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.

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

The Geometry of Frequency Squares

Mavron, Vassili; Jungnickel, D.; McDonough, T.P., (2001) 'The Geometry of Frequency Squares', Journal of Combinatorial Theory, Series A 96, pp.376-387 RAE2008

Genome-wide association with bone mass and geometry in the Framingham Heart Study

BACKGROUND:Osteoporosis is characterized by low bone mass and compromised bone structure, heritable traits that contribute to fracture risk. There have been no genome-wide association and linkage studies for these traits using high-density genotyping platforms.METHODS:We used the Affymetrix 100K SNP GeneChip marker set in the Framingham Heart Study (FHS) to examine genetic associations with ten primary quantitative traits: bone mineral density (BMD), calcaneal ultrasound, and geometric indices of the hip. To test associations with multivariable-adjusted residual trait values, we used additive generalized estimating equation (GEE) and family-based association tests (FBAT) models within each sex as well as sexes combined. We evaluated 70,987 autosomal SNPs with genotypic call rates [greater than or equal to]80%, HWE p [greater than or equal to] 0.001, and MAF [greater than or equal to]10% in up to 1141 phenotyped individuals (495 men and 646 women, mean age 62.5 yrs). Variance component linkage analysis was performed using 11,200 markers.RESULTS:Heritability estimates for all bone phenotypes were 30-66%. LOD scores [greater than or equal to]3.0 were found on chromosomes 15 (1.5 LOD confidence interval: 51,336,679-58,934,236 bp) and 22 (35,890,398-48,603,847 bp) for femoral shaft section modulus. The ten primary phenotypes had 12 associations with 100K SNPs in GEE models at p < 0.000001 and 2 associations in FBAT models at p < 0.000001. The 25 most significant p-values for GEE and FBAT were all less than 3.5 x 10-6 and 2.5 x 10-5, respectively. Of the 40 top SNPs with the greatest numbers of significantly associated BMD traits (including femoral neck, trochanter, and lumbar spine), one half to two-thirds were in or near genes that have not previously been studied for osteoporosis. Notably, pleiotropic associations between BMD and bone geometric traits were uncommon. Evidence for association (FBAT or GEE p < 0.05) was observed for several SNPs in candidate genes for osteoporosis, such as rs1801133 in MTHFR; rs1884052 and rs3778099 in ESR1; rs4988300 in LRP5; rs2189480 in VDR; rs2075555 in COLIA1; rs10519297 and rs2008691 in CYP19, as well as SNPs in PPARG (rs10510418 and rs2938392) and ANKH (rs2454873 and rs379016). All GEE, FBAT and linkage results are provided as an open-access results resource at http://www.ncbi.nlm.nih.gov/projects/gap/cgi-bin/study.cgi?id=phs000007.CONCLUSION:The FHS 100K SNP project offers an unbiased genome-wide strategy to identify new candidate loci and to replicate previously suggested candidate genes for osteoporosis.

Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

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.

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

Case Reports

Measurement of the differential cross section for the reaction gamman-->pi- p from deuterium.

We report a measurement of the differential cross section for the gamman-->pi- p process from the CLAS detector at Jefferson Laboratory in Hall B for photon energies between 1.0 and 3.5 GeV and pion center-of-mass (c.m.) angles (thetac.m.) between 50 degrees and 115 degrees. We confirm a previous indication of a broad enhancement around a c.m. energy ([sqrt]s) of 2.1 GeV at thetac.m.=90 degrees in the scaled differential cross section s7dsigma/dt and a rapid falloff in a center-of-mass energy region of about 400 MeV following the enhancement. Our data show an angular dependence of this enhancement as the suggested scaling region is approached for thetac.m. from 70 degrees to 105 degrees.

Differential inhibition of human immunodeficiency virus type 1 in peripheral blood mononuclear cells and TZM-bl cells by endotoxin-mediated chemokine and gamma interferon production.

Bacterial lipopolysaccharide (endotoxin) is a frequent contaminant of biological specimens and is also known to be a potent inducer of beta-chemokines and other soluble factors that inhibit HIV-1 infection in vitro. Though lipopolysaccharide (LPS) has been shown to stimulate the production of soluble HIV-1 inhibitors in cultures of monocyte-derived macrophages, the ability of LPS to induce similar inhibitors in other cell types is poorly characterized. Here we show that LPS exhibits potent anti-HIV activity in phytohemagglutinin-stimulated peripheral blood mononuclear cells (PBMCs) but has no detectable anti-HIV-1 activity in TZM-bl cells. The anti-HIV-1 activity of LPS in PBMCs was strongly associated with the production of beta-chemokines from CD14-positive monocytes. Culture supernatants from LPS-stimulated PBMCs exhibited potent anti-HIV-1 activity when added to TZM-bl cells but, in this case, the antiviral activity appeared to be related to IFN-gamma rather than to beta-chemokines. These observations indicate that LPS stimulates PBMCs to produce a complex array of soluble HIV-1 inhibitors, including beta-chemokines and IFN-gamma, that differentially inhibit HIV-1 depending on the target cell type. The results also highlight the need to use endotoxin-free specimens to avoid artifacts when assessing HIV-1-specific neutralizing antibodies in PBMC-based assays.

Differential mechanisms of morphine antinociceptive tolerance revealed in (beta)arrestin-2 knock-out mice.

Morphine induces antinociception by activating mu opioid receptors (muORs) in spinal and supraspinal regions of the CNS. (Beta)arrestin-2 (beta)arr2), a G-protein-coupled receptor-regulating protein, regulates the muOR in vivo. We have shown previously that mice lacking (beta)arr2 experience enhanced morphine-induced analgesia and do not become tolerant to morphine as determined in the hot-plate test, a paradigm that primarily assesses supraspinal pain responsiveness. To determine the general applicability of the (beta)arr2-muOR interaction in other neuronal systems, we have, in the present study, tested (beta)arr2 knock-out ((beta)arr2-KO) mice using the warm water tail-immersion paradigm, which primarily assesses spinal reflexes to painful thermal stimuli. In this test, the (beta)arr2-KO mice have greater basal nociceptive thresholds and markedly enhanced sensitivity to morphine. Interestingly, however, after a delayed onset, they do ultimately develop morphine tolerance, although to a lesser degree than the wild-type (WT) controls. In the (beta)arr2-KO but not WT mice, morphine tolerance can be completely reversed with a low dose of the classical protein kinase C (PKC) inhibitor chelerythrine. These findings provide in vivo evidence that the muOR is differentially regulated in diverse regions of the CNS. Furthermore, although (beta)arr2 appears to be the most prominent and proximal determinant of muOR desensitization and morphine tolerance, in the absence of this mechanism, the contributions of a PKC-dependent regulatory system become readily apparent.

Overexpression of the cardiac beta(2)-adrenergic receptor and expression of a beta-adrenergic receptor kinase-1 (betaARK1) inhibitor both increase myocardial contractility but have differential effects on susceptibility to ischemic injury.

Cardiac beta(2)-adrenergic receptor (beta(2)AR) overexpression is a potential contractile therapy for heart failure. Cardiac contractility was elevated in mice overexpressing beta(2)ARs (TG4s) with no adverse effects under normal conditions. To assess the consequences of beta(2)AR overexpression during ischemia, perfused hearts from TG4 and wild-type mice were subjected to 20-minute ischemia and 40-minute reperfusion. During ischemia, ATP and pH fell lower in TG4 hearts than wild type. Ischemic injury was greater in TG4 hearts, as indicated by lower postischemic recoveries of contractile function, ATP, and phosphocreatine. Because beta(2)ARs, unlike beta(1)ARs, couple to G(i) as well as G(s), we pretreated mice with the G(i) inhibitor pertussis toxin (PTX). PTX treatment increased basal contractility in TG4 hearts and abolished the contractile resistance to isoproterenol. During ischemia, ATP fell lower in TG4+PTX than in TG4 hearts. Recoveries of contractile function and ATP were lower in TG4+PTX than in TG4 hearts. We also studied mice that overexpressed either betaARK1 (TGbetaARK1) or a betaARK1 inhibitor (TGbetaARKct). Recoveries of function, ATP, and phosphocreatine were higher in TGbetaARK1 hearts than in wild-type hearts. Despite basal contractility being elevated in TGbetaARKct hearts to the same level as that of TG4s, ischemic injury was not increased. In summary, beta(2)AR overexpression increased ischemic injury, whereas betaARK1 overexpression was protective. Ischemic injury in the beta(2)AR overexpressors was exacerbated by PTX treatment, implying that it was G(s) not G(i) activity that enhanced injury. Unlike beta(2)AR overexpression, basal contractility was increased by betaARK1 inhibitor expression without increasing ischemic injury, thus implicating a safer potential therapy for heart failure.