We study the function spaces ${{Q}_{p}}(R)$ defined on a Riemann surface $R$, which were earlier introduced in the unit disk of the complex plane. The nesting property ${{Q}_{p}}(R)\,\subseteq \,{{Q}_{_{q}}}(R)$ for $0\,<\,p\,<\,q\,<\,\infty $ is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space $\text{AD(}R\text{)}\,\subseteq \,{{Q}_{p}}(R)$ for any $p$, $0\,<\,p\,<\,\infty $, thus sharpening T. Metzger's well-known result $\text{AD(}R\text{)}\,\subseteq \,\text{BMOA}(R)$. Also the first author's result $\text{AD}(R)\,\subseteq \,\text{VMOA}(R)$ for a regular Riemann surface $R$ is sharpened by showing that, in fact, $\text{AD(}R\text{)}\,\subseteq \,{{Q}_{p,0}}(R)$ for all $p$, $0\,<\,p\,<\,\infty $. The relationships between ${{Q}_{p}}(R)$ and various generalizations of the Bloch space on $R$ are considered. Finally we show that ${{Q}_{p}}(R)$ is a Banach space for $0\,<\,p\,<\,\infty $.

There are infinitely many triplets of primes $p,\,q,\,r$ such that the arithmetic means of any two of them, $\frac{p+q}{2},\,\frac{p+r}{2},\,\frac{q+r}{2}$ are also primes. We give an asymptotic formula for the number of such triplets up to a limit. The more involved problem of asking that in addition to the above the arithmetic mean of all three of them, $\frac{p+q+r}{3}$ is also prime seems to be out of reach. We show by combining the Hardy-Littlewood method with the sieve method that there are quite a few triplets for which six of the seven entries are primes and the last is almost prime.

In this paper we construct a bounded strictly positive function $\sigma $ such that the Liouville property fails for the divergence form operator $L\,=\,\nabla ({{\sigma }^{2}}\nabla )$. Since in addition $\Delta \sigma /\sigma $ is bounded, this example also gives a negative answer to a problem of Berestycki, Caffarelli and Nirenberg concerning linear Schrödinger operators.

This paper deals with periodic solutions for the billiard problem in a bounded open set of ${{\mathbb{R}}^{N}}$ which are limits of regular solutions of Lagrangian systems with a potential well. We give a precise link between the Morse index of approximate solutions (regarded as critical points of Lagrangian functionals) and the properties of the bounce trajectory to which they converge.

We study the coordinate rings of $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$ scheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here ${\mu }'$ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi [5] proved a conjecture of Kraft [12] that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer [22, 23]. The famous $q$-Kostka polynomial ${{\tilde{K}}_{\lambda \mu }}(q)$ is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by $\lambda $ in the ring $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$. Lascoux and Schützenberger [15, 13] gave combinatorially a decomposition of ${{\tilde{K}}_{\lambda \mu }}(q)$ as a sum of “atomic” polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model.

Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen [19] imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer [3]. This gives a direct-sum decomposition of the ideals yielding the $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$, and a new proof of the atomic decomposition of the $q$-Kostka polynomials.

The purpose of this note is to provide a simple proof of the sharp polynomial upper bound for the resonance counting function of a Schrödinger operator in odd dimensions. At the same time we generalize the result to the class of superexponentially decreasing potentials.

Cauchy and Poisson integrals over unbounded sets are employed to prove Mittag-Leffler type theorems with massive singularities as well as approximation theorems for holomorphic and harmonic functions.

Consider the variance for the number of primes that are both in the interval $\left[ y,\,y\,+\,h \right]$ for $y\,\in \,[x,\,2x]$ and in an arithmetic progression of modulus $q$. We study the total variance obtained by adding these variances over all the reduced residue classes modulo $q$. Assuming a strong form of the twin prime conjecture and the Riemann Hypothesis one can obtain an asymptotic formula for the total variance in the range when $1\le \,h/q\,\le \,{{x}^{1/2-\in }}$, for any $\varepsilon \,>\,0$. We show that one can still obtain some weaker asymptotic results assuming the Generalized Riemann Hypothesis (GRH) in place of the twin prime conjecture. In their simplest form, our results are that on GRH the same asymptotic formula obtained with the twin prime conjecture is true for “almost all” $q$ in the range $1\le h/q\le {{h}^{1/4-\in }}$, that on averaging over $q$ one obtains an asymptotic formula in the extended range $1\le h/q\le {{h}^{1/2-\in }}$, and that there are lower bounds with the correct order of magnitude for all $q$ in the range $1\le h/q\le {{x}^{1/3-\in }}$.

Let ${{M}_{n}}$ be the variety of spatial polygons $P\,=\,({{a}_{1}},\,{{a}_{2}},...,{{a}_{n}})$ whose sides are vectors ${{a}_{i}}\,\in \,{{\mathbf{R}}^{3}}$ of length $\left| {{a}_{i}} \right|\,=\,1\,(1\,\le \,i\,\le \,n)$, up to motion in ${{\mathbf{R}}^{3}}$. It is known that for odd $n$, ${{M}_{n}}$ is a smooth manifold, while for even $n$, ${{M}_{n}}$ has cone-like singular points. For odd $n$, the rational homology of ${{M}_{n}}$ was determined by Kirwan and Klyachko [6], [9]. The purpose of this paper is to determine the rational homology of ${{M}_{n}}$ for even $n$. For even $n$, let ${{\tilde{M}}_{n}}$ be the manifold obtained from ${{M}_{n}}$ by the resolution of the singularities. Then we also determine the integral homology of ${{\tilde{M}}_{n}}$.

This paper studies conditions on an analytic function that imply it belongs to ${{M}_{\alpha }}$, the set of multipliers of the family of functions given by $f(z)\,=\,{{\int }_{\left| \zeta \right|=1}}\,\frac{1}{{{(1-\overline{\zeta }z)}^{\alpha }}}d\mu (\zeta )\,(\left| z \right|\,<\,1)$ where $\mu $ is a complex Borel measure on the unit circle and $\alpha \,>\,0$. There are two main theorems. The first asserts that if $0\,<\,\alpha \,<\,1$ and ${{\sup }_{\left| \zeta \right|=1}}\,\int_{0}^{1}{}\left| {f}'(r\zeta ) \right|{{(1-r)}^{\alpha -1}}\,dr<\infty \text{then}f\in {{M}_{\alpha }}$. The second asserts that if $0\,<\,\alpha \,\le \,1,f\,\in \,{{H}^{\infty }}$ and $\sup {{}_{t\int_{0}^{\pi }{{}}}}\frac{\left| f({{e}^{i(t+s)}})-2f({{e}^{it}})+f({{e}^{i(t-s)}}) \right|}{{{s}^{2-\alpha }}}\,ds\,<\,\infty$ then $f\in {{M}_{\alpha }}$. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

We define Hardy spaces of conjugate systems of temperature functions on $\mathbb{R}_{+}^{n+1}$. We show that their boundary distributions are the same as the boundary distributions of the usual Hardy spaces of conjugate systems of harmonic functions.

We study ${{\mathbb{Z}}_{p}}$ actions on compact connected Riemann surfaces via their associated Eichler traces.We determine the set of possible Eichler traces and determine the relationship between 2 actions if they have the same trace.

A reverse iterated function system (r.i.f.s.) is defined to be a set of expansive maps $\left\{ {{T}_{1}},...,{{T}_{m}} \right\}$ on a discrete metric space $M$. An invariant set $F$ is defined to be a set satisfying $F\,=\,\bigcup _{j=1}^{m}\,{{T}_{j}}F$, and an invariant measure $\mu $ is defined to be a solution of $\mu \,=\,\sum{_{j=1}^{m}\,{{p}_{j}}\mu {}^\circ T_{j}^{-1}}$ for positive weights ${{p}_{j}}$. The structure and basic properties of such invariant sets and measures is described, and some examples are given. A blowup$\mathcal{F}$ of a self-similar set $F$ in ${{\mathbb{R}}^{n}}$ is defined to be the union of an increasing sequence of sets, each similar to $F$. We give a general construction of blowups, and show that under certain hypotheses a blowup is the sum set of $F$ with an invariant set for a r.i.f.s. Some examples of blowups of familiar fractals are described. If $\mu $ is an invariant measure on ${{\mathbb{Z}}^{+}}$ for a linear r.i.f.s., we describe the behavior of its analytic transform, the power series $\sum{_{n=0}^{\infty }\mu (n){{z}^{n}}}$ on the unit disc.

The aim of this paper is to study small Hankel operators $h$ on the Hardy space or on weighted Bergman spaces,where $\Omega $ is a finite type domain in ${{\mathbb{C}}^{2}}$ or a strictly pseudoconvex domain in ${{\mathbb{C}}^{n}}$. We give a sufficient condition on the symbol $f$ so that $h$ belongs to the Schatten class ${{S}_{p}}$, $1\,\le \,p\,<\,+\infty $.