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'.
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.
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.
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.
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 .
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
- (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
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.
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 .
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.
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 .
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 .