Self-Interactions of Strands and Sheets

Physical strands or sheets that can be modelled as curves or surfaces embedded in three dimensions are ubiquitous in nature, and are of fundamental importance in mathematics, physics, biology, and engineering. Often the physical interpretation dictates that self-avoidance should be enforced in the continuum model, i.e., finite energy configurations should not self-intersect. Current continuum models with self-avoidance frequently employ pairwise repulsive potentials, which are of necessity singular. Moreover the potentials do not have an intrinsic length scale appropriate for modelling the finite thickness of the physical systems. Here we develop a framework for modelling self-avoiding strands and sheets which avoids singularities, and which provides a way to introduce a thickness length scale. In our approach pairwise interaction potentials are replaced by many-body potentials involving three or more points, and the radii of certain associated circles or spheres. Self-interaction energies based on these many-body potentials can be used to describe the statistical mechanics of self-interacting strands and sheets of finite thickness.     

Prediction of critical temperature of n-alkanes and n-alkenes from surface tension vs. temperature data

 The surface tension versus temperature data of homologous series of n-alkanes and n-alkenes is analyzed. Critical temperatures are shown to be predicted from these data, with a very high precision, after some corrections are made. The corrections are shown to arise from the effect of the critical pressure on the extrapolated data to surface tension approaching zero. Received: 29 October 1996 Accepted: 13 December 1996     

Artificial time integration

Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be “close enough” to the dynamics of the continuous system (which is typically easier to analyze) provided that small – hence many – time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought. In this article we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency. AMS subject classification (2000)  65L05, 65M32, 65N21, 65N22, 65D18     

On the excitation of a closely spaced array by a line source

An infinite row of periodically spaced, identical rigid circularcylinders is excited by an acoustic line source which is parallelto the generators of the cylinders. A method for calculatingthe scattered field accurately and efficiently is presented.When the cylinders are sufficiently close together, Rayleigh–Blochsurface waves that propagate energy to infinity along the arrayare excited. An expression is derived which enables the amplitudesof these surface waves to be computed without requiring thesolution to the full scattering problem.     

Contact angle hysteresis of liquid drops as means to measure adhesive energy of zein on solid substrates

Adhesion of zein to solid substrates has been studied using surface energy profiles as indices and by adhesion mapping using atomic force microscopy (AFM). Different plasticizers like glycerol and sorbitol have been used to form mixed films with zein and properties of these films are studied using surface energy profiles. Comparison of the results from the different mixed samples with those from the pure zein films showed that force mapping could identify areas rich in protein. The adhesion maps produced were deconvoluted from sample topography and contrasted with the data obtained from contact angle measurements. A comparison of the two methods shows that the extent of contact angle hysteresis is indicative of both hydrophobicity of the surface as well as the force of adhesion. Mechanical properties and microstructure of zein films prepared by casting from solutions and using Langmuir-Blodgett film technique have been investigated. Pure zein seemed brittle and exhibited an essentially linear relationship between stress and strain. Films with plasticizer were tougher than these films. In general, mixed films showed better mechanical properties than pure films and had higher ultimate tensile strength and increased per cent elongation. Further, the mixed films of zein showed a higher force of adhesion compared to the pure films.     

Vassiliev invariants for braids on surfaces

We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit a universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the surface.

     

Numerical calculations of temperature distribution of double layer metallic surface treated by laser beams

A mathematical model of laser beam treatment of double layer alloys (Ni/Fe, Al/Fe and Cr/Fe systems) describing the effect of laser beam on different physical and geometrical parameters of coated layer system has been adapted. The numerical solutions of the non-homogeneous heat-transport differential equation could estimate the temperature of the treated region.The suggested model allows investigation of the temperature distribution as a function of treated surface and laser parameters. The physical parameters of the treated materials were taken as functions of temperature due to the change in the temperature of the treated double layer materials.     

2‐connected 7‐coverings of 3‐connected graphs on surfaces

An m‐covering of a graph G is a spanning subgraph of G with maximum degree at most m. In this paper, we shall show that every 3‐connected graph on a surface with Euler genus k ≥ 2 with sufficiently large representativity has a 2‐connected 7‐covering with at most 6k ? 12 vertices of degree 7. We also construct, for every surface F² with Euler genus k ≥ 2, a 3‐connected graph G on F² with arbitrarily large representativity each of whose 2‐connected 7‐coverings contains at least 6k ? 12 vertices of degree 7. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 26–36, 2003     

Standardization of Methods for Characterizing the Surface Geometry of Solids

Since a comprehensive survey published in 1999 [1] much work was done in standardizing measuring methods to characterize the surface geometry of dispersed and/or porous solids and to certify reference materials. The present paper is an extension of a short communication [2]. It gives a survey on existing standards and reports on new drafts and proposals.     

Synthesis and adsorption properties of gold nanoparticles within pores of surface‐functional porous polymer microspheres

Mesoporous polymer microspheres with gold (Au) nanoparticles inside their pores were prepared considering their surface functionality and porosity. The Au/polymer composite microspheres prepared were characterized by transmission electron microscope (TEM), X‐ray diffraction (XRD), and Brunauer–Emmett–Teller (BET) techniques. The results showed that the adsorption of Au nanoparticles could be increased by imparting the pore structure and surface‐functional groups into the supporting polymer microspheres (in this study, poly (ethylene glycol dimethacrylate‐co‐acrylonitrile) and poly (EGDMA‐co‐AN) system). Above all, from this study, it was established that the porosity of the polymer microspheres is the most important factor that determines the distribution and adsorption amount of face‐centered cubic (fcc) Au nanoparticles in the final products. Our study showed that the continuous adsorption of Au nanoparticles with the aid of the large surface area and surface interaction sites formed more favorably the Au/polymer composite microspheres. The BET measurements of Au/poly(EGDMA‐co‐AN) composite microspheres reveals that the adsorption of Au nanoparticles into the pores kept the pore structure intact and made it more porous. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 5627–5635, 2004