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.     

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.     

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.     

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.     

Binary Quadratic Forms with Large Discriminants and Sums of Two Squareful Numbers II

Let F = (F₁, ..., F_m) be an m-tuple of primitive positive binaryquadratic forms and let U_F(x) be the number of integers notexceeding x that can be represented simultaneously by all theforms F_j, j = 1, ... , m. Sharp upper and lower bounds for U_F(x)are given uniformly in the discriminants of the quadratic forms. As an application, a problem of Erds is considered. Let V(x)be the number of integers not exceeding x that are representableas a sum of two squareful numbers. Then V(x) = x(log x)^–+o(1)with = 1 – 2^–1/3 = 0.206....     

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.     

Oscillation Properties of a Semi-discrete Approximation to the Beam Equation with a Second Order Term

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx–[p(x)u_x]_x=0, 0< x < L, t > 0, where it is assumed that w, p,and q are positive on the interval [0, L], is approximated bythe method of straight lines. The resulting approximation isa linear system of differential equations with coefficient matrixS. The matrix S is studied under a variety of boundary conditionswhich result in a conservative system. In all cases the matrixS is shown to be similar to an oscillation matrix.     

Kato Class Potentials for Higher Order Elliptic Operators

Our goal in this paper is to determine conditions on a potentialV which ensure that an operator such as H:=(–)^m+V (1) acting on L²(R^N) defines a semigroup in L^p(R^N) for various valuesof p including p=1. The operator is defined as a quadratic formsum. That is, we put for (all integrals are on R^N and are with respect to Lebesgue measure), and note thatthe closure of the form is non-negative and has domain equalto the Sobolev space W^m,2. We then assume that the potentialhas quadratic form bound less than 1 with respect to Q₀, anddefine This form is closed and is associated with a semibounded self-adjointoperator H in L² (see [17, p. 348; 5, Theorem 4.23]). One canthen ask whether the semigroup e^–Ht defined on L² fort0 is extendable to a strongly continuous one-parameter semigroupon L^p for other values of p, and if so whether one can describethe domain and spectrum of its generator.     

One Cubic Diophantine Inequality

Suppose that G(x) is a form, or homogeneous polynomial, of odddegree d in s variables, with real coefficients. Schmidt [15]has shown that there exists a positive integer s₀(d), whichdepends only on the degree d, so that if s s₀(d), then thereis an x Z^s\{0} satisfying the inequality |G(x)|<1. (1) In other words, if there are enough variables, in terms of thedegree only, then there is a nontrivial solution to (1). Lets₀(d) be the minimum integer with the above property. In thecourse of proving this important result, Schmidt did not explicitlygive upper bounds for s₀(d). His methods do indicate how todo so, although not very efficiently. However, in fact muchearlier, Pitman [13] provided explicit bounds in the case whenG is a cubic. We consider a general cubic form F(x) with realcoefficients, in s variables, and look at the inequality |F(x)|<1. (2) Specifically, Pitman showed that if s(1314)²⁵⁶–1, (3) then inequality (2) is non-trivially soluble in integers. Wepresent the following improvement of this bound.     

On the role of separation in compression wave generation by a train entering a tunnel hood with a window

An approximate analytical theory is proposed for calculatingthe compression wave generated when a train enters a tunnelfitted with an entrance hood with an open window. The pressurerise ahead of the entering train causes air to exhaust fromthe window in the form of a high-speed jet. The profile of thecompression wave transmitted into the tunnel is modified by theinteraction of the train nose with the window, by multiple reflectionsof wave energy between the window and the hood portal priorto transmission into the tunnel, and in addition by the productionof a pressure pulse by the jet. The wave generation problemcan be formulated in a quasi-one-dimensional manner, wherebythe pressure field generated in front of and to the sides ofthe train in the absence of the window is assumed to be scatteredby the window. A self-consistent solution is obtained by evaluatingthe jet flow from the window using a nonlinear empirical quationproposed and validated by Cummings (1984, Amer. Inst. Aeron.Astron. J., 22, 786–792; 1986, J. Acoust. Soc. Amer.,79, 942–951) for the velocity in the window-exit plane.Predictions are found to be in excellent agreement with measurementsof compression wave profiles obtained in model scale experimentsreported by Howe et al. (2003, J. Fluid Mech., 487, 211–243)at train speeds 350 km h^–1.     

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.     

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.     

Domain decomposition preconditioners of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions

We develop and analyse Neumann–Neumann methods for hpfinite-element approximations of scalar elliptic problems ongeometrically refined boundary layer meshes in three dimensions.These are meshes that are highly anisotropic where the aspectratio typically grows exponentially with the polynomial degree.The condition number of our preconditioners is shown to be independentof the aspect ratio of the mesh and of potentially large jumpsof the coefficients. In addition, it only grows polylogarithmicallywith the polynomial degree, as in the case of p approximationson shape-regular meshes. This work generalizes our previousone on two-dimensional problems in Toselli & Vasseur (2003a,submitted to Numerische Mathematik, 2003c to appear in Comput.Methods Appl. Mech. Engng.) and the estimates derived here canbe employed to prove condition number bounds for certain typesof FETI methods.     

Serre's Theorem on the Cohomology Algebra of a p-Group

The purpose of this note is to give a proof of a theorem ofSerre, which states that if G is a p-group which is not elementaryabelian, then there exist an integer m and non-zero elementsx₁, ..., x_m H¹ (G, Z/p) such that with ß the Bockstein homomorphism. Denote by m_G thesmallest integer m satisfying the above property. The theoremwas originally proved by Serre [5], without any bound on m_G.Later, in [2], Kroll showed that m_G p^k – 1, with k =dim_Z/pH¹ (G, Z/p). Serre, in [6], also showed that m_G (p^k –1)/(p – 1). In [3], using the Evens norm map, Okuyamaand Sasaki gave a proof with a slight improvement on Serre'sbound; it follows from their proof (see, for example, [1, Theorem4.7.3]) that m_G (p + 1)p^k–2. However, m_G can be sharpenedfurther, as we see below. For convenience, write H^*(G, Z/p) = H^*(G). For every x_i H¹(G),set 1991 Mathematics SubjectClassification 20J06.     

Hypersurface Singularities with 2-Dimensional Critical Locus

Consider an analytic germ f:(C^m, 0)(C, 0) (m3) whose criticallocus is a 2-dimensional complete intersection with an isolatedsingularity (icis). We prove that the homotopy type of the Milnorfiber of f is a bouquet of spheres, provided that the extendedcodimension of the germ f is finite. This result generalizesthe cases when the dimension of the critical locus is zero [8],respectively one [12]. Notice that if the critical locus isnot an icis, then the Milnor fiber, in general, is not homotopicallyequivalent to a wedge of spheres. For example, the Milnor fiberof the germ f:(C⁴, 0)(C, 0), defined by f(x₁, x₂, x₃, x₄) =x₁x₂x₃x₄ has the homotopy type of S¹xS¹xS¹. On the other hand,the finiteness of the extended codimension seems to be the rightgeneralization of the isolated singularity condition; see forexample [9–12, 17, 18]. In the last few years different types of ‘bouquet theorems’have appeared. Some of them deal with germs f:(X, x)(C, 0) wheref defines an isolated singularity. In some cases, similarlyto the Milnor case [8], F has the homotopy type of a bouquetof (dim X–1)-spheres, for example when X is an icis [2],or X is a complete intersection [5]. Moreover, in [13] Siersmaproved that F has a bouquet decomposition FF₀Sⁿ...Sⁿ (whereF₀ is the complex link of (X, x)), provided that both (X, x)and f have an isolated singularity. Actually, Siersma conjecturedand Tibr proved [16] a more general bouquet theorem for thecase when (X, x) is a stratified space and f defines an isolatedsingularity (in the sense of the stratified spaces). In thiscase F_iF_i, where the F_i are repeated suspensions of complexlinks of strata of X. (If (X, x) has the ‘Milnor property’,then the result has been proved by Lê; for details see[6].) In our situation, the space-germ (X, x) is smooth, but f hasbig singular locus. Surprisingly, for dim Sing f^–1(0)2,the Milnor fiber is again a bouquet (actually, a bouquet ofspheres, maybe of different dimensions). This result is in thespirit of Siersma's paper [12], where dim Sing f^–1(0)= 1. In that case, there is only a rather small topologicalobstruction for the Milnor fiber to be homotopically equivalentto a bouquet of spheres (as explained in Corollary 2.4). Inthe present paper, we attack the dim Sing f^–1(0) = 2 case.In our investigation some results of Zaharia are crucial [17,18].     

Logistic Type Equations on RN by a Squeezing Method Involving Boundary Blow-Up Solutions

We study, on the entire space R^N(N 1), the diffusive logisticequation u_t–u=u–u^p, u0 (1.1) and its generalizations. Here p > 1 is a constant. Problem(1.1) plays an important role in understanding various populationmodels and some other problems in applied mathematics. When = 1 and p = 2, it is also known as the Fisher equation andKPP equation, due to the pioneering works of Fisher [8] andKolmogoroff, Petrovsky and Piscounoff [18].     

A Generalised Skolem-Mahler-lech Theorem for Affine Varieties

The Skolem–Mahler–Lech theorem states that if f(n)is a sequence given by a linear recurrence over a field of characteristic0, then the set of m such that f(m) is equal to 0 is the unionof a finite number of arithmetic progressions in m 0 and afinite set. We prove that if X is a subvariety of an affinevariety Y over a field of characteristic 0 and q is a pointin Y, and is an automorphism of Y, then the set of m such that^m(q) lies in X is a union of a finite number of complete doubly-infinitearithmetic progressions and a finite set. We show that thisis a generalisation of the Skolem–Mahler–Lech theorem.     

On the Greatest Prime Divisor of the Sum of Two Squares of Primes

One of the most famous theorems in number theory states thatthere are infinitely many positive prime numbers (namely p =2 and the primes p 1 mod4) that can be represented in the formx²₁+x²₂, where x₁ and x₂ are positive integers. In a recentpaper, Fouvry and Iwaniec [2] have shown that this statementremains valid even if one of the variables, say x₂, is restrictedto prime values only. In the sequel, the letter p, possiblywith an index, is reserved to denote a positive prime number.As p²₁=p²₂ = p is even for p₁, p₂ > 2, it is reasonable toconjecture that the equation p²₁=p²₂ = 2p has an infinity ofsolutions. However, a proof of this statement currently seemsfar beyond reach. As an intermediate step in this direction,one may quantify the problem by asking what can be said aboutlower bounds for the greatest prime divisor, say P(N), of thenumbers p²₁=p²₂, where p₁, p₂ N, as a function of the realparameter N 1. The well-known Chebychev–Hooley methodcombined with the Barban–Davenport–Halberstam theoremalmost immediately leads to the bound P(N) N^1–, if N N_o(); here, denotes some arbitrarily small fixed positivereal number. The first estimate going beyond the exponent 1has been achieved recently by Dartyge [1, Théorème1], who showed that P(N) N^10/9–. Note that Dartyge'sproof provides the more general result that for any irreduciblebinary form f of degree d 2 with integer coefficients the greatestprime divisor of the numbers |f(p₁, p₂)|, p₁, p₂ N, exceedsN^_d–, where _d = 2 – 8/(d = 7). We in particular wantto point out that Dartyge does not make use of the specificfeatures provided by the form x²₁+x²₂. By taking advantage ofsome special properties of this binary form, we are able toimprove upon the exponent ₂ = 10/9 considerably.     

Splendid Derived Equivalences for Blocks of Finite Groups

A central issue in finite group modular representation theoryis the relationship between the p-local structure and the p-modularrepresentation theory of a given finite group. In [5], Brouéposes some startling conjectures. For example, he conjecturesthat if e is a p-block of a finite group G with abelian defectgroup D and if f is the Brauer correspondent block of e of thenormalizer, N_G(D), of D then e and f have equivalent derivedcategories over a complete discrete valuation ring with residuefield of characteristic p. Some evidence for this conjecturehas been obtained using an important Morita analog for derivedcategories of Rickard [11]. This result states that the existenceof a tilting complex is a necessary and sufficient conditionfor the equivalence of two derived categories. In [5], Brouéalso defines an equivalence on the character level between p-blockse and f of finite groups G and H that he calls a ‘perfectisometry’ and he demonstrates that it is a consequenceof a derived category equivalence between e and f. In [5], Brouéalso poses a corresponding perfect isometry conjecture betweena p-block e of a finite group G with an abelian defect groupD and its Brauer correspondent p-block f of N_G(D) and presentsseveral examples of this phenomena. Subsequent research hasprovided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blockse, f of finite groups G, H there are also perfect isometriesbetween p-blocks of p-local subgroups corresponding to e andf and these isometries are compatible in a precise sense. In[5], Broué calls such a family of compatible perfectisometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defininga p-locally compatible family of derived equivalences. In thisimportant paper, he defines a ‘splendid tilting complex’for p-blocks e and f of finite groups G and H with a commonp-subgroup P. Then he demonstrates that if X is such a splendidtilting complex, if P is a Sylow p-subgroup of G and H and ifG and H have the same ‘p-local structure’, thenp-local splendid tilting complexes are obtained from X via theBrauer functor and ‘lifting’. Consequently, in thissituation, we obtain an isotypy when e and f are the principalblocks of G and H. Linckelmann [9] and Puig [10] have also obtained important resultsin this area. In this paper, we refine the methods and program of [11] toobtain variants of some of the results of [11] that have widerapplicability. Indeed, suppose that the blocks e and f of Gand H have a common defect group D. Suppose also that X is asplendid tilting complex for e and f and that the p-local structureof (say) H with respect to D is contained in that of G, thenthe Brauer functor, lifting and ‘cutting’ by blockindempotents applied to X yield local block tilting complexesand consequently an isotypy on the character level. Since thep-local structure containment hypothesis is satisfied, for example,when H is a subgroup of G (as is the case in Broué'sconjectures) our results extend the applicability of these ideasand methods.     

Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on Rⁿ (n2) is the coarsest topology for whichall superharmonic functions on Rⁿ are continuous. We refer toDoob [11, 1.XI] for its basic properties and its relationshipto the notion of thinness. This paper presents several theoremsrelating the fine topology to limits of functions along parallellines. (Results of this nature for the minimal fine topologyhave been given by Doob – see [10, Theorem 3.1] or [11,1.XII.23] – and the second author [15].) In particular,we will establish improvements and generalizations of resultsof Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihland Seidel [6], Schneider [21], Berman [7], and Armitage andNelson [4], and will also solve a problem posed by the latterauthors. An early version of our first result is due to Evans [12, p.234], who proved that, if u is a superharmonic function on R³,then there is a set ER²x{0}, of two-dimensional measure 0, suchthat u(x, y,·) is continuous on R whenever (x, y, 0)E.We denote a typical point of Rⁿ by X=(X' x), where X'R^n–1and xR. Let :RⁿR^n–1x{0} denote the projection map givenby (X', x) = (X', 0). For any function f:Rⁿ[–, +] andpoint X we define the vertical and fine cluster sets of f atX respectively by C_V(f;X)={l[–, +]: there is a sequence (t_m) of numbersin R\{x} such that t_mx and f(X', t_m)l}| and C_F(f;X)={l[–, +]: for each neighbourhood N of l in [–,+], the set f^–1(N) is non-thin at X}. Sets which are open in the fine topology will be called finelyopen, and functions which are continuous with respect to thefine topology will be called finely continuous. Corollary 1(ii)below is an improvement of Evans' result.