Pure Mathematics Research
http://hdl.handle.net/10023/98
Tue, 25 Jul 2017 00:58:43 GMT2017-07-25T00:58:43ZDecision problems for word-hyperbolic semigroups
http://hdl.handle.net/10023/11263
This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These properties do not hold in general for word-hyperbolic semigroups.) It is proved that the uniform word problem for word-hyperbolic semigroup is solvable in polynomial time (improving on the previous exponential-time algorithm). Algorithms are presented for deciding whether a word-hyperbolic semigroup is a monoid, a group, a completely simple semigroup, a Clifford semigroup, or a free semigroup.
Fri, 22 Jul 2016 00:00:00 GMThttp://hdl.handle.net/10023/112632016-07-22T00:00:00ZCain, Alan JamesPfeiffer, Markus JohannesThis paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These properties do not hold in general for word-hyperbolic semigroups.) It is proved that the uniform word problem for word-hyperbolic semigroup is solvable in polynomial time (improving on the previous exponential-time algorithm). Algorithms are presented for deciding whether a word-hyperbolic semigroup is a monoid, a group, a completely simple semigroup, a Clifford semigroup, or a free semigroup.Between primitive and 2-transitive : synchronization and its friends
http://hdl.handle.net/10023/11134
An automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing n-state automaton has a reset word of length at most (n − 1)2 . The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set Ω is said to synchronize a map f if the monoid hG, fi generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.
The second author was supported by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIÊNCIAS UID/Multi/ 04621/2013
Thu, 15 Jun 2017 00:00:00 GMThttp://hdl.handle.net/10023/111342017-06-15T00:00:00ZAraújo, JoãoCameron, Peter JephsonSteinberg, BenjaminAn automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing n-state automaton has a reset word of length at most (n − 1)2 . The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set Ω is said to synchronize a map f if the monoid hG, fi generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.Algorithms for detecting dependencies and rigid subsystems for CAD
http://hdl.handle.net/10023/11110
Automated approaches for detecting dependencies in structures created with Computer Aided Design software are critical for developing robust solvers and providing informative user feedback. We model a set of geometric constraints with a bi-colored multigraph and give a graph-based pebble game algorithm that allows us to determine combinatorially if there are generic dependencies. We further use the pebble game to yield a decomposition of the graph into factor graphs which may be used to give a user detailed feedback about dependent substructures in a specific realization of a system of CAD constraints with non-generic properties.
Louis Theran is partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement No. 247029-SDModels, Academy of Finland (AKA) project COALESCE, and the Hutchcroft fund.
Sat, 01 Oct 2016 00:00:00 GMThttp://hdl.handle.net/10023/111102016-10-01T00:00:00ZFarre, JamesKleinschmidt, HelenaSidman, JessicaJohn, Audrey St.Stark, StephanieTheran, LouisYu, XilinAutomated approaches for detecting dependencies in structures created with Computer Aided Design software are critical for developing robust solvers and providing informative user feedback. We model a set of geometric constraints with a bi-colored multigraph and give a graph-based pebble game algorithm that allows us to determine combinatorially if there are generic dependencies. We further use the pebble game to yield a decomposition of the graph into factor graphs which may be used to give a user detailed feedback about dependent substructures in a specific realization of a system of CAD constraints with non-generic properties.Topological graph inverse semigroups
http://hdl.handle.net/10023/10847
To every directed graph E one can associate a graph inverse semigroup G(E), where elements roughly correspond to possible paths in E . These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph C⁎C⁎-algebras, and Toeplitz C⁎-algebras. We investigate topologies that turn G(E) into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, G(E)∖{0} must be discrete for any directed graph E . On the other hand, G(E) need not be discrete in a Hausdorff semigroup topology, and for certain graphs E , G(E) admits a T1 semigroup topology in which G(E)∖{0} is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G(E) in larger topological semigroups.
Michał Morayne was partially supported by NCN grant DEC-2011/01/B/ST1/01439 while this work was performed.
Mon, 01 Aug 2016 00:00:00 GMThttp://hdl.handle.net/10023/108472016-08-01T00:00:00ZMesyan, Z.Mitchell, J. D.Morayne, M.Péresse, Y. H.To every directed graph E one can associate a graph inverse semigroup G(E), where elements roughly correspond to possible paths in E . These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph C⁎C⁎-algebras, and Toeplitz C⁎-algebras. We investigate topologies that turn G(E) into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, G(E)∖{0} must be discrete for any directed graph E . On the other hand, G(E) need not be discrete in a Hausdorff semigroup topology, and for certain graphs E , G(E) admits a T1 semigroup topology in which G(E)∖{0} is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G(E) in larger topological semigroups.ℤ4-codes and their Gray map images as orthogonal arrays
http://hdl.handle.net/10023/10804
A classic result of Delsarte connects the strength (as orthogonal array) of a linear code with the minimum weight of its dual: the former is one less than the latter.Since the paper of Hammons et al., there is a lot of interest in codes over rings, especially in codes over ℤ4 and their (usually non-linear) binary Gray map images.We show that Delsarte's observation extends to codes over arbitrary finite commutative rings with identity. Also, we show that the strength of the Gray map image of a ℤ4 code is one less than the minimum Lee weight of its Gray map image.
Sat, 01 Jul 2017 00:00:00 GMThttp://hdl.handle.net/10023/108042017-07-01T00:00:00ZCameron, Peter JephsonKusuma, JosephineSolé, PatrickA classic result of Delsarte connects the strength (as orthogonal array) of a linear code with the minimum weight of its dual: the former is one less than the latter.Since the paper of Hammons et al., there is a lot of interest in codes over rings, especially in codes over ℤ4 and their (usually non-linear) binary Gray map images.We show that Delsarte's observation extends to codes over arbitrary finite commutative rings with identity. Also, we show that the strength of the Gray map image of a ℤ4 code is one less than the minimum Lee weight of its Gray map image.Rare events for the Manneville-Pomeau map
http://hdl.handle.net/10023/10742
We prove a dichotomy for Manneville-Pomeau maps ƒ : [0, 1] → [0, 1] : given any point ζ ε [0, 1] , either the Rare Events Point Processes (REPP), counting the number of exceedances, which correspond to entrances in balls around ζ, converge in distribution to a Poisson process; or the point ζ is periodic and the REPP converge in distribution to a compound Poisson process. Our method is to use inducing techniques for all points except 0 and its preimages, extending a recent result [HWZ14], and then to deal with the remaining points separately. The preimages of 0 are dealt with applying recent results in [AFV14]. The point ζ = 0 is studied separately because the tangency with the identity map at this point creates too much dependence, which causes severe clustering of exceedances. The Extremal Index, which measures the intensity of clustering, is equal to 0 at ζ = 0 , which ultimately leads to a degenerate limit distribution for the partial maxima of stochastic processes arising from the dynamics and for the usual normalising sequences. We prove that using adapted normalising sequences we can still obtain non-degenerate limit distributions at ζ = 0 .
Funding: CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.
Tue, 01 Nov 2016 00:00:00 GMThttp://hdl.handle.net/10023/107422016-11-01T00:00:00ZFreitas, Ana Cristina MoreiraFreitas, JorgeTodd, MikeVaienti, SandroWe prove a dichotomy for Manneville-Pomeau maps ƒ : [0, 1] → [0, 1] : given any point ζ ε [0, 1] , either the Rare Events Point Processes (REPP), counting the number of exceedances, which correspond to entrances in balls around ζ, converge in distribution to a Poisson process; or the point ζ is periodic and the REPP converge in distribution to a compound Poisson process. Our method is to use inducing techniques for all points except 0 and its preimages, extending a recent result [HWZ14], and then to deal with the remaining points separately. The preimages of 0 are dealt with applying recent results in [AFV14]. The point ζ = 0 is studied separately because the tangency with the identity map at this point creates too much dependence, which causes severe clustering of exceedances. The Extremal Index, which measures the intensity of clustering, is equal to 0 at ζ = 0 , which ultimately leads to a degenerate limit distribution for the partial maxima of stochastic processes arising from the dynamics and for the usual normalising sequences. We prove that using adapted normalising sequences we can still obtain non-degenerate limit distributions at ζ = 0 .Uniform scaling limits for ergodic measures
http://hdl.handle.net/10023/10724
We provide an elementary proof that ergodic measures on one-sided shift spaces are ‘uniformly scaling’ in the following sense: at almost every point the scenery distributions weakly converge to a common distribution on the space of measures. Moreover, we show how the limiting distribution can be expressed in terms of, and derived from, a 'reverse Jacobian’ function associated with the corresponding measure on the space of left infinite sequences. Finally we specialise to the setting of Gibbs measures, discuss some statistical properties, and prove a Central Limit Theorem for ergodic Markov measures.
J. M. Fraser and M. Pollicott were financially supported in part by the EPSRC grant EP/J013560/1.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10023/107242017-01-01T00:00:00ZFraser, Jonathan MacDonaldPollicott, MarkWe provide an elementary proof that ergodic measures on one-sided shift spaces are ‘uniformly scaling’ in the following sense: at almost every point the scenery distributions weakly converge to a common distribution on the space of measures. Moreover, we show how the limiting distribution can be expressed in terms of, and derived from, a 'reverse Jacobian’ function associated with the corresponding measure on the space of left infinite sequences. Finally we specialise to the setting of Gibbs measures, discuss some statistical properties, and prove a Central Limit Theorem for ergodic Markov measures.Highest rank of a polytope for An
http://hdl.handle.net/10023/10678
We prove that the highest rank of a string C-group constructed from an alternating group An is 3 if n=5, 4 if n=9, 5 if n=10, 6 if n=11, and ⌊(n−1)/2⌋ if n⩾12. Moreover, if n=3,4,6,7, or 8, the group An is not a string C-group. This solves a conjecture made by the last three authors in 2012.
This research was supported by a Marsden grant (UOA1218) of the Royal Society of New Zealand, and by the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.
Tue, 04 Jul 2017 00:00:00 GMThttp://hdl.handle.net/10023/106782017-07-04T00:00:00ZCameron, Peter J.Fernandes, Maria ElisaLeemans, DimitriMixer, MarkWe prove that the highest rank of a string C-group constructed from an alternating group An is 3 if n=5, 4 if n=9, 5 if n=10, 6 if n=11, and ⌊(n−1)/2⌋ if n⩾12. Moreover, if n=3,4,6,7, or 8, the group An is not a string C-group. This solves a conjecture made by the last three authors in 2012.Planar self-affine sets with equal Hausdorff, box and affinity dimensions
http://hdl.handle.net/10023/10634
Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We exhibit some new specific classes of self-affine sets for which these dimensions are equal.
Thu, 20 Oct 2016 00:00:00 GMThttp://hdl.handle.net/10023/106342016-10-20T00:00:00ZFalconer, KennethKempton, Thomas Michael WilliamUsing methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We exhibit some new specific classes of self-affine sets for which these dimensions are equal.Generating sets of finite groups
http://hdl.handle.net/10023/10576
We investigate the extent to which the exchange relation holds in finite groups G. We define a new equivalence relation ≡m, where two elements are equivalent if each can be substituted for the other in any generating set for G. We then refine this to a new sequence ≡(r)/m of equivalence relations by saying that x≡(r)/m y if each can be substituted for the other in any r-element generating set. The relations ≡(r)/m become finer as r increases, and we define a new group invariant ψ(G) to be the value of r at which they stabilise to ≡m. Remarkably, we are able to prove that if G is soluble then ψ(G) ∈ {d(G),d(G)+1}, where d(G) is the minimum number of generators of G, and to classify the finite soluble groups G for which ψ(G)=d(G). For insoluble G, we show that d(G) ≤ ψ(G) ≤ d(G)+5. However, we know of no examples of groups G for which ψ(G) > d(G)+1. As an application, we look at the generating graph of G, whose vertices are the elements of G, the edges being the 2-element generating sets. Our relation ≡(2)m enables us to calculate Aut(Γ(G)) for all soluble groups G of nonzero spread, and give detailed structural information about Aut(Γ(G)) in the insoluble case.
Sun, 02 Apr 2017 00:00:00 GMThttp://hdl.handle.net/10023/105762017-04-02T00:00:00ZCameron, Peter JephsonLucchini, AndreaRoney-Dougal, Colva MaryWe investigate the extent to which the exchange relation holds in finite groups G. We define a new equivalence relation ≡m, where two elements are equivalent if each can be substituted for the other in any generating set for G. We then refine this to a new sequence ≡(r)/m of equivalence relations by saying that x≡(r)/m y if each can be substituted for the other in any r-element generating set. The relations ≡(r)/m become finer as r increases, and we define a new group invariant ψ(G) to be the value of r at which they stabilise to ≡m. Remarkably, we are able to prove that if G is soluble then ψ(G) ∈ {d(G),d(G)+1}, where d(G) is the minimum number of generators of G, and to classify the finite soluble groups G for which ψ(G)=d(G). For insoluble G, we show that d(G) ≤ ψ(G) ≤ d(G)+5. However, we know of no examples of groups G for which ψ(G) > d(G)+1. As an application, we look at the generating graph of G, whose vertices are the elements of G, the edges being the 2-element generating sets. Our relation ≡(2)m enables us to calculate Aut(Γ(G)) for all soluble groups G of nonzero spread, and give detailed structural information about Aut(Γ(G)) in the insoluble case.Mixed moments and local dimensions of measures
http://hdl.handle.net/10023/10568
We analyse the asymptotic behaviour of the mixed moments of Borel probability measures on [0,1]d. In particular, we prove that the asymptotic behaviour of the mixed moments of a measure is intimately related to the local dimensions of the measure.
Thu, 01 Sep 2016 00:00:00 GMThttp://hdl.handle.net/10023/105682016-09-01T00:00:00ZOlsen, Lars Ole RonnowWe analyse the asymptotic behaviour of the mixed moments of Borel probability measures on [0,1]d. In particular, we prove that the asymptotic behaviour of the mixed moments of a measure is intimately related to the local dimensions of the measure.On the generating graph of a simple group
http://hdl.handle.net/10023/10539
The generating graph Γ(H) of a finite group H is the graph defined on the elements of H, with an edge between two vertices if and only if they generate H. We show that if H is a sufficiently large simple group with Γ(G) ≅ Γ(H) for a finite group G, then G ≅ H. We also prove that the generating graph of a symmetric group determines the group.
The authors were supported by Universita di Padova (Progetto di Ricerca di Ateneo: Invariable generation of groups). The second author was also supported by an Alexander von Humboldt Fellowship for Experienced Researchers, by OTKA grants K84233 and K115799, and by the MTA Renyi Lendulet Groups and Graphs Research Group.
Mon, 26 Sep 2016 00:00:00 GMThttp://hdl.handle.net/10023/105392016-09-26T00:00:00ZLucchini, AndreaMaroti, AttilaRoney-Dougal, Colva MaryThe generating graph Γ(H) of a finite group H is the graph defined on the elements of H, with an edge between two vertices if and only if they generate H. We show that if H is a sufficiently large simple group with Γ(G) ≅ Γ(H) for a finite group G, then G ≅ H. We also prove that the generating graph of a symmetric group determines the group.The Assouad dimension of randomly generated fractals
http://hdl.handle.net/10023/10511
We consider several dierent models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdor or packing dimension.
JMF was financially supported by the EPSRC grant EP/J013560/1 whilst employed at the University of Warwick. JJM was partially supported by the NNSF of China (no. 11201152), the Fund for the Doctoral Program of Higher Education of China (no. 20120076120001) and SRF for ROCS, SEM (no. 01207427) ST was financially supported by the EPSRC Doctoral Training Grant EP/K503162/1.
Thu, 22 Sep 2016 00:00:00 GMThttp://hdl.handle.net/10023/105112016-09-22T00:00:00ZFraser, Jonathan MacDonaldMiao, Jun JieTroscheit, SaschaWe consider several dierent models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdor or packing dimension.Linear response for intermittent maps
http://hdl.handle.net/10023/10334
We consider the one parameter family α↦Tα (α∈[0,1)) of Pomeau-Manneville type interval maps Tα(x)=x(1+2αxα) for x∈[0,1/2) and Tα(x)=2x−1 for x∈[1/2,1], with the associated absolutely continuous invariant probability measure μα. For α∈(0,1), Sarig and Gouëzel proved that the system mixes only polynomially with rate n1−1/α (in particular, there is no spectral gap). We show that for any ψ∈Lq, the map α→∫10ψdμα is differentiable on [0,1−1/q), and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For α≥1/2 we need the n−1/α decorrelation obtained by Gouëzel under additional conditions.
Tue, 01 Nov 2016 00:00:00 GMThttp://hdl.handle.net/10023/103342016-11-01T00:00:00ZBaladi, VivianeTodd, Michael JohnWe consider the one parameter family α↦Tα (α∈[0,1)) of Pomeau-Manneville type interval maps Tα(x)=x(1+2αxα) for x∈[0,1/2) and Tα(x)=2x−1 for x∈[1/2,1], with the associated absolutely continuous invariant probability measure μα. For α∈(0,1), Sarig and Gouëzel proved that the system mixes only polynomially with rate n1−1/α (in particular, there is no spectral gap). We show that for any ψ∈Lq, the map α→∫10ψdμα is differentiable on [0,1−1/q), and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For α≥1/2 we need the n−1/α decorrelation obtained by Gouëzel under additional conditions.The classification of partition homogeneous groups with applications to semigroup theory
http://hdl.handle.net/10023/10228
Let λ=(λ1,λ2,...) be a partition of n, a sequence of positive integers in non-increasing order with sum n. Let Ω:={1,...,n}. An ordered partition P=(A1,A2,...) of Ω has type λ if |Ai|=λi.Following Martin and Sagan, we say that G is λ-transitive if, for any two ordered partitions P=(A1,A2,...) and Q=(B1,B2,...) of Ω of type λ, there exists g ∈ G with Aig=Bi for all i. A group G is said to be λ-homogeneous if, given two ordered partitions P and Q as above, inducing the sets P'={A1,A2,...} and Q'={B1,B2,...}, there exists g ∈ G such that P'g=Q'. Clearly a λ-transitive group is λ-homogeneous.The first goal of this paper is to classify the λ-homogeneous groups (Theorems 1.1 and 1.2). The second goal is to apply this classification to a problem in semigroup theory.Let Tn and Sn denote the transformation monoid and the symmetric group on Ω, respectively. Fix a group H<=Sn. Given a non-invertible transformation a in Tn-Sn and a group G<=Sn, we say that (a,G) is an H-pair if the semigroups generated by {a} ∪ H and {a} ∪ G contain the same non-units, that is, {a,G}\G= {a,H}\H. Using the classification of the λ-homogeneous groups we classify all the Sn-pairs (Theorem 1.8). For a multitude of transformation semigroups this theorem immediately implies a description of their automorphisms, congruences, generators and other relevant properties (Theorem 8.5). This topic involves both group theory and semigroup theory; we have attempted to include enough exposition to make the paper self-contained for researchers in both areas. The paper finishes with a number of open problems on permutation and linear groups.
Fri, 15 Apr 2016 00:00:00 GMThttp://hdl.handle.net/10023/102282016-04-15T00:00:00ZAndré, JorgeAraúo, JoāoCameron, Peter JephsonLet λ=(λ1,λ2,...) be a partition of n, a sequence of positive integers in non-increasing order with sum n. Let Ω:={1,...,n}. An ordered partition P=(A1,A2,...) of Ω has type λ if |Ai|=λi.Following Martin and Sagan, we say that G is λ-transitive if, for any two ordered partitions P=(A1,A2,...) and Q=(B1,B2,...) of Ω of type λ, there exists g ∈ G with Aig=Bi for all i. A group G is said to be λ-homogeneous if, given two ordered partitions P and Q as above, inducing the sets P'={A1,A2,...} and Q'={B1,B2,...}, there exists g ∈ G such that P'g=Q'. Clearly a λ-transitive group is λ-homogeneous.The first goal of this paper is to classify the λ-homogeneous groups (Theorems 1.1 and 1.2). The second goal is to apply this classification to a problem in semigroup theory.Let Tn and Sn denote the transformation monoid and the symmetric group on Ω, respectively. Fix a group H<=Sn. Given a non-invertible transformation a in Tn-Sn and a group G<=Sn, we say that (a,G) is an H-pair if the semigroups generated by {a} ∪ H and {a} ∪ G contain the same non-units, that is, {a,G}\G= {a,H}\H. Using the classification of the λ-homogeneous groups we classify all the Sn-pairs (Theorem 1.8). For a multitude of transformation semigroups this theorem immediately implies a description of their automorphisms, congruences, generators and other relevant properties (Theorem 8.5). This topic involves both group theory and semigroup theory; we have attempted to include enough exposition to make the paper self-contained for researchers in both areas. The paper finishes with a number of open problems on permutation and linear groups.On The Lq dimensions of measures on Heuter-Lalley type self-affine sets
http://hdl.handle.net/10023/10165
We study the Lq-dimensions of self-affine measures and the Käenmäki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the Lq-dimensions are equal to the value predicted by Falconer for a range of q. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets.
JMF acknowledges financial support from a Leverhulme Trust Research Fellowship (RF-2016-500).
Sat, 21 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10023/101652017-01-21T00:00:00ZFraser, Jonathan MacDonaldKempton, TomWe study the Lq-dimensions of self-affine measures and the Käenmäki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the Lq-dimensions are equal to the value predicted by Falconer for a range of q. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets.A dynamical definition of f.g. virtually free groups
http://hdl.handle.net/10023/10148
We show that the class of finitely generated virtually free groups is precisely the class of demonstrable subgroups for R. Thompson's group V . The class of demonstrable groups for V consists of all groups which can embed into V with a natural dynamical behaviour in their induced actions on the Cantor space C2 := {0,1}ω. There are also connections with formal language theory, as the class of groups with context-free word problem is also the class of finitely generated virtually free groups, while R. Thompson's group V is a candidate as a universal coCF group by Lehnert's conjecture, corresponding to the class of groups with context free co-word problem (as introduced by Holt, Rees, Röver, and Thomas). Our main results answers a question of Berns-Zieze, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Días, and fits into the larger exploration of the class of coCF groups as it shows that all four of the known properties of the class of coCF groups hold for the set of finitely generation subgroups of V .
Mon, 01 Feb 2016 00:00:00 GMThttp://hdl.handle.net/10023/101482016-02-01T00:00:00ZBennett, DanielBleak, CollinWe show that the class of finitely generated virtually free groups is precisely the class of demonstrable subgroups for R. Thompson's group V . The class of demonstrable groups for V consists of all groups which can embed into V with a natural dynamical behaviour in their induced actions on the Cantor space C2 := {0,1}ω. There are also connections with formal language theory, as the class of groups with context-free word problem is also the class of finitely generated virtually free groups, while R. Thompson's group V is a candidate as a universal coCF group by Lehnert's conjecture, corresponding to the class of groups with context free co-word problem (as introduced by Holt, Rees, Röver, and Thomas). Our main results answers a question of Berns-Zieze, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Días, and fits into the larger exploration of the class of coCF groups as it shows that all four of the known properties of the class of coCF groups hold for the set of finitely generation subgroups of V .On optimality and construction of circular repeated-measurements designs
http://hdl.handle.net/10023/10092
The aim of this paper is to characterize and construct universally optimal designs among the class of circular repeated-measurements designs when the parameters do not permit balance for carry-over effects. It is shown that some circular weakly neighbour balanced designs defined by Filipiak and Markiewicz (2012) are universally optimal repeated-measurements designs. These results extend the work of Magda (1980), Kunert (1984b) and Filipiak and Markiewicz (2012).
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10023/100922017-01-01T00:00:00ZBailey, Rosemary AnneCameron, Peter JephsonFilipiak, KatarzynaKunert, JoachimMarkiewicz, AugustynThe aim of this paper is to characterize and construct universally optimal designs among the class of circular repeated-measurements designs when the parameters do not permit balance for carry-over effects. It is shown that some circular weakly neighbour balanced designs defined by Filipiak and Markiewicz (2012) are universally optimal repeated-measurements designs. These results extend the work of Magda (1980), Kunert (1984b) and Filipiak and Markiewicz (2012).Transience and multifractal analysis
http://hdl.handle.net/10023/10086
We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.
G.I. was partially supported by the Center of Dynamical Systems and Related Fields código ACT1103 and by Proyecto Fondecyt 1150058. T.J. wishes to thank Proyecto Mecesup-0711 for funding his visit to PUC-Chile. M.T. is grateful for the support of Proyecto Fondecyt 1110040 for funding his visit to PUC-Chile and for partial support from NSF grant DMS 1109587.
Mon, 11 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10023/100862016-01-11T00:00:00ZIommi, GodofredoJordan, ThomasTodd, Michael JohnWe study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.Multifractal spectra and multifractal zeta-functions
http://hdl.handle.net/10023/10071
We introduce multifractal zetafunctions providing precise information of a very general class of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these and more general multifractal spectra equal the abscissae of convergence of the associated zeta-functions.
Wed, 01 Feb 2017 00:00:00 GMThttp://hdl.handle.net/10023/100712017-02-01T00:00:00ZMijovic, VuksanOlsen, Lars Ole RonnowWe introduce multifractal zetafunctions providing precise information of a very general class of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these and more general multifractal spectra equal the abscissae of convergence of the associated zeta-functions.Average distances on self-similar sets and higher order average distances of self-similar measures
http://hdl.handle.net/10023/10069
The purpose of this paper is twofold: (1) We study different notions of the average distance between two points of a self-similar subset of ℝ, and (2) we investigate the asymptotic behaviour of higher order average moments of self-similar measures on self-similar subsets of in ℝ.
Thu, 29 Dec 2016 00:00:00 GMThttp://hdl.handle.net/10023/100692016-12-29T00:00:00ZAllen, D.Edwards, H.Harper, S.Olsen, Lars Ole RonnowThe purpose of this paper is twofold: (1) We study different notions of the average distance between two points of a self-similar subset of ℝ, and (2) we investigate the asymptotic behaviour of higher order average moments of self-similar measures on self-similar subsets of in ℝ.The Assouad dimension of self-affine carpets with no grid structure
http://hdl.handle.net/10023/10061
Previous study of the Assouad dimension of planar self-affine sets has relied heavily on the underlying IFS having a `grid structure', thus allowing for the use of approximate squares. We study the Assouad dimension of a class of self-affine carpets which do not have an associated grid structure. We find that the Assouad dimension is related to the box and Assouad dimensions of the (self-similar) projection of the self-affine set onto the first coordinate and to the local dimensions of the projection of a natural Bernoulli measure onto the first coordinate. In a special case we relate the Assouad dimension of the Przytycki-Urbański sets to the lower local dimensions of Bernoulli convolutions.
JMF is financially supported by a Leverhulme Trust Research Fellowship.
Wed, 21 Dec 2016 00:00:00 GMThttp://hdl.handle.net/10023/100612016-12-21T00:00:00ZFraser, Jonathan MacDonaldJordan, ThomasPrevious study of the Assouad dimension of planar self-affine sets has relied heavily on the underlying IFS having a `grid structure', thus allowing for the use of approximate squares. We study the Assouad dimension of a class of self-affine carpets which do not have an associated grid structure. We find that the Assouad dimension is related to the box and Assouad dimensions of the (self-similar) projection of the self-affine set onto the first coordinate and to the local dimensions of the projection of a natural Bernoulli measure onto the first coordinate. In a special case we relate the Assouad dimension of the Przytycki-Urbański sets to the lower local dimensions of Bernoulli convolutions.Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes
http://hdl.handle.net/10023/10058
We present strong versions of Marstrand's projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such that the projection onto any line with direction outside X, of any subset F of E of positive s-dimensional measure, has Hausdorff dimension min(1,s), i.e. the set of exceptional directions is independent of F. Using duality this leads to results on the dimension of sets that intersect families of lines or hyperplanes in positive Lebesgue measure.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10023/100582016-01-01T00:00:00ZFalconer, KennethMattila, PerttiWe present strong versions of Marstrand's projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such that the projection onto any line with direction outside X, of any subset F of E of positive s-dimensional measure, has Hausdorff dimension min(1,s), i.e. the set of exceptional directions is independent of F. Using duality this leads to results on the dimension of sets that intersect families of lines or hyperplanes in positive Lebesgue measure.Observing the formation of flare-driven coronal rain
http://hdl.handle.net/10023/10001
Flare-driven coronal rain can manifest from rapidly cooled plasma condensations near coronal loop-tops in thermally unstable post-flare arcades. We detect 5 phases that characterise the post-flare decay:heating, evaporation, conductive cooling dominance for ~120 s, radiative/ enthalpy cooling dominance for ~4700 s and finally catastrophic cooling occurring within 35-124 s leading to rain strands with s periodicity of 55-70 s. We find an excellent agreement between the observations and model predictions of the dominant cooling timescales and the onset of catastrophic cooling. At the rain formation site we detect co-moving, multi-thermal rain clumps that undergo catastrophic cooling from ~1 MK to ~22000 K. During catastrophic cooling the plasma cools at a maximum rate of 22700 K s-1 in multiple loop-top sources. We calculated the density of the EUV plasma from the DEM of the multi-thermal source employing regularised inversion. Assuming a pressure balance, we estimate the density of the chromospheric component of rain to be 9.21x1011 ±1.76x1011 cm-3 which is comparable with quiescent coronal rain densities. With up to 8 parallel strands in the EUV loop cross section, we calculate the mass loss rate from the post-flare arcade to be as much as 1.98x1012 ± 4.95x1011 g s-1. Finally, we reveal a close proximity between the model predictions of 105.8 K and the observed properties between 105.9 K and 106.2 K, that defines the temperature onset of catastrophic cooling. The close correspondence between the observations and numerical models suggests that indeed acoustic waves (with a sound travel time of 68 s) could play an important role in redistributing energy and sustaining the enthalpy-based radiative cooling.
PA. GV are funded by the European Research Council under the European Union Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement nr. 291058
Tue, 20 Dec 2016 00:00:00 GMThttp://hdl.handle.net/10023/100012016-12-20T00:00:00ZScullion, E.Rouppe Van Der Voort, L.Antolin, P.Wedemeyer, S.Vissers, G.Kontar, E. P.Gallagher, P.Flare-driven coronal rain can manifest from rapidly cooled plasma condensations near coronal loop-tops in thermally unstable post-flare arcades. We detect 5 phases that characterise the post-flare decay:heating, evaporation, conductive cooling dominance for ~120 s, radiative/ enthalpy cooling dominance for ~4700 s and finally catastrophic cooling occurring within 35-124 s leading to rain strands with s periodicity of 55-70 s. We find an excellent agreement between the observations and model predictions of the dominant cooling timescales and the onset of catastrophic cooling. At the rain formation site we detect co-moving, multi-thermal rain clumps that undergo catastrophic cooling from ~1 MK to ~22000 K. During catastrophic cooling the plasma cools at a maximum rate of 22700 K s-1 in multiple loop-top sources. We calculated the density of the EUV plasma from the DEM of the multi-thermal source employing regularised inversion. Assuming a pressure balance, we estimate the density of the chromospheric component of rain to be 9.21x1011 ±1.76x1011 cm-3 which is comparable with quiescent coronal rain densities. With up to 8 parallel strands in the EUV loop cross section, we calculate the mass loss rate from the post-flare arcade to be as much as 1.98x1012 ± 4.95x1011 g s-1. Finally, we reveal a close proximity between the model predictions of 105.8 K and the observed properties between 105.9 K and 106.2 K, that defines the temperature onset of catastrophic cooling. The close correspondence between the observations and numerical models suggests that indeed acoustic waves (with a sound travel time of 68 s) could play an important role in redistributing energy and sustaining the enthalpy-based radiative cooling.Counting chirps : acoustic monitoring of cryptic frogs
http://hdl.handle.net/10023/9921
1 . Global amphibian declines have resulted in a vital need for monitoring programmes that follow population trends. Monitoring using advertisement calls is ideal as choruses are undisturbed during data collection. However, methods currently employed by managers frequently rely on trained observers, and/or do not provide density data on which to base trends. 2 . This study explores the utility of monitoring using acoustic spatially explicit capture-recapture (aSECR) with time of arrival (ToA) and signal strength (SS) as a quantitative monitoring technique to measure call density of a threatened but visually cryptic anuran, the Cape peninsula moss frog Arthroleptella lightfooti. 3 . The relationships between temporal and environmental variables (date, rainfall, temperature) and A. lightfooti call density at three study sites on the Cape peninsula, South Africa were examined. Acoustic data, collected from an array of six microphones over four months during the winter breeding season, provided a time series of call density estimates. 4 . Model selection indicated that call density was primarily associated with seasonality fitted as a quadratic function. Call density peaked mid-breeding season. At the main study site, the lowest recorded mean call density (0·160 calls m-2 min-1) occurred in May and reached its peak mid-July (1·259 calls m-2 min-1). The sites differed in call density, but also the effective sampling area. 5 . Synthesis and applications.The monitoring technique, acoustic spatially explicit capture–recapture (aSCR), quantitatively estimates call density without disturbing the calling animals or their environment, while time of arrival (ToA) and signal strength (SS) significantly add to the accuracy of call localisation, which in turn increases precision of call density estimates without the need for specialist field staff. This technique appears ideally suited to aid the monitoring of visually cryptic, acoustically active species.
Funding for the frog survey was received from the National Geographic Society/Waitt Grants Program (No. W184-11). The EPSRC and NERC helped to fund this research through a PhD grant (No. EP/1000917/1) to D.L.B. R.A. and G.J.M. acknowledge initiative funding from the National Research Foundation of South Africa.
Thu, 01 Jun 2017 00:00:00 GMThttp://hdl.handle.net/10023/99212017-06-01T00:00:00ZMeasey, G. JohnStevenson, Ben C.Scott, TanyaAltwegg, ResBorchers, David L.1 . Global amphibian declines have resulted in a vital need for monitoring programmes that follow population trends. Monitoring using advertisement calls is ideal as choruses are undisturbed during data collection. However, methods currently employed by managers frequently rely on trained observers, and/or do not provide density data on which to base trends. 2 . This study explores the utility of monitoring using acoustic spatially explicit capture-recapture (aSECR) with time of arrival (ToA) and signal strength (SS) as a quantitative monitoring technique to measure call density of a threatened but visually cryptic anuran, the Cape peninsula moss frog Arthroleptella lightfooti. 3 . The relationships between temporal and environmental variables (date, rainfall, temperature) and A. lightfooti call density at three study sites on the Cape peninsula, South Africa were examined. Acoustic data, collected from an array of six microphones over four months during the winter breeding season, provided a time series of call density estimates. 4 . Model selection indicated that call density was primarily associated with seasonality fitted as a quadratic function. Call density peaked mid-breeding season. At the main study site, the lowest recorded mean call density (0·160 calls m-2 min-1) occurred in May and reached its peak mid-July (1·259 calls m-2 min-1). The sites differed in call density, but also the effective sampling area. 5 . Synthesis and applications.The monitoring technique, acoustic spatially explicit capture–recapture (aSCR), quantitatively estimates call density without disturbing the calling animals or their environment, while time of arrival (ToA) and signal strength (SS) significantly add to the accuracy of call localisation, which in turn increases precision of call density estimates without the need for specialist field staff. This technique appears ideally suited to aid the monitoring of visually cryptic, acoustically active species.On the dimensions of a family of overlapping self-affine carpets
http://hdl.handle.net/10023/9835
We consider the dimensions of a family of self-affine sets related to the Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and then randomise the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford-McMullen set up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, M. Hochman's recent work on the dimensions of self-similar sets and measures.
The work of J.M.F. was supported by the EPSRC grant EP/J013560/1 whilst at Warwick and an EPSRC doctoral training grant whilst at St Andrews.
Thu, 01 Dec 2016 00:00:00 GMThttp://hdl.handle.net/10023/98352016-12-01T00:00:00ZFraser, Jonathan MacDonaldShmerkin, PabloWe consider the dimensions of a family of self-affine sets related to the Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and then randomise the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford-McMullen set up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, M. Hochman's recent work on the dimensions of self-similar sets and measures.String C-groups as transitive subgroups of Sn
http://hdl.handle.net/10023/9794
If Γ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sn (other than Sn and the alternating group An), then the rank of Γ is at most n/2+1, with finitely many exceptions (which are classified). It is conjectured that only the symmetric group has to be excluded.
Mon, 01 Feb 2016 00:00:00 GMThttp://hdl.handle.net/10023/97942016-02-01T00:00:00ZCameron, Peter JephsonFernandes, Maria ElisaLeemans, DimitriMixer, MarkIf Γ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sn (other than Sn and the alternating group An), then the rank of Γ is at most n/2+1, with finitely many exceptions (which are classified). It is conjectured that only the symmetric group has to be excluded.The Assouad dimensions of projections of planar sets
http://hdl.handle.net/10023/9725
We consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and self-similar sets. For general sets, the main result is the following: if a set in the plane has Assouad dimension s ∈ [0, 2], then the projections have Assouad dimension at least min{1, s} almost surely. Compared to the famous analogue for Hausdorff dimension – namely Marstrand’s Projection Theorem – a striking difference is that the words ‘at least’cannot be dispensed with: in fact, for many planar self-similar sets of dimension s < 1, we prove that the Assouad dimension of projections can attain both values sand 1 for a set of directions of positive measure. For self-similar sets, our investigation splits naturally into two cases: when the group of rotations is discrete, and when it is dense. In the ‘discrete rotations’ case we prove the following dichotomy for any given projection: either the Hausdorff measure is positive in the Hausdorff dimension, in which case the Hausdorff and Assouad dimensions coincide; or the Hausdorff measure is zero in the Hausdorff dimension,in which case the Assouad dimension is equal to 1. In the ‘dense rotations’ case we prove that every projection has Assouad dimension equal to one, assuming that the planar set is not a singleton. As another application of our results, we show that there is no Falconer’s Theorem for Assouad dimension. More precisely, the Assouad dimension of a self-similar (or self-affine) set is not in general almost surely constant when one randomises the translation vectors.
The first named author is supported by a Leverhulme Trust Research Fellowship and the second named author is supported by the Academy of Finland through the grant Restricted families of projections and connections to Kakeya type problems, grant number 274512.
Wed, 01 Feb 2017 00:00:00 GMThttp://hdl.handle.net/10023/97252017-02-01T00:00:00ZFraser, Jonathan M.Orponen, TuomasWe consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and self-similar sets. For general sets, the main result is the following: if a set in the plane has Assouad dimension s ∈ [0, 2], then the projections have Assouad dimension at least min{1, s} almost surely. Compared to the famous analogue for Hausdorff dimension – namely Marstrand’s Projection Theorem – a striking difference is that the words ‘at least’cannot be dispensed with: in fact, for many planar self-similar sets of dimension s < 1, we prove that the Assouad dimension of projections can attain both values sand 1 for a set of directions of positive measure. For self-similar sets, our investigation splits naturally into two cases: when the group of rotations is discrete, and when it is dense. In the ‘discrete rotations’ case we prove the following dichotomy for any given projection: either the Hausdorff measure is positive in the Hausdorff dimension, in which case the Hausdorff and Assouad dimensions coincide; or the Hausdorff measure is zero in the Hausdorff dimension,in which case the Assouad dimension is equal to 1. In the ‘dense rotations’ case we prove that every projection has Assouad dimension equal to one, assuming that the planar set is not a singleton. As another application of our results, we show that there is no Falconer’s Theorem for Assouad dimension. More precisely, the Assouad dimension of a self-similar (or self-affine) set is not in general almost surely constant when one randomises the translation vectors.On the Lq -spectrum of planar self-affine measures
http://hdl.handle.net/10023/9724
We study the dimension theory of a class of planar self-affine multifractal measures. These measures are the Bernoulli measures supported on box-like self-affine sets, introduced by the author, which are the attractors of iterated function systems consisting of contracting affine maps which take the unit square to rectangles with sides parallel to the axes. This class contains the self-affine measures recently considered by Feng and Wang as well as many other measures. In particular, we allow the defining maps to have non-trivial rotational and reflectional components. Assuming the rectangular open set condition, we compute the Lq-spectrum by means of a q-modified singular value function. A key application of our results is a closed form expression for the Lq-spectrum in the case where there are no mappings that switch the coordinate axes. This is useful for computational purposes and also allows us to prove differentiability of the Lq-spectrum at q=1 in the more difficult `non-multiplicative' situation. This has applications concerning the Hausdorff, packing and entropy dimension of the measure as well as the Hausdorff and packing dimension of the support. Due to the possible inclusion of axis reversing maps, we are led to extend some results of Peres and Solomyak on the existence of the Lq-spectrum of self-similar measures to the graph-directed case.
The author was supported by the EPSRC grant EP/J013560/1. This work was started whilst the author was an EPSRC funded PhD student at the University of St Andrews, and he expresses his gratitude for the support he found there.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10023/97242016-01-01T00:00:00ZFraser, Jonathan M.We study the dimension theory of a class of planar self-affine multifractal measures. These measures are the Bernoulli measures supported on box-like self-affine sets, introduced by the author, which are the attractors of iterated function systems consisting of contracting affine maps which take the unit square to rectangles with sides parallel to the axes. This class contains the self-affine measures recently considered by Feng and Wang as well as many other measures. In particular, we allow the defining maps to have non-trivial rotational and reflectional components. Assuming the rectangular open set condition, we compute the Lq-spectrum by means of a q-modified singular value function. A key application of our results is a closed form expression for the Lq-spectrum in the case where there are no mappings that switch the coordinate axes. This is useful for computational purposes and also allows us to prove differentiability of the Lq-spectrum at q=1 in the more difficult `non-multiplicative' situation. This has applications concerning the Hausdorff, packing and entropy dimension of the measure as well as the Hausdorff and packing dimension of the support. Due to the possible inclusion of axis reversing maps, we are led to extend some results of Peres and Solomyak on the existence of the Lq-spectrum of self-similar measures to the graph-directed case.Finite presentability and isomorphism of Cayley graphs of monoids
http://hdl.handle.net/10023/9711
Two finitely generated monoids are constructed, one finitely presented the other not, whose (directed, unlabelled) Cayley graphs are isomorphic.
Wed, 26 Oct 2016 00:00:00 GMThttp://hdl.handle.net/10023/97112016-10-26T00:00:00ZAwang, Jennifer SylviaPfeiffer, Markus JohannesRuskuc, NikolaTwo finitely generated monoids are constructed, one finitely presented the other not, whose (directed, unlabelled) Cayley graphs are isomorphic.Primitive groups, graph endomorphisms and synchronization
http://hdl.handle.net/10023/9648
Let Ω be a set of cardinality n, G be a permutation group on Ω and f:Ω→Ω be a map that is not a permutation. We say that G synchronizes f if the transformation semigroup ⟨G,f⟩ contains a constant map, and that G is a synchronizing group if G synchronizes every non-permutation. A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it had previously been conjectured that this was essentially the only way in which a primitive group could fail to be synchronizing, in other words, that a primitive group synchronizes every non-uniform transformation. The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every non-uniform transformation of rank at most 4, these examples are of the lowest possible rank. In addition, we produce graphs with primitive automorphism groups that have approximately √n non-synchronizing ranks, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group. The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree n synchronizes every non-uniform transformation of rank n−1 and n−2, and here this is extended to n−3 and n−4. In the process, we will obtain a purely graph-theoretical result showing that, with limited exceptions, in a vertex-primitive graph the union of neighbourhoods of a set of vertices A is bounded below by a function that is asymptotically √|A|. Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.
The third author has been partially supported by the Fundação para a Ciência e a Tecnologia through the project CEMAT-CIÊNCIAS UID/Multi/04621/2013.
Thu, 01 Dec 2016 00:00:00 GMThttp://hdl.handle.net/10023/96482016-12-01T00:00:00ZAraújo, JoãoBentz, WolframCameron, Peter JephsonRoyle, GordonSchaefer, ArturLet Ω be a set of cardinality n, G be a permutation group on Ω and f:Ω→Ω be a map that is not a permutation. We say that G synchronizes f if the transformation semigroup ⟨G,f⟩ contains a constant map, and that G is a synchronizing group if G synchronizes every non-permutation. A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it had previously been conjectured that this was essentially the only way in which a primitive group could fail to be synchronizing, in other words, that a primitive group synchronizes every non-uniform transformation. The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every non-uniform transformation of rank at most 4, these examples are of the lowest possible rank. In addition, we produce graphs with primitive automorphism groups that have approximately √n non-synchronizing ranks, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group. The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree n synchronizes every non-uniform transformation of rank n−1 and n−2, and here this is extended to n−3 and n−4. In the process, we will obtain a purely graph-theoretical result showing that, with limited exceptions, in a vertex-primitive graph the union of neighbourhoods of a set of vertices A is bounded below by a function that is asymptotically √|A|. Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.Bernoulli convolutions and 1D dynamics
http://hdl.handle.net/10023/9629
We describe a family φλ of dynamical systems on the unit interval which preserve Bernoulli convolutions. We show that if there are parameter ranges for which these systems are piecewise convex, then the corresponding Bernoulli convolution will be absolutely continuous with bounded density. We study the systems φλ and give some numerical evidence to suggest values of λ for which φλ may be piecewise convex.
Thu, 08 Oct 2015 00:00:00 GMThttp://hdl.handle.net/10023/96292015-10-08T00:00:00ZKempton, Thomas Michael WilliamPersson, TomasWe describe a family φλ of dynamical systems on the unit interval which preserve Bernoulli convolutions. We show that if there are parameter ranges for which these systems are piecewise convex, then the corresponding Bernoulli convolution will be absolutely continuous with bounded density. We study the systems φλ and give some numerical evidence to suggest values of λ for which φλ may be piecewise convex.Three-dimensional forced-damped dynamical systems with rich dynamics : bifurcations, chaos and unbounded solutions
http://hdl.handle.net/10023/9468
We consider certain autonomous three-dimensional dynamical systems that can arise in mechanical and fluid-dynamical contexts. Extending a previous study in Craik and Okamoto (2002), to include linear forcing and damping, we find that the four-leaf structure discovered in that paper, and unbounded orbits, persist, but may now be accompanied by three distinct period-doubling cascades to chaos, and by orbits that approach stable equilibrium points. This rich structure is investigated both analytically and numerically, distinguishing three main cases determined by the damping and forcing parameter values.
T.M. is supported by the Grant-in-Aid for JSPS Fellow No. 24·5312. H.O. is partially supported by JSPS KAKENHI 24244007.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10023/94682015-01-01T00:00:00ZMiyaji, TomoyukiOkamoto, HisashiCraik, Alexander Duncan DavidsonWe consider certain autonomous three-dimensional dynamical systems that can arise in mechanical and fluid-dynamical contexts. Extending a previous study in Craik and Okamoto (2002), to include linear forcing and damping, we find that the four-leaf structure discovered in that paper, and unbounded orbits, persist, but may now be accompanied by three distinct period-doubling cascades to chaos, and by orbits that approach stable equilibrium points. This rich structure is investigated both analytically and numerically, distinguishing three main cases determined by the damping and forcing parameter values.Recurrence and transience for suspension flows
http://hdl.handle.net/10023/9416
We study the thermodynamic formalism for suspension flows over countable Markov shifts with roof functions not necessarily bounded away from zero. We establish conditions to ensure the existence and uniqueness of equilibrium measures for regular potentials. We define the notions of recurrence and transience of a potential in this setting. We define the renewal flow, which is a symbolic model for a class of flows with diverse recurrence features. We study the corresponding thermodynamic formalism, establishing conditions for the existence of equilibrium measures and phase transitions. Applications are given to suspension flows defined over interval maps having parabolic fixed points.
Funding: Proyecto Fondecyt 1110040 for funding visit to PUC-Chile and partial support from NSF grant DMS 1109587.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10023/94162015-01-01T00:00:00ZIommi, GodofredoJordan, ThomasTodd, Michael JohnWe study the thermodynamic formalism for suspension flows over countable Markov shifts with roof functions not necessarily bounded away from zero. We establish conditions to ensure the existence and uniqueness of equilibrium measures for regular potentials. We define the notions of recurrence and transience of a potential in this setting. We define the renewal flow, which is a symbolic model for a class of flows with diverse recurrence features. We study the corresponding thermodynamic formalism, establishing conditions for the existence of equilibrium measures and phase transitions. Applications are given to suspension flows defined over interval maps having parabolic fixed points.A note on the probability of generating alternating or symmetric groups
http://hdl.handle.net/10023/9348
We improve on recent estimates for the probability of generating the alternating and symmetric groups An and Sn. In particular, we find the sharp lower bound if the probability is given by a quadratic in n−1. This leads to improved bounds on the largest number h(An) such that a direct product of h(An) copies of An can be generated by two elements.
The research of the first author is supported by the Australian Research Council grant DP120100446.
Tue, 01 Sep 2015 00:00:00 GMThttp://hdl.handle.net/10023/93482015-09-01T00:00:00ZMorgan, LukeRoney-Dougal, Colva MaryWe improve on recent estimates for the probability of generating the alternating and symmetric groups An and Sn. In particular, we find the sharp lower bound if the probability is given by a quadratic in n−1. This leads to improved bounds on the largest number h(An) such that a direct product of h(An) copies of An can be generated by two elements.Lengths of words in transformation semigroups generated by digraphs
http://hdl.handle.net/10023/9277
Given a simple digraph D on n vertices (with n≥2), there is a natural construction of a semigroup of transformations ⟨D⟩. For any edge (a, b) of D, let a→b be the idempotent of rank n−1 mapping a to b and fixing all vertices other than a; then, define ⟨D⟩ to be the semigroup generated by a→b for all (a,b)∈E(D). For α∈⟨D⟩, let ℓ(D,α) be the minimal length of a word in E(D) expressing α. It is well known that the semigroup Singn of all transformations of rank at most n−1 is generated by its idempotents of rank n−1. When D=Kn is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate ℓ(Kn,α), for any α∈⟨Kn⟩=Singn; however, no analogous non-trivial results are known when D≠Kn. In this paper, we characterise all simple digraphs D such that either ℓ(D,α) is equal to Howie–Iwahori’s formula for all α∈⟨D⟩, or ℓ(D,α)=n−fix(α) for all α∈⟨D⟩, or ℓ(D,α)=n−rk(α) for all α∈⟨D⟩. We also obtain bounds for ℓ(D,α) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank n−1 of Singn). We finish the paper with a list of conjectures and open problems
The second and third authors were supported by the EPSRC grant EP/K033956/1.
Wed, 01 Feb 2017 00:00:00 GMThttp://hdl.handle.net/10023/92772017-02-01T00:00:00ZCameron, P. J.Castillo-Ramirez, A.Gadouleau, M.Mitchell, J. D.Given a simple digraph D on n vertices (with n≥2), there is a natural construction of a semigroup of transformations ⟨D⟩. For any edge (a, b) of D, let a→b be the idempotent of rank n−1 mapping a to b and fixing all vertices other than a; then, define ⟨D⟩ to be the semigroup generated by a→b for all (a,b)∈E(D). For α∈⟨D⟩, let ℓ(D,α) be the minimal length of a word in E(D) expressing α. It is well known that the semigroup Singn of all transformations of rank at most n−1 is generated by its idempotents of rank n−1. When D=Kn is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate ℓ(Kn,α), for any α∈⟨Kn⟩=Singn; however, no analogous non-trivial results are known when D≠Kn. In this paper, we characterise all simple digraphs D such that either ℓ(D,α) is equal to Howie–Iwahori’s formula for all α∈⟨D⟩, or ℓ(D,α)=n−fix(α) for all α∈⟨D⟩, or ℓ(D,α)=n−rk(α) for all α∈⟨D⟩. We also obtain bounds for ℓ(D,α) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank n−1 of Singn). We finish the paper with a list of conjectures and open problemsIdempotent rank in the endomorphism monoid of a non-uniform partition
http://hdl.handle.net/10023/9275
We calculate the rank and idempotent rank of the semigroup E(X,P) generated by the idempotents of the semigroup T(X,P), which consists of all transformations of the finite set X preserving a non-uniform partition P. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.
Mon, 01 Feb 2016 00:00:00 GMThttp://hdl.handle.net/10023/92752016-02-01T00:00:00ZDolinka, IgorEast, JamesMitchell, James D.We calculate the rank and idempotent rank of the semigroup E(X,P) generated by the idempotents of the semigroup T(X,P), which consists of all transformations of the finite set X preserving a non-uniform partition P. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.Ends of semigroups
http://hdl.handle.net/10023/9254
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 Hopf's 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.
Sat, 01 Oct 2016 00:00:00 GMThttp://hdl.handle.net/10023/92542016-10-01T00:00:00ZCraik, S.Gray, R.Kilibarda, V.Mitchell, J. D.Ruskuc, N.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 Hopf's 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.Dimension conservation for self-similar sets and fractal percolation
http://hdl.handle.net/10023/9253
We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let K be a self-similar subset of R2 with Hausdorff dimension dimHK >1 such that the rotational components of the underlying similarities generate the full rotation group. Then for all ε >0, writing πθ for projection onto the line Lθ in direction θ, the Hausdorff dimensions of the sections satisfy dimH (K ∩ πθ-1x)> dimHK - 1 - ε for a set of x ∈ Lθ of positive Lebesgue measure, for all directions θ except for those in a set of Hausdorff dimension 0. For a class of self-similar sets we obtain a similar conclusion for all directions, but with lower box dimension replacing Hausdorff dimensions of sections. We obtain similar inequalities for the dimensions of sections of Mandelbrot percolation sets.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10023/92532015-01-01T00:00:00ZFalconer, Kenneth JohnJin, XiongWe introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let K be a self-similar subset of R2 with Hausdorff dimension dimHK >1 such that the rotational components of the underlying similarities generate the full rotation group. Then for all ε >0, writing πθ for projection onto the line Lθ in direction θ, the Hausdorff dimensions of the sections satisfy dimH (K ∩ πθ-1x)> dimHK - 1 - ε for a set of x ∈ Lθ of positive Lebesgue measure, for all directions θ except for those in a set of Hausdorff dimension 0. For a class of self-similar sets we obtain a similar conclusion for all directions, but with lower box dimension replacing Hausdorff dimensions of sections. We obtain similar inequalities for the dimensions of sections of Mandelbrot percolation sets.Sixty years of fractal projections
http://hdl.handle.net/10023/9231
Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand’s projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.
Fri, 31 Jul 2015 00:00:00 GMThttp://hdl.handle.net/10023/92312015-07-31T00:00:00ZFalconer, Kenneth JohnFraser, Jonathan MacdonaldJin, XiongSixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand’s projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.Hitting times and periodicity in random dynamics
http://hdl.handle.net/10023/9179
We prove quenched laws of hitting time statistics for random subshifts of finite type. In particular we prove a dichotomy between the law for periodic and for non-periodic points. We show that this applies to random Gibbs measures.
Thu, 01 Oct 2015 00:00:00 GMThttp://hdl.handle.net/10023/91792015-10-01T00:00:00ZTodd, Michael JohnRousseau, JeromeWe prove quenched laws of hitting time statistics for random subshifts of finite type. In particular we prove a dichotomy between the law for periodic and for non-periodic points. We show that this applies to random Gibbs measures.Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph
http://hdl.handle.net/10023/9178
We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph R. As a consequence we show that, for any countable graph Γ, there are uncountably many maximal subgroups of the endomorphism monoid of R isomorphic to the automorphism group of Γ. Further structural information about End R is established including that Aut Γ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.
Sun, 01 May 2016 00:00:00 GMThttp://hdl.handle.net/10023/91782016-05-01T00:00:00ZDolinka, IgorGray, Robert DuncanMcPhee, Jillian DawnMitchell, James DavidQuick, MartynWe establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph R. As a consequence we show that, for any countable graph Γ, there are uncountably many maximal subgroups of the endomorphism monoid of R isomorphic to the automorphism group of Γ. Further structural information about End R is established including that Aut Γ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.On regularity and the word problem for free idempotent generated semigroups
http://hdl.handle.net/10023/9145
The category of all idempotent generated semigroups with a prescribed structure Ɛ of their idempotents E (called the biordered set) has an initial object called the free idempotent generated semigroup over Ɛ, defined by a presentation over alphabet E, and denoted by IG(Ɛ). Recently, much effort has been put into investigating the structure of semigroups of the form IG(Ɛ), especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for IG(Ɛ). We prove two principal results, one positive and one negative. We show that, for a finite biordered set E, it is decidable whether a given word w ∈ E∗ represents a regular element; if in addition one assumes that all maximal subgroups of IG(Ɛ) have decidable word problems, then the word problem in IG(Ɛ) restricted to regular words is decidable. On the other hand, we exhibit a biorder Ɛ arising from a finite idempotent semigroup S, such that the word problem for IG(Ɛ) is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of IG(Ɛ) to the subgroup membership problem infinitely presented groups.
The research of the first author was supported by the Ministry of Education, Science, and Technological Development of the Republic of Serbia through the grant No. 174019, and by the grant No. 0851/2015 of the Secretariat of Science and Technological Development of the Autonomous Province of Vojvodina. The research of the second author was partially supported by the EPSRC-funded project EP/N033353/1 ‘Special inverse monoids: subgroups, structure, geometry, rewriting systems and the word problem’. The research of the third author was supported by the EPSRC-funded project EP/H011978/1 ‘Automata, Languages, Decidability in Algebra’.
Fri, 03 Mar 2017 00:00:00 GMThttp://hdl.handle.net/10023/91452017-03-03T00:00:00ZDolinka, IgorGray, Robert D.Ruskuc, NikolaThe category of all idempotent generated semigroups with a prescribed structure Ɛ of their idempotents E (called the biordered set) has an initial object called the free idempotent generated semigroup over Ɛ, defined by a presentation over alphabet E, and denoted by IG(Ɛ). Recently, much effort has been put into investigating the structure of semigroups of the form IG(Ɛ), especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for IG(Ɛ). We prove two principal results, one positive and one negative. We show that, for a finite biordered set E, it is decidable whether a given word w ∈ E∗ represents a regular element; if in addition one assumes that all maximal subgroups of IG(Ɛ) have decidable word problems, then the word problem in IG(Ɛ) restricted to regular words is decidable. On the other hand, we exhibit a biorder Ɛ arising from a finite idempotent semigroup S, such that the word problem for IG(Ɛ) is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of IG(Ɛ) to the subgroup membership problem infinitely presented groups.The infinite simple group V of Richard J. Thompson : presentations by permutations
http://hdl.handle.net/10023/9143
We show one can naturally describe elements of R. Thompson's infinite finitely presented simple group V, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions. This perspective provides an intuitive explanation towards the simplicity of V and also perhaps indicates a reason as to why it was one of the first discovered infinite finitely presented simple groups; it is (in some basic sense) a relative of the finite alternating groups. We find a natural infinite presentation for V as a group generated by these "transpositions," which presentation bears comparison with Dehornoy's infinite presentation, and which enables us to develop two small presentations for V: a human-interpretable presentation with three generators and eight relations, and a Tietze-derived presentation with two generators and seven relations.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10023/91432015-01-01T00:00:00ZQuick, MartynBleak, Collin PatrickWe show one can naturally describe elements of R. Thompson's infinite finitely presented simple group V, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions. This perspective provides an intuitive explanation towards the simplicity of V and also perhaps indicates a reason as to why it was one of the first discovered infinite finitely presented simple groups; it is (in some basic sense) a relative of the finite alternating groups. We find a natural infinite presentation for V as a group generated by these "transpositions," which presentation bears comparison with Dehornoy's infinite presentation, and which enables us to develop two small presentations for V: a human-interpretable presentation with three generators and eight relations, and a Tietze-derived presentation with two generators and seven relations.The random continued fraction transformation
http://hdl.handle.net/10023/9142
We introduce a random dynamical system related to continued fraction expansions. It uses random combination of the Gauss map and the R\'enyi (or backwards) continued fraction map. We explore the continued fraction expansions that this system produces as well as the dynamical properties of the system.
Wed, 01 Jul 2015 00:00:00 GMThttp://hdl.handle.net/10023/91422015-07-01T00:00:00ZKalle, CharleneKempton, Thomas Michael WilliamVerbitskiy, EvgenyWe introduce a random dynamical system related to continued fraction expansions. It uses random combination of the Gauss map and the R\'enyi (or backwards) continued fraction map. We explore the continued fraction expansions that this system produces as well as the dynamical properties of the system.The scenery flow for self-affine measures
http://hdl.handle.net/10023/9141
We describe the scaling scenery associated to Bernoulli measures supported on separated self-affine sets under the condition that certain projections of the measure are absolutely continuous.
Fri, 01 May 2015 00:00:00 GMThttp://hdl.handle.net/10023/91412015-05-01T00:00:00ZKempton, Thomas Michael WilliamWe describe the scaling scenery associated to Bernoulli measures supported on separated self-affine sets under the condition that certain projections of the measure are absolutely continuous.Computing finite semigroups
http://hdl.handle.net/10023/9138
Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and $0$-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Green's relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate the partial order of the $\mathscr{D}$-classes, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Green's classes of an element of such a subsemigroup without determining the global structure of the semigroup.
Wed, 07 Oct 2015 00:00:00 GMThttp://hdl.handle.net/10023/91382015-10-07T00:00:00ZEast, J.Egri-Nagy, A.Mitchell, J. D.Péresse, Y.Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and $0$-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Green's relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate the partial order of the $\mathscr{D}$-classes, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Green's classes of an element of such a subsemigroup without determining the global structure of the semigroup.Embedding right-angled Artin groups into Brin-Thompson groups
http://hdl.handle.net/10023/9080
We prove that every finitely-generated right-angled Artin group can be embedded into some Brin-Thompson group nV. It follows that many other groups can be embedded into some nV (e.g., any finite extension of any of Haglund and Wise's special groups), and that various decision problems involving subgroups of nV are unsolvable.
7 pages, no figures
Sat, 27 Feb 2016 00:00:00 GMThttp://hdl.handle.net/10023/90802016-02-27T00:00:00ZBelk, JamesBleak, CollinMatucci, FrancescoWe prove that every finitely-generated right-angled Artin group can be embedded into some Brin-Thompson group nV. It follows that many other groups can be embedded into some nV (e.g., any finite extension of any of Haglund and Wise's special groups), and that various decision problems involving subgroups of nV are unsolvable.Universal sequences for the order-automorphisms of the rationals
http://hdl.handle.net/10023/9024
In this paper, we consider the group Aut(Q,≤) of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Khélif states that every countable subset of Aut(Q,≤) is contained in an N-generated subgroup of Aut(Q,≤) for some fixed N ∈ N. We show that the least such N is 2. Moreover, for every countable subset of Aut(Q,≤), we show that every element can be given as a prescribed product of two generators without using their inverses. More precisely, suppose that a and b freely generate the free semigroup {a,b}+ consisting of the non-empty words over a and b. Then we show that there exists a sequence of words w1, w2,... over {a,b} such that for every sequence f1, f2, ... ∈ Aut(Q,≤) there is a homomorphism φ : {a,b}+ → Aut(Q,≤) where (wi)φ=fi for every i. The main theorem in this paper provides an alternative proof of a result of Droste and Holland showing that the strong cofinality of Aut(Q,≤) is uncountable, or equivalently that Aut(Q,≤) has uncountable cofinality and Bergman's property.
Mon, 01 Aug 2016 00:00:00 GMThttp://hdl.handle.net/10023/90242016-08-01T00:00:00ZHyde, J.Jonusas, J.Mitchell, J. D.Peresse, Y. H.In this paper, we consider the group Aut(Q,≤) of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Khélif states that every countable subset of Aut(Q,≤) is contained in an N-generated subgroup of Aut(Q,≤) for some fixed N ∈ N. We show that the least such N is 2. Moreover, for every countable subset of Aut(Q,≤), we show that every element can be given as a prescribed product of two generators without using their inverses. More precisely, suppose that a and b freely generate the free semigroup {a,b}+ consisting of the non-empty words over a and b. Then we show that there exists a sequence of words w1, w2,... over {a,b} such that for every sequence f1, f2, ... ∈ Aut(Q,≤) there is a homomorphism φ : {a,b}+ → Aut(Q,≤) where (wi)φ=fi for every i. The main theorem in this paper provides an alternative proof of a result of Droste and Holland showing that the strong cofinality of Aut(Q,≤) is uncountable, or equivalently that Aut(Q,≤) has uncountable cofinality and Bergman's property.From one-dimensional fields to Vlasov equilibria : Theory and application of Hermite Polynomials
http://hdl.handle.net/10023/8992
We consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a Vlasov-Maxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans' Theorem, the equilibria are expressed as functions of the constants of motion, in the form of a Maxwellian multiplied by an unknown function of the canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite Polynomials. Sufficient conditions are found which guarantee the convergence,boundedness and non-negativity of the candidate solution, when satisfied. These conditions are obtained by elementary means, and it is clear how to put them into practice. Illustrative examples of the use of this method with both force-free and non force-free macroscopic equilibria are presented, including the full verification of a recently derived distribution function for the Force-Free Harris Sheet (Allanson et al. (2015)). In the effort to model equilibria with lower values of the plasma beta, solutions for the same macroscopic equilibrium in a new gauge are calculated, with numerical results presented for βpl = 0:05.
Wed, 01 Jun 2016 00:00:00 GMThttp://hdl.handle.net/10023/89922016-06-01T00:00:00ZAllanson, Oliver DouglasNeukirch, ThomasTroscheit, SaschaWilson, FionaWe consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a Vlasov-Maxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans' Theorem, the equilibria are expressed as functions of the constants of motion, in the form of a Maxwellian multiplied by an unknown function of the canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite Polynomials. Sufficient conditions are found which guarantee the convergence,boundedness and non-negativity of the candidate solution, when satisfied. These conditions are obtained by elementary means, and it is clear how to put them into practice. Illustrative examples of the use of this method with both force-free and non force-free macroscopic equilibria are presented, including the full verification of a recently derived distribution function for the Force-Free Harris Sheet (Allanson et al. (2015)). In the effort to model equilibria with lower values of the plasma beta, solutions for the same macroscopic equilibrium in a new gauge are calculated, with numerical results presented for βpl = 0:05.An efficient acoustic density estimation method with human detectors applied to gibbons in Cambodia
http://hdl.handle.net/10023/8842
Some animal species are hard to see but easy to hear. Standard visual methods for estimating population density for such species are often ineffective or inefficient, but methods based on passive acoustics show more promise. We develop spatially explicit capture-recapture (SECR) methods for territorial vocalising species, in which humans act as an acoustic detector array. We use SECR and estimated bearing data from a single-occasion acoustic survey of a gibbon population in northeastern Cambodia to estimate the density of calling groups. The properties of the estimator are assessed using a simulation study, in which a variety of survey designs are also investigated. We then present a new form of the SECR likelihood for multi-occasion data which accounts for the stochastic availability of animals. In the context of gibbon surveys this allows model-based estimation of the proportion of groups that produce territorial vocalisations on a given day, thereby enabling the density of groups, instead of the density of calling groups, to be estimated. We illustrate the performance of this new estimator by simulation. We show that it is possible to estimate density reliably from human acoustic detections of visually cryptic species using SECR methods. For gibbon surveys we also show that incorporating observers' estimates of bearings to detected groups substantially improves estimator performance. Using the new form of the SECR likelihood we demonstrate that estimates of availability, in addition to population density and detection function parameters, can be obtained from multi-occasion data, and that the detection function parameters are not confounded with the availability parameter. This acoustic SECR method provides a means of obtaining reliable density estimates for territorial vocalising species. It is also efficient in terms of data requirements since it only requires routine survey data. We anticipate that the low-tech field requirements will make this method an attractive option in many situations where populations can be surveyed acoustically by humans.
D. Kidney was supported by an Engineering and Physical Sciences Research Council (EPSRC) Doctoral Training Grant studentship (EPSRC grant EP/P505097/1). B. Stevenson was supported by a studentship jointly funded by the University of St Andrews and EPSRC, through the National Centre for Statistical Ecology (EPSRC grant EP/I000917/1).
Thu, 19 May 2016 00:00:00 GMThttp://hdl.handle.net/10023/88422016-05-19T00:00:00ZKidney, DarrenRawson, Benjamin M.Borchers, David LouisStevenson, BenMarques, Tiago A.Thomas, LenSome animal species are hard to see but easy to hear. Standard visual methods for estimating population density for such species are often ineffective or inefficient, but methods based on passive acoustics show more promise. We develop spatially explicit capture-recapture (SECR) methods for territorial vocalising species, in which humans act as an acoustic detector array. We use SECR and estimated bearing data from a single-occasion acoustic survey of a gibbon population in northeastern Cambodia to estimate the density of calling groups. The properties of the estimator are assessed using a simulation study, in which a variety of survey designs are also investigated. We then present a new form of the SECR likelihood for multi-occasion data which accounts for the stochastic availability of animals. In the context of gibbon surveys this allows model-based estimation of the proportion of groups that produce territorial vocalisations on a given day, thereby enabling the density of groups, instead of the density of calling groups, to be estimated. We illustrate the performance of this new estimator by simulation. We show that it is possible to estimate density reliably from human acoustic detections of visually cryptic species using SECR methods. For gibbon surveys we also show that incorporating observers' estimates of bearings to detected groups substantially improves estimator performance. Using the new form of the SECR likelihood we demonstrate that estimates of availability, in addition to population density and detection function parameters, can be obtained from multi-occasion data, and that the detection function parameters are not confounded with the availability parameter. This acoustic SECR method provides a means of obtaining reliable density estimates for territorial vocalising species. It is also efficient in terms of data requirements since it only requires routine survey data. We anticipate that the low-tech field requirements will make this method an attractive option in many situations where populations can be surveyed acoustically by humans.Embeddings into Thompson's group V and coCF groups
http://hdl.handle.net/10023/8747
It is shown in Lehnert and Schweitzer (‘The co-word problem for the Higman–Thompson group is context-free’, Bull. London Math. Soc. 39 (2007) 235–241) that R. Thompson's group V is a co-context-free (coCF) group, thus implying that all of its finitely generated subgroups are also coCF groups. Also, Lehnert shows in his thesis that V embeds inside the coCF group QAut(T2,c), which is a group of particular bijections on the vertices of an infinite binary 2-edge-coloured tree, and he conjectures that QAut(T2,c) is a universal coCF group. We show that QAut(T2,c) embeds into V, and thus obtain a new form for Lehnert's conjecture. Following up on these ideas, we begin work to build a representation theory into R. Thompson's group V. In particular, we classify precisely which Baumslag–Solitar groups embed into V.
Sat, 01 Oct 2016 00:00:00 GMThttp://hdl.handle.net/10023/87472016-10-01T00:00:00ZBleak, CollinMatucci, FrancescoNeunhöffer, MaxIt is shown in Lehnert and Schweitzer (‘The co-word problem for the Higman–Thompson group is context-free’, Bull. London Math. Soc. 39 (2007) 235–241) that R. Thompson's group V is a co-context-free (coCF) group, thus implying that all of its finitely generated subgroups are also coCF groups. Also, Lehnert shows in his thesis that V embeds inside the coCF group QAut(T2,c), which is a group of particular bijections on the vertices of an infinite binary 2-edge-coloured tree, and he conjectures that QAut(T2,c) is a universal coCF group. We show that QAut(T2,c) embeds into V, and thus obtain a new form for Lehnert's conjecture. Following up on these ideas, we begin work to build a representation theory into R. Thompson's group V. In particular, we classify precisely which Baumslag–Solitar groups embed into V.Copulae on products of compact Riemannian manifolds
http://hdl.handle.net/10023/8672
Abstract One standard way of considering a probability distribution on the unit n -cube, [0 , 1]n , due to Sklar (1959), is to decompose it into its marginal distributions and a copula, i.e. a probability distribution on [0 , 1]n with uniform marginals. The definition of copula was extended by Jones et al. (2014) to probability distributions on products of circles. This paper defines a copula as a probability distribution on a product of compact Riemannian manifolds that has uniform marginals. Basic properties of such copulae are established. Two fairly general constructions of copulae on products of compact homogeneous manifolds are given; one is based on convolution in the isometry group, the other using equivariant functions from compact Riemannian manifolds to their spaces of square integrable functions. Examples illustrate the use of copulae to analyse bivariate spherical data and bivariate rotational data.
Tue, 01 Sep 2015 00:00:00 GMThttp://hdl.handle.net/10023/86722015-09-01T00:00:00ZJupp, P.E.Abstract One standard way of considering a probability distribution on the unit n -cube, [0 , 1]n , due to Sklar (1959), is to decompose it into its marginal distributions and a copula, i.e. a probability distribution on [0 , 1]n with uniform marginals. The definition of copula was extended by Jones et al. (2014) to probability distributions on products of circles. This paper defines a copula as a probability distribution on a product of compact Riemannian manifolds that has uniform marginals. Basic properties of such copulae are established. Two fairly general constructions of copulae on products of compact homogeneous manifolds are given; one is based on convolution in the isometry group, the other using equivariant functions from compact Riemannian manifolds to their spaces of square integrable functions. Examples illustrate the use of copulae to analyse bivariate spherical data and bivariate rotational data.Constructing flag-transitive, point-imprimitive designs
http://hdl.handle.net/10023/8546
We give a construction of a family of designs with a specified point-partition and determine the subgroup of automorphisms leaving invariant the point-partition. We give necessary and sufficient conditions for a design in the family to possess a flag-transitive group of automorphisms preserving the specified point-partition. We give examples of flag-transitive designs in the family, including a new symmetric 2-(1408,336,80) design with automorphism group 2^12:((3⋅M22):2) and a construction of one of the families of the symplectic designs (the designs S^−(n) ) exhibiting a flag-transitive, point-imprimitive automorphism group.
Wed, 04 May 2016 00:00:00 GMThttp://hdl.handle.net/10023/85462016-05-04T00:00:00ZCameron, Peter JephsonPraeger, Cheryl E.We give a construction of a family of designs with a specified point-partition and determine the subgroup of automorphisms leaving invariant the point-partition. We give necessary and sufficient conditions for a design in the family to possess a flag-transitive group of automorphisms preserving the specified point-partition. We give examples of flag-transitive designs in the family, including a new symmetric 2-(1408,336,80) design with automorphism group 2^12:((3⋅M22):2) and a construction of one of the families of the symplectic designs (the designs S^−(n) ) exhibiting a flag-transitive, point-imprimitive automorphism group.Permutation groups and transformation semigroups : results and problems
http://hdl.handle.net/10023/8532
J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics. This paper surveys some recent results on the transformation semigroup generated by a permutation group G and a single non-permutation a. Our particular concern is the influence that properties of G (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of S=<G,a> such as regularity and generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances S contains an element of rank 1. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided. These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie's elegance and value tests in a new and fascinating way.
Thu, 01 Oct 2015 00:00:00 GMThttp://hdl.handle.net/10023/85322015-10-01T00:00:00ZAraujo, JoaoCameron, Peter JephsonJ.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics. This paper surveys some recent results on the transformation semigroup generated by a permutation group G and a single non-permutation a. Our particular concern is the influence that properties of G (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of S=<G,a> such as regularity and generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances S contains an element of rank 1. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided. These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie's elegance and value tests in a new and fascinating way.Guessing games on triangle-free graphs
http://hdl.handle.net/10023/8518
The guessing game introduced by Riis is a variant of the "guessing your own hats" game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markström introduced a method to bound the value of the guessing number from below using the fractional clique cover number kappa_f(G). In particular they showed gn(G) >= |V(G)| - kappa_f(G). Moreover, it is pointed out that equality holds in this bound if the underlying undirected graph G falls into one of the following categories: perfect graphs, cycle graphs or their complement. In this paper, we show that there are triangle-free graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous triangle-free Higman-Sims graph has guessing number at least 77 and at most 78, while the bound given by fractional clique cover is 50.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10023/85182016-01-01T00:00:00ZCameron, Peter JephsonDang, AnhRiis, SorenThe guessing game introduced by Riis is a variant of the "guessing your own hats" game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markström introduced a method to bound the value of the guessing number from below using the fractional clique cover number kappa_f(G). In particular they showed gn(G) >= |V(G)| - kappa_f(G). Moreover, it is pointed out that equality holds in this bound if the underlying undirected graph G falls into one of the following categories: perfect graphs, cycle graphs or their complement. In this paper, we show that there are triangle-free graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous triangle-free Higman-Sims graph has guessing number at least 77 and at most 78, while the bound given by fractional clique cover is 50.Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V
http://hdl.handle.net/10023/8508
Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group 2V is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite order. A modification of the construction gives other undecidability results about the dynamics of the action of elements of 2V on Cantor Space. Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely presented group with solvable word problem and unsolvable torsion problem. To our knowledge, 2V furnishes the first concrete example of such a group, and gives an example of a direct undecidability result in the extended family of R. Thompson type groups.
Mon, 01 May 2017 00:00:00 GMThttp://hdl.handle.net/10023/85082017-05-01T00:00:00ZBelk, JamesBleak, CollinUsing a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group 2V is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite order. A modification of the construction gives other undecidability results about the dynamics of the action of elements of 2V on Cantor Space. Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely presented group with solvable word problem and unsolvable torsion problem. To our knowledge, 2V furnishes the first concrete example of such a group, and gives an example of a direct undecidability result in the extended family of R. Thompson type groups.Effects of thermal conduction and compressive viscosity on the period ratio of the slow mode
http://hdl.handle.net/10023/8423
Aims: Increasing observational evidence of wave modes brings us to a closer understanding of the solar corona. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P1/2P2, between the period P1 of the fundamental mode and the period P2 of its first overtone, is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. Methods: We consider analytically the effects of thermal conduction and compressive viscosity on the period ratio for a longitudinally propagating sound wave. Results: For coronal values of thermal conduction the effect on the period ratio is negligible. For compressive viscosity the effect on the period ratio may become important for some short hot loops. Conclusions: Damping typically has a small effect on the period ratio, suggesting that longitudinal structuring remains the most significant effect.
C.K.M. acknowledges financial support from the CarnegieTrust. Discussions with Dr. I. De Moortel and Prof. A. W. Hood are gratefully acknowledged
Tue, 01 Jun 2010 00:00:00 GMThttp://hdl.handle.net/10023/84232010-06-01T00:00:00ZMacnamara, Cicely KrystynaRoberts, BernardAims: Increasing observational evidence of wave modes brings us to a closer understanding of the solar corona. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P1/2P2, between the period P1 of the fundamental mode and the period P2 of its first overtone, is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. Methods: We consider analytically the effects of thermal conduction and compressive viscosity on the period ratio for a longitudinally propagating sound wave. Results: For coronal values of thermal conduction the effect on the period ratio is negligible. For compressive viscosity the effect on the period ratio may become important for some short hot loops. Conclusions: Damping typically has a small effect on the period ratio, suggesting that longitudinal structuring remains the most significant effect.Wild attractors and thermodynamic formalism
http://hdl.handle.net/10023/8394
Fibonacci unimodal maps can have a wild Cantor attractor, and hence be Lebesgue dissipative, depending on the order of the critical point. We present a one-parameter family ƒλ of countably piecewise linear unimodal Fibonacci maps in order to study the thermodynamic formalism of dynamics where dissipativity of Lebesgue (and conformal) measure is responsible for phase transitions. We show that for the potential φt = -t log |ƒλ'|, there is a unique phase transition at some t1 ≤ 1, and the pressure P(φt ) is analytic (with unique equilibrium state) elsewhere. The pressure is majorised by a non-analytic C∞ curve (with all derivatives equal to 0 at t1 < 1) at the emergence of a wild attractor, whereas the phase transition at t1 = 1 can be of any finite order for those λ for which ƒλ is Lebesgue conservative. We also obtain results on the existence of conformal measures and equilibrium states, as well as the hyperbolic dimension and the dimension of the basin of ω(c).
MT was partially supported by NSF Grants DMS 0606343 and DMS 0908093.
Tue, 01 Sep 2015 00:00:00 GMThttp://hdl.handle.net/10023/83942015-09-01T00:00:00ZBruin, HenkTodd, Michael JohnFibonacci unimodal maps can have a wild Cantor attractor, and hence be Lebesgue dissipative, depending on the order of the critical point. We present a one-parameter family ƒλ of countably piecewise linear unimodal Fibonacci maps in order to study the thermodynamic formalism of dynamics where dissipativity of Lebesgue (and conformal) measure is responsible for phase transitions. We show that for the potential φt = -t log |ƒλ'|, there is a unique phase transition at some t1 ≤ 1, and the pressure P(φt ) is analytic (with unique equilibrium state) elsewhere. The pressure is majorised by a non-analytic C∞ curve (with all derivatives equal to 0 at t1 < 1) at the emergence of a wild attractor, whereas the phase transition at t1 = 1 can be of any finite order for those λ for which ƒλ is Lebesgue conservative. We also obtain results on the existence of conformal measures and equilibrium states, as well as the hyperbolic dimension and the dimension of the basin of ω(c).Well quasi-order in combinatorics : embeddings and homomorphisms
http://hdl.handle.net/10023/7963
The notion of well quasi-order (wqo) from the theory of ordered sets often arises naturally in contexts where one deals with infinite collections of structures which can somehow be compared, and it then represents a useful discriminator between ‘tame’ and ‘wild’ such classes. In this article we survey such situations within combinatorics, and attempt to identify promising directions for further research. We argue that these are intimately linked with a more systematic and detailed study of homomorphisms in combinatorics.
Wed, 01 Jul 2015 00:00:00 GMThttp://hdl.handle.net/10023/79632015-07-01T00:00:00ZHuczynska, SophieRuskuc, NikThe notion of well quasi-order (wqo) from the theory of ordered sets often arises naturally in contexts where one deals with infinite collections of structures which can somehow be compared, and it then represents a useful discriminator between ‘tame’ and ‘wild’ such classes. In this article we survey such situations within combinatorics, and attempt to identify promising directions for further research. We argue that these are intimately linked with a more systematic and detailed study of homomorphisms in combinatorics.Coprime invariable generation and minimal-exponent groups
http://hdl.handle.net/10023/7910
A finite group G is coprimely invariably generated if there exists a set of generators {g1,. .,gu} of G with the property that the orders |g1|,. .,|gu| are pairwise coprime and that for all x1,. .,xu∈G the set {g1x1,. .,guxu} generates G.We show that if G is coprimely invariably generated, then G can be generated with three elements, or two if G is soluble, and that G has zero presentation rank. As a corollary, we show that if G is any finite group such that no proper subgroup has the same exponent as G, then G has zero presentation rank. Furthermore, we show that every finite simple group is coprimely invariably generated by two elements, except for O8+(2) which requires three elements.Along the way, we show that for each finite simple group S, and for each partition π1,. .,πu of the primes dividing |S|, the product of the number kπi(S) of conjugacy classes of πi-elements satisfies. ∏i=1ukπi(S)≤|S|2|OutS|.
Colva Roney-Dougal acknowledges the support of EPSRC grant EP/I03582X/1.
Sat, 01 Aug 2015 00:00:00 GMThttp://hdl.handle.net/10023/79102015-08-01T00:00:00ZDetomi, EloisaLucchini, AndreaRoney-Dougal, C.M.A finite group G is coprimely invariably generated if there exists a set of generators {g1,. .,gu} of G with the property that the orders |g1|,. .,|gu| are pairwise coprime and that for all x1,. .,xu∈G the set {g1x1,. .,guxu} generates G.We show that if G is coprimely invariably generated, then G can be generated with three elements, or two if G is soluble, and that G has zero presentation rank. As a corollary, we show that if G is any finite group such that no proper subgroup has the same exponent as G, then G has zero presentation rank. Furthermore, we show that every finite simple group is coprimely invariably generated by two elements, except for O8+(2) which requires three elements.Along the way, we show that for each finite simple group S, and for each partition π1,. .,πu of the primes dividing |S|, the product of the number kπi(S) of conjugacy classes of πi-elements satisfies. ∏i=1ukπi(S)≤|S|2|OutS|.Speed of convergence for laws of rare events and escape rates
http://hdl.handle.net/10023/7837
We obtain error terms on the rate of convergence to Extreme Value Laws, and to the asymptotic Hitting Time Statistics, for a general class of weakly dependent stochastic processes. The dependence of the error terms on the ‘time’ and ‘length’ scales is very explicit. Specialising to data derived from a class of dynamical systems we find even more detailed error terms, one application of which is to consider escape rates through small holes in these systems.
MT was partially supported by NSF grant DMS 1109587. All authors are supported by FCT (Portugal) projects PTDC/MAT/099493/2008 and PTDC/MAT/120346/2010, which are financed by national and European structural funds through the programs FEDER and COMPETE. All three authors were also supported by CMUP, which is financed by FCT (Portugal) through the programs POCTI and POSI, with national and European structural funds, under the project PEst-C/MAT/UI0144/2013.
Wed, 01 Apr 2015 00:00:00 GMThttp://hdl.handle.net/10023/78372015-04-01T00:00:00ZFreitas, AnaFreitas, JorgeTodd, Michael JohnWe obtain error terms on the rate of convergence to Extreme Value Laws, and to the asymptotic Hitting Time Statistics, for a general class of weakly dependent stochastic processes. The dependence of the error terms on the ‘time’ and ‘length’ scales is very explicit. Specialising to data derived from a class of dynamical systems we find even more detailed error terms, one application of which is to consider escape rates through small holes in these systems.On simultaneous local dimension functions of subsets of Rd
http://hdl.handle.net/10023/7778
For a subset E ⊑ Rd and x ∈ Rd, the local Hausdorff dimension function of E at x and the local packing dimension function of E at x are defined by (Formula presented.) where dimH and dimP denote the Hausdorff dimension and the packing dimension, respectively. In this note we give a short and simple proof showing that for any pair of continuous functions f,g: Rd → [0, d] with f ≤ g, it is possible to choose a set E that simultaneously has f as its local Hausdorff dimension function and g as its local packing dimension function.
Date of Acceptance: 04/05/2015
Wed, 30 Sep 2015 00:00:00 GMThttp://hdl.handle.net/10023/77782015-09-30T00:00:00ZOlsen, Lars Ole RonnowFor a subset E ⊑ Rd and x ∈ Rd, the local Hausdorff dimension function of E at x and the local packing dimension function of E at x are defined by (Formula presented.) where dimH and dimP denote the Hausdorff dimension and the packing dimension, respectively. In this note we give a short and simple proof showing that for any pair of continuous functions f,g: Rd → [0, d] with f ≤ g, it is possible to choose a set E that simultaneously has f as its local Hausdorff dimension function and g as its local packing dimension function.Near-threshold electron injection in the laser-plasma wakefield accelerator leading to femtosecond bunches
http://hdl.handle.net/10023/7750
The laser-plasma wakefield accelerator is a compact source of high brightness, ultra-short duration electron bunches. Self-injection occurs when electrons from the background plasma gain sufficient momentum at the back of the bubble-shaped accelerating structure to experience sustained acceleration. The shortest duration and highest brightness electron bunches result from self-injection close to the threshold for injection. Here we show that in this case injection is due to the localized charge density build-up in the sheath crossing region at the rear of the bubble, which has the effect of increasing the accelerating potential to above a critical value. Bunch duration is determined by the dwell time above this critical value, which explains why single or multiple ultra-short electron bunches with little dark current are formed in the first bubble. We confirm experimentally, using coherent optical transition radiation measurements, that single or multiple bunches with femtosecond duration and peak currents of several kiloAmpere, and femtosecond intervals between bunches, emerge from the accelerator.
We gratefully acknowledge the support of the UK EPSRC (grant no. EP/J018171/1), the EU FP7 programmes: the Extreme Light Infrastructure (ELI) project, the Laserlab-Europe (no. 284464), and the EUCARD-2 project (no. 312453).
Thu, 17 Sep 2015 00:00:00 GMThttp://hdl.handle.net/10023/77502015-09-17T00:00:00ZIslam, M.R.Brunetti, E.Shanks, R.P.Ersfeld, B.Issac, R.C.Cipiccia, S.Anania, M.P.Welsh, G.H.Wiggins, S.M.Noble, A.Cairns, R AlanRaj, G.Jaroszynski, D.A.The laser-plasma wakefield accelerator is a compact source of high brightness, ultra-short duration electron bunches. Self-injection occurs when electrons from the background plasma gain sufficient momentum at the back of the bubble-shaped accelerating structure to experience sustained acceleration. The shortest duration and highest brightness electron bunches result from self-injection close to the threshold for injection. Here we show that in this case injection is due to the localized charge density build-up in the sheath crossing region at the rear of the bubble, which has the effect of increasing the accelerating potential to above a critical value. Bunch duration is determined by the dwell time above this critical value, which explains why single or multiple ultra-short electron bunches with little dark current are formed in the first bubble. We confirm experimentally, using coherent optical transition radiation measurements, that single or multiple bunches with femtosecond duration and peak currents of several kiloAmpere, and femtosecond intervals between bunches, emerge from the accelerator.Digit frequencies and Bernoulli convolutions
http://hdl.handle.net/10023/7719
It is well known that when β is a Pisot number, the corresponding Bernoulli convolution ν(β) has Hausdorff dimension less than 1, i.e. that there exists a set A(β) with (ν(β))(A(β))=1 and dim_H(A(β))<1. We show explicitly how to construct for each Pisot number β such a set A(β).
This work was supported partly by the Dutch Organisation for Scientific Research (NWO) grant number 613.001.022 and partly by the Engineering and Physical Sciences Research Council grant number EP/K029061/1.
Fri, 27 Jun 2014 00:00:00 GMThttp://hdl.handle.net/10023/77192014-06-27T00:00:00ZKempton, Thomas Michael WilliamIt is well known that when β is a Pisot number, the corresponding Bernoulli convolution ν(β) has Hausdorff dimension less than 1, i.e. that there exists a set A(β) with (ν(β))(A(β))=1 and dim_H(A(β))<1. We show explicitly how to construct for each Pisot number β such a set A(β).Self-affine sets with positive Lebesgue measure
http://hdl.handle.net/10023/7718
Using techniques introduced by C. Gunturk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous β-expansion’ of different numbers in different bases.
Fri, 27 Jun 2014 00:00:00 GMThttp://hdl.handle.net/10023/77182014-06-27T00:00:00ZDajani, KarmaJiang, KanKempton, Thomas Michael WilliamUsing techniques introduced by C. Gunturk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous β-expansion’ of different numbers in different bases.An exact collisionless equilibrium for the Force-Free Harris Sheet with low plasma beta
http://hdl.handle.net/10023/7691
We present a first discussion and analysis of the physical properties of a new exact collisionless equilibrium for a one-dimensional nonlinear force-free magnetic field, namely, the force-free Harris sheet. The solution allows any value of the plasma beta, and crucially below unity, which previous nonlinear force-free collisionless equilibria could not. The distribution function involves infinite series of Hermite polynomials in the canonical momenta, of which the important mathematical properties of convergence and non-negativity have recently been proven. Plots of the distribution function are presented for the plasma beta modestly below unity, and we compare the shape of the distribution function in two of the velocity directions to a Maxwellian distribution.
Funding: STFC Consolidated Grant [ST/K000950/1] (OA, TN & FW) and a Doctoral Training Grant [ST/K502327/1] (OA). EPSRC Doctoral Training Grant [EP/K503162/1] (ST).
Thu, 01 Oct 2015 00:00:00 GMThttp://hdl.handle.net/10023/76912015-10-01T00:00:00ZAllanson, Oliver DouglasNeukirch, ThomasWilson, FionaTroscheit, SaschaWe present a first discussion and analysis of the physical properties of a new exact collisionless equilibrium for a one-dimensional nonlinear force-free magnetic field, namely, the force-free Harris sheet. The solution allows any value of the plasma beta, and crucially below unity, which previous nonlinear force-free collisionless equilibria could not. The distribution function involves infinite series of Hermite polynomials in the canonical momenta, of which the important mathematical properties of convergence and non-negativity have recently been proven. Plots of the distribution function are presented for the plasma beta modestly below unity, and we compare the shape of the distribution function in two of the velocity directions to a Maxwellian distribution.Homomorphic image orders on combinatorial structures
http://hdl.handle.net/10023/7679
Combinatorial structures have been considered under various orders, including substructure order and homomorphism order. In this paper, we investigate the homomorphic image order, corresponding to the existence of a surjective homomorphism between two structures. We distinguish between strong and induced forms of the order and explore how they behave in the context of different common combinatorial structures. We focus on three aspects: antichains and partial well-order, the joint preimage property and the dual amalgamation property. The two latter properties are natural analogues of the well-known joint embedding property and amalgamation property, and are investigated here for the first time.
Wed, 01 Jul 2015 00:00:00 GMThttp://hdl.handle.net/10023/76792015-07-01T00:00:00ZHuczynska, SophieRuskuc, NikCombinatorial structures have been considered under various orders, including substructure order and homomorphism order. In this paper, we investigate the homomorphic image order, corresponding to the existence of a surjective homomorphism between two structures. We distinguish between strong and induced forms of the order and explore how they behave in the context of different common combinatorial structures. We focus on three aspects: antichains and partial well-order, the joint preimage property and the dual amalgamation property. The two latter properties are natural analogues of the well-known joint embedding property and amalgamation property, and are investigated here for the first time.A Hölder-type inequality on a regular rooted tree
http://hdl.handle.net/10023/7658
We establish an inequality which involves a non-negative function defined on the vertices of a finite m-ary regular rooted tree. The inequality may be thought of as relating an interaction energy defined on the free vertices of the tree summed over automorphisms of the tree, to a product of sums of powers of the function over vertices at certain levels of the tree. Conjugate powers arise naturally in the inequality, indeed, Hölder's inequality is a key tool in the proof which uses induction on subgroups of the automorphism group of the tree.
Sun, 15 Mar 2015 00:00:00 GMThttp://hdl.handle.net/10023/76582015-03-15T00:00:00ZFalconer, Kenneth JohnWe establish an inequality which involves a non-negative function defined on the vertices of a finite m-ary regular rooted tree. The inequality may be thought of as relating an interaction energy defined on the free vertices of the tree summed over automorphisms of the tree, to a product of sums of powers of the function over vertices at certain levels of the tree. Conjugate powers arise naturally in the inequality, indeed, Hölder's inequality is a key tool in the proof which uses induction on subgroups of the automorphism group of the tree.Higher moments for random multiplicative measures
http://hdl.handle.net/10023/7474
We obtain a condition for the Lq-convergence of martingales generated by random multiplicative cascade measures for q>1 without any self-similarity requirements on the cascades.
Sat, 01 Aug 2015 00:00:00 GMThttp://hdl.handle.net/10023/74742015-08-01T00:00:00ZFalconer, Kenneth JohnWe obtain a condition for the Lq-convergence of martingales generated by random multiplicative cascade measures for q>1 without any self-similarity requirements on the cascades.Generalized energy inequalities and higher multifractal moments
http://hdl.handle.net/10023/7095
We present a class of generalized energy inequalities and indicate their use in investigating higher multifractal moments, in particular Lq-dimensions of images of measures under Brownian processes, Lq-dimensions of almost self-aﬃne measures, and moments of random cascade measures
Sat, 02 Aug 2014 00:00:00 GMThttp://hdl.handle.net/10023/70952014-08-02T00:00:00ZFalconer, Kenneth JohnWe present a class of generalized energy inequalities and indicate their use in investigating higher multifractal moments, in particular Lq-dimensions of images of measures under Brownian processes, Lq-dimensions of almost self-aﬃne measures, and moments of random cascade measuresInflations of geometric grid classes of permutations
http://hdl.handle.net/10023/6862
Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than κ ≈ 2.20557 (a specific algebraic integer at which infinite antichains first appear). Using language- and order-theoretic methods, we prove that the substitution closures of geometric grid classes are well partially ordered, finitely based, and that all their subclasses have algebraic generating functions. We go on to show that the inflation of a geometric grid class by a strongly rational class is well partially ordered, and that all its subclasses have rational generating functions. This latter fact allows us to conclude that every permutation class with growth rate less than κ has a rational generating function. This bound is tight as there are permutation classes with growth rate κ which have nonrational generating functions.
All three authors were partially supported by EPSRC via the grant EP/J006440/1.
Sun, 01 Feb 2015 00:00:00 GMThttp://hdl.handle.net/10023/68622015-02-01T00:00:00ZAlbert, M.D.Ruskuc, NikVatter, V.Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than κ ≈ 2.20557 (a specific algebraic integer at which infinite antichains first appear). Using language- and order-theoretic methods, we prove that the substitution closures of geometric grid classes are well partially ordered, finitely based, and that all their subclasses have algebraic generating functions. We go on to show that the inflation of a geometric grid class by a strongly rational class is well partially ordered, and that all its subclasses have rational generating functions. This latter fact allows us to conclude that every permutation class with growth rate less than κ has a rational generating function. This bound is tight as there are permutation classes with growth rate κ which have nonrational generating functions.Subalgebras of FA-presentable algebras
http://hdl.handle.net/10023/6852
Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First, an example is given to show that the class of finitely generated FA-presentable algebras is not closed under forming finitely generated subalgebras, even within the class of algebras with only unary operations. In contrast, a finitely generated subalgebra of an FA-presentable algebra with a single unary operation is itself FA-presentable. Furthermore, it is proven that the class of unary FA-presentable algebras is closed under forming finitely generated subalgebras and that the membership problem for such subalgebras is decidable.
Sun, 01 Jun 2014 00:00:00 GMThttp://hdl.handle.net/10023/68522014-06-01T00:00:00ZCain, A.J.Ruskuc, NikAutomatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First, an example is given to show that the class of finitely generated FA-presentable algebras is not closed under forming finitely generated subalgebras, even within the class of algebras with only unary operations. In contrast, a finitely generated subalgebra of an FA-presentable algebra with a single unary operation is itself FA-presentable. Furthermore, it is proven that the class of unary FA-presentable algebras is closed under forming finitely generated subalgebras and that the membership problem for such subalgebras is decidable.Most switching classes with primitive automorphism groups contain graphs with trivial groups
http://hdl.handle.net/10023/6429
The operation of switching a graph Gamma with respect to a subset X of the vertex set interchanges edges and non-edges between X and its complement, leaving the rest of the graph unchanged. This is an equivalence relation on the set of graphs on a given vertex set, so we can talk about the automorphism group of a switching class of graphs. It might be thought that switching classes with many automorphisms would have the property that all their graphs also have many automorphisms. But the main theorem of this paper shows a different picture: with finitely many exceptions, if a non-trivial switching class S has primitive automorphism group, then it contains a graph whose automorphism group is trivial. We also find all the exceptional switching classes; up to complementation, there are just six.
Mon, 01 Jun 2015 00:00:00 GMThttp://hdl.handle.net/10023/64292015-06-01T00:00:00ZCameron, Peter JephsonSpiga, PabloThe operation of switching a graph Gamma with respect to a subset X of the vertex set interchanges edges and non-edges between X and its complement, leaving the rest of the graph unchanged. This is an equivalence relation on the set of graphs on a given vertex set, so we can talk about the automorphism group of a switching class of graphs. It might be thought that switching classes with many automorphisms would have the property that all their graphs also have many automorphisms. But the main theorem of this paper shows a different picture: with finitely many exceptions, if a non-trivial switching class S has primitive automorphism group, then it contains a graph whose automorphism group is trivial. We also find all the exceptional switching classes; up to complementation, there are just six.On residual finiteness of monoids, their Schützenberger groups and associated actions
http://hdl.handle.net/10023/6310
In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M ; (RFSG) residual finiteness of Schützenberger groups of M ; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table.
RG was supported by an EPSRC Postdoctoral Fellowship EP/E043194/1 held at the University of St Andrews, Scotland.
Sun, 01 Jun 2014 00:00:00 GMThttp://hdl.handle.net/10023/63102014-06-01T00:00:00ZGray, RRuskuc, NikIn this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M ; (RFSG) residual finiteness of Schützenberger groups of M ; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table.Codimension formulae for the intersection of fractal subsets of Cantor spaces
http://hdl.handle.net/10023/6030
We examine the dimensions of the intersection of a subset E of an m-ary Cantor space Cm with the image of a subset F under a random isometry with respect to a natural metric. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. The dimensions of the intersections are typically max{dim E +dim F -dim Cm, 0}, akin to other codimension theorems. The upper estimates come from the expected sizes of coverings, whilst the lower estimate is more intricate, using martingales to define a random measure on the intersection to facilitate a potential theoretic argument.
Mon, 01 Feb 2016 00:00:00 GMThttp://hdl.handle.net/10023/60302016-02-01T00:00:00ZDonoven, CaseyFalconer, Kenneth JohnWe examine the dimensions of the intersection of a subset E of an m-ary Cantor space Cm with the image of a subset F under a random isometry with respect to a natural metric. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. The dimensions of the intersections are typically max{dim E +dim F -dim Cm, 0}, akin to other codimension theorems. The upper estimates come from the expected sizes of coverings, whilst the lower estimate is more intricate, using martingales to define a random measure on the intersection to facilitate a potential theoretic argument.Hölder differentiability of self-conformal devil's staircases
http://hdl.handle.net/10023/5980
In this paper we consider the probability distribution function of a Gibbs measure supported on a self-conformal set given by an iterated function system (devil's staircase) applied to a compact subset of ℝ. We use thermodynamic multifractal formalism to calculate the Hausdorff dimension of the sets Sα 0, Sα ∞ and Sα, the set of points at which this function has, respectively, Hölder derivative 0, ∞ or no derivative in the general sense. This extends recent work by Darst, Dekking, Falconer, Kesseböhmer and Stratmann, and Yao, Zhang and Li by considering arbitrary such Gibbs measures given by a potential function independent of the geometric potential.
Sat, 01 Mar 2014 00:00:00 GMThttp://hdl.handle.net/10023/59802014-03-01T00:00:00ZTroscheit, S.In this paper we consider the probability distribution function of a Gibbs measure supported on a self-conformal set given by an iterated function system (devil's staircase) applied to a compact subset of ℝ. We use thermodynamic multifractal formalism to calculate the Hausdorff dimension of the sets Sα 0, Sα ∞ and Sα, the set of points at which this function has, respectively, Hölder derivative 0, ∞ or no derivative in the general sense. This extends recent work by Darst, Dekking, Falconer, Kesseböhmer and Stratmann, and Yao, Zhang and Li by considering arbitrary such Gibbs measures given by a potential function independent of the geometric potential.Assouad type dimensions and homogeneity of fractals
http://hdl.handle.net/10023/5941
We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural 'dimension pair'. In particular, we compute these dimensions for certain classes of self-affine sets and quasi-self-similar sets and study their relationships with other notions of dimension, such as the Hausdorff dimension for example. We also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity.
The author was supported by an EPSRC Doctoral Training Grant
Mon, 01 Dec 2014 00:00:00 GMThttp://hdl.handle.net/10023/59412014-12-01T00:00:00ZFraser, Jonathan M.We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural 'dimension pair'. In particular, we compute these dimensions for certain classes of self-affine sets and quasi-self-similar sets and study their relationships with other notions of dimension, such as the Hausdorff dimension for example. We also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity.Negative ion sound solitary waves revisited
http://hdl.handle.net/10023/5845
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.
Sun, 01 Dec 2013 00:00:00 GMThttp://hdl.handle.net/10023/58452013-12-01T00:00:00ZCairns, 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)
http://hdl.handle.net/10023/5819
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.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10023/58192015-01-01T00:00:00ZButton, JackRoney-Dougal, ColvaHelfgott 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
http://hdl.handle.net/10023/5802
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.
This work was supported by EPSRC Grant No. EP/G04239X/1.
Tue, 07 Oct 2014 00:00:00 GMThttp://hdl.handle.net/10023/58022014-10-07T00:00:00ZSpeirs, 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
http://hdl.handle.net/10023/5793
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.
Sun, 01 Mar 2015 00:00:00 GMThttp://hdl.handle.net/10023/57932015-03-01T00:00:00ZEast, J.Mitchell, James DavidPéresse, 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.Computing in permutation groups without memory
http://hdl.handle.net/10023/5727
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.
Funding: UK Engineering and Physical Sciences Research Council (EP/K033956/1)
Sun, 02 Nov 2014 00:00:00 GMThttp://hdl.handle.net/10023/57272014-11-02T00:00:00ZCameron, Peter JephsonFairbairn, BenGadouleau, MaximilienMemoryless 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
http://hdl.handle.net/10023/5715
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.
Funding: UK Engineering and Physical Sciences Research Council award EP/K033956/1
Sun, 02 Nov 2014 00:00:00 GMThttp://hdl.handle.net/10023/57152014-11-02T00:00:00ZCameron, Peter JephsonFairbairn, BenGadouleau, MaximilienMemoryless 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
http://hdl.handle.net/10023/5658
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.
Fri, 01 Nov 2013 00:00:00 GMThttp://hdl.handle.net/10023/56582013-11-01T00:00:00ZMenezes, Nina EmmaQuick, MartynRoney-Dougal, Colva MaryWe 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
http://hdl.handle.net/10023/5580
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.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10023/55802015-01-01T00:00:00ZBabai, LaszloCameron, Peter JephsonWe 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
http://hdl.handle.net/10023/5514
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.
Wed, 01 Oct 2014 00:00:00 GMThttp://hdl.handle.net/10023/55142014-10-01T00:00:00ZFalconer, KennethJin, XiongWe 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
http://hdl.handle.net/10023/5459
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.
Tue, 10 Sep 2013 00:00:00 GMThttp://hdl.handle.net/10023/54592013-09-10T00:00:00ZPontin, 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
http://hdl.handle.net/10023/5393
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.
Fri, 21 Feb 2014 00:00:00 GMThttp://hdl.handle.net/10023/53932014-02-21T00:00:00ZMacTaggart, DavidHaynes, Andrew LewisWe 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
http://hdl.handle.net/10023/5318
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.
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.
Tue, 20 May 2014 00:00:00 GMThttp://hdl.handle.net/10023/53182014-05-20T00:00:00ZRachmeler, 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
http://hdl.handle.net/10023/5303
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).
Sun, 01 Sep 2013 00:00:00 GMThttp://hdl.handle.net/10023/53032013-09-01T00:00:00ZJanjic, VladimirBrown, Christopher MarkNeunhoeffer, MaxHammond, KevinLinton, Stephen AlexanderLoidl, Hans-WolfgangOrbit 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
http://hdl.handle.net/10023/5291
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.
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.
Tue, 01 Jul 2014 00:00:00 GMThttp://hdl.handle.net/10023/52912014-07-01T00:00:00ZKliem, 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
http://hdl.handle.net/10023/5271
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.
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).
Thu, 01 May 2014 00:00:00 GMThttp://hdl.handle.net/10023/52712014-05-01T00:00:00ZPlatten, 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
http://hdl.handle.net/10023/5237
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.
Fri, 01 Nov 2013 00:00:00 GMThttp://hdl.handle.net/10023/52372013-11-01T00:00:00ZBleak, Collin PatrickSalazar-Diaz, OlgaWe 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
http://hdl.handle.net/10023/5234
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.
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.
Mon, 13 May 2013 00:00:00 GMThttp://hdl.handle.net/10023/52342013-05-13T00:00:00ZForbes, 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
http://hdl.handle.net/10023/5232
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.
Mon, 23 Sep 2013 00:00:00 GMThttp://hdl.handle.net/10023/52322013-09-23T00:00:00ZSutherland, 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
http://hdl.handle.net/10023/5186
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.
Wed, 07 May 2014 00:00:00 GMThttp://hdl.handle.net/10023/51862014-05-07T00:00:00ZKing, 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
http://hdl.handle.net/10023/5185
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.
Wed, 07 May 2014 00:00:00 GMThttp://hdl.handle.net/10023/51852014-05-07T00:00:00ZMcConville, 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
http://hdl.handle.net/10023/5184
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.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10023/51842014-01-01T00:00:00ZGillespie, 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
http://hdl.handle.net/10023/5183
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.
Wed, 07 May 2014 00:00:00 GMThttp://hdl.handle.net/10023/51832014-05-07T00:00:00ZSpeirs, 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
http://hdl.handle.net/10023/5180
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.
Sat, 01 Feb 2014 00:00:00 GMThttp://hdl.handle.net/10023/51802014-02-01T00:00:00ZCairns, 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
http://hdl.handle.net/10023/5173
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.
Authors KH, RT, DCS, RAC, RB were supported by EPSRC grant EP/G04239X/1.
Tue, 01 Oct 2013 00:00:00 GMThttp://hdl.handle.net/10023/51732013-10-01T00:00:00ZHumphrey, 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
http://hdl.handle.net/10023/4986
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.
Fri, 07 Feb 2014 00:00:00 GMThttp://hdl.handle.net/10023/49862014-02-07T00:00:00ZHuczynska, SophieFor 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
http://hdl.handle.net/10023/4626
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.
This research was supported through EPSRC grant EP/I03582X/1. The APC was paid through RCUK open access block grant funds.
Sun, 01 Jun 2014 00:00:00 GMThttp://hdl.handle.net/10023/46262014-06-01T00:00:00ZDetomi, EloisaLucchini, AndreaRoney-Dougal, Colva MaryA 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
http://hdl.handle.net/10023/4319
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.
26 pages
Sat, 15 Feb 2014 00:00:00 GMThttp://hdl.handle.net/10023/43192014-02-15T00:00:00ZFalconer, KennethXiao, YiminWe 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
http://hdl.handle.net/10023/4061
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.
Mon, 01 Oct 2012 00:00:00 GMThttp://hdl.handle.net/10023/40612012-10-01T00:00:00ZFalconer, Kenneth JohnHu, JiaxinSun, YuhuaWe 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
http://hdl.handle.net/10023/3933
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.
Fri, 01 Jun 2012 00:00:00 GMThttp://hdl.handle.net/10023/39332012-06-01T00:00:00ZKesseboehmer, MarcMunday, SaraStratmann, 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
http://hdl.handle.net/10023/3902
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.
This work is supported by EPSRC Doctoral Training Grants
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10023/39022013-01-01T00:00:00ZBalka, RichardFarkas, AbelFraser, 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.Multistable processes and localizability
http://hdl.handle.net/10023/3560
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.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10023/35602012-01-01T00:00:00ZFalconer, Kenneth JohnLiu, LiningWe 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
http://hdl.handle.net/10023/3383
Let ΩΩ be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of ΩΩ, then we write U≈V if there exists a finite subset F of ΩΩ such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in ΩΩ is the least cardinality of a subset A of ΩΩ such that the union of U and A generates ΩΩ. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω. The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in ΩΩ where d is the minimum cardinality of a dominating family for NN. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in ΩΩ are 0, 1, 2, and d. We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2ℵ0.
Wed, 01 Sep 2010 00:00:00 GMThttp://hdl.handle.net/10023/33832010-09-01T00:00:00ZMitchell, James DavidMorayne, MichalPeresse, Yann HamonQuick, MartynLet 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.Every group is a maximal subgroup of the free idempotent generated semigroup over a band
http://hdl.handle.net/10023/3342
Given an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area.
Wed, 01 May 2013 00:00:00 GMThttp://hdl.handle.net/10023/33422013-05-01T00:00:00ZDolinka, IRuskuc, NikGiven an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area.On disjoint unions of finitely many copies of the free monogenic semigroup
http://hdl.handle.net/10023/3341
Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.
Thu, 01 Aug 2013 00:00:00 GMThttp://hdl.handle.net/10023/33412013-08-01T00:00:00ZAbughazalah, NabilahRuskuc, NikEvery semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.Ideals and finiteness conditions for subsemigroups
http://hdl.handle.net/10023/3335
In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub- or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, D=J and minimal conditions on ideals.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10023/33352014-01-01T00:00:00ZGray, Robert DuncanMaltcev, VictorMitchell, James DavidRuskuc, N.In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub- or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, D=J and minimal conditions on ideals.Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure
http://hdl.handle.net/10023/3237
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.
"GCB was supported by an EPSRC Doctoral Training Grant whilst undertaking this work"
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10023/32372013-01-01T00:00:00ZBoore, GraemeFalconer, Kenneth JohnFor 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.Growth of generating sets for direct powers of classical algebraic structures
http://hdl.handle.net/10023/3058
For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.
Sun, 01 Aug 2010 00:00:00 GMThttp://hdl.handle.net/10023/30582010-08-01T00:00:00ZQuick, MartynRuskuc, NikFor 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.Green index in semigroups : generators, presentations and automatic structures
http://hdl.handle.net/10023/2760
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).
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10023/27602012-01-01T00:00:00ZCain, A.J.Gray, RRuskuc, NikThe 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
http://hdl.handle.net/10023/2756
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.
JMF was supported by an EPSRC grant whilst undertaking this work.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10023/27562013-01-01T00:00:00ZFalconer, Kenneth JohnFraser, Jonathan MacdonaldGiven 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.Substitution-closed pattern classes
http://hdl.handle.net/10023/2149
The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classied by listing them as a set of explicit families.
Tue, 01 Feb 2011 00:00:00 GMThttp://hdl.handle.net/10023/21492011-02-01T00:00:00ZAtkinson, M.D.Ruskuc, NikSmith, RThe substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classied by listing them as a set of explicit families.Automatic presentations and semigroup constructions
http://hdl.handle.net/10023/2148
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.
Sun, 01 Aug 2010 00:00:00 GMThttp://hdl.handle.net/10023/21482010-08-01T00:00:00ZCain, Alan J.Oliver, GrahamRuskuc, NikThomas, 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
http://hdl.handle.net/10023/2147
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.
Special Issue: 2nd International Conference on Language and Automata Theory and Applications (LATA 2008)
Sun, 01 Nov 2009 00:00:00 GMThttp://hdl.handle.net/10023/21472009-11-01T00:00:00ZCain, Alan JamesOliver, GrahamRuskuc, NikThomas, 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
http://hdl.handle.net/10023/2146
It is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general.
Tue, 01 Sep 2009 00:00:00 GMThttp://hdl.handle.net/10023/21462009-09-01T00:00:00ZGray, R.Ruskuc, NikIt is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general.The Bergman property for semigroups
http://hdl.handle.net/10023/2145
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.
Sat, 01 Aug 2009 00:00:00 GMThttp://hdl.handle.net/10023/21452009-08-01T00:00:00ZMaltcev, 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
http://hdl.handle.net/10023/2144
Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. If the complement S \ T has finitely many strong orbits by both these actions we say that T has finite Green index in S. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents. (c) 2008 Elsevier Inc. All rights reserved.
Wed, 15 Oct 2008 00:00:00 GMThttp://hdl.handle.net/10023/21442008-10-15T00:00:00ZGray, Robert DuncanRuskuc, NikLet S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. If the complement S \ T has finitely many strong orbits by both these actions we say that T has finite Green index in S. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents. (c) 2008 Elsevier Inc. All rights reserved.Properties of the subsemigroups of the bicyclic monoid
http://hdl.handle.net/10023/2142
In this paper we study some properties of the subsemigroups of the bicyclic monoid B, by using a recent description of its subsemigroups. We start by giving necessary and sufficient conditions for a subsemigroup to be finitely generated. Then we show that all finitely generated subsemigroups are automatic and finitely presented. Finally we prove that a subsemigroup of B is residually finite if and only if it does not contain a copy of B.
Sun, 01 Jun 2008 00:00:00 GMThttp://hdl.handle.net/10023/21422008-06-01T00:00:00ZDescalco, L.Ruskuc, NikIn this paper we study some properties of the subsemigroups of the bicyclic monoid B, by using a recent description of its subsemigroups. We start by giving necessary and sufficient conditions for a subsemigroup to be finitely generated. Then we show that all finitely generated subsemigroups are automatic and finitely presented. Finally we prove that a subsemigroup of B is residually finite if and only if it does not contain a copy of B.Pattern classes of permutations via bijections between linearly ordered sets
http://hdl.handle.net/10023/2140
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.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10023/21402008-01-01T00:00:00ZHuczynska, SophieRuskuc, NikolaA 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
http://hdl.handle.net/10023/2138
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.).
Fri, 01 Feb 2008 00:00:00 GMThttp://hdl.handle.net/10023/21382008-02-01T00:00:00ZCain, Alan JamesRobertson, Edmund FrederickRuskuc, NikIt 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)
http://hdl.handle.net/10023/2137
Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates.
Fri, 22 Oct 2010 00:00:00 GMThttp://hdl.handle.net/10023/21372010-10-22T00:00:00ZAlbert, M.H.Atkinson, M.D.Brignall, RRuskuc, NikSmith, RWest, JPattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates.On generators and presentations of semidirect products in inverse semigroups
http://hdl.handle.net/10023/2136
In this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented.
Mon, 01 Jun 2009 00:00:00 GMThttp://hdl.handle.net/10023/21362009-06-01T00:00:00ZDombi, Erzsebet RitaRuskuc, NikIn 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
http://hdl.handle.net/10023/2134
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.
Tue, 01 May 2012 00:00:00 GMThttp://hdl.handle.net/10023/21342012-05-01T00:00:00ZGray, RRuskuc, NikLet Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying |im (e)| = r ≤ n − 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr.Generators and relations for subsemigroups via boundaries in Cayley graphs
http://hdl.handle.net/10023/2131
Given a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.
Tue, 01 Nov 2011 00:00:00 GMThttp://hdl.handle.net/10023/21312011-11-01T00:00:00ZGray, RRuskuc, NikGiven a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.On the growth of generating sets for direct powers of semigroups
http://hdl.handle.net/10023/2129
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.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10023/21292012-01-01T00:00:00ZHyde, James ThomasLoughlin, NicholasQuick, MartynRuskuc, NikWallis, AlistairFor 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
http://hdl.handle.net/10023/2128
We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister–Schreier type rewriting.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10023/21282012-01-01T00:00:00ZGray, RRuskuc, NikWe 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.On convex permutations
http://hdl.handle.net/10023/2000
A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem.
Sun, 01 May 2011 00:00:00 GMThttp://hdl.handle.net/10023/20002011-05-01T00:00:00ZAlbert, M.H.Linton, Stephen AlexanderRuskuc, NikVatter, VWaton, SA selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem.Presentations of inverse semigroups, their kernels and extensions
http://hdl.handle.net/10023/1998
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.
"Part of this work was done while Gray was an EPSRC Postdoctoral Research Fellow at the University of St Andrews, Scotland"
Wed, 01 Jun 2011 00:00:00 GMThttp://hdl.handle.net/10023/19982011-06-01T00:00:00ZCarvalho, C.A.Gray, RRuskuc, NikLet S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.Simple extensions of combinatorial structures
http://hdl.handle.net/10023/1997
An interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible.
Fri, 01 Jul 2011 00:00:00 GMThttp://hdl.handle.net/10023/19972011-07-01T00:00:00ZBrignall, RRuskuc, NikVatter, VAn interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible.The horizon problem for prevalent surfaces
http://hdl.handle.net/10023/1956
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.
JMF was supported by an EPSRC Doctoral Training Grant whilst undertaking this work.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10023/19562011-01-01T00:00:00ZFalconer, Kenneth JohnFraser, Jonathan MacdonaldWe 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.Primitive free cubics with specified norm and trace
http://hdl.handle.net/10023/1615
The existence of a primitive free (normal) cubic x(3) ax(2) + cx b over a finite field F with arbitrary specified values of a (not equal 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.
Fri, 01 Aug 2003 00:00:00 GMThttp://hdl.handle.net/10023/16152003-08-01T00:00:00ZHuczynska, SophieCohen, SDThe existence of a primitive free (normal) cubic x(3) ax(2) + cx b over a finite field F with arbitrary specified values of a (not equal 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.Subsemigroups of virtually free groups : finite Malcev presentations and testing for freeness
http://hdl.handle.net/10023/1561
This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.
Sat, 01 Jul 2006 00:00:00 GMThttp://hdl.handle.net/10023/15612006-07-01T00:00:00ZCain, AJRobertson, Edmund FrederickRuskuc, NikolaThis paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.Generating the full transformation semigroup using order preserving mappings
http://hdl.handle.net/10023/1553
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.
Mon, 01 Sep 2003 00:00:00 GMThttp://hdl.handle.net/10023/15532003-09-01T00:00:00ZHiggins, PMMitchell, James DavidRuskuc, NikolaFor 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
http://hdl.handle.net/10023/1442
Let G be a group, and let <A \ R> be a finite group presentation for G with \R\ greater than or equal to \A\. Then there exists a, finite semigroup, presentation <B \ Q> for G such that \Q\ - \B\ = \R\ - \A\. Moreover, B is either the same generating set or else it contains one additional generator.
Mon, 01 Jul 2002 00:00:00 GMThttp://hdl.handle.net/10023/14422002-07-01T00:00:00ZCampbell, Colin MatthewMitchell, James DavidRuskuc, NikolaLet 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.