Ramanujan Graphs with Small Girth

We construct an innite family of (q + 1)-regular Ramanujan graphs X _n of girth 1. We also give covering maps X _n+1 X _n such that the minimal common covering of all the X _n s is the universal covering tree.     

Sub-second 2D NMR spectroscopy at sub-millimolar concentrations

A recently proposed protocol enables the acquisition of two-dimensional nuclear magnetic resonance (2D NMR) spectra within a single scan. A promising application opened up by this new data acquisition mode concerns its combination with active nuclear polarization methods, whereby spectroscopy is carried out on analytes whose spin magnetizations have been significantly enhanced over their Boltzmann thermal values. The present paper explores the potential of such combination, with the acquisition of peptide and protein 2D NMR 1H correlation spectra recorded after the samples had been subject to laser-driven chemically induced dynamic nuclear polarization (CIDNP). It is demonstrated that the speed and sensitivity enhancement afforded by these combined processes enables the acquisition of quality 2D NMR data sets within a fraction of a second, at analyte concentrations that are under 1 mM.     

Pareto optimality in multiobjective problems

In this study, the optimization theory of Dubovitskii and Milyutin is extended to multiobjective optimization problems, producing new necessary conditions for local Pareto optima. Cones of directions of decrease, cones of feasible directions and a cone of tangent directions, as well as, a new cone of directions of nonincrease play an important role here. The dual cones to the cones of direction of decrease and to the cones of directions of nonincrease are characterized for convex functionals without differentiability, with the aid of their subdifferential, making the optimality theorems applicable. The theory is applied to vector mathematical programming, giving a generalized Fritz John theorem, and other applications are mentioned. It turns out that, under suitable convexity and regularity assumptions, the necessary conditions for local Pareto optima are also necessary and sufficient for global Pareto optimum. With the aid of the theory presented here, a result is obtained for the, so-called, scalarization problem of multiobjective optimization.The author's work in this area is now supported by NIH grants HL 18968 and HL 4664 and NCI contract NO1-CB-5386.     

A multiprojection algorithm using Bregman projections in a product space

Generalized distances give rise to generalized projections into convex sets. An important question is whether or not one can use within the same projection algorithm different types of such generalized projections. This question has practical consequences in the area of signal detection and image recovery in situations that can be formulated mathematically as a convex feasibility problem. Using an extension of Pierra's product space formalism, we show here that a multiprojection algorithm converges. Our algorithm is fully simultaneous, i.e., it uses in each iterative stepall sets of the convex feasibility problem. Different multiprojection algorithms can be derived from our algorithmic scheme by a judicious choice of the Bregman functions which govern the process. As a by-product of our investigation we also obtain blockiterative schemes for certain kinds of linearly constraned optimization problems.     

Reconstructing graphs from size and degree properties of their induced k-subgraphs

As a variant of the famous graph reconstruction problem we characterize classes of graphs of order n such that all their induced subgraphs on k?n vertices satisfy some property related to the number of edges or to the vertex degrees.We give complete solutions for the properties (i) to be regular, (ii) to be regular modulo m?2 or (iii) to have one of two possible numbers of edges. Furthermore, for an order n large enough, we give solutions for the properties (iv) to be bi-regular or (v) to have a bounded difference between the maximum and the minimum degree.     

Spin-flip Raman scattering and magnetization measurements on (Cd,Mn)S

Spin-flip Raman scattering and magnetization measurements are reported for the semimagnetic semiconductor Cd_1-xMn_xS, x = 0.023. The data at 1.9 K and high fields (near saturation) give the exchange energy αN_o = 217 ±11 meV. At low fields, the spin-flip energy for donor-bound electrons contains the predicted contribution from the excess local magnetization near the donors. Magnetoresistance and Hall data are also reported.     

Magnetic phase diagram of MnP: Field parallel to the hard direction

The phase diagram of MnP, for magnetic field parallel to the a axis, was determined. Three ordered phases were found. They are tentatively identified as ferromagnetic, fan, and cone phases. The triple point where the ferro, fan, and paramagnetic phases meet has features which are expected for a Lifshitz point.