Far downstream analysis for the Blasius boundary-layer stability problem

In this paper, we examine the large Reynolds number (Re) asymptoticstructure of the wave number in the Orr–Sommerfeld regionfor the Blasius boundary layer on a semi-infinite flat plategiven by Goldstein (1983, J. Fluid Mech., 127, 59–81).We show that the inclusion of the term which contains the leading-ordernon-parallel effects, at O(Re^{– 1/2}), leads to a non-uniformexpansion. By considering the far downstream form of each termin the asymptotic expansion, we derive a length scale at whichthe non-uniformity appears, and compare this position with theposition seen in plots of the wave number.     

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.     

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.     

On the Number of Solutions of an Equation Over a Finite Field

The Stöhr–Voloch approach is used to obtain a newbound for the number of solutions in (F_q)² of an equation f(X,Y) = 0, where f(X, Y) is an absolutely irreducible polynomialwith coefficients in a finite field F_q.     

Computations of form and stability of rotating drops with finite elements

^** Email: claus{at}mathematik.uni-freiburg.de We consider the numerical computation of equilibrium shapesof rotating drops and their bifurcations, depending on the angularvelocity. The drops are subject to centrifugal forces and surfacetension alone. We present a path-tracking algorithm which isbased on the discretisation of a continuously formulated Newton'siteration. Our algorithm uses parametric finite elements andavoids artificial symmetry assumptions. The results of our numericalexperiments extend those published by Brown & Scriven (1980,The shape and stability of rotating liquid drops. Proc. R. Soc.Lond. Ser. A, 371, 331– 357) and cover drops of spheroidalas well as annular shape.     

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.     

The Kissing Numbers of Convex Bodies -- A Brief Survey

Nearly four hundred years ago, the cubic close-packing of equalspheres in R³ was discovered by Kepler [21], in which each spheretouches 12 others. In 1694, Gregory and Newton discussed thefollowing thirteen spheres problem. Can a rigid material spherebe brought into contact with 13 other such spheres of the samesize? Gregory believed ‘yes’, while Newton thought‘no’. 1991 Mathematics Subject Classication 11H31,52C17.     

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 division of polynomial matrices

The main purpose of this paper is to determine two new algorithmsfor the division of the polynomial matrix B(s) R[s]^pxq by A(s) R[s]^pxp (a) based on the Laurent matrix expansion at s = =of the inverse of A(s), i.e. A(s)^–1, and (b) in a waysimilar to the one presented by Gantmacher (1959).     

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.     

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.     

An Euler-Type Version of the Local Steiner Formula for Convex Bodies

The purpose of this note is to establish a new version of thelocal Steiner formula and to give an application to convex bodiesof constant width. This variant of the Steiner formula generalizesresults of Hann [3] and Hug [6], who use much less elementarytechniques than the methods of this paper. In fact, Hann askedfor a simpler proof of these results [4, Problem 2, p. 900].We remark that our formula can be considered as a Euclideananalogue of a spherical result proved in [2, p. 46], and thatour method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensionalMinkowski spaces, see Hann [5, p. 363]. For further information about the notions used below, we referto Schneider's book [9]. Let Kⁿ be the set of all convex bodiesin Euclidean space Rⁿ, that is, the set of all compact, convex,non-empty subsets of Rⁿ. Let S^n–1 be the unit sphere.For KKⁿ, let NorK be the set of all support elements of K, thatis, the pairs (x, u)RⁿxS^n–1 such that x is a boundarypoint of K and u is an outer unit normal vector of K at thepoint x. The support measures (or generalized curvature measures)of K, denoted by ₀(K.), ..., _n–1(K.), are the unique Borelmeasures on RⁿxS^n–1 that are concentrated on NorK andsatisfy [formula] for all integrable functions f:RⁿR; here denotes the Lebesguemeasure on Rⁿ. Equation (1), which is a consequence and a slightgeneralization of Theorem 4.2.1 in Schneider [9], is calledthe local Steiner formula. Our main result is the following.1991 Mathematics Subject Classification 52A20, 52A38, 52A55.     

On the Discreteness and Convergence in n-Dimensional Mobius Groups

Throughout this paper, we adopt the same notations as in [1,6, 8] such as the Möbius group M(Rⁿ), the Clifford algebraC_n–1, the Clifford matrix group SL(2, _n), the Cliffordnorm of ||A||=(|a|²+|b|²+|c|²+|d|²) (1) and the Clifford metric of SL(2, _n) or of the Möbius groupM(Rⁿ) d(A₁,A₂)=||A₁–A₂||(|a₁–a₂|²+|b₁–b₂|²+|c₁–c₂|²+|d₁–d₂|²)(2) where |·| is the norm of a Clifford number and represents f_i M(), i = 1,2, and so on. In addition, we adopt some notions in [6, 12]:the elementary group, the uniformly bounded torsion, and soon. For example, the definition of the uniformly bounded torsionis as follows.     

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].     

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.     

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.     

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.     

Euler Characteristics of Real Varieties

In this paper we show how to associate to any real projectivealgebraic variety Z RP^n–1 a real polynomial F₁:Rⁿ,0 R, 0 with an algebraically isolated singularity, having theproperty that (Z) = (1 – deg (grad F₁), where deg (gradF₁ is the local real degree of the gradient grad F₁:Rⁿ, 0 Rⁿ,0. This degree can be computed algebraically by the method ofEisenbud and Levine, and Khimshiashvili [5]. The variety Z neednot be smooth. This leads to an expression for the Euler characteristic ofany compact algebraic subset of Rⁿ, and the link of a quasihomogeneousmapping f: Rⁿ, 0 Rⁿ, 0 again in terms of the local degree ofa gradient with algebraically isolated singularity. Similar expressions for the Euler characteristic of an arbitraryalgebraic subset of Rⁿ and the link of any polynomial map aregiven in terms of the degrees of algebraically finite gradientmaps. These maps do involve ‘sufficiently small’constants, but the degrees involved ar (theoretically, at least)algebraically computable.     

Intertwining Maps from Certain Group Algebras

In [17, 18, 19], we began to investigate the continuity propertiesof homomorphisms from (non-abelian) group algebras. Alreadyin [19], we worked with general intertwining maps [3, 12]. Thesemaps not only provide a unified approach to both homomorphismsand derivations, but also have some significance in their ownright in connection with the cohomology comparison problem [4]. The present paper is a continuation of [17, 18, 19]; this timewe focus on groups which are connected or factorizable in thesense of [26]. In [26], G. A. Willis showed that if G is a connectedor factorizable, locally compact group, then every derivationfrom L¹(G) into a Banach L¹(G)-module is automatically continuous.For general intertwining maps from L¹(G), this conclusion isfalse: if G is connected and, for some nN, has an infinite numberof inequivalent, n-dimensional, irreducible unitary representations,then there is a discontinuous homomorphism from L¹(G into aBanach algebra by [18, Theorem 2.2] (provided that the continuumhypothesis is assumed). Hence, for an arbitrary intertwiningmap from L¹(G), the best we can reasonably hope for is a resultasserting the continuity of on a ‘large’, preferablydense subspace of L¹(G). Even if the target space of is a Banachmodule (which implies that the continuity ideal I() of is closed),it is not a priori evident that is automatically continuous:the proofs of the automatic continuity theorems in [26] relyon the fact that we can always confine ourselves to restrictionsto L¹(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4].It is not clear if this strategy still works for an arbitraryintertwining map from L¹(G) into a Banach L¹(G)-module.     

Gromov-Witten Invariants of Flag Manifolds, VIA D-Modules

We present a method for computing the 3-point genus zero Gromov–Witteninvariants of the complex flag manifold G/B from the relationsof the small quantum cohomology algebra QH^*G/B (G is a complexsemisimple Lie group and B is a Borel subgroup). In [3] and[9], at least in the case G = GL_nC, two algebraic/combinatoricmethods have been proposed, based on suitably designed axioms.Our method is quite different, being differential geometricin nature; it is based on the approach to quantum cohomologydescribed in [7], which is in turn based on the integrable systemspoint of view of Dubrovin and Givental.