Double Arrays,Triple Arrays and Balanced Grids

Triple arrays are a class of designs introduced by Agrawal in 1966 for two-way elimination of heterogeneity in experiments. In this paper we investigate their existence and their connection to other classes of designs, including balanced incomplete block designs and balanced grids.AMS Classification:05B05, 05B30     

Solution of the Room square existence problem

We prove that if there is a Room square of side r with a subsquare of side s, then there is a Room square of side 5(r ? s) + s. This enables us to construct a Room square of side 257, thus proving that Room squares exist for all odd positive integer sides except 3 and 5.     

Surface vibrations on face centered cubic metal surfaces: the (111) surfaces

We have studied the dynamics of atom motion at face centered metal (111) surfaces. The interactions between atoms are modelled with central forces and angle bending forces. Results for frequencies and polarizations of surface modes are presented for the following metals: Ag, Al, Au, Cu, Ni, Pd and Pt. In addition, the results obtained for phonon spectral densities are presented for nickel.     

Theory of surface polaritons in n-type silicon in the presence of a DC current

A theoretical investigation has been carried out on the effect of a drift current on surface polaritons in n-type silicon. The current direction is taken to be parallel to the direction of propagation of the surface polaritons. From the dispersion curves, there is evidence that an interaction takes place between the current and the polaritons which gives rise to polariton instability or amplification for certain frequency ranges. These instabilities are believed to be due to the presence of the surface.     

On the existence of Hadamard matrices

Given any natural number q > 3 we show there exists an integer t ? [2log₂(q ? 3)] such that an Hadamard matrix exists for every order 2^sq where s > t. The Hadamard conjecture is that s = 2.This means that for each q there is a finite number of orders 2^υq for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q.In particular it is already known that an Hadamard matrix exists for each 2^sq where if q = 2^m ? 1 then s ? m, if q = 2^m + 3 (a prime power) then s ? m, if q = 2^m + 1 (a prime power) then s ? m + 1.It is also shown that all orthogonal designs of types (a, b, m ? a ? b) and (a, b), 0 ? a + b ? m, exist in orders m = 2^t and 2^t+2 · 3, t ? 1 a positive integer.     

Surface plasma waves in silver and gold

The dispersion curves for surface plasma waves (SPW) in Ag have been determined from calculated reflectivity minima as exhibited by attenuated total reflection (ATR spectra) for the prism-air-metal (PAM) configuration and from the direct calculation of the dispersion relation for the same configuration. The dispersion curves for Au have been determined by measuring the ATR spectra for the prism-dielectric-metal (PDM) configuration, by calculating the ATR spectra from published optical constants and from the direct calculation of the despersion relation for the PDM configuration. We have found two general types of solutions from the direct calculation of the dispersion relation for both configurations. The two solutions are the surface or Brewster modes and the virtual modes. The characteristics of both modes are discussed. The effect of electronic damping upon the dispersion curves for Ag which exhibits low electronic damping and Au which exhibits moderate electronic damping is demonstrated. Finally the perturbing effect of the dielectric (referred herein as dielectric shift) upon the displacement of the dispersion curves to higher wave number for the PAM configuration for Ag and for the PDM configuration for Au is shown.     

Higher-Order Temporal and Spatial Plasma Wave Echoes

The collisionsless Boltzmann equation is solved in two ways by the method of characteristics. The solution is used for an exact treatment of higher-order temporal and spatial plasma wave echoes in the absence of steady external fields. The second-order spatial echo is treated separately and analyzed numerically.     

Book Reviews

Methods of Intermediate Problems for Eigenvalues: Theory and Ramifications, by Alexander Weinstein and William Stenger. Academic Press, 1971. ($16.00)

Combinatorics, Proceedings of the British Combinatorial Conference 1973, Edited by T.P. McDnough and V.C. Mavron, London Mathematical Society Lecture Note Series 13, Cambridge University Press, London, New York, 1974. 204pp.($9.95)