The Kirwan Map for Singular Symplectic Quotients

Let M be a Hamiltonian K-space with proper moment map µ.The symplectic quotient X = µ^–1(0)/K is a singularstratified space with a symplectic structure on the strata.In this paper we generalise the Kirwan map, which maps the Kequivariant cohomology of µ^–1(0) to the middle perversityintersection cohomology of X, to this symplectic setting. The key technical results which allow us to do this are Meinrenken'sand Sjamaar's partial desingularisation of singular symplecticquotients and a decomposition theorem, proved in Section 2 ofthis paper, exhibiting the intersection cohomology of a ‘symplecticblowup’ of the singular quotient X along a maximal depthstratum as a direct sum of terms including the intersectioncohomology of X.     

Exotic smooth structures on Formula

Motivated by Stipsicz and Szabó's exotic 4-manifoldswith b₂⁺ = 3 and b₂^– = 8, we construct a family of simplyconnected smooth 4-manifolds with b₂⁺ = 3 and b₂^– = 8.As a corollary, we conclude that the topological 4-manifold     

The Volume of Hyperbolic Alternating Link Complements

If a hyperbolic link has a prime alternating diagram D, thenwe show that the link complement's volume can be estimated directlyfrom D. We define a very elementary invariant of the diagramD, its twist number t(D), and show that the volume lies betweenv₃(t(D) – 2)/2 and v₃(10t(D) – 10), where v₃ isthe volume of a regular hyperbolic ideal 3-simplex. As a consequence,the set of all hyperbolic alternating and augmented alternatinglink complements is a closed subset of the space of all completefinite-volume hyperbolic 3-manifolds, in the geometric topology.2000 Mathematics Subject Classification 57M25, 57N10.     

Non-Complex Symplectic 4-Manifolds with Formula

Simply connected closed symplectic 4-manifolds with and K² = 0 are investigated. As aresult, it is confirmed that most of homotopy elliptic surfaces{E(1)_k|K is a fibred knot in S³} constructed by R. Fintusheland R. Stern in Invent. Math. 134 (1998) 363–400 are simplyconnected closed minimal symplectic 4-manifolds that do notadmit a complex structure. 2000 Mathematics Subject Classification57R17, 57R57 (primary), 14J26 (secondary).     

The Global Mckay-Ruan Correspondence Via Motivic Integration

The purpose of this paper is to show how the methods of motivicintegration of Kontsevich, Denef–Loeser (Invent. Math.135 (1999) 201–232 and Compositio Math. 131 (2002) 267–290)and Looijenga (Astérisque 276 (2002) 267–297) canbe adapted to prove the McKay–Ruan correspondence, a generalizationof the McKay–Reid correspondence to orbifolds that arenot necessarily global quotients. 2000 Mathematics Subject Classification14A20, 14E15, 14F43.     

On the Mislin Genus of Symplectic Groups

In this paper we give lower bounds for the Mislin genus of thesymplectic groups Sp(m). This result appears to be the exactanalogue of Zabrodsky's theorem concerning the special unitarygroups SU(n). It is achieved by the determination of the stablegenus of the quasi-projective quaternionic spaces QH(m), followingthe approach of McGibbon. It leads to a symplectic version ofZabrodsky's conjecture, saying that these lower bounds are infact the exact cardinality of the genus sets. The genus of Sp(2)is well known to contain exactly two elements. We show thatthe genus of Sp(3) has exactly 32 elements and see that theconjecture is true in these two cases. Independently, we also show that any homotopy type in the genusof Sp(m) fibers over the sphere S⁴m–1 with fiber in thegenus of Sp(m–1), and that any homotopy type in the genusof SU(n) fibers over the sphere S²n–1 with fiber in thegenus of SU(n–1). Moreover, these fibrations are principalwith respect to some appropriate loop structures on the fibers.These constructions permit us to produce particular spaces realizingthe lower bounds obtained. 2000 Mathematics Subject Classification55P60 (primary), 55P15, 55R35 (secondary)     

Banach Spaces Whose Algebras of Operators are Unitary: A Holomorphic Approach

An element u of a norm-unital Banach algebra A is said to beunitary if u is invertible in A and satisfies ||u|| = ||u^–1||= 1. The norm-unital Banach algebra A is called unitary if theconvex hull of the set of its unitary elements is norm-densein the closed unit ball of A. If X is a complex Hilbert space,then the algebra BL(X) of all bounded linear operators on Xis unitary by the Russo–Dye theorem. The question of whetherthis property characterizes complex Hilbert spaces among complexBanach spaces seems to be open. Some partial affirmative answersto this question are proved here. In particular, a complex Banachspace X is a Hilbert space if (and only if) BL(X) is unitaryand, for Y equal to X, X* or X** there exists a biholomorphicautomorphism of the open unit ball of Y that cannot be extendedto a surjective linear isometry on Y. 2000 Mathematics SubjectClassification 46B04, 46B10, 46B20.     

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.     

Another Proof for the Presentation of the Quantum Cohomology of the Moduli of Bundles over a Riemann Surface

The presentation of the quantum cohomology of the moduli spaceof stable vector bundles of rank two and odd degree with fixeddeterminant over a Riemann surface of genus g > 2 is obtained.The argument avoids the use of gauge theory, providing an alternativeproof to that given by the author in Duke Math. J. 98 (1999)525–540. 2000 Mathematics Subject Classification 14N35(primary); 14H60, 53D45 (secondary).     

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.     

Bounds For Gradient Trajectories and Geodesic Diameter of Real Algebraic Sets

Let M Rⁿ be a connected component of an algebraic set ^–1(0),where is a polynomial of degree d. Assume that M is containedin a ball of radius r. We prove that the geodesic diameter ofM is bounded by 2rv(n)d(4d–5)^n–2, where v(n) =(1/2)((n+1)/2)(n/2)^–1.This estimate is based on the bound rv(n)d(4d–5)^n–2for the length of the gradient trajectories of a linear projectionrestricted to M. 2000 Mathematics Subject Classification 32Bxx,34Cxx (primary), 32Sxx, 14P10 (secondary).     

On Smooth Maps with Finitely Many Critical Points: Addendum

The paper is an addendum to D. Andrica and L. Funar, ‘Onsmooth maps with finitely many critical points’, J. LondonMath. Soc. (2) 69 (2004) 783–800.     

Curves which Intersect Lines in Finite Sets

We study closed subsets in the plane which intersect each linein at least m points and at most n points, for which we tryto minimize the difference n – m. It is known that m cannotbe equal to n. The results in this paper show that for everyeven number n there exist closed sets in the plane for whichm = n – 2.     

Funk-Hecke Formula for Orthogonal Polynomials on Spheres and on Balls

Analogues of the Funk–Hecke formula for spherical harmonicsare proved for Dunkl's h-harmonics associated to the reflectiongroups, and for orthogonal polynomials related to h-harmonicson the unit ball. In particular, an analogue and its applicationare discussed for the weight function (1–|x|²)^µ–1/2on the unit ball in R^d. 2000 Mathematics Subject Classification33C50, 33C55, 42C10.     

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.     

Cohen-Macaulay Local Rings of Embedding Dimension e + d - 3

In this paper we determine the possible Hilbert functions ofa Cohen–Macaulay local ring of dimension d and multiplicitye, in the case where the embedding dimension v satisfies v =e + d – 3 and the Cohen–Macaulay type is less thanor equal to e – 3. 1991 Mathematics Subject Classification:primary 13D40; secondary 13P99.     

The Chern-Simons Invariants of Hyperbolic Manifolds Via Covering Spaces

One important invariant of a closed Riemannian 3-manifold isthe Chern–Simons invariant [1]. The concept was generalizedto hyperbolic 3-manifolds with cusps in [11], and to geometric(spherical, euclidean or hyperbolic) 3-orbifolds, as particularcases of geometric cone-manifolds, in [7]. In this paper, westudy the behaviour of this generalized invariant under changeof orientation, and we give a method to compute it for hyperbolic3-manifolds using virtually regular coverings [10]. We confineourselves to virtually regular coverings because a coveringof a geometric orbifold is a geometric manifold if and onlyif the covering is a virtually regular covering of the underlyingspace of the orbifold, branched over the singular locus. Thereforeour work is the most general for the applications in mind; namely,computing volumes and Chern–Simons invariants of hyperbolicmanifolds, using the computations for cone-manifolds for whicha convenient Schläfli formula holds (see [7]). Among otherresults, we prove that every hyperbolic manifold obtained asa virtually regular covering of a figure-eight knot hyperbolicorbifold has rational Chern–Simons invariant. We giveexplicit examples with computations of volumes and Chern–Simonsinvariants for some hyperbolic 3-manifolds, to show the efficiencyof our method. We also give examples of different hyperbolicmanifolds with the same volume, whose Chern–Simons invariants(mod ) differ by a rational number, as well as pairs of differenthyperbolic manifolds with the same volume and the same Chern–Simonsinvariant (mod ). (Examples of this type were also obtainedin [12] and [9], but using mutation and surgery techniques,respectively, instead of coverings as we do here.) 1991 MathematicsSubject Classification 57M50, 51M10, 51M25.     

The Heat Flow of the CCR Algebra

Let Pf(x) = –if'(x) and Qf(x) = xf(x) be the canonicaloperators acting on an appropriate common dense domain in L²(R).The derivations D_P(A) = i(PA–AP) and D_Q(A) = i(QA–AQ)act on the *-algebra A of all integral operators having smoothkernels of compact support, for example, and one may considerthe noncommutative ‘Laplacian’, L = + , as a linear mapping of A into itself. L generates a semigroup of normal completely positive linearmaps on B(L₂(R)), and this paper establishes some basic propertiesof this semigroup and its minimal dilation to an E₀-semigroup.In particular, the author shows that its minimal dilation ispure and has no normal invariant states, and he discusses thesignificance of those facts for the interaction theory introducedin a previous paper. There are similar results for the canonical commutation relationswith n degrees of freedom, where 1 n < . 2000 MathematicsSubject Classification 46L57 (primary), 46L53, 46L65 (secondary).     

Long-Time Behavior of Solutions of the Fast Diffusion Equations with Critical Absorption Terms

This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation u_t = (u^m)– u^p in Rⁿ x (0,) with (1 – 2/n)₊ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)^–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of u_t = (u^m) with an additional logarithmicanomalous decay exponent in time as t .     

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.