Analysis of a finite volume method for a cross-diffusion model in population dynamics

The main goal of this paper is to propose a convergent finite volume method for a reactionâeuro"diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then the standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time $L^1$ compactness argument that mimics the compactness lemma due to Kruzhkov. The proofs of these results are given in the Appendix.

Nonstandard Finite Difference Schemes with Application to Finance: Option Pricing

2000 Mathematics Subject Classification: 65M06, 65M12.

POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.

Transport properties in nontwist area-preserving maps

Nontwist systems, common in the dynamical descriptions of fluids and plasmas, possess a shearless curve with a concomitant transport barrier that eliminates or reduces chaotic transport, even after its breakdown. In order to investigate the transport properties of nontwist systems, we analyze the barrier escape time and barrier transmissivity for the standard nontwist map, a paradigm of such systems. We interpret the sensitive dependence of these quantities upon map parameters by investigating chaotic orbit stickiness and the associated role played by the dominant crossing of stable and unstable manifolds. (C) 2009 American Institute of Physics. [doi: 10.1063/1.3247349]

POSITIVITY PROPERTIES OF THE FOURIER TRANSFORM AND THE STABILITY OF PERIODIC TRAVELLING-WAVE SOLUTIONS

In this paper we establish a method to obtain the stability of periodic travelling-wave solutions for equations of Korteweg-de Vries-type u(t) + u(p)u(x) - Mu(x) = 0, with M being a general pseudodifferential operator and where p >= 1 is an integer. Our approach uses the theory of totally positive operators, the Poisson summation theorem, and the theory of Jacobi elliptic functions. In particular we obtain the stability of a family of periodic travelling waves solutions for the Benjamin Ono equation. The present technique gives a new way to obtain the existence and stability of cnoidal and dnoidal waves solutions associated with the Korteweg-de Vries and modified Korteweg-de Vries equations, respectively. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.

A technique for preserving pigmentation in some capsalid monogeneans for taxonomic purposes

A technique is described to preserve the pigment found in the bodies and the intestine of some brightly coloured and darkly pigmented benedeniine capsalid monogeneans. Previous studies of these pigmented capsalids have proven difficult because the pigmentation usually disappears when the worms are fixed using preservatives containing concentrations of formalin over 5% and/or ethanol, acetic acid, chromic acid, picric acid and mercuric chloride. The technique developed here uses a fixative comprising glycerol, acetone and formalin (GAF). After fixation under light coverslip compression for three minutes, specimens are transferred to absolute acetone for three minutes and cleared in a mixture of nine parts cedar wood oil and one part absolute acetone before mounting in Canada balsam. Processing must be carried out quickly, as these chemicals will cause the pigments to fade if the specimens are exposed to them for too long. Pigmented benedeniines processed using this technique retain the distribution, intensity and colour observed in live worms. The colour and distribution of pigmentation in monogeneans may be of taxonomic importance and this technique aids preparation of whole-mounts suitable for registration as type-material.

Riddling and invariance for discontinuous maps preserving Lebesgue measure

In this paper we use the mixture of topological and measure-theoretic dynamical approaches to consider riddling of invariant sets for some discontinuous maps of compact regions of the plane that preserve two-dimensional Lebesgue measure. We consider maps that are piecewise continuous and with invertible except on a closed zero measure set. We show that riddling is an invariant property that can be used to characterize invariant sets, and prove results that give a non-trivial decomposion of what we call partially riddled invariant sets into smaller invariant sets. For a particular example, a piecewise isometry that arises in signal processing (the overflow oscillation map), we present evidence that the closure of the set of trajectories that accumulate on the discontinuity is fully riddled. This supports a conjecture that there are typically an infinite number of periodic orbits for this system.

Recording and preserving legal history

The lack of a systematic approach in Australia to recording and preserving legal history was highlighted in the article by Catherine Hutchins in the August 2002 issue of the Australian Bar Review. This article describes the modest, albeit continuing, efforts on the part of the Supreme Court Library to meet this need in the State of Queensland. The efforts began in 1983 and have recently escalated with a series of exhibitions, talks, displays, interviews and preservations of documents (from online publication).

Congruences on monoids of transformations preserving the orientation on a finite chain

Journal of Algebra, 321 (2009), p. 743–757

Preserving the validity of the two-Higgs-doublet model up to the Planck scale

We examine the constraints on the two Higgs doublet model (2HDM) due to the stability of the scalar potential and absence of Landau poles at energy scales below the Planck scale. We employ the most general 2HDM that incorporates an approximately Standard Model (SM) Higgs boson with a flavor aligned Yukawa sector to eliminate potential tree-level Higgs-mediated flavor changing neutral currents. Using basis independent techniques, we exhibit robust regimes of the 2HDM parameter space with a 125 GeV SM-like Higgs boson that is stable and perturbative up to the Planck scale. Implications for the heavy scalar spectrum are exhibited.

Congruences on monoids of order-preserving or order-reversing transformations on a finite chain

Glasgow Mathematical Journal, nº 47 (2005), pg. 413-424

Abelian kernels of monoids of order-preserving maps and of some of its extensions

Semigroup Forum vol. 68 (2004), p. 335–356

Occurrence of positivity for Trypanosoma cruzi in triatomine from municipalities in Southeastern Brazil, from 2002 to 2004

INTRODUCTION: from an epidemiological point of view, more than 120 species of triatomine (Hemiptera, Reduviidae) are known. The occurrence and positivity for Trypanosoma cruzi in triatomines in 16 municipalities of the Triângulo Mineiro and Alto Paranaíba were evaluated from January 2002 to December 2004. METHODS: the triatomines were captured basically according to the classic norms of the National Health Foundation. The parasitological exams of the triatomines were conducted according to the technique described by the Ministry of Health. During the study period, 990 specimens of triatomines were captured and of these, 771 could be examined. RESULTS: five species were identified: Triatoma sordida, Panstrongylus diasi, Panstrongylus megistus, Panstrongylus geniculatus and Rhodnius neglectus. Triatoma sordida represented 71.5% of all the triatomines captured, followed by Panstrongylus megistus (18%), Rhodnius neglectus (9.3%), Panstrongylus diasi (0.8%) and Panstrongylus geniculatus (0.4%). Of the total number of triatomines examined, 2.7% were positive for Trypanosoma cruzi. Panstrongylus megistus was the species that presented the highest rates of infection by Trypanosoma cruzi (8.3%), followed by Rhodnius neglectus (2.9%) and Triatoma sordida (1.4%). CONCLUSIONS: there is a need to adapt to new circumstances in epidemiology, with greater emphasis on entomological surveillance, since the potential for adaptation of secondary species of triatomines exists, especially where Chagas' disease is already under control.