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.     

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.     

Weakly Almost Periodic Functions on N with a Negative Base

Weakly almost periodic compactifications have been seriouslystudied for over 30 years. In the pioneering papers of de Leeuwand Glicksberg [4] and [5], the approach adopted was operator-theoretic.The current definition is more likely to be created from theperspective of universal algebra (see [1, Chapter 3]). For adiscrete group or semigroup S, the weakly almost periodic compactificationwS is the largest compact semigroup which (i) contains S asa dense subsemigroup, and (ii) has multiplication continuousin each variable separately (where largest means that any othercompact semigroup with the properties (i) and (ii) is a quotientof wS). A third viewpoint is to envisage wS as the Gelfand spaceof the C*-algebra of bounded weakly almost periodic functionson S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup(N, +). The three approaches described above give three methodsof obtaining information about wN. An early striking resultabout wN, that it contains more than one idempotent, was obtainedby T. T. West using operator theory [13]. He considered theweak operator closure of the semigroup {T, T², T³, ...} of iteratesof a single operator T on the Hilbert space L²(µ) fora particular measure µ on [0, 1]. Brown and Moran, ina series of papers culminating in [2], used sophisticated techniquesfrom harmonic analysis to produce measures µ that permittedthe detection of further structure in wN; in particular, theyfound 2^cdistinct idempotents. However, for many years, no otherway of showing the existence of more than one idempotent inwN was found. The breakthrough came in 1991, and it was made by Ruppert [11].In his paper, he created a direct construction of a family ofweakly almost periodic functions which could detect 2^c differentidempotents in wN. His method was very ingenious (he used aunique variant of the p-adic expansion of integers) and rathercomplicated. Our main aim in this paper is to construct weaklyalmost periodic functions which are easy to describe and soappear more ‘natural’ than Ruppert's. We also showthat there are enough functions of our type to distinguish 2^cidempotentsin wN.     

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.     

On Fusion in Unipotent Blocks

The context of this note is as follows. One considers a connectedreductive group G and a Frobenius endomorphism F: G G definingG over a finite field of order q. One denotes by G^F the associated(finite) group of fixed points. Let l be a prime not dividing q. We are interested in the l-blocksof the finite group G^F. Such a block is called unipotent ifthere is a unipotent character (see, for instance, [6, Definition12.1]) among its representations in characteristic zero. Roughlyspeaking, it is believed that the study of arbitrary blocksof G^F might be reduced to unipotent blocks (see [2, Théorème2.3], [5, Remark 3.6]). In view of certain conjectures aboutblocks (see, for instance, [9]), it would be interesting tofurther reduce the study of unipotent blocks to the study ofprincipal blocks (blocks containing the trivial character).Our Theorem 7 is a step in that direction: we show that thelocal structure of any unipotent block of G^F is very close tothat of a principal block of a group of related type (notionof ‘control of fusion’, see [13, 49]). 1991 MathematicsSubject Classification 20Cxx.     

Tensor Products of C*-Algebras Over Abelian Subalgebras

Suppose that A is a C*-algebra and C is a unital abelian C*-subalgebrawhich is isomorphic to a unital subalgebra of the centre ofM(A), the multiplier algebra of A. Letting = , so that we maywrite C = C(), we call A a C()-algebra (following Blanchard[7]). Suppose that B is another C()-algebra, then we form A_CB, the algebraic tensor product of A with B over C as follows:A B is the algebraic tensor product over C, I_C = {ⁿ_i–1(f_i 1–1f_i)x|f_iC, xAB} is the ideal in AB generated by f1–1f|fC,and A _CB = AB/I_C. Then A_CB is an involutive algebra over C,and we shall be interested in deciding when A_CB is a pre-C*-algebra;that is, when is there a C*-norm on A_C B? There is a C*-semi-norm,which we denote by ||·||_C-min, which is minimal in thesense that it is dominated by any semi-norm whose kernel containsthe kernel of ||·||_C-min. Moreover, if A _C B has a C*-norm,then ||·||_C-min is a C*-norm on A_C B. The problem isto decide when ||·||_C-min is a norm. It was shown byBlanchard [7, Proposition 3.1] that when A and B are continuousfields and C is separable, then ||·||_C-min is a norm.In this paper we show that ||·||_C-min is a norm whenC is a von Neumann algebra, and then we examine some consequences.     

Mappings Preserving Submodules of Hilbert C*-Modules

A Hilbert module over a C*-algebra B is a right B-module X,equipped with an inner product ·, · which is linearover B in the second factor, such that X is a Banach space withthe norm ||x||:=||x, x||^1/2. (We refer to [8] for the basictheory of Hilbert modules; the basic example for us will beX=B with the inner product x, y=x*y.) We denote by B(X) thealgebra of all bounded linear operators on X, and we denoteby L(X) the C*-algebra of all adjointable operators. (In thebasic example X=B, L(X) is just the multiplier algebra of B.)Let A be a C*-subalgebra of L(X), so that X is an A-B-bimodule.We always assume that A is nondegenerate in the sense that [AX]=X,where [AX] denotes the closed linear span of AX. Denote by A_X the algebra of all mappings on X of the form (1.1) where m is an integer and a_iA, b_iB for all i. Mappings of form(1.1) will be called elementary, and this paper is concernedwith the question of which mappings on X can be approximatedby elementary mappings in the point norm topology.     

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]).     

Beardon's Diophantine Equations and Non-Free Mobius Groups

Let µ be a real number. The Möbius group G_µis the matrix group generated by It is known that G_µ is free if |µ| 2 (see [1])or if µ is transcendental (see [3, 8]). Moreover, thereis a set of irrational algebraic numbers µ which is densein (–2, 2) and for which G_µ is non-free [2, p. 528].We may assume that µ > 0, and in this paper we considerrational µ in (0, 2). The following problem is difficult. Let G^nf denote the set of all rational numbers µ in (0,2) for which G_µ is non-free. In 1969 Lyndon and Ullman[8] proved that G^nf contains the elements of the forms p/(p²+ 1) and 1/(p + 1), where p = 1, 2, ..., and that if µ₀ G^nf, then µ₀/p G^nf for p = 1, 2, .... In 1993 Beardon[2] studied problem (P) by means of the words of the form A^rB^s A^t and A^r B^s A^t B^u A^v, and he obtained a sufficient conditionfor solvability of (P), included implicitly in [2, pp. 530–531],by means of the following Diophantine equations: 1991 Mathematics SubjectClassification 20E05, 20H20, 11D09.     

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.     

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.     

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).     

SIMULTANEOUS DIOPHANTINE APPROXIMATION BY SQUARE-FREE NUMBERS

Let ₁, ...,_r R be ‘not very well approximable’,for example, Q-linearly independent real algebraic numbers.Then there are infinitely many positive square-free integersn such that ||n_i|| << n^–(2/3r)+(1 i r), where||·|| denotes distance to the nearest integer.     

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].     

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.     

On Simultaneously Badly Approximable Numbers

For any pair i,j 0 with i+j=1 let Bad(i,j) denote the set ofpairs (,ß) R² for which max{||q||^1/i||qß|^1/j}>c/qfor all q N. Here c=c(,ß) is a positive constant.If i=0 the set Bad(0, 1) is identified with RxBad where Badis the set of badly approximable numbers. That is, Bad(0, 1)consists of pairs (, ß) with R and ß Bad If j=0 the roles of and ß are reversed. It isproved that the set Bad(1,0)Bad (0,1) Bad(i,j) has Hausdorffdimension 2, that is, full dimension. The method easily generalizesto give analogous statements in higher dimensions.     

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.     

Rangs Stables De Certaines Extensions

M. A. Rieffel [24] a introduit le rang stable topologique (tsr),pour généraliser aux C*-algèbres, le conceptde dimension de recouvrement pour les espaces compacts, affirmantainsi le principe selon lequel une C*-algèbre est ‘unespace localement compact non commutatif’. II a montréque l'on a tsr ((A)) = [dim (Â)] + 1, pour toute C*-algèbre commutative A etque trs (B/J) tsr (B), pour toute C*-algèbre B et pourtout idéal bilatère fermé J dans B (généralisantle fait que, si X est un espace compact et F un sous-ensemblefermé dans X, alors on a dim (F) dim (X), oùdim(X) est la dimension de recouvrement de X [19]). D'autrepart, le rang stable topologique peut être utilisépour obtenir des théorèmes de ‘cancellation’pour les modules projectifs, comme ceci est fait dans [25, 2].Un peu plus tard, R. H. Herman et L. N. Vaserstein [14] ontmontré que pour toute C*-algèbre unitaire A, lerang stable topologique de A et le rang stable de Basse de Acoincident, done, pour toute C*-algèbre unitaire A, onnote sr(A) cette valeur commune appelée rang stable deA. Les C*-algèbres unitaires de rang stable 1 ont étéétudiée géométriquement par M. Rørdam[27], il a montré que l'on a sr(A) = 1 si et seulementsi l'enveloppe convexe des unitaires de A est égale àla boule unité fermé de A. D'autre part, Rieffel[24] avait introduit le rang stable connexe (csr) d'une C*-algèbre,sur lequel V. Nistor [18] a publié un article trésintéressant. Mon travail dans ce papier consiste àcompléter certains travaux déjà entreprisdans les articles qui sont cités ci-dessus.     

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.