On band spectrum of Schroedinger operator in periodic system of domains coupled by small windows

We consider a periodic system of domains coupled by small windows. In such domain we study the band spectrum of a Schroedinger operator subject to Neumann condition. We show that near each isolated eigenvalue of the similar operator but in the periodicity cell, there are several non-intersecting bands of the spectrum for the perturbed operator. We also discuss the position of the points at which the band functions attain the edges of each band

Some Interesting Integer Polynomial Maps

We introduce three simple polynomial maps with integer coefficients that have interesting dynamical properties modulo primes.Comment: 2 page

On a Question of Craven and a Theorem of Belyi

In this elementary note we prove that a polynomial with rational coefficients divides the derivative of some polynomial which splits in \Q if and only if all of its irrational roots are real and simple. This provides an answer to a question posed by Thomas Craven. Similar ideas also lead to a variation of the proof of Belyi's theorem that every algebraic curve defined over an algebraic number field admits a map to $P^1$ which is only ramified above three points. As it turned out, this variation was noticed previously by G. Belyi himself and Leonardo Zapponi.Comment: Main theorem was generalized to include reducible polynomials. To appear in Proceedings of AM

Asymptotics for the solutions of elliptic systems with fast oscillating coefficients

We consider a singularly perturbed second order elliptic system in the whole space. The coefficients of the systems fast oscillate and depend both of slow and fast variables. We obtain the homogenized operator and in the uniform norm sense we construct the leading terms of the asymptotics expansion for the resolvent of the operator described by the system. The convergence of the spectrum is established. The convergence of the spectrum is established. The examples are given.Comment: I have excluded the results on the eigenvalues asymptotitcs from the article. The reason is that I've obtained the referee report and he is demanded me to do it. As a result, I shortened this preprint. The results on the eigenvalue asymptotics will be written as an independent article, and the preprint will be submitted to arXi

Convolution structures and arithmetic cohomology

In this paper we construct arithmetic analogs of the Riemann-Roch theorem and Serre's duality for line bundles. This improves on the works of Tate and van der Geer - Schoof. We define $H^0(L)$ and $H^1(L)$ as some convolution of measures structures. The $H^1$ is defined by a procedure very similar to the usual Cech cohomology. We get Serre's duality as Pontryagin duality of convolution structures. We get separately Riemann-Roch formula and Serre's duality. Instead of using the Poisson summation formula, we basically reprove it. The whole theory is pretty much parallel to the geometric case.Comment: Extra section on harmonic analysis included to make the paper more accessible for arithmetic geometers. Also, the ghost-spaces of second kind are treated somewhat differentl

Convex lattice polytopes and cones with few lattice points inside, from a birational geometry viewpoint

It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P in R^k of positive finite volume, such that intersection of P and Z^k consists of the point 0 and vertices of P. Likewise, Q-factorial terminal toric singularities essentially correspond to lattice simplexes with no lattice points inside or on the boundary (except the vertices). There have been a lot work, especially in the last 20 years or so on classification of these objects. The main goal of this paper is to bring together these and related results, that are currently scattered in the literature. We also want to emphasize the deep similarity between the problems of classification of toric Fano varieties and classification of Q-factorial toric singularities.Comment: 13 pages, 43 reference

Quotient singularities, integer ratios of factorials and the Riemann Hypothesis

The goal of this paper is to reveal a close connection between the following three subjects that have not been studied together in the past: terminal and canonical cyclic quotient singularities, integer ratios of factorials, Nyman's approach to the Riemann Hypothesis. In particular, we notice that the constructions of P.A. Picon are relevant for the study of singularities and possibly the Riemann Hypothesis. The list of the 29 stable quintuples of Mori-Morrison-Morrison coincides, up to the choice of notation, with the list of the 29 step functions with five terms of Vasyunin. We also reformulate and generalize a conjecture of Vasyunin.Comment: 14 pages, no figure

Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties

We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces.Comment: 6 pages, Latex fil

Boundedness theorem for Fano log-threefolds

The main purpose of this article is to prove that the family of all Fano threefolds with log-terminal singularities with bounded index is bounded.Comment: 16 pages, LaTeX v2.09, abstract correcte

On the spectrum of two quantum layers coupled by a window

We consider the Dirichlet Laplacian in a domain two three-dimensional parallel layers having common boundary and coupled by a window. The window produces the bound states below the essential spectrum; we obtain two-sided estimates for them. It is also shown that the eigenvalues emerge from the threshold of essential spectrum as the window passes through certain critical shapes. We prove the necessary condition for the window to be of critical shape. Under an additional assumption we show that this condition is sufficient and obtain the asymptotic expansion for the emerging eigenvalue as well as for the associated eigenfunction