Delaunay graphs are almost as good as complete graphs

LetS be any set ofN points in the plane and let DT(S) be the graph of the Delaunay triangulation ofS. For all pointsa andb ofS, letd(a, b) be the Euclidean distance froma tob and let DT(a, b) be the length of the shortest path in DT(S) froma tob. We show that there is a constantc (≤((1+√5)/2) π≈5.08) independent ofS andN such that $$\frac{{DT(a,b)}}{{d(a,b)}}< c.$$      

The Model Plasma Problem: Existence of a Solution Branch

We are interested in the model plasma problem –u = u⁺in ,u = –d on , _au⁺ dx=j where is a bounded domain in with boundary ; here, j isa given positive number, the function u and the positive number are the unknowns of the problem, and d is a real parameter.Using a variant of the implicit function theorem, we can provethe existence of a global solution branch parametrized by d.The method has the advantage that it can be used for analysingthe approximation of the above problem by a finite-element method.     

Consistency of decision processes

If a statistical or a voting decision procedure is used by several subpopulations and if each reaches an identical conclusion, then one might expect this conclusion to be the outcome for the full group. It is shown that this property fails to hold for large classes of decision procedures. The geometric reasons why the consistency does not hold are described. A general theorem is given to characterize the procedures that satisfy this property of weak consistency.This research was supported in part by NSF Grant IRI-8803505 and a Guggenheim Fellowship. Also, the author thanks a referee for some comments that stimulated certain revisions.     

Reduction of operators with almost periodic symbols to operators overC *-Algebras on sections of associated bundles over a torus

A decomposition of any pseudodifferential operator (D) onR ⁿ with almost periodic symbol as 113-1 1 is obtained in the paper, where _A (D) is a pseudodifferential operator over a certainC ^*-algebraA acting on sections of a vector bundle over a torusT ⁿ whose fibre isA. The coincidence of spectra sp (D) = sp _A (D) is proved for all (D) either bounded or elliptic.     

On the long term behavior of some finite particle systems

Summary We consider the problem of comparing large finite and infinite systems with locally interacting components, and present a general comparison scheme for the case when the infinite system is nonergodic. We show that this scheme holds for some specific models. One of these is critical branching random walk onZ ^d . Let _t denote this system, and let _t ^N denote a finite version of _t defined on the torus [–N,N] ^d Z ^d . Ford3 we prove that for stationary, shift ergodic initial measures with density , that ifT(N) andT(N)/(2N+1)^d s[0,] asN, then {v }, 0 is the set of extremal invariant measures for the infinite system _t andQ _s is the transition function of Feller's branching diffusion. We prove several extensions and refinements of this result. The other systems we consider are the voter model and the contact process.Work supported in part by the National Science Foundation under Grant DMS-8802055, by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University and by the Deutsche Forschungsgemeinschaft through the SFB 123 at the Universität Heidelberg     

Variations on the theme of repeated distances

We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ^{d, d}4. We also give asymptotically tight upper bounds on the total number of occurrences of the favourite distances fromn points in ^{d, d}4. Related results are proved for distances determined byn disjoint compact convex sets in ².At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.     

A numerical analysis of chaotic behaviour in Bianchi IX models

In this paper we investigate the chaotic behaviour of the Bianchi IX cosmological models using techniques developed in the study of dynamical systems and chaotic behaviour. We numerically calculate the Lyapunov exponent, , and show that instead of converging to a constant value, it decreases steadily. We study this effect further by studying the Lyapunov exponent using short-time averages. We show that the usual method of calculating is invalid in the case of a cosmological model.     

Primitive points on constant elliptic curves over function fields

Letf:CE be a non-constant rational map between curves over a finite field, whereE is elliptic. We estimate the number of rational points ofC whose image underf generate the group of rational points ofE.     

Water 1H spin-lattice relaxation as a fingerprint of porous media

The 1H spin-lattice relaxation curves of water in samples of natural porous media can be thought of as "fingerprints" of the porous samples. Also the whole of traditional petrophysical properties (permeability, irreducible water saturation, etc.) can be thought of as "fingerprints" of the porous samples. The characteristics of the pore space determine on one hand the relaxation curve shape, and on the other the petrophysical properties. The understanding of this correspondence can contribute to a better definition of the concept of the architecture of a porous medium. At this purpose we have obtained 1H spin-lattice relaxation curves from a collection of standard sandstone cores of known petrophysical properties and characterized by the same surface properties. The results corroborate the idea that the structure of relaxation curves contains information on the distance scale and on the architecture of the pore space, even if it is difficult to extract it without ambiguities. Different methods of curve fitting were performed and compared with the aim of getting the maximum information from the relaxation curves. Several aspects of this kind of investigation indicate the analogies between 1H response of water confined in porous media and in biological tissues.     

The clinical status of low energy laser therapy in 1989

Low energy laser therapy has gained varying acceptance as a treatment for a broad range of soft tissue, musculoskeletal and neurological conditions. A controversial, but surprisingly large body of research with cell cultures suggests that laser irradiation can nondestructively alter cellular processes. Unfortunately, animal and human studies are often contradictory and difficult to evaluate due to differing study designs. As a result, the clinical effectiveness of low energy laser therapy remains debatable. Nevertheless, the findings are intriguing and deserve further investigation.