On the Characterization of Expansion Maps for Self-Affine Tilings

We consider self-affine tilings in ℝ ⁿ with expansion matrix φ and address the question which matrices φ can arise this way. In one dimension, λ is an expansion factor of a self-affine tiling if and only if |λ| is a Perron number, by a result of Lind. In two dimensions, when φ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex λ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for φ to be an expansion matrix for any n, assuming only that φ is diagonalizable over ℂ. We conjecture that this condition on φ is also sufficient for the existence of a self-affine tiling.     

Solution properties of ethylene-vinyl acetate copolymers

High-pressure ethylene–vinyl acetate copolymers of four different chemical compositions(9%, 15%, 45%, and 70% VA) were characterized to determine molecular weight and distribution. The four samples were fractionated by solvent–nonsolvent precipitation methods. Light-scattering, osmometry, and viscosity measurements were made on these fractionated copolymers to determine weight-average molecular weight \documentclass{article}\pagestyle{empty}\begin{document}$ \overline {M_w } $\end{document}, number-average molecular weight \documentclass{article}\pagestyle{empty}\begin{document}$ \overline {M_n } $\end{document}, molecular size in solution, and interaction constants. Dilute solution viscosity was measured on the fractions to determine intrinsic viscosity and Huggins' constant k′. Viscosity–molecular weight equations were established for the four copolymer compositions. The log intrinsic viscosity versus log molecular weight diagrams were analyzed and the average length of branches calculated. The composition of the polymer fractions, determined by C and H combustion analysis, was found not to vary significantly with molecular weight. The uniformly random character of the E/VA copolymers was thereby confirmed. The density of the fractions was determined by density-gradient column method. Chain sequence distribution of monomer units for the four copolymers was calculated by using IBM 704 computations involving the actual monomer reactivity ratios. Long sequences of either ethylene or vinyl acetate are improbable, except at the extremes of copolymer composition.     

Synthesis,chiroptical properties and absolute configuration of α-phenylglycidic acid

The (+)- and (-) enantiomers of potassium α-phenylglycidate, an irreversible inhibitor of the enzyme mandelate racemase, were synthesized by resolution of the diastereomeric esters with R-(-)-2-octanol. Base-catalyzed ring-opening of the resolved α-phenylglycidate esters gave the enantiomers of 2,3-dihydroxy-2-phenylpropanoic acid, also obtained by resolution of the racemic dihydroxy acid using ephedrine. A comparison of the chiroptical properties of the esters of α-phenylglycidic and 2,3-dihydroxy-2-phenylpropanoic acids with those of the structurally similar atrolactic and mandelic acids and their 2-methoxy-derivatives showed that the (-)-methyl 2,3-dihydroxy-2-phenylpropanoate corresponding to the (+)-enantiomer of potassium α-phenylglycidate, as well as the esters of α-phenylglycidic acid derived from the same (+)-potassium salt, were all configurationally related to S-(+)-atrolactic and mandelic acids. The configurational assignments made on the basis of the chiroptical data were confirmed by lithium aluminum hydride reduction of the (-)-2-octyl S- and R-α-phenylglycidates, which led exclusively to the R-(-)- and S-(+)-2-phenyl-1, 2-propanediols, respectively, previously related configurationally to R-(-)- and S-(+)-atrolactic acids.     

Approximating the number of monomer-dimer coverings of a lattice

We study the problem of counting the number of coverings of ad-dimensional rectangular lattice by a specified number of monomers and dimers. This problem arises in several models in statistical physics, and has been widely studied. A classical technique due to Fisher, Kasteleyn, and Temperley solves the problem exactly in two dimensions when the number of monomers is zero (the dimer covering problem), but is not applicable in higher dimensions or in the presence of monomers. This paper presents the first provably polynomial-time approximation algorithms for computing the number of coverings with any specified number of monomers ind-dimensional rectangular lattices with periodic boundaries, for any fixed dimensiond, and in two-dimensional lattices with fixed boundaries. The algorithms are based on Monte Carlo simulation of a suitable Markov chain, and, in constrast to most Monte Carlo algorithms in statistical physics, have rigorously derived performance guarantees that do not rely on any assumptions. The method generalizes to counting coverings of any finite vertex-transitive graph, a class which includes most natural finite lattices with periodic boundary conditions.     

On boolean decision trees with faulty nodes

We consider the problem of computing with faulty components in the context of the Boolean decision tree model, in which cost is measured by the number of input bits queried, and the responses to queries are faulty with a fixed probability. We show that if f can be represented in k-DNF form and in j-CNF form, then O(n log(min(k, j)/q)) queries suffice to compute f with error probability less than q, where n is the number of input bits. © 1994 John Wiley & Sons, Inc.     

How to take short cuts

We present an elementary construction for adding short cuts on a rectifiable curve of the plane, so that any two points on the curve are linked by a path along the curve and short cuts, of total length at most a constant times their Euclidean distance. The total length of the short cuts is a constant times the perimeter of the curve. This work was done while Claire Kenyon was at Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France, and while Richard Kenyon was at IHES, 35 route de Chartres, 91440 Bures-sur-Yvette, France.     

Limit shapes and the complex Burgers equation

In this paper we study surfaces in R ³ that arise as limit shapes in random surface models related to planar dimers. These limit shapes are surface tension minimizers, that is, they minimize a functional of the form ∫σ(∇h) dx dy among all Lipschitz functions h taking given values on the boundary of the domain. The surface tension σ has singularities and is not strictly convex, which leads to formation of facets and edges in the limit shapes. We find a change of variables that reduces the Euler–Lagrange equation for the variational problem to the complex inviscid Burgers equation (complex Hopf equation). The equation can thus be solved in terms of an arbitrary holomorphic function, which is somewhat similar in spirit to Weierstrass parametrization of minimal surfaces. We further show that for a natural dense set of boundary conditions, the holomorphic function in question is, in fact, algebraic. The tools of algebraic geometry can thus be brought in to study the minimizers and, especially, the formation of their singularities. This is illustrated by several explicitly computed examples.