Mathematical Sciences- Journal Articles

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When is a finite ring a field?
(Irish Mathematical Society, 1996-12) MacHale, Desmond
Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise
(Springer, 2000-04) Lindsay, J. Martin; Wills, Stephen J.
Quantum stochastic differential equations of the form govern stochastic flows on a C *-algebra ?. We analyse this class of equation in which the matrix of fundamental quantum stochastic integrators Λ is infinite dimensional, and the coefficient matrix θ consists of bounded linear operators on ?. Weak and strong forms of solution are distinguished, and a range of regularity conditions on the mapping matrix θ are considered, for investigating existence and uniqueness of solutions. Necessary and sufficient conditions on θ are determined, for any sufficiently regular weak solution k to be completely positive. The further conditions on θ for k to also be a contraction process are found; and when ? is a von Neumann algebra and the components of θ are normal, these in turn imply sufficient regularity for the equation to have a strong solution. Weakly multiplicative and *-homomorphic solutions and their generators are also investigated. We then consider the right and left Hudson-Parthasarathy equations: in which F is a matrix of bounded Hilbert space operators. Their solutions are interchanged by a time reversal operation on processes. The analysis of quantum stochastic flows is applied to obtain characterisations of the generators F of contraction, isometry and coisometry processes. In particular weak solutions that are contraction processes are shown to have bounded generators, and to be necessarily strong solutions.
Mild solutions of quantum stochastic differential equations
(Institute of Mathematical Statistics and Bernoulli Society, 2000-11) Fagnola, Franco; Wills, Stephen J.
We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the correspondence between our definition and similar ideas in the theory of classical stochastic differential equations. The conditions that a process must satisfy in order for it to be a mild solution are shown to be strictly weaker than those for it to be a strong solution by exhibiting a class of coefficient matrices for which a mild unitary solution can be found, but for which no strong solution exists.
Markovian cocycles on operator algebras adapted to a Fock filtration
(Elsevier, 2000-12-20) Lindsay, J. Martin; Wills, Stephen J.
The introduction of a Feller-type condition allows the study of Markovian, cocycles adapted to a Fock filtration to be extended from von Neumann algebras to C*-algebras. It is shown that every such cocycle which, along with its adjoint cocycle, is pointwise strongly continuous and whose associated semigroups are norm continuous, weakly satisfies a quantum stochastic differential equation (QSDE). The matrix of coefficients of this equation may thereby be considered as the generator of the cocycle. The QSDE is satisfied strongly in any of the following cases: when the cocycle is completely positive and contractive, or the driving quantum noise is finite dimensional, or the C*-algebra is finite dimensional and the cocycle generator is bounded. Applying the algebra results to Fock-adapted Markovian cocycles on a Hilbert space we obtain similar characterisations. In particular a contraction cocycle whose Markov semigroup is norm continuous strongly satisfies a QSDE. A representation of cocycles in terms of a family of associated semigroups is central to the present analysis, providing the connection with QSDEs through a parallel work (2000, J. M. Lindsay and S. J. Wills, Probab. Theory Related Fields 116, 505–543).
A stochastic Stinespring theorem
(Springer Nature Ltd., 2001-01) Goswami, Debashish; Lindsay, J. Martin; Wills, Stephen J.; Indian National Board for Higher Mathematics; Department of Atomic Energy, Government of India; Indian National Science Academy; Royal Society
Completely positive Markovian cocycles on a von Neumann algebra, adapted to a Fock filtration, are realised as conjugations of ∗-homomorphic Markovian cocycles. The conjugating processes are affiliated to the algebra, and are governed by quantum stochastic differential equations whose coefficients evolve according to the ∗-homomorphic process. Some perturbation theory for quantum stochastic flows is developed in order to achieve the above Stinespring decomposition.
Homomorphic Feller cocycles on a C*-algebra
(John Wiley & Sons, Inc., 2003-01) Lindsay, J. Martin; Wills, Stephen J.; Lloyd’s Tercentenary Research Foundation
When a Fock-adapted Feller cocycle on a C*-algebra is regular, completely positive and contractive, it possesses a stochastic generator that is necessarily completely bounded. Necessary and sufficient conditions are given, in the form of a sequence of identities, for a completely bounded map to generate a weakly multiplicative cocycle. These are derived from a product formula for iterated quantum stochastic integrals. Under two alternative assumptions, one of which covers all previously considered cases, the first identity in the sequence is shown to imply the rest.
Dilation of Markovian cocycles on a von Neumann algebra
(Mathematical Sciences Publishers (MSP), 2003-10-01) Goswami, Debashish; Lindsay, J. M.; Sinha, Kalyan B.; Wills, Stephen J.
We consider normal Markovian cocycles on a von Neumann algebra which are adapted to a Fock filtration. Every such cocycle k which is Markov-regular and consists of completely positive contractions is realised as a conditioned ∗-homomorphic cocycle. This amounts to a stochastic generalisation of a recent dilation result for norm-continuous normal completely positive contraction semigroups. To achieve this stochastic dilation we use the fact that k is governed by a quantum stochastic differential equation whose coefficient matrix has a specific structure, and extend a technique for obtaining stochastic flow generators from Markov semigroup generators, to the context of cocycles. Number/exchange-free dilatability is seen to be related to locality in the case where the cocycle is a Markovian semigroup. In the same spirit unitary dilations of Markov-regular contraction cocycles on a Hilbert space are also described. The paper ends with a discussion of connections with measure-valued diffusion.
Construction of some quantum stochastic operator cocycles by the semigroup method
(Indian Academy of Sciences; Springer, 2006-11) Lindsay, J. Martin; Wills, Stephen J.; European Commission
A new method for the construction of Fock-adapted quantum stochastic operator cocycles is outlined, and its use is illustrated by application to a number of examples arising in physics and probability. The construction uses the Trotter-Kato theorem and a recent characterisation of such cocycles in terms of an associated family of contraction semigroups.
Excitability in a quantum dot semiconductor laser with optical injection
(American Physical Society, 2007) Goulding, David; Hegarty, Stephen P.; Rasskazov, Oleg; Melnik, Sergey; Hartnett, Mark C.; Greene, G.; McInerney, John G.; Rachinskii, Dmitrii; Huyet, Guillaume
We experimentally analyze the dynamics of a quantum dot semiconductor laser operating under optical injection. We observe the appearance of single- and double-pulse excitability at one boundary of the locking region. Theoretical considerations show that these pulses are related to a saddle-node bifurcation on a limit cycle as in the Adler equation. The double pulses are related to a period-doubling bifurcation and occur on the same homoclinic curve as the single pulses.
On the generators of quantum stochastic operator cocycles
(Polymat, 2007-01) Wills, Stephen J.
The stochastic generators of Markov-regular operator cocycles on symmetric Fock space are studied in a variety of cases: positive cocycles, projection cocycles, and partially isometric cocycles. Moreover a class of transformations of positive contraction cocycles is exhibited which leads to a polar decomposition result.
Bounds on the distribution of the number of gaps when circles and lines are covered by fragments: theory and practical application to genomic and metagenomic projects
(BioMed Central, 2007-03-02) Moriarty, John; Marchesi, Julian R.; Metcalfe, Anthony; Science Foundation Ireland; Irish Government
Background: The question of how a circle or line segment becomes covered when random arcs are marked off has arisen repeatedly in bioinformatics. The number of uncovered gaps is of particular interest. Approximate distributions for the number of gaps have been given in the literature, one motivation being ease of computation. Error bounds for these approximate distributions have not been given. Results: We give bounds on the probability distribution of the number of gaps when a circle is covered by fragments of fixed size. The absolute error in the approximation is typically on the order of 0.1% at 10× coverage depth. The method can be applied to coverage problems on the interval, including edge effects, and applications are given to metagenomic libraries and shotgun sequencing.
Quantum stochastic operator cocycles via associated semigroups
(Cambridge University Press, 2007-05-01) Lindsay, J. Martin; Wills, Stephen J.
A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation.
Prevalence of overweight and obesity on the island of Ireland: results from the North South Survey of Children's Height, Weight and Body Mass Index, 2002
(BioMed Central Ltd., 2007-07-31) Whelton, Helen; Harrington, Janas M.; Crowley, Evelyn; Kelleher, Virginia; Cronin, Michael S.; Perry, Ivan J.
Background: Childhood obesity is emerging as a major public health problem in developed and developing countries worldwide. The aim of this survey was to establish baseline data on the prevalence and correlates of overweight and obesity among children and adolescents in the Republic of Ireland (RoI) and Northern Ireland (NI). Methods: The heights and weights of 19,617 school-going children and adolescents aged between 4 and 16 years in NI and RoI were measured using standardised and calibrated scales and measures. The participants were a representative cross-sectional sample of children randomly selected on the basis of age, gender and geographical location of the school attended. Overweight and obesity were classified according to standard IOTF criteria. Results: Males were taller than females, children in RoI were taller than those in NI and the more affluent were taller than the less well off. The overall prevalence of overweight and obesity was higher among females than males in both jurisdictions. Overall, almost one in four boys (23% RoI and NI) and over one in four girls (28% RoI, 25% NI) were either overweight or obese. In RoI, the highest prevalence of overweight was among 13 year old girls (32%) and obesity among 7 year old girls (11%). In NI the highest prevalence of overweight and obesity were found among 11 and 8 year old girls respectively (33% and 13%). Conclusion: These figures confirm the emergence of the obesity epidemic among children in Ireland, a wealthy country with the European Union. The results serve to underpin the urgency of implementing broad intersectoral measures to reduce calorie intake and increase levels of physical activity, particularly among children.
On the deep-water Stokes wave flow
(Oxford University Press, 2008-01-01) Henry, David; Science Foundation Ireland
We prove a new result detailing the monotonicity of the horizontal velocity component of deep-water Stokes waves along streamlines.
Development of multiple strain competitive index assays for Listeria monocytogenes using pIMC; a new site-specific integrative vector
(BioMed Central, 2008-06) Monk, Ian R.; Casey, Pat G.; Cronin, Michael S.; Gahan, Cormac G.; Hill, Colin; Irish Research Council for Science, Engineering and Technology
The foodborne, gram-positive pathogen, Listeria monocytogenes, is capable of causing lethal infections in compromised individuals. In the post genomic era of L. monocytogenes research, techniques are required to identify and validate genes involved in the pathogenicity and environmental biology of the organism. The aim here was to develop a widely applicable method to tag L. monocytogenes strains, with a particular emphasis on the development of multiple strain competitive index assays.
E-semigroups subordinate to CCR flows
(Serial Publications, 2010-01-12) Wills, Stephen J.
The subordinate E-semigroups of a fixed E-semigroup α are in one-to-one correspondence with local projection-valued cocycles of α. For the CCR flow we characterise these cocycles in terms of their stochastic generators, that is, in terms of the coefficient driving the quantum stochastic differential equation of Hudson-Parthasarathy type that such cocycles necessarily satisfy. In addition various equivalence relations and order-type relations on E-semigroups are considered, and shown to work especially well in the case of those semigroups subordinate to the CCR flows by exploiting our characterisation.
Living longer and feeling better: healthy lifestyle, self-rated health, obesity and depression in Ireland
(Oxford University Press, 2010-02-01) Harrington, Janas M.; Perry, Ivan J.; Lutomski, Jennifer E.; Fitzgerald, Anthony P.; Shiely, Frances; McGee, Hannah; Barry, Margaret M.; Van Lente, Eric; Morgan, Karen; Shelley, Emer; Department of Health and Children, Ireland
Background: The combination of four protective lifestyle behaviours (being physically active, a non-smoker, a moderate alcohol consumer and having adequate fruit and vegetable intake) has been estimated to increase life expectancy by 14 years. However, the effect of adopting these lifestyle behaviours on general health, obesity and mental health is less defined. We examined the combined effect of these behaviours on self-rated health, overweight/obesity and depression. Methods: Using data from the Survey of Lifestyle Attitudes and Nutrition (SLÁN) 2007 (), a protective lifestyle behaviour (PLB) score was constructed for 10 364 men and women (>18 years), and representative of the Republic of Ireland adult population (response rate 62%). Respondents scored a maximum of four points, one point each for being physically active, consuming five or more fruit and vegetable servings daily, a non-smoker and a moderate drinker. Results: One-fifth of respondents (20%) adopted four PLBs, 35% adopted three, 29% two, 13% one and 2% adopted none. Compared to those with zero PLBs, those with four were seven times more likely to rate their general health as excellent/very good [OR 6.8 95% CI (3.64–12.82)] and four times more likely to have better mental health [OR 4.4 95% CI (2.34–8.22)]. Conclusions: Adoption of core protective lifestyle factors known to increase life expectancy is associated with positive self-rated health, healthier weight and better mental health. These lifestyles have the potential to add quality and quantity to life.
Emerging meso- and macroscales from synchronization of adaptive networks
(American Physical Society, 2011) Gutierrez, R.; Amann, Andreas; Assenza, S.; Gomez-Gardenes, J.; Latora, V.; Boccaletti, S.; Science Foundation Ireland; Ministerio de Ciencia e Innovación
We consider a set of interacting phase oscillators, with a coupling between synchronized nodes adaptively reinforced, and the constraint of a limited resource for a node to establish connections with the other units of the network. We show that such a competitive mechanism leads to the emergence of a rich modular structure underlying cluster synchronization, and to a scale-free distribution for the connection strengths of the units.
The impact of a workplace catering initiative on dietary intakes of salt and other nutrients: a pilot study
(Cambridge University Press, 2011-01) Geaney, Fiona; Harrington, Janas M.; Fitzgerald, Anthony P.; Perry, Ivan J.; Health Research Board
Owing to modern lifestyles, individuals are dependent on out-of-home eating. The catering sector can have a pivotal role in influencing our food choices. The objective of the present study was to examine the impact of a structured catering initiative on food choices in a public sector workplace setting.
Self-management education for cystic fibrosis
(John Wiley & Sons, Ltd., 2011-07) Savage, Eileen; Beirne, Paul V.; Ní Chróinín, Muireann; Duff, Alistair J.; Fitzgerald, Anthony P.; Farrell, Dawn; Health Research Board
Self-management education may help patients with cystic fibrosis and their families to choose, monitor and adjust treatment requirements for their illness, and also to manage the effects of illness on their lives. Although self-management education interventions have been developed for cystic fibrosis, no previous systematic review of the evidence of effectiveness of these interventions has been conducted.

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