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.     

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝ ^d by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe. The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.      

X-ray investigations of sulfur-containing fungicide: Structure of the benzoylphenyl(4-tolylazo)methyl 4-tolyl sulfone

Crystal and molecular structure of the of benzoylphenyl(4-tolylazo)methyl 4-tolyl sulfone was determined by x-ray analysis; crystal system, monoclinic; space group P2 ₁/c; a = 12.807(1) Å, b = 17.512(1) Å, c = 11.879(1) Å, β = 111.23(1)°. Conformation of the title compound is constrained by attraction of the positively charged sulfur and β-carbon atoms to the negatively charged oxygen atoms. Characteristic distribution of bond lengths in the central part of a molecule suggests that electrostatic interactions are strengthened by the stereoelectronic donation of electron density from the α-azo N2 towards the sulfonyl S and β-carbonyl C2 atoms.     

Boundary layer in compressible flow of wet water vapour

In this paper the influence of small droplets, with radius $\text{10}{}^{\mathrm{?8}}\text{m}\text{< r < 10}{}^{\mathrm{?6}}\text{m}$, on laminar and turbulent boundary layer behavior is considered. It is found that the laminar boundary layer in a two-phase flow with strongly dispersed liquid retains dissipation energy and that the recovery factor of enthalpy is greater than unity. In turbulent boundary layers small droplets are transported by turbulent diffusion and this leads to the recovery factor being less than unity. Its value in both cases depends mainly on the nondimensional number $\text{Ds}\text{= C}{}_{\mathrm{Le}}\text{L/(U}{}_{\mathrm{e}}{}^2\text{/2)}$. The laminar boundary layer solution for non-equilibrium two-phase flow is obtained. Profiles of the droplet mass fraction, vapour and droplets temperatures and droplet radius are computed for the case of a steady two-dimensional flow. The turbulent boundary layer is treated using a semi-empirical theory assuming thermodynamic equilibrium.