We complete the classical Schoenberg representation theorem for radial positive definite functions. We apply this result to study spectral properties of self-adjoint realizations of two- and three-dimensional Schr\"odinger operators with point interactions on a finite set. In particular, we prove that any realization has purely absolutely continuous non-negative spectrum.

N. Goloshchapova M. Malamud V. Zastavnyi

Let $U$ be the quantum group with divided powers in $p-$th root of unity for prime $p$. For any two-sided cell $A$ in the corresponding affine Weyl group one associates tensor ideal in the category of tilting modules over $U$. In this note we show that for any cell $A$ there exists tilting module $T$ from the corresponding tensor ideal such that biggest power of $p$ which divides $dim T$ is $p^{a(A)}$ where $a(A)$ is Lusztig's $a-$function. In new version some typos are corrected and exposition is improved following suggestions of the referee.

Viktor Ostrik

The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial $p \in \mathbb{N}[x]$, and any directed finite bipartite graph can be considered as a polynomial $p\in\mathbb{N}[x,y]$, and vise verse. We also show that the multiplication in semirings $\mathbb{N}[x]$, $\mathbb{N}[x,y]$ correspondences to a operations of the corresponding graphs which looks like a ``perturbed'' products of graphs. As an application, we give a new point of view to dividing in semirings $\mathbb{N}[x]$, $\mathbb{N}[x,y]$. Finally, we endow the set of all bipartite graphs with the Zariski topology.

Andrey Grinblat Viktor Lopatkin

The new method of investigation of strongly correlated electronic system is presented. Using this method we have derived thee superstring from 2D Hubbard model. The novel expression for generalized supercoherent state has been calculated.

V. M. Zharkov

We study how the superfluidity depends on the resonance between one- and two-particle series. The frequency of the spectrum of two-particle solutions in an interval is calculated.

V. P. Maslov

In the theory of superfluidity and superconductivity, a jump of the free energy was discovered theoretically and was naturally called a {\it zeroth-order phase transition}. We present an example of an exactly solvable problem in which such a phase transition occurs.

V. P. Maslov

We introduce the notion of negative topological dimension and the notion of weight for the asymptotic topological dimension. Quantizing of spaces of negative dimension is applied to linguistic statistics.

V. P. Maslov

In this paper, we briefly discuss a mathematical concept that can be used in economics.

V. P. Maslov

We provide a correction to the Stefan--Boltzmann law and discuss the problem of a phase transition from the superfluid state into the normal state.

V. P. Maslov

In this paper, we present theorems specifying the critical values for series associated with debts arranged in the order of their duration.

V. P. Maslov