The Poincaré Map of Randomly Perturbed Periodic Motion

A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincaré map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends to zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of the stability of periodic motion to random perturbations. In addition, we show that the geometric distribution of the first exit times translates into statistical properties of solutions of important differential equation models in applications. To this end, we demonstrate three distinct examples from mathematical neuroscience featuring complex oscillatory patterns characterized by the geometric distribution. We show that in each of these models the statistical properties of emerging oscillations are fully explained by the general properties of randomly perturbed periodic motions identified in this paper.     

A Characterization of Finitely Decidable Congruence Modular Varieties

For every finitely generated, congruence modular variety of finite type we find a finite family of finite rings such that the variety is finitely decidable if and only if is congruence permutable and residually small, all solvable congruences in finite algebras from are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra from is comparable with all congruences of , each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring from , the variety of -modules is finitely decidable.

     

DFT study of Co-C bond cleavage in the neutral and one-electron-reduced alkyl-cobalt(III) phthalocyanines

Density functional theory (DFT) has been applied to the analysis of the structural and electronic properties of the alkyl-cobalt(III) phthalocyanine complexes, [CoIIIPc]-R (Pc = phthalocyanine, R = Me or Et), and their pyridine adducts. The BP86/6-31G(d) level of theory shows good reliability for the optimized axial bond lengths and bond dissociation energies (BDEs). The mechanism of the reductive cleavage was probed for the [CoIIIPc]-Me complex which is known as a highly effective methyl group donor. In the present analysis, which follows a recent study on the reductive Co-C bond cleavage in methylcobalamin (J. Phys. Chem. B 2007, 111, 7638-7645), it is demonstrated that addition of an electron and formation of the pi-anion radical [CoIII(Pc*)]-Me- significantly lowers the energetic barrier required for homolytic Co-C bond dissociation. Such BDE lowering in [CoIII(Pc*)]-Me- arises from the involvement of two electronic states: upon electron addition, a quasi-degenerate pi*Pc state is initially formed, but when the cobalt-carbon bond is stretched, the unpaired electron moves to a sigma*Co-C state and the final cleavage involves the three-electron (sigma)2(sigma*)1 bond. As in corrin complexes, the pi*Pc-sigma*Co-C states crossing does not take place at the equilibrium geometry of [CoIII(Pc*)]-Me- but only when the Co-C bond is stretched to approximately 2.3 A. The DFT computed Co-C BDE of 23.3 kcal/mol in the one-electron-reduced phthalocyanine species, [CoIII(Pc*)]-Me-, is lowered by approximately 37% compared to the neutral Py-[CoIIIPc]-Me complex where BDE = 36.8 kcal/mol. A similar comparison for the corrin-containing complexes shows that a DFT computed BDE of 20.4 kcal/mol for [CoIII(corrin*)]-Me leads to approximately 45% bond strength reduction, in comparison to 37.0 kcal/mol for Im-[CoIII(corrin)]-Me+. These results suggest some preference by the alkylcorrinoids for the reductive cleavage mechanism.     

Direct catalytic asymmetric three-component Mannich reactions with dihydroxyacetone: enantioselective synthesis of amino sugar derivatives

Highly enantioselective, amino acid-catalyzed, one-pot three-component asymmetric Mannich reactions between dihydroxyacetone, p-anisidine, and aldehydes are presented. The reactions proceeded with high chemo- and stereoselectivity and furnished the corresponding α,α′-dihydroxy-β-aminoketones in high yields with 82-95% ee.     

One-pot catalytic enantioselective domino nitro-michael/michael synthesis of cyclopentanes with four stereocenters

A highly enantioselective organocatalytic one-pot synthesis of nitro-, formyl-, and ester-functionalized cyclopentanes with four stereocenters is presented. The cyclopentanes were formed as a predominant diasteroisomer and isolated in high yields with 97-99 % ee.     

Self-assembled monolayer of alkanephosphoric acid on nanotextured Ti

Surface modification of titanium and its alloys is of great importance for their practical application as biomedical implants. We have studied and compared assembly of dodecylphosphoric acid on commercial polished and on nanostructured titanium disks. The latter were produced by chemical etching that created nanoscale pits of typical size of about 20 nm. Enhanced hydrophobicity and high molecular density were obtained after functionalization of the nanotextured substrate. Aging tests showed a lifetime of the organic films of about one month in phosphate buffer. The samples were characterized by means of infrared spectroscopy, contact angle measurements, ellipsometry, and atomic force and scanning tunneling microscopies.     

Electrophoretic assay for penicillinase: substrate specificity screening by parallel CE with an active pixel sensor

We report application of a new UV imaging detector incorporating an active pixel sensor in an electrophoretic enzyme assay for penicillinase (beta-lactamase) with multiple substrates. The method based on electrophoretically mediated microanalysis was developed on a standard CE system with a single-point diode array detector and 200 nm UV wavelength, then transferred to a parallel capillary setup with the UV imaging detector for screening of penicillinase substrate specificity. One capillary is used for the assay and the other for reference, with an enzyme solution plug introduced into the first at the same time as a water plug into the second capillary. A mixture of antibiotics and markers is subsequently introduced as a sample plug to both capillaries, and driven through the enzyme (or water) plug by application of voltage. Most individual reactant and product peaks were separated and compounds amenable to beta-lactam hydrolysis could readily be identified and the extent of the reaction quantified within a single electrophoretic run.     

Greedy wavelet projections are bounded on BV

Let be the space of functions of bounded variation on with . Let , , be a wavelet system of compactly supported functions normalized in , i.e., , . Each has a unique wavelet expansion with convergence in . If is the set of indicies for which are largest (with ties handled in an arbitrary way), then is called a greedy approximation to . It is shown that with a constant independent of . This answers in the affirmative a conjecture of Meyer (2001).

     

Electrostatically Self-Assembled Multilayers of Chitosan and Xanthan Studied by Atomic Force Microscopy and Micro-Interferometry

The thickness and surface morphology of electrostatically self-assembled films of chitosan and xanthan (persistence length of ∼120nm) have been studied using dual-wavelength Reflection Interference Contrast Microscopy (DW-RICM) and tapping mode Atomic Force Microscopy (AFM). The multilayers were prepared at two ionic strengths (5mM and 150mM). When the multilayers were assembled at 150 mM a network like morphology was observed after one bilayer. This structure was found to be of large influence in the further growth of the multilayers, with the same kind of network structure being observed at all number of bilayers. A lack of swelling behaviour, as well as the network structure and the poresize of the network, is suggested to originate from the high chain stiffness of xanthan.     

On upper bounds for high order Neumann eigenvalues of convex domains in Euclidean space

We derive sharp upper bounds for eigenvalues of the Laplacian under Neumann boundary conditions on convex domains with given diameter in Euclidean space. We use the Brunn-Minkowski theorem in order to reduce the problem to a question about eigenvalues of certain classes of Sturm-Liouville problems.