Centre for Interdisciplinary Research in Computational Algebra (CIRCA) Research
http://hdl.handle.net/10023/197
20170524T09:45:23Z

Z4codes 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 nonlinear) 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.
20170701T00:00:00Z
Cameron, Peter Jephson
Kusuma, Josephine
Solé, Patrick
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 nonlinear) 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.

The Missing Link! A new skeleton for evolutionary multiagent systems in Erlang
http://hdl.handle.net/10023/10680
Evolutionary multiagent systems (EMAS) play a critical role in many artificial intelligence applications that are in use today. In this paper, we present a new generic skeleton in Erlang for parallel EMAS computations. The skeleton enables us to capture a wide variety of concrete evolutionary computations that can exploit the same underlying parallel implementation. We demonstrate the use of our skeleton on two different evolutionary computing applications: (1) computing the minimum of the Rastrigin function; and (2) solving an urban traffic optimisation problem. We show that we can obtain very good speedups (up to 142.44 ×× the sequential performance) on a variety of different parallel hardware, while requiring very little parallelisation effort.
20170426T00:00:00Z
Stypka, Jan
Turek, Wojciech
Byrski, Aleksander
KisielDorohinicki, Marek
Barwell, Adam David
Brown, Christopher Mark
Hammond, Kevin
Janjic, Vladimir
Evolutionary multiagent systems (EMAS) play a critical role in many artificial intelligence applications that are in use today. In this paper, we present a new generic skeleton in Erlang for parallel EMAS computations. The skeleton enables us to capture a wide variety of concrete evolutionary computations that can exploit the same underlying parallel implementation. We demonstrate the use of our skeleton on two different evolutionary computing applications: (1) computing the minimum of the Rastrigin function; and (2) solving an urban traffic optimisation problem. We show that we can obtain very good speedups (up to 142.44 ×× the sequential performance) on a variety of different parallel hardware, while requiring very little parallelisation effort.

Highest rank of a polytope for An
http://hdl.handle.net/10023/10678
We prove that the highest rank of a string Cgroup constructed from an alternating group Altn is 0 if n = 3, 4, 6, 7, 8; 3 if n = 5; 4 if n = 9; 5 if n = 10; 6 if n = 11; and the floor of (n1)/2 if n ≥ 12. 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 (FCTFundação para a Ciência e a Tecnologia), through CIDMA  Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.
20170428T00:00:00Z
Cameron, Peter Jephson
Fernandes, Maria Elisa
Leemans, Dimitri
Mixer, Mark
We prove that the highest rank of a string Cgroup constructed from an alternating group Altn is 0 if n = 3, 4, 6, 7, 8; 3 if n = 5; 4 if n = 9; 5 if n = 10; 6 if n = 11; and the floor of (n1)/2 if n ≥ 12. This solves a conjecture made by the last three authors in 2012.

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 relement 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 2element 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.
20170402T00:00:00Z
Cameron, Peter Jephson
Lucchini, Andrea
RoneyDougal, Colva Mary
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 relement 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 2element 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.

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.
20160926T00:00:00Z
Lucchini, Andrea
Maroti, Attila
RoneyDougal, Colva Mary
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.

Persistent identification and citation of software
http://hdl.handle.net/10023/10237
Software underpins the academic research process across disciplines. To be able to understand, use/reuse and preserve data, the software code that generated, analysed or presented the data will need to be retained and executed. An important part of this process is being able to persistently identify the software concerned. This paper discusses the reasons for doing so and introduces a model of software entities to enable better identification of what is being identified. The DataCite metadata schema provides a persistent identification scheme and we consider how this scheme can be applied to software. We then explore examples of persistent identification and reuse. The examples show the differences and similarities of software used in academic research, which has been written and reused at different scales. The key concepts of being able to identify what precisely is being used and provide a mechanism for appropriate credit are important to both of them.
This work has been funded by Jisc in the Research@Risk scheme.
20160101T00:00:00Z
Jones, Catherine Mary
Matthews, Brian
Gent, Ian
Griffin, Tom
Tedds, Jonathan
Software underpins the academic research process across disciplines. To be able to understand, use/reuse and preserve data, the software code that generated, analysed or presented the data will need to be retained and executed. An important part of this process is being able to persistently identify the software concerned. This paper discusses the reasons for doing so and introduces a model of software entities to enable better identification of what is being identified. The DataCite metadata schema provides a persistent identification scheme and we consider how this scheme can be applied to software. We then explore examples of persistent identification and reuse. The examples show the differences and similarities of software used in academic research, which has been written and reused at different scales. The key concepts of being able to identify what precisely is being used and provide a mechanism for appropriate credit are important to both of them.

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 nonincreasing 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 λtransitive 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 λhomogeneous 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 λtransitive group is λhomogeneous.The first goal of this paper is to classify the λhomogeneous 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 noninvertible transformation a in TnSn and a group G<=Sn, we say that (a,G) is an Hpair if the semigroups generated by {a} ∪ H and {a} ∪ G contain the same nonunits, that is, {a,G}\G= {a,H}\H. Using the classification of the λhomogeneous groups we classify all the Snpairs (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 selfcontained for researchers in both areas. The paper finishes with a number of open problems on permutation and linear groups.
20160415T00:00:00Z
André, Jorge
Araúo, Joāo
Cameron, Peter Jephson
Let λ=(λ1,λ2,...) be a partition of n, a sequence of positive integers in nonincreasing 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 λtransitive 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 λhomogeneous 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 λtransitive group is λhomogeneous.The first goal of this paper is to classify the λhomogeneous 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 noninvertible transformation a in TnSn and a group G<=Sn, we say that (a,G) is an Hpair if the semigroups generated by {a} ∪ H and {a} ∪ G contain the same nonunits, that is, {a,G}\G= {a,H}\H. Using the classification of the λhomogeneous groups we classify all the Snpairs (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 selfcontained for researchers in both areas. The paper finishes with a number of open problems on permutation and linear groups.

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 contextfree 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 coword problem (as introduced by Holt, Rees, Röver, and Thomas). Our main results answers a question of BernsZieze, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and SalazarDías, 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 .
20160201T00:00:00Z
Bennett, Daniel
Bleak, Collin
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 contextfree 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 coword problem (as introduced by Holt, Rees, Röver, and Thomas). Our main results answers a question of BernsZieze, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and SalazarDías, 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 .

PAEAN : portable and scalable runtime support for parallel Haskell dialects
http://hdl.handle.net/10023/10103
Over time, several competing approaches to parallel Haskell programming have emerged. Different approaches support parallelism at various different scales, ranging from small multicores to massively parallel highperformance computing systems. They also provide varying degrees of control, ranging from completely implicit approaches to ones providing full programmer control. Most current designs assume a shared memory model at the programmer, implementation and hardware levels. This is, however, becoming increasingly divorced from the reality at the hardware level. It also imposes significant unwanted runtime overheads in the form of garbage collection synchronisation etc. What is needed is an easy way to abstract over the implementation and hardware levels, while presenting a simple parallelism model to the programmer. The PArallEl shAred Nothing runtime system design aims to provide a portable and highlevel sharednothing implementation platform for parallel Haskell dialects. It abstracts over major issues such as work distribution and data serialisation, consolidating existing, successful designs into a single framework. It also provides an optional virtual sharedmemory programming abstraction for (possibly) sharednothing parallel machines, such as modern multicore/manycore architectures or cluster/cloud computing systems. It builds on, unifies and extends, existing welldeveloped support for sharedmemory parallelism that is provided by the widely used GHC Haskell compiler. This paper summarises the stateoftheart in sharednothing parallel Haskell implementations, introduces the PArallEl shAred Nothing abstractions, shows how they can be used to implement three distinct parallel Haskell dialects, and demonstrates that good scalability can be obtained on recent parallel machines.
20160101T00:00:00Z
Berthold, Jost
Loidl, HansWolfgang
Hammond, Kevin
Over time, several competing approaches to parallel Haskell programming have emerged. Different approaches support parallelism at various different scales, ranging from small multicores to massively parallel highperformance computing systems. They also provide varying degrees of control, ranging from completely implicit approaches to ones providing full programmer control. Most current designs assume a shared memory model at the programmer, implementation and hardware levels. This is, however, becoming increasingly divorced from the reality at the hardware level. It also imposes significant unwanted runtime overheads in the form of garbage collection synchronisation etc. What is needed is an easy way to abstract over the implementation and hardware levels, while presenting a simple parallelism model to the programmer. The PArallEl shAred Nothing runtime system design aims to provide a portable and highlevel sharednothing implementation platform for parallel Haskell dialects. It abstracts over major issues such as work distribution and data serialisation, consolidating existing, successful designs into a single framework. It also provides an optional virtual sharedmemory programming abstraction for (possibly) sharednothing parallel machines, such as modern multicore/manycore architectures or cluster/cloud computing systems. It builds on, unifies and extends, existing welldeveloped support for sharedmemory parallelism that is provided by the widely used GHC Haskell compiler. This paper summarises the stateoftheart in sharednothing parallel Haskell implementations, introduces the PArallEl shAred Nothing abstractions, shows how they can be used to implement three distinct parallel Haskell dialects, and demonstrates that good scalability can be obtained on recent parallel machines.

On optimality and construction of circular repeatedmeasurements 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 repeatedmeasurements designs when the parameters do not permit balance for carryover effects. It is shown that some circular weakly neighbour balanced designs defined by Filipiak and Markiewicz (2012) are universally optimal repeatedmeasurements designs. These results extend the work of Magda (1980), Kunert (1984b) and Filipiak and Markiewicz (2012).
20170101T00:00:00Z
Bailey, Rosemary Anne
Cameron, Peter Jephson
Filipiak, Katarzyna
Kunert, Joachim
Markiewicz, Augustyn
The aim of this paper is to characterize and construct universally optimal designs among the class of circular repeatedmeasurements designs when the parameters do not permit balance for carryover effects. It is shown that some circular weakly neighbour balanced designs defined by Filipiak and Markiewicz (2012) are universally optimal repeatedmeasurements designs. These results extend the work of Magda (1980), Kunert (1984b) and Filipiak and Markiewicz (2012).

Fertility preservation in prepubertal girls with cancer : the role of ovarian tissue cryopreservation
http://hdl.handle.net/10023/10029
With increasing numbers of survivors of cancer in young people future fertility and ovarian function are important considerations that should be discussed before treatment commences. Some young people, by nature of the treatment they will receive, are at high risk of premature ovarian insufficiency and infertility. For them, ovarian tissue cryopreservation (OTC) is one approach to fertility preservation that remains both invasive and for young patients experimental. There are important ethical and consent issues that need to be explored and accepted before OTC can be considered established in children with cancer. In this review we have discussed a framework for patient selection which has been shown to be effective in identifying those patients at high risk of premature ovarian insufficiency (POI) and who can be offered OTC safely.
Copyright © 2016 American Society for Reproductive Medicine. Published by Elsevier Inc. All rights reserved.
20160101T00:00:00Z
Wallace, W Hamish B
Kelsey, Thomas W
Anderson, Richard A
With increasing numbers of survivors of cancer in young people future fertility and ovarian function are important considerations that should be discussed before treatment commences. Some young people, by nature of the treatment they will receive, are at high risk of premature ovarian insufficiency and infertility. For them, ovarian tissue cryopreservation (OTC) is one approach to fertility preservation that remains both invasive and for young patients experimental. There are important ethical and consent issues that need to be explored and accepted before OTC can be considered established in children with cancer. In this review we have discussed a framework for patient selection which has been shown to be effective in identifying those patients at high risk of premature ovarian insufficiency (POI) and who can be offered OTC safely.

Estimability of variance components when all model matrices commute
http://hdl.handle.net/10023/9983
This paper deals with estimability of variance components in mixed models when all model matrices commute. In this situation, it is well known that the best linear unbiased estimators of fixed effects are the ordinary least squares estimators. If, in addition, the family of possible variancecovariance matrices forms an orthogonal block structure, then there are the same number of variance components as strata, and the variance components are all estimable if and only if there are nonzero residual degrees of freedom in each stratum. We investigate the case where the family of possible variancecovariance matrices, while still commutative, no longer forms an orthogonal block structure. Now the variance components may or may not all be estimable, but there is no clear link with residual degrees of freedom. Whether or not they are all estimable, there may or may not be uniformly best unbiased quadratic estimators of those that are estimable. Examples are given to demonstrate all four possibilities.
This work was partially supported by national funds of FCT  Foundation for Science and Technology under UID/MAT/00212/2013.
20160301T00:00:00Z
Bailey, Rosemary Anne
Ferreira, Sandra S.
Ferreira, Dario
Nunes, Celia
This paper deals with estimability of variance components in mixed models when all model matrices commute. In this situation, it is well known that the best linear unbiased estimators of fixed effects are the ordinary least squares estimators. If, in addition, the family of possible variancecovariance matrices forms an orthogonal block structure, then there are the same number of variance components as strata, and the variance components are all estimable if and only if there are nonzero residual degrees of freedom in each stratum. We investigate the case where the family of possible variancecovariance matrices, while still commutative, no longer forms an orthogonal block structure. Now the variance components may or may not all be estimable, but there is no clear link with residual degrees of freedom. Whether or not they are all estimable, there may or may not be uniformly best unbiased quadratic estimators of those that are estimable. Examples are given to demonstrate all four possibilities.

Timing properties and correctness for structured parallel programs on x8664 multicores
http://hdl.handle.net/10023/9935
This paper determines correctness and timing properties for structured parallel programs on x8664 multicores. Multicore architectures are increasingly common, but real architectures have unpredictable timing properties, and even correctness is not obvious above the relaxedmemory concurrency models that are enforced by commonlyused hardware. This paper takes a rigorous approach to correctness and timing properties, examining common locking protocols from first principles, and extending this through queues to structured parallel constructs. We prove functional correctness and derive simple timing models, and both extend for the first time from lowlevel primitives to highlevel parallel patterns. Our derived highlevel timing models for structured parallel programs allow us to accurately predict upper bounds on program execution times on x8664 multicores.
20160101T00:00:00Z
Hammond, Kevin
Brown, Christopher Mark
Sarkar, Susmit
This paper determines correctness and timing properties for structured parallel programs on x8664 multicores. Multicore architectures are increasingly common, but real architectures have unpredictable timing properties, and even correctness is not obvious above the relaxedmemory concurrency models that are enforced by commonlyused hardware. This paper takes a rigorous approach to correctness and timing properties, examining common locking protocols from first principles, and extending this through queues to structured parallel constructs. We prove functional correctness and derive simple timing models, and both extend for the first time from lowlevel primitives to highlevel parallel patterns. Our derived highlevel timing models for structured parallel programs allow us to accurately predict upper bounds on program execution times on x8664 multicores.

MapMySmoke–a context aware mobile phone application targeted at smoking cessation
http://hdl.handle.net/10023/9872
20161122T00:00:00Z
Schick, Robert Schilling
Humphris, Gerald Michael
Kelsey, Thomas William
Marston, John
Sampson, Kay

Farms, pipes, streams and reforestation : reasoning about structured parallel processes using types and hylomorphisms
http://hdl.handle.net/10023/9810
The increasing importance of parallelism has motivated the creation of better abstractions for writing parallel software, including structured parallelism using nested algorithmic skeletons. Such approaches provide highlevel abstractions that avoid common problems, such as race conditions, and often allow strong cost models to be defined. However, choosing a combination of algorithmic skeletons that yields good parallel speedups for a program on some specific parallel architecture remains a difficult task. In order to achieve this, it is necessary to simultaneously reason both about the costs of different parallel structures and about the semantic equivalences between them. This paper presents a new typebased mechanism that enables strong static reasoning about these properties. We exploit wellknown properties of a very general recursion pattern, hylomorphisms, and give a denotational semantics for structured parallel processes in terms of these hylomorphisms. Using our approach, it is possible to determine formally whether it is possible to introduce a desired parallel structure into a program without altering its functional behaviour, and also to choose a version of that parallel structure that minimises some given cost model.
20160904T00:00:00Z
Castro, David
Hammond, Kevin
Sarkar, Susmit
The increasing importance of parallelism has motivated the creation of better abstractions for writing parallel software, including structured parallelism using nested algorithmic skeletons. Such approaches provide highlevel abstractions that avoid common problems, such as race conditions, and often allow strong cost models to be defined. However, choosing a combination of algorithmic skeletons that yields good parallel speedups for a program on some specific parallel architecture remains a difficult task. In order to achieve this, it is necessary to simultaneously reason both about the costs of different parallel structures and about the semantic equivalences between them. This paper presents a new typebased mechanism that enables strong static reasoning about these properties. We exploit wellknown properties of a very general recursion pattern, hylomorphisms, and give a denotational semantics for structured parallel processes in terms of these hylomorphisms. Using our approach, it is possible to determine formally whether it is possible to introduce a desired parallel structure into a program without altering its functional behaviour, and also to choose a version of that parallel structure that minimises some given cost model.

String Cgroups as transitive subgroups of Sn
http://hdl.handle.net/10023/9794
If Γ is a string Cgroup 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.
20160201T00:00:00Z
Cameron, Peter Jephson
Fernandes, Maria Elisa
Leemans, Dimitri
Mixer, Mark
If Γ is a string Cgroup 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.

Habitat complexity in aquatic microcosms affects processes driven by detrivores
http://hdl.handle.net/10023/9749
Habitat complexity can influence predation rates (e.g. by providing refuge) but other ecosystem processes and species interactions might also be modulated by the properties of habitat structure. Here, we focussed on how complexity of artificial habitat (plastic plants), in microcosms, influenced shortterm processes driven by three aquatic detritivores. The effects of habitat complexity on leaf decomposition, production of fine organic matter and pH levels were explored by measuring complexity in three ways: 1. as the presence vs. absence of habitat structure; 2. as the amount of structure (3 or 4.5 g of plastic plants); and 3. as the spatial configuration of structures (measured as fractal dimension). The experiment also addressed potential interactions among the consumers by running all possible species combinations. In the experimental microcosms, habitat complexity influenced how species performed, especially when comparing structure present vs. structure absent. Treatments with structure showed higher fine particulate matter production and lower pH compared to treatments without structures and this was probably due to higher digestion and respiration when structures were present. When we explored the effects of the different complexity levels, we found that the amount of structure added explained more than the fractal dimension of the structures. We give a detailed overview of the experimental design, statistical models and R codes, because our statistical analysis can be applied to other study systems (and disciplines such as restoration ecology). We further make suggestions of how to optimise statistical power when artificially assembling, and analysing, ‘habitat complexity’ by not confounding complexity with the amount of structure added. In summary, this study highlights the importance of habitat complexity for energy flow and the maintenance of ecosystem processes in aquatic ecosystems.
LF was supported in part by the Spanish Ministry of Economy and Competitiveness through the project SCARCE ConsoliderIngenio CSD200900065.
20161101T00:00:00Z
Flores, Lorea
Bailey, R. A.
Elosegi, Arturo
Larrañaga, Aitor
Reiss, Julia
Habitat complexity can influence predation rates (e.g. by providing refuge) but other ecosystem processes and species interactions might also be modulated by the properties of habitat structure. Here, we focussed on how complexity of artificial habitat (plastic plants), in microcosms, influenced shortterm processes driven by three aquatic detritivores. The effects of habitat complexity on leaf decomposition, production of fine organic matter and pH levels were explored by measuring complexity in three ways: 1. as the presence vs. absence of habitat structure; 2. as the amount of structure (3 or 4.5 g of plastic plants); and 3. as the spatial configuration of structures (measured as fractal dimension). The experiment also addressed potential interactions among the consumers by running all possible species combinations. In the experimental microcosms, habitat complexity influenced how species performed, especially when comparing structure present vs. structure absent. Treatments with structure showed higher fine particulate matter production and lower pH compared to treatments without structures and this was probably due to higher digestion and respiration when structures were present. When we explored the effects of the different complexity levels, we found that the amount of structure added explained more than the fractal dimension of the structures. We give a detailed overview of the experimental design, statistical models and R codes, because our statistical analysis can be applied to other study systems (and disciplines such as restoration ecology). We further make suggestions of how to optimise statistical power when artificially assembling, and analysing, ‘habitat complexity’ by not confounding complexity with the amount of structure added. In summary, this study highlights the importance of habitat complexity for energy flow and the maintenance of ecosystem processes in aquatic ecosystems.

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.
20161026T00:00:00Z
Awang, Jennifer Sylvia
Pfeiffer, Markus Johannes
Ruskuc, Nikola
Two 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 nonpermutation. A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every nonsynchronizing 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 nonuniform transformation. The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific nonuniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every nonuniform 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 nonsynchronizing ranks, thus refuting another conjecture on the number of nonsynchronizing 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 nonuniform transformation of that rank. It has previously been shown that a primitive group of degree n synchronizes every nonuniform 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 graphtheoretical result showing that, with limited exceptions, in a vertexprimitive 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 nonuniform 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 CEMATCIÊNCIAS UID/Multi/04621/2013.
20161201T00:00:00Z
Araújo, João
Bentz, Wolfram
Cameron, Peter Jephson
Royle, Gordon
Schaefer, Artur
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 nonpermutation. A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every nonsynchronizing 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 nonuniform transformation. The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific nonuniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every nonuniform 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 nonsynchronizing ranks, thus refuting another conjecture on the number of nonsynchronizing 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 nonuniform transformation of that rank. It has previously been shown that a primitive group of degree n synchronizes every nonuniform 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 graphtheoretical result showing that, with limited exceptions, in a vertexprimitive 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 nonuniform 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.

Automatically improving SAT encoding of constraint problems through common subexpression elimination in Savile Row
http://hdl.handle.net/10023/9589
The formulation of a Propositional Satisfiability (SAT) problem instance is vital to efficient solving. This has motivated research on preprocessing, and inprocessing techniques where reformulation of a SAT instance is interleaved with solving. Preprocessing and inprocessing are highly effective in extending the reach of SAT solvers, however they necessarily operate on the lowest level representation of the problem, the raw SAT clauses, where higherlevel patterns are difficult and/or costly to identify. Our approach is different: rather than reformulate the SAT representation directly, we apply automated reformulations to a higher level representation (a constraint model) of the original problem. Common Subexpression Elimination (CSE) is a family of techniques to improve automatically the formulation of constraint satisfaction problems, which are often highly beneficial when using a conventional constraint solver. In this work we demonstrate that CSE has similar benefits when the reformulated constraint model is encoded to SAT and solved using a stateoftheart SAT solver. In some cases we observe speed improvements of over 100 times.
20151001T00:00:00Z
Nightingale, Peter
Spracklen, Patrick
Miguel, Ian James
The formulation of a Propositional Satisfiability (SAT) problem instance is vital to efficient solving. This has motivated research on preprocessing, and inprocessing techniques where reformulation of a SAT instance is interleaved with solving. Preprocessing and inprocessing are highly effective in extending the reach of SAT solvers, however they necessarily operate on the lowest level representation of the problem, the raw SAT clauses, where higherlevel patterns are difficult and/or costly to identify. Our approach is different: rather than reformulate the SAT representation directly, we apply automated reformulations to a higher level representation (a constraint model) of the original problem. Common Subexpression Elimination (CSE) is a family of techniques to improve automatically the formulation of constraint satisfaction problems, which are often highly beneficial when using a conventional constraint solver. In this work we demonstrate that CSE has similar benefits when the reformulated constraint model is encoded to SAT and solved using a stateoftheart SAT solver. In some cases we observe speed improvements of over 100 times.

Towards semiautomatic datatype translation for parallelism in Erlang
http://hdl.handle.net/10023/9545
As part of our ongoing research programme into programmerintheloop parallelisation, we are studying the problem of introducing alternative data structures to support parallelism. Automated support for data structure transformations makes it easier to produce the best parallelisation for some given program, or even to make parallelisation feasible. We use a refactoring approach to choose and introduce these transformations for specific algorithmic skeletons, structured forms of parallelism that capture common patterns of parallelism. Our approach integrates with the Wrangler refactoring tool for Erlang, and uses the advanced Skel [4] skeleton library for Erlang. This library has previously been shown to give good parallelisations for a number of applications, including a multiagent system [1] where we have achieved speedups of up to 142.44 on a 61core machine with 244 threads. We have investigated three widelyused Erlang data structures: lists, binary structures and ETS (Erlang Term Storage) tables. In general, we have found that ETS tables deliver the best parallel performance for the examples that we have considered. However, our results show that simple lists may deliver similar performance to the use of ETS tables, and better performance than using binary structures. This means that we cannot blindly choose to implement a single optimisation as part of the compilation process. Our approach also allows the use of new (possibly userdefined) data structures and other transformations in future, giving a high level of flexibility and generality.
20160923T00:00:00Z
Barwell, Adam David
Brown, Christopher Mark
Castro, David
Hammond, Kevin
As part of our ongoing research programme into programmerintheloop parallelisation, we are studying the problem of introducing alternative data structures to support parallelism. Automated support for data structure transformations makes it easier to produce the best parallelisation for some given program, or even to make parallelisation feasible. We use a refactoring approach to choose and introduce these transformations for specific algorithmic skeletons, structured forms of parallelism that capture common patterns of parallelism. Our approach integrates with the Wrangler refactoring tool for Erlang, and uses the advanced Skel [4] skeleton library for Erlang. This library has previously been shown to give good parallelisations for a number of applications, including a multiagent system [1] where we have achieved speedups of up to 142.44 on a 61core machine with 244 threads. We have investigated three widelyused Erlang data structures: lists, binary structures and ETS (Erlang Term Storage) tables. In general, we have found that ETS tables deliver the best parallel performance for the examples that we have considered. However, our results show that simple lists may deliver similar performance to the use of ETS tables, and better performance than using binary structures. This means that we cannot blindly choose to implement a single optimisation as part of the compilation process. Our approach also allows the use of new (possibly userdefined) data structures and other transformations in future, giving a high level of flexibility and generality.

Cloud benchmarking for maximising performance of scientific applications
http://hdl.handle.net/10023/9538
How can applications be deployed on the cloud to achieve maximum performance? This question is challenging to address with the availability of a wide variety of cloud Virtual Machines (VMs) with different performance capabilities. The research reported in this paper addresses the above question by proposing a six step benchmarking methodology in which a user provides a set of weights that indicate how important memory, local communication, computation and storage related operations are to an application. The user can either provide a set of four abstract weights or eight fine grain weights based on the knowledge of the application. The weights along with benchmarking data collected from the cloud are used to generate a set of two rankings  one based only on the performance of the VMs and the other takes both performance and costs into account. The rankings are validated on three case study applications using two validation techniques. The case studies on a set of experimental VMs highlight that maximum performance can be achieved by the three top ranked VMs and maximum performance in a costeffective manner is achieved by at least one of the top three ranked VMs produced by the methodology.
This research was pursued under the EPSRC grant, EP/K015745/1, a Royal Society Industry Fellowship and an AWS Education Research grant.
20160826T00:00:00Z
Varghese, Blesson
Akgun, Ozgur
Miguel, Ian James
Thai, Long Thanh
Barker, Adam David
How can applications be deployed on the cloud to achieve maximum performance? This question is challenging to address with the availability of a wide variety of cloud Virtual Machines (VMs) with different performance capabilities. The research reported in this paper addresses the above question by proposing a six step benchmarking methodology in which a user provides a set of weights that indicate how important memory, local communication, computation and storage related operations are to an application. The user can either provide a set of four abstract weights or eight fine grain weights based on the knowledge of the application. The weights along with benchmarking data collected from the cloud are used to generate a set of two rankings  one based only on the performance of the VMs and the other takes both performance and costs into account. The rankings are validated on three case study applications using two validation techniques. The case studies on a set of experimental VMs highlight that maximum performance can be achieved by the three top ranked VMs and maximum performance in a costeffective manner is achieved by at least one of the top three ranked VMs produced by the methodology.

POSIX lexing with derivatives of regular expressions (proof pearl)
http://hdl.handle.net/10023/9472
Brzozowski introduced the notion of derivatives for regular expressions. They can be used for a very simple regular expression matching algorithm. Sulzmann and Lu cleverly extended this algorithm in order to deal with POSIX matching, which is the underlying disambiguation strategy for regular expressions needed in lexers. Sulzmann and Lu have made available online what they call a “rigorous proof” of the correctness of their algorithm w.r.t. their specification; regrettably, it appears to us to have unfillable gaps. In the first part of this paper we give our inductive definition of what a POSIX value is and show (i) that such a value is unique (for given regular expression and string being matched) and (ii) that Sulzmann and Lu’s algorithm always generates such a value (provided that the regular expression matches the string). We also prove the correctness of an optimised version of the POSIX matching algorithm. Our definitions and proof are much simpler than those by Sulzmann and Lu and can be easily formalised in Isabelle/HOL. In the second part we analyse the correctness argument by Sulzmann and Lu and explain why the gaps in this argument cannot be filled easily.
20160101T00:00:00Z
Ausaf, Fahad
Dyckhoff, Roy
Urban, Christian
Brzozowski introduced the notion of derivatives for regular expressions. They can be used for a very simple regular expression matching algorithm. Sulzmann and Lu cleverly extended this algorithm in order to deal with POSIX matching, which is the underlying disambiguation strategy for regular expressions needed in lexers. Sulzmann and Lu have made available online what they call a “rigorous proof” of the correctness of their algorithm w.r.t. their specification; regrettably, it appears to us to have unfillable gaps. In the first part of this paper we give our inductive definition of what a POSIX value is and show (i) that such a value is unique (for given regular expression and string being matched) and (ii) that Sulzmann and Lu’s algorithm always generates such a value (provided that the regular expression matches the string). We also prove the correctness of an optimised version of the POSIX matching algorithm. Our definitions and proof are much simpler than those by Sulzmann and Lu and can be easily formalised in Isabelle/HOL. In the second part we analyse the correctness argument by Sulzmann and Lu and explain why the gaps in this argument cannot be filled easily.

Exploiting short supports for improved encoding of arbitrary constraints into SAT
http://hdl.handle.net/10023/9436
Encoding to SAT and applying a highly efficient modern SAT solver is an increasingly popular method of solving finitedomain constraint problems. In this paper we study encodings of arbitrary constraints where unit propagation on the encoding provides strong reasoning. Specifically, unit propagation on the encoding simulates generalised arc consistency on the original constraint. To create compact and efficient encodings we use the concept of short support. Short support has been successfully applied to create efficient propagation algorithms for arbitrary constraints. A short support of a constraint is similar to a satisfying tuple however a short support is not required to assign every variable in scope. Some variables are left free to take any value. In some cases a short support representation is smaller than the table of satisfying tuples by an exponential factor. We present two encodings based on short supports and evaluate them on a set of benchmark problems, demonstrating a substantial improvement over the state of the art.
20160101T00:00:00Z
Akgün, Özgur
Gent, Ian Philip
Jefferson, Christopher Anthony
Miguel, Ian James
Nightingale, Peter William
Encoding to SAT and applying a highly efficient modern SAT solver is an increasingly popular method of solving finitedomain constraint problems. In this paper we study encodings of arbitrary constraints where unit propagation on the encoding provides strong reasoning. Specifically, unit propagation on the encoding simulates generalised arc consistency on the original constraint. To create compact and efficient encodings we use the concept of short support. Short support has been successfully applied to create efficient propagation algorithms for arbitrary constraints. A short support of a constraint is similar to a satisfying tuple however a short support is not required to assign every variable in scope. Some variables are left free to take any value. In some cases a short support representation is smaller than the table of satisfying tuples by an exponential factor. We present two encodings based on short supports and evaluate them on a set of benchmark problems, demonstrating a substantial improvement over the state of the art.

Automatically generating streamlined constraint models with ESSENCE and CONJURE
http://hdl.handle.net/10023/9369
Streamlined constraint reasoning is the addition of uninferred constraints to a constraint model to reduce the search space, while retaining at least one solution. Previously, effective streamlined models have been constructed by hand, requiring an expert to examine closely solutions to small instances of a problem class and identify regularities. We present a system that automatically generates many conjectured regularities for a given Essence specification of a problem class by examining the domains of decision variables present in the problem specification. These conjectures are evaluated independently and in conjunction with one another on a set of instances from the specified class via an automated modelling toolchain comprising of Conjure, Savile Row and Minion. Once the system has identified effective conjectures they are used to generate streamlined models that allow instances of much larger scale to be solved. Our results demonstrate good models can be identified for problems in combinatorial design, Ramsey theory, graph theory and group theory  often resulting in order of magnitude speedups.
20150813T00:00:00Z
Wetter, James
Akgün, Özgür
Miguel, Ian
Streamlined constraint reasoning is the addition of uninferred constraints to a constraint model to reduce the search space, while retaining at least one solution. Previously, effective streamlined models have been constructed by hand, requiring an expert to examine closely solutions to small instances of a problem class and identify regularities. We present a system that automatically generates many conjectured regularities for a given Essence specification of a problem class by examining the domains of decision variables present in the problem specification. These conjectures are evaluated independently and in conjunction with one another on a set of instances from the specified class via an automated modelling toolchain comprising of Conjure, Savile Row and Minion. Once the system has identified effective conjectures they are used to generate streamlined models that allow instances of much larger scale to be solved. Our results demonstrate good models can be identified for problems in combinatorial design, Ramsey theory, graph theory and group theory  often resulting in order of magnitude speedups.

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.
20150901T00:00:00Z
Morgan, Luke
RoneyDougal, Colva Mary
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.

HPCGAP : engineering a 21stcentury HighPerformance Computer algebra system
http://hdl.handle.net/10023/9325
Symbolic computation has underpinned a number of key advances in Mathematics and Computer Science. Applications are typically large and potentially highly parallel, making them good candidates for parallel execution at a variety of scales from multicore to highperformance computing systems. However, much existing work on parallel computing is based around numeric rather than symbolic computations. In particular, symbolic computing presents particular problems in terms of varying granularity and irregular task sizes that do not match conventional approaches to parallelisation. It also presents problems in terms of the structure of the algorithms and data. This paper describes a new implementation of the free opensource GAP computational algebra system that places parallelism at the heart of the design, dealing with the key scalability and crossplatform portability problems. We provide three system layers that deal with the three most important classes of hardware: individual shared memory multicore nodes, midscale distributed clusters of (multicore) nodes, and full blown HPC systems, comprising largescale tightlyconnected networks of multicore nodes. This requires us to develop new crosslayer programming abstractions in the form of new domainspecific skeletons that allow us to seamlessly target different hardware levels. Our results show that, using our approach, we can achieve good scalability and speedups for two realistic exemplars, on highperformance systems comprising up to 32,000 cores, as well as on ubiquitous multicore systems and distributed clusters. The work reported here paves the way towards full scale exploitation of symbolic computation by highperformance computing systems, and we demonstrate the potential with two major case studies.
20160910T00:00:00Z
Behrends, Reimer
Hammond, Kevin
Janjic, Vladimir
Konovalov, Alexander
Linton, Stephen Alexander
Loidl, HansWolfgang
Maier, Patrick
Trinder, Philip
Symbolic computation has underpinned a number of key advances in Mathematics and Computer Science. Applications are typically large and potentially highly parallel, making them good candidates for parallel execution at a variety of scales from multicore to highperformance computing systems. However, much existing work on parallel computing is based around numeric rather than symbolic computations. In particular, symbolic computing presents particular problems in terms of varying granularity and irregular task sizes that do not match conventional approaches to parallelisation. It also presents problems in terms of the structure of the algorithms and data. This paper describes a new implementation of the free opensource GAP computational algebra system that places parallelism at the heart of the design, dealing with the key scalability and crossplatform portability problems. We provide three system layers that deal with the three most important classes of hardware: individual shared memory multicore nodes, midscale distributed clusters of (multicore) nodes, and full blown HPC systems, comprising largescale tightlyconnected networks of multicore nodes. This requires us to develop new crosslayer programming abstractions in the form of new domainspecific skeletons that allow us to seamlessly target different hardware levels. Our results show that, using our approach, we can achieve good scalability and speedups for two realistic exemplars, on highperformance systems comprising up to 32,000 cores, as well as on ubiquitous multicore systems and distributed clusters. The work reported here paves the way towards full scale exploitation of symbolic computation by highperformance computing systems, and we demonstrate the potential with two major case studies.

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 nontrivial 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.
20170201T00:00:00Z
Cameron, P. J.
CastilloRamirez, 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 nontrivial 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

Idempotent rank in the endomorphism monoid of a nonuniform 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 nonuniform 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.
20160201T00:00:00Z
Dolinka, Igor
East, James
Mitchell, 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 nonuniform 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.
20161001T00:00:00Z
Craik, 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.

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.
20160501T00:00:00Z
Dolinka, Igor
Gray, Robert Duncan
McPhee, Jillian Dawn
Mitchell, James David
Quick, Martyn
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.

Ten Simple Rules for taking advantage of Git and GitHub
http://hdl.handle.net/10023/9146
A 'Ten Simple Rules' guide to Git and GitHub. We describe and provide examples on how to use these software to track projects, as users, teams and organizations. We document collaborative development using branching and forking, interaction between collaborators using issues and continuous integration and automation using, for example, Travis CI and codevoc. We also describe dissemination and social aspects of GitHub such as GitHub pages, following and watching repositories, and give advice on how to make code citable.
This study was supported by Wellcome Trust [grant number WT101477MA] (http://www.wellcome.ac.uk/), BBSRC [grant numbers BB/K01997X/1, BB/I00095X/1, BB/L024225/1 and BB/L002817/1] (http://www.bbsrc.ac.uk/), BMBF grant de.NBI  German Network for Bioinformatics Infrastructure (FKZ031 A 534A) (https://www.denbi.de/), NIH grant numbers R01GM094231 and R01EB017205 (http://www.nih.gov/), EPSRC [reference EP/M022641/1] (https://www.epsrc.ac.uk), NSF grant number 1252893 (http://www.nsf.gov/), and Novo Nordisk Foundation (http://www.novonordiskfonden.dk/en).
20160714T00:00:00Z
PerezRiverol, Yasset
Gatto, Laurent
Wang, Rui
Sachsenberg, Timo
Uszkoreit, Julian
da Veiga Leprevost, Felipe
Fufezan, Christian
Ternent, Tobias
Eglen, Stephen J.
Katz, Daniel S.
Pollard, Tom J.
Konovalov, Alexander
Flight, Robert M.
Blin, Kai
Vizcaíno, Juan Antonio
A 'Ten Simple Rules' guide to Git and GitHub. We describe and provide examples on how to use these software to track projects, as users, teams and organizations. We document collaborative development using branching and forking, interaction between collaborators using issues and continuous integration and automation using, for example, Travis CI and codevoc. We also describe dissemination and social aspects of GitHub such as GitHub pages, following and watching repositories, and give advice on how to make code citable.

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 humaninterpretable presentation with three generators and eight relations, and a Tietzederived presentation with two generators and seven relations.
20150101T00:00:00Z
Quick, Martyn
Bleak, Collin Patrick
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 humaninterpretable presentation with three generators and eight relations, and a Tietzederived presentation with two generators and seven relations.

Towards ‘Metaheuristics in the Large’
http://hdl.handle.net/10023/9139
There is a pressing need for a higherlevel architectural per spective in metaheuristics research. This article proposes a purely functional collection of component signatures as a basis for the scalable and automatic construction of meta heuristics. We claim that this is an important step for sci entific progress because: i). It is increasingly accepted that newlyproposed meta heuristics should be grounded in terms of welldefined frameworks and components. Standardized descrip tions help to distinguish novelty from minor variation. ii). Greater reproducibility is needed, particularly to facil itate comparison with the stateoftheart. iii). Interoperable descriptions are a prerequisite for a data model supporting largescale knowledge discovery across frameworks and problems. A key obstacle is that metaheuristic components suffer from an intrinsic lack of modularity, so we present some design op tions for dealing with this and use this to provide a roadmap for addressing the above issues.
20150607T00:00:00Z
Swann, Jerry
Hammond, Kevin
There is a pressing need for a higherlevel architectural per spective in metaheuristics research. This article proposes a purely functional collection of component signatures as a basis for the scalable and automatic construction of meta heuristics. We claim that this is an important step for sci entific progress because: i). It is increasingly accepted that newlyproposed meta heuristics should be grounded in terms of welldefined frameworks and components. Standardized descrip tions help to distinguish novelty from minor variation. ii). Greater reproducibility is needed, particularly to facil itate comparison with the stateoftheart. iii). Interoperable descriptions are a prerequisite for a data model supporting largescale knowledge discovery across frameworks and problems. A key obstacle is that metaheuristic components suffer from an intrinsic lack of modularity, so we present some design op tions for dealing with this and use this to provide a roadmap for addressing the above issues.

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.
20151007T00:00:00Z
East, J.
EgriNagy, 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 rightangled Artin groups into BrinThompson groups
http://hdl.handle.net/10023/9080
We prove that every finitelygenerated rightangled Artin group can be embedded into some BrinThompson 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
20160227T00:00:00Z
Belk, James
Bleak, Collin
Matucci, Francesco
We prove that every finitelygenerated rightangled Artin group can be embedded into some BrinThompson 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 orderautomorphisms of the rationals
http://hdl.handle.net/10023/9024
In this paper, we consider the group Aut(Q,≤) of orderautomorphisms 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 Ngenerated 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 nonempty 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.
20160801T00:00:00Z
Hyde, J.
Jonusas, J.
Mitchell, J. D.
Peresse, Y. H.
In this paper, we consider the group Aut(Q,≤) of orderautomorphisms 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 Ngenerated 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 nonempty 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.

A validated normative model for human uterine volume from birth to age 40 years
http://hdl.handle.net/10023/8973
Transabdominal pelvic ultrasound and/or pelvic Magnetic Resonance Imaging are safe, accurate and non  invasive means of determining the size and configuration of the internal female genitalia. The assessment of uterine size and volume is helpful in the assessment of many conditions including disorders of sex development, precocious or delayed puberty, infertility and menstrual disorders. Using our own data from the assessment of MRI scans in healthy young females and data extracted from four studies that assessed uterine volume using transabdominal ultrasound in healthy females we have derived and validated a normative model of uterine volume from birth to age 40 years. This shows that uterine volume increases across childhood, with a faster increase in adolescence reflecting the influence of puberty, followed by a slow but progressive rise during adult life. The model suggests that around 84% of the variation in uterine volumes in the healthy population up to age 40 is due to age alone . The derivation of a validated normative model for uterine volume from birth to age 40 years has important clinical applications by providing agerelated reference values for uterine volume.
20160613T00:00:00Z
Kelsey, Thomas W
Ginbey, Eleanor
Chowdhury, Moti M
Bath, Louise E
Anderson, Richard A
Wallace, W Hamish B
Transabdominal pelvic ultrasound and/or pelvic Magnetic Resonance Imaging are safe, accurate and non  invasive means of determining the size and configuration of the internal female genitalia. The assessment of uterine size and volume is helpful in the assessment of many conditions including disorders of sex development, precocious or delayed puberty, infertility and menstrual disorders. Using our own data from the assessment of MRI scans in healthy young females and data extracted from four studies that assessed uterine volume using transabdominal ultrasound in healthy females we have derived and validated a normative model of uterine volume from birth to age 40 years. This shows that uterine volume increases across childhood, with a faster increase in adolescence reflecting the influence of puberty, followed by a slow but progressive rise during adult life. The model suggests that around 84% of the variation in uterine volumes in the healthy population up to age 40 is due to age alone . The derivation of a validated normative model for uterine volume from birth to age 40 years has important clinical applications by providing agerelated reference values for uterine volume.

Interoperability in the OpenDreamKit project : the MathintheMiddle approach
http://hdl.handle.net/10023/8918
OpenDreamKit  "Open Digital Research Environment Toolkit for the Advancement of Mathematics"  is an H2020 EU Research Infrastructure project that aims at supporting, over the period 20152019, the ecosystem of opensource mathematical software systems. OpenDreamKit will deliver a flexible toolkit enabling research groups to set up Virtual Research Environments, customised to meet the varied needs of research projects in pure mathematics and applications. An important step in the OpenDreamKit endeavor is to foster the interoperability between a variety of systems, ranging from computer algebra systems over mathematical databases to frontends. This is the mission of the integration work package. We report on experiments and future plans with the MathintheMiddle approach. This architecture consists of a central mathematical ontology that documents the domain and xes a joint vocabulary, or even a language, going beyond existing systems such as OpenMath, combined with specifications of the functionalities of the various systems. Interaction between systems can then be enriched by pivoting around this architecture.
20160506T00:00:00Z
Dehaye, PaulOlivier
Iancu, Mihnea
Kohlhase, Michael
Konovalov, Alexander
Lelièvre, Samuel
Müller, Dennis
Pfeiffer, Markus
Rabe, Florian
Thiéry, Nicolas M.
Wiesling, Tom
OpenDreamKit  "Open Digital Research Environment Toolkit for the Advancement of Mathematics"  is an H2020 EU Research Infrastructure project that aims at supporting, over the period 20152019, the ecosystem of opensource mathematical software systems. OpenDreamKit will deliver a flexible toolkit enabling research groups to set up Virtual Research Environments, customised to meet the varied needs of research projects in pure mathematics and applications. An important step in the OpenDreamKit endeavor is to foster the interoperability between a variety of systems, ranging from computer algebra systems over mathematical databases to frontends. This is the mission of the integration work package. We report on experiments and future plans with the MathintheMiddle approach. This architecture consists of a central mathematical ontology that documents the domain and xes a joint vocabulary, or even a language, going beyond existing systems such as OpenMath, combined with specifications of the functionalities of the various systems. Interaction between systems can then be enriched by pivoting around this architecture.

Intuitionistic decision procedures since Gentzen
http://hdl.handle.net/10023/8824
Gentzen solved the decision problem for intuitionistic propositional logic in his doctoral thesis [31]; this paper reviews some of the subsequent progress. Solutions to the problem are of importance both for general philosophical reasons and because of their use in implementations of proof assistants (such as Coq [4], widely used in software verification) based on intuitionistic logic.
20160505T00:00:00Z
Dyckhoff, Roy
Gentzen solved the decision problem for intuitionistic propositional logic in his doctoral thesis [31]; this paper reviews some of the subsequent progress. Solutions to the problem are of importance both for general philosophical reasons and because of their use in implementations of proof assistants (such as Coq [4], widely used in software verification) based on intuitionistic logic.

Adult dental anxiety : recent assessment approaches and psychological management in a dental practice setting
http://hdl.handle.net/10023/8821
Dental Anxiety of patients is a common feature of the everyday experience of dental practice. This article advocates the use of regular assessment of this psychological construct to assist in patient management. Various tools, such as the Modified Dental Anxiety Scale (MDAS), are available to monitor dental anxiety that are quick to complete and easy to interpret. Patient burden is low. A new mobile phone assessment system (DENTANX) is being developed for distribution. This application and other psychological interventions are being investigated to assist patients to receive dental care routinely.
20160501T00:00:00Z
Humphris, Gerald Michael
Spyt, James
Herbison, Alice
Kelsey, Tom
Dental Anxiety of patients is a common feature of the everyday experience of dental practice. This article advocates the use of regular assessment of this psychological construct to assist in patient management. Various tools, such as the Modified Dental Anxiety Scale (MDAS), are available to monitor dental anxiety that are quick to complete and easy to interpret. Patient burden is low. A new mobile phone assessment system (DENTANX) is being developed for distribution. This application and other psychological interventions are being investigated to assist patients to receive dental care routinely.

Embeddings into Thompson's group V and coCF groups
http://hdl.handle.net/10023/8747
It is shown in Lehnert and Schweitzer (‘The coword problem for the Higman–Thompson group is contextfree’, Bull. London Math. Soc. 39 (2007) 235–241) that R. Thompson's group V is a cocontextfree (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 2edgecoloured 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.
20161001T00:00:00Z
Bleak, Collin
Matucci, Francesco
Neunhöffer, Max
It is shown in Lehnert and Schweitzer (‘The coword problem for the Higman–Thompson group is contextfree’, Bull. London Math. Soc. 39 (2007) 235–241) that R. Thompson's group V is a cocontextfree (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 2edgecoloured 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.

Lapedo : hybrid skeletons for programming heterogeneous multicore machines in Erlang
http://hdl.handle.net/10023/8678
We describe Lapedo, a novel library of hybrid parallel skeletons for programming heterogeneous multicore/manycore CPU/GPU sys tems in Erlang. Lapedo’s hybrid skeletons comprise a mixture of CPU and GPU components, allowing skeletons to be flexibly and dynamically mapped to available resources. We also describe a model for deriving nearoptimal division of work between CPUs and GPUs, ensuring load balancing between resources. Finally, we evaluate the effectiveness of Lapedo on three realistic use cases from different domains, demonstrating significant speedups compared to executing the same application on only CPU cores or a GPU.
20160401T00:00:00Z
Janjic, Vladimir
Brown, Christopher Mark
Hammond, Kevin
We describe Lapedo, a novel library of hybrid parallel skeletons for programming heterogeneous multicore/manycore CPU/GPU sys tems in Erlang. Lapedo’s hybrid skeletons comprise a mixture of CPU and GPU components, allowing skeletons to be flexibly and dynamically mapped to available resources. We also describe a model for deriving nearoptimal division of work between CPUs and GPUs, ensuring load balancing between resources. Finally, we evaluate the effectiveness of Lapedo on three realistic use cases from different domains, demonstrating significant speedups compared to executing the same application on only CPU cores or a GPU.

AntiMüllerian hormone serum concentrations of women with germline BRCA1 or BRCA2 mutations
http://hdl.handle.net/10023/8649
Study question: Do women with BRCA1 and BRCA2 mutations have reduced ovarian reserve, as measured by circulating antimüllerian hormone (AMH) concentration? Summary answer: Women with a germline mutation in BRCA1 have reduced ovarian reserve as measured by AMH. What is known already: The DNA repair enzymes encoded by BRCA1 and BRCA2 are implicated in reproductive aging. Circulating AMH is a biomarker of ovarian reserve and hence reproductive lifespan. Study design, size, duration: Crosssectional study of AMH concentrations of 693 women at the time of enrolment into the Kathleen Cuningham Foundation Consortium for research into Familial Breast Cancer (kConFab) cohort study (recruitment from 19/08/1997 until 18/9/2012). AMH was measured on stored plasma samples between November 2014 and January 2015 using an electrochemiluminescence immunoassay platform. Participants/materials, setting, methods: Eligible women were from families segregating BRCA1 or BRCA2 mutations and had known mutation status. Participants were aged 25 to 45 years, had no personal history of cancer, retained both ovaries and were not pregnant or breastfeeding at the time of plasma storage. Circulating AMH was measured for 172 carriers and 216 noncarriers from families carrying BRCA1 mutations, and 147 carriers and 158 noncarriers from families carrying BRCA2 mutations. Associations between plasma AMH concentration and carrier status were tested by linear regression, adjusted for age at plasma storage, oral contraceptive use, body mass index and cigarette smoking. Main results and the role of chance: Mean AMH concentration was negatively associated with age (P < 0.001). Mutation carriers were younger at blood draw than noncarriers (P ≤ 0.031). BRCA1 mutation carriers had, on average, 25% (95% CI: 5%  41%, P = 0.02) lower AMH concentrations than noncarriers and were more likely to have AMH concentrations in the lowest quartile for age (OR 1.84, 95% CI: 1.11303, P=0.02). There was no evidence of an association between AMH concentration and BRCA2 mutation status (P = 0.94). Limitations, reasons for caution: The clinical implications of the lower AMH concentrations seen in BRCA1 mutation carriers cannot be assessed by this study design. Wider implications of the findings: Women with a germline mutation in BRCA1 may have reduced ovarian reserve. This is consistent with other smaller studies in the literature and has potential implications for fertility and reproductive lifespan.
Study funding/competing interest(s): kConFab is supported by a grant from the Australian National Breast Cancer Foundation, and previously by the National Health and Medical Research Council (NHMRC), the Queensland Cancer Fund, the Cancer Councils of New South Wales, Victoria, Tasmania and South Australia, and the Cancer Foundation of Western Australia. KAP is an Australian National Breast Cancer Foundation Practitioner Fellow. JLH is a NHMRC Senior Principal Research Fellow. MH is a NHMRC Practitioner Fellow. RA reports personal fees from Roche Diagnostics & Beckman Coulter outside the submitted work and CS reports other from Melbourne IVF outside the submitted work. The remaining authors have nothing to declare and no conflicts of interest.
20160501T00:00:00Z
Phillips, KA
Collins, I M
Milne, R L
McLachlan, S A
Friedlander, M
Hickey, M
Stern, C
Hopper, J L
Fisher, R
Kannemeyer, G
Picken, S
Smith, C D
Kelsey, Thomas William
Anderson, R A
Study question: Do women with BRCA1 and BRCA2 mutations have reduced ovarian reserve, as measured by circulating antimüllerian hormone (AMH) concentration? Summary answer: Women with a germline mutation in BRCA1 have reduced ovarian reserve as measured by AMH. What is known already: The DNA repair enzymes encoded by BRCA1 and BRCA2 are implicated in reproductive aging. Circulating AMH is a biomarker of ovarian reserve and hence reproductive lifespan. Study design, size, duration: Crosssectional study of AMH concentrations of 693 women at the time of enrolment into the Kathleen Cuningham Foundation Consortium for research into Familial Breast Cancer (kConFab) cohort study (recruitment from 19/08/1997 until 18/9/2012). AMH was measured on stored plasma samples between November 2014 and January 2015 using an electrochemiluminescence immunoassay platform. Participants/materials, setting, methods: Eligible women were from families segregating BRCA1 or BRCA2 mutations and had known mutation status. Participants were aged 25 to 45 years, had no personal history of cancer, retained both ovaries and were not pregnant or breastfeeding at the time of plasma storage. Circulating AMH was measured for 172 carriers and 216 noncarriers from families carrying BRCA1 mutations, and 147 carriers and 158 noncarriers from families carrying BRCA2 mutations. Associations between plasma AMH concentration and carrier status were tested by linear regression, adjusted for age at plasma storage, oral contraceptive use, body mass index and cigarette smoking. Main results and the role of chance: Mean AMH concentration was negatively associated with age (P < 0.001). Mutation carriers were younger at blood draw than noncarriers (P ≤ 0.031). BRCA1 mutation carriers had, on average, 25% (95% CI: 5%  41%, P = 0.02) lower AMH concentrations than noncarriers and were more likely to have AMH concentrations in the lowest quartile for age (OR 1.84, 95% CI: 1.11303, P=0.02). There was no evidence of an association between AMH concentration and BRCA2 mutation status (P = 0.94). Limitations, reasons for caution: The clinical implications of the lower AMH concentrations seen in BRCA1 mutation carriers cannot be assessed by this study design. Wider implications of the findings: Women with a germline mutation in BRCA1 may have reduced ovarian reserve. This is consistent with other smaller studies in the literature and has potential implications for fertility and reproductive lifespan.

Randomizationbased models for multitiered experiments : I. a chain of randomizations
http://hdl.handle.net/10023/8636
We derive randomizationbased models for experiments with a chain of randomizations. Estimation theory for these models leads to formulae for the estimators of treatment effects, their standard errors, and expected mean squares in the analysis of variance. We discuss the practicalities in fitting these models and outline the difficulties that can occur, many of which do not arise in twotiered experiments.
20160601T00:00:00Z
Bailey, Rosemary Anne
Brien, C. J.
We derive randomizationbased models for experiments with a chain of randomizations. Estimation theory for these models leads to formulae for the estimators of treatment effects, their standard errors, and expected mean squares in the analysis of variance. We discuss the practicalities in fitting these models and outline the difficulties that can occur, many of which do not arise in twotiered experiments.

A normative model of serum inhibin B in young males
http://hdl.handle.net/10023/8617
Inhibin B has been identified as a potential marker of Sertoli cell function in males. The aim of this study is to produce a normative model of serum inhibin B in males from birth to seventeen years. We used a welldefined search strategy to identify studies containing data that can contribute to a larger approximation of the healthy population. We combined data from four published studies (n = 709) and derived an internally validated model with high goodnessoffit and normally distributed residuals. Our results show that inhibin B increases following birth to a postnatal peak of 270 pg/mL (IQR 210–335 pg/mL) and then decreases during childhood followed by a rise at around 8 years, peaking at a mean 305 pg/mL (IQR 240–445 pg/mL) at around age 17. Following this peak there is a slow decline to the standard mature adult normal range of 170 pg/mL (IQR 125–215 pg/mL). This normative model suggests that 35% of the variation in Inhibin B levels in young males is due to age alone, provides an agespecific reference range for inhibin B in the young healthy male population, and will be a powerful tool in evaluating the potential of inhibin B as a marker of Sertoli cell function in prepubertal boys.
RTM is supported by a Wellcome Trust Intermediate Clinical Fellowship (Grant No: 098522).
20160414T00:00:00Z
Kelsey, Thomas William
Miles, Amy
Mitchell, Rod T.
Anderson, Richard
Wallace, W. Hamish B.
Inhibin B has been identified as a potential marker of Sertoli cell function in males. The aim of this study is to produce a normative model of serum inhibin B in males from birth to seventeen years. We used a welldefined search strategy to identify studies containing data that can contribute to a larger approximation of the healthy population. We combined data from four published studies (n = 709) and derived an internally validated model with high goodnessoffit and normally distributed residuals. Our results show that inhibin B increases following birth to a postnatal peak of 270 pg/mL (IQR 210–335 pg/mL) and then decreases during childhood followed by a rise at around 8 years, peaking at a mean 305 pg/mL (IQR 240–445 pg/mL) at around age 17. Following this peak there is a slow decline to the standard mature adult normal range of 170 pg/mL (IQR 125–215 pg/mL). This normative model suggests that 35% of the variation in Inhibin B levels in young males is due to age alone, provides an agespecific reference range for inhibin B in the young healthy male population, and will be a powerful tool in evaluating the potential of inhibin B as a marker of Sertoli cell function in prepubertal boys.

Typebased allocation analysis for corecursion in lazy functional languages
http://hdl.handle.net/10023/8612
This paper presents a novel typeandeffect analysis for predicting upperbounds on memory allocation costs for corecursive definitions in a simple lazilyevaluated functional language. We show thesoundness of this system against an instrumented variant of Launchbury’s semantics for lazy evaluation which serves as a formal cost model.Our soundness proof requires an intermediate semantics employing indirections. Our proof of correspondence between these semantics that weprovide is thus a crucial part of this work.The analysis has been implemented as an automatic inference system.We demonstrate its effectiveness using several example programs thatpreviously could not be automatically analysed.
20150101T00:00:00Z
Vasconcelos, Pedro Baltazar
Jost, Steffen
Florido, Mario
Hammond, Kevin
This paper presents a novel typeandeffect analysis for predicting upperbounds on memory allocation costs for corecursive definitions in a simple lazilyevaluated functional language. We show thesoundness of this system against an instrumented variant of Launchbury’s semantics for lazy evaluation which serves as a formal cost model.Our soundness proof requires an intermediate semantics employing indirections. Our proof of correspondence between these semantics that weprovide is thus a crucial part of this work.The analysis has been implemented as an automatic inference system.We demonstrate its effectiveness using several example programs thatpreviously could not be automatically analysed.

Constructing flagtransitive, pointimprimitive designs
http://hdl.handle.net/10023/8546
We give a construction of a family of designs with a specified pointpartition and determine the subgroup of automorphisms leaving invariant the pointpartition. We give necessary and sufficient conditions for a design in the family to possess a flagtransitive group of automorphisms preserving the specified pointpartition. We give examples of flagtransitive 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 flagtransitive, pointimprimitive automorphism group.
20160504T00:00:00Z
Cameron, Peter Jephson
Praeger, Cheryl E.
We give a construction of a family of designs with a specified pointpartition and determine the subgroup of automorphisms leaving invariant the pointpartition. We give necessary and sufficient conditions for a design in the family to possess a flagtransitive group of automorphisms preserving the specified pointpartition. We give examples of flagtransitive 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 flagtransitive, pointimprimitive 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 nonpermutation 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.
20151001T00:00:00Z
Araujo, Joao
Cameron, Peter Jephson
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 nonpermutation 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 trianglefree 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 trianglefree graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous trianglefree HigmanSims graph has guessing number at least 77 and at most 78, while the bound given by fractional clique cover is 50.
20160101T00:00:00Z
Cameron, Peter Jephson
Dang, Anh
Riis, Soren
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 trianglefree graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous trianglefree HigmanSims 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 BrinThompson 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 BrinThompson 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.
20170501T00:00:00Z
Belk, James
Bleak, Collin
Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the BrinThompson 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.

Kindergarten Cop : dynamic nursery resizing for GHC
http://hdl.handle.net/10023/8432
Generational garbage collectors are among the most popular garbage collectors used in programming language runtime systems. Their performance is known to depend heavily on choosing the appropriate size of the area where new objects are allocated (the nursery). In imperative languages, it is usual to make the nursery as large as possible, within the limits imposed by the heap size. Functional languages, however, have quite different memory behaviour. In this paper, we study the effect that the nursery size has on the performance of lazy functional programs, through the interplay between cache locality and the frequency of collections. We demonstrate that, in contrast with imperative programs, having large nurseries is not always the best solution. Based on these results, we propose two novel algorithms for dynamic nursery resizing that aim to achieve a compromise between good cache locality and the frequency of garbage collections. We present an implementation of these algorithms in the stateoftheart GHC compiler for the functional language Haskell, and evaluate them using an extensive benchmark suite. In the best case, we demonstrate a reduction in total execution times of up to 88.5%, or an 8.7 overall speedup, compared to using the production GHC garbage collector. On average, our technique gives an improvement of 9.3% in overall performance across a standard suite of 63 benchmarks for the production GHC compiler.
20160317T00:00:00Z
Ferreiro, Henrique
Castro, Laura
Janjic, Vladimir
Hammond, Kevin
Generational garbage collectors are among the most popular garbage collectors used in programming language runtime systems. Their performance is known to depend heavily on choosing the appropriate size of the area where new objects are allocated (the nursery). In imperative languages, it is usual to make the nursery as large as possible, within the limits imposed by the heap size. Functional languages, however, have quite different memory behaviour. In this paper, we study the effect that the nursery size has on the performance of lazy functional programs, through the interplay between cache locality and the frequency of collections. We demonstrate that, in contrast with imperative programs, having large nurseries is not always the best solution. Based on these results, we propose two novel algorithms for dynamic nursery resizing that aim to achieve a compromise between good cache locality and the frequency of garbage collections. We present an implementation of these algorithms in the stateoftheart GHC compiler for the functional language Haskell, and evaluate them using an extensive benchmark suite. In the best case, we demonstrate a reduction in total execution times of up to 88.5%, or an 8.7 overall speedup, compared to using the production GHC garbage collector. On average, our technique gives an improvement of 9.3% in overall performance across a standard suite of 63 benchmarks for the production GHC compiler.

Some remarks on prooftheoretic semantics
http://hdl.handle.net/10023/8155
This is a tripartite work. The first part is a brief discussion of what it is to be a logical constant, rejecting a view that allows a particular selfreferential “constant” • to be such a thing in favour of a view that leads to strong normalisation results. The second part is a commentary on the flattened version of Modus Ponens, and its relationship with rules of type theory. The third part is a commentary on work (joint with Nissim Francez) on “general elimination rules” and harmony, with a retraction of one of the main ideas of that work, i.e. the use of “flattened” general elimination rules for situations with discharge of assumptions. We begin with some general background on general elimination rules.
20160101T00:00:00Z
Dyckhoff, Roy
This is a tripartite work. The first part is a brief discussion of what it is to be a logical constant, rejecting a view that allows a particular selfreferential “constant” • to be such a thing in favour of a view that leads to strong normalisation results. The second part is a commentary on the flattened version of Modus Ponens, and its relationship with rules of type theory. The third part is a commentary on work (joint with Nissim Francez) on “general elimination rules” and harmony, with a retraction of one of the main ideas of that work, i.e. the use of “flattened” general elimination rules for situations with discharge of assumptions. We begin with some general background on general elimination rules.

Accuracy of circulating adiponectin for predicting gestational diabetes : a systematic review and metaanalysis
http://hdl.handle.net/10023/8130
Aims/hypothesis Universal screening for gestational diabetes mellitus (GDM) has not been implemented, and this has had substantial clinical implications. Biomarkerdirected targeted screening might be feasible. We sought to determine the accuracy of circulating adiponectin for early prediction of GDM. Methods A systematic review and metaanalysis of the literature to May 2015 identified studies in which circulating adiponectin was measured prior to a diagnosis of GDM. Data on diagnostic accuracy were synthesised by bivariate mixed effects and hierarchical summary receiver operating characteristic (HSROC) models. Results Thirteen studies met the eligibility criteria, 11 of which (2,865 women; 794 diagnosed with GDM) had extractable data. Circulating adiponectin had a pooled diagnostic odds ratio (DOR) of 6.4 (95% CI 4.1, 9.9), a summary sensitivity of 64.7% (95% CI 51.0%, 76.4%) and a specificity of 77.8% (95% CI 66.4%, 86.1%) for predicting future GDM. The AUC of the HSROC was 0.78 (95% CI 0.74, 0.81). First trimester adiponectin had a pooled sensitivity of 60.3% (95% CI 46.0%, 73.1%), a specificity of 81.3% (95% CI 71.6%, 88.3%) and a DOR of 6.6 (95% CI 3.6, 12.1). The AUC was 0.79 (95% CI 0.75, 0.82). Pooled estimates were similar after adjustment for age, BMI or specific GDM diagnostic threshold. Conclusions/interpretation Prepregnancy and early pregnancy measurement of circulating adiponectin may improve the detection of women at high risk of developing GDM. Prospective evaluation of the combination of adiponectin and maternal characteristics for early identification of those who do and do not require OGTT is warranted.
20160401T00:00:00Z
Iliodromiti, S
Sassarini, J
Kelsey, Tom
Lindsay, R
Sattar, N
Nelson, S
Aims/hypothesis Universal screening for gestational diabetes mellitus (GDM) has not been implemented, and this has had substantial clinical implications. Biomarkerdirected targeted screening might be feasible. We sought to determine the accuracy of circulating adiponectin for early prediction of GDM. Methods A systematic review and metaanalysis of the literature to May 2015 identified studies in which circulating adiponectin was measured prior to a diagnosis of GDM. Data on diagnostic accuracy were synthesised by bivariate mixed effects and hierarchical summary receiver operating characteristic (HSROC) models. Results Thirteen studies met the eligibility criteria, 11 of which (2,865 women; 794 diagnosed with GDM) had extractable data. Circulating adiponectin had a pooled diagnostic odds ratio (DOR) of 6.4 (95% CI 4.1, 9.9), a summary sensitivity of 64.7% (95% CI 51.0%, 76.4%) and a specificity of 77.8% (95% CI 66.4%, 86.1%) for predicting future GDM. The AUC of the HSROC was 0.78 (95% CI 0.74, 0.81). First trimester adiponectin had a pooled sensitivity of 60.3% (95% CI 46.0%, 73.1%), a specificity of 81.3% (95% CI 71.6%, 88.3%) and a DOR of 6.6 (95% CI 3.6, 12.1). The AUC was 0.79 (95% CI 0.75, 0.82). Pooled estimates were similar after adjustment for age, BMI or specific GDM diagnostic threshold. Conclusions/interpretation Prepregnancy and early pregnancy measurement of circulating adiponectin may improve the detection of women at high risk of developing GDM. Prospective evaluation of the combination of adiponectin and maternal characteristics for early identification of those who do and do not require OGTT is warranted.

Well quasiorder in combinatorics : embeddings and homomorphisms
http://hdl.handle.net/10023/7963
The notion of well quasiorder (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.
20150701T00:00:00Z
Huczynska, Sophie
Ruskuc, Nik
The notion of well quasiorder (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.

Cutelimination, substitution and normalisation
http://hdl.handle.net/10023/7962
We present a proof (of the main parts of which there is a formal version, checked with the Isabelle proof assistant) that, for a G3style calculus covering all of intuitionistic zeroorder logic, with an associated term calculus, and with a particular strongly normalising and confluent system of cutreduction rules, every reduction step has, as its natural deduction translation, a sequence of zero or more reduction steps (detour reductions, permutation reductions or simplifications). This complements and (we believe) clarifies earlier work by (e.g.) Zucker and Pottinger on a question raised in 1971 by Kreisel.
Date of Acceptance: 01/2015
20150101T00:00:00Z
Dyckhoff, Roy
We present a proof (of the main parts of which there is a formal version, checked with the Isabelle proof assistant) that, for a G3style calculus covering all of intuitionistic zeroorder logic, with an associated term calculus, and with a particular strongly normalising and confluent system of cutreduction rules, every reduction step has, as its natural deduction translation, a sequence of zero or more reduction steps (detour reductions, permutation reductions or simplifications). This complements and (we believe) clarifies earlier work by (e.g.) Zucker and Pottinger on a question raised in 1971 by Kreisel.

Coprime invariable generation and minimalexponent 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 πielements satisfies. ∏i=1ukπi(S)≤S2OutS.
Colva RoneyDougal acknowledges the support of EPSRC grant EP/I03582X/1.
20150801T00:00:00Z
Detomi, Eloisa
Lucchini, Andrea
RoneyDougal, 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 πielements satisfies. ∏i=1ukπi(S)≤S2OutS.

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 wellorder, the joint preimage property and the dual amalgamation property. The two latter properties are natural analogues of the wellknown joint embedding property and amalgamation property, and are investigated here for the first time.
20150701T00:00:00Z
Huczynska, Sophie
Ruskuc, Nik
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 wellorder, the joint preimage property and the dual amalgamation property. The two latter properties are natural analogues of the wellknown joint embedding property and amalgamation property, and are investigated here for the first time.

Cancer treatment and gonadal function : experimental and established strategies for fertility preservation in children and young adults
http://hdl.handle.net/10023/7641
Preservation of gonadal function,is an important priority for the longterm health of cancer survivors of both sexes and all ages at treatment.. The loss of an opportunity for fertility is a prime concern in both male and female cancer survivors, however the endocrine consequences of gonadal damage are also central to longterm health and wellbeing. Some fertility preservation techniques, such as semen and embryo cryopreservation for the adult man and woman respectively, are established and successful and the recent development of oocyte vitrification has greatly improved the potential to cryopreserve unfertilised oocytes from women. Despite being recommended for all pubertal males, sperm banking is not universally practised in Paediatric Oncology centres, and there are very few ‘adolescentfriendly’ facilities. All approaches to fertility preservation have particular challenges in children and teenagers, including ethical, practical and scientific issues. For the young female, cryopreservation of ovarian cortical tissue with later replacement has now resulted in at least 35 live births, but is still regarded as experimental in most countries. For prepubertal males, testicular biopsy cryopreservation is offered in some centres, but it is unclear how that tissue might be used in the future, and to date there is no evidence that fertility can be restored. For both sexes these approaches require an invasive procedure, and there is an uncertain risk of tissue contamination in haematological and other malignancies. Decision making for all these approaches requires an assessment of the individual’s risk of loss of fertility, and is being made at a time of emotional distress. The development of this field requires better provision of information for patients and their medical teams as well as improvements in service provision, to match technical and scientific advances. Search strategy and selection criteria We searched Medline between Jan 1, 1990, and Sept 1, 2014, for reports published in English using the search terms “fertility preservation”, “cancer”, “childhood cancer”, “gonadotoxic”, and “cancer treatment” in several disjunctive and conjunctive combinations. We mainly selected publications in English from the past 5 years, but did not exclude older, significant publications. We also checked the reference lists of articles identified by this search strategy.
Copyright © 2015 Elsevier Ltd. All rights reserved.
20150701T00:00:00Z
Anderson, Richard A
Mitchell, Rod T
Kelsey, Thomas W
Spears, Norah
Telfer, Evelyn E
Wallace, W Hamish B
Preservation of gonadal function,is an important priority for the longterm health of cancer survivors of both sexes and all ages at treatment.. The loss of an opportunity for fertility is a prime concern in both male and female cancer survivors, however the endocrine consequences of gonadal damage are also central to longterm health and wellbeing. Some fertility preservation techniques, such as semen and embryo cryopreservation for the adult man and woman respectively, are established and successful and the recent development of oocyte vitrification has greatly improved the potential to cryopreserve unfertilised oocytes from women. Despite being recommended for all pubertal males, sperm banking is not universally practised in Paediatric Oncology centres, and there are very few ‘adolescentfriendly’ facilities. All approaches to fertility preservation have particular challenges in children and teenagers, including ethical, practical and scientific issues. For the young female, cryopreservation of ovarian cortical tissue with later replacement has now resulted in at least 35 live births, but is still regarded as experimental in most countries. For prepubertal males, testicular biopsy cryopreservation is offered in some centres, but it is unclear how that tissue might be used in the future, and to date there is no evidence that fertility can be restored. For both sexes these approaches require an invasive procedure, and there is an uncertain risk of tissue contamination in haematological and other malignancies. Decision making for all these approaches requires an assessment of the individual’s risk of loss of fertility, and is being made at a time of emotional distress. The development of this field requires better provision of information for patients and their medical teams as well as improvements in service provision, to match technical and scientific advances. Search strategy and selection criteria We searched Medline between Jan 1, 1990, and Sept 1, 2014, for reports published in English using the search terms “fertility preservation”, “cancer”, “childhood cancer”, “gonadotoxic”, and “cancer treatment” in several disjunctive and conjunctive combinations. We mainly selected publications in English from the past 5 years, but did not exclude older, significant publications. We also checked the reference lists of articles identified by this search strategy.

Cloudbased eInfrastructure for scheduling astronomical observations
http://hdl.handle.net/10023/7605
Gravitational microlensing exploits a transient phenomenon where an observed star is brightened due to deflection of its light by the gravity of an intervening foreground star. It is conjectured that this technique can be used to measure the abundance of planets throughout the Milky Way. In order to undertake efficient gravitational microlensing an observation schedule must be constructed such that various targets are observed while undergoing a microlensing event. In this paper, we propose a cloudbased eInfrastructure that currently supports four methods to compute candidate schedules via the application of local search and probabilistic metaheuristics. We then validate the feasibility of the eInfrastructure by evaluating the methods on historic data. The experiments demonstrate that the use of ondemand cloud resources for the eInfrastructure can allow better schedules to be found more rapidly.
This research was pursued under the EPSRC grant ‘Working Together: Constraint Programming and Cloud Computing’ (EP/K015745/1) and an Amazon Web Services (AWS) Education Research Grant.
20150831T00:00:00Z
Wetter, James Patrick
Akgun, Ozgur
Barker, Adam David
Dominik, Martin
Miguel, Ian James
Varghese, Blesson
Gravitational microlensing exploits a transient phenomenon where an observed star is brightened due to deflection of its light by the gravity of an intervening foreground star. It is conjectured that this technique can be used to measure the abundance of planets throughout the Milky Way. In order to undertake efficient gravitational microlensing an observation schedule must be constructed such that various targets are observed while undergoing a microlensing event. In this paper, we propose a cloudbased eInfrastructure that currently supports four methods to compute candidate schedules via the application of local search and probabilistic metaheuristics. We then validate the feasibility of the eInfrastructure by evaluating the methods on historic data. The experiments demonstrate that the use of ondemand cloud resources for the eInfrastructure can allow better schedules to be found more rapidly.

The physiology and clinical utility of antiMüllerian hormone in women
http://hdl.handle.net/10023/7488
BACKGROUND The measurement of circulating antiMüllerian hormone (AMH) has been applied to a wide array of clinical applications, mainly based on its ability to reflect the number of antral and preantral follicles present in the ovaries. AMH has been suggested to predict the ovarian response to hyperstimulation of the ovaries for IVF and the timing of menopause, and to indicate iatrogenic damage to the ovarian follicle reserve. It has also been proposed as a surrogate for antral follicle count (AFC) in the diagnosis of polycystic ovary syndrome (PCOS). METHODS This paper is a summary of presentations at a European Society of Human Reproduction and Embryology campus workshop on AMH, with literature cited until September 2013. Published peerreviewed medical literature about AMH was searched through MEDLINE and was subjected to systematic review and critical assessment by the panel of authors. RESULTS Physiologically, recent data confirm that AMH is a follicular gatekeeper limiting follicle growth initiation, and subsequently estradiol production from small antral follicles prior to selection. AMH assays continue to evolve and technical issues remain; the absence of an international standard is a key issue. The dynamics of circulating AMH levels throughout life can be split into several distinct phases, with a peak in the early 20s before a decline to the menopause, with a strong and positive correlation with nongrowing follicle recruitment. There is a more complex rise during childhood and adolescence, which is likely to be more reflective of different stages of follicle development. AMH shows limited shortterm variability, but the influence of states such as prolonged oral contraceptive use need to be considered in clinical assessment. There are only very limited data on relationships between AMH and natural fertility at different stages of reproductive life, and while it has a relationship to age at menopause the marked variability in this needs further exploration. AMH may be useful in assessing the need for fertility preservation strategies and detecting postchemotherapy or surgical damage to the ovarian reserve. Longterm followup of patients to ascertain fully the value of postcancer serum AMH in predicting longterm ovarian function is required. There is a linear relationship between AMH and oocyte yield after ovarian stimulation, which is of value in predicting ovarian hyperstimulation. AMH can also identify 'poor responders', but it seems inappropriate at present to withhold IVF purely on this basis. Women with PCOS show markedly raised AMH levels, due to both the increased number of small antral follicles and intrinsic characteristics of those granulosa cells, and this may contribute to anovulation. The value of AMH in the diagnosis of PCOS remains controversial, but it may replace AFC in the future. CONCLUSIONS For the first time in female reproductive biology, it is possible to measure the submerged part of the iceberg of follicle growth, i.e. the intrinsic, socalled 'acyclic' ovarian activity. An international standard for AMH and improved assay validity are urgently needed to maximize the clinical utility of this very promising biomarker of ovarian function in a large array of clinical situations, both in childhood and adulthood.
This paper is a summary of the presentations at the ESHRE campus workshop on AMH in Lille, France, on 10–11 May 2012, with literature update until September 2013. We are grateful to Ronnie Grant for assistance with the ﬁgures.
20140101T00:00:00Z
Dewailly, Didier
Andersen, Claus Yding
Balen, Adam
Broekmans, Frank
Dilaver, Nafi
Fanchin, Renato
Griesinger, Georg
Kelsey, Tom W
La Marca, Antonio
Lambalk, Cornelius
Mason, Helen
Nelson, Scott M
Visser, Jenny A
Wallace, W Hamish
Anderson, Richard A
BACKGROUND The measurement of circulating antiMüllerian hormone (AMH) has been applied to a wide array of clinical applications, mainly based on its ability to reflect the number of antral and preantral follicles present in the ovaries. AMH has been suggested to predict the ovarian response to hyperstimulation of the ovaries for IVF and the timing of menopause, and to indicate iatrogenic damage to the ovarian follicle reserve. It has also been proposed as a surrogate for antral follicle count (AFC) in the diagnosis of polycystic ovary syndrome (PCOS). METHODS This paper is a summary of presentations at a European Society of Human Reproduction and Embryology campus workshop on AMH, with literature cited until September 2013. Published peerreviewed medical literature about AMH was searched through MEDLINE and was subjected to systematic review and critical assessment by the panel of authors. RESULTS Physiologically, recent data confirm that AMH is a follicular gatekeeper limiting follicle growth initiation, and subsequently estradiol production from small antral follicles prior to selection. AMH assays continue to evolve and technical issues remain; the absence of an international standard is a key issue. The dynamics of circulating AMH levels throughout life can be split into several distinct phases, with a peak in the early 20s before a decline to the menopause, with a strong and positive correlation with nongrowing follicle recruitment. There is a more complex rise during childhood and adolescence, which is likely to be more reflective of different stages of follicle development. AMH shows limited shortterm variability, but the influence of states such as prolonged oral contraceptive use need to be considered in clinical assessment. There are only very limited data on relationships between AMH and natural fertility at different stages of reproductive life, and while it has a relationship to age at menopause the marked variability in this needs further exploration. AMH may be useful in assessing the need for fertility preservation strategies and detecting postchemotherapy or surgical damage to the ovarian reserve. Longterm followup of patients to ascertain fully the value of postcancer serum AMH in predicting longterm ovarian function is required. There is a linear relationship between AMH and oocyte yield after ovarian stimulation, which is of value in predicting ovarian hyperstimulation. AMH can also identify 'poor responders', but it seems inappropriate at present to withhold IVF purely on this basis. Women with PCOS show markedly raised AMH levels, due to both the increased number of small antral follicles and intrinsic characteristics of those granulosa cells, and this may contribute to anovulation. The value of AMH in the diagnosis of PCOS remains controversial, but it may replace AFC in the future. CONCLUSIONS For the first time in female reproductive biology, it is possible to measure the submerged part of the iceberg of follicle growth, i.e. the intrinsic, socalled 'acyclic' ovarian activity. An international standard for AMH and improved assay validity are urgently needed to maximize the clinical utility of this very promising biomarker of ovarian function in a large array of clinical situations, both in childhood and adulthood.

Circular designs balanced for neighbours at distances one and two
http://hdl.handle.net/10023/7454
We define three types of neighbourbalanced designs for experiments where the units are arranged in a circle or single line in space or time. The designs are balanced with respect to neighbours at distance one and at distance two. The variants come from allowing or forbidding selfneighbours, and from considering neighbours to be directed or undirected. For two of the variants, we give a method of constructing a design for all values of the number of treatments, except for some small values where it is impossible. In the third case, we give a partial solution that covers all sizes likely to be used in practice.
20141201T00:00:00Z
Aldred, R. E. L.
Bailey, R. A.
Mckay, Brendan D.
Wanless, Ian M.
We define three types of neighbourbalanced designs for experiments where the units are arranged in a circle or single line in space or time. The designs are balanced with respect to neighbours at distance one and at distance two. The variants come from allowing or forbidding selfneighbours, and from considering neighbours to be directed or undirected. For two of the variants, we give a method of constructing a design for all values of the number of treatments, except for some small values where it is impossible. In the third case, we give a partial solution that covers all sizes likely to be used in practice.

Bayesian spatial NBDA for diffusion data with homebase coordinates
http://hdl.handle.net/10023/6981
Networkbased diffusion analysis (NBDA) is a statistical method that allows the researcher to identify and quantify a social influence on the spread of behaviour through a population. Hitherto, NBDA analyses have not directly modelled spatial population structure. Here we present a spatial extension of NBDA, applicable to diffusion data where the spatial locations of individuals in the population, or of their home bases or nest sites, are available. The method is based on the estimation of interindividual associations (for association matrix construction) from the mean interpoint distances as represented on a spatial point pattern of individuals, nests or home bases. We illustrate the method using a simulated dataset, and show how environmental covariates (such as that obtained from a satellite image, or from direct observations in the study area) can also be included in the analysis. The analysis is conducted in a Bayesian framework, which has the advantage that prior knowledge of the rate at which the individuals acquire a given task can be incorporated into the analysis. This method is especially valuable for studies for which detailed spatially structured data, but no other association data, is available. Technological advances are making the collection of such data in the wild more feasible: for example, biologging facilitates the collection of a wide range of variables from animal populations in the wild. We provide an R package, spatialnbda, which is hosted on the Comprehensive R Archive Network (CRAN). This package facilitates the construction of association matrices with the spatial x and y coordinates as the input arguments, and spatial NBDA analyses.
20150702T00:00:00Z
Nightingale, Glenna Faith
Laland, Kevin Neville
Hoppitt, William John Edward
Nightingale, Peter
Networkbased diffusion analysis (NBDA) is a statistical method that allows the researcher to identify and quantify a social influence on the spread of behaviour through a population. Hitherto, NBDA analyses have not directly modelled spatial population structure. Here we present a spatial extension of NBDA, applicable to diffusion data where the spatial locations of individuals in the population, or of their home bases or nest sites, are available. The method is based on the estimation of interindividual associations (for association matrix construction) from the mean interpoint distances as represented on a spatial point pattern of individuals, nests or home bases. We illustrate the method using a simulated dataset, and show how environmental covariates (such as that obtained from a satellite image, or from direct observations in the study area) can also be included in the analysis. The analysis is conducted in a Bayesian framework, which has the advantage that prior knowledge of the rate at which the individuals acquire a given task can be incorporated into the analysis. This method is especially valuable for studies for which detailed spatially structured data, but no other association data, is available. Technological advances are making the collection of such data in the wild more feasible: for example, biologging facilitates the collection of a wide range of variables from animal populations in the wild. We provide an R package, spatialnbda, which is hosted on the Comprehensive R Archive Network (CRAN). This package facilitates the construction of association matrices with the spatial x and y coordinates as the input arguments, and spatial NBDA analyses.

Inflations 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 ordertheoretic 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.
20150201T00:00:00Z
Albert, M.D.
Ruskuc, Nik
Vatter, 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 ordertheoretic 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 FApresentable algebras
http://hdl.handle.net/10023/6852
Automatic presentations, also called FApresentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FApresentable algebras. First, an example is given to show that the class of finitely generated FApresentable 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 FApresentable algebra with a single unary operation is itself FApresentable. Furthermore, it is proven that the class of unary FApresentable algebras is closed under forming finitely generated subalgebras and that the membership problem for such subalgebras is decidable.
20140601T00:00:00Z
Cain, A.J.
Ruskuc, Nik
Automatic presentations, also called FApresentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FApresentable algebras. First, an example is given to show that the class of finitely generated FApresentable 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 FApresentable algebra with a single unary operation is itself FApresentable. Furthermore, it is proven that the class of unary FApresentable algebras is closed under forming finitely generated subalgebras and that the membership problem for such subalgebras is decidable.

Geometrisation of firstorder logic
http://hdl.handle.net/10023/6818
That every firstorder theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently wellknown nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary firstorder theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms.
20150601T00:00:00Z
Dyckhoff, Roy
Negri, Sara
That every firstorder theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently wellknown nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary firstorder theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms.

An externally validated agerelated model of mean follicle density in the cortex of the human ovary
http://hdl.handle.net/10023/6772
The population of nongrowing follicles present in the ovary is defined as the ovarian reserve. This underpins the reproductive lifespan in women, with its depletion determining age at loss of fertility and the menopause. Data amassed from published results of indirect invasive and noninvasive procedures has resulted in the generation of predictive models which estimate the ovarian reserve from conception throughout adult life. The distribution of follicles in the ovary is not uniform, with the great majority of NGFs located in the cortex, which is the region normally biopsied and used for fertility preservation. Previous models have however analysed whole ovary NGF populations and ovarian volumes, but not cortical NGF density. In this study we compared mean nongrowing follicle density values obtained from tissue samples from 13 ovarian cortical biopsies (1637 years) against age matched modelpredicted values generated from population and ovarian volume models, taking into account the proportion of the ovary that is cortex. A mean nongrowing follicle density was calculated for each patient by counting all follicles in a given volume of freshly biopsied ovarian cortical tissue. These values were compared to agematched model generated densities and the correlation between data sets tested. Nongrowing follicle density values obtained from fresh biopsied ovarian cortex samples closely matched model generated data with low mean difference, tight agreement limits and no proportional error between the observed and predicted results. These findings validate the use of the population and ovarian volume models to accurately predict mean follicle density in the ovarian cortex of adult women.
20150701T00:00:00Z
McLaughlin, Marie
Kelsey, Tom
Wallace, W Hamish B
Anderson, Richard A
Telfer, Evelyn E
The population of nongrowing follicles present in the ovary is defined as the ovarian reserve. This underpins the reproductive lifespan in women, with its depletion determining age at loss of fertility and the menopause. Data amassed from published results of indirect invasive and noninvasive procedures has resulted in the generation of predictive models which estimate the ovarian reserve from conception throughout adult life. The distribution of follicles in the ovary is not uniform, with the great majority of NGFs located in the cortex, which is the region normally biopsied and used for fertility preservation. Previous models have however analysed whole ovary NGF populations and ovarian volumes, but not cortical NGF density. In this study we compared mean nongrowing follicle density values obtained from tissue samples from 13 ovarian cortical biopsies (1637 years) against age matched modelpredicted values generated from population and ovarian volume models, taking into account the proportion of the ovary that is cortex. A mean nongrowing follicle density was calculated for each patient by counting all follicles in a given volume of freshly biopsied ovarian cortical tissue. These values were compared to agematched model generated densities and the correlation between data sets tested. Nongrowing follicle density values obtained from fresh biopsied ovarian cortex samples closely matched model generated data with low mean difference, tight agreement limits and no proportional error between the observed and predicted results. These findings validate the use of the population and ovarian volume models to accurately predict mean follicle density in the ovarian cortex of adult women.

Identifying long cycles in finite alternating and symmetric groups acting on subsets
http://hdl.handle.net/10023/6762
Let H be a permutation group on a set Λ, which is permutationally isomorphic to a finite alternating or symmetric group An or Sn acting on the kelement subsets of points from {1, . . . , n}, for some arbitrary but fixed k. Suppose moreover that no isomorphism with this action is known. We show that key elements of H needed to construct such an isomorphism ϕ, such as those whose image under ϕ is an ncycle or (n − 1)cycle, can be recognised with high probability by the lengths of just four of their cycles in Λ.
20150501T00:00:00Z
Linton, Stephen Alexander
Niemeyer, Alice C.
Praeger, Cheryl E.
Let H be a permutation group on a set Λ, which is permutationally isomorphic to a finite alternating or symmetric group An or Sn acting on the kelement subsets of points from {1, . . . , n}, for some arbitrary but fixed k. Suppose moreover that no isomorphism with this action is known. We show that key elements of H needed to construct such an isomorphism ϕ, such as those whose image under ϕ is an ncycle or (n − 1)cycle, can be recognised with high probability by the lengths of just four of their cycles in Λ.

The relation between variation in size of the primordial follicle pool and age at natural menopause
http://hdl.handle.net/10023/6589
Context: Tumors producing insulinlike growth factor 2 (IGF2oma) are a major cause of spontaneous hypoglycemia. The treatment mainstay is surgical resection. Many case reports note resolution of hypoglycemia after IGF2oma resection; however, outcomes are variable according to tumor type. We report a case of resolving hypoglycemia, observed on continuous glucose monitoring (CGM), after resection of an IGF2producing solitary fibrous tumor, of pleura and review the current literature. Case Report: A 69yearold woman presented with impaired consciousness because of hypoglycemia. An IGF2oma was diagnosed as the cause for hypoglycemia because of decreased serum insulin and IGF1, presence of a pleural tumor, and a highmolecularweight form of serum IGF2 detected by western immunoblot. Surgical resection was performed; pathological examination demonstrated a solitary fibrous tumor with lowgrade malignancy. CGM showed reversal of hypoglycemia after tumor resection. Approximately 2 years after resection, the patient has no signs of tumor recurrence or hypoglycemia. Conclusions: An IGF2producing solitary fibrous tumor of pleura in this case caused hypoglycemia. From a search of the literature of 2004–2014, 32 cases of IGF2oma with hypoglycemia that underwent radical surgery were identified; in 19 (59%) patients, hypoglycemia was reversed and there was no subsequent recurrence. The remaining 13 (41%) experienced tumor recurrence or metastasis and recurrence of hypoglycemia average 43 months after initial tumor resection. The tumor of the present case was a lowgrade malignancy. Regular followup with biomarkermonitoring of glucose metabolism and assessment of hypoglycemic symptomatology, in conjunction with imaging tests, is important for detecting possible tumor recurrence and metastasis.
20150601T00:00:00Z
Depmann, M
Faddy, M J
van der Schoew, Y T
Peeters, P H M
Broer, S L
Kelsey, Tom
Nelson, S M
Broekmans, F J M
Context: Tumors producing insulinlike growth factor 2 (IGF2oma) are a major cause of spontaneous hypoglycemia. The treatment mainstay is surgical resection. Many case reports note resolution of hypoglycemia after IGF2oma resection; however, outcomes are variable according to tumor type. We report a case of resolving hypoglycemia, observed on continuous glucose monitoring (CGM), after resection of an IGF2producing solitary fibrous tumor, of pleura and review the current literature. Case Report: A 69yearold woman presented with impaired consciousness because of hypoglycemia. An IGF2oma was diagnosed as the cause for hypoglycemia because of decreased serum insulin and IGF1, presence of a pleural tumor, and a highmolecularweight form of serum IGF2 detected by western immunoblot. Surgical resection was performed; pathological examination demonstrated a solitary fibrous tumor with lowgrade malignancy. CGM showed reversal of hypoglycemia after tumor resection. Approximately 2 years after resection, the patient has no signs of tumor recurrence or hypoglycemia. Conclusions: An IGF2producing solitary fibrous tumor of pleura in this case caused hypoglycemia. From a search of the literature of 2004–2014, 32 cases of IGF2oma with hypoglycemia that underwent radical surgery were identified; in 19 (59%) patients, hypoglycemia was reversed and there was no subsequent recurrence. The remaining 13 (41%) experienced tumor recurrence or metastasis and recurrence of hypoglycemia average 43 months after initial tumor resection. The tumor of the present case was a lowgrade malignancy. Regular followup with biomarkermonitoring of glucose metabolism and assessment of hypoglycemic symptomatology, in conjunction with imaging tests, is important for detecting possible tumor recurrence and metastasis.

Nested rowcolumn designs for nearfactorial experiments with two treatment factors and one control treatment
http://hdl.handle.net/10023/6556
This paper presents some methods of designing experiments in a block design with nested rows and columns. The treatments consist of all combinations of levels of two treatment factors, with an additional control treatment.
The authors also thank Queen Mary, University of London, the University of St Andrews and the Poznan University of Life Sciences for financial support. The second author was also supported by the BritishPolish Young Scientists Programme, grant WAR/342/116.
20151001T00:00:00Z
Bailey, Rosemary Anne
Lacka, Agnieszka
This paper presents some methods of designing experiments in a block design with nested rows and columns. The treatments consist of all combinations of levels of two treatment factors, with an additional control treatment.

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 nonedges 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 nontrivial 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.
20150601T00:00:00Z
Cameron, Peter Jephson
Spiga, Pablo
The operation of switching a graph Gamma with respect to a subset X of the vertex set interchanges edges and nonedges 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 nontrivial 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 Lclasses. 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 Jclass of M has finitely many R and Lclasses; 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.
20140601T00:00:00Z
Gray, R
Ruskuc, Nik
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 Lclasses. 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 Jclass of M has finitely many R and Lclasses; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table.

Breaking conditional symmetry in automated constraint modelling with CONJURE
http://hdl.handle.net/10023/6174
Many constraint problems contain symmetry, which can lead to redundant search. If a partial assignment is shown to be invalid, we are wasting time if we ever consider a symmetric equivalent of it. A particularly important class of symmetries are those introduced by the constraint modelling process: model symmetries. We present a systematic method by which the automated constraint modelling tool CONJURE can break conditional symmetry as it enters a model during refinement. Our method extends, and is compatible with, our previous work on automated symmetry breaking in CONJURE. The result is the automatic and complete removal of model symmetries for the entire problem class represented by the input specification. This applies to arbitrarily nested conditional symmetries and represents a significant step forward for automated constraint modelling.
This work was supported by UK EPSRC EP/K015745/1. Jefferson is supported by a Royal Society University Research Fellowship.
20140101T00:00:00Z
Akgun, O.
Gent, I.P.
Jefferson, C.
Miguel, I.
Nightingale, P.
Many constraint problems contain symmetry, which can lead to redundant search. If a partial assignment is shown to be invalid, we are wasting time if we ever consider a symmetric equivalent of it. A particularly important class of symmetries are those introduced by the constraint modelling process: model symmetries. We present a systematic method by which the automated constraint modelling tool CONJURE can break conditional symmetry as it enters a model during refinement. Our method extends, and is compatible with, our previous work on automated symmetry breaking in CONJURE. The result is the automatic and complete removal of model symmetries for the entire problem class represented by the input specification. This applies to arbitrarily nested conditional symmetries and represents a significant step forward for automated constraint modelling.

Mapping parallel programs to heterogeneous CPU/GPU architectures using a Monte Carlo Tree Search
http://hdl.handle.net/10023/6157
The single core processor, which has dominated for over 30 years, is now obsolete with recent trends increasing towards parallel systems, demanding a huge shift in programming techniques and practices. Moreover, we are rapidly moving towards an age where almost all programming will be targeting parallel systems. Parallel hardware is rapidly evolving, with large heterogeneous systems, typically comprising a mixture of CPUs and GPUs, becoming the mainstream. Additionally, with this increasing heterogeneity comes increasing complexity: not only does the programmer have to worry about where and how to express the parallelism, they must also express an efficient mapping of resources to the available system. This generally requires indepth expert knowledge that most application programmers do not have. In this paper we describe a new technique that derives, automatically, optimal mappings for an application onto a heterogeneous architecture, using a Monte Carlo Tree Search algorithm. Our technique exploits highlevel design patterns, targeting a set of wellspecified parallel skeletons. We demonstrate that our MCTS on a convolution example obtained speedups that are within 5% of the speedups achieved by a handtuned version of the same application.
20130620T00:00:00Z
Goli, Mehdi
McCall, John
Brown, Christopher Mark
Janjic, Vladimir
Hammond, Kevin
The single core processor, which has dominated for over 30 years, is now obsolete with recent trends increasing towards parallel systems, demanding a huge shift in programming techniques and practices. Moreover, we are rapidly moving towards an age where almost all programming will be targeting parallel systems. Parallel hardware is rapidly evolving, with large heterogeneous systems, typically comprising a mixture of CPUs and GPUs, becoming the mainstream. Additionally, with this increasing heterogeneity comes increasing complexity: not only does the programmer have to worry about where and how to express the parallelism, they must also express an efficient mapping of resources to the available system. This generally requires indepth expert knowledge that most application programmers do not have. In this paper we describe a new technique that derives, automatically, optimal mappings for an application onto a heterogeneous architecture, using a Monte Carlo Tree Search algorithm. Our technique exploits highlevel design patterns, targeting a set of wellspecified parallel skeletons. We demonstrate that our MCTS on a convolution example obtained speedups that are within 5% of the speedups achieved by a handtuned version of the same application.

Cloud benchmarking for performance
http://hdl.handle.net/10023/6107
How can applications be deployed on the cloud to achieve maximum performance? This question has become significant and challenging with the availability of a wide variety of Virtual Machines (VMs) with different performance capabilities in the cloud. The above question is addressed by proposing a six step benchmarking methodology in which a user provides a set of four weights that indicate how important each of the following groups: memory, processor, computation and storage are to the application that needs to be executed on the cloud. The weights along with cloud benchmarking data are used to generate a ranking of VMs that can maximise performance of the application. The rankings are validated through an empirical analysis using two case study applications; the first is a financial risk application and the second is a molecular dynamics simulation, which are both representative of workloads that can benefit from execution on the cloud. Both case studies validate the feasibility of the methodology and highlight that maximum performance can be achieved on the cloud by selecting the top ranked VMs produced by the methodology.
Date of Acceptance: 20/09/2014
20141215T00:00:00Z
Varghese, Blesson
Akgun, Ozgur
Miguel, Ian
Thai, Long
Barker, Adam
How can applications be deployed on the cloud to achieve maximum performance? This question has become significant and challenging with the availability of a wide variety of Virtual Machines (VMs) with different performance capabilities in the cloud. The above question is addressed by proposing a six step benchmarking methodology in which a user provides a set of four weights that indicate how important each of the following groups: memory, processor, computation and storage are to the application that needs to be executed on the cloud. The weights along with cloud benchmarking data are used to generate a ranking of VMs that can maximise performance of the application. The rankings are validated through an empirical analysis using two case study applications; the first is a financial risk application and the second is a molecular dynamics simulation, which are both representative of workloads that can benefit from execution on the cloud. Both case studies validate the feasibility of the methodology and highlight that maximum performance can be achieved on the cloud by selecting the top ranked VMs produced by the methodology.

Optimal deployment of geographically distributed workflow engines on the Cloud
http://hdl.handle.net/10023/6106
When orchestrating Web service workflows, the geographical placement of the orchestration engine(s) can greatly affect workflow performance. Data may have to be transferred across long geographical distances, which in turn increases execution time and degrades the overall performance of a workflow. In this paper, we present a framework that, given a DAGbased workflow specification, computes the op timal Amazon EC2 cloud regions to deploy the orchestration engines and execute a workflow. The framework incorporates a constraint model that solves the workflow deployment problem, which is generated using an automated constraint modelling system. The feasibility of the framework is evaluated by executing different sample workflows representative of sci entific workloads. The experimental results indicate that the framework reduces the workflow execution time and provides a speed up of 1.3x2.5x over centralised approaches.
This research was pursued under the EPSRC ‘Working Together: Constraint Programming and Cloud Computing’ grant, a Royal Society Industry Fellowship ‘Bringing Science to the Cloud’, and an Amazon Web Services Education Research Grant. Date of Acceptance: 02/09/2014
20141030T00:00:00Z
Thai, Long
Barker, Adam
Varghese, Blesson
Akgun, Ozgur
Miguel, Ian
When orchestrating Web service workflows, the geographical placement of the orchestration engine(s) can greatly affect workflow performance. Data may have to be transferred across long geographical distances, which in turn increases execution time and degrades the overall performance of a workflow. In this paper, we present a framework that, given a DAGbased workflow specification, computes the op timal Amazon EC2 cloud regions to deploy the orchestration engines and execute a workflow. The framework incorporates a constraint model that solves the workflow deployment problem, which is generated using an automated constraint modelling system. The feasibility of the framework is evaluated by executing different sample workflows representative of sci entific workloads. The experimental results indicate that the framework reduces the workflow execution time and provides a speed up of 1.3x2.5x over centralised approaches.

Resource Analyses for Parallel and Distributed Coordination
http://hdl.handle.net/10023/6039
Predicting the resources that are consumed by a program component is crucial for many parallel or distributed systems. In this context, the main resources of interest are execution time, space and communication/synchronisation costs. There has recently been significant progress in resource analysis technology, notably in typebased analyses and abstract interpretation. At the same time, parallel and distributed computing are becoming increasingly important. This paper synthesises progress in both areas to survey the stateoftheart in resource analysis for parallel and distributed computing. We articulate a general model of resource analysis and describe parallel/distributed resource analysis together with the relationship to sequential analysis. We use three parallel or distributed resource analyses as examples and provide a critical evaluation of the analyses. We investigate why the chosen analysis is effective for each application and identify general principles governing why the resource analysis is effective.
20130301T00:00:00Z
Trinder, Phil
Cole, Murray
Hammond, Kevin
Loidl, HansWolfgang
Michaelson, Greg
Predicting the resources that are consumed by a program component is crucial for many parallel or distributed systems. In this context, the main resources of interest are execution time, space and communication/synchronisation costs. There has recently been significant progress in resource analysis technology, notably in typebased analyses and abstract interpretation. At the same time, parallel and distributed computing are becoming increasingly important. This paper synthesises progress in both areas to survey the stateoftheart in resource analysis for parallel and distributed computing. We articulate a general model of resource analysis and describe parallel/distributed resource analysis together with the relationship to sequential analysis. We use three parallel or distributed resource analyses as examples and provide a critical evaluation of the analyses. We investigate why the chosen analysis is effective for each application and identify general principles governing why the resource analysis is effective.

Higher biodiversity is required to sustain multiple ecosystem processes across temperature regimes
http://hdl.handle.net/10023/5975
Biodiversity loss is occurring rapidly worldwide, yet it is uncertain whether few or many species are required to sustain ecosystem functioning in the face of environmental change. The importance of biodiversity might be enhanced when multiple ecosystem processes (termed multifunctionality) and environmental contexts are considered, yet no studies have quantified this explicitly to date. We measured five key processes and their combined multifunctionality at three temperatures (5, 10 and 15 °C) in freshwater aquaria containing different animal assemblages (14 benthic macroinvertebrate species). For single processes, biodiversity effects were weak and were best predicted by additivebased models, i.e. polyculture performances represented the sum of their monoculture parts. There were, however, significant effects of biodiversity on multifunctionality at the low and the high (but not the intermediate) temperature. Variation in the contribution of species to processes across temperatures meant that greater biodiversity was required to sustain multifunctionality across different temperatures than was the case for single processes. This suggests that previous studies might have underestimated the importance of biodiversity in sustaining ecosystem functioning in a changing environment.
The authors thank the Natural Environment Research Council for financial support awarded to G. W. (Grant reference: NE/D013305/1) that funded D. M. P.'s research. Accepted 11 July 2014.
20150101T00:00:00Z
Perkins, D.M.
Bailey, R.A.
Dossena, M.
Gamfeldt, L.
Reiss, J.
Trimmer, M.
Woodward, G.
Biodiversity loss is occurring rapidly worldwide, yet it is uncertain whether few or many species are required to sustain ecosystem functioning in the face of environmental change. The importance of biodiversity might be enhanced when multiple ecosystem processes (termed multifunctionality) and environmental contexts are considered, yet no studies have quantified this explicitly to date. We measured five key processes and their combined multifunctionality at three temperatures (5, 10 and 15 °C) in freshwater aquaria containing different animal assemblages (14 benthic macroinvertebrate species). For single processes, biodiversity effects were weak and were best predicted by additivebased models, i.e. polyculture performances represented the sum of their monoculture parts. There were, however, significant effects of biodiversity on multifunctionality at the low and the high (but not the intermediate) temperature. Variation in the contribution of species to processes across temperatures meant that greater biodiversity was required to sustain multifunctionality across different temperatures than was the case for single processes. This suggests that previous studies might have underestimated the importance of biodiversity in sustaining ecosystem functioning in a changing environment.

Repeating history : execution replay for Parallel Haskell programs
http://hdl.handle.net/10023/5895
Parallel profiling tools, such as ThreadScope for Parallel Haskell, allow programmers to obtain information about the performance of their parallel programs. However, the information they provide is not always sufficiently detailed to precisely pinpoint the cause of some per formance problems. Often, this is because the cost of obtaining that information would be prohibitive for a complete program execution. In this paper, we adapt the wellknown technique of execution replay to make it possible to simulate a previous run of a program. We ensure that the nondeterministic parallel behaviour of the application is prop erly emulated while the deterministic functional code is run unmodified. In this way, we can gather additional data about the behaviour of a par allel program by replaying some parts of it with more detailed profiling information. We exploit this ability to identify performance bottlenecks in a quicksort implementation, and to derive a version that gives better speedups on multicore machines.
20130101T00:00:00Z
Ferrerio, Henrique
Janjic, Vladimir
Castro, Laura
Hammond, Kevin
Parallel profiling tools, such as ThreadScope for Parallel Haskell, allow programmers to obtain information about the performance of their parallel programs. However, the information they provide is not always sufficiently detailed to precisely pinpoint the cause of some per formance problems. Often, this is because the cost of obtaining that information would be prohibitive for a complete program execution. In this paper, we adapt the wellknown technique of execution replay to make it possible to simulate a previous run of a program. We ensure that the nondeterministic parallel behaviour of the application is prop erly emulated while the deterministic functional code is run unmodified. In this way, we can gather additional data about the behaviour of a par allel program by replaying some parts of it with more detailed profiling information. We exploit this ability to identify performance bottlenecks in a quicksort implementation, and to derive a version that gives better speedups on multicore machines.

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 ≥S1+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.
20150101T00:00:00Z
Button, Jack
RoneyDougal, Colva
Helfgott proved that there exists a δ>0 such that if S is a symmetric generating subset of SL(2,p) containing 1 then either S3=SL(2,p) or S3 ≥S1+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.

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 nonempty 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 nonempty finite set, the almost stabiliser of a finite partition, and the stabiliser of an ultrafilter are maximal subsemigroups of the symmetric group.
20150301T00:00:00Z
East, J.
Mitchell, James David
Péresse, Y.
We classify the maximal subsemigroups of the semigroup ΩΩ of all mappings on an infinite set Ω that contain one of the following groups: the symmetric group on Ω, the setwise stabilizer of a nonempty 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 nonempty finite set, the almost stabiliser of a finite partition, and the stabiliser of an ultrafilter are maximal subsemigroups of the symmetric group.

A validated agerelated normative model for male total testosterone shows increasing variance but no decline after age 40 years
http://hdl.handle.net/10023/5775
The diagnosis of hypogonadism in human males includes identification of low serum testosterone levels, and hence there is an underlying assumption that normal ranges of testosterone for the healthy population are known for all ages. However, to our knowledge, no such reference model exists in the literature, and hence the availability of an applicable biochemical reference range would be helpful for the clinical assessment of hypogonadal men. In this study, using model selection and validation analysis of data identified and extracted from thirteen studies, we derive and validate a normative model of total testosterone across the lifespan in healthy men. We show that total testosterone peaks [mean (2.597.5 percentile)] at 15.4 (7.231.1) nmol/L at an average age of 19 years, and falls in the average case [mean (2.597.5 percentile)] to 13.0 (6.625.3) nmol/L by age 40 years, but we find no evidence for a further fall in mean total testosterone with increasing age through to old age. However we do show that there is an increased variation in total testosterone levels with advancing age after age 40 years. This model provides the age related reference ranges needed to support research and clinical decision making in males who have symptoms that may be due to hypogonadism.
20141008T00:00:00Z
Kelsey, Thomas W
Li, Lucy Q
Mitchell, Rod T
Whelan, Ashley
Anderson, Richard A
Wallace, W Hamish B
The diagnosis of hypogonadism in human males includes identification of low serum testosterone levels, and hence there is an underlying assumption that normal ranges of testosterone for the healthy population are known for all ages. However, to our knowledge, no such reference model exists in the literature, and hence the availability of an applicable biochemical reference range would be helpful for the clinical assessment of hypogonadal men. In this study, using model selection and validation analysis of data identified and extracted from thirteen studies, we derive and validate a normative model of total testosterone across the lifespan in healthy men. We show that total testosterone peaks [mean (2.597.5 percentile)] at 15.4 (7.231.1) nmol/L at an average age of 19 years, and falls in the average case [mean (2.597.5 percentile)] to 13.0 (6.625.3) nmol/L by age 40 years, but we find no evidence for a further fall in mean total testosterone with increasing age through to old age. However we do show that there is an increased variation in total testosterone levels with advancing age after age 40 years. This model provides the age related reference ranges needed to support research and clinical decision making in males who have symptoms that may be due to hypogonadism.

Optimal crossover designs for full interaction models
http://hdl.handle.net/10023/5768
We consider repeated measurement designs when a residual or carryover effect may be present in at most one later period. Since assuming an additive model may be unrealistic for some applications and leads to biased estimation of treatment effects, we consider a model with interactions between carryover and direct treatment effects. When the aim of the experiment is to study the effects of a treatment used alone, we obtain universally optimal approximate designs. We also propose some efficient designs with a reduced number of subjects.
July 2014
20141101T00:00:00Z
Bailey, Rosemary Anne
Druilhet, Pierre
We consider repeated measurement designs when a residual or carryover effect may be present in at most one later period. Since assuming an additive model may be unrealistic for some applications and leads to biased estimation of treatment effects, we consider a model with interactions between carryover and direct treatment effects. When the aim of the experiment is to study the effects of a treatment used alone, we obtain universally optimal approximate designs. We also propose some efficient designs with a reduced number of subjects.

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)
20141102T00:00:00Z
Cameron, Peter Jephson
Fairbairn, Ben
Gadouleau, Maximilien
Memoryless computation is a new technique to compute any function of a set of registers by updating one register at a time while using no memory. Its aim is to emulate how computations are performed in modern cores, since they typically involve updates of single registers. The memoryless computation model can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we consider how efficiently permutations can be computed without memory. We determine the minimum number of basic updates required to compute any permutation, or any even permutation. The small number of required instructions shows that very small instruction sets could be encoded on cores to perform memoryless computation. We then start looking at a possible compromise between the size of the instruction set and the length of the resulting programs. We consider updates only involving a limited number of registers. In particular, we show that binary instructions are not enough to compute all permutations without memory when the alphabet size is even. These results, though expressed as properties of special generating sets of the symmetric or alternating groups, provide guidelines on the implementation of memoryless computation.

Computing in matrix groups without memory
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
20141102T00:00:00Z
Cameron, Peter Jephson
Fairbairn, Ben
Gadouleau, Maximilien
Memoryless computation is a novel means of computing any function of a set of registers by updating one register at a time while using no memory. We aim to emulate how computations are performed on modern cores, since they typically involve updates of single registers. The computation model of memoryless computation can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we view registers as elements of a finite field and we compute linear permutations without memory. We first determine the maximum complexity of a linear function when only linear instructions are allowed. We also determine which linear functions are hardest to compute when the field in question is the binary field and the number of registers is even. Secondly, we investigate some matrix groups, thus showing that the special linear group is internally computable but not fast. Thirdly, we determine the smallest set of instructions required to generate the special and general linear groups. These results are important for memoryless computation, for they show that linear functions can be computed very fast or that very few instructions are needed to compute any linear function. They thus indicate new advantages of using memoryless computation.

The probability of generating a finite simple group
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.
20131101T00:00:00Z
Menezes, Nina Emma
Quick, Martyn
RoneyDougal, Colva Mary
We study the probability of generating a finite simple group, together with its generalisation PG,socG(d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ socG. We prove that PG,socG(2) ⩾ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for PG,socG(3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.

Most primitive groups are full automorphism groups of edgetransitive 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.
20150101T00:00:00Z
Babai, Laszlo
Cameron, Peter Jephson
We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,YG) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. Klin. Moreover, we give an upper bound n1/2+ε for the minimum size of the edges in such a hypergraph. This is essentially best possible.

Fertility preservation for girls and young women with cancer : Populationbased validation of criteria for ovarian tissue cryopreservation
http://hdl.handle.net/10023/5511
Background: Ovarian tissue cryopreservation with later reimplantation has been shown to preserve fertility in adult women, but this approach remains unproven and experimental in children and adolescents. We aimed to assess the use of the Edinburgh selection criteria for ovarian tissue cryopreservation in girls and young women with cancer to determine whether we are offering this invasive procedure to the patients who are most at risk of premature ovarian insufficiency. Methods: Cryopreservation of ovarian tissue has been selectively offered to girls and young women with cancer who met the Edinburgh selection criteria since 1996. Between Jan 1, 1996, and June 30, 2012, 410 female patients younger than 18 years at diagnosis were treated for cancer (including leukaemia and brain tumours) at the Edinburgh Children's Cancer Centre, which serves the whole South East of Scotland region. We determined the ovarian status of these patients from review of clinical records and classified them as having premature ovarian insufficiency or not, or as unable to be determined. Patients younger than 12 years at time of data cutoff (Jan 31, 2013) were excluded from the analysis. Findings: 34 (8%) of the 410 patients met the Edinburgh selection criteria and were offered ovarian tissue cryopreservation before starting cancer treatment. 13 patients declined the procedure and 21 consented, and the procedure was completed successfully in 20 patients. Of the 20 patients who had ovarian tissue successfully cryopreserved, 14 were available for assessment of ovarian function. Of the 13 patients who had declined the procedure, six were available for assessment of ovarian function. Median age at the time of followup for the 20 assessable patients was 16·9 years (IQR 15·521·8). Of the 14 assessable patients who had successfully undergone ovarian cryopreservation, six had developed premature ovarian insufficiency at a median age of 13·4 years (IQR 12·514·6), one of whom also had a natural pregnancy. Of the six assessable patients who had declined the procedure, one had developed premature ovarian insufficiency. Assessment of ovarian function was possible for 141 of the 376 patients who were not offered cryopreservation; one of these patients had developed premature ovarian insufficiency. The cumulative probability of developing premature ovarian insufficiency after treatment was completed was significantly higher for patients who met the criteria for ovarian tissue cryopreservation than for those who did not (15year probability 35% [95% CI 1053] vs 1% [02]; p
This study was partly funded by the UK Medical Research Council grant G1100357 (to RAA). Open Access funded by Department of Health UK.
20140901T00:00:00Z
Wallace, W Hamish B
Smith, Alice Grove
Kelsey, Thomas W
Edgar, Angela E
Anderson, Richard A
Background: Ovarian tissue cryopreservation with later reimplantation has been shown to preserve fertility in adult women, but this approach remains unproven and experimental in children and adolescents. We aimed to assess the use of the Edinburgh selection criteria for ovarian tissue cryopreservation in girls and young women with cancer to determine whether we are offering this invasive procedure to the patients who are most at risk of premature ovarian insufficiency. Methods: Cryopreservation of ovarian tissue has been selectively offered to girls and young women with cancer who met the Edinburgh selection criteria since 1996. Between Jan 1, 1996, and June 30, 2012, 410 female patients younger than 18 years at diagnosis were treated for cancer (including leukaemia and brain tumours) at the Edinburgh Children's Cancer Centre, which serves the whole South East of Scotland region. We determined the ovarian status of these patients from review of clinical records and classified them as having premature ovarian insufficiency or not, or as unable to be determined. Patients younger than 12 years at time of data cutoff (Jan 31, 2013) were excluded from the analysis. Findings: 34 (8%) of the 410 patients met the Edinburgh selection criteria and were offered ovarian tissue cryopreservation before starting cancer treatment. 13 patients declined the procedure and 21 consented, and the procedure was completed successfully in 20 patients. Of the 20 patients who had ovarian tissue successfully cryopreserved, 14 were available for assessment of ovarian function. Of the 13 patients who had declined the procedure, six were available for assessment of ovarian function. Median age at the time of followup for the 20 assessable patients was 16·9 years (IQR 15·521·8). Of the 14 assessable patients who had successfully undergone ovarian cryopreservation, six had developed premature ovarian insufficiency at a median age of 13·4 years (IQR 12·514·6), one of whom also had a natural pregnancy. Of the six assessable patients who had declined the procedure, one had developed premature ovarian insufficiency. Assessment of ovarian function was possible for 141 of the 376 patients who were not offered cryopreservation; one of these patients had developed premature ovarian insufficiency. The cumulative probability of developing premature ovarian insufficiency after treatment was completed was significantly higher for patients who met the criteria for ovarian tissue cryopreservation than for those who did not (15year probability 35% [95% CI 1053] vs 1% [02]; p

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 sharedmemory 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 64core sharedmemory machine. Our results show that we are able to obtain good speedups over sequential GAP programs (up to 25.27 on 64 cores).
20130901T00:00:00Z
Janjic, Vladimir
Brown, Christopher Mark
Neunhoeffer, Max
Hammond, Kevin
Linton, Stephen Alexander
Loidl, HansWolfgang
Orbit enumerations represent an important class of mathematical algorithms which is widely used in computational discrete mathematics. In this paper, we present a new sharedmemory 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 64core sharedmemory machine. Our results show that we are able to obtain good speedups over sequential GAP programs (up to 25.27 on 64 cores).

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.
20131101T00:00:00Z
Bleak, Collin Patrick
SalazarDiaz, Olga
We investigate some product structures in R. Thompson's group $ V$, primarily by studying the topological dynamics associated with $ V$'s action on the Cantor set C. We draw attention to the class D(V,C) of groups which have embeddings as demonstrative subgroups of V whose class can be used to assist in forming various products. Note that D(V,C) contains all finite groups, the free group on two generators, and Q/Z, and is closed under passing to subgroups and under taking direct products of any member by any finite member. If G≤V and H ∈ D(V,C), then G~H embeds into V. Finally, if G, H ∈ D(V,C), then G*H embeds in V. Using a dynamical approach, we also show the perhaps surprising result that Z2 * Z does not embed in V, even though V has many embedded copies of Z2 and has many embedded copies of free products of various pairs of its subgroups.

Can AntiMüllerian hormone predict the diagnosis of polycystic ovary syndrome? : A systematic review and metaanalysis of extracted data
http://hdl.handle.net/10023/5171
Context: Existing biochemical tests for polycystic ovary syndrome (PCOS) have poor sensitivity and specificity. Many women with PCOS have high antiMüllerian hormone (AMH) concentrations; thus, this may be a useful addition to the diagnostic criteria. Objective: A systematic literature review was performed to assess the true accuracy of AMH in the prediction of PCOS and to determine the optimal diagnostic threshold. Data Sources: Published and gray literature were searched for all years until January 2013. Study Selection: Observational studies defining PCOS according to the Rotterdam criteria and assessing the value of AMH in diagnosing PCOS were selected. Ten studies of the initial 314 hits reporting AMH values in the diagnosis of PCOS were included in the metaanalysis and the construction of the summary receiveroperating characteristic curve. Four studies that plotted individual AMH serum levels of women with PCOS and controls on graphs were selected for individual data extraction. Data Extraction: Two researchers independently assessed the abstracts resulted from the initial search against the inclusion criteria, graded the papers for selection and verification biases, and selected the papers that assessed the value of AMH in diagnosing PCOS. Data were extracted from 4 studies with the plotted individual data on graphs with the help of computer software. Data Synthesis: The metaanalysis of the extracted data demonstrated the specificity and sensitivity in diagnosing PCOS in the symptomatic women of 79.4% and 82.8%, respectively, for a cutoff value of AMH of 4.7 ng/mL. The area under the curve was 0.87 (95% confidence interval 0.83–0.92), identical with the area under the curve of 0.87 for the summary receiveroperating characteristic curve involving 10 separate studies. Conclusions: AMH may be a useful initial diagnostic test for PCOS subject to validation in prospective population cohorts.
20130801T00:00:00Z
Iliodromiti, Stamatina
Kelsey, Tom
Anderson, Richard
Nelson, Scott
Context: Existing biochemical tests for polycystic ovary syndrome (PCOS) have poor sensitivity and specificity. Many women with PCOS have high antiMüllerian hormone (AMH) concentrations; thus, this may be a useful addition to the diagnostic criteria. Objective: A systematic literature review was performed to assess the true accuracy of AMH in the prediction of PCOS and to determine the optimal diagnostic threshold. Data Sources: Published and gray literature were searched for all years until January 2013. Study Selection: Observational studies defining PCOS according to the Rotterdam criteria and assessing the value of AMH in diagnosing PCOS were selected. Ten studies of the initial 314 hits reporting AMH values in the diagnosis of PCOS were included in the metaanalysis and the construction of the summary receiveroperating characteristic curve. Four studies that plotted individual AMH serum levels of women with PCOS and controls on graphs were selected for individual data extraction. Data Extraction: Two researchers independently assessed the abstracts resulted from the initial search against the inclusion criteria, graded the papers for selection and verification biases, and selected the papers that assessed the value of AMH in diagnosing PCOS. Data were extracted from 4 studies with the plotted individual data on graphs with the help of computer software. Data Synthesis: The metaanalysis of the extracted data demonstrated the specificity and sensitivity in diagnosing PCOS in the symptomatic women of 79.4% and 82.8%, respectively, for a cutoff value of AMH of 4.7 ng/mL. The area under the curve was 0.87 (95% confidence interval 0.83–0.92), identical with the area under the curve of 0.87 for the summary receiveroperating characteristic curve involving 10 separate studies. Conclusions: AMH may be a useful initial diagnostic test for PCOS subject to validation in prospective population cohorts.

Beyond sumfree 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 sumfree 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 rvalues, constructive existence results and structural characterizations for sets attaining extremal and nearextremal values.
20140207T00:00:00Z
Huczynska, Sophie
For an interval [1,N]⊆N, sets S⊆[1,N] with the property that {(x,y)∈S2:x+y∈S}=0, known as sumfree 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 rvalues, constructive existence results and structural characterizations for sets attaining extremal and nearextremal values.

Proliferating cell nuclear antigen (PCNA) allows the automatic identification of follicles in microscopic images of human ovarian tissue
http://hdl.handle.net/10023/4961
Background: Human ovarian reserve is defined by the population of nongrowing follicles (NGFs) in the ovary. Direct estimation of ovarian reserve involves the identification of NGFs in prepared ovarian tissue. Previous studies involving human tissue have used hematoxylin and eosin (HE) stain, with NGF populations estimated by human examination either of tissue under a microscope, or of images taken of this tissue. Methods: In this study we replaced HE with proliferating cell nuclear antigen (PCNA), and automated the identification and enumeration of NGFs that appear in the resulting microscopic images. We compared the automated estimates to those obtained by human experts, with the “gold standard” taken to be the average of the conservative and liberal estimates by three human experts. Results: The automated estimates were within 10% of the “gold standard”, for images at both 100× and 200× magnifications. Automated analysis took longer than human analysis for several hundred images, not allowing for breaks from analysis needed by humans. Conclusion: Our results both replicate and improve on those of previous studies involving rodent ovaries, and demonstrate the viability of largescale studies of human ovarian reserve using a combination of immunohistochemistry and computational image analysis techniques.
TWK is supported by EPSRC grants EP/CS23229/1 and EP/H004092/1.
20100724T00:00:00Z
Kelsey, Thomas William
Caserta, B
Castillo, L
Wallace, W H B
Coppola, F
Background: Human ovarian reserve is defined by the population of nongrowing follicles (NGFs) in the ovary. Direct estimation of ovarian reserve involves the identification of NGFs in prepared ovarian tissue. Previous studies involving human tissue have used hematoxylin and eosin (HE) stain, with NGF populations estimated by human examination either of tissue under a microscope, or of images taken of this tissue. Methods: In this study we replaced HE with proliferating cell nuclear antigen (PCNA), and automated the identification and enumeration of NGFs that appear in the resulting microscopic images. We compared the automated estimates to those obtained by human experts, with the “gold standard” taken to be the average of the conservative and liberal estimates by three human experts. Results: The automated estimates were within 10% of the “gold standard”, for images at both 100× and 200× magnifications. Automated analysis took longer than human analysis for several hundred images, not allowing for breaks from analysis needed by humans. Conclusion: Our results both replicate and improve on those of previous studies involving rodent ovaries, and demonstrate the viability of largescale studies of human ovarian reserve using a combination of immunohistochemistry and computational image analysis techniques.

Casimir forces for inhomogeneous planar media
http://hdl.handle.net/10023/4758
Casimir forces arise from vacuum uctuations. They are fully understood only for simple models, and are important in nano and microtechnologies. We report our experience of computer algebra calculations towards the Casimir force for models involving inhomogeneous dielectrics. We describe a methodology that greatly increases condence in any results obtained, and use this methodology to demonstrate that the analytic derivation of scalar Green's functions is at the boundary of current computer algebra technology. We further demonstrate that Lifshitz theory of electromagnetic vacuum energy can not be directly applied to calculate the Casimir stress for models of this type, and produce results that have led to alternative regularisations. Using a combination of our new computational framework and the new theory based on our results, we provide specic calculations of Casimir forces for planar dielectrics having permittivity that declines exponentially. We discuss the relative strengths and weaknesses of computer algebra systems when applied to this type of problem, and describe a combined numerical and symbolic computational framework for calculating Casimir forces for arbitrary planar models.
20130125T00:00:00Z
Xiong, Chun
Kelsey, Tom
Linton, Stephen Alexander
Leonhardt, Ulf
Casimir forces arise from vacuum uctuations. They are fully understood only for simple models, and are important in nano and microtechnologies. We report our experience of computer algebra calculations towards the Casimir force for models involving inhomogeneous dielectrics. We describe a methodology that greatly increases condence in any results obtained, and use this methodology to demonstrate that the analytic derivation of scalar Green's functions is at the boundary of current computer algebra technology. We further demonstrate that Lifshitz theory of electromagnetic vacuum energy can not be directly applied to calculate the Casimir stress for models of this type, and produce results that have led to alternative regularisations. Using a combination of our new computational framework and the new theory based on our results, we provide specic calculations of Casimir forces for planar dielectrics having permittivity that declines exponentially. We discuss the relative strengths and weaknesses of computer algebra systems when applied to this type of problem, and describe a combined numerical and symbolic computational framework for calculating Casimir forces for arbitrary planar models.

Ovarian volume correlates strongly with the number of nongrowing follicles in the human ovary
http://hdl.handle.net/10023/4683
A reliable indirect measure of ovarian reserve for the individual woman remains a challenge for reproductive specialists. Using descriptive statistics from a largescale study of ovarian volumes, we have developed a normative model for healthy females for ages 25 through 85. For average values, this model has a strong and positive correlation (r=0.89) with our recent model of nongrowing follicles (NGFs) in the human ovary for ages 25 through 51. When both models are logadjusted, the correlation increases to r=0.99, over the full range of ovarian volume. Furthermore we can deduce that an ovary of 3 cm3 volume (or less) contains approximately 1000 NGF (or fewer). These strong correlations indicate that ovarian volume is a useful factor in the indirect estimation of human ovarian reserve for the individual woman.
20120101T00:00:00Z
Kelsey, Tom
Wallace, W Hamish B
A reliable indirect measure of ovarian reserve for the individual woman remains a challenge for reproductive specialists. Using descriptive statistics from a largescale study of ovarian volumes, we have developed a normative model for healthy females for ages 25 through 85. For average values, this model has a strong and positive correlation (r=0.89) with our recent model of nongrowing follicles (NGFs) in the human ovary for ages 25 through 51. When both models are logadjusted, the correlation increases to r=0.99, over the full range of ovarian volume. Furthermore we can deduce that an ovary of 3 cm3 volume (or less) contains approximately 1000 NGF (or fewer). These strong correlations indicate that ovarian volume is a useful factor in the indirect estimation of human ovarian reserve for the individual woman.

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 2generated 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.
20140601T00:00:00Z
Detomi, Eloisa
Lucchini, Andrea
RoneyDougal, Colva Mary
A group L is primitive monolithic if L has a unique minimal normal subgroup, N , and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L , given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 2generated 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.

Generating custom propagators for arbitrary constraints
http://hdl.handle.net/10023/4566
Constraint Programming (CP) is a proven set of techniques for solving complex combinatorial problems from a range of disciplines. The problem is specified as a set of decision variables (with finite domains) and constraints linking the variables. Local reasoning (propagation) on the constraints is central to CP. Many constraints have efficient constraintspecific propagation algorithms. In this work, we generate custom propagators for constraints. These custom propagators can be very efficient, even approaching (and in some cases exceeding) the efficiency of handoptimised propagators. Given an arbitrary constraint, we show how to generate a custom propagator that establishes GAC in small polynomial time. This is done by precomputing the propagation that would be performed on every relevant subdomain. The number of relevant subdomains, and therefore the size of the generated propagator, is potentially exponential in the number and domain size of the constrained variables. The limiting factor of our approach is the size of the generated propagators. We investigate symmetry as a means of reducing that size. We exploit the symmetries of the constraint to merge symmetric parts of the generated propagator. This extends the reach of our approach to somewhat larger constraints, with a small runtime penalty. Our experimental results show that, compared with optimised implementations of the table constraint, our techniques can lead to an order of magnitude speedup. Propagation is so fast that the generated propagators compare well with handwritten carefully optimised propagators for the same constraints, and the time taken to generate a propagator is more than repaid. © 2014 PublishedbyElsevierB.V.
Open Access funded by Engineering and Physical Sciences Research Council.
20140601T00:00:00Z
Gent, I.P.
Jefferson, C.
Linton, S.
Miguel, I.
Nightingale, P.
Constraint Programming (CP) is a proven set of techniques for solving complex combinatorial problems from a range of disciplines. The problem is specified as a set of decision variables (with finite domains) and constraints linking the variables. Local reasoning (propagation) on the constraints is central to CP. Many constraints have efficient constraintspecific propagation algorithms. In this work, we generate custom propagators for constraints. These custom propagators can be very efficient, even approaching (and in some cases exceeding) the efficiency of handoptimised propagators. Given an arbitrary constraint, we show how to generate a custom propagator that establishes GAC in small polynomial time. This is done by precomputing the propagation that would be performed on every relevant subdomain. The number of relevant subdomains, and therefore the size of the generated propagator, is potentially exponential in the number and domain size of the constrained variables. The limiting factor of our approach is the size of the generated propagators. We investigate symmetry as a means of reducing that size. We exploit the symmetries of the constraint to merge symmetric parts of the generated propagator. This extends the reach of our approach to somewhat larger constraints, with a small runtime penalty. Our experimental results show that, compared with optimised implementations of the table constraint, our techniques can lead to an order of magnitude speedup. Propagation is so fast that the generated propagators compare well with handwritten carefully optimised propagators for the same constraints, and the time taken to generate a propagator is more than repaid. © 2014 PublishedbyElsevierB.V.

Scrucial and bicrucial permutations with respect to squares
http://hdl.handle.net/10023/4495
A permutation is squarefree if it does not contain two consecutive factors of length two or more that are orderisomorphic. A permutation is bicrucial with respect to squares if it is squarefree but any extension of it to the right or to the left by any element gives a permutation that is not squarefree. Bicrucial permutations with respect to squares were studied by Avgustinovich et al., who proved that there exist bicrucial permutations of lengths 8k + 1, 8k + 5, 8k + 7 for k ≥ 1. It was left as open questions whether bicrucial permutations of even length, or such permutations of length 8k +3 exist. In this paper, we provide an encoding of orderings which allows us, using the constraint solver Minion, to show that bicrucial permutations of even length exist, and the smallest such permutations are of length 32. To show that 32 is the minimum length in question, we establish a result on leftcrucial (that is, not extendable to the left) squarefree permutations which begin with three elements in monotone order. Also, we show that bicrucial permutations of length 8k + 3 exist for k = 2, 3 and they do not exist for k = 1. Further, we generalise the notions of rightcrucial, leftcrucial, and bicrucial permutations studied in the literature in various contexts, by introducing the notion of Pcrucial permutations that can be extended to the notion of Pcrucial words. In Scrucial permutations, a particular case of Pcrucial permutations, we deal with permutations that avoid prohibitions, but whose extensions in any position contain a prohibition. We show that Scrucial permutations exist with respect to squares, and minimal such permutations are of length 17. Finally, using our software, we generate much of relevant data showing, for example, that there are 162,190,472 bicrucial squarefree permutations of length 19.
20140214T00:00:00Z
Gent, Ian
Kitaev, Sergey
Konovalov, Alexander
Linton, Steve
Nightingale, Peter
A permutation is squarefree if it does not contain two consecutive factors of length two or more that are orderisomorphic. A permutation is bicrucial with respect to squares if it is squarefree but any extension of it to the right or to the left by any element gives a permutation that is not squarefree. Bicrucial permutations with respect to squares were studied by Avgustinovich et al., who proved that there exist bicrucial permutations of lengths 8k + 1, 8k + 5, 8k + 7 for k ≥ 1. It was left as open questions whether bicrucial permutations of even length, or such permutations of length 8k +3 exist. In this paper, we provide an encoding of orderings which allows us, using the constraint solver Minion, to show that bicrucial permutations of even length exist, and the smallest such permutations are of length 32. To show that 32 is the minimum length in question, we establish a result on leftcrucial (that is, not extendable to the left) squarefree permutations which begin with three elements in monotone order. Also, we show that bicrucial permutations of length 8k + 3 exist for k = 2, 3 and they do not exist for k = 1. Further, we generalise the notions of rightcrucial, leftcrucial, and bicrucial permutations studied in the literature in various contexts, by introducing the notion of Pcrucial permutations that can be extended to the notion of Pcrucial words. In Scrucial permutations, a particular case of Pcrucial permutations, we deal with permutations that avoid prohibitions, but whose extensions in any position contain a prohibition. We show that Scrucial permutations exist with respect to squares, and minimal such permutations are of length 17. Finally, using our software, we generate much of relevant data showing, for example, that there are 162,190,472 bicrucial squarefree permutations of length 19.

Pretreatment antiMüllerian hormone predicts for loss of ovarian function after chemotherapy for early breast cancer
http://hdl.handle.net/10023/4452
Aim: Improving survival for women with early breast cancer (eBC) requires greater attention to the consequences of treatment, including risk to ovarian function. We have assessed whether biochemical markers of the ovarian reserve might improve prediction of chemotherapy related amenorrhoea. Methods: Women (n = 59, mean age 42.6 years [(range 23.3–52.5]) with eBC were recruited before any treatment. Pretreatment ovarian reserve markers (antiMüllerian hormone [AMH], folliclestimulating hormone [FSH], inhibin B) were analysed in relation to ovarian status at 2 years. Results: Pretreatment AMH was significantly lower in women with amenorrhoea at 2 years (4.0 ± 0.9 pmol/L versus 17.2 ± 2.5, P < 0.0001), but FSH and inhibin B did not differ between groups. By logistic regression, pretreatment AMH, but not age, FSH or inhibin B, was an independent predictor of ovarian status at 2 years (P = 0.005; odds ratio 0.013). We combined these data with a similar cohort (combined n = 75); receiver–operator characteristic analysis for AMH gave area under curve (AUC) of 0.90 (95% confidence interval (CI) 0.82–0.97)). A crossvalidated classification tree analysis resulted in a binary classification schema with sensitivity 98.2% and specificity 80.0% for correct classification of amenorrhoea. Conclusion: Pretreatment AMH is a useful predictor of long term post chemotherapy loss of ovarian function in women with eBC, adding significantly to the only previously established individualising predictor, i.e. age. AMH measurement may assist decisionmaking regarding treatment options and fertility preservation procedures.
20131101T00:00:00Z
Anderson, Richard
Rosendahl, Mikkel
Kelsey, Tom
Cameron, David
Aim: Improving survival for women with early breast cancer (eBC) requires greater attention to the consequences of treatment, including risk to ovarian function. We have assessed whether biochemical markers of the ovarian reserve might improve prediction of chemotherapy related amenorrhoea. Methods: Women (n = 59, mean age 42.6 years [(range 23.3–52.5]) with eBC were recruited before any treatment. Pretreatment ovarian reserve markers (antiMüllerian hormone [AMH], folliclestimulating hormone [FSH], inhibin B) were analysed in relation to ovarian status at 2 years. Results: Pretreatment AMH was significantly lower in women with amenorrhoea at 2 years (4.0 ± 0.9 pmol/L versus 17.2 ± 2.5, P < 0.0001), but FSH and inhibin B did not differ between groups. By logistic regression, pretreatment AMH, but not age, FSH or inhibin B, was an independent predictor of ovarian status at 2 years (P = 0.005; odds ratio 0.013). We combined these data with a similar cohort (combined n = 75); receiver–operator characteristic analysis for AMH gave area under curve (AUC) of 0.90 (95% confidence interval (CI) 0.82–0.97)). A crossvalidated classification tree analysis resulted in a binary classification schema with sensitivity 98.2% and specificity 80.0% for correct classification of amenorrhoea. Conclusion: Pretreatment AMH is a useful predictor of long term post chemotherapy loss of ovarian function in women with eBC, adding significantly to the only previously established individualising predictor, i.e. age. AMH measurement may assist decisionmaking regarding treatment options and fertility preservation procedures.

Optimal implementation of watched literals and more general techniques
http://hdl.handle.net/10023/4132
I prove that an implementation technique for scanning lists in backtracking search algorithms is optimal. The result applies to a simple general framework, which I present: applications include watched literal unit propagation in SAT and a number of examples in constraint satisfaction. Techniques like watched literals are known to be highly space efficient and effective in practice. When implemented in the 'circular' approach described here, these techniques also have optimal run time per branch in bigO terms when amortized across a search tree. This also applies when multiple list elements must be found. The constant factor overhead of the worst case is only 2. Replacing the existing nonoptimal implementation of unit propagation in MiniSat speeds up propagation by 29%, though this is not enough to improve overall run time significantly.
Includes 2 appendixes: one with additional proofs and one with code, scripts and data.
20131001T00:00:00Z
Gent, Ian Philip
I prove that an implementation technique for scanning lists in backtracking search algorithms is optimal. The result applies to a simple general framework, which I present: applications include watched literal unit propagation in SAT and a number of examples in constraint satisfaction. Techniques like watched literals are known to be highly space efficient and effective in practice. When implemented in the 'circular' approach described here, these techniques also have optimal run time per branch in bigO terms when amortized across a search tree. This also applies when multiple list elements must be found. The constant factor overhead of the worst case is only 2. Replacing the existing nonoptimal implementation of unit propagation in MiniSat speeds up propagation by 29%, though this is not enough to improve overall run time significantly.

Ovarian volume throughout life : a validated normative model
http://hdl.handle.net/10023/4094
The measurement of ovarian volume has been shown to be a useful indirect indicator of the ovarian reserve in women of reproductive age, in the diagnosis and management of a number of disorders of puberty and adult reproductive function, and is under investigation as a screening tool for ovarian cancer. To date there is no normative model of ovarian volume throughout life. By searching the published literature for ovarian volume in healthy females, and using our own data from multiple sources (combined n = 59,994) we have generated and robustly validated the first model of ovarian volume from conception to 82 years of age. This model shows that 69% of the variation in ovarian volume is due to age alone. We have shown that in the average case ovarian volume rises from 0.7 mL (95% CI 0.4–1.1 mL) at 2 years of age to a peak of 7.7 mL (95% CI 6.5–9.2 mL) at 20 years of age with a subsequent decline to about 2.8 mL (95% CI 2.7–2.9 mL) at the menopause and smaller volumes thereafter. Our model allows us to generate normal values and ranges for ovarian volume throughout life. This is the first validated normative model of ovarian volume from conception to old age; it will be of use in the diagnosis and management of a number of diverse gynaecological and reproductive conditions in females from birth to menopause and beyond.
20130903T00:00:00Z
Kelsey, Tom
Dodwell, Sarah
Wilkinson, Graham
Greve, Tine
Andersen, Claus
Anderson, Richard
Wallace, Hamish
The measurement of ovarian volume has been shown to be a useful indirect indicator of the ovarian reserve in women of reproductive age, in the diagnosis and management of a number of disorders of puberty and adult reproductive function, and is under investigation as a screening tool for ovarian cancer. To date there is no normative model of ovarian volume throughout life. By searching the published literature for ovarian volume in healthy females, and using our own data from multiple sources (combined n = 59,994) we have generated and robustly validated the first model of ovarian volume from conception to 82 years of age. This model shows that 69% of the variation in ovarian volume is due to age alone. We have shown that in the average case ovarian volume rises from 0.7 mL (95% CI 0.4–1.1 mL) at 2 years of age to a peak of 7.7 mL (95% CI 6.5–9.2 mL) at 20 years of age with a subsequent decline to about 2.8 mL (95% CI 2.7–2.9 mL) at the menopause and smaller volumes thereafter. Our model allows us to generate normal values and ranges for ovarian volume throughout life. This is the first validated normative model of ovarian volume from conception to old age; it will be of use in the diagnosis and management of a number of diverse gynaecological and reproductive conditions in females from birth to menopause and beyond.

A validated model of serum antiMüllerian hormone from conception to menopause
http://hdl.handle.net/10023/4056
Background AntiMüllerian hormone (AMH) is a product of growing ovarian follicles. The concentration of AMH in blood may also reflect the nongrowing follicle (NGF) population, i.e. the ovarian reserve, and be of value in predicting reproductive lifespan. A full description of AMH production up to the menopause has not been previously reported. Methodology/Principal Findings By searching the published literature for AMH concentrations in healthy premenopausal females, and using our own data (combined ) we have generated and robustly validated the first model of AMH concentration from conception to menopause. This model shows that 34% of the variation in AMH is due to age alone. We have shown that AMH peaks at age 24.5 years, followed by a decline to the menopause. We have also shown that there is a neonatal peak and a potential prepubertal peak. Our model allows us to generate normative data at all ages. Conclusions/Significance These data highlight key inflection points in ovarian follicle dynamics. This first validated model of circulating AMH in healthy females describes a transition period in early adulthood, after which AMH reflects the progressive loss of the NGF pool. The existence of a neonatal increase in gonadal activity is confirmed for females. An improved understanding of the relationship between circulating AMH and age will lead to more accurate assessment of ovarian reserve for the individual woman.
20110715T00:00:00Z
Kelsey, Tom
Wright, Phoebe
Nelson, Scott
Anderson, Richard
Wallace, Hamish
Background AntiMüllerian hormone (AMH) is a product of growing ovarian follicles. The concentration of AMH in blood may also reflect the nongrowing follicle (NGF) population, i.e. the ovarian reserve, and be of value in predicting reproductive lifespan. A full description of AMH production up to the menopause has not been previously reported. Methodology/Principal Findings By searching the published literature for AMH concentrations in healthy premenopausal females, and using our own data (combined ) we have generated and robustly validated the first model of AMH concentration from conception to menopause. This model shows that 34% of the variation in AMH is due to age alone. We have shown that AMH peaks at age 24.5 years, followed by a decline to the menopause. We have also shown that there is a neonatal peak and a potential prepubertal peak. Our model allows us to generate normative data at all ages. Conclusions/Significance These data highlight key inflection points in ovarian follicle dynamics. This first validated model of circulating AMH in healthy females describes a transition period in early adulthood, after which AMH reflects the progressive loss of the NGF pool. The existence of a neonatal increase in gonadal activity is confirmed for females. An improved understanding of the relationship between circulating AMH and age will lead to more accurate assessment of ovarian reserve for the individual woman.

Minimal and random generation of permutation and matrix groups
http://hdl.handle.net/10023/3823
We prove explicit bounds on the numbers of elements needed to generate various types of finite permutation groups and finite completely reducible matrix groups, and present examples to show that they are sharp in all cases. The bounds are linear in the degree of the permutation or matrix group in general, and logarithmic when the group is primitive. They can be combined with results of Lubotzky to produce explicit bounds on the number of random elements required to generate these groups with a specified probability. These results have important applications to computational group theory. Our proofs are inductive and largely theoretical, but we use computer calculations to establish the bounds in a number of specific small cases.
20130801T00:00:00Z
Holt, Derek
RoneyDougal, Colva Mary
We prove explicit bounds on the numbers of elements needed to generate various types of finite permutation groups and finite completely reducible matrix groups, and present examples to show that they are sharp in all cases. The bounds are linear in the degree of the permutation or matrix group in general, and logarithmic when the group is primitive. They can be combined with results of Lubotzky to produce explicit bounds on the number of random elements required to generate these groups with a specified probability. These results have important applications to computational group theory. Our proofs are inductive and largely theoretical, but we use computer calculations to establish the bounds in a number of specific small cases.

Short and long supports for constraint propagation
http://hdl.handle.net/10023/3503
Specialpurpose constraint propagation algorithms frequently make implicit use of short supports  by examining a subset of the variables, they can infer support (a justification that a variablevalue pair may still form part of an assignment that satisfies the constraint) for all other variables and values and save substantial work  but short supports have not been studied in their own right. The two main contributions of this paper are the identification of short supports as important for constraint propagation, and the introduction of HaggisGAC, an efficient and effective general purpose propagation algorithm for exploiting short supports. Given the complexity of HaggisGAC, we present it as an optimised version of a simpler algorithm ShortGAC. Although experiments demonstrate the efficiency of ShortGAC compared with other generalpurpose propagation algorithms where a compact set of short supports is available, we show theoretically and experimentally that HaggisGAC is even better. We also find that HaggisGAC performs better than GACSchema on fulllength supports. We also introduce a variant algorithm HaggisGACStable, which is adapted to avoid work on backtracking and in some cases can be faster and have significant reductions in memory use. All the proposed algorithms are excellent for propagating disjunctions of constraints. In all experiments with disjunctions we found our algorithms to be faster than Constructive Or and GACSchema by at least an order of magnitude, and up to three orders of magnitude.
20130101T00:00:00Z
Nightingale, Peter
Gent, Ian Philip
Jefferson, Christopher Anthony
Miguel, Ian James
Specialpurpose constraint propagation algorithms frequently make implicit use of short supports  by examining a subset of the variables, they can infer support (a justification that a variablevalue pair may still form part of an assignment that satisfies the constraint) for all other variables and values and save substantial work  but short supports have not been studied in their own right. The two main contributions of this paper are the identification of short supports as important for constraint propagation, and the introduction of HaggisGAC, an efficient and effective general purpose propagation algorithm for exploiting short supports. Given the complexity of HaggisGAC, we present it as an optimised version of a simpler algorithm ShortGAC. Although experiments demonstrate the efficiency of ShortGAC compared with other generalpurpose propagation algorithms where a compact set of short supports is available, we show theoretically and experimentally that HaggisGAC is even better. We also find that HaggisGAC performs better than GACSchema on fulllength supports. We also introduce a variant algorithm HaggisGACStable, which is adapted to avoid work on backtracking and in some cases can be faster and have significant reductions in memory use. All the proposed algorithms are excellent for propagating disjunctions of constraints. In all experiments with disjunctions we found our algorithms to be faster than Constructive Or and GACSchema by at least an order of magnitude, and up to three orders of magnitude.

Decomposition tables for experiments : II. Two–one randomizations
http://hdl.handle.net/10023/3479
We investigate structure for pairs of randomizations that do not follow each other in a chain. These are unrandomizedinclusive, independent, coincident or double randomizations. This involves taking several structures that satisfy particular relations and combining them to form the appropriate orthogonal decomposition of the data space for the experiment. We show how to establish the decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated. This leads to recommendations for when the different types of multiple randomization should be used.
20101001T00:00:00Z
Brien, C. J.
Bailey, Rosemary Anne
We investigate structure for pairs of randomizations that do not follow each other in a chain. These are unrandomizedinclusive, independent, coincident or double randomizations. This involves taking several structures that satisfy particular relations and combining them to form the appropriate orthogonal decomposition of the data space for the experiment. We show how to establish the decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated. This leads to recommendations for when the different types of multiple randomization should be used.

Decomposition tables for experiments : I. A chain of randomizations
http://hdl.handle.net/10023/3478
One aspect of evaluating the design for an experiment is the discovery of the relationships between subspaces of the data space. Initially we establish the notation and methods for evaluating an experiment with a single randomization. Starting with two structures, or orthogonal decompositions of the data space, we describe how to combine them to form the overall decomposition for a singlerandomization experiment that is "structure balanced." The relationships between the two structures are characterized using efficiency factors. The decomposition is encapsulated in a decomposition table. Then, for experiments that involve multiple randomizations forming a chain, we take several structures that pairwise are structure balanced and combine them to establish the form of the orthogonal decomposition for the experiment. In particular, it is proven that the properties of the design for Such an experiment are derived in a straightforward manner from those of the individual designs. We show how to formulate an extended decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated.
20091201T00:00:00Z
Brien, C. J.
Bailey, Rosemary Anne
One aspect of evaluating the design for an experiment is the discovery of the relationships between subspaces of the data space. Initially we establish the notation and methods for evaluating an experiment with a single randomization. Starting with two structures, or orthogonal decompositions of the data space, we describe how to combine them to form the overall decomposition for a singlerandomization experiment that is "structure balanced." The relationships between the two structures are characterized using efficiency factors. The decomposition is encapsulated in a decomposition table. Then, for experiments that involve multiple randomizations forming a chain, we take several structures that pairwise are structure balanced and combine them to establish the form of the orthogonal decomposition for the experiment. In particular, it is proven that the properties of the design for Such an experiment are derived in a straightforward manner from those of the individual designs. We show how to formulate an extended decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated.

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

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.
20130501T00:00:00Z
Dolinka, I
Ruskuc, Nik
Given an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area.

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

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

Interfacing Coq + SSReflect with GAP
http://hdl.handle.net/10023/3175
We report on an extendable implementation of the communication interface connecting Coq proof assistant to the computational algebra system GAP using the Symbolic Computation Software Composability Protocol (SCSCP). It allows Coq to issue OpenMath requests to a local or remote GAP instances and represent server responses as Coq terms.
Presentation slides and preprint both provided by author. Preprint published in Electronic Notes in Theoretical Computer Science: Proceedings of the 9th International Workshop On User Interfaces for Theorem Provers (UITP10).
20120919T00:00:00Z
Komendantsky, Vladimir
Konovalov, Alexander
Linton, Stephen Alexander
We report on an extendable implementation of the communication interface connecting Coq proof assistant to the computational algebra system GAP using the Symbolic Computation Software Composability Protocol (SCSCP). It allows Coq to issue OpenMath requests to a local or remote GAP instances and represent server responses as Coq terms.

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

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

Behind and beyond a theorem on groups related to trivalent graphs
http://hdl.handle.net/10023/2462
In 2006 we completed the proof of a fivepart conjecture that was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2generator, 2relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theoretical, but based upon explicit proofs produced by machine for individual cases. Here we explain how we derived the general proofs from specific cases. The conjecture essentially addressed only the finite groups in the family. Here we extend the results to infinite groups, effectively determining when members of this family of finitely presented groups are simply isomorphic to a specific quotient.
20081201T00:00:00Z
Havas, George
Robertson, Edmund F.
Sutherland, Dale C.
In 2006 we completed the proof of a fivepart conjecture that was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2generator, 2relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theoretical, but based upon explicit proofs produced by machine for individual cases. Here we explain how we derived the general proofs from specific cases. The conjecture essentially addressed only the finite groups in the family. Here we extend the results to infinite groups, effectively determining when members of this family of finitely presented groups are simply isomorphic to a specific quotient.

Geometric grid classes of permutations
http://hdl.handle.net/10023/2450
A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ±1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.
20131101T00:00:00Z
Albert, M.H.
Atkinson, M.D.
Bouvel, M.
Ruskuc, Nik
Vatter, V.
A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ±1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.

Human ovarian reserve from conception to the menopause
http://hdl.handle.net/10023/2449
The human ovary contains a fixed number of nongrowing follicles (NGFs) established before birth that decline with increasing age culminating in the menopause at 5051 years. The objective of this study is to model the agerelated population of NGFs in the human ovary from conception to menopause. Data were taken from eight separate quantitative histological studies (n = 325) in which NGF populations at known ages from seven weeks post conception to 51 years ( median 32 years) were calculated. The data set was fitted to 20 peak function models, with the results ranked by obtained r(2) correlation coefficient. The highest ranked model was chosen. Our model matches the logadjusted NGF population from conception to menopause to a fiveparameter asymmetric double Gaussian cumulative (ADC) curve (r(2) = 0.81). When restricted to ages up to 25 years, the ADC curve has r(2) = 0.95. We estimate that for 95% of women by the age of 30 years only 12% of their maximum prebirth NGF population is present and by the age of 40 years only 3% remains. Furthermore, we found that the rate of NGF recruitment towards maturation for most women increases from birth until approximately age 14 years then decreases towards the menopause. To our knowledge, this is the first model of ovarian reserve from conception to menopause. This model allows us to estimate the number of NGFs present in the ovary at any given age, suggests that 81% of the variance in NGF populations is due to age alone, and shows for the first time, to our knowledge, that the rate of NGF recruitment increases from birth to age 14 years then declines with age until menopause. An increased understanding of the dynamics of human ovarian reserve will provide a more scientific basis for fertility counselling for both healthy women and those who have survived gonadotoxic cancer treatments.
20100127T00:00:00Z
Wallace, W. Hamish B.
Kelsey, Tom
The human ovary contains a fixed number of nongrowing follicles (NGFs) established before birth that decline with increasing age culminating in the menopause at 5051 years. The objective of this study is to model the agerelated population of NGFs in the human ovary from conception to menopause. Data were taken from eight separate quantitative histological studies (n = 325) in which NGF populations at known ages from seven weeks post conception to 51 years ( median 32 years) were calculated. The data set was fitted to 20 peak function models, with the results ranked by obtained r(2) correlation coefficient. The highest ranked model was chosen. Our model matches the logadjusted NGF population from conception to menopause to a fiveparameter asymmetric double Gaussian cumulative (ADC) curve (r(2) = 0.81). When restricted to ages up to 25 years, the ADC curve has r(2) = 0.95. We estimate that for 95% of women by the age of 30 years only 12% of their maximum prebirth NGF population is present and by the age of 40 years only 3% remains. Furthermore, we found that the rate of NGF recruitment towards maturation for most women increases from birth until approximately age 14 years then decreases towards the menopause. To our knowledge, this is the first model of ovarian reserve from conception to menopause. This model allows us to estimate the number of NGFs present in the ovary at any given age, suggests that 81% of the variance in NGF populations is due to age alone, and shows for the first time, to our knowledge, that the rate of NGF recruitment increases from birth to age 14 years then declines with age until menopause. An increased understanding of the dynamics of human ovarian reserve will provide a more scientific basis for fertility counselling for both healthy women and those who have survived gonadotoxic cancer treatments.

Unary FApresentable semigroups
http://hdl.handle.net/10023/2375
Automatic presentations, also called FApresentations, were introduced to extend nite model theory to innite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classication of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a dicult problem in general. A restricted problem, also of signicant interest, is to ask this question for unary automatic presentations: automatic presentations over a oneletter alphabet. This paper studies unary FApresentable semigroups. We prove the following: Every unary FApresentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a oneletter alphabet. Unary FApresentable semigroups are locally nite, but nonnitely generated unary FApresentable semigroups may be innite. Every unary FApresentable semigroup satises some Burnside identity.We describe the Green's relations in unary FApresentable semigroups. We investigate the relationship between the class of unary FApresentable semigroups and various semigroup constructions. A classication is given of the unary FApresentable completely simple semigroups.
20120608T00:00:00Z
Cain, Alan James
Ruskuc, Nik
Thomas, R.M.
Automatic presentations, also called FApresentations, were introduced to extend nite model theory to innite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classication of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a dicult problem in general. A restricted problem, also of signicant interest, is to ask this question for unary automatic presentations: automatic presentations over a oneletter alphabet. This paper studies unary FApresentable semigroups. We prove the following: Every unary FApresentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a oneletter alphabet. Unary FApresentable semigroups are locally nite, but nonnitely generated unary FApresentable semigroups may be innite. Every unary FApresentable semigroup satises some Burnside identity.We describe the Green's relations in unary FApresentable semigroups. We investigate the relationship between the class of unary FApresentable semigroups and various semigroup constructions. A classication is given of the unary FApresentable completely simple semigroups.

Substitutionclosed 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.
20110201T00:00:00Z
Atkinson, M.D.
Ruskuc, Nik
Smith, R
The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classied by listing them as a set of explicit families.

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

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

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

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

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

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

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

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

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

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

Maximal subgroups of free idempotentgenerated semigroups over the full transformation monoid
http://hdl.handle.net/10023/2134
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.
20120501T00:00:00Z
Gray, R
Ruskuc, Nik
Let Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying im (e) = r ≤ n − 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr.

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

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

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

The primitive permutation groups of degree less than 4096
http://hdl.handle.net/10023/2045
In this paper we use the Classification of the Finite Simple Groups, the O’Nan– Scott Theorem and Aschbacher’s theorem to classify the primitive permutation groups of degree less than 4096. The results will be added to the primitive groups databases of GAP and Magma.
The first author is supported by an EPSRC doctoral training grant. The second and third authors acknowledge the support of EPSRC grant number EP/C523229/1.
20111014T00:00:00Z
Coutts, Hannah Jane
Quick, Martyn
RoneyDougal, Colva Mary
In this paper we use the Classification of the Finite Simple Groups, the O’Nan– Scott Theorem and Aschbacher’s theorem to classify the primitive permutation groups of degree less than 4096. The results will be added to the primitive groups databases of GAP and Magma.

Groups with the basis property
http://hdl.handle.net/10023/2044
We study finite groups for which every minimal generating set has the same cardinality. A group has the basis property if it and every subgroup satisfies this condition on minimal generating sets. We classify all finite groups with the basis property.
"The ﬁrst author is supported by an EPSRC Doctoral Training Grant"
20111115T00:00:00Z
McDougallBagnall, Jonathan M.
Quick, Martyn
We study finite groups for which every minimal generating set has the same cardinality. A group has the basis property if it and every subgroup satisfies this condition on minimal generating sets. We classify all finite groups with the basis property.

Finite groups are big as semigroups
http://hdl.handle.net/10023/2004
We prove that a finite group G occurs as a maximal proper subsemigroup of an infinite semigroup (in the terminology of Freese, Ježek, and Nation, G is a big semigroup) if and only if G ≥ 3. In fact, any finite semigroup whose minimal ideal contains a subgroup with at least three elements is big.
20110901T00:00:00Z
Dolinka, Igor
Ruskuc, Nik
We prove that a finite group G occurs as a maximal proper subsemigroup of an infinite semigroup (in the terminology of Freese, Ježek, and Nation, G is a big semigroup) if and only if G ≥ 3. In fact, any finite semigroup whose minimal ideal contains a subgroup with at least three elements is big.

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

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

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

Dominion : an architecturedriven approach to generating efficient constraint solvers
http://hdl.handle.net/10023/1967
Constraints are used to solve combinatorial problems in a variety of industrial and academic disciplines. However most constraint solvers are designed to be general and monolithic, leading to problems with efficiency, scalability and extensibility. We propose a novel, architecturedriven constraint solver generation framework called Dominion to tackle these issues. For any given problem, Dominion generates a lean and efficient solver tailored to that problem. In this paper, we outline the Dominion approach and its implications for software architecture specification of the solver.
This work is supported by the EPSRC grant “A Constraint Solver Synthesiser” (EP/H004092/1) and SICSA studentships.
20110601T00:00:00Z
Balasubramaniam, Dharini
De Silva, Lakshitha Ramesh
Jefferson, Christopher Anthony
Kotthoff, Lars
Miguel, Ian James
Nightingale, Peter
Constraints are used to solve combinatorial problems in a variety of industrial and academic disciplines. However most constraint solvers are designed to be general and monolithic, leading to problems with efficiency, scalability and extensibility. We propose a novel, architecturedriven constraint solver generation framework called Dominion to tackle these issues. For any given problem, Dominion generates a lean and efficient solver tailored to that problem. In this paper, we outline the Dominion approach and its implications for software architecture specification of the solver.

Generating continuous mappings with Lipschitz mappings
http://hdl.handle.net/10023/1616
If X is a metric space, then CX and LX denote the semigroups of continuous and Lipschitz mappings, respectively, from X to itself. The relative rank of CX modulo LX is the least cardinality of any set U\LX where U generates CX. For a large class of separable metric spaces X we prove that the relative rank of CX modulo LX is uncountable. When X is the Baire space NN, this rank is N1. A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.
20070501T00:00:00Z
Cichon, J
Mitchell, James David
Morayne, M
If X is a metric space, then CX and LX denote the semigroups of continuous and Lipschitz mappings, respectively, from X to itself. The relative rank of CX modulo LX is the least cardinality of any set U\LX where U generates CX. For a large class of separable metric spaces X we prove that the relative rank of CX modulo LX is uncountable. When X is the Baire space NN, this rank is N1. A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

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.
20030801T00:00:00Z
Huczynska, Sophie
Cohen, SD
The existence of a primitive free (normal) cubic x(3) ax(2) + cx b over a finite field F with arbitrary specified values of a (not equal 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.

Counting cases in substitope algorithms
http://hdl.handle.net/10023/1594
We describe how to count the cases that arise in a family of visualization techniques, including Marching Cubes, Sweeping Simplices, Contour Meshing, Interval Volumes, and Separating Surfaces. Counting the cases is the first step toward developing a generic visualization algorithm to produce substitopes ( geometric substitutions of polytopes). We demonstrate the method using "GAP," a software system for computational group theory. The casecounts are organized into a table that provides a taxonomy of members of the family; numbers in the table are derived from actual lists of cases, which are computed by our methods. The calculations confirm previously reported casecounts for four dimensions that are too large to check by hand and predict the number of cases that will arise in substitope algorithms that have not yet been invented. We show how Polya theory produces a closedform upper bound on the case counts.
20040701T00:00:00Z
Banks, D.C.
Linton, Stephen Alexander
Stockmeyer, P.K.
We describe how to count the cases that arise in a family of visualization techniques, including Marching Cubes, Sweeping Simplices, Contour Meshing, Interval Volumes, and Separating Surfaces. Counting the cases is the first step toward developing a generic visualization algorithm to produce substitopes ( geometric substitutions of polytopes). We demonstrate the method using "GAP," a software system for computational group theory. The casecounts are organized into a table that provides a taxonomy of members of the family; numbers in the table are derived from actual lists of cases, which are computed by our methods. The calculations confirm previously reported casecounts for four dimensions that are too large to check by hand and predict the number of cases that will arise in substitope algorithms that have not yet been invented. We show how Polya theory produces a closedform upper bound on the case counts.

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.
20060701T00:00:00Z
Cain, AJ
Robertson, Edmund Frederick
Ruskuc, Nikola
This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.

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

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

Symmetric subgroups in modular group algebras
http://hdl.handle.net/10023/1417
Let V(KG) be a normalised unit group of the modular group algebra of a finite pgroup G over the field K of p elements. We introduce a notion of symmetric subgroups in V(KG) as subgroups invariant under the action of the classical involution of the group algebra KG. We study properties of symmetric subgroups and construct a counterexample to the conjecture by V.Bovdi, which states that V(KG)=<G,S*>, where S* is a set of symmetric units of V(KG).
This preprint is translated from the original journal publication in Russian: A. Konovalov and A. Tsapok, Symmetric subgroups of the normalised unit group of the modular group algebra of a finite pgroup, Nauk. Visn. Uzhgorod. Univ., Ser. Mat., 9 (2004), 20–24.
20080105T00:00:00Z
Konovalov, Alexander
Krivokhata, A. G.
Let V(KG) be a normalised unit group of the modular group algebra of a finite pgroup G over the field K of p elements. We introduce a notion of symmetric subgroups in V(KG) as subgroups invariant under the action of the classical involution of the group algebra KG. We study properties of symmetric subgroups and construct a counterexample to the conjecture by V.Bovdi, which states that V(KG)=<G,S*>, where S* is a set of symmetric units of V(KG).