Fredholmness and invertibility of Toeplitz operators with matrix almost periodic symbols

We consider Toeplitz operators with symbols that are almost periodic matrix functions of several variables. It is shown that under certain conditions on the group generated by the Fourier support of the symbol, a Toeplitz operator is Fredholm if and only if it is invertible.

     

The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Let F _k be a free group of rank k ≥ 2 with a fixed set of free generators. We associate to any homomorphism φ from F _k to a group G with a left-invariant semi-norm a generic stretching factor, λ(φ), which is a noncommutative generalization of the translation number. We concentrate on the situation where φ: F _k → Aut(X) corresponds to a free action of F _k on a simplicial tree X, in particular, where φ corresponds to the action of F _k on its Cayley graph via an automorphism of F _k . In this case we are able to obtain some detailed “arithmetic” information about the possible values of λ = λ(φ). We show that λ ≥ 1 and is a rational number with 2kλ ∈ ℤ[1/(2k − 1)] for every φ ∈ Aut(F _k ). We also prove that the set of all λ(φ), where φ varies over Aut(F _k ), has a gap between 1 and 1+(2k−3)/(2k ²−k), and the value 1 is attained only for “trivial” reasons. Furthermore, there is an algorithm which, when given φ, calculates λ(φ). The second and the third author were supported by the NSF grant DMS#0404991 and the NSA grant DMA#H98230-04-1-0115.     

Nonlinear convective flows in multilayer fluid system

The nonlinear thermocapillary and buoyant-thermocapillary flows in a three-layer system, filling a closed cavity and subjected to a temperature gradient directed along the interfaces, are investigated. The nonlinear simulations of convective regimes are performed by the finite-difference method. The process of transition of unicell structures into multicell structures is studied.     

Polymerization of ethylene oxide with potassium-graphite inclusion compounds

The polymerization of ethylene oxide (EO) in tetrahydrofuran (THF) initiated by inclusion compounds of potassium in graphite (C₂₄K) at 50°C was studied. During the initial slow stage of the polymerization the catalyst efficiency increases, while during the second rapid stage the reaction proceeds at a constant concentration of the active centers. The influence of the catalyst and monomer concentrations on the concentration of the active centers and on the catalyst efficiency was studied. The existence of the slow stage is explained by diffusion of ethylene oxide between the graphite planes, where both initiation and initial chain growth take place. The diffusion of the growing chains from the catalyst into the solution brings about the formation of a necessary concentration of active centers in the solution, which ensures the rapid growth of the polymer chains under homogeneous conditions. An x-ray study of the catalyst after polymerization shows a strongly disturbed crystal structure of the graphite. Poly-(ethylene oxide) obtained at a higher degree of conversion, does not contain low molecular weight fractions.     

Uranium oligomerization in chloride-based high temperature melts: in situ XAS studies

In situ EXAFS spectroscopic studies of uranium compounds in high temperature alkali chloride melts indicate the presence of oligomeric species. An investigation into UCl(3) and UCl(4) dissolved in LiCl reveals long range ordering of uranium atoms in the molten state which is not maintained on quenching. Studies of uranium dioxide dissolved in LiCl-KCl eutectic with HCl exhibit long range ordering in both molten and quenched states, and the EXAFS data can be modeled using multiple coordination shells.     

Formation of long-lived fluorofullerene trianions in collisions with Na

The first experimental observation of long-lived triply charged fluorofullerene anions in the gas phase obtained from C60F48 is reported. The existence of a Coloumb barrier trapping the third electron in the trianion is supposed to be responsible for detection of the species which is estimated to have negative third electron affinity.