``Best possible' upper and lower bounds for the zeros of the Bessel function

Let denote the -th positive zero of the Bessel function . In this paper, we prove that for and , 2, 3, ,

These bounds coincide with the first few terms of the well-known asymptotic expansion

as , being fixed, where is the -th negative zero of the Airy function , and so are ``best possible'.

     

Integrals, partitions, and cellular automata

We prove that

where is the decreasing function that satisfies , for . When is an integer and we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.

     

A generalization of Dahlberg's theorem concerning the regularity of harmonic Green potentials

Let be the solution operator for in , Tr on , where is a bounded domain in . B. E. J. Dahlberg proved that for a bounded Lipschitz domain maps boundedly into weak- and that there exists such that is bounded for . In this paper, we generalize this result by addressing two aspects. First we are also able to treat the solution operator corresponding to Neumann boundary conditions and, second, we prove mapping properties for these operators acting on Sobolev (rather than Lebesgue) spaces.

     

Bochner-Riesz means with respect to a rough distance function

The generalized Bochner-Riesz operator may be defined as

where is an appropriate distance function and is the inverse Fourier transform. The behavior of on is described for , a rough distance function. We conjecture that this operator is bounded on when and , and unbounded when . This conjecture is verified for large ranges of .

     

Differentiability of spectral functions for symmetric -stable processes

Let be a signed Radon measure in the Kato class and define a Schrödinger type operator on . We show that its spectral bound is differentiable if and is Green-tight.

     

Combinatorial congruences modulo prime powers

Let be any prime, and let and be nonnegative integers. Let and . We establish the congruence

(motivated by a conjecture arising from algebraic topology) and obtain the following vast generalization of Lucas' theorem: If is greater than one, and are nonnegative integers with , then

We also present an application of the first congruence to Bernoulli polynomials and apply the second congruence to show that a -adic order bound given by the authors in a previous paper can be attained when .

     

Projective metrics and mixing properties on towers

We study the decay of correlations for towers. Using Birkhoff's projective metrics, we obtain a rate of mixing of the form:

where goes to zero in a way related to the asymptotic mass of upper floors, is some Lipschitz norm and is some norm. The fact that the dependence on is given by an norm is useful to study asymptotic laws of successive entrance times.

     

Random variable dilation equation and multidimensional prescale functions

A random variable satisfying the random variable dilation equation , where is a discrete random variable independent of with values in a lattice and weights and is an expanding and -preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density which will satisfy a dilation equation

We have obtained necessary and sufficient conditions for the existence of the density and a simple sufficient condition for 's existence in terms of the weights Wavelets in can be generated in several ways. One is through a multiresolution analysis of generated by a compactly supported prescale function . The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when is a prescale function.

     

Explicit merit factor formulae for Fekete and Turyn polynomials

We give explicit formulas for the norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials

where is the Legendre symbol. For example for an odd prime,

where is the class number of . Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of . This is the sequence that has the largest known asymptotic merit factor. Explicitly,

where denotes the nearest integer, satisfies

where

Indeed we derive a closed form for the norm of all shifted Fekete polynomials

Namely

and if .

     

-bounds for the Beurling-Ahlfors transform

Let denote the Beurling-Ahlfors transform defined on , . The celebrated conjecture of T. Iwaniec states that its norm where . In this paper the new upper estimate

is found.

     

The average norm of polynomials of fixed height

Let be any integer and let

be the set of all polynomials of height 1 and degree . Let

Here is the power of the norm on the boundary of the unit disc. So is the average of the power of the norm over

In this paper we give exact formulae for for various values of . We also give a variety of related results for different classes of polynomials including polynomials of fixed height H, polynomials with coefficients and reciprocal polynomials. The results are surprisingly precise. Typical of the results we get is the following.

Theorem 0.1. For , we have

and

     

Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations

We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space with and . On the scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for the following generalisation of the KP II equation:

for , , and . We deduce global well-posedness for , and real valued initial data.

     

Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers

This paper gives exact rates of quadratic approximations to an infinite class of cubic Pisot numbers. We show that for any cubic Pisot number , with minimal polynomial , such that , and where has only one real root, then there exists a , explicitly given here, such that:

(1)

For all 0$">, all but finitely many integer quadratics satisfy

where is the height function.

(2)

For all 0$"> there exists a sequence of integer quadratics such that

Furthermore, for all in this class of cubic Pisot numbers. What is surprising about this result is how precise it is, giving exact upper and lower bounds for these approximations.

     

Theta lifting of holomorphic discrete series: The case of

Let be a reductive dual pair in the stable range. We investigate theta lifts to of unitary characters and holomorphic discrete series representations of , in relation to the geometry of nilpotent orbits. We give explicit formulas for their -type decompositions. In particular, for the theta lifts of unitary characters, or holomorphic discrete series with a scalar extreme -type, we show that the structure of the resulting representations of is almost identical to the -module structure of the regular function rings on the closure of the associated nilpotent -orbits in , where is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.

     

The Hausdorff dimension of visible sets of planar continua

For a compact set and a point , we define the visible part of from to be the set

(Here denotes the closed line segment joining to .)

In this paper, we use energies to show that if is a compact connected set of Hausdorff dimension greater than one, then for (Lebesgue) almost every point , the Hausdorff dimension of is strictly less than the Hausdorff dimension of . In fact, for almost every ,

We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension greater than for some .

     

Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree

We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function , the number of bound states of the operator in below . Here is a bounded potential behaving asymptotically like where is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator on the sphere has negative eigenvalues less than , we prove that may be estimated as

Thus, in particular, if there are no such negative eigenvalues, then has a finite discrete spectrum. Moreover, under some additional assumptions including the fact that and that there is exactly one eigenvalue less than , with all others , we show that the negative spectrum is asymptotic to a geometric progression with ratio .

     

A counterexample concerning the relation between decoupling constants and -constants

For Banach spaces and and a bounded linear operator
we let such that

for all finitely supported and all , where is the sequence of Haar functions. We construct an operator , where is superreflexive and of type 2, with such that there is no constant with

In particular it turns out that the decoupling constants , where is the identity of a Banach space , fail to be equivalent up to absolute multiplicative constants to the usual -constants. As a by-product we extend the characterization of the non-superreflexive Banach spaces by the finite tree property using lower 2-estimates of sums of martingale differences.

     

Scrollar syzygies of general canonical curves with genus

We prove that for a general canonical curve of genus , the space of th (last) scrollar syzygies is isomorphic to the Brill-Noether locus . Schreyer has conjectured that these scrollar syzygies span the space of all th (last) syzygies of . Using Mukai varieties we prove this conjecture for genus , and .

     

Geometry of chain complexes and outer automorphisms under derived equivalence

The two main theorems proved here are as follows: If is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization of the family of finite -module complexes with fixed sequence of dimensions and an ``almost projective' complex , there exists a canonical vector space embedding

where is the pertinent product of general linear groups acting on , tangent spaces at are denoted by , and is identified with its image in the derived category .

     

Gauss sums and Kloosterman sums over residue rings of algebraic integers

Let denote the ring of integers of an algebraic number field of degree which is totally and tamely ramified at the prime . Write , where . We evaluate the twisted Kloosterman sum

where and denote trace and norm, and where is a Dirichlet character (mod ). This extends results of Salié for and of Yangbo Ye for prime dividing Our method is based upon our evaluation of the Gauss sum

which extends results of Mauclaire for .