On a conjecture of Schmutz

In this work we discuss Schmutz’s conjecture that in dimension 2 to 8 the distinct norms that occur in the lattices with the best known sphere packings are strictly greater than those in any other lattice of the same covolume. We see that the ternary conjecture is not true. However, it seems that there is but one exception: one lattice, where for one length the conjecture fails. Received: 11 February 2008, Revised: 20 May 2008     

Spectral Radius Algebras of Idempotents

We show that the spectral radius algebras of certain quadratic operators possess nontrivial invariant subspaces. Additionally, such algebras properly contain the operator’s commutant, so that the invariant subspaces are in some sense beyond hyperinvariant. The spectral radius algebras of idempotents are completely described and, as a consequence, it is shown that every intransitive collection of operators must be contained in a norm-closed proper spectral radius algebra.      

A p-adic analogue of a formula of Ramanujan

During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured p-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures. Received: 11 April 2008     

Estimates for Littlewood-Paley Functions and Extrapolation

We prove certain L ^p -estimates for Littlewood-Paley functions arising from rough kernels. The estimates are useful for extrapolation to prove L ^p -boundedness of the Littlewood-Paley functions under a sharp kernel condition.      

Compact Hankel Forms on Planar Domains

A Hankel form on a Hilbert function space is a bounded, symmetric, bilinear form [., .] satisfying [fx, y] = [x, fy] for a class of multipliers f. We prove analogs of Weyl–Horn and Ky Fan inequalities for compact Hankel forms, and apply them to estimate the related eigenvalues, both for Hardy–Smirnov and Bergman spaces norms associated to multiply connected planar domains. In the case of the unit disk, we investigate the asymptotic of some measures constructed by eigenfunctions of Hankel operators with certain Markov functions as symbols. Submitted: May 2, 2008. Accepted: June 28, 2008.     

A Clifford Algebraic Framework for $$\mathfrak{sp}(m)$$-Invariant Differential Operators

We introduce a framework for studying differential operators which are invariant with respect to the real (complex) symplectic Lie algebra ( ), associated to a quaternionic structure on a vector space . To do so, these algebras are realized within the orthogonal Lie algebra . This leads in a natural way to a refinement of the recently introduced notion of complex Hermitean Clifford analysis, in which four variations of the classical Dirac operator play a dominant role. David Eelbode: Postdoctoral fellow supported by the F.W.O. Vlaanderen (Belgium).     

Hardy-Littlewood system and representations of integers by an invariant polynomial

Let f be an integral homogeneous polynomial of degree d, and let be the level set for each . For a compact subset in ), set

Distance sets corresponding to convex bodies

Suppose that is a 0-symmetric convex body which denes the usual norm

On totally reducible binary forms: I

Letv(n) be the number of positive numbers up to a large limit n that are expressible in essentially more than one way by a binary formf that is a product ofl > 2 distinct linear factors with integral coefficients. We prove that

Quadrature Domains with Infinite Volume

We discuss quadrature domains for subharmonic functions and prove the existence of core quadrature domains for certain positive measures. The core quadrature domains are the smallest quadrature domains as measures and inherit good properties from quadrature domains with finite volume. We next discuss new balayage for the class of harmonic functions integrable in a neighborhood of ∞. We give several estimates of balayage measures. The new balayage is introduced to construct quadrature domains for harmonic functions. Submitted: June 26, 2008. Accepted: July 24, 2008.     

Maximal subnear-rings of functions

For a group G, let M(G) denote the near-ring of functions on G. We characterize all maximal subnear-rings of M(G) and show that for many classes of groups, E(G), the near-ring generated by the semigroup, End(G) of G, is never maximal as a subnear-ring of M ₀ (G). Received: 25 April 2008     

On the Airy Reproducing Kernel, Sampling Series, and Quadrature Formula

We determine the class of entire functions for which the Airy kernel (of random matrix theory) is a reproducing kernel. We deduce an Airy sampling series and quadrature formula. Our results are analogues of well known ones for the Bessel kernel. The need for these arises in investigating universality limits for random matrices at the soft edge of the spectrum. Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353.     

Infinite Classes of Dihedral Snarks

Flower snarks and Goldberg snarks are two infinite families of cyclically 5–edge–connected cubic graphs with girth at least five and chromatic index four. For any odd integer k, k > 3, there is a Flower snark, say J _k , of order 4k and a Goldberg snark, say B _k , of order 8k. We determine the automorphism groups of J _k and B _k for every k and prove that they are isomorphic to the dihedral group D _4k of order 4k. Research performed within the activity of INdAM–GNSAGA with the financial support of the Italian Ministry MIUR, project “Strutture Geometriche, Combinatoria e loro Applicazioni”.     

Theta series and trace operators for Г<Subscript>n</Subscript>[q]

We consider the action of suitable trace operators on non homogeneous theta series that are Siegel modular forms for the principal congruence subgroups of the symplectic group of odd levelq: Г _n [q]. This is used for investigating whether modular forms forГ _n [N], withN|q, which are linear combination of such theta series, can be expressed as combination of theta series that are modular forms with respect toГ _n [N].     

Spectral measures corresponding to orthogonal polynomials with unbounded recurrence coefficients

Sufficient conditions are presented for the existence of an absolutely continuous part for the measure of orthogonality for a system of polynomials with unbounded recurrence coefficients. Results are obtained by analyzing the spectral measure of a related self-adjoint operator.Communicated by Paul Nevai.     

Non-Commutative Harmonic and Subharmonic Polynomials

The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x = (x ₁,…, x _g ). The Laplacian Lap[p, h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[p, h] = h ²Δ _x p where Δ _x p is the usual Laplacian. A symmetric polynomial in symmetric variables will be called harmonic if Lap[p, h] = 0 and subharmonic if the polynomial q(x, h) :=  Lap[p, h] takes positive semidefinite matrix values whenever matrices X ₁,…, X _g , H are substituted for the variables x ₁,…, x _g , h. In this paper we classify all homogeneous symmetric harmonic and subharmonic polynomials in two symmetric variables. We find there are not many of them: for example, the span of all such subharmonics of any degree higher than 4 has dimension 2 (if odd degree) and 3 (if even degree). Hopefully, the approach here will suggest ways of defining and analyzing other partial differential equations and inequalities. Dedicated to Israel Gohberg on the occasion of his 80^th birthday. All authors were partially supported by J.W. Helton’s grants from the NSF and the Ford Motor Co. and J. A. Hernandez was supported by a McNair Fellowship.     

On a Bernoulli-integrable magnetohydrodynamic system

A geometric procedure is used to isolate four Bernoulli integrals for a nonlinear, two-dimensional magnetohydrodynamic system. The general solution of the latter may thereby be constructed.     

Pencils on double coverings of curves

Let X be a smooth curve of genus g. When and d ≥ π−2g+1 we show the existence of a double covering where C a smooth curve of genus π with a base-point-free pencil of degree d which is not the pull-back of a pencil on X. Received: 7 February 2007; Revised: 1 July 2008     

On the geometric character of stress in continuum mechanics

This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other fundamental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented. Jerrold G. Marsden: Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932.     

Singular Integrals and Potential Theory (II)

In this note I give an extension of the T1-Theorem of David and Journé. In this extension the standard estimates on the kernel are replaced by “average standard estimates” but the classical proof is essentially identical. I then use that extension to prove the L^p-boundedness of some natural rough singular integrals. The applications of these singular integrals in Potential Theory has already been described in my previous paper in the area. Received: September 2008