On a problem with nonperiodic frequent alternation of boundary condition imposed on fast oscillating sets

We consider singular perturbed eigenvalue problem for Laplace operator in a cylinder with frequent and nonperiodic alternation of boundary conditions imposed on narrow strips lying in the lateral surface. The width of strips depends on a small parameter in a arbitrary way and may oscillate fast, moreover, the nature of oscillation is arbitrary, too. We obtain two-sided estimates for degree of convergences of the perturbed eigenvalues.Comment: This preprint is a short version; the bigger version with more results will appear in "Computational Mathematics and Mathematical Physics

Hindman's finite sums theorem and its application to topologizations of algebras

The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent ultrafilters in ultrafilter extensions of semigroups. The second, main part of the paper is devoted to the topologizability problem of a wide class of algebraic structures called polyrings; this class includes Abelian groups, rings, modules, algebras over a ring, differential rings, and others. We show that the Zariski topology on such an algebra is always non-discrete. Actually, a much stronger fact holds: if $K$ is an infinite polyring, $n$ a natural number, and a map $F$ of $K^n$ into $K$ is defined by a term in $n$ variables, then $F$ is a closed nowhere dense subset of the space $K^{n+1}$ with its Zariski topology. In particular, $K^n$ is a closed nowhere dense subset of $K^{n+1}$. The proof essentially uses a multidimensional version of Hindman's finite sums theorem established by Bergelson and Hindman. The third part of the paper lists several problems concerning topologization of various algebraic structures, their Zariski topologies, and some related questions. This paper is an extended version of the lecture at Journ\'ees sur les Arithm\'etiques Faibles 36: \`a l'occasion du 70\`eme anniversaire de Yuri Matiyasevich, delivered on 7th July, 2017, in Saint Petersburg.Comment: The main result of the paper, Theorem 2.4.1, was proved around 2010 but not published until 2017 though presented at several seminars and conferences, e.g. Colloquium Logicum 2012 in Paderborn, and included in author's course lectured at the Steklov Mathematical Institute in 201

Ultrafilter extensions of linear orders

It was recently shown that arbitrary first-order models canonically extend to models (of the same language) consisting of ultrafilters. The main precursor of this construction was the extension of semigroups to semigroups of ultrafilters, a technique allowing to obtain significant results in algebra and dynamics. Here we consider another particular case where the models are linearly ordered sets. We explicitly calculate the extensions of a given linear order and the corresponding operations of minimum and maximum on a set. We show that the extended relation is not more an order however is close to the natural linear ordering of nonempty half-cuts of the set and that the two extended operations define a skew lattice structure on the set of ultrafilters

The Hanbury Brown Twiss effect for atomic matter waves

This paper discusses our recent work on developing the matter wave analogs to the Hanbury Brown Twiss experiment. We discuss experiments using cold atoms, both bosons and fermions, both coherent and incoherent. Simple concepts from classical and quantum optics suffice to understand most of the results, but the ideas can also be traced back to the work of Einstein on the thermodynamics of Bose gases.Comment: also available at http://pos.sissa.it