An integral equation-fast Fourier transform (IE-FFT) algorithm is applied to the electromagnetic solutions of the combined field integral equation (CFIE) for scattering problems by an arbitrary-shaped three-dimensional perfect electric conducting object. The IE-FFT with CFIE uses a Cartesian grid for known Green's function to considerably reduce memory storage and speed up CPU time for both matrix fill-in and matrix vector multiplication when used with a generalized minimal residual method. The uniform interpolation of the Green's function on an equally spaced Cartesian grid allows a global FFT for field interaction terms. However, the near interaction terms do not take care for the singularity of the Green's function and should be adequately corrected. The IE-FFT with CFIE does not always require a suitable preconditioner for electrically large problems. It is shown that the complexity of the IE-FFT with CFIE is found to be approximately O(N1.5) and O(N1.5log N) for memory and CPU time, respectively.

We prove that positive solutions of an integral equation of Wolff type are radially symmetric and decreasing about some point in $R^{n}$. The hypotheses allow a wider range of exponents and are easier to apply than those in previous work.

Drawdown (respectively, drawup) of a stochastic process, also referred as the reflected process at its supremum (respectively, infimum), has wide applications in many areas including financial risk management, actuarial mathematics, and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (respectively, drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a Lévy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to study various drawdown quantities for the general class of time-homogeneous Markov processes.

This article explores recursive and integral equations for ruin probabilities of generalised risk processes, under rates of interest with homogenous Markov chain claims and homogenous Markov chain premiums. We assume that claim and premium take a countable number of non-negative values. Generalised Lundberg inequalities for the ruin probabilities of these processes are derived via a recursive technique. Recursive equations for finite time ruin probabilities and an integral equation for the ultimate ruin probability are presented, from which corresponding probability inequalities and upper bounds are obtained. An illustrative numerical example is discussed.

This paper concerns the electromagnetic scattering by arbitrary shaped three dimensional imperfectly conducting objects modeled with non-constant Leontovitch impedance boundary condition. It has two objectives. Firstly, the intrinsically well-conditioned integral equation (noted GCSIE) proposed in [30] is described focusing on its discretization. Secondly, we highlight the potential of this method by comparison with two other methods, the first being a two currents formulation in which the impedance condition is implicitly imposed and whose the convergence is quasi-optimal for Lipschitz polyhedron, the second being a CFIE-like formulation [14]. In particular, we prove that the new approach is less costly in term of CPU time and gives a more accurate solution than that obtained from the CFIE formulation. Finally, as expected, It is demonstrated that no preconditioner is needed for this formulation.

In this paper we deal with an M/G/1 vacation system with the sojourn time (wait plus service) limit and two typical vacation rules, i.e. multiple and single vacation rules. Using the level crossing approach, we derive recursive equations for the steady-state distributions of the virtual waiting times in M/G/1 vacation systems with a general vacation time and two vacation rules.

We show how to apply convolution quadrature (CQ) to approximate the time domain electric field integral equation (EFIE) for electromagnetic scattering. By a suitable choice of CQ, we prove that the method is unconditionally stable and has the optimal order of convergence. Surprisingly, the resulting semi discrete EFIE is dispersive and dissipative, and we analyze this phenomena. Finally, we present numerical results supporting and extending our convergence analysis.

In this paper we deal with an M/G/1 vacation system with the sojourn time (wait plus service) limit and two typical vacation rules, i.e. multiple and single vacation rules. Using the level crossing approach, explicit expressions for the steady-state distributions of the virtual waiting times are obtained in vacation systems with exponential and constant service times, a general vacation time, and two vacation rules.

We study the biharmonic equation Δ2u = u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn, n ≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

The two dimensional punch problem for planar anisotropic elastic half-plane is revisited using the Lekhnitskii's formulation with aid of the Fourier transform and boundary integral equation. Four different conditions of contact problem for the rigid punch are analyzed in this study. From the combination of surface Green's function of half-plane and Hooke's law of anisotropic material, a set of Fredholm integral equations are obtained for mixed boundary value problems. After solving the integral equation according to specified contact condition, the explicit distributions of surface traction under the punch are obtained in closed-form. From the surface traction and Green's function of anisotropic half-plane, the full-field solutions of stresses are constructed. Numerical calculations of surface traction under the rigid punch are presented base on the analysis and are discussed in detail.

We analyze the mean cost of the partial match queries in random two-dimensional quadtrees. The method is based on fragmentation theory. The convergence is guaranteed by a coupling argument of Markov chains, whereas the value of the limit is computed as the fixed point of an integral equation.

The effective method of carbon nanotubes antennas' parameters calculation has been developed. The frequency dependencies of input impedance of CNT in dielectric medium have been investigated. It is shown that an increase in the length of a nanotube length does not lead to the appearance of resonances in the centimeter wavelength range.

On exhibe dans cette note une paramétrix (au sens faible) de l'opérateur sous-jacent à l'équation CFIE de l'électromagnétisme. L'intérêt de cetteparamétrix est de se prêter à différentes stratégies de discrétisationet ainsi de pouvoir être utilisée comme préconditionneur de la CFIE.On montre aussi que l'opérateur sous-jacent à la CFIE satisfait une conditionInf-Sup discrète uniforme, applicable aux espaces de discrétisation usuellement rencontrésen électromagnétisme, et qui permet d'établir un résultat inédit de convergencenumérique de la CFIE.

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.

This paper studies the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The finance applications include pricing barrier and lookback options.

A new system of integral equations for the exterior 2D time harmonicscattering problem is investigated. This system was first proposed by B. Després in [11]. Two new derivations of this system are given:one from elementary manipulationsof classical equations, the other based on a minimization of a quadratic functional. Numerical issues are addressed to investigate the potentialof the method.

When claims in the compound Poisson risk model are from a heavy-tailed distribution (such as the Pareto or the lognormal), traditional techniques used to compute the probability of ultimate ruin converge slowly to desired probabilities. Thus, faster and more accurate methods are needed. Product integration can be used in such situations to yield fast and accurate estimates of ruin probabilities because it uses quadrature weights that are suited to the underlying distribution. Tables of ruin probabilities for the Pareto and lognormal distributions are provided.

Using Laplace transforms and the notion of a pseudo compound Poisson distribution, some risk theoretical results are revisited. A well-known theorem by Feller (1968) and Van Harn (1978) on infinitely divisible distributions is generalized. The result may be used for the efficient evaluation of convolutions for some distributions. In the particular arithmetic case, alternate formulae to those recently proposed by De Pril (1985) are derived and shown more adequate in some cases. The individual model of risk theory is shown to be pseudo compound Poisson. It is thus computable using numerical tools from the theory of integral equations in the continuous case, a formula of Panjer type or the Fast Fourier transform in the arithmetic case. In particular our results contain some of De Pril's (1986/89) recursive formulae for the individual life model with one and multiple causes of decrement. As practical illustration of the continuous case we construct a new two-parametric family of claim size density functions whose corresponding compound Poisson distributions are analytical finite sum expressions. Analytical expressions for the finite and infinite time ruin probabilities are also derived.

We consider a risk generating claims for a period of N consecutive years (after which it expires), N being an integer valued random variable. Let Xk denote the total claims generated in the kth year, k ≥ 1. The Xk's are assumed to be independent and identically distributed random variables, and are paid at the end of the year. The aggregate discounted claims generated by the risk until it expires is defined as where υ is the discount factor. An integral equation similar to that given by Panjer (1981) is developed for the pdf of SN(υ). This is accomplished by assuming that N belongs to a new class of discrete distributions called annuity distributions. The probabilities in annuity distributions satisfy the following recursion:

where an is the present value of an n-year immediate annuity.

This paper is concerned with a model for the effect of erosion on crop production. Crop yield in the year n is given by X(n) = YnLn, where is a sequence of strictly positive i.i.d. random variables such that E{Y1} <∞, and is a Markov chain with stationary transition probabilities, independent of . When suitably normalized, leads to a martingale which converges to 0 almost everywhere (a.e.) as n → ∞. In addition, for large n, the distribution of Ln is approximately lognormal. The conditional expectations and probabilities of , given the past history of the process, are determined. Finally, the asymptotic behaviour of the total crop yield is discussed. It is established that under certain regularity conditions Sn converges a.e. to a finite-valued random variable S whose Laplace transform can be obtained as the solution of a Volterra-type linear integral equation.