Additivity for the parametrized topological Euler characteristic and Reidemeister torsion

Dwyer, Weiss, and Williams have recently defined the notions of the parametrized topological Euler characteristic and the parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and the Reidemeister torsion of CW-complexes.     

On the Joint Path Length Distribution in Random Binary Trees

During the 10th Seminar on Analysis of Algorithms , MSRI, Berkeley, June 2004, Knuth posed the problem of analyzing the left and the right path length in a random binary tree. In particular, Knuth asked about properties of the generating function of the joint distribution of the left and the right path lengths. In this paper, we mostly focus on the asymptotic properties of the distribution of the difference between the left and the right path lengths. Among other things, we show that the Laplace transform of the appropriately normalized moment generating function of the path difference satisfies the first Painlevé transcendent . This is a nonlinear differential equation that has appeared in many modern applications, from nonlinear waves to random matrices. Surprisingly, we find out that the difference between path lengths is of the order n ^5/4 where n is the number of nodes in the binary tree. This was also recently observed by Marckert and Janson. We present precise asymptotics of the distribution's tails and moments. We will also discuss the joint distribution of the left and right path lengths. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the Wentzel, Kramers, Brillouin (WKB) method.     

Algebraicity of Global Real Analytic Hypersurfaces

Let X be an algebraic manifold without compact component and let V be a compact coherent analytic hypersurface in X, with finite singular set. We prove that V is diffeotopic (in X) to an algebraic hypersurface in X if and only if the homology class represented by V is algebraic and singularities are locally analytically equivalent to Nash singularities. This allows us to construct algebraic hypersurfaces in X with prescribed Nash singularities.     

Concerning the Basicity of 4-Dimethylaminoquinazoline Derivatives

 Two series of selected 4-N,N-dimethylamino-2-phenylquinazoline and 2-N,N-diethylamino-4-N,N-dimethylamino-quinazoline derivatives obtained in the reaction of substituted N-phenylbenzimidoyl chlorides or N¹,N¹-diethyl-N²-phenylchlorocarboxyamidines with N,N-dimethylcyanamide in the presence of TiCl₄ were examined in order to detect the protonation centre. The untypical correlations between pK _a and σ were supported by MNDO calculations and single crystal X-ray diffraction data and point to a protonation of the ring-N-atoms with delocalization of the positive charge into the N,N-dimethylamino group.     

Chemical and enzymatic stability of the oleanoyl oxime ibuprofenate

The stress tests as well as photostability analysis in solutions and the solid phase of the oleanoyl oxime ibuprofenate (Ibu-OxOA) were determined according the International Conference on Harmonization guidelines. For observation of the degradation of tested compounds, the reversed phase high-performance liquid chromatography (RP-HPLC) method was used. The study included the effect of temperature, water, H⁺ and OH⁻ ions, hydrogen peroxide, and light (6.0 × 10⁶, 1.2 × 10⁶ lux·h) on the stability of the tested hybrid. Studies have shown that these compounds are not stable in a neutral, acidic, or oxidizing medium; very unstable in an alkaline medium; and photostable in a solution and photostable in the solid phase. Studies on the enzymatic stability of the tested hybrid showed that it is not susceptible to degradation in the presence of plasma enzymes as well as in the presence of esterase (37°C, 80% human plasma). The degradation of this compound was observed in the presence of lipase (from Candida antarctica), and the linear dependence of the determined kinetic parameters of the reaction on the enzyme activity was demonstrated.     

One Dimensional Enhanced Anhydrous Proton Conduction in Well Defined Molecular Columns Induced by Non-Covalent Interactions

1D anhydrous proton conduction is enhanced significantly in ionic channels created by self-assembly of functionalized organic phosphonic acid and aromatic heterocyclic 1,2,4-triazole molecules. This study reveals high proton conduction in one dimension through a well-defined supramolecular architecture in which two different molecules undergo host–guest synergy and self-assemble to provide two-fold advantages: 1) formation of the ionic channels and 2) higher proton conduction in the absence of water. A clear correlation is found between the phenomena of ionic channels and anhydrous conductivity in the absolute dry state and we demonstrate that the one-dimensional conductivity can be as high as that recorded for 3D channels in, for instance, Nafion.