We consider in $L^2(ℝ^{d}),d≥2$, the perturbed polyharmonic operator $H=(−Δ)^{l}+V, l > 0$, with a function $V$ periodic with respect to a lattice in $ℝ^d$. We prove that the number of gaps in the spectrum of $H$ is finite if $6 l > d+2$. Previously the finiteness of the number of gaps was known for $4 l > d+1$. The proof is based on arithmetic properties of the lattice and elementary perturbation theory.

Two of our principal results (in a simplified form) are as follows.

THEOREM

For $p$ prime, the number of solutions of the equation $$c_{1}a_{1}+\cdots+c_{k}a_ {k}=\lambda,\quad a_{j}\in A_{j},$$ where $c_j ∈ \mathbb{F}_p^x$ and $λ ∈\mathbb{F}_p$ are fixed coefficients, and the variables $a_j$ range over sets $A_j ⊆ \mathbb{F}_p$, does not exceed the number of solutions of the equation $$a_{1}+\cdots+a^{k}=0,\quad a_ {j}\in\overline {A}_{j},$$ where the variables $a_j$ range over arithmetic progressions $\overline {A}_j⊆\mathbb{F}_p$ of cardinalities $|\overline {A}_j|=|A_j|$, balanced around zero.

This readily implies an integer version, which strengthens a result of R. Gabriel, G. Hardy, and J. Littlewood.

THEOREM

Let $A$ be a set of $n=|A|$ residues modulo a prime $p$. For $z∈\mathbb{F}_p$, write $$S_{A}(z)=\sum_{a\in A}e^{2\pi i(az/p)}.$$ Then for $ε>0$ we have $$\#\lbrace z\in\mathbb {F}_{p}^{x} : |S_{A}(z)>(1-\varepsilon)n\rbrace\leq\frac {2\sqrt {6}}{\pi}\frac {p}{n}\varepsilon^{1/2}(1+o(1)),$$ provided $n→∞$ and $ε→0$. Equality is attained when $A$ is an arithmetic progression modulo $p$.

This implies an integer version that is due to A. A. Yudin.

We give a formula representing a duality of Macdonald-Koornwinder polynomials. Using this formula, an integral representation of Macdonald-Koornwinder polynomials is derived, a special case of which is the conjectural formula stated in [22]. We also present the corresponding formula to Heckman and Opdam's Jacobi polynomials of type $BC_m$.

The 1-skeleton of a $G$-manifold $M$ is the set of points $p ∈ M$, where $\dim G_{p}≥ \dim G−1$, and $M$ is a GKM manifold if the dimension of this 1-skeleton is 2. M. Goresky, R. Kottwitz, and R. MacPherson show that for such a manifold this 1-skeleton has the structure of a “labeled” graph, $(Γ,α)$, and that the equivariant cohomology ring of $M$ is isomorphic to the “cohomology ring” of this graph. Hence, if $M$ is symplectic, one can show that this ring is a free module over the symmetric algebra $\mathbb{S}(\mathfrak{g}^*)$, with $b_{2i}(Γ)$ generators in dimension $2 i, b_{2i}(Γ)$ being the “combinatorial” $2i$th Betti number of $Γ$. In this article we show that this “topological” result is, in fact, a combinatorial result about graphs.

The aim of this paper is to study oscillatory solutions of nonlinear hyperbolic systems in the framework developed during the last decade by J.-L. Joly, G. Métivier, and J. Rauch. Here we focus mainly on rectification effects, that is, the interaction of oscillations with a mean field created by the nonlinearity. A real interaction can occur only under some geometric conditions described in [JMR1] and [L1] that are generally not satisfied by the physical models except in the 1-dimensional case. We introduce here a new type of ansatz that allows us to obtain rectification effects under weaker assumptions. We obtain a new class of profile equations and construct solutions for a subclass. Finally, the stability of the asymptotic expansion is proved in the context of Maxwell-Bloch-type systems.

In this short paper, we apply estimates and ideas from [CM4] to study the ends of a properly embedded complete minimal surface $Σ^{2} ⊂\mathbf{R}^{3}$ with finite topology. The main result is that any complete properly embedded minimal annulus that lies above a sufficiently narrow downward sloping cone must have finite total curvature.