Mathematical Sciences

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On classical and other methods of discriminant analysis and estimation of log-odds
(University College Cork, 1981) Murphy, Brendan J.; Moran, M. A.
For two multinormal populations with equal covariance matrices the likelihood ratio discriminant function, an alternative allocation rule to the sample linear discriminant function when n1 ≠ n2 ,is studied analytically. With the assumption of a known covariance matrix its distribution is derived and the expectation of its actual and apparent error rates evaluated and compared with those of the sample linear discriminant function. This comparison indicates that the likelihood ratio allocation rule is robust to unequal sample sizes. The quadratic discriminant function is studied, its distribution reviewed and evaluation of its probabilities of misclassification discussed. For known covariance matrices the distribution of the sample quadratic discriminant function is derived. When the known covariance matrices are proportional exact expressions for the expectation of its actual and apparent error rates are obtained and evaluated. The effectiveness of the sample linear discriminant function for this case is also considered. Estimation of true log-odds for two multinormal populations with equal or unequal covariance matrices is studied. The estimative, Bayesian predictive and a kernel method are compared by evaluating their biases and mean square errors. Some algebraic expressions for these quantities are derived. With equal covariance matrices the predictive method is preferable. Where it derives this superiority is investigated by considering its performance for various levels of fixed true log-odds. It is also shown that the predictive method is sensitive to n1 ≠ n2. For unequal but proportional covariance matrices the unbiased estimative method is preferred. Product Normal kernel density estimates are used to give a kernel estimator of true log-odds. The effect of correlation in the variables with product kernels is considered. With equal covariance matrices the kernel and parametric estimators are compared by simulation. For moderately correlated variables and large dimension sizes the product kernel method is a good estimator of true log-odds.
Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations
(University College Cork, 1993) Guo, Wen; Stynes, Martin
This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.
Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations
(University College Cork, 1993) Sun, Guangfu; Stynes, Martin
This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.
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).
Generalized metrics and topology in logic programming semantics
(University College Cork, 2001) Hitzler, Pascal; Seda, Tony; Enterprise Ireland; University College Cork
Many fixed-point theorems are essentially topological in nature. Among them are the Banach contraction mapping theorem on metric spaces and the fixed-point theorem for Scott-continuous mappings on complete partial orders. The latter theorem is fundamental in denotational semantics since semantic operators in most programming language paradigms satisfy its requirements. The use of negation in logic programming and non-monotonic reasoning, however, renders some semantic operators to be non-monotonic, hence discontinuous with respect to the Scott topology, and therefore invalidates the standard approach, so that alternative methods have to be sought. In this thesis, we investigate topological methods, including generalized metric fixed-point theorems, and their applicability to the analysis of semantic operators in logic programming and non-monotonic reasoning. In the first part of the thesis, we present weak versions of the Banach contraction mapping theorem for single-valued and multivalued mappings, and investigate relationships between the underlying spaces. In the second part, we apply the obtained results to several semantic paradigms in logic programming and non-monotonic reasoning. These investigations will also lead to a clearer understanding of some of the relationships between these semantic paradigms and of the general topological structures which underly the behaviour of the corresponding semantic operators. We will also obtain some results related to termination properties of normal logic programs, clarify some of the relationships between different semantic approaches in non-monotonic reasoning, and will establish some results concerning the conversion of logic programs into artificial neural networks.
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.
Gröbner basis techniques for certain problems in coding and systems theory
(University College Cork, 2003-10) O'Keeffe, Henry; Fitzpatrick, Patrick
There is much common ground between the areas of coding theory and systems theory. Fitzpatrick has shown that a Göbner basis approach leads to efficient algorithms in the decoding of Reed-Solomon codes and in scalar interpolation and partial realization. This thesis simultaneously generalizes and simplifies that approach and presents applications to discrete-time modeling, multivariable interpolation and list decoding. Gröbner basis theory has come into its own in the context of software and algorithm development. By generalizing the concept of polynomial degree, term orders are provided for multivariable polynomial rings and free modules over polynomial rings. The orders are not, in general, unique and this adds, in no small way, to the power and flexibility of the technique. As well as being generating sets for ideals or modules, Gröbner bases always contain a element which is minimal with respect tot the corresponding term order. Central to this thesis is a general algorithm, valid for any term order, that produces a Gröbner basis for the solution module (or ideal) of elements satisfying a sequence of generalized congruences. These congruences, based on shifts and homomorphisms, are applicable to a wide variety of problems, including key equations and interpolations. At the core of the algorithm is an incremental step. Iterating this step lends a recursive/iterative character to the algorithm. As a consequence, not all of the input to the algorithm need be available from the start and different "paths" can be taken to reach the final solution. The existence of a suitable chain of modules satisfying the criteria of the incremental step is a prerequisite for applying the algorithm.
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.

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