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     

A General Framework of Multivariate Wavelets with Duals

A comprehensive development of multivariate wavelets along with their duals is presented in this paper. The basic ingredients, such as duality relations, reconstruction and decomposition formulas, and the notion of infinite direct sums, are formulated and established in the general nonorthogonal and multivariate setting. Special emphases include duality criteria and stability conditions. As an application, new results are contributed, particularly in dual wavelets, to the existing literature on low-dimensional wavelets. In addition, the special case when the dilation matrix has determinant 2 is studied in some detail.     

Interpolation of scattered data: Distance matrices and conditionally positive definite functions

Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.