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 $$\frac{{DT(a,b)}}{{d(a,b)}}< c.$$      

Factorizations of the Complete Graph into C 5-Factors and 1-Factors

In this paper, we show that K_10n can be factored into C₅-factors and 1-factors for all non-negative integers and satisfying 2+=10n–1.Research partially supported by an NSF-AWM Mentoring Travel Grant     

The phase structure of the two-dimensionalN=2 Wess-Zumino model

We construct a convergent cluster expansion for the two-dimensionalN=2 Wess-Zumino model, in a region of parameter space where there are multiple phase. As a result of this expansion, we are able to construct the infinite volume field theory and demonstrate exponential decay of correlations. We are also able to investigate the different phases of the model, develop the phase diagram, and show that the free energy of each phase vanishes.Supported in part by National Science Foundation grants DMS 90-08827, PHY/DMS 88-16214 and DMS 88-58073Supported in part by National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291     

Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity

We study the conformal scalar curvature problem

where is a continuous function. We show that a necessary and sufficient condition on for this problem to have positive solutions which are arbitrarily large at is that be less than 1 on a sequence of points in which tends to .

     

Nuclear magnetic resonance and spin relaxation in biological systems

Proton nuclear spin-lattice relaxation in biological systems is generally distinguished from that in inorganic systems such as rocks by the presence of locally disordered macromolecular environments. Rapid exchange of readily observed labile small molecules among differently oriented macromolecular sites generally nearly averages the spectral anisotropies in the small molecule resonances. The biological tissue is generally distinguished from the inorganic matrix by the presence of a significant population of protons in the solid components that are well connected by dipolar spin couplings. Magnetic coupling between the solid and the liquid components generally dominates the magnetic field dependence of the spin-lattice relaxation rates observed in the small molecule components which is generally described by a power law in the Larmor frequency. Recent theory involving a modification of the spin-phonon class of relaxation mechanism provides a quantitative understanding of these data in terms of the dynamics of the chain molecules generally present in the solid spin systems, folded proteins for example.     

A vanishing theorem for supersymmetric quantum field theory and finite size effects in multiphase cluster expansions

We apply cluster expansion methods to to theN=2 Wess-Zumino models in finite volume, in two space-time dimensions. We show that in the region of convergence of the cluster expansion, a vanishing theorem holds for the supercharge of the theory; that is, the dimension of the kernel of the Hamiltonian is equal to the index of the supercharge.Supported in part by National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship DMS 90-07206Supported in part by National Science Foundation Mathematical Sciences Postodoctoral Research Fellowship DmS 88-07291     

Recasting the (Synchrosqueezed) Short-Time Fourier Transform as an Instantaneous Spectrum

In a previous work, we proposed a time-frequency analysis called instantaneous spectral analysis (ISA), which generalizes the notion of the Fourier spectrum and in which instantaneous frequency is utilized to the fullest extent. In this paper, we recast both the Fourier transform (FT) and filterbank (FB) interpretations of the short-time Fourier transform (STFT) as instantaneous spectra. We show that to recast the FB interpretation of STFT as an instantaneous spectrum with valid structure, frequency reassignment is a fundamental necessity, thus demonstrating that this IS is closely related to the synchrosqueezed STFT. This result provides a new theoretical motivation for the synchrosqueezed STFT. Finally, we illustrate through example the instantaneous spectra corresponding to the FT and FB interpretations of STFT using two closed-form examples.     

Implantation of ytterbium ions into group II-VI compounds

50 keV ytterbium ions have been implanted into ZnS, CdS, ZnSe, CdTe, CdSe and ZnTe and, after subsequent annealing treatments of varying degree, the cathodoluminescence emission spectra produced by 10 keV or 100 keV electron excitation have been studied.