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In this paper we study various "abscissae" which one can associate to a given function f, or rather to the power moments of f. These are motivated by long-standing open problems in analytic number theory. We show how these abscissae connect to the distribution of values of f in a very elegant way using convex conjugates. This connection allows us to show which abscissae are realizable for both general and more specific arithmetical functions. Further it may give a new approach to, for example, Dirichlet's divisor problem. [ABSTRACT FROM AUTHOR]

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of R(ζ, q), where R(z, q) is the two-variable generating function of Dyson's rank function and ζ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon, and McIntosh, and motivated by Dyson's question, Bringmann, Ono, and Rhoades studied transformation properties of R(ζ, q). In this paper we strengthen and extend the results of Bringmann, Rhoades, and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo 5 from the Lost Notebook has an analogue for all primes greater than 3. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod 7. [ABSTRACT FROM AUTHOR]

In this paper, we derive a function field version of theWaldspurger formula for the central critical values of the Rankin-Selberg L-functions. This formula states that the central critical L-values in question can be expressed as the "ratio" of the global toric period integral to the product of the local toric period integrals. Consequently, this result provides a necessary and sufficient criterion for the non-vanishing of these central critical L-values, and supports the Gross-Prasad conjecture for SO(3) over function fields. [ABSTRACT FROM AUTHOR]

In this paper we first construct a two-variable p-adic L-function for the standard representation associated with a Hida family of parallel weight genus g Siegel forms, using a method developed by Böcherer-Schmidt in one variable. When a form f has weight g + 1 a non-crystalline trivial zero could appear. In this case, using the two-variable p-adic L-function we have constructed, we can apply the method of Greenberg-Stevens to calculate the first derivative of the p-adic L-function for f and show that it has the form predicted by a conjecture of Greenberg on trivial zeros. [ABSTRACT FROM AUTHOR]

Iterated function systems have been most extensively studied when the functions are affine transformations of Euclidean space and, more recently, projective transformations on real projective space. This paper investigates iterated function systems consisting of Möbius transformations on the extended complex plane or, equivalently, on the Riemann sphere. The main result is a characterization, in terms of topological, geometric, and dynamical properties, of Möbius iterated function systems that possess an attractor. The paper also includes results on the duality between the attractor and repeller of a Möbius iterated function system. [ABSTRACT FROM AUTHOR]

In this paper we study entire solutions of the Allen-Cahn equation Δu - F'(u) = 0, where F is an even, bistable function. We are particularly interested in the description of the moduli space of solutions which have some special structure at infinity. The solutions we are interested in have their zero set asymptotic to 2k, k&#8805 2 oriented affine half-lines at infinity and, along each of these affine half-lines, the solutions are asymptotic to the one-dimensional heteroclinic solution: such solutions are called multiple-end solutions, and their set is denoted by MM2k. The main result of our paper states that if u ∈ M2k is nondegenerate, then locally near u the set of solutions is a smooth manifold of dimension 2k. This paper is part of a program whose aim is to classify all 2k-ended solutions of the Allen-Cahn equation in dimension 2, for k ≥ 2. [ABSTRACT FROM AUTHOR]

Unlike the case of surfaces of topologically finite type, there are several different Teichmüller spaces that are associated to a surface of topologically infinite type. These Teichmüller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, given the choice of a point of view (hyperbolic or conformal), on the choice of a distance function on Teichmüller space. Examples of distance functions that appear naturally in the hyperbolic setting are the length spectrum distance and the bi-Lipschitz distance, and there are other useful distance functions. The Teichmüller spaces also depend on the choice of a basepoint. The aim of this paper is to present some examples, results and questions on the Teichmüller theory of surfaces of infinite topological type that do not appear in the setting of the Teichmüller theory of surfaces of finite type. In particular, we point out relations and differences between the various Teichmüller spaces associated to a given surface of topologically infinite type. [ABSTRACT FROM AUTHOR]

In this paper we deal with second angular derivatives at Denjoy-Wolff points for parabolic functions in the unit disc. Namely, we study and analyze the existence and the dynamical meaning of this second angular derivative. For instance, we provide several characterizations of that existence in terms of the so-called Koenigs function. It is worth pointing out that there are two quite different classes of parabolic iteration: those with positive hyperbolic step and those with zero hyperbolic step. In the first case, the Koenigs function is in the Carathéodory class but, in the second case, it is even unknown if it is normal. Therefore, the ideas and techniques to approach these two cases are really different. In the end, we also present several rigidity results related to the second angular derivatives at Denjoy-Wolff points. [ABSTRACT FROM AUTHOR]

In this paper, we obtain asymptotic formulas for an infinite class of rank generating functions. As an application, we solve a conjecture of Andrews and Lewis on inequalities between certain ranks. [ABSTRACT FROM AUTHOR]

We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two types of forms which does not exist in lower dimensions. Finally, we produce some counterexamples concerning Carleson measures on the infinite polydisc. [ABSTRACT FROM AUTHOR]

For the Dunkl transforms associated with the weight functions Hk²(x) = ∏j=1d ..., k1, ⋯, kd ≥ 0 on ℝd, it is proved that if p ≥ 2 + 1/λk and λk := d-1/2 + ∑j=1dkj, the maximal Bochner-Riesz operator B*δ(hk²; f) order δ > 0 is bounded on the space Lp(ℝd; hk² dx) if and only if δ > δk(p) := max{(2λk +1)(1/2 -- 1/p ) -- 1/2, 0}. This extends a well known result of M. Christ for the classical Fourier transforms (Proc. Amer. Math. Soc. 95 (1985), 16-20). The proof relies on a new local restriction theorem for the Dunkl transforms, which is stronger than the corresponding global restriction theorem, but significantly more difficult to prove. [ABSTRACT FROM AUTHOR]

Adams' inequality for bounded domains Ω ⊂ R4 states that the supremum of ... dx over all functions u ∈ W²0, ²(Ω) with ||Δu||2 ≤ 1 is bounded by a constant depending on Ω only. This bound becomes infinite for unbounded domains and in particular for R4. We prove that if ||Δu||2 is replaced by a suitable norm, namely ||u||= ||- Δu + u||2, then the supremum of ... dx over all functions u ∈ W²0, ²(Ω) with ||u|| ≤ 1 is bounded by a constant independent of the domain Ω. Furthermore, we generalize this result to any Wm.n/m0 (Ω) with Ω ⊆ Rn and m an even integer less than n. [ABSTRACT FROM AUTHOR]