Universal lattices and property tau

We prove that the universal lattices – the groups G=SL_d(R) where R=ℤ[x₁,...,x_k], have property τ for d≥3. This provides the first example of linear groups with τ which do not come from arithmetic groups. We also give a lower bound for the τ-constant with respect to the natural generating set of G. Our methods are based on bounded elementary generation of the finite congruence images of G, a generalization of a result by Dennis and Stein on K₂ of some finite commutative rings and a relative property T of . Mathematics Subject Classification (2000) 20F69, 13M05, 19C20, 20G05, 20H05     

The Influence of HCl Concentration on the Rate of the Hydrolysis–Condensation Reaction of Phenyltrichlorosilane and the Yield of (Tetrahydroxy)(Tetraphenyl)Cyclotetrasiloxanes,Synthesis of All Its Geometrical Isomers and Thermal Self-Condensation of Them under “Pseudo”-Equilibrium Conditions

The rate of hydrolysis–condensation reaction of phenyltrichlorosilane in water-acetone solutions and the product yields were shown to significantly depend on the concentration of HCl (C_HCl) in the solutions. The main product of the reaction was all-cis-(tetrahydroxy)(tetraphenyl)cyclotetrasiloxane. This was different from the earlier published results of analogous reactions of m-tolylSiCl₃, m-ClPhSiCl₃, and α-naphtylSiCl, in which some products of other types were formed. For example, trans-1,1,3,3-tetrahydroxy-1,3-di-α-naphtyldisiloxane was obtained in the case of α-naphtylSiCl₃. All-cis-(tetrahydroxy)(tetraphenyl)cyclotetrasiloxane was treated in acetone with HCl to give the other three geometric isomers (cis-cis-trans-, cis-trans-, and all-trans-). The thermal self-condensation of these four isomers under “pseudo”-equilibrium conditions (under atmospheric pressure) was investigated in different solvents, in quartz or molybdenum glass flasks. The compositions of the products were monitored by APCI-MS and ²⁹Si NMR spectroscopy. It was shown that all-cis- and cis-cis-trans-isomers in toluene or anisole mostly gave the cage-like Ph-T_8,10,12,14 and uncompleted cage-like Ph-T_10,12OSi(HO)Ph compounds. In contrast to these two isomers, the cis-trans–isomer in toluene mainly formed dimers with the loss of one or two molecules of water. However, in acetonitrile, significant amounts of Ph-T_10,12 and Ph-T_10,12OSi(HO)Ph species were formed along with the dimers. All-trans-isomer did not enter into the reaction at all.     

On Linearly Kleiman Groups

Let V denote a finite-dimensional vector space over some field. We say that a linear group G ≤ GL(V) is a linearly Kleiman group if, for every pair of linear subspaces v, u ≤ V, there is an element g ∈ G such that the subspaces g(v), u are in general position. The main result of this paper is the classification of connected linear algebraic groups over a field of characteristic zero which are linearly Kleiman. We also consider some properties of linearly Kleiman groups.     

A description of Baer-Suzuki type of the solvable radical of a finite group

We obtain the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g∈G such that for any 3 elements a₁,a₂,a₃∈G the subgroup generated by the elements , i=1,2,3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2,3}^′-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups).     

Photoconductivity of thin organic films

Thin organic films were deposited on silicon oxide surfaces with golden interdigitated electrodes (interelectrode gap was 2 μm), and the film resistivities were measured in dark and under white light illumination. The compounds selected for the measurements include molecules widely used in solar cell applications, such as polythiophene (PHT), fullerene (C₆₀), pyrelene tetracarboxylic diimide (PTCDI) and copper phthalocyanine (CuPc), as well as molecules potentially interesting for photovoltaic applications, e.g. porphyrin-fullerene dyads. The films were deposited using thermal evaporation (e.g. for C₆₀ and CuPc films), spin coating for PHT, and Langmuir-Schaeffer for the layer-by-layer deposition of porphyrin-fullerene dyads. The most conducting materials in the series are films of PHT and CuPc with resistivities 1.2 × 10³ Ω m and 3 × 10⁴ Ω m, respectively. Under light illumination resistivity of all films decreases, with the strongest light effect observed for PTCDI, for which resistivity decreases by 100 times, from 3.2 × 10⁸ Ω m in dark to 3.1 × 10⁶ Ω m under the light.     

A Symplectic Generalization of the Peradzyński Helicity Theorem and Some Applications

Symplectic and symmetry analysis for studying MHD superfluid flows is devised, a new version of the Z. Peradzyński (Int. J. Theor. Phys. 29(11):1277–1284, [1990]) helicity theorem based on differential-geometric and group-theoretical methods is derived. Having reanalyzed the Peradzyński helicity theorem within the modern symplectic theory of differential-geometric structures on manifolds, a new unified proof and a new generalization of this theorem for the case of compressible MHD superfluid flow are proposed. As a by-product, a sequence of nontrivial helicity type local and global conservation laws for the case of incompressible superfluid flow, playing a crucial role for studying the stability problem under suitable boundary conditions, is constructed.     

Highly efficient nonlinear filter for femtosecond pulse contrast enhancement and pulse shortening

We propose a highly efficient scheme for temporal filters devoted to femtosecond pulse contrast enhancement. The filter is based on cross-polarized wave generation with a spatially suger-Gaussian-shaped beam. In a single nonlinear crystal scheme the energy conversion to the cross-polarized pulse can reach 28%. We demonstrate that the process enables a significant spectral broadening. For an efficiency of 23% the pulse shortening is estimated to 2.2, leading to an intensity transmission of the nonlinear filter of 50%.     

Stationary Configurations of Point Vortices on a Cylinder

In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation Γ₁ and Γ₂ (Γ₂ = ?μΓ₁) are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained.     

Design and Synthesis of Highly Photostable Yellow–Green Emitting 1,8-Naphthalimides as Fluorescent Sensors for Metal Cations and Protons

Two highly photostable yellow–green emitting 1,8-naphthalimides 5 and 6, containing both N-linked hindered amine moiety and a secondary or tertiary cation receptor, were synthesized for the first time. Novel compounds were configured as “fluorophore–spacer–receptor” systems based on photoinduced electron transfer. Photophysical characteristics of the dyes were investigated in DMF and water/DMF (4:1, v/v) solution. The ability of the new compounds to detect cations was evaluated by the changes in their fluorescence intensity in the presence of metal ions (Cu²⁺, Pb²⁺, Zn²⁺, Ni²⁺, Co²⁺) and protons. The presence of metal ions and protons was found to disallow a photoinduced electron transfer leading to an enhancement in the dye fluorescence intensity. Compound 5, containing secondary amine receptor, displayed a good sensor activity towards metal ions and protons. However the sensor activity of dye 6, containing a tertiary amine receptor and a shorter hydrocarbon spacer, was substantially higher. The results obtained indicate the potential of the novel compounds as highly photostable and efficient “off–on” pH switchers and fluorescent detectors for metal ions with pronounced selectivity towards Cu²⁺ ions.