Automatic Qualitative Analysis of Ordinary Differential Equations Using Piecewise Linear Approximations

This paper explores automating the qualitative analysis of physical systems. It describes a program, called PLR, that takes parameterized ordinary differential equations as input and produces a qualitative description of the solutions for all initial values. PLR approximates intractable nonlinear systems with piecewise linear ones, analyzes the approximations, and draws conclusions about the original systems. It chooses approximations that are accurate enough to reproduce the essential properties of their nonlinear prototypes, yet simple enough to be analyzed completely and efficiently. It derives additional properties, such as boundedness or periodicity, by theoretical methods. I demonstrate PLR on several common nonlinear systems and on published examples from mechanical engineering.

Il Sistema di Revenue Management strumento per lo sviluppo turistico sostenibile delle destinazioni

Businesses interact constantly with the environment, realizing several and heterogeneous exchanges. Organizations can be considered a system of different interests, frequently conflicting and the satisfaction of different stakeholders is a condition of success and survival. National and international literature attempts to explain the complex connection between companies and environment. In particular, the Stakeholder Theory considers crucial for businesses the identification of different stakeholders and their involvement in decision-making process. In this context, profit can not be considered the only purpose of companies existence and business aims become more numerous and different. The Stakeholder Theory is often utilized as framework for tourism studies, in particular in Sustainable Tourism Development research. In fact, authors consider sustainable the tourism development able to satisfy interests of different stakeholders, traditionally identified as local community and government, businesses, tourists and natural environment. Tourism businesses have to guarantee the optimal use of natural resources, the respect of socio-cultural tradition of local community and the creation of socio-economic benefits for all stakeholders in destinations. An obstacle to sustainable tourism development that characterizes a number of destinations worldwide is tourism demand seasonality. In fact, its negative impact on the environment, economy and communities may be highly significant. Pollution, difficulties in the use of public services, stress for residents, seasonal incomes, are all examples of the negative effects of seasonality. According to the World Tourism Organization (2004) the limitation of seasonality can favour the sustainability of tourism. Literature suggests private and public strategies to minimize the negative effects of tourism seasonality, as diversification of tourism products, identification of new market segments, launching events, application of public instruments like eco-taxes and use of differential pricing policies. Revenue Management is a managerial system based on differential pricing and able to affect price sensitive tourists. This research attempts to verify if Revenue Management, created to maximize profits in tourism companies, can also mitigate the seasonality of tourism demand, producing benefits for different stakeholders of destinations and contributing to Sustainable Tourism Development. In particular, the study attempts to answer the following research questions: 1) Can Revenue Management control the flow of tourist demand? 2) Can Revenue Management limit seasonality, producing benefits for different stakeholders of a destination? 3) Can Revenue Management favor the development of Sustainable Tourism? The literature review on Stakeholder Theory, Sustainable Tourism Development, tourism seasonality and Revenue Management forms the foundation of the research, based on a case study approach looking at a significant destination located in the Southern coast of Sardinia, Italy. A deductive methodology was applied and qualitative and quantitative methods were utilized. This study shows that Revenue Management has the potential to limit tourism seasonality, to mitigate negative impacts occurring from tourism activities, producing benefits for local community and to contribute to Sustainable Tourism Development.

Public service employment and the public-private wage differential in British regions

Thomas, Dennis, Henley, Andrew, 'Public service employment and the public-private wage differential in British regions', Regional Studies (2001) 35(3) pp.229-240 RAE2008

Singular perturbation of quantum stochastic differential equations with coupling through an oscillator mode

Gough, John; Van Handel, R., (2007) 'Singular perturbation of quantum stochastic differential equations with coupling through an oscillator mode', Journal of Statistical Physics 127(3) pp.575-607 RAE2008

Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

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.

Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

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.

Variations in total and differential milk somatic cell counts and plasmin levels and their role in proteolysis and quality of milk and cheese

Increased plasmin and plasminogen levels and elevated somatic cell counts (SCC) and polymorphonuclear leucocyte levels (PMN) were evident in late lactation milk. Compositional changes in these milks were associated with increased SCC. The quality of late lactation milks was related to nutritional status of herds, with milks from herds on a high plane of nutrition having composition and clotting properties similar to, or superior to, early-mid lactation milks. Nutritionally-deficient cows had elevated numbers of polymorphonuclear leucocytes (PMNs) in their milk, elevated plasmin levels and increased overall proteolytic activity. The dominant effect of plasmin on proteolysis in milks of low SCC was established. When present in elevated numbers, somatic cells and PMNs in particular had a more significant influence on the proteolysis of both raw and pasteurised milks than plasmin. PMN protease action on the caseins showed proteolysis products of two specific enzymes, cathepsin B and elastase, which were also shown in high SCC milk. Crude extracts of somatic cells had a high specificity on αs1-casein. Cheeses made from late lactation milks had increased breakdown of αs1-casein, suggestive of the action of somatic cell proteinases, which may be linked to textural defects in cheese. Late lactation cheeses also showed decreased production of small peptides and amino acids, the reason for which is unknown. Plasmin, which is elevated in activity in late lactation milk, accelerated the ripening of Gouda-type cheese, but was not associated with defects of texture or flavour. The retention of somatic cell enzymes in cheese curd was confirmed, and a potential role in production of bitter peptides identified. Cheeses made from milks containing high levels of PMNs had accelerated αs1-casein breakdown relative to cheeses made from low PMN milk of the same total SCC, consistent with the demonstrated action of PMN proteinases. The two types of cheese were determined significantly different by blind triangle testing.

Differential and numerical models of hysteretic systems with stochastic and deterministic inputs

Many deterministic models with hysteresis have been developed in the areas of economics, finance, terrestrial hydrology and biology. These models lack any stochastic element which can often have a strong effect in these areas. In this work stochastically driven closed loop systems with hysteresis type memory are studied. This type of system is presented as a possible stochastic counterpart to deterministic models in the areas of economics, finance, terrestrial hydrology and biology. Some price dynamics models are presented as a motivation for the development of this type of model. Numerical schemes for solving this class of stochastic differential equation are developed in order to examine the prototype models presented. As a means of further testing the developed numerical schemes, numerical examination is made of the behaviour near equilibrium of coupled ordinary differential equations where the time derivative of the Preisach operator is included in one of the equations. A model of two phenotype bacteria is also presented. This model is examined to explore memory effects and related hysteresis effects in the area of biology. The memory effects found in this model are similar to that found in the non-ideal relay. This non-ideal relay type behaviour is used to model a colony of bacteria with multiple switching thresholds. This model contains a Preisach type memory with a variable Preisach weight function. Shown numerically for this multi-threshold model is a pattern formation for the distribution of the phenotypes among the available thresholds.

Financial modelling with 2-EPT probability density functions

The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.

An exploration of the positive and negative relationships associated with the development of asthma and atopic disorders in primary school children in Cork

Childhood asthma, allergic rhinitis and eczema are complex heterogenic chronic inflammatory allergic disorders which constitute a major burden to children, their families. The prevalence of childhood allergic disorders is increasing worldwide and merely rudimentary understanding exists regarding causality, or the influence of the environment on disease expression. Phase Three of the International Study of Asthma and Allergy in Childhood (ISAAC) reported that Irish adolescents had the 4th highest eczema and rhinoconjunctivitis prevalence and 3rd highest asthma prevalence in the world. There are no ISAAC data pertaining to young Irish children. In 2002, Sturley reported a high prevalence of current asthma in Cork primary school children aged 6-9 years. This thesis comprises of three cross-sectional studies which examined the prevalence of and associations with childhood allergy and a quasi-retrospective cohort study which observed the natural history of allergy from 6-9 until 11-13 years. Although not part of ISAAC, data was attained by parentally completed ISAAC-based questionnaires, using the ISAAC protocol. The prevalence, natural history and risk factors of childhood allergy in Ireland, as described in this thesis, echo those in worldwide allergy research. The variations of prevalence in different populations worldwide and the recurring themes of associations between childhood allergy and microbial exposures, from farming environments and/or gastrointestinal infections, as shown in this thesis, strengthen the mounting evidence that microbial exposure on GALT may hold the key to the mechanisms of allergy development. In this regard, probiotics may be an area of particular interest in allergy modification. Although their effects in relation to allergy, have been investigated now for several years, our knowledge of their diversity, complex functions and interactions with gut microflora, remain rudimentary. Birth cohort studies which include genomic and microbiomic research are recommended in order to examine the underlying mechanisms and the natural course of allergic diseases.

Identification of novel innate immune mechanisms regulating inflammation in the gastrointestinal tract

The Gastro-Intestinal (GI) tract is a unique region in the body. Our innate immune system retains a fine homeostatic balance between avoiding inappropriate inflammatory responses against the myriad commensal microbes residing in the gut while also remaining active enough to prevent invasive pathogenic attack. The intestinal epithelium represents the frontline of this interface. It has long been known to act as a physical barrier preventing the lumenal bacteria of the gastro-intestinal tract from activating an inflammatory immune response in the immune cells of the underlying mucosa. However, in recent years, an appreciation has grown surrounding the role played by the intestinal epithelium in regulating innate immune responses, both in the prevention of infection and in maintaining a homeostatic environment through modulation of innate immune signalling systems. The aim of this thesis was to identify novel innate immune mechanisms regulating inflammation in the GI tract. To achieve this aim, we chose several aspects of regulatory mechanisms utilised in this region by the innate immune system. We identified several commensal strains of bacteria expressing proteins containing signalling domains used by Pattern Recognition Receptors (PRRs) of the innate immune system. Three such bacterial proteins were studied for their potentially subversive roles in host innate immune signalling as a means of regulating homeostasis in the GI tract. We also examined differential responses to PRR activation depending on their sub-cellular localisation. This was investigated based on reports that apical Toll-Like Receptor (TLR) 9 activation resulted in abrogation of inflammatory responses mediated by other TLRs in Intestinal Epithelial Cells (IECs) such as basolateral TLR4 activation. Using the well-studied invasive intra-cellular pathogen Listeria monocytogenes as a model for infection, we also used a PRR siRNA library screening technique to identify novel PRRs used by IECs in both inhibition and activation of inflammatory responses. Many of the PRRs identified in this screen were previously believed not to be expressed in IECs. Furthermore, the same study has led to the identification of the previously uncharacterised TLR10 as a functional inflammatory receptor of IECs. Further analysis revealed a similar role in macrophages where it was shown to respond to intracellular and motile pathogens such as Gram-positive L.monocytogenes and Gram negative Salmonella typhimurium. TLR10 expression in IECs was predominantly intracellular. This is likely in order to avoid inappropriate inflammatory activation through the recognition of commensal microbial antigens on the apical cell surface of IECs. Moreover, these results have revealed a more complex network of innate immune signalling mechanisms involved in both activating and inhibiting inflammatory responses in IECs than was previously believed. This contribution to our understanding of innate immune regulation in this region has several direct and indirect benefits. The identification of several novel PRRs involved in activating and inhibiting inflammation in the GI tract may be used as novel therapeutic targets in the treatment of disease; both for inducing tolerance and reducing inflammation, or indeed, as targets for adjuvant activation in the development of oral vaccines against pathogenic attack.

Novel insights into the expression and function of p38δ MAPK in oesophageal squamous cell carcinoma

Oesophageal cancer is an aggressive malignancy which is resistant to conventional therapy and has a poor prognosis. A greater understanding of the underlying molecular biology of oesophageal cancer and the identification of novel targets is necessary for the future treatment of this disease. This thesis focuses specifically on the ill-defined and understudied p38δ mitogen-activated protein kinase (MAPK) and its function(s) in oesophageal squamous cell carcinoma (OESCC). In contrast to the three other p38 isoforms (p38α, -β and –γ which have to-date been relatively well-studied), p38δ MAPK signalling is poorly understood. Thus, this research elucidates some of the role(s) played by p38δ MAPK in cancer progression. This work outlines how loss of p38δ MAPK expression confers greater tumourigenicity in oesophageal cancer. Restoration of p38δ MAPK expression, however, has anti-proliferative and anti-migratory effects and decreases OESCC capacity for anchorageindependent growth. Using a novel application of an enzyme-substrate fusion approach, the effect of phosphorylated p38δ (p-p38δ) MAPK expression is also considered. The work goes onto describe the effect(s) of p38δ MAPK status on the chemosensitivity of OESCC to conventional cisplatin and 5-fluorouracil (CF) versus the effectiveness of doxorubicin, cisplatin and 5-fluorouracil (ACF). ACF treatment of p38δ MAPK-negative OESCC results in decreased proliferation, migration and recovery, and increased apoptosis when compared with CF treatment. This thesis examines the potential mechanisms by which p38δ MAPK expression is lost in OESCC and identifies epigenetic regulation as the probable cause of differential p38δ MAPK expression. Also analysed is the role p38δ MAPK and p-p38δ MAPK play in the cell cycle. In summary, this research identifies p38δ MAPK as a possible molecular target and a potential predictor of response to chemotherapy in OESCC patients.

Differential inducibility of HLA class I and II antigens by r-IFN gamma in type III bare lymphocyte syndrome.

Case Reports

Interstitial insertion of varying amounts of ABL-containing genetic material into chromosome 22 in Ph-negative CML.

We studied the cells from three selected patients with Ph-chromosome-negative chronic myeloid leukemia (CML) by Southern blotting, polymerase chain reaction, and in situ hybridization of informative probes to metaphase chromosomes. All three patients had rearrangement of M-BCR sequences in the BCR gene and expression of one or other of the mRNA species characteristic of Ph-positive CML. Leukemic metaphases studied after trypsin-Giemsa banding were indistinguishable from normal. The ABL probe localized both to chromosome 9 and 22 in each case. A probe containing 3' M-BCR sequences localized only to chromosome 22, and not to chromosome 9 as would be expected in Ph-positive CML. Two new probes that recognize different polymorphic regions distal to the ABL gene on chromosome 9 in normal subjects localized exclusively to chromosome 9 in two patients and to both chromosomes 9 and 22 in one patient. These results show that Ph-negative CML with BCR rearrangement is associated with insertion of a variable quantity of chromosome 9 derived material into chromosome 22q11; there is no evidence for reciprocal translocation of material from chromosome 22 to chromosome 9.

Design Optimization and Security For Communication Networks

In this work we introduce a new mathematical tool for optimization of routes, topology design, and energy efficiency in wireless sensor networks. We introduce a vector field formulation that models communication in the network, and routing is performed in the direction of this vector field at every location of the network. The magnitude of the vector field at every location represents the density of amount of data that is being transited through that location. We define the total communication cost in the network as the integral of a quadratic form of the vector field over the network area. With the above formulation, we introduce a mathematical machinery based on partial differential equations very similar to the Maxwell's equations in electrostatic theory. We show that in order to minimize the cost, the routes should be found based on the solution of these partial differential equations. In our formulation, the sensors are sources of information, and they are similar to the positive charges in electrostatics, the destinations are sinks of information and they are similar to negative charges, and the network is similar to a non-homogeneous dielectric media with variable dielectric constant (or permittivity coefficient). In one of the applications of our mathematical model based on the vector fields, we offer a scheme for energy efficient routing. Our routing scheme is based on changing the permittivity coefficient to a higher value in the places of the network where nodes have high residual energy, and setting it to a low value in the places of the network where the nodes do not have much energy left. Our simulations show that our method gives a significant increase in the network life compared to the shortest path and weighted shortest path schemes. Our initial focus is on the case where there is only one destination in the network, and later we extend our approach to the case where there are multiple destinations in the network. In the case of having multiple destinations, we need to partition the network into several areas known as regions of attraction of the destinations. Each destination is responsible for collecting all messages being generated in its region of attraction. The complexity of the optimization problem in this case is how to define regions of attraction for the destinations and how much communication load to assign to each destination to optimize the performance of the network. We use our vector field model to solve the optimization problem for this case. We define a vector field, which is conservative, and hence it can be written as the gradient of a scalar field (also known as a potential field). Then we show that in the optimal assignment of the communication load of the network to the destinations, the value of that potential field should be equal at the locations of all the destinations. Another application of our vector field model is to find the optimal locations of the destinations in the network. We show that the vector field gives the gradient of the cost function with respect to the locations of the destinations. Based on this fact, we suggest an algorithm to be applied during the design phase of a network to relocate the destinations for reducing the communication cost function. The performance of our proposed schemes is confirmed by several examples and simulation experiments. In another part of this work we focus on the notions of responsiveness and conformance of TCP traffic in communication networks. We introduce the notion of responsiveness for TCP aggregates and define it as the degree to which a TCP aggregate reduces its sending rate to the network as a response to packet drops. We define metrics that describe the responsiveness of TCP aggregates, and suggest two methods for determining the values of these quantities. The first method is based on a test in which we drop a few packets from the aggregate intentionally and measure the resulting rate decrease of that aggregate. This kind of test is not robust to multiple simultaneous tests performed at different routers. We make the test robust to multiple simultaneous tests by using ideas from the CDMA approach to multiple access channels in communication theory. Based on this approach, we introduce tests of responsiveness for aggregates, and call it CDMA based Aggregate Perturbation Method (CAPM). We use CAPM to perform congestion control. A distinguishing feature of our congestion control scheme is that it maintains a degree of fairness among different aggregates. In the next step we modify CAPM to offer methods for estimating the proportion of an aggregate of TCP traffic that does not conform to protocol specifications, and hence may belong to a DDoS attack. Our methods work by intentionally perturbing the aggregate by dropping a very small number of packets from it and observing the response of the aggregate. We offer two methods for conformance testing. In the first method, we apply the perturbation tests to SYN packets being sent at the start of the TCP 3-way handshake, and we use the fact that the rate of ACK packets being exchanged in the handshake should follow the rate of perturbations. In the second method, we apply the perturbation tests to the TCP data packets and use the fact that the rate of retransmitted data packets should follow the rate of perturbations. In both methods, we use signature based perturbations, which means packet drops are performed with a rate given by a function of time. We use analogy of our problem with multiple access communication to find signatures. Specifically, we assign orthogonal CDMA based signatures to different routers in a distributed implementation of our methods. As a result of orthogonality, the performance does not degrade because of cross interference made by simultaneously testing routers. We have shown efficacy of our methods through mathematical analysis and extensive simulation experiments.