A symmetrized Hückel analysis of carbon cages of Ih point group symmetry

Some new aspects of qualitative molecular-orbital theory are developed within the context of group theory and a symmetrized Hückel approximation. Conclusions concerning the eigenvalue sums for sets of eigenfunctions of the same symmetries, the occurrence of repetitions of symmetry-adapted projections and the relations between eigenvalues of Γ_g and Γ_u symmetries are exemplified using the ₁₂₀ cage structure of I_h point symmetry. The implications of the orbit-by-orbit approach are outlined for larger clusters. This revised version was published online in July 2006 with corrections to the Cover Date.     

On Travelling Waves for the Stochastic Fisher–Kolmogorov–Petrovsky–Piscunov Equation

This paper is concerned with properties of the wave speed for the stochastically perturbed Fisher–Kolmogorov–Petrovsky–Piscunov (FKPP) equation. It was shown in the classical 1937 paper by Kolmogorov, Petrovsky and Piscunov that the large time behavior of the solution to the FKPP equation with Heaviside initial data is a travelling wave. In a seminal 1995 paper Mueller and Sowers proved that this also holds for a stochastically perturbed FKPP equation. The wave speed depends on the strength σ of the noise. In this paper bounds on the asymptotic behavior of the wave speed c(σ) as σ→0 and σ→∞ are obtained.     

Crystals and quasicrystals: A continuum model

We construct the first model of particles in the plane with completely symmetric, short range, two body interactions which has quasiperiodic, but no periodic, ground states.Supported in part by NSF Grant No. DMS-8501911     

Sign patterns that require real, nonreal or pure imaginary eigenvalues

We characterize the n-by-n sign pattern matrices that require all real, all nonreal, and all pure imaginary eigenvalues. Characterization of sign patterns that allow a real eigenvalue and those that allow a nonreal eigenvalue then follow. Some related specialized results and a characterization of sign patterns that allow a positive real eigenvalue are included.     

The small octagon with longest perimeter

The convex octagon with unit diameter and maximum perimeter is determined. This answers an open question dating from 1922. The proof uses geometric reasoning and an interval arithmetic based global optimization algorithm to solve a series of non-linear and non-convex programs involving trigonometric functions.     

An algebraic characterization of planar graphs

A cycle in a graph is a set of edges that covers each vertex an even number of times. A cocycle is a collection of edges that intersects each cycle in an even number of edges. A bicycle is a collection of edges that is both a cycle and a cocycle. The cycles, cocycles, and bicycles each form a vector space over the integers modulo two when addition is defined as symmetric difference of sets. In this paper we examine the relationship between the left-right paths in a planar graph and the cycle space, cocylce space, and bicycle space. We show that planar graphs are characterized by the existence of a diagonal—a double cover by tours that interacts with the cycle space, cocycle space, and bicycle space in a special manner. This generalizes a result of Rosenstiehl and Read that characterized those planar graphs with no nonempty bicycles. © 1995 John Wiley & Sons, Inc.     

Molecular weight control of polyacrylamide with sodium formate as a chain‐transfer agent: Characterization via size exclusion chromatography/multi‐angle laser light scattering and determination of chain‐transfer constant

We discuss the synthesis and characterization of polyacrylamide (PAM) homopolymers with carefully controlled molecular weights (MWs). PAM was synthesized via free‐radical solution polymerization under conditions that yield highly linear polymer with minimal levels of hydrolysis. The MW of the PAM homopolymers was controlled by the addition of sodium formate (NaOOCH) to the polymerization medium as a conventional chain‐transfer agent. MWs and polydispersity indices (PDIs) were determined via size exclusion chromatography/multi‐angle laser light scattering analysis; for polymerizations carried out to high conversion, PAM MWs ranged from 0.23 to 6.19 × 10⁶ g/mol, with most samples having PDI ≈2.0. Zero‐shear intrinsic viscosities of the polymers were determined via low‐shear viscometry in 0.514 M NaCl at 25 °C. Data derived from the polymer characterization were used to determine the chain‐transfer constant to NaOOCH under the given polymerization conditions and to calculate Mark–Houwink–Sakurada K and a values for PAM in 0.514 M NaCl at 25 °C. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 560–568, 2003