DSpace Community:
http://hdl.handle.net/10023/196
20150705T13:20:05Z

Inflations of geometric grid classes of permutations
http://hdl.handle.net/10023/6862
Abstract: 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.
Description: 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
Abstract: 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
Abstract: 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.
Description: Date of Acceptance: 31/03/2015
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
Abstract: 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.
Description: Date of Acceptance: 15/05/2015
20150101T00:00:00Z
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
Abstract: 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.
Description: Date of Acceptance: 22/04/2015
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.

Data underpinning : Bayesian Spatial NBDA for diffusion data with homebase coordinates
http://hdl.handle.net/10023/6713
20150101T00:00:00Z
Nightingale, Glenna Faith
Laland, Kevin Neville
Nightingale, Peter
Hoppitt, William

The relation between variation in size of the primordial follicle pool and age at natural menopause
http://hdl.handle.net/10023/6589
Abstract: 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.
Description: Date of Acceptance: 20/04/2015
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
Abstract: 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.
Description: 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. Date of Acceptance: 07/04/2015
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
Abstract: 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.
Description: Date of Acceptance: 20/02/2015
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
Abstract: 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.
Description: 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
Abstract: 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.
Description: 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
Abstract: 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.
Description: Date of Acceptance: 22/04/2013
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
Abstract: 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.
Description: 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
Abstract: 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.
Description: 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
Abstract: 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.
Description: Date of Acceptance: 03/10/2011
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
Abstract: 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.
Description: 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
Abstract: 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
Abstract: Helfgott proved that there exists a δ>0 such that if S is a symmetric generating subset of SL(2,p) containing 1 then either S3=SL(2,p) or S3 ≥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
Abstract: We classify the maximal subsemigroups of the semigroup ΩΩ of all mappings on an infinite set Ω that contain one of the following groups: the symmetric group on Ω, the setwise stabilizer of a 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.
D. Mitchell, J.
Péresse, Y.
We classify the maximal subsemigroups of the semigroup ΩΩ of all mappings on an infinite set Ω that contain one of the following groups: the symmetric group on Ω, the setwise stabilizer of a 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
Abstract: 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
Abstract: 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.
Description: 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
Abstract: Memoryless computation is a new technique to compute any function of a set of registers by updating one register at a time while using no memory. Its aim is to emulate how computations are performed in modern cores, since they typically involve updates of single registers. The memoryless computation model can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we consider how efficiently permutations can be computed without memory. We determine the minimum number of basic updates required to compute any permutation, or any even permutation. The small number of required instructions shows that very small instruction sets could be encoded on cores to perform memoryless computation. We then start looking at a possible compromise between the size of the instruction set and the length of the resulting programs. We consider updates only involving a limited number of registers. In particular, we show that binary instructions are not enough to compute all permutations without memory when the alphabet size is even. These results, though expressed as properties of special generating sets of the symmetric or alternating groups, provide guidelines on the implementation of memoryless computation.
Description: Funding: UK Engineering and Physical Sciences Research Council (EP/K033956/1)
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
Abstract: Memoryless computation is a novel means of computing any function of a set of registers by updating one register at a time while using no memory. We aim to emulate how computations are performed on modern cores, since they typically involve updates of single registers. The computation model of memoryless computation can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we view registers as elements of a finite field and we compute linear permutations without memory. We first determine the maximum complexity of a linear function when only linear instructions are allowed. We also determine which linear functions are hardest to compute when the field in question is the binary field and the number of registers is even. Secondly, we investigate some matrix groups, thus showing that the special linear group is internally computable but not fast. Thirdly, we determine the smallest set of instructions required to generate the special and general linear groups. These results are important for memoryless computation, for they show that linear functions can be computed very fast or that very few instructions are needed to compute any linear function. They thus indicate new advantages of using memoryless computation.
Description: Funding: UK Engineering and Physical Sciences Research Council award EP/K033956/1
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
Abstract: We study the probability of generating a finite simple group, together with its generalisation PG,socG(d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ socG. We prove that PG,socG(2) ⩾ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for PG,socG(3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.
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
Abstract: We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,YG) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. Klin. Moreover, we give an upper bound n1/2+ε for the minimum size of the edges in such a hypergraph. This is essentially best possible.
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
Abstract: 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.
Description: 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
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.

Space exploration using parallel orbits : a study in parallel symbolic computing
http://hdl.handle.net/10023/5303
Abstract: Orbit enumerations represent an important class of mathematical algorithms which is widely used in computational discrete mathematics. In this paper, we present a new 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
Abstract: We investigate some product structures in R. Thompson's group $ V$, primarily by studying the topological dynamics associated with $ V$'s action on the Cantor set C. We draw attention to the class D(V,C) of groups which have embeddings as demonstrative subgroups of V whose class can be used to assist in forming various products. Note that D(V,C) contains all finite groups, the free group on two generators, and Q/Z, and is closed under passing to subgroups and under taking direct products of any member by any finite member. If G≤V and H ∈ D(V,C), then G~H embeds into V. Finally, if G, H ∈ D(V,C), then G*H embeds in V. Using a dynamical approach, we also show the perhaps surprising result that Z2 * Z does not embed in V, even though V has many embedded copies of Z2 and has many embedded copies of free products of various pairs of its subgroups.
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
Abstract: 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
Abstract: 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
Abstract: 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.
Description: 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
Abstract: 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
Abstract: 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
Abstract: A group L is primitive monolithic if L has a unique minimal normal subgroup, N , and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L , given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 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.
Description: 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
Abstract: 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.
Description: 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
Abstract: 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
Abstract: 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
Abstract: 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.
Description: 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
Abstract: 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
Abstract: 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
Abstract: 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
Abstract: 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
Abstract: 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
Abstract: 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
Abstract: Let ΩΩ be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of ΩΩ, then we write U≈V if there exists a finite subset F of ΩΩ such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in ΩΩ is the least cardinality of a subset A of ΩΩ such that the union of U and A generates ΩΩ. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω. The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in ΩΩ where d is the minimum cardinality of a dominating family for NN. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in ΩΩ are 0, 1, 2, and d. We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2ℵ0.
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
Abstract: Given an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area.
20130501T00:00:00Z
Dolinka, I
Ruskuc, Nik
Given an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area.

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

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

Interfacing Coq + SSReflect with GAP
http://hdl.handle.net/10023/3175
Abstract: 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.
Description: 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
Abstract: For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.
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
Abstract: The Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S n T under the natural actions of T on S via right and left multiplication. This partitions the complement S nT into Trelative H classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schützenberger group. If the Rees index ΙS n TΙ is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows how generating sets for S may be used to obtain generating sets for T and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of groupembeddable semigroups).
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
Abstract: 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
Abstract: 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
Abstract: 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
Abstract: 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
Abstract: The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classied by listing them as a set of explicit families.
20110201T00:00:00Z
Atkinson, M.D.
Ruskuc, Nik
Smith, R
The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classied by listing them as a set of explicit families.

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

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

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

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

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

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

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

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

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

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

Maximal subgroups of free idempotentgenerated semigroups over the full transformation monoid
http://hdl.handle.net/10023/2134
Abstract: Let Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying im (e) = r ≤ n − 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr.
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
Abstract: Given a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.
20111101T00:00:00Z
Gray, R
Ruskuc, Nik
Given a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.

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

On maximal subgroups of free idempotent generated semigroups
http://hdl.handle.net/10023/2128
Abstract: We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister–Schreier type rewriting.
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
Abstract: 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.
Description: 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
Abstract: 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.
Description: "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
Abstract: 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
Abstract: A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem.
20110501T00:00:00Z
Albert, M.H.
Linton, Stephen Alexander
Ruskuc, Nik
Vatter, V
Waton, S
A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem.

Presentations of inverse semigroups, their kernels and extensions
http://hdl.handle.net/10023/1998
Abstract: Let S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.
Description: "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
Abstract: An interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible.
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
Abstract: 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.
Description: 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
Abstract: 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
Abstract: The existence of a primitive free (normal) cubic x(3) ax(2) + cx b over a finite field F with arbitrary specified values of a (not equal 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.
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
Abstract: 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
Abstract: This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.
20060701T00:00:00Z
Cain, AJ
Robertson, Edmund Frederick
Ruskuc, Nikola
This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.

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

On defining groups efficiently without using inverses
http://hdl.handle.net/10023/1442
Abstract: Let G be a group, and let <A \ R> be a finite group presentation for G with \R\ greater than or equal to \A\. Then there exists a, finite semigroup, presentation <B \ Q> for G such that \Q\  \B\ = \R\  \A\. Moreover, B is either the same generating set or else it contains one additional generator.
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
Abstract: 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).
Description: 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
G. Krivokhata, A.
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).