Maximal Holonomy of Almost Bieberbach Groups for Heis5

Let Heis _2n+1 be the Heisenberg group of dimension 2n + 1 and M an infra-nilmanifold with Heis _2n+1-geometry. The fundamental group of M contains a cocompact lattice of Heis _2n+1 with index bounded above by a universal constant I _n+1, i.e., I _n+1 is the maximal order of the holonomy groups. We prove that I ₃ = 24. As an application we give an estimate for the volumes of finite volume non-compact complex hyperbolic 3-manifolds.     

The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum

We realize the lamplighter group /2 as a group defined by a 2-state automaton. We study the corresponding action of this group on a binary tree and on its boundary. The final goal is the computation for a special system of generators of the spectrum of the Markov (or the random walk) operator which is [–1,1] in this case and of the spectral measure which is a discrete measure concentrated on a dense countable set of points in [–1,1] (a new effect unseen before for Markovian operators on groups which leads to a counterexample to the Strong Atiyah Conjecture). This is done by the computation of spectra of finite-dimensional approximations of the operator and uses an idea of fractalness in a similar way it was used by Bartholdi and Grigorchuk for the computation of the spectra of some branch groups. We also obtain the asymptotic of type e^–1/1–x of the spectral measure in the neighborhood of 1 and show that Følner sets grow exponentially.     

The yellow cake

In this paper we consider the following property:

For every function there are functions

(for ) such that

We show that, despite some expectation suggested by S. Shelah (1997), does not imply . Next, we introduce cardinal characteristics of the continuum responsible for the failure of .

     

Some descriptive set-theoretic properties of the isomorphism relation between Banach spaces

Consider the space of closed linear subspaces of a separable Banach space equipped with the standard Effros Borel structure. The isomorphism relation between Banach spaces being elements of  determines a partition of  . In this note we prove a result describing the complexity of analytic subsets of  intersecting a large enough number of the above-mentioned parts of  .

     

Self-organizing neural networks for modeling robust 3D and 4D QSAR: application to dihydrofolate reductase inhibitors

We have used SOM and grid 3D and 4D QSAR schemes for modeling the activity of a series of dihydrofolate reductase inhibitors. Careful analysis of the performance and external predictivities proves that this method can provide an efficient inhibition model.     

Organic spin clusters: macrocyclic-macrocyclic polyarylmethyl polyradicals with very high spin S = 5-13

Synthesis and magnetic studies of a new class of organic spin clusters, possessing alternating connectivity of unequal spins, are described. Polyarylmethyl polyether precursors to the spin clusters, with linear and branched connectivity between calix[4]arene-based macrocycles, are prepared via modular, multistep syntheses. Their molecular connectivity and stereoisomerism are analyzed using NMR spectroscopy. The absolute masses (4-10 kDa) are determined by FABMS and GPC/MALS. Small angle neutron scattering (SANS) provides the radii of gyration of 1.2-1.8 nm. The corresponding polyradicals with 15, 22, and 36 triarylmethyls, which are prepared and studied as solutions in tetrahydrofuran-d(8), may be described as S' = 7/2, 1/2, 7/2 spin trimer (average S = 5-6), S' = 7/2, 1/2, 6/2, 1/2, 7/2 spin pentamer (average S = 7-9), and spin nonamer (average S = 11-13), respectively, as determined by SQUID magnetometry and numerical fits to linear combinations of the Brillouin functions. For spin trimer and pentamer, the quantitative magnetization data are fit to new percolation models, based upon random distributions of chemical defects and ferromagnetic vs antiferromagnetic couplings. The value of S = 13 is the highest for an organic molecule.     

The Deletion Method For Upper Tail Estimates

We present a new method to show concentration of the upper tail of random variables that can be written as sums of variables with plenty of independence. We compare our method with the martingale method by Kim and Vu, which often leads to similar results.Some applications are given to the number X _G of copies of a graph G in the random graph (n,p). In particular, for G = K ₄ and G = C ₄ we improve the earlier known upper bounds on —ln (X _G 2 X _G ) in some range of p = p(n).* Research of the second author supported by KBN grants 2 P03A 032 16 and 2 P03A 015 23.     

Drastic reduction of plasmon damping in gold nanorods

The dephasing of particle plasmons is investigated using light-scattering spectroscopy on individual gold nanoparticles. We find a drastic reduction of the plasmon dephasing rate in nanorods as compared to small nanospheres due to a suppression of interband damping. The rods studied here also show very little radiation damping, due to their small volumes. These findings imply large local-field enhancement factors and relatively high light-scattering efficiencies, making metal nanorods extremely interesting for optical applications. Comparison with theory shows that pure dephasing and interface damping give negligible contributions to the total plasmon dephasing rate.