Approximate Carleman theorems and a Denjoy-Carleman maximum principle

We give an extension of the Denjoy-Carleman theorem, which leads to a generalization of Carleman's theorem on the unique determination of probability measures by their moments. We also give complex versions of Carleman's theorem extending Theorem 4.1 of [2]

Applications of Group Representation Theory to the Easier Waring Problem

AbstractWe show how Rao and Vaserstein′s identities may be related to the groups S22 and S3. We then develop a theory that enables us to produce various identities, for any given pair (G, ϵ) of a group G and a character ϵ defined on G. When ϵ is ± 1-valued, these identities may be used to obtain upper bounds for the easier Waring problem over Z and Q. This approach may be considered as an alternative to the Tarry-Escott problem

Logarithmic and complex constant term identities

In recent work on the representation theory of vertex algebras related to the Virasoro minimal models M(2,p), Adamovic and Milas discovered logarithmic analogues of (special cases of) the famous Dyson and Morris constant term identities. In this paper we show how the identities of Adamovic and Milas arise naturally by differentiating as-yet-conjectural complex analogues of the constant term identities of Dyson and Morris. We also discuss the existence of complex and logarithmic constant term identities for arbitrary root systems, and in particular prove complex and logarithmic constant term identities for the root system G_2.Comment: 26 page

Random matrix theory, the exceptional Lie groups, and L-functions

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are here extended to non-classical groups. We focus on an explicit example: the exceptional Lie group G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G_2, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the 7-dimensional representation of G_2. The random matrix calculations extend to all exceptional Lie groupsComment: 14 page

On the Additive Completion of Polynomial Sets

AbstractLet P be a polynomial of degree k ≥ 2 with nonnegative coefficients. Let B be a set of nonnegative numbers such that every integer n ≤ N can be written as n = b + P(λ) for some integer λ and some b in B. Then, given a positive real number ϵ, we show that |B| P−1(N) > ((1 − 1/k)−1 sin(π/k)/(π/k) − ϵ)N for sufficiently large N