Error expansion for a first-order upwind difference scheme applied to a model convection-diffusion problem

A singularly perturbed convection–diffusion problem isconsidered. The problem is discretized using a simple first-orderupwind difference scheme on general meshes. We derive an expansionof the error of the scheme that enables uniform error boundswith respect to the perturbation parameter in the discrete maximumnorm for both a defect correction method and the Richardsonextrapolation technique. This generalizes and simplifies resultsobtained in earlier publications by Fröhner et al.(2001,Numer. Algorithms, 26, 281–299) and by Natividad &Stynes (2003, Appl. Numer. Math., 45, 315–329). Numericalexperiments complement our theoretical results.     

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.     

Asymptotic expansions of the generalized Epstein-Hubbell integral

The generalized Epstein–Hubbell integral recently introducedby Kalla & Tuan (Comput. Math. Applic. 32, 1996) is consideredfor values of the variable k close to its upper limit k = 1.Distributional approach is used for deriving two convergentexpansions of this integral in increasing powers of 1 –k². For certain values of the parameters, one of these expansionsinvolves also a logarithmic term in the asymptotic variable1 – k². Coefficients of these expansions are given interms of the Appell function and its derivative. All the expansionsare accompanied by an error bound at any order of the approximation.Numerical experiments show that this bound is considerably accurate.     

On the asymptotic behaviour for an electromagnetic system with a dissipative boundary condition

In this work we study some properties of solutions for the systemdescribing a three-dimensional non-homogeneous non-conductingdielectric with a general boundary condition with memory. Wefirst show the existence of the inverse of this boundary condition,which allows us to introduce a boundary free energy, similarto the one considered by Fabrizio & Morro (1996, Arch. Rat.Mech. Anal., 136, 359–381). Then, we prove existence anduniqueness theorems for weak and strong solutions of the evolutiveproblem in a finite time interval. Moreover, following Rivera& Olivera (1997, Boll. U.M.I., 11-A, 115–127), weexamine some dissipative properties of the boundary conditionand of its inverse and we give a useful energy estimate. Finally,when there is no memory in the boundary condition the exponentialdecay of the solution is proved.     

Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay

In this note we propose a method for the integration of y'(t) = f(t, y(t), y(rt)), 0 t t_f y(0) = y₀, where 0 < r < 1, by a superconvengent s-stage continuousRK method of discrete global order p and continuous uniformorder q < p – 1 for the approximation of the delayedterm y(rt). We prove that, although the maximum attainable orderof the method on an arbitrary mesh is q' = min{p, q + 1}, byusing a quasi-geometric mesh, introduced by Bellen et al. (1997,Appl. Numer. Math. 24, 1997, 279–293), the optimal accuracyorder p is preserved.     

Multiple Positive Solutions of Semilinear Differential Equations with Singularities

The existence of positive solutions of a second order differentialequation of the form z'+g(t)f(z)=0 (1.1) with the separated boundary conditions: z(0) – ßz'(0)= 0 and z(1)+z'(1) = 0 has proved to be important in physicsand applied mathematics. For example, the Thomas–Fermiequation, where f = z^3/2 and g = t^–1/2 (see [12, 13, 24]),so g has a singularity at 0, was developed in studies of atomicstructures (see for example, [24]) and atomic calculations [6].The separated boundary conditions are obtained from the usualThomas–Fermi boundary conditions by a change of variableand a normalization (see [22, 24]). The generalized Emden–Fowlerequation, where f = z^p, p > 0 and g is continuous (see [24,28]) arises in the fields of gas dynamics, nuclear physics,chemically reacting systems [28] and in the study of multipoletoroidal plasmas [4]. In most of these applications, the physicalinterest lies in the existence and uniqueness of positive solutions.     

Optimal control for cooperative parabolic systems governed by Schrodinger operator with control constraints

^** Email: Bahaa_gm{at}hotmail.com A distributed control problem for cooperative parabolic systemsgoverned by Schrödinger operator is considered. The performanceindex is more general than the quadratic one and has an integralform. Constraints on controls are imposed. Making use of theDubovitskii–Milyutin theorem given by Walczak (1984, Onesome control problems. Acta Univ. Lod. Folia Math., 1, 187–196),the optimality conditions are derived for the Dirichlet problem.     

A geometric approach to fingering instabilities for reaction fronts in fully coupled geochemical systems

A geometrical approach described by Grindrod (1995, Proc. R.Soc. Lond. A 449, 123–38) is applied to analyse spontaneoussymmetry breaking of planar reaction fronts in fully coupledreaction-diffusion-advection problems arising in geochemistry.This method yields stability results qualitatively similar tothose of Ortoleva et al. (1987, Am. J. Sci287, 1008–40)and Chen & Ortoleva (1990, Earth Sci. Rev. 29, 183–98;1992, Modelling and Analysis of Diffusive and Advective Processesin Geosciences, SIAM), yet distinct in the treatment of large-wavenumberperturbations. The analysis is verified numerically.     

Multilevel flexible specification of the production function in health economics

^** Email: grassetti{at}stat.unipd.it^*** Email: e.gori{at}dss.uniud.it^**** Email: simona.minotti{at}unicatt.it Previous studies on hospitals' efficiency often refer to quiterestrictive functional forms for the technology (Aigner et al.,1977, J. Econom., 6, 21–37). In this paper, referringto a study about some hospitals in Lombardy, we formulate convenientcorrectives to a statistical model based on the translogarithmicfunction—the most widely used flexible functional form(Christensen et al., 1973, Rev. Econ. Stat., 55, 28–45).More specifically, in order to take into consideration the hierarchicalstructure of the data (as in Gori et al., 2002, Stat. Appl.,14, 247–275), we propose a multilevel model, ignoringfor the moment the one-side error specification, typical ofstochastic frontier analysis (Aigner et al., 1977, J. Econom.,6, 21–37). Given this simplification, however, we areeasily able to take into account some typical econometric problemsas, e.g. heteroscedasticity. The estimated production functioncan be used to identify the technical inefficiency of hospitals(as already seen in previous works), but also to draw some economicconsiderations about scale elasticity, scale efficiency andoptimal resource allocation of the productive units. We willshow, in fact, that for the translogarithmic specification itis possible to obtain the elasticity of the output (regardingan input) at hospital level as a weighted sum of elasticitiesat ward level. Analogous results can be achieved for scale elasticity,which measures how output changes in response to simultaneousinputs variation. In addition, referring to scale efficiencyand to optimal resource allocation, we will consider the resultsof Ray (1998, J. Prod. Anal., 11, 183–194) to our context.The interpretation of the results is surely an interesting administrativeinstrument for decision makers in order to analyse the productiveconditions of each hospital and its single wards and also todecide the preferable interventions.     

Hierarchical Morse—Smale Complexes for Piecewise Linear 2-Manifolds

   Abstract. We present algorithms for constructing a hierarchy of increasingly coarse Morse—Smale complexes that decompose a piecewise linear 2-manifold. While these complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability. We then simplify Morse—Smale complexes by canceling pairs of critical points in order of increasing persistence.     

Modular Form Congruences and Selmer Groups

Motivated by Cremona and Mazur's notion of visibility of elementsin Shafarevich–Tate groups [6, 27], there have been anumber of recent works which test its compatibility with theBirch and Swinnerton–Dyer conjecture and the Bloch–Katoconjecture. These conjectures provide formulas for the ordersof Shafarevich–Tate groups in terms of values of L-functions.For example, one may see recent work of Agashe, Dummigan, Steinand Watkins [1, 2, 10, 11]. In their examples, they find thatthe presence of visible elements agrees with the expected divisibilityproperties of the relevant L-values.     

Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkinfinite-element method (hp-DGFEM) for second-order linear reaction–diffusionequations. To the best of our knowledge, the sharpest knownerror bounds for the hp-DGFEM are due to Rivière et al.(1999,Comput. Geosci., 3, 337–360) and Houston et al.(2002,SIAM J. Numer. Anal., 99, 2133–2163). These are optimalwith respect to the meshsize h but suboptimal with respect tothe polynomial degree p by half an order of p. We present improvederror bounds in the energy norm, by introducing a new functionspace framework. More specifically, assuming that the solutionsbelong element-wise to an augmented Sobolev space, we deducefully hp-optimal error bounds.     

Optimal control of parabolic equation with an infinite number of variables for non-standard functional and time delay

A distributed control problem for the parabolic operator withan infinite number of variables and time delay is considered.The performance index has an integral form. Constraints on controlsare imposed. To obtain optimality conditions for the Neumannproblem, the generalization of the Dubovitskii–Milyutintheorem given by Walczak in WALCZAK, S. Folia Mathematics, 1,187–196 and WALCZAK, S. J. Optim. Theory Appl., 42, 561–582was applied.     

Capillary waves past a flat plate in water of finite depth

Free-surface flow past a semi-infinite flat plate in a channelof finite depth is considered. The fluid is assumed to be inviscidand incompressible, and the flow to be two-dimensional and irrotational.Surface tension is included in the dynamic boundary conditionbut the effects of gravity are neglected. It is shown that thereis a three-parameter family of solutions with waves in the farfield and a discontinuity in slope at the separation point.This family includes as particular cases the solutions previouslycomputed by Osborn & Stump (2001, Phys. Fluids, 13, 616–623)and by Andersson & Vanden-Broeck (1996, Proc. R. Soc., 452,1985–1997).     

On the global stability of a temporal discretization scheme for the Navier-Stokes equations

The time discretization by a linear backward Euler scheme forthe non-stationary viscous incompressible Navier–Stokesequations with a non-zero external force in a bounded 2D domainwith no-slip boundary condition or periodic boundary conditionis studied. Improved global stability results are obtained. The boundedness of the solution sequence in V and D(A) normsuniform with respect to &t for t [0, ) is proved. A similarresult in the V norm was previously obtained by (Geveci, 1989Math. Comp., 53, 43–53) for the non-forced system. A differentapproach is used here. As a corollary, the global attractorfor the approximation scheme is proved to exist, which is boundedin both V and D(A) spaces, thus compact in both H and V spaces.Applying the same techniques developed here, we are able toimprove the main result of (Hill and Süli 2000 IMA J. Numer.Anal., 20, 633–667) by showing that besides the existenceof a global attractor, the whole solution sequence is uniformlybounded in V as well, which is of significance from the pointof view of computing. As a corollary of local convergence results,upper semi-continuity of the attractor with respect to the numericalperturbation induced by the linear scheme is also establishedin both H and V spaces. Finally, some preliminary estimates,which are to our knowledge the first of their kind, on the dimensionsof the attractors in H and V spaces are also obtained.     

A Functional Relation for Accessory Parameters for Genus 2 Algebraic Curves with an Order 4 Automorphism

Let C be a genus 2 algebraic curve defined by an equation ofthe form y² = x(x² – 1)(x – a)(x – 1/a). Asis well known, the five accessory parameters for such an equationcan all be expressed in terms of a and the accessory parameter b corresponding to a. The main result of the paper is thatif a' = 1 – a², which in general yields a non-isomorphiccurve C', then b'a'(a'² – 1) = – – ba(a²– 1). This is proven by it being shown how the uniformizing functionfrom the unit disk to C' can be explicitly described in termsof the uniformizing function for C.     

Shape-preserving interpolation by G1 and G2 PH quintic splines

The interpolation of a planar sequence of points p₀, ..., p_Nby shape-preserving G¹ or G² PH quintic splines with specifiedend conditions is considered. The shape-preservation propertyis secured by adjusting ‘tension’ parameters thatarise upon relaxing parametric continuity to geometric continuity.In the G² case, the PH spline construction is based on applyingNewton–Raphson iterations to a global system of equations,commencing with a suitable initialization strategy—thisgeneralizes the construction described previously in NumericalAlgorithms 27, 35–60 (2001). As a simpler and cheaperalternative, a shape-preserving G¹ PH quintic spline schemeis also introduced. Although the order of continuity is lower,this has the advantage of allowing construction through purelylocal equations.     

Locally Nilpotent p-Groups whose Proper Subgroups are Hypercentral or Nilpotent-by-Chernikov

Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for ‘hypercentral’or ‘nilpotent-by-Chernikov’. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [1–3, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]).     

Time-optimal control problem for parabolic equations with control constraints and infinite number of variables

A time optimal control problem for parabolic equations withan infinite number of variables is considered. A time optimalcontrol problem is replaced by an equivalent one with a performanceindex in the form of integral form. Constraints on controlsare assumed. To obtain the optimality conditions for the Neumannproblem, the generalization of the Dubovitskii–Milyutintheorem given by Walczak (1984, Acta Universitatis LodziensisFolia Mathematica, 187–196; 1984, J. Optim. Theor. Appl.,42, 561–582) was applied.     

Parameter-free H(div) preconditioning for a mixed finite element formulation of diffusion problems

^** Email: c.powell{at}manchester.ac.uk Mixed finite element formulations of generalised diffusion problemsyield linear systems with ill-conditioned, symmetric and indefinitecoefficient matrices. Preconditioners with optimal work complexitythat do not rely on artificial parameters are essential. Weimplement lowest order Raviart–Thomas elements and analysepractical issues associated with so-called ‘H(div) preconditioning’.Properties of the exact scheme are discussed in Powell &Silvester (2003, SIAM J. Matrix Anal. Appl., 25, 718–738).We extend the discussion, here, to practical implementation,the components of which are any available multilevel solverfor a weighted H(div) operator and a pressure mass matrix. Anew bound is established for the eigenvalue spectrum of thepreconditioned system matrix and extensive numerical resultsare presented.