The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .     

On Cyclotomic Numbers and the Reduction Map for the K-Theory of the Integers

We apply the recently proven compatibility of Beilinson and Soulé elements in K-theory to investigate density of rational primes p, for which the reduction map K _2n+1() K_{2n+1}(F_p)is nontrivial. Here n is an even, positive integer and F_p denotes the field of p elements. In the proof we use arithmetic of cyclotomic numbers which come from Soulé elements. Divisibility properties of the numbers are related to the Vandiver conjecture on the class group of cyclotomic fields. Using the K-theory of the integers, we compute an upper bound on the divisibility of these cyclotomic numbers.     

Type factors generated by purely infinite simple C*-algebras associated with free groups

Let be a free product of at least two but at most countably many cyclic groups. With each such group we associate a family of C*-algebras, denoted and generated by the reduced group C*-algebra and a collection of projections onto the -spaces over certain subsets of . We determine , the weak closure of in , and use this result to show that many of the C*-algebras in question are non-nuclear.

     

Enhancement of second-order optical susceptibilities of Er doped germanate glasses

During bicolor optical treatment by two coherent laser beams from 25 ps Erbium-glass laser (λ = 1.54 μm), we have established substantial increase of effective optical second-order susceptibility near the critical temperature points corresponding to the glass structural transformation near 435 °C. The increasing Er³⁺ content favors substantial increase of the photoinduced optical second harmonic generation (SHG). This temperature dependence of the second-order optical susceptibilities detected by the SHG correlates well with the differential scanning calorimetric (DSC) measurements. At the same time, the effect exists only in a narrow temperature range (up to 12 °C) . There was not temperature hysteresis. Possible physical mechanisms of the phenomenon observed are discussed.     

Experimental and computational study on the reactivity of 2,3-bis[(3-pyridylmethyl)amino]-2(Z)-butene-1,4-dinitrile, a key intermediate for the synthesis of tribenzoporphyrazine bearing peripheral methyl(3-pyridylmethyl)amino substituents

Abstract  

An earlier developed alkylating path leading to tetraalkylated diaminomaleonitrile derivatives was explored. Attempts to explain the reactivity of the representative dialkylated diaminomaleonitrile 2,3-bis[(3-pyridylmethyl)amino]-2(Z)-butene-1,4-dinitrile during the alkylation reaction were performed using X-ray and density functional theory (DFT) studies. The condensed Fukui functions accompanied by softness indices were found to be useful in explaining its reactivity observed during the reaction. The values of the Fukui functions and condensed softness for electrophilic attack calculated from Mulliken, L?wdin, and natural population analyses closely corresponded to the experimental observations. When 2,3-bis[(3-pyridylmethyl)amino]-2(Z)-butene-1,4-dinitrile disodium salt was treated with dimethyl sulfate at lower temperatures the alkylation reaction prevailed, whereas at higher temperatures the alkylating agent acted as a hydride anion acceptor, which favored the elimination reaction. The tetraalkylated dinitrile 2,3-bis[methyl(3-pyridylmethyl)amino]-2(Z)-butene-1,4-dinitrile was used in the synthesis of tribenzoporphyrazine bearing methyl(3-pyridylmethyl)amino groups, which was subsequently subjected to solvatochromic and metallation studies. The changes observed during metallation seem to result from the coordination of the 3-pyridyl group by a palladium ion. This could influence the configuration of the methyl(3-pyridylmethyl)amino moiety, causing more effective donation of a lone pair of electrons from peripheral nitrogen to the macrocyclic ring.     

The Conley index, cup-length and bifurcation

A module structure of the cohomology Conley index is used to define a relative cup-length. This invariant is applied then to prove a multiplicity theorem for periodic solutions to Hamiltonian systems.     

Spatial distribution of radioactive fallout in Poland

The spatial distribution of the post-Chernobyl radioactive fallout in Poland is presented. These results can be compared with results of computer simulations.