Runge–Kutta convolution quadrature for operators arising in wave propagation

An error analysis of Runge–Kutta convolution quadrature is presented for a class of non-sectorial operators whose Laplace transform satisfies, besides the standard assumptions of analyticity in a half-plane Re s > σ ₀ and a polynomial bound \operatornameO(|s|^m₁){\operatorname{O}(|s|^{\mu_1})} there, the stronger polynomial bound \operatornameO(s^m₂){\operatorname{O}(s^{\mu_2})} in convex sectors of the form |\operatorname*arg s| ￡ p/2-q{|\operatorname*{arg} s| \leq \pi/2-\theta} for θ > 0. The order of convergence of the Runge–Kutta convolution quadrature is determined by μ ₂ and the underlying Runge–Kutta method, but is independent of μ ₁. Time domain boundary integral operators for wave propagation problems have Laplace transforms that satisfy bounds of the above type. Numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour of Runge–Kutta convolution quadrature for this class of applications. Our results show in particular that the full classical order of the Runge–Kutta method is attained away from the scattering boundary.     

Transient Dynamics in the Random Growth and Reset Model

A mean-field type model with random growth and reset terms is considered. The stationary distributions resulting from the corresponding master equation are relatively easy to obtain; however, for practical applications one also needs to know the convergence to stationarity. The present work contributes to this direction, studying the transient dynamics in the discrete version of the model by two different approaches. The first method is based on mathematical induction by the recursive integration of the coupled differential equations for the discrete states. The second method transforms the coupled ordinary differential equation system into a partial differential equation for the generating function. We derive analytical results for some important, practically interesting cases and discuss the obtained results for the transient dynamics.     

Maximum directed cuts in digraphs with degree restriction

For integers m, k≥1, we investigate the maximum size of a directed cut in directed graphs in which there are m edges and each vertex has either indegree at most k or outdegree at most k. © 2009 Wiley Periodicals, Inc. J Graph Theory     

224 nm segmented hollow-cathode silver ion laser

The segmented hollow-cathode discharge arrangement is used the first time to excite the 224 nm Ag II laser transition. Quasi-continuous output power of 45 mW is obtained during the 300 s current pulses at optimal discharge conditions (10 hPa of He+4% Ar buffer gas) for discharge current of 3 A. No power saturation is observed up to this current value. An average output power of 0.75 mW is reached using pulse repetition frequency of 190 Hz. The longitudinal mode structure of the TEM₀₀ transversal mode is measured by means of a scanning confocal Fabry–Perot interferometer. Two-mode operation is found to be dominant at high current values. Attempts and suggestions are made to prolong the lifetime of the laser tube. PACS 42.55.Lt; 42.60.Lh; 52.80.-s     

Efficient long-time computations of time-domain boundary integrals for 2D and dissipative wave equation

Linear hyperbolic partial differential equations in a homogeneous medium, e.g., the wave equation describing the propagation and scattering of acoustic waves, can be reformulated as time-domain boundary integral equations. We propose an efficient implementation of a numerical discretization of such equations when the strong Huygens’ principle does not hold.For the numerical discretization, we make use of convolution quadrature in time and standard Galerkin boundary element method in space. The quadrature in time results in a discrete convolution of weights W_j with the boundary density evaluated at equally spaced time points. If the strong Huygens’ principle holds, W_j converge to 0 exponentially quickly for large enough j. If the strong Huygens’ principle does not hold, e.g., in even space dimensions or when some damping is present, the weights are never zero, thereby presenting a difficulty for efficient numerical computation.In this paper we prove that the kernels of the convolution weights approximate in a certain sense the time domain fundamental solution and that the same holds if both are differentiated in space. The tails of the fundamental solution being very smooth, this implies that the tails of the weights are smooth and can efficiently be interpolated. Further, we hint on the possibility to apply the fast and oblivious convolution quadrature algorithm of Schädle et al. to further reduce memory requirements for long-time computation. We discuss the efficient implementation of the whole numerical scheme and present numerical experiments.     

An Asymptotic Ratio in the Complete Binary Tree

Let T _n be the complete binary tree of height n considered as the Hasse-diagram of a poset with its root 1 _n as the maximum element. For a rooted tree T, define two functions counting the embeddings of T into T _n as follows A(n;T)=|{S T _n  : 1 _n ∈S, S≅T}|, and B(n;T)=|{S T _n :1 _n ∉S, S≅T}|. In this paper we investigate the asymptotic behavior of the ratio A(n;T)/B(n;T), and we show that lim  _n→∞[A(n;T)/B(n;T)]=2^ℓ;−1−1, for any tree T with ℓ leaves. This revised version was published online in June 2006 with corrections to the Cover Date.     

Equilibrium sublimation and thermodynamic properties of SnS

Knudsen effusion studies of the sublimation of polycrystalline SnS, prepared by annealing and chemical vapor transport, have been performed employing vacuum micro-balance techniques in the temperature range 733–944 K and at pressures ranging from about 6 × 10^?3 to 11 Pa.The third-law heats of sublimation and second-law entropy of reaction SnS(s) = SnS(g) were determined to be ΔH⁰₂₉₈ = 220.4 ± 3.0 kJ mole^? and ΔS⁰₂₉₈ = 162.4 ± 4.5 J K^?1 mole^?1. From these data the standard heat of formation and absolute entropy of SnS(s) were calculated to be ?102.9 ± 4.0 kJ mole^?1 and 79.9 ± 6.0 J K^?1, respectively.     

Linewidth studies on the Kr+ 469.4 nm and 473.9 nm lines excited in a helium-krypton hollow cathode discharge

Collisional and Doppler linewidths (v _C and v _D ) of the 469.4 nm and 473.9 nm Kr ion lines were measured in a He-Kr hollow cathode discharge using Fabry-Perot technique. A linear dependence of v _C on He pressure was found for both lines. Significant differences were found between the temperature values deduced from the v _D -s of the two lines, and an unexpected temperature dependence of the broadening parameter for the Kr⁺ 469 nm line was also observed. The temperature difference between the two lines is explained by excitation of the upper level of the 469 nm line by second kind collisions between metastable He atoms and ground-state Kr ions, while the temperature dependence of the broadening parameter of the Kr⁺ 469 nm line is suggested to be due to the inverse process.     

Effective on-line coloring ofP 5-free graphs

An on-line algorithm is given that colors anyP ₅-free graph withf() colors, wheref is a function of the clique number of the graph.