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.     

The reaction of optically active α-aminocarboxylic acid hydrazides with triethyl orthoesters

New derivatives of 2-(1-amino-1-phenylmethyl)-1,3,4-oxadiazole and 1,2,4-triazin-6-one were synthesised in the reactions of optically active α-aminocarboxylic acid hydrazides and triethyl orthoesters in xylene. The electronic and steric effects of substituents at the α position influencing the formation of five- or six-membered products are discussed.     

How does mechanism of biosorption determine the differences between the initial and equilibrium adsorption states?

The analysis of various models assumed to represent the influence of pH on heavy metals biosorption equilibrium is presented. It shows that all of them lead to the same mathematical expressions (e.g. the Langmuir or the Flory adsorption isotherm equations) when the pH effects are neglected. Even if considering the pH effects, some of them (competitive adsorption and ion-exchange models, for instance) still lead to analogical expressions for sorption isotherm equations. The accepted mechanism of biosorption may, however, influence strongly the differences between the initial and equilibrium states of biosorption system.     

A DFT study of isomeric conjugated,cross-conjugated and semi-conjugated six-membered heterocyclic mesomeric betaines

The structures and electronic properties of seven isomeric six-membered heterocyclic mesomeric betaines (HMBs) and four fully-covalent isomers have been calculated using B3LYP/6-311++G(p,d) methodology. The aromaticity indexes Bird I₆, HOMA, NICS(1) and aromatic stabilisation energy (ASE) have been calculated and compared. In agreement with an earlier study employing connectivity-matrix analysis, the previously unrecognised and little known semi-conjugated HMBs (Class 3) have calculated properties that are quite distinct from the isomeric conjugated HMBs (Class 1) and cross-conjugated HMBs (Class 2). In particular semi-conjugated systems have a higher degree of both classical and magnetic aromaticity than the isomers. In agreement with the calculated vertical ionisation potentials and electron affinities, the six-membered semi-conjugated species have high energy HOMOs and low energy LUMOs and these observations are consistent with the limited experimental data available. The results support the view that semi-conjugated rings should be recognised as a discrete class of dipolar heterocycle.