Representations of Ariki-Koike Algebras and Grobner-Shirshov Bases

In this paper, we investigate the structure of Ariki–Koikealgebras and their Specht modules using Gröbner–Shirshovbasis theory and combinatorics of Young tableaux. For a multipartition, we find a presentation of the Specht module S given by generatorsand relations, and determine its Gröbner–Shirshovpair. As a consequence, we obtain a linear basis of S consistingof standard monomials with respect to the Gröbner–Shirshovpair. We show that this monomial basis can be canonically identifiedwith the set of cozy tableaux of shape . 2000 Mathematics SubjectClassification 16Gxx, 05Exx.     

Stability of uniformly Morse-Smale gradient-like numerical methods for flows

In Garay (1996, Numer. Math., 72, 449–479) and Li (1997b,SIAM J. Math. Anal., 28, 381–388), it was shown that thequalitative properties of a Morse–Smale gradient-likeflow are preserved by its numerical approximations. In thispaper, we show that the qualitative properties of a family ofuniformly Morse–Smale gradient-like numerical methodsare preserved by the approximated flow. The techniques usedin the study of the structural stability theorem for diffeomorphismsare the main tools for this work.     

Pattern formation in a 2D simple chemical system with general orders of autocatalysis and decay

In this paper, we investigate pattern formation in a coupledsystem of reaction–diffusion equations in two spatialdimensions. These equations arise as a model of isothermal chemicalautocatalysis with termination in which the orders of autocatalysisand termination, m and n, respectively, are such that 1 <n < m. We build on the preliminary work by Leach & Wei(2003, Physica D, 180, 185–209) for this coupled systemin one spatial dimension, by presenting rigorous stability analysisand detailed numerical simulations for the coupled system intwo spatial dimensions. We demonstrate that spotty patternsare observed over a wide parameter range.     

Lax-Wendroff-type schemes of arbitrary order in several space dimensions

^{The second-order accurate Lax–Wendroff scheme is basedon the first three terms of a Taylor expansion in time in whichthe time derivatives are replaced by space derivatives usingthe governing evolution equations. The space derivatives arethen approximated by central differencing. In this paper, weextend this idea and truncate the Taylor expansion at an arbitraryorder. One main building block is the so-called Cauchy–Kovalevskayaprocedure to replace all the time derivatives by space derivativeswhich can be formulated for a general system of linear equationswith arbitrary order and in two- or three-space dimensions.The linear case is the main focus of this paper because theproposed high-order schemes are good candidates for the approximationof linear wave motion over long distances and times with importantapplications in aeroacoustics and electromagnetics. The stabilityand the efficiency of Lax–Wendroff-type schemes are examined.The numerical results are compared with a standard scheme foraeroacoustical applications with respect to their quality andthe computational effort. The extensions of the schemes to generalgrids, nonconstant and nonlinear cases are alsoaddressed.}     

On the formation of acceleration and reaction-diffusion wavefronts in autocatalytic-type reaction-diffusion systems

^** Email: d.j.needham{at}reading.ac.uk We consider generalization of the theory for the evolution ofreaction–diffusion and accelerating wavefronts in KPP-typesystems as developed in Needham (2004, Proc. R. Soc. Lond. A,460, 1921–1934) (DN). These generalizations allow forthe removal of a number of technical restrictions imposed inthe paper of DN.     

Finite element approximation of an Allen-Cahn/Cahn-Hilliard system

We consider an Allen–Cahn/Cahn–Hilliard system witha non-degenerate mobility and (i) a logarithmic free energyand (ii) a non-smooth free energy (the deep quench limit). Thissystem arises in the modelling of phase separation and orderingin binary alloys. In particular we prove in each case that thereexists a unique solution for sufficiently smooth initial data.Further, we prove an error bound for a fully practical piecewiselinear finite element approximation of (i) and (ii) in one andtwo space dimensions (and three space dimensions for constantmobility). The error bound being optimal in the deep quenchlimit. In addition an iterative scheme for solving the resultingnonlinear discrete system is analysed. Finally some numericalexperiments are presented.     

Cohen-Macaulay Properties of Thom-Boardman Strata I: Morin's Ideal

Thom–Boardman strata ^I are fundamental tools in studyingsingularities of maps. The Zariski closures of the strata ^Iare components of the set of zeros of the ideals ^I defined by B. Morin using iterated jacobian extensions in his paper‘Calcul jacobien’ (Ann. Sci. École Norm.Sup.} 8 (1975) 1–98). In this paper, we consider the questionof when the Morin ideals ^I define Cohen–Macaulay spaces.We determine all I=(i₁...,i_k) such that ^I defines a Cohen–Macaulayspace alongthe stratum. 1991 Mathematics Subject Classification: 13D25, 14B05, 14M12, 58C25.     

Global existence and blow-up for the solutions to nonlinear parabolic-elliptic system modelling chemotaxis

In this paper we study the global in-time and blow-up solutionsfor the simplified Keller–Segel system modelling chemotaxis.We prove that there is a critical number which determines theoccurrence of blowup in the two-dimensional case for 1 <p < 2. In three- or higher-dimensional cases, we show thatthe radial symmetrical solution will blow up if 1 < p <N/N–2 (N 3) for non-negative initial value.     

Improving the accuracy of the mini-element approximation to Navier-Stokes equations

^** Email: blanca{at}imati.cnr.it^*** Email: frutos{at}mac.cie.uva.es^**** Corresponding author. Email: julia.novo{at}uam.es A technique to improve the accuracy of the mini-element approximationto incompressible the Navier–Stokes equations is introduced.Once the mini-element approximation has been computed at a fixedtime, the linear part of this approximation is postprocessedby solving a discrete Stokes problem. The bubble functions neededto stabilize the approximation to the Navier–Stokes equationsare not used at the postprocessing step. This postprocessingprocedure allows us to increase by one unit (up to a logarithmicterm) the H¹ norm rate of convergence of the velocity and correspondinglythe L² norm of the pressure. An error analysis of the algorithmis performed.     

All but Finitely Many Non-Trivial Zeros of the Approximations of the Epstein Zeta Function are Simple and on the Critical Line

The Chowla–Selberg formula is applied in approximatinga given Epstein zeta function. Partial sums of the series derivefrom the Chowla–Selberg formula, and although these partialsums satisfy a functional equation, as does an Epstein zetafunction, they do not possess an Euler product. What we callpartial sums throughout this paper may be considered as specialcases concerning a more general function satisfying a functionalequation only. In this article we study the distribution ofzeros of the function. We show that in any strip containingthe critical line, all but finitely many zeros of the functionare simple and on the critical line. For many Epstein zeta functionswe show that all but finitely many non-trivial zeros of partialsums in the Chowla–Selberg formula are simple and on thecritical line. 2000 Mathematics Subject Classification 11M26.     

Partial Theta Functions. I. Beyond the Lost Notebook

It is shown how many of the partial theta function identitiesin Ramanujan's lost notebook can be generalized to infinitefamilies of such identities. Key in our construction is theBailey lemma and a new generalization of the Jacobi triple productidentity. By computing residues around the poles of our identitieswe find a surprising connection between partial theta functionidentities and Garrett–Ismail–Stanton-type extensionsof multisum Rogers–Ramanujan identities. 2000 MathematicsSubject Classification 05A30, 33D15, 33D90.     

Equilibrium attractivity of Runge-Kutta methods

For dissipative differential equations y' = f (y) it is knownthat contractivity of the exact solution is reproduced by algebraicallystable Runge–Kutta methods. In this paper we investigatewhether a different property of the exact solution also holdsfor Runge–Kutta solutions. This property, called equilibriumattractivity, means that the norm of the righthand side f neverincreases. It is a property dual to algebraic stability sinceneither is sufficient for the other, in general. We derive sufficientalgebraic conditions for Runge–Kutta methods and proveequilibrium attractivity of the high-order algebraically stableRadau-IIA and Lobatto-IIIC methods and the Lobatto-IIIA collocationmethods (which are not algebraically stable). No smoothnessassumptions on f and no stepsize restrictions are required but,except for some simple cases, f has to satisfy certain additionalproperties which are generalizations of the simple one-sidedLipschitz condition using more than two argument points. Thesemultipoint conditions are discussed in detail.     

A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem

A fully discrete stabilized finite-element method is presentedfor the two-dimensional time-dependent Navier–Stokes problem.The spatial discretization is based on a finite-element spacepair (X_h, M_h) for the approximation of the velocity and thepressure, constructed by using the Q₁ – P₀ quadrilateralelement or the P₁ – P₀ triangular element; the time discretizationis based on the Euler semi-implicit scheme. It is shown thatthe proposed fully discrete stabilized finite-element methodresults in the optimal order error bounds for the velocity andthe pressure.     

Error estimates for stochastic differential games: the adverse stopping case

^** Email: frederic.bonnans{at}inria.fr^*** Email: stefania.maroso{at}inria.fr^**** Email: zidani{at}ensta.fr We obtain error bounds for monotone approximation schemes ofa particular Isaacs equation. This is an extension of the theoryfor estimating errors for the Hamilton–Jacobi–Bellmanequation. To obtain the upper error bound, we consider the ‘Krylovregularization’ of the Isaacs equation to build an approximatesub-solution of the scheme. To get the lower error bound, weextend the method of Barles & Jakobsen (2005, SIAM J. Numer.Anal.) which consists in introducing a switching system whosesolutions are local super-solutions of the Isaacs equation.     

Finite-difference approximation of a one-dimensional Hamilton-Jacobi/elliptic system arising in superconductivity

Finite-difference approximations to an elliptic–hyperbolicsystem arising in vortex density models for type II superconductorsare studied. The problem can be formulated as a non-local Hamilton–Jacobiequation on a bounded domain with zero Neumann boundary conditions.Monotone schemes are defined and shown to be stable. An L errorbound is proved for the approximations of the unique viscositysolution.     

Multi-symplectic Runge-Kutta-type methods for Hamiltonian wave equations

^** Email: kzhang{at}math.cuhk.edu.hk The non-linear wave equation is taken as a model problem forthe investigation. Different multi-symplectic reformulationsof the equation are discussed. Multi-symplectic Runge–Kuttamethods and multi-symplectic partitioned Runge–Kutta methodsare explored based on these different reformulations. Some popularand efficient multi-symplectic schemes are collected and constructed.Stability analyses are performed for these schemes.     

Sharp error estimates for discretizations of the 1D convection-diffusion equation with Dirac initial data

This paper derives sharp estimates of the error arising fromexplicit and implicit approximations of the constant-coefficient1D convection–diffusion equation with Dirac initial data.The error analysis is based on Fourier analysis and asymptoticapproximation of the integrals resulting from the inverse Fouriertransform. This research is motivated by applications in computationalfinance and the desire to prove convergence of approximationsto adjoint partial differential equations.     

The efficient computation of periodic states of cyclically operated chemical processes

In this paper we present a shooting method in combination witha hybrid Newton–Picard method to compute efficiently theperiodic states of cyclically operated chemical processes. Themerits of the method are investigated by considering the computationof the periodic states of a rapid pressure swing adsorptionsystem and of a cooled reverse-flow reactor.We show that theseperiodically forced chemical processes exhibit low-dimensionaldynamics. The Newton–Picard method is designed to makeefficient use of this low-dimensional dynamics of the chemicalprocesses. In addition the Newton–Picard method is ableto determine the stability of the computed periodic states.As an illustration of the efficiency of the method, we computea branch of periodic states for the reverse flow reactor.     

Semi-discrete central-upwind schemes with reduced dissipation for Hamilton-Jacobi equations

We introduce a new family of Godunov-type semi-discrete centralschemes for multidimensional Hamilton–Jacobi equations.These schemes are a less dissipative generalization of the central-upwindschemes that have been recently proposed in Kurganov, Noelleand Petrova (2001, SIAM J. Sci. Comput., 23, pp. 707–740).We provide the details of the new family of methods in one,two, and three space dimensions, and then verify their expectedlow-dissipative property in a variety of examples.