Automorphisms of Relatively Free Groups in the Variety N2A {wedge} AN2 {wedge} Nc

For positive integers n and c, with n 2, let G_{n, c} be a relativelyfree group of finite rank n in the variety N₂A AN₂ N_c. Itis shown that the subgroup of the automorphism group Aut(G_n,c) of G_{n, c} generated by the tame automorphisms and an explicitlydescribed finite set of IA-automorphisms of G_{n, c} has finiteindex in Aut(G_{n, c}). Furthermore, it is proved that there areno non-trivial elements of G_{n, c} fixed by every tame automorphismof G_{n, c}.     

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.     

Differential Operators and the Steenrod Algebra

The Novikov-Landweber algebra and the Steenrod algebra are setup in terms of the primitive differential operators acting in the usual way on the integralpolynomial ring Z[x₁,... ,x_n,...]. A commutative wedge productV for differential operators is introduced and it is shown thatthe iterated wedge product is divisible by r! as an integral operator. The divided differentialoperator algebra D is generated over the integers by thedividedoperators under the wedge product. D is additively isomorphic to the abelian group ofsymmetric functions in the variables x_i. Furthermore D is closedunder composition of operators and admits a natural coproductwhich makes it a Hopf algebra in two ways, with respect to thecomposition and wedge products. Under composition D is isomorphicto the Landweber-Novikov algebra. A Hopf sub-algebra is generatedunder composition by the integral Steenrod squares and reduces mod 2 to the Steenrod algebra. An explicitproduct formula for two wedge expressions is developed and usedto derive Milnor's product formula for his basis elements inthe Steenrod algebra. The hit problem in the Steenrod algebrais reformulated in terms of partial differential operators.1991 Mathematics Subject Classification: 55S10.     

Decomposition of elasticity tensors and tensors that are structurally invariant in three dimensions

The elastic stiffness or compliance is a fourth-order tensorthat can be expressed in terms of two second-order symmetrictensors A and B and a fourth-order completely symmetric andtraceless tensor Z (or z). It is shown that the parts associatedwith A, B and Z (or z) are all structurally invariant undera three-dimensional transformation. Thus a linear combinationof the three parts gives a general expression for three-dimensionalstructural invariants. All three-dimensional structural invariantsavailable in the literature are shown to be special cases ofthis general expression. Invariants that are inherited by eachstructural invariant are presented.     

Simplices of Maximal Volume or Minimal Total Edge Length in Hyperbolic Space

In this paper, we are mainly concerned with n-dimensional simplicesin hyperbolic space Hⁿ. We will also consider simplices withideal vertices, and we suggest that the reader keeps the Poincaréunit ball model of hyperbolic space in mind, in which the sphereat infinity Hⁿ() corresponds to the bounding sphere of radius1. It is known that all hyperbolic simplices (even the idealones) have finite volume. However, explicit calculation of theirvolume is generally a very difficult problem (see, for example,[1] or [16]). Our first theorem states that, amongst all simplicesin a closed geodesic ball, the simplex of maximal volume isregular. We call a simplex regular if every permutation of itsvertices can be realized by an isometry of Hⁿ. A correspondingresult for simplices in the sphere has been proved by Böröczky[4].     

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.     

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].     

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.     

Confined Subgroups of Simple Locally Finite Groups and Ideals of their Group Rings

We are concerned in this paper with the ideal structure of grouprings of infinite simple locally finite groups over fields ofcharacteristic zero, and its relation with certain subgroupsof the groups, called confined subgroups. The systematic studyof the ideals in these group rings was initiated by the secondauthor in[15], although some results had been obtained previously(see [3, 1]). Let G be an infinite simple locally finite groupand K a field of characteristic zero. It is expected that inmost cases, the group ring KG will have the smallest possiblenumber of ideals, namely three, (KG itself, {0} and the augmentationideal), and this has been verified in some cases. In some interestingcases, however, the situation is different, and there are moreideals. We mention in particular the infinite alternating groups[3] and the stable special linear groups [9], in which the ideallattice has been completely determined. The second author hasconjectured that the presence of ideals in KG, other than thethree unavoidable ones, is synonymous with the presence in thegroup of proper confined subgroups. Here a subgroup H of a locallyfinite group G is called confined, if there exists a finitesubgroup F of G such that H^gF1 for all gG. This amounts to sayingthat F has no regular orbit in the permutation representationof G on the cosets of H.     

Interfacial instabilities and fingering formation in Hele-Shaw flow

The interfacial instability of Hele-Shaw flow has been a crucialissue for the understanding of the pattern formation of viscousfingers in a Hele-Shaw cell. By using a unified asymptotic approach,we derive two different types of instability mechanisms for‘slightly’ time-dependent finger solutions; namely,(i) the global-trapped-wave (GTW) instability; and (ii) thezero-frequency (null-f) instability. On the basis of these instabilitymechanisms, the selection of viscous finger formation is clarified;the apparent contradiction between the previous linearstabilityanalysis by Tanveer (1987, Phys. Fluid 30, 1589) and othersand the numerical simulations by DeGregoria & Schwartz (1986,J. Fluid Mech. 164, 383)and the experimental evidence is reconciled.     

The Ubiquity of Thompson's Group F in Groups of Piecewise Linear Homeomorphisms of the Unit Interval

In the 1960s, Richard J. Thompson introduced a triple of groupsF T G which, among them, supplied the first examples of infinite,finitely presented, simple groups [14] (see [6] for publisheddetails), a technique for constructing an elementary exampleof a finitely presented group with an unsolvable word problem[12], the universal obstruction to a problem in homotopy theory[8], and the first examples of torsion free groups of type FPand not of type FP [5]. In abstract measure theory, it has beensuggested by Geoghegan (see [3] or [9, Question 13]) that Fmight be a counterexample to the conjecture that any finitelypresented group with no non-cyclic free subgroup is amenable(admits a bounded, non-trivial, finitely additive measure onall subsets that is invariant under left multiplication). Recently,F has arisen in the theory of groups of diagrams over semigrouppresentations [10], and as the object of questions in the algebraof string rewriting systems [7]. For more extensive bibliographiesand more results on Thompson's groups and their generalizationssee [1, 4, 6]. A persistent peculiarity of Thompson's groups is their abilityto pop up in diverse areas of mathematics. This suggests thatthere might be something very natural about Thompson's groups.We support this idea by showing (Theorem 1.1 below) that PL_o(I),the group of piecewise linear (finitely many changes of slope),orientation-preserving, self-homeomorphisms of the unit interval,is riddled with copies of F: a very weak criterion implies thata subgroup of PL_o(I) must contain an isomorphic copy of F.     

Linear Spaces on Cubic Hypersurfaces, and Pairs of Homogeneous Cubic Equations

A remarkable theorem of Birch [2] shows that a system of homogeneouspolynomials with rational coefficients has a non-trivial zero,provided only that these polynomials are of odd degree, andthe system has sufficiently many variables in terms of the numberand degrees of these polynomials. Despite four decades of effort,the problem of obtaining a reasonable bound for the latter numberof variables has proved to be one of great difficulty. Whenthe system consists of a single cubic form, Davenport [4] hassucceeded in showing that 16 variables suffice, and Schmidt[17, 18, 19, 20] has devoted a series of papers to systems ofcubic forms, showing in particular that 5140 variables sufficefor pairs of cubic forms, and that (10r)⁵ variables sufficefor systems of r cubic forms. The current state of knowledgefor forms of higher degree is, by comparison, extremely weak(but see [21, 22]), and so it seems worthwhile expending furthereffort on the case of systems of cubic forms. In this paperwe improve on Schmidt's result for pairs of cubic forms. Incontrast with the sophisticated versions of the Hardy–Littlewoodmethod employed by Davenport and Schmidt, our approach is basedon an elementary idea of Lewis [12], and is applicable in arbitrarynumber fields. This method also has consequences for the existenceof linear spaces of rational solutions on cubic hypersurfaces,thereby improving on work of Lewis and Schulze-Pillot [14] onthis topic. 1991 Mathematics Subject Classification 11D72, 11E76.     

A LOCALLY FINITE VARIETY OF RINGS WITH AN UNDECIDABLE EQUATIONAL THEORY

We prove that the non-finitely based system of polynomial identitiesover an arbitrary field of characteristic 2 given by Gupta and Krasilnikov in [A non-finitelybased system of polynomial identities which contains the identityx⁶ = 0, Quart. J. Math. 53 (2002), 173–183] can, by aslight modification, be turned into an independent system. Thishas important consequences for algorithmic problems in algebra:there is a locally finite-dimensional variety of associativealgebras (and, in particular, a locally finite variety of rings)that has an undecidable equational theory. For such a variety,the uniform word problem is unsolvable, and yet the word problemis recursively solvable for each individual finitely presentedalgebra (ring) in that variety.     

Rayleigh waves having generalised lateral dependence

It is shown that the surface-guided elastic waves found by Kiselevfor isotropic materials and having displacements depending linearlyupon the Cartesian coordinate orthogonal to the sagittal planemay be generalised in many ways. For surface waves on any anisotropichalf-space, a simple procedure applied to the displacementswithin the standard surface wave having dependence eⁱ, where k · x – t and k is the (surface) wave vector,yields displacements depending linearly upon the surface cartesiancoordinate orthogonal to the group velocity vector. Moreover,repeated application of this (differentiation) procedure yieldsa hierarchy of waves having algebraic dependence of successivelyincreasing degree. For isotropic materials, substantial simplificationand generalization are possible. Solutions of all algebraicdegrees have identical depth dependence. This allows the solutionsto be constructed iteratively and motivates a search for generalsolutions having depth dependence of the normal displacementthe same as in the standard surface wave. The procedure givesa new derivation of the solutions found by Achenbach havingamplitude of the normal displacement of the surface given byany solution to the two-dimensional Helmholtz equation. Furthermore,exploiting the scale invariance (a property of surface waveson any homogeneous half-space) shows that in every surface-guideddisturbance of an elastic half-space, the elevation of the freesurface is a solution of the wave equation in two dimensions(the membrane equation). Using the paraxial approximation tothe membrane equation, high-frequency Rayleigh waves propagatingas narrow beams are described in terms of a scalar Gaussianbeam.     

On Power Series Having Sections with Multiply Positive Coefficients and a Theorem of Polya

Let (0.1) be a formal power series. In 1913, G. Pólya [7] provedthat if, for all sufficiently large n, the sections (0.2) have real negative zeros only, then the series (0.1) convergesin the whole complex plane C, and its sum f(z) is an entirefunction of order 0. Since then, formal power series with restrictionson zeros of their sections have been deeply investigated byseveral mathematicians. We cannot present an exhaustive bibliographyhere, and restrict ourselves to the references [1, 2, 3], wherethe reader can find detailed information. In this paper, we propose a different kind of generalisationof Pólya's theorem. It is based on the concept of multiplepositivity introduced by M. Fekete in 1912, and it has beentreated in detail by S. Karlin [4].     

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.     

On the Representations of a Number as the Sum of Four Fifth Powers

It is known from Vaughan and Wooley's work on Waring's problemthat every sufficiently large natural number is the sum of atmost 17 fifth powers [13]. It is also known that at least sixfifth powers are required to be able to express every sufficientlylarge natural number as a sum of fifth powers (see, for instance,[5, Theorem 394]). The techniques of [13] allow one to showthat almost all natural numbers are the sum of nine fifth powers.A problem of related interest is to obtain an upper bound forthe number of representations of a number as a sum of a fixednumber of powers. Let R(n) denote the number of representationsof the natural number n as a sum of four fifth powers. In thispaper, we establish a non-trivial upper bound for R(n), whichis expressed in the following theorem.     

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.     

Residually Finite Groups with all Subgroups Subnormal

The following result is established. THEOREM. Let G be a periodic, residually finite group with allsubgroups sub-normal. Then G is nilpotent. The well-known groups of Heineken and Mohamed [1] show thatthe hypothesis of residual finiteness cannot be omitted here,while examples in [5] show that a residually finite group withall subgroups subnormal need not be nilpotent. The proof ofthe Theorem will use the results of Möhres that a groupwith all subgroups subnormal is soluble [3] and that a periodichypercentral group with all subgroups subnormal is nilpotent[4]. Borrowing an idea from [2], the plan is to construct certainsubgroups H and K that intersect trivially, and to show thatthe subnormality of both leads to a contradiction. 1991 MathematicsSubject Classification 20E15.     

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.