Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

Graphs with many r-cliques have large complete r-partite subgraphs

Let r2 and c>0. Every graph on n vertices with at least cn^rcliques on r vertices contains a complete r-partite subgraphwith r–1 parts of size c^rlog n and one part of size greaterthan n^1–cr–1. This result implies a quantitativeform of the Erdös–Stone theorem.     

A spectral condition for odd cycles in graphs

Let G be a graph of sufficiently large order n, and let the largest eigenvalue μ(G) of its adjacency matrix satisfies . Then G contains a cycle of length t for every t?n/320This condition is sharp: the complete bipartite graph T₂(n) with parts of size ⌊n/2⌋ and ⌈n/2⌉ contains no odd cycles and its largest eigenvalue is equal to .This condition is stable: if μ(G) is close to and G fails to contain a cycle of length t for some t?n/321, then G resembles T₂(n).     

Plasma sputtering of water molecules from the liquid phase by low-energy ions: Molecular dynamics simulation

The ion bombardment-assisted transfer of the components of a solution to the plasma zone in an atmospheric-pressure glow discharge with a liquid cathode plays an important role. The dynamics of the impact of a 50–500 eV ion on the surface of liquid water was studied by molecular dynamics simulation. Data on the amount of water molecules transferred by ion impact to the gas phase are presented. It was shown that the sputtering yield of water can reach 450 molecules per ion at an energy consumption of 0.75 eV for sputtering. Structural changes occurring in the liquid phase under ion bombardment were analyzed on the basis of the dynamics of degradation of hydrogen bonds.     

Nondestructive diagnostics of charged particle beams in accelerators

The basic techniques for nondestructive diagnostics and detection of losses of charged particle beams used in accelerator engineering are reviewed. The data provided may help choose the systems for diagnostics and detection of losses of beams and give a qualitative picture of the operation principles of such devices. Quantitative characteristics that define the limits of applicability of each diagnostic technique are outlined.     

Extraordinary Magnetic Properties of Ca1–xEuxS and Sr1–xEuxS Solid Solutions

The electron paramagnetic resonance (EPR) spectra and the stationary magnetic susceptibility are investigated for Ca_1–x Eu _x S and Sr_1–x Eu _x S solid solutions (0.00005 < x < 0.0032). The EPR spectrum of Eu²⁺ in both matrices contains a wide structureless line in addition to the narrow lines of the superfine structure characteristic of ions with high local symmetry. Because of the extraordinary temperature dependence of this line in CaS:Eu, it is associated with strongly interacting magnetic centers, namely, exchange-coupling pairs and chains or more complex clusters of magnetic ions. At the same time, the dependence of the stationary magnetic susceptibility in the temperature range 4.2–50 K has no peculiarities for CaS:Eu and SrS:Eu and obeys the Curie law.     

Raman scattering of Ge dot superlattices

Self-organised Ge dot superlattices grown by molecular beam epitaxy of Ge and Si layers utilizing Stranski-Krastanov growth mode were investigated by Raman spectroscopy. An average size of Ge quantum dots was obtained from transmission electron microscopy measurements. The strain and interdiffusion of Ge and Si atoms in Ge quantum dots were estimated from the analysis of frequency positions of optical phonons observed in the Raman spectra. Raman scattering by folded longitudinal acoustic phonons in the Ge dot superlattices was observed and explained using of elastic continuum theory. Received 25 January 2000     

Electronic structure of semiconductor solutions of cadmium chalcogenides

The electronic structure of the semiconductor solid solutions CdS_xSe_1−x x=(0, 0.17, 0.33, 0.5, 0.67, 0.83, 1.0) is investigated. The experimental x-ray and x-ray-induced electronic spectra of sulfur in CdS are compared with computed spectra. The calculations are performed using the FEFF (Version 7) program and a package of programs that implements a cluster version of the local-coherent-potential approximation. The effect of an electron vacancy in the 1s level on the x-ray absorption spectrum is investigated. The band gap as a function of the anion concentration in the solution is estimated. Fiz. Tverd. Tela (St. Petersburg) 41, 41–43 (January 1999)     

Conservative methods for the Toda lattice equations

We are concerned with the numerical integration of the Todalattice equations by using different conservative methods. Numericalexperiments suggest that the global error for isospectral schemesdecreases exponentially with time but it is almost constantfor either symplectic or more general integrators. We providea theoretical explanation for these experimental findings.