DSpace Community:
http://hdl.handle.net/10023/94
2014-12-18T08:37:12Z
2014-12-18T08:37:12Z
Negative ion sound solitary waves revisited
Cairns, R. A.
http://hdl.handle.net/10023/5845
2014-12-01T00:01:36Z
2013-12-01T00:00:00Z
Abstract: Some years ago, a group including the present author and Padma Shukla showed that a suitable non-thermal electron distribution allows the formation of ion sound solitary waves with either positive or negative density perturbations, whereas with Maxwellian electrons only a positive density perturbation is possible. The present paper discusses the qualitative features of this distribution allowing the negative waves and shared with suitable two-temperature distributions.
2013-12-01T00:00:00Z
Cairns, R. A.
Some years ago, a group including the present author and Padma Shukla showed that a suitable non-thermal electron distribution allows the formation of ion sound solitary waves with either positive or negative density perturbations, whereas with Maxwellian electrons only a positive density perturbation is possible. The present paper discusses the qualitative features of this distribution allowing the negative waves and shared with suitable two-temperature distributions.
An explicit upper bound for the Helfgott delta in SL(2,p)
Button, Jack
Roney-Dougal, Colva
http://hdl.handle.net/10023/5819
2014-11-20T16:31:03Z
2015-01-01T00:00:00Z
Abstract: Helfgott proved that there exists a δ>0 such that if S is a symmetric generating subset of SL(2,p) containing 1 then either S3=SL(2,p) or |S3| ≥|S|1+δ. It is known that δ ≥ 1/3024. Here we show that δ ≤(log2(7)-1)/6 ≈ 0.3012 and we present evidence suggesting that this might be the true value of δ.
2015-01-01T00:00:00Z
Button, Jack
Roney-Dougal, Colva
Helfgott proved that there exists a δ>0 such that if S is a symmetric generating subset of SL(2,p) containing 1 then either S3=SL(2,p) or |S3| ≥|S|1+δ. It is known that δ ≥ 1/3024. Here we show that δ ≤(log2(7)-1)/6 ≈ 0.3012 and we present evidence suggesting that this might be the true value of δ.
Backward wave cyclotron-maser emission in the auroral magnetosphere
Speirs, D. C.
Bingham, R.
Cairns, R. A.
Vorgul, I.
Kellett, B. J.
Phelps, A. D. R.
Ronald, K.
http://hdl.handle.net/10023/5802
2014-11-19T17:01:04Z
2014-10-07T00:00:00Z
Abstract: In this Letter, we present theory and particle-in-cell simulations describing cyclotron radio emission from Earth's auroral region and similar phenomena in other astrophysical environments. In particular, we find that the radiation, generated by a down-going electron horseshoe distribution is due to a backward wave cyclotron-maser emission process. The backward wave nature of the radiation contributes to upward refraction of the radiation that is also enhanced by a density inhomogeneity. We also show that the radiation is preferentially amplified along the auroral oval rather than transversely. The results are in agreement with recent Cluster observations.
Description: This work was supported by EPSRC Grant No. EP/G04239X/1.
2014-10-07T00:00:00Z
Speirs, D. C.
Bingham, R.
Cairns, R. A.
Vorgul, I.
Kellett, B. J.
Phelps, A. D. R.
Ronald, K.
In this Letter, we present theory and particle-in-cell simulations describing cyclotron radio emission from Earth's auroral region and similar phenomena in other astrophysical environments. In particular, we find that the radiation, generated by a down-going electron horseshoe distribution is due to a backward wave cyclotron-maser emission process. The backward wave nature of the radiation contributes to upward refraction of the radiation that is also enhanced by a density inhomogeneity. We also show that the radiation is preferentially amplified along the auroral oval rather than transversely. The results are in agreement with recent Cluster observations.
Maximal subsemigroups of the semigroup of all mappings on an infinite set
East, J.
D. Mitchell, J.
Péresse, Y.
http://hdl.handle.net/10023/5793
2014-11-19T10:01:08Z
2014-01-01T00:00:00Z
Abstract: We classify the maximal subsemigroups of the semigroup ΩΩ of all mappings on an infinite set Ω that contain one of the following groups: the symmetric group on Ω, the setwise stabilizer of a non-empty finite subset of Ω, the stabilizer of a finite partition of Ω, or the stabilizer of an ultrafilter on Ω. If G is any of these groups, then we also characterise the mappings f,g ∈ ΩΩ such that the semigroup G, f, g generated by G ∪ {f,g} equals ΩΩ. We also show that the setwise stabiliser of a non-empty finite set, the almost stabiliser of a finite partition, and the stabiliser of an ultrafilter are maximal subsemigroups of the symmetric group.
2014-01-01T00:00:00Z
East, J.
D. Mitchell, J.
Péresse, Y.
We classify the maximal subsemigroups of the semigroup ΩΩ of all mappings on an infinite set Ω that contain one of the following groups: the symmetric group on Ω, the setwise stabilizer of a non-empty finite subset of Ω, the stabilizer of a finite partition of Ω, or the stabilizer of an ultrafilter on Ω. If G is any of these groups, then we also characterise the mappings f,g ∈ ΩΩ such that the semigroup G, f, g generated by G ∪ {f,g} equals ΩΩ. We also show that the setwise stabiliser of a non-empty finite set, the almost stabiliser of a finite partition, and the stabiliser of an ultrafilter are maximal subsemigroups of the symmetric group.
The nature of separator current layers in MHS equilibria : I. Current parallel to the separator
E. H. Stevenson, J.
E. Parnell, C.
R. Priest, E.
L. Haynes, A.
http://hdl.handle.net/10023/5785
2014-11-18T13:01:03Z
2014-10-31T00:00:00Z
Abstract: Separators, which are in many ways the three-dimensional equivalent to two-dimensional nulls, are important sites for magnetic reconnection. Magnetic reconnection occurs in strong current layers which have very short length scales. The aim of this work is to explore the nature of current layers around separators. A separator is a special field line which lies along the intersection of two separatrix surfaces and forms the boundary between four topologically distinct flux domains. In particular, here the current layer about a separator that joins two 3D nulls and lies along the intersection of their separatrix surfaces is investigated. A magnetic configuration containing a single separator embedded in a uniform plasma with a uniform electric current parallel to the separator is considered. This initial magnetic setup, which is not in equilibrium, relaxes in a non-resistive manner to form an equilibrium. The relaxation is achieved using the 3D MHD code, Lare3d, with resistivity set to zero. A series of experiments with varying initial current are run to investigate the characteristics of the resulting current layers present in the final (quasi-) equilibrium states. In each experiment, the separator collapses and a current layer forms along it. The dimensions and strength of the current layer increase with initial current. It is found that separator current layers formed from current parallel to the separator are twisted. Also the collapse of the separator is a process that evolves like an infinite-time singularity where the length, width and peak current in the layer grow slowly whilst the depth of the current layer decreases.
Description: JEHS would like to thank STFC for financial support during her Ph.D and CEP acknowledges support from the STFC consolidated grant.
2014-10-31T00:00:00Z
E. H. Stevenson, J.
E. Parnell, C.
R. Priest, E.
L. Haynes, A.
Separators, which are in many ways the three-dimensional equivalent to two-dimensional nulls, are important sites for magnetic reconnection. Magnetic reconnection occurs in strong current layers which have very short length scales. The aim of this work is to explore the nature of current layers around separators. A separator is a special field line which lies along the intersection of two separatrix surfaces and forms the boundary between four topologically distinct flux domains. In particular, here the current layer about a separator that joins two 3D nulls and lies along the intersection of their separatrix surfaces is investigated. A magnetic configuration containing a single separator embedded in a uniform plasma with a uniform electric current parallel to the separator is considered. This initial magnetic setup, which is not in equilibrium, relaxes in a non-resistive manner to form an equilibrium. The relaxation is achieved using the 3D MHD code, Lare3d, with resistivity set to zero. A series of experiments with varying initial current are run to investigate the characteristics of the resulting current layers present in the final (quasi-) equilibrium states. In each experiment, the separator collapses and a current layer forms along it. The dimensions and strength of the current layer increase with initial current. It is found that separator current layers formed from current parallel to the separator are twisted. Also the collapse of the separator is a process that evolves like an infinite-time singularity where the length, width and peak current in the layer grow slowly whilst the depth of the current layer decreases.
Computing in permutation groups without memory
Cameron, Peter Jephson
Fairbairn, Ben
Gadouleau, Maximilien
http://hdl.handle.net/10023/5727
2014-11-11T12:31:01Z
2014-11-02T00:00:00Z
Abstract: Memoryless computation is a new technique to compute any function of a set of registers by updating one register at a time while using no memory. Its aim is to emulate how computations are performed in modern cores, since they typically involve updates of single registers. The memoryless computation model can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we consider how efficiently permutations can be computed without memory. We determine the minimum number of basic updates required to compute any permutation, or any even permutation. The small number of required instructions shows that very small instruction sets could be encoded on cores to perform memoryless computation. We then start looking at a possible compromise between the size of the instruction set and the length of the resulting programs. We consider updates only involving a limited number of registers. In particular, we show that binary instructions are not enough to compute all permutations without memory when the alphabet size is even. These results, though expressed as properties of special generating sets of the symmetric or alternating groups, provide guidelines on the implementation of memoryless computation.
Description: Funding: UK Engineering and Physical Sciences Research Council (EP/K033956/1)
2014-11-02T00:00:00Z
Cameron, Peter Jephson
Fairbairn, Ben
Gadouleau, Maximilien
Memoryless computation is a new technique to compute any function of a set of registers by updating one register at a time while using no memory. Its aim is to emulate how computations are performed in modern cores, since they typically involve updates of single registers. The memoryless computation model can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we consider how efficiently permutations can be computed without memory. We determine the minimum number of basic updates required to compute any permutation, or any even permutation. The small number of required instructions shows that very small instruction sets could be encoded on cores to perform memoryless computation. We then start looking at a possible compromise between the size of the instruction set and the length of the resulting programs. We consider updates only involving a limited number of registers. In particular, we show that binary instructions are not enough to compute all permutations without memory when the alphabet size is even. These results, though expressed as properties of special generating sets of the symmetric or alternating groups, provide guidelines on the implementation of memoryless computation.
Computing in matrix groups without memory
Cameron, Peter Jephson
Fairbairn, Ben
Gadouleau, Maximilien
http://hdl.handle.net/10023/5715
2014-11-11T11:31:02Z
2014-11-02T00:00:00Z
Abstract: Memoryless computation is a novel means of computing any function of a set of registers by updating one register at a time while using no memory. We aim to emulate how computations are performed on modern cores, since they typically involve updates of single registers. The computation model of memoryless computation can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we view registers as elements of a finite field and we compute linear permutations without memory. We first determine the maximum complexity of a linear function when only linear instructions are allowed. We also determine which linear functions are hardest to compute when the field in question is the binary field and the number of registers is even. Secondly, we investigate some matrix groups, thus showing that the special linear group is internally computable but not fast. Thirdly, we determine the smallest set of instructions required to generate the special and general linear groups. These results are important for memoryless computation, for they show that linear functions can be computed very fast or that very few instructions are needed to compute any linear function. They thus indicate new advantages of using memoryless computation.
Description: Funding: UK Engineering and Physical Sciences Research Council award EP/K033956/1
2014-11-02T00:00:00Z
Cameron, Peter Jephson
Fairbairn, Ben
Gadouleau, Maximilien
Memoryless computation is a novel means of computing any function of a set of registers by updating one register at a time while using no memory. We aim to emulate how computations are performed on modern cores, since they typically involve updates of single registers. The computation model of memoryless computation can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we view registers as elements of a finite field and we compute linear permutations without memory. We first determine the maximum complexity of a linear function when only linear instructions are allowed. We also determine which linear functions are hardest to compute when the field in question is the binary field and the number of registers is even. Secondly, we investigate some matrix groups, thus showing that the special linear group is internally computable but not fast. Thirdly, we determine the smallest set of instructions required to generate the special and general linear groups. These results are important for memoryless computation, for they show that linear functions can be computed very fast or that very few instructions are needed to compute any linear function. They thus indicate new advantages of using memoryless computation.
The probability of generating a finite simple group
Menezes, Nina Emma
Quick, Martyn
Roney-Dougal, Colva Mary
http://hdl.handle.net/10023/5658
2014-11-01T00:01:25Z
2013-11-01T00:00:00Z
Abstract: We study the probability of generating a finite simple group, together with its generalisation PG,socG(d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ socG. We prove that PG,socG(2) ⩾ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for PG,socG(3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.
2013-11-01T00:00:00Z
Menezes, Nina Emma
Quick, Martyn
Roney-Dougal, Colva Mary
We study the probability of generating a finite simple group, together with its generalisation PG,socG(d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ socG. We prove that PG,socG(2) ⩾ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for PG,socG(3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.
Most primitive groups are full automorphism groups of edge-transitive hypergraphs
Babai, Laszlo
Cameron, Peter Jephson
http://hdl.handle.net/10023/5580
2014-10-29T15:31:07Z
2015-01-01T00:00:00Z
Abstract: We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,YG) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. Klin. Moreover, we give an upper bound n1/2+ε for the minimum size of the edges in such a hypergraph. This is essentially best possible.
2015-01-01T00:00:00Z
Babai, Laszlo
Cameron, Peter Jephson
We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,YG) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. Klin. Moreover, we give an upper bound n1/2+ε for the minimum size of the edges in such a hypergraph. This is essentially best possible.
Exact dimensionality and projections of random self-similar measures and sets
Falconer, Kenneth
Jin, Xiong
http://hdl.handle.net/10023/5514
2014-10-02T16:01:00Z
2014-09-01T00:00:00Z
Abstract: We study the geometric properties of random multiplicative cascade measures defined on self-similar sets. We show that such measures and their projections and sections are almost surely exact-dimensional, generalizing Feng and Hu's result for self-similar measures. This, together with a compact group extension argument, enables us to generalize Hochman and Shmerkin's theorems on projections of deterministic self-similar measures to these random measures without requiring any separation conditions on the underlying sets. We give applications to self-similar sets and fractal percolation, including new results on projections, C1-images and distance sets.
2014-09-01T00:00:00Z
Falconer, Kenneth
Jin, Xiong
We study the geometric properties of random multiplicative cascade measures defined on self-similar sets. We show that such measures and their projections and sections are almost surely exact-dimensional, generalizing Feng and Hu's result for self-similar measures. This, together with a compact group extension argument, enables us to generalize Hochman and Shmerkin's theorems on projections of deterministic self-similar measures to these random measures without requiring any separation conditions on the underlying sets. We give applications to self-similar sets and fractal percolation, including new results on projections, C1-images and distance sets.
On the nature of reconnection at a solar coronal null point above a separatrix dome
Pontin, D. I.
Priest, E. R.
Galsgaard, K.
http://hdl.handle.net/10023/5459
2014-10-09T12:01:01Z
2013-09-10T00:00:00Z
Abstract: Three-dimensional magnetic null points are ubiquitous in the solar corona and in any generic mixed-polarity magnetic field. We consider magnetic reconnection at an isolated coronal null point whose fan field lines form a dome structure. Using analytical and computational models, we demonstrate several features of spine-fan reconnection at such a null, including the fact that substantial magnetic flux transfer from one region of field line connectivity to another can occur. The flux transfer occurs across the current sheet that forms around the null point during spine-fan reconnection, and there is no separator present. Also, flipping of magnetic field lines takes place in a manner similar to that observed in the quasi-separatrix layer or slip-running reconnection.
2013-09-10T00:00:00Z
Pontin, D. I.
Priest, E. R.
Galsgaard, K.
Three-dimensional magnetic null points are ubiquitous in the solar corona and in any generic mixed-polarity magnetic field. We consider magnetic reconnection at an isolated coronal null point whose fan field lines form a dome structure. Using analytical and computational models, we demonstrate several features of spine-fan reconnection at such a null, including the fact that substantial magnetic flux transfer from one region of field line connectivity to another can occur. The flux transfer occurs across the current sheet that forms around the null point during spine-fan reconnection, and there is no separator present. Also, flipping of magnetic field lines takes place in a manner similar to that observed in the quasi-separatrix layer or slip-running reconnection.
On magnetic reconnection and flux rope topology in solar flux emergence
MacTaggart, David
Haynes, Andrew Lewis
http://hdl.handle.net/10023/5393
2014-11-06T22:31:00Z
2014-02-21T00:00:00Z
Abstract: We present an analysis of the formation of atmospheric flux ropes in a magnetohydrodynamic solar flux emergence simulation. The simulation domain ranges from the top of the solar interior to the low corona. A twisted magnetic flux tube emerges from the solar interior and into the atmosphere where it interacts with the ambient magnetic field. By studying the connectivity of the evolving magnetic field, we are able to better understand the process of flux rope formation in the solar atmosphere. In the simulation, two flux ropes are produced as a result of flux emergence. Each has a different evolution resulting in different topological structures. These are determined by plasma flows and magnetic reconnection. As the flux rope is the basic structure of the coronal mass ejection, we discuss the implications of our findings for solar eruptions.
2014-02-21T00:00:00Z
MacTaggart, David
Haynes, Andrew Lewis
We present an analysis of the formation of atmospheric flux ropes in a magnetohydrodynamic solar flux emergence simulation. The simulation domain ranges from the top of the solar interior to the low corona. A twisted magnetic flux tube emerges from the solar interior and into the atmosphere where it interacts with the ambient magnetic field. By studying the connectivity of the evolving magnetic field, we are able to better understand the process of flux rope formation in the solar atmosphere. In the simulation, two flux ropes are produced as a result of flux emergence. Each has a different evolution resulting in different topological structures. These are determined by plasma flows and magnetic reconnection. As the flux rope is the basic structure of the coronal mass ejection, we discuss the implications of our findings for solar eruptions.
Observations of a hybrid double-streamer/pseudostreamer in the solar corona
Rachmeler, L.A.
Platten, S.J.
Bethge, C.
Seaton, D.B.
Yeates, A.R.
http://hdl.handle.net/10023/5318
2014-10-13T11:01:03Z
2014-05-20T00:00:00Z
Abstract: We report on the first observation of a single hybrid magnetic structure that contains both a pseudostreamer and a double streamer. This structure was originally observed by the SWAP instrument on board the PROBA2 satellite between 2013 May 5 and 10. It consists of a pair of filament channels near the south pole of the Sun. On the western edge of the structure, the magnetic morphology above the filaments is that of a side-by-side double streamer, with open field between the two channels. On the eastern edge, the magnetic morphology is that of a coronal pseudostreamer without the central open field. We investigated this structure with multiple observations and modeling techniques. We describe the topology and dynamic consequences of such a unified structure.
Description: D.B.S. and L.A.R. acknowledge support from the Belgian Federal Science Policy Office (BELSPO) through the ESA-PRODEX program, grant No. 4000103240. S.J.P. acknowledges the financial support of the Isle of Man Government.
2014-05-20T00:00:00Z
Rachmeler, L.A.
Platten, S.J.
Bethge, C.
Seaton, D.B.
Yeates, A.R.
We report on the first observation of a single hybrid magnetic structure that contains both a pseudostreamer and a double streamer. This structure was originally observed by the SWAP instrument on board the PROBA2 satellite between 2013 May 5 and 10. It consists of a pair of filament channels near the south pole of the Sun. On the western edge of the structure, the magnetic morphology above the filaments is that of a side-by-side double streamer, with open field between the two channels. On the eastern edge, the magnetic morphology is that of a coronal pseudostreamer without the central open field. We investigated this structure with multiple observations and modeling techniques. We describe the topology and dynamic consequences of such a unified structure.
Space exploration using parallel orbits : a study in parallel symbolic computing
Janjic, Vladimir
Brown, Christopher Mark
Neunhoeffer, Max
Hammond, Kevin
Linton, Stephen Alexander
Loidl, Hans-Wolfgang
http://hdl.handle.net/10023/5303
2014-09-01T14:31:04Z
2013-09-01T00:00:00Z
Abstract: Orbit enumerations represent an important class of mathematical algorithms which is widely used in computational discrete mathematics. In this paper, we present a new shared-memory implementation of a generic Orbit skeleton in the GAP computer algebra system [5]. By defining a skeleton, we are easily able to capture a wide variety of concrete Orbit enumerations that can exploit the same underlying parallel implementation. We also propose a generic cost model for predicting the speedups that our Orbit skeleton will deliver for a given application on a given parallel system. We demonstrate the scalability of our implementation on a 64-core shared-memory machine. Our results show that we are able to obtain good speedups over sequential GAP programs (up to 25.27 on 64 cores).
2013-09-01T00:00:00Z
Janjic, Vladimir
Brown, Christopher Mark
Neunhoeffer, Max
Hammond, Kevin
Linton, Stephen Alexander
Loidl, Hans-Wolfgang
Orbit enumerations represent an important class of mathematical algorithms which is widely used in computational discrete mathematics. In this paper, we present a new shared-memory implementation of a generic Orbit skeleton in the GAP computer algebra system [5]. By defining a skeleton, we are easily able to capture a wide variety of concrete Orbit enumerations that can exploit the same underlying parallel implementation. We also propose a generic cost model for predicting the speedups that our Orbit skeleton will deliver for a given application on a given parallel system. We demonstrate the scalability of our implementation on a 64-core shared-memory machine. Our results show that we are able to obtain good speedups over sequential GAP programs (up to 25.27 on 64 cores).
Catastrophe versus instability for the eruption of a toroidal solar magnetic flux rope
Kliem, B.
Lin, J.
Forbes, T. G.
Priest, E. R.
Toeroek, T.
http://hdl.handle.net/10023/5291
2014-10-09T14:31:00Z
2014-07-01T00:00:00Z
Abstract: The onset of a solar eruption is formulated here as either a magnetic catastrophe or as an instability. Both start with the same equation of force balance governing the underlying equilibria. Using a toroidal flux rope in an external bipolar or quadrupolar field as a model for the current-carrying flux, we demonstrate the occurrence of a fold catastrophe by loss of equilibrium for several representative evolutionary sequences in the stable domain of parameter space. We verify that this catastrophe and the torus instability occur at the same point; they are thus equivalent descriptions for the onset condition of solar eruptions.
Description: B.K. acknowledges support by the Chinese Academy of Sciences under grant No. 2012T1J0017. He also acknowledges support by the DFG, the STFC, and the NSF. J.L.'s work was supported by 973 Program grants 2013CB815103 and 2011CB811403, NSFC grants 11273055, and 11333007, and CAS grant KJCX2-EW-T07 to Yunnan Observatory. E.R.P. is grateful to the Leverhulme Trust for financial support. The contribution of T.T. was supported by NASA's HTP, LWS, and SR&T programs and by NSF.
2014-07-01T00:00:00Z
Kliem, B.
Lin, J.
Forbes, T. G.
Priest, E. R.
Toeroek, T.
The onset of a solar eruption is formulated here as either a magnetic catastrophe or as an instability. Both start with the same equation of force balance governing the underlying equilibria. Using a toroidal flux rope in an external bipolar or quadrupolar field as a model for the current-carrying flux, we demonstrate the occurrence of a fold catastrophe by loss of equilibrium for several representative evolutionary sequences in the stable domain of parameter space. We verify that this catastrophe and the torus instability occur at the same point; they are thus equivalent descriptions for the onset condition of solar eruptions.
The solar cycle variation of topological structures in the global solar corona
Platten, S.J.
Parnell, C.E.
Haynes, A.L.
Priest, E.R.
MacKay, D.H.
http://hdl.handle.net/10023/5271
2014-10-10T09:01:04Z
2014-05-01T00:00:00Z
Abstract: Context. The complicated distribution of magnetic flux across the solar photosphere results in a complex web of coronal magnetic field structures. To understand this complexity, the magnetic skeleton of the coronal field can be calculated. The skeleton highlights the (separatrix) surfaces that divide the field into topologically distinct regions, allowing open-field regions on the solar surface to be located. Furthermore, separatrix surfaces and their intersections with other separatrix surfaces (i.e., separators) are important likely energy release sites. Aims. The aim of this paper is to investigate, throughout the solar cycle, the nature of coronal magnetic-field topologies that arise under the potential-field source-surface approximation. In particular, we characterise the typical global fields at solar maximum and minimum. Methods. Global magnetic fields are extrapolated from observed Kitt Peak and SOLIS synoptic magnetograms, from Carrington rotations 1645 to 2144, using the potential-field source-surface model. This allows the variations in the coronal skeleton to be studied over three solar cycles. Results. The main building blocks which make up magnetic fields are identified and classified according to the nature of their separatrix surfaces. The magnetic skeleton reveals that, at solar maximum, the global coronal field involves a multitude of topological structures at all latitudes criss-crossing throughout the atmosphere. Many open-field regions exist originating anywhere on the photosphere. At solar minimum, the coronal topology is heavily influenced by the solar magnetic dipole. A strong dipole results in a simple large-scale structure involving just two large polar open-field regions, but, at short radial distances between ± 60° latitude, the small-scale topology is complex. If the solar magnetic dipole if weak, as in the recent minimum, then the low-latitude quiet-sun magnetic fields may be globally significant enough to create many disconnected open-field regions between ± 60° latitude, in addition to the two polar open-field regions.
Description: S.J.P. acknowledges financial support from the Isle of Man Government. E.R.P. is grateful to the Leverhulme Trust for his emeritus fellowship. The research leading to these results has received funding from the European Commission’s Seventh Framework Programme (FP7/2007-2013) under the grant agreement SWIFF (project No. 263340, www.swiff.eu).
2014-05-01T00:00:00Z
Platten, S.J.
Parnell, C.E.
Haynes, A.L.
Priest, E.R.
MacKay, D.H.
Context. The complicated distribution of magnetic flux across the solar photosphere results in a complex web of coronal magnetic field structures. To understand this complexity, the magnetic skeleton of the coronal field can be calculated. The skeleton highlights the (separatrix) surfaces that divide the field into topologically distinct regions, allowing open-field regions on the solar surface to be located. Furthermore, separatrix surfaces and their intersections with other separatrix surfaces (i.e., separators) are important likely energy release sites. Aims. The aim of this paper is to investigate, throughout the solar cycle, the nature of coronal magnetic-field topologies that arise under the potential-field source-surface approximation. In particular, we characterise the typical global fields at solar maximum and minimum. Methods. Global magnetic fields are extrapolated from observed Kitt Peak and SOLIS synoptic magnetograms, from Carrington rotations 1645 to 2144, using the potential-field source-surface model. This allows the variations in the coronal skeleton to be studied over three solar cycles. Results. The main building blocks which make up magnetic fields are identified and classified according to the nature of their separatrix surfaces. The magnetic skeleton reveals that, at solar maximum, the global coronal field involves a multitude of topological structures at all latitudes criss-crossing throughout the atmosphere. Many open-field regions exist originating anywhere on the photosphere. At solar minimum, the coronal topology is heavily influenced by the solar magnetic dipole. A strong dipole results in a simple large-scale structure involving just two large polar open-field regions, but, at short radial distances between ± 60° latitude, the small-scale topology is complex. If the solar magnetic dipole if weak, as in the recent minimum, then the low-latitude quiet-sun magnetic fields may be globally significant enough to create many disconnected open-field regions between ± 60° latitude, in addition to the two polar open-field regions.
Free products in R. Thompson’s group V
Bleak, Collin Patrick
Salazar-Diaz, Olga
http://hdl.handle.net/10023/5237
2014-08-26T09:01:01Z
2013-11-01T00:00:00Z
Abstract: We investigate some product structures in R. Thompson's group $ V$, primarily by studying the topological dynamics associated with $ V$'s action on the Cantor set C. We draw attention to the class D(V,C) of groups which have embeddings as demonstrative subgroups of V whose class can be used to assist in forming various products. Note that D(V,C) contains all finite groups, the free group on two generators, and Q/Z, and is closed under passing to subgroups and under taking direct products of any member by any finite member. If G≤V and H ∈ D(V,C), then G~H embeds into V. Finally, if G, H ∈ D(V,C), then G*H embeds in V. Using a dynamical approach, we also show the perhaps surprising result that Z2 * Z does not embed in V, even though V has many embedded copies of Z2 and has many embedded copies of free products of various pairs of its subgroups.
2013-11-01T00:00:00Z
Bleak, Collin Patrick
Salazar-Diaz, Olga
We investigate some product structures in R. Thompson's group $ V$, primarily by studying the topological dynamics associated with $ V$'s action on the Cantor set C. We draw attention to the class D(V,C) of groups which have embeddings as demonstrative subgroups of V whose class can be used to assist in forming various products. Note that D(V,C) contains all finite groups, the free group on two generators, and Q/Z, and is closed under passing to subgroups and under taking direct products of any member by any finite member. If G≤V and H ∈ D(V,C), then G~H embeds into V. Finally, if G, H ∈ D(V,C), then G*H embeds in V. Using a dynamical approach, we also show the perhaps surprising result that Z2 * Z does not embed in V, even though V has many embedded copies of Z2 and has many embedded copies of free products of various pairs of its subgroups.
Indeterminacy and instability in Petschek reconnection
Forbes, T.G.
Priest, E.R.
Seaton, D.B.
Litvinenko, Y.E.
http://hdl.handle.net/10023/5234
2014-12-07T02:01:45Z
2013-05-13T00:00:00Z
Abstract: We explain two puzzling aspects of Petschek's model for fast reconnection. One is its failure to occur in plasma simulations with uniform resistivity. The other is its inability to provide anything more than an upper limit for the reconnection rate. We have found that previously published analytical solutions based on Petschek's model are structurally unstable if the electrical resistivity is uniform. The structural instability is associated with the presence of an essential singularity at the X-line that is unphysical. By requiring that such a singularity does not exist, we obtain a formula that predicts a specific rate of reconnection. For uniform resistivity, reconnection can only occur at the slow, Sweet-Parker rate. For nonuniform resistivity, reconnection can occur at a much faster rate provided that the resistivity profile is not too flat near the X-line. If this condition is satisfied, then the scale length of the nonuniformity determines the reconnection rate.
Description: This work was supported by NSF Grants ATM-0734032 and AGS-0962698, NASA Grants NNX08AG44G and NNX-10AC04G to the University of New Hampshire, and subcontract SVT-7702 from the Smithsonian Astrophysical Observatory in support of their NASA Grants NNM07AA02C and NNM07AB07C. D. B. Seaton was supported by PRODEX Grant C90193 managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office, and by Grant FP7/2007-2013 from the European Commission's Seventh Framework Program under the agreement eHeroes (Project No. 284461). Additional support was provided by the Leverhulme Trust to E. R. Priest.
2013-05-13T00:00:00Z
Forbes, T.G.
Priest, E.R.
Seaton, D.B.
Litvinenko, Y.E.
We explain two puzzling aspects of Petschek's model for fast reconnection. One is its failure to occur in plasma simulations with uniform resistivity. The other is its inability to provide anything more than an upper limit for the reconnection rate. We have found that previously published analytical solutions based on Petschek's model are structurally unstable if the electrical resistivity is uniform. The structural instability is associated with the presence of an essential singularity at the X-line that is unphysical. By requiring that such a singularity does not exist, we obtain a formula that predicts a specific rate of reconnection. For uniform resistivity, reconnection can only occur at the slow, Sweet-Parker rate. For nonuniform resistivity, reconnection can occur at a much faster rate provided that the resistivity profile is not too flat near the X-line. If this condition is satisfied, then the scale length of the nonuniformity determines the reconnection rate.
The effect of slip length on vortex rebound from a rigid boundary
Sutherland, D.
Macaskill, C.
Dritschel, D.G.
http://hdl.handle.net/10023/5232
2014-10-10T10:01:01Z
2013-09-23T00:00:00Z
Abstract: The problem of a dipole incident normally on a rigid boundary, for moderate to large Reynolds numbers, has recently been treated numerically using a volume penalisation method by Nguyen van yen, Farge, and Schneider [Phys. Rev. Lett.106, 184502 (2011)]. Their results indicate that energy dissipating structures persist in the inviscid limit. They found that the use of penalisation methods intrinsically introduces some slip at the boundary wall, where the slip approaches zero as the Reynolds number goes to infinity, so reducing to the no-slip case in this limit. We study the same problem, for both no-slip and partial slip cases, using compact differences on a Chebyshev grid in the direction normal to the wall and Fourier methods in the direction along the wall. We find that for the no-slip case there is no indication of the persistence of energy dissipating structures in the limit as viscosity approaches zero and that this also holds for any fixed slip length. However, when the slip length is taken to vary inversely with Reynolds number then the results of Nguyen van yen et al. are regained. It therefore appears that the prediction that energy dissipating structures persist in the inviscid limit follows from the two limits of wall slip length going to zero, and viscosity going to zero, not being treated independently in their use of the volume penalisation method.
2013-09-23T00:00:00Z
Sutherland, D.
Macaskill, C.
Dritschel, D.G.
The problem of a dipole incident normally on a rigid boundary, for moderate to large Reynolds numbers, has recently been treated numerically using a volume penalisation method by Nguyen van yen, Farge, and Schneider [Phys. Rev. Lett.106, 184502 (2011)]. Their results indicate that energy dissipating structures persist in the inviscid limit. They found that the use of penalisation methods intrinsically introduces some slip at the boundary wall, where the slip approaches zero as the Reynolds number goes to infinity, so reducing to the no-slip case in this limit. We study the same problem, for both no-slip and partial slip cases, using compact differences on a Chebyshev grid in the direction normal to the wall and Fourier methods in the direction along the wall. We find that for the no-slip case there is no indication of the persistence of energy dissipating structures in the limit as viscosity approaches zero and that this also holds for any fixed slip length. However, when the slip length is taken to vary inversely with Reynolds number then the results of Nguyen van yen et al. are regained. It therefore appears that the prediction that energy dissipating structures persist in the inviscid limit follows from the two limits of wall slip length going to zero, and viscosity going to zero, not being treated independently in their use of the volume penalisation method.
Progress towards numerical and experimental simulations of fusion relevant beam instabilities
King, M.
Bryson, R.
Ronald, K.
Cairns, R. A.
McConville, S. L.
Speirs, D. C.
Phelps, A. D. R.
Bingham, R.
Gillespie, K. M.
Cross, A. W.
Vorgul, I.
Trines, R.
http://hdl.handle.net/10023/5186
2014-10-13T16:01:03Z
2014-01-01T00:00:00Z
Abstract: In certain plasmas, non-thermal electron distributions can produce instabilities. These instabilities may be useful or potentially disruptive. Therefore the study of these instabilities is of importance in a variety of fields including fusion science and astrophysics. Following on from previous work conducted at the University of Strathclyde on the cyclotron resonance maser instability that was relevant to astrophysical radiowave generation, further instabilities are being investigated. Particular instabilities of interest are the anomalous Doppler instability which can occur in magnetic confinement fusion plasmas and the two-stream instability that is of importance in fast-ignition inertial confinement fusion. To this end, computational simulations have been undertaken to investigate the behaviour of both the anomalous Doppler and two-stream instabilities with the goal of designing an experiment to observe these behaviours in a laboratory.
2014-01-01T00:00:00Z
King, M.
Bryson, R.
Ronald, K.
Cairns, R. A.
McConville, S. L.
Speirs, D. C.
Phelps, A. D. R.
Bingham, R.
Gillespie, K. M.
Cross, A. W.
Vorgul, I.
Trines, R.
In certain plasmas, non-thermal electron distributions can produce instabilities. These instabilities may be useful or potentially disruptive. Therefore the study of these instabilities is of importance in a variety of fields including fusion science and astrophysics. Following on from previous work conducted at the University of Strathclyde on the cyclotron resonance maser instability that was relevant to astrophysical radiowave generation, further instabilities are being investigated. Particular instabilities of interest are the anomalous Doppler instability which can occur in magnetic confinement fusion plasmas and the two-stream instability that is of importance in fast-ignition inertial confinement fusion. To this end, computational simulations have been undertaken to investigate the behaviour of both the anomalous Doppler and two-stream instabilities with the goal of designing an experiment to observe these behaviours in a laboratory.
Scaled Experiment to Investigate Auroral Kilometric Radiation Mechanisms in the Presence of Background Electrons
McConville, S. L.
Ronald, K.
Speirs, D. C.
Gillespie, K. M.
Phelps, A. D. R.
Cross, A. W.
Bingham, R.
Robertson, C. W.
Whyte, C. G.
He, W.
King, M.
Bryson, R.
Vorgul, I.
Cairns, R. A.
Kellett, B. J.
http://hdl.handle.net/10023/5185
2014-10-21T13:31:00Z
2014-01-01T00:00:00Z
Abstract: Auroral Kilometric Radiation (AKR) emissions occur at frequencies similar to 300kHz polarised in the X-mode with efficiencies similar to 1-2% [1,2] in the auroral density cavity in the polar regions of the Earth's magnetosphere, a region of low density plasma similar to 3200km above the Earth's surface, where electrons are accelerated down towards the Earth whilst undergoing magnetic compression. As a result of this magnetic compression the electrons acquire a horseshoe distribution function in velocity space. Previous theoretical studies have predicted that this distribution is capable of driving the cyclotron maser instability. To test this theory a scaled laboratory experiment was constructed to replicate this phenomenon in a controlled environment, [3-5] whilst 2D and 3D simulations are also being conducted to predict the experimental radiation power and mode, [6-9]. The experiment operates in the microwave frequency regime and incorporates a region of increasing magnetic field as found at the Earth's pole using magnet solenoids to encase the cylindrical interaction waveguide through which an initially rectilinear electron beam (12A) was accelerated by a 75keV pulse. Experimental results showed evidence of the formation of the horseshoe distribution function. The radiation was produced in the near cut-off TE01 mode, comparable with X-mode characteristics, at 4.42GHz. Peak microwave output power was measured similar to 35kW and peak efficiency of emission similar to 2%, [3]. A Penning trap was constructed and inserted into the interaction waveguide to enable generation of a background plasma which would lead to closer comparisons with the magnetospheric conditions. Initial design and measurements are presented showing the principle features of the new geometry.
2014-01-01T00:00:00Z
McConville, S. L.
Ronald, K.
Speirs, D. C.
Gillespie, K. M.
Phelps, A. D. R.
Cross, A. W.
Bingham, R.
Robertson, C. W.
Whyte, C. G.
He, W.
King, M.
Bryson, R.
Vorgul, I.
Cairns, R. A.
Kellett, B. J.
Auroral Kilometric Radiation (AKR) emissions occur at frequencies similar to 300kHz polarised in the X-mode with efficiencies similar to 1-2% [1,2] in the auroral density cavity in the polar regions of the Earth's magnetosphere, a region of low density plasma similar to 3200km above the Earth's surface, where electrons are accelerated down towards the Earth whilst undergoing magnetic compression. As a result of this magnetic compression the electrons acquire a horseshoe distribution function in velocity space. Previous theoretical studies have predicted that this distribution is capable of driving the cyclotron maser instability. To test this theory a scaled laboratory experiment was constructed to replicate this phenomenon in a controlled environment, [3-5] whilst 2D and 3D simulations are also being conducted to predict the experimental radiation power and mode, [6-9]. The experiment operates in the microwave frequency regime and incorporates a region of increasing magnetic field as found at the Earth's pole using magnet solenoids to encase the cylindrical interaction waveguide through which an initially rectilinear electron beam (12A) was accelerated by a 75keV pulse. Experimental results showed evidence of the formation of the horseshoe distribution function. The radiation was produced in the near cut-off TE01 mode, comparable with X-mode characteristics, at 4.42GHz. Peak microwave output power was measured similar to 35kW and peak efficiency of emission similar to 2%, [3]. A Penning trap was constructed and inserted into the interaction waveguide to enable generation of a background plasma which would lead to closer comparisons with the magnetospheric conditions. Initial design and measurements are presented showing the principle features of the new geometry.
3D PiC code investigations of Auroral Kilometric Radiation mechanisms
Gillespie, K. M.
McConville, S. L.
Speirs, D. C.
Ronald, K.
Phelps, A. D. R.
Bingham, R.
Cross, A. W.
Robertson, C. W.
Whyte, C. G.
He, W.
Vorgul, I.
Cairns, R. A.
Kellett, B. J.
http://hdl.handle.net/10023/5184
2014-10-21T12:31:00Z
2014-01-01T00:00:00Z
Abstract: Efficient (similar to 1%) electron cyclotron radio emissions are known to originate in the X mode from regions of locally depleted plasma in the Earths polar magnetosphere. These emissions are commonly referred to as the Auroral Kilometric Radiation (AKR). AKR occurs naturally in these polar regions where electrons are accelerated by electric fields into the increasing planetary magnetic dipole. Here conservation of the magnetic moment converts axial to rotational momentum forming a horseshoe distribution in velocity phase space. This distribution is unstable to cyclotron emission with radiation emitted in the X-mode. Initial studies were conducted in the form of 2D PiC code simulations [1] and a scaled laboratory experiment that was constructed to reproduce the mechanism of AKR. As studies progressed, 3D PiC code simulations were conducted to enable complete investigation of the complex interaction dimensions. A maximum efficiency of 1.25% is predicted from these simulations in the same mode and frequency as measured in the experiment. This is also consistent with geophysical observations and the predictions of theory.
2014-01-01T00:00:00Z
Gillespie, K. M.
McConville, S. L.
Speirs, D. C.
Ronald, K.
Phelps, A. D. R.
Bingham, R.
Cross, A. W.
Robertson, C. W.
Whyte, C. G.
He, W.
Vorgul, I.
Cairns, R. A.
Kellett, B. J.
Efficient (similar to 1%) electron cyclotron radio emissions are known to originate in the X mode from regions of locally depleted plasma in the Earths polar magnetosphere. These emissions are commonly referred to as the Auroral Kilometric Radiation (AKR). AKR occurs naturally in these polar regions where electrons are accelerated by electric fields into the increasing planetary magnetic dipole. Here conservation of the magnetic moment converts axial to rotational momentum forming a horseshoe distribution in velocity phase space. This distribution is unstable to cyclotron emission with radiation emitted in the X-mode. Initial studies were conducted in the form of 2D PiC code simulations [1] and a scaled laboratory experiment that was constructed to reproduce the mechanism of AKR. As studies progressed, 3D PiC code simulations were conducted to enable complete investigation of the complex interaction dimensions. A maximum efficiency of 1.25% is predicted from these simulations in the same mode and frequency as measured in the experiment. This is also consistent with geophysical observations and the predictions of theory.
Numerical simulation of unconstrained cyclotron resonant maser emission
Speirs, D. C.
Gillespie, K. M.
Ronald, K.
McConville, S. L.
Phelps, A. D. R.
Cross, A. W.
Bingham, R.
Kellett, B. J.
Cairns, R. A.
Vorgul, I.
http://hdl.handle.net/10023/5183
2014-10-13T16:01:01Z
2014-01-01T00:00:00Z
Abstract: When a mainly rectilinear electron beam is subject to significant magnetic compression, conservation of magnetic moment results in the formation of a horseshoe shaped velocity distribution. It has been shown that such a distribution is unstable to cyclotron emission and may be responsible for the generation of Auroral Kilometric Radiation (AKR) an intense rf emission sourced at high altitudes in the terrestrial auroral magnetosphere. PiC code simulations have been undertaken to investigate the dynamics of the cyclotron emission process in the absence of cavity boundaries with particular consideration of the spatial growth rate, spectral output and rf conversion efficiency. Computations reveal that a well-defined cyclotron emission process occurs albeit with a low spatial growth rate compared to waveguide bounded simulations. The rf output is near perpendicular to the electron beam with a slight backward-wave character reflected in the spectral output with a well defined peak at 2.68GHz, just below the relativistic electron cyclotron frequency. The corresponding rf conversion efficiency of 1.1% is comparable to waveguide bounded simulations and consistent with the predictions of kinetic theory that suggest efficient, spectrally well defined radiation emission can be obtained from an electron horseshoe distribution in the absence of radiation boundaries.
2014-01-01T00:00:00Z
Speirs, D. C.
Gillespie, K. M.
Ronald, K.
McConville, S. L.
Phelps, A. D. R.
Cross, A. W.
Bingham, R.
Kellett, B. J.
Cairns, R. A.
Vorgul, I.
When a mainly rectilinear electron beam is subject to significant magnetic compression, conservation of magnetic moment results in the formation of a horseshoe shaped velocity distribution. It has been shown that such a distribution is unstable to cyclotron emission and may be responsible for the generation of Auroral Kilometric Radiation (AKR) an intense rf emission sourced at high altitudes in the terrestrial auroral magnetosphere. PiC code simulations have been undertaken to investigate the dynamics of the cyclotron emission process in the absence of cavity boundaries with particular consideration of the spatial growth rate, spectral output and rf conversion efficiency. Computations reveal that a well-defined cyclotron emission process occurs albeit with a low spatial growth rate compared to waveguide bounded simulations. The rf output is near perpendicular to the electron beam with a slight backward-wave character reflected in the spectral output with a well defined peak at 2.68GHz, just below the relativistic electron cyclotron frequency. The corresponding rf conversion efficiency of 1.1% is comparable to waveguide bounded simulations and consistent with the predictions of kinetic theory that suggest efficient, spectrally well defined radiation emission can be obtained from an electron horseshoe distribution in the absence of radiation boundaries.
Laminar shocks in high power laser plasma interactions
Cairns, R. A.
Bingham, R.
Norreys, P.
Trines, R.
http://hdl.handle.net/10023/5180
2014-10-09T14:01:00Z
2014-02-01T00:00:00Z
Abstract: We propose a theory to describe laminar ion sound structures in a collisionless plasma. Reflection of a small fraction of the upstream ions converts the well known ion acoustic soliton into a structure with a steep potential gradient upstream and with downstream oscillations. The theory provides a simple interpretation of results dating back more than forty years but, more importantly, is shown to provide an explanation for recent observations on laser produced plasmas relevant to inertial fusion and to ion acceleration. (C) 2014 AIP Publishing LLC.
2014-02-01T00:00:00Z
Cairns, R. A.
Bingham, R.
Norreys, P.
Trines, R.
We propose a theory to describe laminar ion sound structures in a collisionless plasma. Reflection of a small fraction of the upstream ions converts the well known ion acoustic soliton into a structure with a steep potential gradient upstream and with downstream oscillations. The theory provides a simple interpretation of results dating back more than forty years but, more importantly, is shown to provide an explanation for recent observations on laser produced plasmas relevant to inertial fusion and to ion acceleration. (C) 2014 AIP Publishing LLC.
Effect of collisions on amplification of laser beams by Brillouin scattering in plasmas
Humphrey, K. A.
Trines, R. M. G. M.
Fiuza, F.
Speirs, D. C.
Norreys, P.
Cairns, R. A.
Silva, L. O.
Bingham, R.
http://hdl.handle.net/10023/5173
2014-08-15T14:01:01Z
2013-10-01T00:00:00Z
Abstract: We report on particle in cell simulations of energy transfer between a laser pump beam and a counter-propagating seed beam using the Brillouin scattering process in uniform plasma including collisions. The results presented show that the ion acoustic waves excited through naturally occurring Brillouin scattering of the pump field are preferentially damped without affecting the driven Brillouin scattering process resulting from the beating of the pump and seed fields together. We find that collisions, including the effects of Landau damping, allow for a more efficient transfer of energy between the laser beams, and a significant reduction in the amount of seed pre-pulse produced.
Description: Authors KH, RT, DCS, RAC, RB were supported by EPSRC grant EP/G04239X/1.
2013-10-01T00:00:00Z
Humphrey, K. A.
Trines, R. M. G. M.
Fiuza, F.
Speirs, D. C.
Norreys, P.
Cairns, R. A.
Silva, L. O.
Bingham, R.
We report on particle in cell simulations of energy transfer between a laser pump beam and a counter-propagating seed beam using the Brillouin scattering process in uniform plasma including collisions. The results presented show that the ion acoustic waves excited through naturally occurring Brillouin scattering of the pump field are preferentially damped without affecting the driven Brillouin scattering process resulting from the beating of the pump and seed fields together. We find that collisions, including the effects of Landau damping, allow for a more efficient transfer of energy between the laser beams, and a significant reduction in the amount of seed pre-pulse produced.
Beyond sum-free sets in the natural numbers
Huczynska, Sophie
http://hdl.handle.net/10023/4986
2014-10-07T16:01:00Z
2014-02-07T00:00:00Z
Abstract: For an interval [1,N]⊆N, sets S⊆[1,N] with the property that |{(x,y)∈S2:x+y∈S}|=0, known as sum-free sets, have attracted considerable attention. In this paper, we generalize this notion by considering r(S)=|{(x,y)∈S2:x+y∈S}|, and analyze its behaviour as S ranges over the subsets of [1,N]. We obtain a comprehensive description of the spectrum of attainable r-values, constructive existence results and structural characterizations for sets attaining extremal and near-extremal values.
2014-02-07T00:00:00Z
Huczynska, Sophie
For an interval [1,N]⊆N, sets S⊆[1,N] with the property that |{(x,y)∈S2:x+y∈S}|=0, known as sum-free sets, have attracted considerable attention. In this paper, we generalize this notion by considering r(S)=|{(x,y)∈S2:x+y∈S}|, and analyze its behaviour as S ranges over the subsets of [1,N]. We obtain a comprehensive description of the spectrum of attainable r-values, constructive existence results and structural characterizations for sets attaining extremal and near-extremal values.
On the probability of generating a monolithic group
Detomi, Eloisa
Lucchini, Andrea
Roney-Dougal, Colva Mary
http://hdl.handle.net/10023/4626
2014-04-28T15:01:35Z
2014-06-01T00:00:00Z
Abstract: A group L is primitive monolithic if L has a unique minimal normal subgroup, N , and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L , given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N . Information aboutPL,N(k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups.
Description: This research was supported through EPSRC grant EP/I03582X/1. The APC was paid through RCUK open access block grant funds.
2014-06-01T00:00:00Z
Detomi, Eloisa
Lucchini, Andrea
Roney-Dougal, Colva Mary
A group L is primitive monolithic if L has a unique minimal normal subgroup, N , and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L , given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N . Information aboutPL,N(k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups.
Generalized dimensions of images of measures under Gaussian processes
Falconer, Kenneth
Xiao, Yimin
http://hdl.handle.net/10023/4319
2014-04-02T14:01:01Z
2014-02-15T00:00:00Z
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 q-dimension μX is the image of the measure μ under X. In particular this holds for index-α fractional Brownian motion, for fractional Riesz–Bessel motions and for certain infinity scale fractional Brownian motions.
Description: 26 pages
2014-02-15T00:00:00Z
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 q-dimension μX is the image of the measure μ under X. In particular this holds for index-α fractional Brownian motion, for fractional Riesz–Bessel motions and for certain infinity scale fractional Brownian motions.
Inhomogeneous parabolic equations on unbounded metric measure spaces
Falconer, Kenneth John
Hu, Jiaxin
Sun, Yuhua
http://hdl.handle.net/10023/4061
2014-05-22T15:31:00Z
2012-10-01T00:00:00Z
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 Harnack-type inequalities in time t and some powerful estimates, we give sufficient conditions for non-existence, local existence and global existence of weak solutions, depending on the value of p relative to a critical exponent.
2012-10-01T00:00:00Z
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 Harnack-type inequalities in time t and some powerful estimates, we give sufficient conditions for non-existence, 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 alpha-Farey and alpha-Luroth systems
Kesseboehmer, Marc
Munday, Sara
Stratmann, Bernd O.
http://hdl.handle.net/10023/3933
2013-08-20T09:50:45Z
2012-06-01T00:00:00Z
Abstract: In this paper, we introduce and study the alpha-Farey map and its associated jump transformation, the alpha-Luroth 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 alpha-sum-level sets for the alpha-Luroth 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 alpha-Farey map and the alpha-Luroth 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.
2012-06-01T00:00:00Z
Kesseboehmer, Marc
Munday, Sara
Stratmann, Bernd O.
In this paper, we introduce and study the alpha-Farey map and its associated jump transformation, the alpha-Luroth 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 alpha-sum-level sets for the alpha-Luroth 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 alpha-Farey map and the alpha-Luroth 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
Balka, Richard
Farkas, Abel
Fraser, Jonathan M.
Hyde, James T.
http://hdl.handle.net/10023/3902
2014-06-12T14:31:02Z
2013-01-01T00:00:00Z
Abstract: We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, X, into R-n. 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 real-valued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux.
Description: This work is supported by EPSRC Doctoral Training Grants
2013-01-01T00:00:00Z
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 R-n. 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 real-valued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux.
Dimension theory and fractal constructions based on self-affine carpets
Fraser, Jonathan M.
http://hdl.handle.net/10023/3869
2014-04-18T14:17:19Z
2013-11-29T00:00:00Z
Abstract: The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student.
The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012.
In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'.
In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions.
In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation.
Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.
2013-11-29T00:00:00Z
Fraser, Jonathan M.
The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student.
The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012.
In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'.
In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions.
In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation.
Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.
Finiteness conditions for unions of semigroups
Abu-Ghazalh, Nabilah Hani
http://hdl.handle.net/10023/3687
2014-03-24T15:54:28Z
2013-06-28T00:00:00Z
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.
2013-06-28T00:00:00Z
Abu-Ghazalh, 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
Pfeiffer, Markus Johannes
http://hdl.handle.net/10023/3671
2014-11-06T10:52:29Z
2013-06-28T00:00:00Z
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 so-called iteration lemmas.
We define a hierarchy of the considered classes of semigroups and connect our original results with previous research.
2013-06-28T00:00:00Z
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 so-called iteration lemmas.
We define a hierarchy of the considered classes of semigroups and connect our original results with previous research.
Ends of semigroups
Craik, Simon
http://hdl.handle.net/10023/3590
2014-03-24T15:49:19Z
2013-01-01T00:00:00Z
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 quasi-isometric digraphs have isomorphic end sets. We also prove that vertex transitive digraphs have 1, 2 or infinitely many ends and construct a topology that reflects the properties of the ends of a digraph.
2013-01-01T00:00:00Z
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 quasi-isometric digraphs have isomorphic end sets. We also prove that vertex transitive digraphs have 1, 2 or infinitely many ends and construct a topology that reflects the properties of the ends of a digraph.
Random generation and chief length of finite groups
Menezes, Nina E.
http://hdl.handle.net/10023/3578
2014-02-05T16:19:59Z
2013-01-01T00:00:00Z
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 2-generated almost simple group, P[subscript(G,Soc(G))](2) 53≥90,
with equality if and only if G = A₆ or S₆. Furthermore P[subscript(G,Soc(G))](2) 9≥10
except for 30 almost simple groups G, and we specify this list and provide
exact values for P[subscript(G,Soc(G))](2) in these cases. We conclude Part I by showing
that for all almost simple groups P[subscript(G,Soc(G))](3)≥139/150.
In Part II we consider a related notion. Given a probability ε, we wish
to determine d[superscript(ε)] (G), the number of random elements needed to generate a finite group G with failure probabilty at most ε. A generalisation of a result
of Lubotzky bounds d[superscript(ε)](G) in terms of l(G), the chief length of G, and d(G),
the minimal number of generators needed to generate G. We obtain bounds
on the chief length of permutation groups in terms of the degree n, and
bounds on the chief length of completely reducible matrix groups in terms
of the dimension and field size. Combining these with existing bounds on
d(G), we obtain bounds on d[superscript(ε)] (G) for permutation groups and completely
reducible matrix groups.
2013-01-01T00:00:00Z
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 2-generated almost simple group, P[subscript(G,Soc(G))](2) 53≥90,
with equality if and only if G = A₆ or S₆. Furthermore P[subscript(G,Soc(G))](2) 9≥10
except for 30 almost simple groups G, and we specify this list and provide
exact values for P[subscript(G,Soc(G))](2) in these cases. We conclude Part I by showing
that for all almost simple groups P[subscript(G,Soc(G))](3)≥139/150.
In Part II we consider a related notion. Given a probability ε, we wish
to determine d[superscript(ε)] (G), the number of random elements needed to generate a finite group G with failure probabilty at most ε. A generalisation of a result
of Lubotzky bounds d[superscript(ε)](G) in terms of l(G), the chief length of G, and d(G),
the minimal number of generators needed to generate G. We obtain bounds
on the chief length of permutation groups in terms of the degree n, and
bounds on the chief length of completely reducible matrix groups in terms
of the dimension and field size. Combining these with existing bounds on
d(G), we obtain bounds on d[superscript(ε)] (G) for permutation groups and completely
reducible matrix groups.
Multistable processes and localizability
Falconer, Kenneth John
Liu, Lining
http://hdl.handle.net/10023/3560
2014-05-21T14:31:00Z
2012-01-01T00:00:00Z
Abstract: We use characteristic functions to construct alpha-multistable 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 alpha-multistable 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.
2012-01-01T00:00:00Z
Falconer, Kenneth John
Liu, Lining
We use characteristic functions to construct alpha-multistable 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 alpha-multistable 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
Mitchell, James David
Morayne, Michal
Peresse, Yann Hamon
Quick, Martyn
http://hdl.handle.net/10023/3383
2014-05-20T12:01:02Z
2010-09-01T00:00:00Z
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.
2010-09-01T00:00:00Z
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
McPhee, Jillian Dawn
http://hdl.handle.net/10023/3358
2014-03-24T15:47:39Z
2012-11-30T00:00:00Z
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 H-classes, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for which we answer this question include the random graph
R, the random directed graph D, the random tournament T, the random bipartite graph B, Henson's graphs G[subscript n] (for n greater or equal to 3) and the total order Q. The maximal subgroups of End(Ω) are closely connected to the automorphism groups of the relational structures induced by the images of idempotents from End(Ω). It has been shown that the relational structure induced by the image of an idempotent from End(Ω) is algebraically closed. Accordingly, we investigate which groups can be realised as the automorphism group of an algebraically closed relational structure in order to
determine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph and Ω = R,D,B,
then there exist 2[superscript aleph-naught] maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsets
of Q which are the image of an idempotent from End(Q). We call these subsets retracts of Q and show that if Ω is a total order and f is an embedding of Ω into Q such that im f is a retract of Q, then there exist 2[superscript aleph-naught] maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Zⁿ for some natural number n. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2[superscript aleph-naught] regular D-classes of End(Ω) and when Ω = R,D,B there exist 2[superscript aleph-naught] J-classes of End(Ω). Additionally we show
that if Ω = R,D then all regular D-classes contain 2[superscript aleph-naught] group H-classes. On the other hand, we show that when
Ω = B,Q there exist regular D-classes
which contain countably many group H-classes.
2012-11-30T00:00:00Z
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 H-classes, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for which we answer this question include the random graph
R, the random directed graph D, the random tournament T, the random bipartite graph B, Henson's graphs G[subscript n] (for n greater or equal to 3) and the total order Q. The maximal subgroups of End(Ω) are closely connected to the automorphism groups of the relational structures induced by the images of idempotents from End(Ω). It has been shown that the relational structure induced by the image of an idempotent from End(Ω) is algebraically closed. Accordingly, we investigate which groups can be realised as the automorphism group of an algebraically closed relational structure in order to
determine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph and Ω = R,D,B,
then there exist 2[superscript aleph-naught] maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsets
of Q which are the image of an idempotent from End(Q). We call these subsets retracts of Q and show that if Ω is a total order and f is an embedding of Ω into Q such that im f is a retract of Q, then there exist 2[superscript aleph-naught] maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Zⁿ for some natural number n. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2[superscript aleph-naught] regular D-classes of End(Ω) and when Ω = R,D,B there exist 2[superscript aleph-naught] J-classes of End(Ω). Additionally we show
that if Ω = R,D then all regular D-classes contain 2[superscript aleph-naught] group H-classes. On the other hand, we show that when
Ω = B,Q there exist regular D-classes
which contain countably many group H-classes.
Every group is a maximal subgroup of the free idempotent generated semigroup over a band
Dolinka, I
Ruskuc, Nik
http://hdl.handle.net/10023/3342
2014-04-28T15:01:26Z
2013-05-01T00:00:00Z
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.
2013-05-01T00:00:00Z
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
Abughazalah, Nabilah
Ruskuc, Nik
http://hdl.handle.net/10023/3341
2014-04-28T15:01:26Z
2013-08-01T00:00:00Z
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.
2013-08-01T00:00:00Z
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
Gray, Robert Duncan
Maltcev, Victor
D. Mitchell, J.
Ruskuc, N.
http://hdl.handle.net/10023/3335
2014-04-28T15:01:24Z
2014-01-01T00:00:00Z
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.
2014-01-01T00:00:00Z
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
Boore, Graeme
Falconer, Kenneth John
http://hdl.handle.net/10023/3237
2014-05-23T16:01:01Z
2013-01-01T00:00:00Z
Abstract: For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known.
Description: "GCB was supported by an EPSRC Doctoral Training Grant whilst undertaking this work"
2013-01-01T00:00:00Z
Boore, Graeme
Falconer, Kenneth John
For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known.
Topics in combinatorial semigroup theory
Maltcev, Victor
http://hdl.handle.net/10023/3226
2014-03-24T15:55:42Z
2012-11-30T00:00:00Z
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; word-hyperbolic semigroups; decision problems for finitely presented
and one-relator monoids. First, in order to show that general ideas from Combinatorial Semigroup Theory can apply to uncountable semigroups, at the beginning of the thesis we discuss semigroups with Bergman’s property. We prove that an automaton semigroup generated by a Cayley machine
of a finite semigroup S is itself finite if and only if S is aperiodic, which yields a new characterisation of finite aperiodic monoids. Using this, we derive some further results about Cayley automaton semigroups.
We investigate how various semigroup finiteness conditions, linked to
the notion of ideal, are preserved under finite Rees and Green indices. We
obtain a surprising result that J = D is preserved by supersemigroups of finite Green index, but it is not preserved by subsemigroups of finite Rees index even in the finitely generated case. We also consider the question of preservation of hopficity for finite Rees index. We prove that in general hopficity is preserved neither by finite Rees index subsemigroups, nor by finite Rees index extensions. However, under finite generation assumption,
hopficity is preserved by finite Rees index extensions. Still, there is
an example of a finitely generated hopfian semigroup with a non-hopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent context-free monadic rewriting systems are word-hyperbolic, and provide an example of such a monoid, which does not admit a word-hyperbolic structure with uniqueness.
This answers in the negative a question of Duncan & Gilman. We initiate in this thesis a study of Markov semigroups. We investigate
how the property of being Markov is preserved under finite Rees and
Green indices. For various semigroup properties P we examine whether P , ¬P are Markov properties, and whether P is decidable for finitely presented and
one-relator monoids.
2012-11-30T00:00:00Z
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; word-hyperbolic semigroups; decision problems for finitely presented
and one-relator monoids. First, in order to show that general ideas from Combinatorial Semigroup Theory can apply to uncountable semigroups, at the beginning of the thesis we discuss semigroups with Bergman’s property. We prove that an automaton semigroup generated by a Cayley machine
of a finite semigroup S is itself finite if and only if S is aperiodic, which yields a new characterisation of finite aperiodic monoids. Using this, we derive some further results about Cayley automaton semigroups.
We investigate how various semigroup finiteness conditions, linked to
the notion of ideal, are preserved under finite Rees and Green indices. We
obtain a surprising result that J = D is preserved by supersemigroups of finite Green index, but it is not preserved by subsemigroups of finite Rees index even in the finitely generated case. We also consider the question of preservation of hopficity for finite Rees index. We prove that in general hopficity is preserved neither by finite Rees index subsemigroups, nor by finite Rees index extensions. However, under finite generation assumption,
hopficity is preserved by finite Rees index extensions. Still, there is
an example of a finitely generated hopfian semigroup with a non-hopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent context-free monadic rewriting systems are word-hyperbolic, and provide an example of such a monoid, which does not admit a word-hyperbolic structure with uniqueness.
This answers in the negative a question of Duncan & Gilman. We initiate in this thesis a study of Markov semigroups. We investigate
how the property of being Markov is preserved under finite Rees and
Green indices. For various semigroup properties P we examine whether P , ¬P are Markov properties, and whether P is decidable for finitely presented and
one-relator monoids.
Growth of generating sets for direct powers of classical algebraic structures
Quick, Martyn
Ruskuc, Nik
http://hdl.handle.net/10023/3058
2014-05-20T12:01:02Z
2010-08-01T00:00:00Z
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.
2010-08-01T00:00:00Z
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
Ayik, Hayrullah
http://hdl.handle.net/10023/2843
2012-06-22T15:39:04Z
1998-01-01T00:00:00Z
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 O-direct union of semigroups with zero. In Chapter 3 we investigate
finite generation and presentability of Rees matrix semigroups over semigroups.
We find necessary and sufficient conditions for finite generation and presentability.
In Chapter 4 we investigate some other finiteness conditions for Rees matrix
semigroups.
In Chapter 5 we consider groups as semigroups and investigate their semigroup
efficiency. In Chapter 6 we look at "proper" semigroups, that is semigroups
that are not groups. We first give examples of efficient and inefficient "proper"
semigroups by computing their homology and finding their minimal presentations.
In Chapter 7 we compute the second homology of finite simple semigroups and
find a "small" presentation for them. If that "small" presentation has a special
relation, we prove that finite simple semigroups are efficient. Finally, in Chapter
8, we investigate the efficiency of wreath products of finite groups as groups and
as semigroups. We give more examples of efficient groups and inefficient groups.
1998-01-01T00:00:00Z
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 O-direct union of semigroups with zero. In Chapter 3 we investigate
finite generation and presentability of Rees matrix semigroups over semigroups.
We find necessary and sufficient conditions for finite generation and presentability.
In Chapter 4 we investigate some other finiteness conditions for Rees matrix
semigroups.
In Chapter 5 we consider groups as semigroups and investigate their semigroup
efficiency. In Chapter 6 we look at "proper" semigroups, that is semigroups
that are not groups. We first give examples of efficient and inefficient "proper"
semigroups by computing their homology and finding their minimal presentations.
In Chapter 7 we compute the second homology of finite simple semigroups and
find a "small" presentation for them. If that "small" presentation has a special
relation, we prove that finite simple semigroups are efficient. Finally, in Chapter
8, we investigate the efficiency of wreath products of finite groups as groups and
as semigroups. We give more examples of efficient groups and inefficient groups.
Semigroups of order-decreasing transformations
Umar, Abdullahi
http://hdl.handle.net/10023/2834
2012-06-21T13:44:10Z
1992-01-01T00:00:00Z
Abstract: Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial one-to-one) transformations of X. In this
Thesis the study of order-increasing full (partial, partial one-to-one) transformations
has been reduced to that of order-decreasing full (partial, partial one-to-one)
transformations and the study of order-decreasing partial transformations to that of
order-decreasing full transformations for both the finite and infinite cases.
For the finite order-decreasing full (partial one-to-one) transformation
semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden
(1990) concerning products of idempotents (quasi-idempotents), and concerning
combinatorial and rank properties. By contrast with the semigroups of order-preserving
transformations and the full transformation semigroup, the semigroups of orderdecreasing
full (partial one-to-one) transformations and their Rees quotient semigroups
are not regular. They are, however, abundant (type A) semigroups in the sense of
Fountain (1982,1979). An explicit characterisation of the minimum semilattice
congruence on the finite semigroups of order-decreasing transformations and their Rees
quotient semigroups is obtained.
If X is an infinite chain then the semigroup S of order-decreasing full
transformations need not be abundant. A necessary and sufficient condition on X is
obtained for S to be abundant. By contrast, for every chain X the semigroup of
order-decreasing partial one-to-one transformations is type A.
The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing
full (partial one-to-one) transformations have been investigated.
1992-01-01T00:00:00Z
Umar, Abdullahi
Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial one-to-one) transformations of X. In this
Thesis the study of order-increasing full (partial, partial one-to-one) transformations
has been reduced to that of order-decreasing full (partial, partial one-to-one)
transformations and the study of order-decreasing partial transformations to that of
order-decreasing full transformations for both the finite and infinite cases.
For the finite order-decreasing full (partial one-to-one) transformation
semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden
(1990) concerning products of idempotents (quasi-idempotents), and concerning
combinatorial and rank properties. By contrast with the semigroups of order-preserving
transformations and the full transformation semigroup, the semigroups of orderdecreasing
full (partial one-to-one) transformations and their Rees quotient semigroups
are not regular. They are, however, abundant (type A) semigroups in the sense of
Fountain (1982,1979). An explicit characterisation of the minimum semilattice
congruence on the finite semigroups of order-decreasing transformations and their Rees
quotient semigroups is obtained.
If X is an infinite chain then the semigroup S of order-decreasing full
transformations need not be abundant. A necessary and sufficient condition on X is
obtained for S to be abundant. By contrast, for every chain X the semigroup of
order-decreasing partial one-to-one transformations is type A.
The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing
full (partial one-to-one) transformations have been investigated.
Semigroup presentations
Ruškuc, Nik
http://hdl.handle.net/10023/2821
2014-03-24T15:52:25Z
1995-01-01T00:00:00Z
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 O-simple semigroups, transformation semigroups,
matrix semigroups and various endomorphism semigroups. In Chapter 6
we find presentations for the following semi group constructions: wreath product,
Bruck-Reilly extension, Schiitzenberger product, strong semilattices of monoids,
Rees matrix semigroups, ideal extensions and subsemigroups. We investigate in
more detail presentations for subsemigroups in Chapters 7 and 10, where we prove
a number of Reidemeister-Schreier type results for semigroups. In Chapter 9
we examine the connection between the semi group and the group defined by the
same presentation. The general results from Chapters 6, 7, 9 and 10 are applied
in Chapters 8, 11, 12 and 13 to subsemigroups of free semigroups, Fibonacci
semigroups, semigroups defined by Coxeter type presentations and one relator
products of cyclic groups. Finally, in Chapter 14 we describe the Todd-Coxeter
enumeration procedure and introduce three modifications of this procedure.
1995-01-01T00:00:00Z
Ruškuc, Nik
In this thesis we consider in detail the following two fundamental problems for
semigroup presentations:
1. Given a semigroup find a presentation defining it.
2. Given a presentation describe the semigroup defined by it.
We also establish two links between these two approaches: semigroup constructions
and computational methods.
After an introduction to semigroup presentations in Chapter 3, in Chapters 4
and 5 we consider the first of the two approaches. The semigroups we examine in
these two chapters include completely O-simple semigroups, transformation semigroups,
matrix semigroups and various endomorphism semigroups. In Chapter 6
we find presentations for the following semi group constructions: wreath product,
Bruck-Reilly extension, Schiitzenberger product, strong semilattices of monoids,
Rees matrix semigroups, ideal extensions and subsemigroups. We investigate in
more detail presentations for subsemigroups in Chapters 7 and 10, where we prove
a number of Reidemeister-Schreier type results for semigroups. In Chapter 9
we examine the connection between the semi group and the group defined by the
same presentation. The general results from Chapters 6, 7, 9 and 10 are applied
in Chapters 8, 11, 12 and 13 to subsemigroups of free semigroups, Fibonacci
semigroups, semigroups defined by Coxeter type presentations and one relator
products of cyclic groups. Finally, in Chapter 14 we describe the Todd-Coxeter
enumeration procedure and introduce three modifications of this procedure.
Green index in semigroups : generators, presentations and automatic structures
Cain, A.J.
Gray, R
Ruskuc, Nik
http://hdl.handle.net/10023/2760
2014-05-22T13:01:01Z
2012-01-01T00:00:00Z
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 T-relative 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 group-embeddable semigroups).
2012-01-01T00:00:00Z
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 T-relative 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 group-embeddable semigroups).
The visible part of plane self-similar sets
Falconer, Kenneth John
Fraser, Jonathan Macdonald
http://hdl.handle.net/10023/2756
2014-05-21T14:31:00Z
2013-01-01T00:00:00Z
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 half-line 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 self-similar 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.
2013-01-01T00:00:00Z
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 half-line 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 self-similar 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
Coutts, Hannah Jane
http://hdl.handle.net/10023/2561
2013-05-07T09:56:02Z
2011-11-01T00:00:00Z
Abstract: Part I of this thesis presents methods for finding the primitive permutation
groups of degree d, where 2500 ≤ d < 4096, using the O'Nan-Scott Theorem
and Aschbacher's theorem. Tables of the groups G are given for each O'Nan-Scott class. For the non-affine groups, additional information is given: the
degree d of G, the shape of a stabiliser in G of the primitive action, the
shape of the normaliser N in S[subscript(d)] of G and the rank of N.
Part II presents a new algorithm NormaliserGL for computing the normaliser
in GL[subscript(n)](q) of a group G ≤ GL[subscript(n)](q). The algorithm is implemented in
the computational algebra system MAGMA and employs Aschbacher's theorem
to break the problem into several cases. The attached CD contains the
code for the algorithm as well as several test cases which demonstrate the
improvement over MAGMA's existing algorithm.
2011-11-01T00:00:00Z
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'Nan-Scott Theorem
and Aschbacher's theorem. Tables of the groups G are given for each O'Nan-Scott class. For the non-affine groups, additional information is given: the
degree d of G, the shape of a stabiliser in G of the primitive action, the
shape of the normaliser N in S[subscript(d)] of G and the rank of N.
Part II presents a new algorithm NormaliserGL for computing the normaliser
in GL[subscript(n)](q) of a group G ≤ GL[subscript(n)](q). The algorithm is implemented in
the computational algebra system MAGMA and employs Aschbacher's theorem
to break the problem into several cases. The attached CD contains the
code for the algorithm as well as several test cases which demonstrate the
improvement over MAGMA's existing algorithm.
Generation problems for finite groups
McDougall-Bagnall, Jonathan M.
http://hdl.handle.net/10023/2529
2012-03-29T12:05:29Z
2011-11-30T00:00:00Z
Abstract: It can be deduced from the Burnside Basis Theorem that if G is a finite p-group with d(G)=r then given any generating set A for G there exists a subset of A of size r that generates G. We have denoted this property B. A group is said to have the basis property if all subgroups have property B. This thesis is a study into the nature of these two properties. Note all groups are finite unless stated otherwise.
We begin this thesis by providing examples of groups with and without property B and several results on the structure of groups with property B, showing that under certain conditions property B is inherited by quotients. This culminates with a result which shows that groups with property B that can be expressed as direct products are exactly those arising from the Burnside Basis Theorem.
We also seek to create a class of groups which have property B. We provide a method for constructing groups with property B and trivial Frattini subgroup using finite fields. We then classify all groups G where the quotient of G by the Frattini subgroup is isomorphic to this construction. We finally note that groups arising from this construction do not in general have the basis property.
Finally we look at groups with the basis property. We prove that groups with the basis property are soluble and consist only of elements of prime-power order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.
2011-11-30T00:00:00Z
McDougall-Bagnall, Jonathan M.
It can be deduced from the Burnside Basis Theorem that if G is a finite p-group with d(G)=r then given any generating set A for G there exists a subset of A of size r that generates G. We have denoted this property B. A group is said to have the basis property if all subgroups have property B. This thesis is a study into the nature of these two properties. Note all groups are finite unless stated otherwise.
We begin this thesis by providing examples of groups with and without property B and several results on the structure of groups with property B, showing that under certain conditions property B is inherited by quotients. This culminates with a result which shows that groups with property B that can be expressed as direct products are exactly those arising from the Burnside Basis Theorem.
We also seek to create a class of groups which have property B. We provide a method for constructing groups with property B and trivial Frattini subgroup using finite fields. We then classify all groups G where the quotient of G by the Frattini subgroup is isomorphic to this construction. We finally note that groups arising from this construction do not in general have the basis property.
Finally we look at groups with the basis property. We prove that groups with the basis property are soluble and consist only of elements of prime-power order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.
Substitution-closed pattern classes
Atkinson, M.D.
Ruskuc, Nik
Smith, R
http://hdl.handle.net/10023/2149
2014-05-22T13:31:00Z
2011-02-01T00:00:00Z
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.
2011-02-01T00:00:00Z
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
Cain, Alan J.
Oliver, Graham
Ruskuc, Nik
Thomas, Richard M.
http://hdl.handle.net/10023/2148
2014-04-28T15:01:10Z
2010-08-01T00:00:00Z
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 FA-presentable. 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, Bruck-Reilly extensions, zero-direct unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FA-presentable semigroups under that construction is considered, as is the question of whether the FA-presentability of the semigroup obtained from such a construction implies the FA-presentability of the original semigroup[s]. Classifications are also given of the FA-presentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0-simple semigroups.
2010-08-01T00:00:00Z
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 FA-presentable. 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, Bruck-Reilly extensions, zero-direct unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FA-presentable semigroups under that construction is considered, as is the question of whether the FA-presentability of the semigroup obtained from such a construction implies the FA-presentability of the original semigroup[s]. Classifications are also given of the FA-presentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0-simple semigroups.
Automatic presentations for semigroups
Cain, Alan James
Oliver, Graham
Ruskuc, Nik
Thomas, Richard M.
http://hdl.handle.net/10023/2147
2014-11-09T01:31:32Z
2009-11-01T00:00:00Z
Abstract: This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We give a complete list of FA-presentable one-relation semigroups and compare the classes of FA-presentable 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)
2009-11-01T00:00:00Z
Cain, Alan James
Oliver, Graham
Ruskuc, Nik
Thomas, Richard M.
This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We give a complete list of FA-presentable one-relation semigroups and compare the classes of FA-presentable semigroups and automatic semigroups. (C) 2009 Elsevier Inc. All rights reserved.
On residual finiteness of direct products of algebraic systems
Gray, R.
Ruskuc, Nik
http://hdl.handle.net/10023/2146
2014-04-28T15:01:08Z
2009-09-01T00:00:00Z
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.
2009-09-01T00:00:00Z
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
Maltcev, V.
Mitchell, J. D.
Ruskuc, N.
http://hdl.handle.net/10023/2145
2014-07-27T01:01:03Z
2009-08-01T00:00:00Z
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 Baer-Levi semigroup does not have the Bergman property.
2009-08-01T00:00:00Z
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 Baer-Levi semigroup does not have the Bergman property.
Green index and finiteness conditions for semigroups
Gray, Robert Duncan
Ruskuc, Nik
http://hdl.handle.net/10023/2144
2014-05-18T00:31:21Z
2008-10-15T00:00:00Z
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.
2008-10-15T00:00:00Z
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
Descalco, L.
Ruskuc, Nik
http://hdl.handle.net/10023/2142
2014-04-28T15:01:06Z
2008-06-01T00:00:00Z
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.
2008-06-01T00:00:00Z
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
Huczynska, Sophie
Ruskuc, Nikola
http://hdl.handle.net/10023/2140
2014-05-08T15:31:01Z
2008-01-01T00:00:00Z
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 order-theoretic 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.
2008-01-01T00:00:00Z
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 order-theoretic 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
Cain, Alan James
Robertson, Edmund Frederick
Ruskuc, Nik
http://hdl.handle.net/10023/2138
2014-04-28T15:01:05Z
2008-02-01T00:00:00Z
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, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.).
2008-02-01T00:00:00Z
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, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.).
Growth rates for subclasses of Av(321)
Albert, M.H.
Atkinson, M.D.
Brignall, R
Ruskuc, Nik
Smith, R
West, J
http://hdl.handle.net/10023/2137
2014-05-22T13:31:00Z
2010-10-22T00:00:00Z
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.
2010-10-22T00:00:00Z
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
Dombi, Erzsebet Rita
Ruskuc, Nik
http://hdl.handle.net/10023/2136
2014-04-28T15:01:07Z
2009-06-01T00:00:00Z
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.
2009-06-01T00:00:00Z
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 idempotent-generated semigroups over the full transformation monoid
Gray, R
Ruskuc, Nik
http://hdl.handle.net/10023/2134
2014-05-26T09:31:00Z
2012-05-01T00:00:00Z
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.
2012-05-01T00:00:00Z
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
Gray, R
Ruskuc, Nik
http://hdl.handle.net/10023/2131
2014-05-22T13:01:02Z
2011-11-01T00:00:00Z
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.
2011-11-01T00:00:00Z
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
Hyde, James Thomas
Loughlin, Nicholas
Quick, Martyn
Ruskuc, Nik
Wallis, Alistair
http://hdl.handle.net/10023/2129
2014-05-22T14:31:00Z
2012-01-01T00:00:00Z
Abstract: For a semigroup S its d-sequence 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.
2012-01-01T00:00:00Z
Hyde, James Thomas
Loughlin, Nicholas
Quick, Martyn
Ruskuc, Nik
Wallis, Alistair
For a semigroup S its d-sequence 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
Gray, R
Ruskuc, Nik
http://hdl.handle.net/10023/2128
2014-05-22T13:01:02Z
2012-01-01T00:00:00Z
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.
2012-01-01T00:00:00Z
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
Boore, Graeme C.
http://hdl.handle.net/10023/2109
2014-09-22T13:59:11Z
2011-11-30T00:00:00Z
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 2-vertex directed graph IFSs have attractors
that cannot be the attractors of standard (1-vertex directed graph) IFSs, with
or without separation conditions. We also calculate their exact Hausdorff measure.
Thus we are able to identify a new class of attractors for which the exact Hausdorff
measure is known.
We give a constructive algorithm for calculating the set of gap lengths of any
attractor as a finite union of cosets of finitely generated semigroups of positive real
numbers. The generators of these semigroups are contracting similarity ratios of
simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ
with no limit on the number of vertices in the directed graph, provided a separation
condition holds.
The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We
obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth
packing moment of μᵤ, the self-similar measure at a vertex u, for the non-lattice case,
with a corresponding limit for the lattice case. We do this
(i) for any q ∈ ℝ if the strong separation condition holds,
(ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative
matrix associated with the system is irreducible.
In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum
of μᵤ to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper
multifractal q box-dimension with respect to μᵤ, of the set consisting of all the intersections
of the components of Fᵤ, is strictly less than the multifractal q Hausdorff
dimension with respect to μᵤ of Fᵤ.
2011-11-30T00:00:00Z
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 2-vertex directed graph IFSs have attractors
that cannot be the attractors of standard (1-vertex directed graph) IFSs, with
or without separation conditions. We also calculate their exact Hausdorff measure.
Thus we are able to identify a new class of attractors for which the exact Hausdorff
measure is known.
We give a constructive algorithm for calculating the set of gap lengths of any
attractor as a finite union of cosets of finitely generated semigroups of positive real
numbers. The generators of these semigroups are contracting similarity ratios of
simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ
with no limit on the number of vertices in the directed graph, provided a separation
condition holds.
The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We
obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth
packing moment of μᵤ, the self-similar measure at a vertex u, for the non-lattice case,
with a corresponding limit for the lattice case. We do this
(i) for any q ∈ ℝ if the strong separation condition holds,
(ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative
matrix associated with the system is irreducible.
In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum
of μᵤ to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper
multifractal q box-dimension with respect to μᵤ, of the set consisting of all the intersections
of the components of Fᵤ, is strictly less than the multifractal q Hausdorff
dimension with respect to μᵤ of Fᵤ.
On convex permutations
Albert, M.H.
Linton, Stephen Alexander
Ruskuc, Nik
Vatter, V
Waton, S
http://hdl.handle.net/10023/2000
2014-05-22T13:01:04Z
2011-05-01T00:00:00Z
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.
2011-05-01T00:00:00Z
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
Carvalho, C.A.
Gray, R
Ruskuc, Nik
http://hdl.handle.net/10023/1998
2014-05-22T13:01:02Z
2011-06-01T00:00:00Z
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.
Description: "Part of this work was done while Gray was an EPSRC Postdoctoral Research Fellow at the University of St Andrews, Scotland"
2011-06-01T00:00:00Z
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
Brignall, R
Ruskuc, Nik
Vatter, V
http://hdl.handle.net/10023/1997
2014-05-22T13:01:03Z
2011-07-01T00:00:00Z
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.
2011-07-01T00:00:00Z
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
Falconer, Kenneth John
Fraser, Jonathan Macdonald
http://hdl.handle.net/10023/1956
2014-06-01T00:31:19Z
2011-01-01T00:00:00Z
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.
Description: JMF was supported by an EPSRC Doctoral Training Grant whilst undertaking this work.
2011-01-01T00:00:00Z
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
Samuel, Anthony
http://hdl.handle.net/10023/1710
2014-04-18T14:16:58Z
2010-01-01T00:00:00Z
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 non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S.
Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}}^{{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.
2010-01-01T00:00:00Z
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 non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S.
Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}}^{{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.
The lesser names : the teachers of the Edinburgh Mathematical Society and other aspects of Scottish mathematics, 1867–1946
Hartveit, Marit
http://hdl.handle.net/10023/1700
2011-07-05T11:50:44Z
2011-06-22T00:00:00Z
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.
2011-06-22T00:00:00Z
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
Huczynska, Sophie
Cohen, SD
http://hdl.handle.net/10023/1615
2014-04-28T15:01:03Z
2003-08-01T00:00:00Z
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.
2003-08-01T00:00:00Z
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
Cain, AJ
Robertson, Edmund Frederick
Ruskuc, Nikola
http://hdl.handle.net/10023/1561
2014-05-08T16:01:00Z
2006-07-01T00:00:00Z
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.
2006-07-01T00:00:00Z
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
Higgins, PM
Mitchell, James David
Ruskuc, Nikola
http://hdl.handle.net/10023/1553
2014-07-27T00:31:06Z
2003-09-01T00:00:00Z
Abstract: For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings O-X 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 <O-X boolean OR A> = T-X. When X is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when X = R (the set of real numbers) it is uncountable.
2003-09-01T00:00:00Z
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 O-X 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 <O-X boolean OR A> = T-X. When X is countably infinite or well-ordered (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
Campbell, Colin Matthew
Mitchell, James David
Ruskuc, Nikola
http://hdl.handle.net/10023/1442
2014-05-04T00:31:03Z
2002-07-01T00:00:00Z
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.
2002-07-01T00:00:00Z
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
Liu, Lining
http://hdl.handle.net/10023/948
2014-04-18T14:16:28Z
2010-06-23T00:00:00Z
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 α-stable random measures restricted
to small intervals with α constant on each interval.
We then use the multistable random integrals to define multistable random
processes on R and study the localisability of these processes. Thus we find conditions
that ensure that a process locally ‘looks like’ a given stochastic process
under enlargement and appropriate scaling. We give many examples of multistable
random processes and examine their local forms.
Finally, we examine the dimensions of graphs of α-stable random functions
defined by series with α-stable random variables as coefficients.
2010-06-23T00:00:00Z
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 α-stable random measures restricted
to small intervals with α constant on each interval.
We then use the multistable random integrals to define multistable random
processes on R and study the localisability of these processes. Thus we find conditions
that ensure that a process locally ‘looks like’ a given stochastic process
under enlargement and appropriate scaling. We give many examples of multistable
random processes and examine their local forms.
Finally, we examine the dimensions of graphs of α-stable random functions
defined by series with α-stable random variables as coefficients.
Classification and enumeration of finite semigroups
Distler, Andreas
http://hdl.handle.net/10023/945
2014-03-24T15:53:55Z
2010-06-23T00:00:00Z
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 anti-isomorphism of any order
are the main results in the theoretical part of this thesis. Further
studies concern the classification of nilpotent semigroups by rank,
leading to a full classification for large ranks.
In the computational part, a method to find and
enumerate multiplication tables of semigroups and subclasses is
presented. The approach combines the advantages of computer algebra
and constraint satisfaction, to allow for an efficient and fast
search. The problem of avoiding isomorphic and anti-isomorphic
semigroups is dealt with by supporting standard methods from
constraint satisfaction with structural knowledge about the semigroups
under consideration. The approach is adapted to various
problems, and realised using the computer algebra system GAP and the
constraint solver Minion. New results include the numbers of
semigroups of order 9, and of monoids and bands of order 10. Up to
isomorphism and anti-isomorphism there are 52,989,400,714,478 semigroups
with 9 elements, 52,991,253,973,742 monoids with 10 elements, and
7,033,090 bands with 10 elements. That constraint satisfaction can also
be utilised for the analysis of algebraic objects is demonstrated by
determining the automorphism groups of all semigroups with 9 elements.
A classification of the semigroups of orders 1 to 8 is made available
as a data library in form of the GAP package Smallsemi. Beyond the
semigroups themselves a large amount of precomputed properties is
contained in the library. The package as well as the code used to
obtain the enumeration results are available on the attached DVD.
2010-06-23T00:00:00Z
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 anti-isomorphism of any order
are the main results in the theoretical part of this thesis. Further
studies concern the classification of nilpotent semigroups by rank,
leading to a full classification for large ranks.
In the computational part, a method to find and
enumerate multiplication tables of semigroups and subclasses is
presented. The approach combines the advantages of computer algebra
and constraint satisfaction, to allow for an efficient and fast
search. The problem of avoiding isomorphic and anti-isomorphic
semigroups is dealt with by supporting standard methods from
constraint satisfaction with structural knowledge about the semigroups
under consideration. The approach is adapted to various
problems, and realised using the computer algebra system GAP and the
constraint solver Minion. New results include the numbers of
semigroups of order 9, and of monoids and bands of order 10. Up to
isomorphism and anti-isomorphism there are 52,989,400,714,478 semigroups
with 9 elements, 52,991,253,973,742 monoids with 10 elements, and
7,033,090 bands with 10 elements. That constraint satisfaction can also
be utilised for the analysis of algebraic objects is demonstrated by
determining the automorphism groups of all semigroups with 9 elements.
A classification of the semigroups of orders 1 to 8 is made available
as a data library in form of the GAP package Smallsemi. Beyond the
semigroups themselves a large amount of precomputed properties is
contained in the library. The package as well as the code used to
obtain the enumeration results are available on the attached DVD.
Generating uncountable transformation semigroups
Péresse, Yann
http://hdl.handle.net/10023/867
2014-03-24T15:48:44Z
2009-01-01T00:00:00Z
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.
2009-01-01T00:00:00Z
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 self-affine fractals
Miao, Jun Jie
http://hdl.handle.net/10023/838
2009-12-16T15:37:03Z
2008-01-01T00:00:00Z
Abstract: In this thesis we study the dimension theory of self-affine sets. We begin by
introducing a number of notions from fractal geometry, in particular, dimensions,
measure properties and iterated functions systems. We give a review of existing
work on self-affine sets. We then develop a variety of new results on self-affine
sets and their dimensional properties.
This work falls into three parts:
Firstly, we look at the dimension formulae for a class of self-affine sets generated
by upper triangular matrices. In this case, we simplify the affine dimension
formula into equations only involving the diagonal elements of the matrices.
Secondly, since the Hausdorff dimensions of self-affine sets depend not only
on the linear parts of the contractions but also on the translation parameters, we
obtain an upper bound for the dimensions of exceptional sets, that is, the set of
parameters such that the Hausdorff dimension of the attractor is smaller than the
affine dimension.
Thirdly, we investigate dimensions of a class of random self-affine sets, aiming
to extend the ‘almost sure’ formula for random self-similar sets to random self-affine
sets.
2008-01-01T00:00:00Z
Miao, Jun Jie
In this thesis we study the dimension theory of self-affine sets. We begin by
introducing a number of notions from fractal geometry, in particular, dimensions,
measure properties and iterated functions systems. We give a review of existing
work on self-affine sets. We then develop a variety of new results on self-affine
sets and their dimensional properties.
This work falls into three parts:
Firstly, we look at the dimension formulae for a class of self-affine sets generated
by upper triangular matrices. In this case, we simplify the affine dimension
formula into equations only involving the diagonal elements of the matrices.
Secondly, since the Hausdorff dimensions of self-affine sets depend not only
on the linear parts of the contractions but also on the translation parameters, we
obtain an upper bound for the dimensions of exceptional sets, that is, the set of
parameters such that the Hausdorff dimension of the attractor is smaller than the
affine dimension.
Thirdly, we investigate dimensions of a class of random self-affine sets, aiming
to extend the ‘almost sure’ formula for random self-similar sets to random self-affine
sets.
Intersection problems in combinatorics
Brunk, Fiona
http://hdl.handle.net/10023/765
2014-03-24T15:54:50Z
2009-11-30T00:00:00Z
Abstract: With the publication of the famous Erdős-Ko-Rado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is t-intersecting if any two of its elements mutually t-intersect, where the latter concept needs to be specified separately in each instance. This thesis is split into two parts; the first is concerned with intersecting injections while the second investigates intersecting posets.
We classify maximum 1-intersecting families of injections from {1, ..., k} to {1, ..., n}, a generalisation of the corresponding result on permutations from the early 2000s. Moreover, we obtain classifications in the general t>1 case for different parameter limits:
if n is large in terms of k and t, then the so-called fix-families, consisting of all injections which map some fixed set of t points to the same image points, are the only t-intersecting injection families of maximal size. By way of contrast, fixing the differences k-t and n-k while increasing k leads to optimal families which are equivalent to one of the so-called saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=|_ (k-t)/2 _|. Furthermore we demonstrate that, among injection families with t-intersecting and left-compressed fixed point sets, for some value of r the saturation family has maximal size .
The concept that two posets intersect if they share a comparison is new. We begin by classifying maximum intersecting families in several isomorphism classes of posets which are linear, or almost linear. Then we study the union of the almost linear classes, and derive a bound for an intersecting family by adapting Katona's elegant cycle method to posets. The thesis ends with an investigation of the intersection structure of poset classes whose elements are close to the antichain.
The overarching theme of this thesis is fixing versus saturation: we compare the sizes and structures of intersecting families obtained from these two distinct principles in the context of various classes of combinatorial objects.
2009-11-30T00:00:00Z
Brunk, Fiona
With the publication of the famous Erdős-Ko-Rado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is t-intersecting if any two of its elements mutually t-intersect, where the latter concept needs to be specified separately in each instance. This thesis is split into two parts; the first is concerned with intersecting injections while the second investigates intersecting posets.
We classify maximum 1-intersecting families of injections from {1, ..., k} to {1, ..., n}, a generalisation of the corresponding result on permutations from the early 2000s. Moreover, we obtain classifications in the general t>1 case for different parameter limits:
if n is large in terms of k and t, then the so-called fix-families, consisting of all injections which map some fixed set of t points to the same image points, are the only t-intersecting injection families of maximal size. By way of contrast, fixing the differences k-t and n-k while increasing k leads to optimal families which are equivalent to one of the so-called saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=|_ (k-t)/2 _|. Furthermore we demonstrate that, among injection families with t-intersecting and left-compressed fixed point sets, for some value of r the saturation family has maximal size .
The concept that two posets intersect if they share a comparison is new. We begin by classifying maximum intersecting families in several isomorphism classes of posets which are linear, or almost linear. Then we study the union of the almost linear classes, and derive a bound for an intersecting family by adapting Katona's elegant cycle method to posets. The thesis ends with an investigation of the intersection structure of poset classes whose elements are close to the antichain.
The overarching theme of this thesis is fixing versus saturation: we compare the sizes and structures of intersecting families obtained from these two distinct principles in the context of various classes of combinatorial objects.
Resonances for graph directed Markov systems, and geometry of infinitely generated dynamical systems
Hille, Martial R.
http://hdl.handle.net/10023/719
2012-10-24T13:38:13Z
2009-06-24T00:00:00Z
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(Im|w|)} and card{w \in\ Q(k) | zeta(w)=0} << k^{delta(S)}.
Here, Q(k)\subset\%C$ denotes a certain box of height k, and
delta(S) refers to the Hausdorff dimension of the limit set of S.
In the second part of this thesis we show that in any dimension
m \in\$N$ there are GDMSs for which the Hausdorff dimension of the uniformly radial limit set
is equal to a given arbitrary number d \in\(0,m) and the Hausdorff dimension of the Jørgensen limit set
is equal to a given arbitrary number j \in\ [0,m).
Furthermore, we derive various relations between the exponents of
convergence and the Hausdorff dimensions of certain different types of limit sets for iterated function systems (IFS), GDMSs, pseudo GDMSs and normal subsystems
of finitely generated GDMSs.
Finally, we apply our results to Kleinian groups and generalise
a result of Patterson by showing that in any dimension m \in\$N$ there
are Kleinian groups for which the Hausdorff dimension of their uniformly
radial limit set is less than a given arbitrary number d \in\ (0,m) and the Hausdorff dimension
of their Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m).
2009-06-24T00:00:00Z
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(Im|w|)} and card{w \in\ Q(k) | zeta(w)=0} << k^{delta(S)}.
Here, Q(k)\subset\%C$ denotes a certain box of height k, and
delta(S) refers to the Hausdorff dimension of the limit set of S.
In the second part of this thesis we show that in any dimension
m \in\$N$ there are GDMSs for which the Hausdorff dimension of the uniformly radial limit set
is equal to a given arbitrary number d \in\(0,m) and the Hausdorff dimension of the Jørgensen limit set
is equal to a given arbitrary number j \in\ [0,m).
Furthermore, we derive various relations between the exponents of
convergence and the Hausdorff dimensions of certain different types of limit sets for iterated function systems (IFS), GDMSs, pseudo GDMSs and normal subsystems
of finitely generated GDMSs.
Finally, we apply our results to Kleinian groups and generalise
a result of Patterson by showing that in any dimension m \in\$N$ there
are Kleinian groups for which the Hausdorff dimension of their uniformly
radial limit set is less than a given arbitrary number d \in\ (0,m) and the Hausdorff dimension
of their Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m).
Inhomogeneous self-similar sets and measures
Snigireva, Nina
http://hdl.handle.net/10023/682
2010-08-06T09:28:56Z
2008-01-01T00:00:00Z
Abstract: The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and
measures. In particular, we show that these sets and measures
are natural generalizations of the well known self-similar sets and
measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal
dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary)
self-similar sets. In the third chapter we investigate the L^{q}
spectra and the Renyi dimensions of inhomogeneous self-similar
measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case.
Namely, we show that inhomogeneous self-similar measures may
have phase transitions which is in sharp contrast to the
behaviour of the
L^{q} spectra
of (ordinary) self-similar
measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra
of inhomogeneous self-similar measures. We show that
the multifractal spectra
of
inhomogeneous self-similar
measures
may be non-concave which is again in sharp contrast to the
behaviour of the
multifractal spectra
of (ordinary) self-similar
measures satisfying the Open Set Condition. Then we present a number of
applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of
inhomogeneous self-similar measures and again we present a
number of applications of our results, in particular to non-linear
self-similar measures.
2008-01-01T00:00:00Z
Snigireva, Nina
The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and
measures. In particular, we show that these sets and measures
are natural generalizations of the well known self-similar sets and
measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal
dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary)
self-similar sets. In the third chapter we investigate the L^{q}
spectra and the Renyi dimensions of inhomogeneous self-similar
measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case.
Namely, we show that inhomogeneous self-similar measures may
have phase transitions which is in sharp contrast to the
behaviour of the
L^{q} spectra
of (ordinary) self-similar
measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra
of inhomogeneous self-similar measures. We show that
the multifractal spectra
of
inhomogeneous self-similar
measures
may be non-concave which is again in sharp contrast to the
behaviour of the
multifractal spectra
of (ordinary) self-similar
measures satisfying the Open Set Condition. Then we present a number of
applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of
inhomogeneous self-similar measures and again we present a
number of applications of our results, in particular to non-linear
self-similar measures.
Simplicity in relational structures and its application to permutation classes
Brignall, Robert
http://hdl.handle.net/10023/431
2014-03-24T15:52:44Z
2007-11-30T00:00:00Z
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 "query-complete set" is enumerated by an algebraic generating function. Such properties include being an even or alternating permutation, or avoiding generalised (blocked or barred) permutations. We further indicate that membership of a permutation class containing only finitely many simple permutations can be computed in linear time.
Using the decomposition of simple permutations, we establish, by representing pin sequences as a language over an eight-letter alphabet, that it is decidable if a permutation class given by a finite basis contains only finitely many simple permutations. We also discuss possible approaches to the same question for other relational structures, in particular the difficulties that arise for graphs. The pin sequence technology provides a further result relating to the wreath product of two permutation classes, namely that C â D is finitely based whenever D does not admit arbitrarily long pin sequences. As a partial converse, we also exhibit a number of explicit examples of wreath products that are not finitely based.
2007-11-30T00:00:00Z
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 "query-complete set" is enumerated by an algebraic generating function. Such properties include being an even or alternating permutation, or avoiding generalised (blocked or barred) permutations. We further indicate that membership of a permutation class containing only finitely many simple permutations can be computed in linear time.
Using the decomposition of simple permutations, we establish, by representing pin sequences as a language over an eight-letter alphabet, that it is decidable if a permutation class given by a finite basis contains only finitely many simple permutations. We also discuss possible approaches to the same question for other relational structures, in particular the difficulties that arise for graphs. The pin sequence technology provides a further result relating to the wreath product of two permutation classes, namely that C â D is finitely based whenever D does not admit arbitrarily long pin sequences. As a partial converse, we also exhibit a number of explicit examples of wreath products that are not finitely based.
On permutation classes defined by token passing networks, gridding matrices and pictures : three flavours of involvement
Waton, Stephen D.
http://hdl.handle.net/10023/237
2014-03-24T15:52:06Z
2007-06-19T00:00:00Z
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.
2007-06-19T00:00:00Z
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.