3869
Abstract: The aim of this thesis is to develop the dimension theory of selfaffine carpets in several directions. Selfaffine carpets are an important class of planar selfaffine 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 wellunderstood world of selfsimilar sets and the far from understood world of general selfaffine 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 selfsimilar 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 selfaffine 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 selfaffine carpets, which we call boxlike selfaffine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on selfaffine 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 selfaffine 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 quasiconformal 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 46 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1variable attractors, with the aim of developing the dimension theory of selfaffine carpets in these directions.
In order to put our work into context, in Chapter 4 we consider inhomogeneous selfsimilar 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 selfaffine setting and, in Chapter 5, investigate the dimensions of inhomogeneous selfaffine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of selfsimilar 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 selfaffine 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 `biLipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random selfsimilar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random selfaffine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.
Fri, 29 Nov 2013 00:00:00 GMT
http://hdl.handle.net/10023/3869
20131129T00:00:00Z
Fraser, Jonathan M.
The aim of this thesis is to develop the dimension theory of selfaffine carpets in several directions. Selfaffine carpets are an important class of planar selfaffine 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 wellunderstood world of selfsimilar sets and the far from understood world of general selfaffine 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 selfsimilar 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 selfaffine 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 selfaffine carpets, which we call boxlike selfaffine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on selfaffine 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 selfaffine 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 quasiconformal 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 46 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1variable attractors, with the aim of developing the dimension theory of selfaffine carpets in these directions.
In order to put our work into context, in Chapter 4 we consider inhomogeneous selfsimilar 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 selfaffine setting and, in Chapter 5, investigate the dimensions of inhomogeneous selfaffine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of selfsimilar 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 selfaffine 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 `biLipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random selfsimilar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random selfaffine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.

Finiteness conditions for unions of semigroups
http://hdl.handle.net/10023/3687
Abstract: 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.
Fri, 28 Jun 2013 00:00:00 GMT
http://hdl.handle.net/10023/3687
20130628T00:00:00Z
AbuGhazalh, Nabilah Hani
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.

Adventures in applying iteration lemmas
http://hdl.handle.net/10023/3671
Abstract: 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 socalled iteration lemmas.
We define a hierarchy of the considered classes of semigroups and connect our original results with previous research.
Fri, 28 Jun 2013 00:00:00 GMT
http://hdl.handle.net/10023/3671
20130628T00:00:00Z
Pfeiffer, Markus Johannes
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 socalled iteration lemmas.
We define a hierarchy of the considered classes of semigroups and connect our original results with previous research.

Ends of semigroups
http://hdl.handle.net/10023/3590
Abstract: 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 quasiisometric 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.
Tue, 01 Jan 2013 00:00:00 GMT
http://hdl.handle.net/10023/3590
20130101T00:00:00Z
Craik, Simon
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 quasiisometric 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.

Random generation and chief length of finite groups
http://hdl.handle.net/10023/3578
Abstract: 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 2generated 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.
Tue, 01 Jan 2013 00:00:00 GMT
http://hdl.handle.net/10023/3578
20130101T00:00:00Z
Menezes, Nina E.
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 2generated 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.

Multistable processes and localizability
http://hdl.handle.net/10023/3560
Abstract: We use characteristic functions to construct alphamultistable measures and integrals, where the measures behave locally like stable measures, but with the stability index alpha(x) varying with x. This enables us to construct alphamultistable processes on R, that is processes whose scaling limit at time t is an alpha(t)stable process. We present several examples of such multistable processes and examine their localisability.
Sun, 01 Jan 2012 00:00:00 GMT
http://hdl.handle.net/10023/3560
20120101T00:00:00Z
Falconer, Kenneth John
Liu, Lining
We use characteristic functions to construct alphamultistable measures and integrals, where the measures behave locally like stable measures, but with the stability index alpha(x) varying with x. This enables us to construct alphamultistable processes on R, that is processes whose scaling limit at time t is an alpha(t)stable process. We present several examples of such multistable processes and examine their localisability.

Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances
http://hdl.handle.net/10023/3383
Abstract: Let ΩΩ be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of ΩΩ, then we write U≈V if there exists a finite subset F of ΩΩ such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in ΩΩ is the least cardinality of a subset A of ΩΩ such that the union of U and A generates ΩΩ. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω. The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in ΩΩ where d is the minimum cardinality of a dominating family for NN. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in ΩΩ are 0, 1, 2, and d. We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2ℵ0.
Wed, 01 Sep 2010 00:00:00 GMT
http://hdl.handle.net/10023/3383
20100901T00:00:00Z
Mitchell, James David
Morayne, Michal
Peresse, Yann Hamon
Quick, Martyn
Let ΩΩ be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of ΩΩ, then we write U≈V if there exists a finite subset F of ΩΩ such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in ΩΩ is the least cardinality of a subset A of ΩΩ such that the union of U and A generates ΩΩ. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω. The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in ΩΩ where d is the minimum cardinality of a dominating family for NN. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in ΩΩ are 0, 1, 2, and d. We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2ℵ0.

Endomorphisms of Fraïssé limits and automorphism groups of algebraically closed relational structures
http://hdl.handle.net/10023/3358
Abstract: 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 Hclasses, 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 alephnaught] 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 alephnaught] 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 alephnaught] regular Dclasses of End(Ω) and when Ω = R,D,B there exist 2[superscript alephnaught] Jclasses of End(Ω). Additionally we show
that if Ω = R,D then all regular Dclasses contain 2[superscript alephnaught] group Hclasses. On the other hand, we show that when
Ω = B,Q there exist regular Dclasses
which contain countably many group Hclasses.
Fri, 30 Nov 2012 00:00:00 GMT
http://hdl.handle.net/10023/3358
20121130T00:00:00Z
McPhee, Jillian Dawn
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 Hclasses, 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 alephnaught] 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 alephnaught] 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 alephnaught] regular Dclasses of End(Ω) and when Ω = R,D,B there exist 2[superscript alephnaught] Jclasses of End(Ω). Additionally we show
that if Ω = R,D then all regular Dclasses contain 2[superscript alephnaught] group Hclasses. On the other hand, we show that when
Ω = B,Q there exist regular Dclasses
which contain countably many group Hclasses.

Every group is a maximal subgroup of the free idempotent generated semigroup over a band
http://hdl.handle.net/10023/3342
Abstract: Given an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area.
Wed, 01 May 2013 00:00:00 GMT
http://hdl.handle.net/10023/3342
20130501T00:00:00Z
Dolinka, I
Ruskuc, Nik
Given an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area.

On disjoint unions of finitely many copies of the free monogenic semigroup
http://hdl.handle.net/10023/3341
Abstract: Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.
Thu, 01 Aug 2013 00:00:00 GMT
http://hdl.handle.net/10023/3341
20130801T00:00:00Z
Abughazalah, Nabilah
Ruskuc, Nik
Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.

Ideals and finiteness conditions for subsemigroups
http://hdl.handle.net/10023/3335
Abstract: In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, D=J and minimal conditions on ideals.
Wed, 01 Jan 2014 00:00:00 GMT
http://hdl.handle.net/10023/3335
20140101T00:00:00Z
Gray, Robert Duncan
Maltcev, Victor
D. Mitchell, J.
Ruskuc, N.
In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, D=J and minimal conditions on ideals.

Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure
http://hdl.handle.net/10023/3237
Abstract: For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2vertex directed graph IFSs have attractors that cannot be the attractors of standard (1vertex 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.
Tue, 01 Jan 2013 00:00:00 GMT
http://hdl.handle.net/10023/3237
20130101T00:00:00Z
Boore, Graeme
Falconer, Kenneth John
For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2vertex directed graph IFSs have attractors that cannot be the attractors of standard (1vertex 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.

Topics in combinatorial semigroup theory
http://hdl.handle.net/10023/3226
Abstract: 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; wordhyperbolic semigroups; decision problems for finitely presented
and onerelator 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 nonhopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent contextfree monadic rewriting systems are wordhyperbolic, and provide an example of such a monoid, which does not admit a wordhyperbolic 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
onerelator monoids.
Fri, 30 Nov 2012 00:00:00 GMT
http://hdl.handle.net/10023/3226
20121130T00:00:00Z
Maltcev, Victor
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; wordhyperbolic semigroups; decision problems for finitely presented
and onerelator 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 nonhopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent contextfree monadic rewriting systems are wordhyperbolic, and provide an example of such a monoid, which does not admit a wordhyperbolic 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
onerelator monoids.

Growth of generating sets for direct powers of classical algebraic structures
http://hdl.handle.net/10023/3058
Abstract: For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.
Sun, 01 Aug 2010 00:00:00 GMT
http://hdl.handle.net/10023/3058
20100801T00:00:00Z
Quick, Martyn
Ruskuc, Nik
For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.

Presentations and efficiency of semigroups
http://hdl.handle.net/10023/2843
Abstract: 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 Odirect 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.
Thu, 01 Jan 1998 00:00:00 GMT
http://hdl.handle.net/10023/2843
19980101T00:00:00Z
Ayik, Hayrullah
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 Odirect 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.

Semigroups of orderdecreasing transformations
http://hdl.handle.net/10023/2834
Abstract: Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial onetoone) transformations of X. In this
Thesis the study of orderincreasing full (partial, partial onetoone) transformations
has been reduced to that of orderdecreasing full (partial, partial onetoone)
transformations and the study of orderdecreasing partial transformations to that of
orderdecreasing full transformations for both the finite and infinite cases.
For the finite orderdecreasing full (partial onetoone) transformation
semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden
(1990) concerning products of idempotents (quasiidempotents), and concerning
combinatorial and rank properties. By contrast with the semigroups of orderpreserving
transformations and the full transformation semigroup, the semigroups of orderdecreasing
full (partial onetoone) 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 orderdecreasing transformations and their Rees
quotient semigroups is obtained.
If X is an infinite chain then the semigroup S of orderdecreasing 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
orderdecreasing partial onetoone transformations is type A.
The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing
full (partial onetoone) transformations have been investigated.
Wed, 01 Jan 1992 00:00:00 GMT
http://hdl.handle.net/10023/2834
19920101T00:00:00Z
Umar, Abdullahi
Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial onetoone) transformations of X. In this
Thesis the study of orderincreasing full (partial, partial onetoone) transformations
has been reduced to that of orderdecreasing full (partial, partial onetoone)
transformations and the study of orderdecreasing partial transformations to that of
orderdecreasing full transformations for both the finite and infinite cases.
For the finite orderdecreasing full (partial onetoone) transformation
semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden
(1990) concerning products of idempotents (quasiidempotents), and concerning
combinatorial and rank properties. By contrast with the semigroups of orderpreserving
transformations and the full transformation semigroup, the semigroups of orderdecreasing
full (partial onetoone) 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 orderdecreasing transformations and their Rees
quotient semigroups is obtained.
If X is an infinite chain then the semigroup S of orderdecreasing 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
orderdecreasing partial onetoone transformations is type A.
The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing
full (partial onetoone) transformations have been investigated.

Semigroup presentations
http://hdl.handle.net/10023/2821
Abstract: 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 Osimple semigroups, transformation semigroups,
matrix semigroups and various endomorphism semigroups. In Chapter 6
we find presentations for the following semi group constructions: wreath product,
BruckReilly 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 ReidemeisterSchreier 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 ToddCoxeter
enumeration procedure and introduce three modifications of this procedure.
Sun, 01 Jan 1995 00:00:00 GMT
http://hdl.handle.net/10023/2821
19950101T00:00:00Z
Ruskuc, Nikola
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 Osimple semigroups, transformation semigroups,
matrix semigroups and various endomorphism semigroups. In Chapter 6
we find presentations for the following semi group constructions: wreath product,
BruckReilly 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 ReidemeisterSchreier 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 ToddCoxeter
enumeration procedure and introduce three modifications of this procedure.

Green index in semigroups : generators, presentations and automatic structures
http://hdl.handle.net/10023/2760
Abstract: The Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S n T under the natural actions of T on S via right and left multiplication. This partitions the complement S nT into Trelative H classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schützenberger group. If the Rees index ΙS n TΙ is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows how generating sets for S may be used to obtain generating sets for T and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of groupembeddable semigroups).
Sun, 01 Jan 2012 00:00:00 GMT
http://hdl.handle.net/10023/2760
20120101T00:00:00Z
Cain, A.J.
Gray, R
Ruskuc, Nik
The Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S n T under the natural actions of T on S via right and left multiplication. This partitions the complement S nT into Trelative H classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schützenberger group. If the Rees index ΙS n TΙ is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows how generating sets for S may be used to obtain generating sets for T and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of groupembeddable semigroups).

The visible part of plane selfsimilar sets
http://hdl.handle.net/10023/2756
Abstract: Given a compact subset F of R2, the visible part VθF of F from direction θ is the set of x in F such that the halfline from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1 then dimH VθF = 1 for almost all θ , where dimH denotes Hausdorff dimension. We conrm this when F is a selfsimilar set satisfying the convex open set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected.
Description: JMF was supported by an EPSRC grant whilst undertaking this work.
Tue, 01 Jan 2013 00:00:00 GMT
http://hdl.handle.net/10023/2756
20130101T00:00:00Z
Falconer, Kenneth John
Fraser, Jonathan Macdonald
Given a compact subset F of R2, the visible part VθF of F from direction θ is the set of x in F such that the halfline from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1 then dimH VθF = 1 for almost all θ , where dimH denotes Hausdorff dimension. We conrm this when F is a selfsimilar set satisfying the convex open set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected.

Topics in computational group theory : primitive permutation groups and matrix group normalisers
http://hdl.handle.net/10023/2561
Abstract: Part I of this thesis presents methods for finding the primitive permutation
groups of degree d, where 2500 ≤ d < 4096, using the O'NanScott Theorem
and Aschbacher's theorem. Tables of the groups G are given for each O'NanScott class. For the nonaffine 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.
Tue, 01 Nov 2011 00:00:00 GMT
http://hdl.handle.net/10023/2561
20111101T00:00:00Z
Coutts, Hannah Jane
Part I of this thesis presents methods for finding the primitive permutation
groups of degree d, where 2500 ≤ d < 4096, using the O'NanScott Theorem
and Aschbacher's theorem. Tables of the groups G are given for each O'NanScott class. For the nonaffine 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.

Generation problems for finite groups
http://hdl.handle.net/10023/2529
Abstract: It can be deduced from the Burnside Basis Theorem that if G is a finite pgroup 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 primepower order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.
Wed, 30 Nov 2011 00:00:00 GMT
http://hdl.handle.net/10023/2529
20111130T00:00:00Z
McDougallBagnall, Jonathan M.
It can be deduced from the Burnside Basis Theorem that if G is a finite pgroup 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 primepower order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.

Substitutionclosed pattern classes
http://hdl.handle.net/10023/2149
Abstract: The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classied by listing them as a set of explicit families.
Tue, 01 Feb 2011 00:00:00 GMT
http://hdl.handle.net/10023/2149
20110201T00:00:00Z
Atkinson, M.D.
Ruskuc, Nik
Smith, R
The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classied by listing them as a set of explicit families.

Automatic presentations and semigroup constructions
http://hdl.handle.net/10023/2148
Abstract: An automatic presentation for a relational structure is, informally, an abstract representation of the elements of that structure by means of a regular language such that the relations can all be recognized by finite automata. A structure admitting an automatic presentation is said to be FApresentable. This paper studies the interaction of automatic presentations and certain semigroup constructions, namely: direct products, free products, finite Rees index extensions and subsemigroups, strong semilattices of semigroups, Rees matrix semigroups, BruckReilly extensions, zerodirect unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FApresentable semigroups under that construction is considered, as is the question of whether the FApresentability of the semigroup obtained from such a construction implies the FApresentability of the original semigroup[s]. Classifications are also given of the FApresentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0simple semigroups.
Sun, 01 Aug 2010 00:00:00 GMT
http://hdl.handle.net/10023/2148
20100801T00:00:00Z
Cain, Alan J.
Oliver, Graham
Ruskuc, Nik
Thomas, Richard M.
An automatic presentation for a relational structure is, informally, an abstract representation of the elements of that structure by means of a regular language such that the relations can all be recognized by finite automata. A structure admitting an automatic presentation is said to be FApresentable. This paper studies the interaction of automatic presentations and certain semigroup constructions, namely: direct products, free products, finite Rees index extensions and subsemigroups, strong semilattices of semigroups, Rees matrix semigroups, BruckReilly extensions, zerodirect unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FApresentable semigroups under that construction is considered, as is the question of whether the FApresentability of the semigroup obtained from such a construction implies the FApresentability of the original semigroup[s]. Classifications are also given of the FApresentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0simple semigroups.

Automatic presentations for semigroups
http://hdl.handle.net/10023/2147
Abstract: This paper applies the concept of FApresentable structures to semigroups. We give a complete classification of the finitely generated FApresentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FApresentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FApresentable. We give a complete list of FApresentable onerelation semigroups and compare the classes of FApresentable semigroups and automatic semigroups. (C) 2009 Elsevier Inc. All rights reserved.
Description: Special Issue: 2nd International Conference on Language and Automata Theory and Applications (LATA 2008)
Sun, 01 Nov 2009 00:00:00 GMT
http://hdl.handle.net/10023/2147
20091101T00:00:00Z
Cain, Alan James
Oliver, Graham
Ruskuc, Nik
Thomas, Richard M.
This paper applies the concept of FApresentable structures to semigroups. We give a complete classification of the finitely generated FApresentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FApresentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FApresentable. We give a complete list of FApresentable onerelation semigroups and compare the classes of FApresentable semigroups and automatic semigroups. (C) 2009 Elsevier Inc. All rights reserved.

On residual finiteness of direct products of algebraic systems
http://hdl.handle.net/10023/2146
Abstract: It is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general.
Tue, 01 Sep 2009 00:00:00 GMT
http://hdl.handle.net/10023/2146
20090901T00:00:00Z
Gray, R.
Ruskuc, Nik
It is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general.

The Bergman property for semigroups
http://hdl.handle.net/10023/2145
Abstract: In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa. Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the BaerLevi semigroup does not have the Bergman property.
Sat, 01 Aug 2009 00:00:00 GMT
http://hdl.handle.net/10023/2145
20090801T00:00:00Z
Maltcev, V.
Mitchell, J. D.
Ruskuc, N.
In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa. Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the BaerLevi semigroup does not have the Bergman property.

Green index and finiteness conditions for semigroups
http://hdl.handle.net/10023/2144
Abstract: Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. If the complement S \ T has finitely many strong orbits by both these actions we say that T has finite Green index in S. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents. (c) 2008 Elsevier Inc. All rights reserved.
Wed, 15 Oct 2008 00:00:00 GMT
http://hdl.handle.net/10023/2144
20081015T00:00:00Z
Gray, Robert Duncan
Ruskuc, Nik
Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. If the complement S \ T has finitely many strong orbits by both these actions we say that T has finite Green index in S. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents. (c) 2008 Elsevier Inc. All rights reserved.

Properties of the subsemigroups of the bicyclic monoid
http://hdl.handle.net/10023/2142
Abstract: In this paper we study some properties of the subsemigroups of the bicyclic monoid B, by using a recent description of its subsemigroups. We start by giving necessary and sufficient conditions for a subsemigroup to be finitely generated. Then we show that all finitely generated subsemigroups are automatic and finitely presented. Finally we prove that a subsemigroup of B is residually finite if and only if it does not contain a copy of B.
Sun, 01 Jun 2008 00:00:00 GMT
http://hdl.handle.net/10023/2142
20080601T00:00:00Z
Descalco, L.
Ruskuc, Nik
In this paper we study some properties of the subsemigroups of the bicyclic monoid B, by using a recent description of its subsemigroups. We start by giving necessary and sufficient conditions for a subsemigroup to be finitely generated. Then we show that all finitely generated subsemigroups are automatic and finitely presented. Finally we prove that a subsemigroup of B is residually finite if and only if it does not contain a copy of B.

Pattern classes of permutations via bijections between linearly ordered sets
http://hdl.handle.net/10023/2140
Abstract: A pattern class is a set of permutations closed under pattern involvement or, equivalently, defined by certain subsequence avoidance conditions. Any pattern class X which is atomic, i.e. indecomposable as a union of proper subclasses, has a representation as the set of subpermutations of a bijection between two countable (or finite) linearly ordered sets A and B. Concentrating on the situation where A is arbitrary and B = N, we demonstrate how the ordertheoretic properties of A determine the structure of X and we establish results about independence, contiguousness and subrepresentations for classes admitting multiple representations of this form.
Tue, 01 Jan 2008 00:00:00 GMT
http://hdl.handle.net/10023/2140
20080101T00:00:00Z
Huczynska, Sophie
Ruskuc, Nikola
A pattern class is a set of permutations closed under pattern involvement or, equivalently, defined by certain subsequence avoidance conditions. Any pattern class X which is atomic, i.e. indecomposable as a union of proper subclasses, has a representation as the set of subpermutations of a bijection between two countable (or finite) linearly ordered sets A and B. Concentrating on the situation where A is arbitrary and B = N, we demonstrate how the ordertheoretic properties of A determine the structure of X and we establish results about independence, contiguousness and subrepresentations for classes admitting multiple representations of this form.

Cancellative and Malcev presentations for finite Rees index subsemigroups and extensions
http://hdl.handle.net/10023/2138
Abstract: It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, leftcancellative and rightcancellative presentations. (A Malcev (respectively, cancellative, leftcancellative, rightcancellative) presentation is a presentation of a special type that can be used to define any groupembeddable (respectively, cancellative, leftcancellative, rightcancellative) semigroup.).
Fri, 01 Feb 2008 00:00:00 GMT
http://hdl.handle.net/10023/2138
20080201T00:00:00Z
Cain, Alan James
Robertson, Edmund Frederick
Ruskuc, Nik
It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, leftcancellative and rightcancellative presentations. (A Malcev (respectively, cancellative, leftcancellative, rightcancellative) presentation is a presentation of a special type that can be used to define any groupembeddable (respectively, cancellative, leftcancellative, rightcancellative) semigroup.).

Growth rates for subclasses of Av(321)
http://hdl.handle.net/10023/2137
Abstract: Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates.
Fri, 22 Oct 2010 00:00:00 GMT
http://hdl.handle.net/10023/2137
20101022T00:00:00Z
Albert, M.H.
Atkinson, M.D.
Brignall, R
Ruskuc, Nik
Smith, R
West, J
Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates.

On generators and presentations of semidirect products in inverse semigroups
http://hdl.handle.net/10023/2136
Abstract: In this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented.
Mon, 01 Jun 2009 00:00:00 GMT
http://hdl.handle.net/10023/2136
20090601T00:00:00Z
Dombi, Erzsebet Rita
Ruskuc, Nik
In this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented.

Maximal subgroups of free idempotentgenerated semigroups over the full transformation monoid
http://hdl.handle.net/10023/2134
Abstract: Let Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying im (e) = r ≤ n − 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr.
Sun, 01 Jan 2012 00:00:00 GMT
http://hdl.handle.net/10023/2134
20120101T00:00:00Z
Gray, R
Ruskuc, Nik
Let Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying im (e) = r ≤ n − 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr.

Generators and relations for subsemigroups via boundaries in Cayley graphs
http://hdl.handle.net/10023/2131
Abstract: Given a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.
Tue, 01 Nov 2011 00:00:00 GMT
http://hdl.handle.net/10023/2131
20111101T00:00:00Z
Gray, R
Ruskuc, Nik
Given a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.

On the growth of generating sets for direct powers of semigroups
http://hdl.handle.net/10023/2129
Abstract: For a semigroup S its dsequence is d(S) = (d1, d2, d3, . . .), where di is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.
Sun, 01 Jan 2012 00:00:00 GMT
http://hdl.handle.net/10023/2129
20120101T00:00:00Z
Hyde, James Thomas
Loughlin, Nicholas
Quick, Martyn
Ruskuc, Nik
Wallis, Alistair
For a semigroup S its dsequence is d(S) = (d1, d2, d3, . . .), where di is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.

On maximal subgroups of free idempotent generated semigroups
http://hdl.handle.net/10023/2128
Abstract: We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister–Schreier type rewriting.
Sun, 01 Jan 2012 00:00:00 GMT
http://hdl.handle.net/10023/2128
20120101T00:00:00Z
Gray, R
Ruskuc, Nik
We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister–Schreier type rewriting.

Directed graph iterated function systems
http://hdl.handle.net/10023/2109
Abstract: 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 2vertex directed graph IFSs have attractors
that cannot be the attractors of standard (1vertex 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 μ[subscript(u)], the selfsimilar measure at a vertex u, for the nonlattice 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 nonnegative
matrix associated with the system is irreducible.
In the nonlattice case this enables the rate of convergence of the packing L[superscript(q)]spectrum
of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper
multifractal q boxdimension with respect to μ[subscript(u)], of the set consisting of all the intersections
of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff
dimension with respect to μ[subscript(u)] of F[subscript(u)].
Wed, 30 Nov 2011 00:00:00 GMT
http://hdl.handle.net/10023/2109
20111130T00:00:00Z
Boore, Graeme C.
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 2vertex directed graph IFSs have attractors
that cannot be the attractors of standard (1vertex 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 μ[subscript(u)], the selfsimilar measure at a vertex u, for the nonlattice 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 nonnegative
matrix associated with the system is irreducible.
In the nonlattice case this enables the rate of convergence of the packing L[superscript(q)]spectrum
of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper
multifractal q boxdimension with respect to μ[subscript(u)], of the set consisting of all the intersections
of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff
dimension with respect to μ[subscript(u)] of F[subscript(u)].

On convex permutations
http://hdl.handle.net/10023/2000
Abstract: A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem.
Sun, 01 May 2011 00:00:00 GMT
http://hdl.handle.net/10023/2000
20110501T00:00:00Z
Albert, M.H.
Linton, Stephen Alexander
Ruskuc, Nik
Vatter, V
Waton, S
A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem.

Presentations of inverse semigroups, their kernels and extensions
http://hdl.handle.net/10023/1998
Abstract: Let S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.
Wed, 01 Jun 2011 00:00:00 GMT
http://hdl.handle.net/10023/1998
20110601T00:00:00Z
Carvalho, C.A.
Gray, R
Ruskuc, Nik
Let S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.

Simple extensions of combinatorial structures
http://hdl.handle.net/10023/1997
Abstract: An interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible.
Fri, 01 Jul 2011 00:00:00 GMT
http://hdl.handle.net/10023/1997
20110701T00:00:00Z
Brignall, R
Ruskuc, Nik
Vatter, V
An interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible.

The horizon problem for prevalent surfaces
http://hdl.handle.net/10023/1956
Abstract: We investigate the box dimensions of the horizon of a fractal surface defined by a function $f \in C[0,1]^2 $. In particular we show that a prevalent surface satisfies the `horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most $\alpha$, for $\alpha \in [2,3)$. In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.
Sat, 01 Jan 2011 00:00:00 GMT
http://hdl.handle.net/10023/1956
20110101T00:00:00Z
Falconer, Kenneth John
Fraser, Jonathan Macdonald
We investigate the box dimensions of the horizon of a fractal surface defined by a function $f \in C[0,1]^2 $. In particular we show that a prevalent surface satisfies the `horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most $\alpha$, for $\alpha \in [2,3)$. In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.

A commutative noncommutative fractal geometry
http://hdl.handle.net/10023/1710
Abstract: 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 nonempty 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 boxcounting dimension can be recovered. We show that for a selfsimilar 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 selfsimilar 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 AntonescuIvan and Christensen, we construct a family of (1, +)summable spectral triples for a onesided topologically exact subshift of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the PerronFrobeniusRuelle operator, whose potential function is nonarithemetic and Hölder continuous. We show that the Connes' pseudometric, 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.
Fri, 01 Jan 2010 00:00:00 GMT
http://hdl.handle.net/10023/1710
20100101T00:00:00Z
Samuel, Anthony
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 nonempty 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 boxcounting dimension can be recovered. We show that for a selfsimilar 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 selfsimilar 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 AntonescuIvan and Christensen, we construct a family of (1, +)summable spectral triples for a onesided topologically exact subshift of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the PerronFrobeniusRuelle operator, whose potential function is nonarithemetic and Hölder continuous. We show that the Connes' pseudometric, 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.

The lesser names : the teachers of the Edinburgh Mathematical Society and other aspects of Scottish mathematics, 1867–1946
http://hdl.handle.net/10023/1700
Abstract: 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.
Wed, 22 Jun 2011 00:00:00 GMT
http://hdl.handle.net/10023/1700
20110622T00:00:00Z
Hartveit, Marit
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.

Primitive free cubics with specified norm and trace
http://hdl.handle.net/10023/1615
Abstract: The existence of a primitive free (normal) cubic x(3) ax(2) + cx b over a finite field F with arbitrary specified values of a (not equal 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.
Fri, 01 Aug 2003 00:00:00 GMT
http://hdl.handle.net/10023/1615
20030801T00:00:00Z
Huczynska, Sophie
Cohen, SD
The existence of a primitive free (normal) cubic x(3) ax(2) + cx b over a finite field F with arbitrary specified values of a (not equal 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.

Subsemigroups of virtually free groups : finite Malcev presentations and testing for freeness
http://hdl.handle.net/10023/1561
Abstract: This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.
Sat, 01 Jul 2006 00:00:00 GMT
http://hdl.handle.net/10023/1561
20060701T00:00:00Z
Cain, AJ
Robertson, Edmund Frederick
Ruskuc, Nikola
This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.

Generating the full transformation semigroup using order preserving mappings
http://hdl.handle.net/10023/1553
Abstract: For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings OX on X modulo the full transformation semigroup Ex. In other words, we ask what is the smallest cardinality of a set A of mappings such that <OX boolean OR A> = TX. When X is countably infinite or wellordered (of arbitrary cardinality) we show that this number is one, while when X = R (the set of real numbers) it is uncountable.
Mon, 01 Sep 2003 00:00:00 GMT
http://hdl.handle.net/10023/1553
20030901T00:00:00Z
Higgins, PM
Mitchell, James David
Ruskuc, Nikola
For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings OX on X modulo the full transformation semigroup Ex. In other words, we ask what is the smallest cardinality of a set A of mappings such that <OX boolean OR A> = TX. When X is countably infinite or wellordered (of arbitrary cardinality) we show that this number is one, while when X = R (the set of real numbers) it is uncountable.

On defining groups efficiently without using inverses
http://hdl.handle.net/10023/1442
Abstract: Let G be a group, and let <A \ R> be a finite group presentation for G with \R\ greater than or equal to \A\. Then there exists a, finite semigroup, presentation <B \ Q> for G such that \Q\  \B\ = \R\  \A\. Moreover, B is either the same generating set or else it contains one additional generator.
Mon, 01 Jul 2002 00:00:00 GMT
http://hdl.handle.net/10023/1442
20020701T00:00:00Z
Campbell, Colin Matthew
Mitchell, James David
Ruskuc, Nikola
Let G be a group, and let <A \ R> be a finite group presentation for G with \R\ greater than or equal to \A\. Then there exists a, finite semigroup, presentation <B \ Q> for G such that \Q\  \B\ = \R\  \A\. Moreover, B is either the same generating set or else it contains one additional generator.

Stable and multistable processes and localisability
http://hdl.handle.net/10023/948
Abstract: 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 αstable 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 αstable random functions
defined by series with αstable random variables as coefficients.
Wed, 23 Jun 2010 00:00:00 GMT
http://hdl.handle.net/10023/948
20100623T00:00:00Z
Liu, Lining
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 αstable 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 αstable random functions
defined by series with αstable random variables as coefficients.

Classification and enumeration of finite semigroups
http://hdl.handle.net/10023/945
Abstract: 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 antiisomorphism 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 antiisomorphic
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 antiisomorphism 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.
Wed, 23 Jun 2010 00:00:00 GMT
http://hdl.handle.net/10023/945
20100623T00:00:00Z
Distler, Andreas
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 antiisomorphism 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 antiisomorphic
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 antiisomorphism 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.

Generating uncountable transformation semigroups
http://hdl.handle.net/10023/867
Abstract: 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.
Thu, 01 Jan 2009 00:00:00 GMT
http://hdl.handle.net/10023/867
20090101T00:00:00Z
Péresse, Yann
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.

The geometry of selfaffine fractals
http://hdl.handle.net/10023/838
Abstract: In this thesis we study the dimension theory of selfaffine 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 selfaffine sets. We then develop a variety of new results on selfaffine
sets and their dimensional properties.
This work falls into three parts:
Firstly, we look at the dimension formulae for a class of selfaffine 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 selfaffine 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 selfaffine sets, aiming
to extend the ‘almost sure’ formula for random selfsimilar sets to random selfaffine
sets.
Tue, 01 Jan 2008 00:00:00 GMT
http://hdl.handle.net/10023/838
20080101T00:00:00Z
Miao, Jun Jie
In this thesis we study the dimension theory of selfaffine 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 selfaffine sets. We then develop a variety of new results on selfaffine
sets and their dimensional properties.
This work falls into three parts:
Firstly, we look at the dimension formulae for a class of selfaffine 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 selfaffine 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 selfaffine sets, aiming
to extend the ‘almost sure’ formula for random selfsimilar sets to random selfaffine
sets.

Intersection problems in combinatorics
http://hdl.handle.net/10023/765
Abstract: With the publication of the famous ErdősKoRado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is tintersecting if any two of its elements mutually tintersect, 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 1intersecting 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 socalled fixfamilies, consisting of all injections which map some fixed set of t points to the same image points, are the only tintersecting injection families of maximal size. By way of contrast, fixing the differences kt and nk while increasing k leads to optimal families which are equivalent to one of the socalled saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=_ (kt)/2 _. Furthermore we demonstrate that, among injection families with tintersecting and leftcompressed 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.
Mon, 30 Nov 2009 00:00:00 GMT
http://hdl.handle.net/10023/765
20091130T00:00:00Z
Brunk, Fiona
With the publication of the famous ErdősKoRado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is tintersecting if any two of its elements mutually tintersect, 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 1intersecting 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 socalled fixfamilies, consisting of all injections which map some fixed set of t points to the same image points, are the only tintersecting injection families of maximal size. By way of contrast, fixing the differences kt and nk while increasing k leads to optimal families which are equivalent to one of the socalled saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=_ (kt)/2 _. Furthermore we demonstrate that, among injection families with tintersecting and leftcompressed 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.

Resonances for graph directed Markov systems, and geometry of infinitely generated dynamical systems
http://hdl.handle.net/10023/719
Abstract: 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(Imw)} 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).
Wed, 24 Jun 2009 00:00:00 GMT
http://hdl.handle.net/10023/719
20090624T00:00:00Z
Hille, Martial R.
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(Imw)} 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).

Inhomogeneous selfsimilar sets and measures
http://hdl.handle.net/10023/682
Abstract: The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous selfsimilar sets and
measures. In particular, we show that these sets and measures
are natural generalizations of the well known selfsimilar 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 selfsimilar sets and compare our results with the wellknown results for (ordinary)
selfsimilar sets. In the third chapter we investigate the L^{q}
spectra and the Renyi dimensions of inhomogeneous selfsimilar
measures and prove that new multifractal phenomena, not exhibited by (ordinary) selfsimilar measures, appear in the inhomogeneous case.
Namely, we show that inhomogeneous selfsimilar measures may
have phase transitions which is in sharp contrast to the
behaviour of the
L^{q} spectra
of (ordinary) selfsimilar
measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra
of inhomogeneous selfsimilar measures. We show that
the multifractal spectra
of
inhomogeneous selfsimilar
measures
may be nonconcave which is again in sharp contrast to the
behaviour of the
multifractal spectra
of (ordinary) selfsimilar
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 nontrivial bounds) for the multifractal spectra of selfsimilar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain nontrivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of selfsimilar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of
inhomogeneous selfsimilar measures and again we present a
number of applications of our results, in particular to nonlinear
selfsimilar measures.
Tue, 01 Jan 2008 00:00:00 GMT
http://hdl.handle.net/10023/682
20080101T00:00:00Z
Snigireva, Nina
The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous selfsimilar sets and
measures. In particular, we show that these sets and measures
are natural generalizations of the well known selfsimilar 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 selfsimilar sets and compare our results with the wellknown results for (ordinary)
selfsimilar sets. In the third chapter we investigate the L^{q}
spectra and the Renyi dimensions of inhomogeneous selfsimilar
measures and prove that new multifractal phenomena, not exhibited by (ordinary) selfsimilar measures, appear in the inhomogeneous case.
Namely, we show that inhomogeneous selfsimilar measures may
have phase transitions which is in sharp contrast to the
behaviour of the
L^{q} spectra
of (ordinary) selfsimilar
measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra
of inhomogeneous selfsimilar measures. We show that
the multifractal spectra
of
inhomogeneous selfsimilar
measures
may be nonconcave which is again in sharp contrast to the
behaviour of the
multifractal spectra
of (ordinary) selfsimilar
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 nontrivial bounds) for the multifractal spectra of selfsimilar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain nontrivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of selfsimilar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of
inhomogeneous selfsimilar measures and again we present a
number of applications of our results, in particular to nonlinear
selfsimilar measures.

Simplicity in relational structures and its application to permutation classes
http://hdl.handle.net/10023/431
Abstract: 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 FraÃ¯ssÃ©, 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 "querycomplete 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 eightletter 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.
Fri, 30 Nov 2007 00:00:00 GMT
http://hdl.handle.net/10023/431
20071130T00:00:00Z
Brignall, Robert
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 FraÃ¯ssÃ©, 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 "querycomplete 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 eightletter 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.

On permutation classes defined by token passing networks, gridding matrices and pictures : three flavours of involvement
http://hdl.handle.net/10023/237
Abstract: 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.
Tue, 19 Jun 2007 00:00:00 GMT
http://hdl.handle.net/10023/237
20070619T00:00:00Z
Waton, Stephen D.
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.