Externally definable sets and dependent pairs

We prove that externally definable sets in first order NIP theories have honest definitions, giving a new proof of Shelah’s expansion theorem. Also we discuss a weak notion of stable embeddedness true in this context. Those results are then used to prove a general theorem on dependent pairs, which in particular answers a question of Baldwin and Benedikt on naming an indiscernible sequence.     

Fluorescence of semiconductor nanorods in liquid-crystal composites

The fluorescence of CdSe/ZnS nanorods in liquid-crystal and anisotropic polymer composites is studied. A high degree of the fluorescence anisotropy is shown. It is found that a solubilizer strongly affects the CdSe/ZnS orientation, which opens new possibilities for creating electrically controlled liquid-crystal composites.     

On diagonal actions of branch groups and the corresponding characters

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acting on a regular rooted tree acts absolutely non-freely on the boundary of the tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II₁-factor representations, and infinitely many continuous ergodic invariant random subgroups.     

Isomorphic embedding of ℓ p n , 1<p<2, into ℓ 1 (1+ε)n

In this paper we partially answer a question posed by V. Milman and G. Schechtman by proving that ℓ _p ⁿ , (C logn)^1/q(1+1/ε)-embeds into ℓ ₁ ^(1+ε)n , where 1<p<2 and 1/p+1/q=1. Supported by ISF.     

Parametric analysis of the oscillatory solutions to stochastic differential equations with the Wiener and Poisson components by the Monte Carlo method

Using the Monte Carlo method, we address the influence of the Wiener and Poisson random noises on the behavior of oscillatory solutions to systems of stochastic differential equations (SDEs). For the linear and Van der Pol oscillators, we study the accuracy of estimates of the functionals of numerical solutions to SDEs obtained by the generalized explicit Euler method. For a linear oscillator, we obtain the exact analytical expressions for the mathematical expectation and the variance of the SDE solution. These expressions allow us to investigate the dependence of the accuracy of estimates of the solution moments on the values of SDE parameters, the size of meshsize, and the ensemble of simulated trajectories of the solution. For the Van der Pol oscillator, we study the dependence of the frequency and the damping rate of the oscillations of the mathematical expectation of SDE solution on the values of parameters of the Poisson component. The results of the numerical experiments are presented.     

Image of the Spectral Measure of a Jacobi Field and the Corresponding Operators

By definition, a Jacobi field is a family of commuting selfadjoint three-diagonal operators in the Fock space The operators J(ϕ) are indexed by the vectors of a real Hilbert space H₊. The spectral measure ρ of the field J is defined on the space H₋ of functionals over H₊. The image of the measure ρ under a mapping is a probability measure ρ_K on T₋. We obtain a family J_K of operators whose spectral measure is equal to ρ_K. We also obtain the chaotic decomposition for the space L²(T₋, d_{ρ K}).     

Electrochemical scanning tunnelling spectroscopy of a ferrocene-modified n-Si(111)-surface: electrolyte gating and ambipolar FET behaviour

As revealed for the first time by in situ scanning tunnelling spectroscopy (STS), ferrocene-modified Si(111) substrates show ambipolar field effect transistor (FET) behaviour upon electrolyte gating.