The distance between homotopy classes of <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-valued maps in multiply connected domains

Certain Sobolev spaces of S ¹-valued functions can be written as a disjoint union of homotopy classes. The problem of finding the distance between different homotopy classes in such spaces is considered. In particular, several types of one-dimensional and two-dimensional domains are studied. Lower bounds are derived for these distances. Furthermore, in many cases it is shown that the lower bounds are sharp but are not achieved. The first author’s work of was supported in part by NSF grant 0503887. The second author’s research of was supported by G.S. Elkin research fund.     

On the solid state of hydrogen fluoride: A self-consistent crystal field study

A recently developed self-consistent crystal field (SCCF ) method is applied to two suggested crystal structures of solid hydrogen fluoride, polar and nonpolar, respectively. Constrained geometry optimizations are performed and the results are compared with experiment and previous theoretical studies. Comparisons are also made with a previous SCCF study on hydrogen cyanide. The Pauli repulsion, dispersion, and “classical” crystal field contributions to the lattice energy are calculated. © 1993 John Wiley & Sons, Inc.     

Survival of the weak in hyperbolic spaces, a remark on competition and geometry

A simple competition model is presented. While in Euclidean spaces the weak will ``die out', in the presence of hyperbolicity, coexistence take place.

     

Inclusion complex formation in a particular geometry by a water-soluble paracyclophane in aqueous solution — NMR Studies —

NMR studies (¹H and ¹³C) demonstrating an inclusion complex formation in a particular geometry between a water-soluble paracyclophane, CP44, and hydrophobic substrates in acidic aqueous solution are described.     

Bi-Lipschitz bijection between the Boolean cube and the Hamming ball

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n ∈ N there exists an explicit bijection ψ: {0, 1}ⁿ → {x ∈ {0, 1}ⁿ⁺¹ : |x| > n/2} such that for every x ≠ y ∈ {0, 1}ⁿ, \(\frac{1}{5} \leqslant \frac{{dis\tan ce\left( {\psi \left( x \right),\psi \left( y \right)} \right)}}{{dis\tan ce\left( {x,y} \right)}} \leqslant 4,\) where distance(·, ·) denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive.     

On a minimization problem with a mass constraint involving a potential vanishing on two curves

We study a singular perturbation type minimization problem with a mass constraint over a domain in ℝ ^N , involving a potential vanishing on two curves in the plane. We demonstrate how the behavior of the minimizers depends on the geometry of the domain and, more precisely, on its isoperimetric profile.     

Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points

We study the pioneer points of the simple random walk on the uniform infinite planar quadrangulation (UIPQ) using an adaptation of the peeling procedure of Angel (Geom Funct Anal 13:935–974, 2003) to the quadrangulation case. Our main result is that, up to polylogarithmic factors, n ³ pioneer points have been discovered before the walk exits the ball of radius n in the UIPQ. As a result we verify the KPZ relation Knizhnik et al. (Modern Phys Lett A 3:819–826, 1988) in the particular case of the pioneer exponent and prove that the walk is subdiffusive with exponent less than 1/3. Along the way, new geometric controls on the UIPQ are established.