A Multidimensional Superposition Principle: Classical Solitons IV

This article continues the series of the works of 1998–2007 years devoted to the Multidimensional Superposition Principle, the concept easily explaining both classical soliton and more complex wave interactions in nonlinear PDEs and allowing one, in particular, to construct the general Superposition Formulae for nonlinear wave interactions. In the present research the technique of multiexpansions with constraints is considered for finding the above SFs and investigation of the related solitons. (The simplest case of such expansions technically is analogous to the so-called invariant truncated singular expansions.) As the applications, the soliton SFs of the MKdV_±, Kaup-Kupershmidt and new A_± equation are obtained for the bell-shape exponential solitons of the various families, algebraic solitons, and the configuration of the two noninteracting kinks. The linearized, parameterized versions of these SFs are investigated then, and the related analysis of the interactions is presented. The obtained results allow one to consider the one soliton solutions mentioned as the strong bound states of the simpler solitons. Concerning the results for the above concrete nonlinear PDEs, the approach being developed made it possible both to obtain the new results and to reveal new moments for the already known ones.     

General Interpolation Formulae for Barycentric Blending Interpolation

General interpolation formulae for barycentric interpolation and barycentric rational Hermite interpolation are established by introducing multiple parameters,which include many kinds of barycentric interpolation and barycentric rational Hermite interpolation. We discussed the interpolation theorem, dual interpolation and special cases. Numerical example is given to show the effectiveness of the method.     

An optimization approach for the numerical approximation of differential equations

An alternative approach for the numerical approximation of ODEs is presented in this article. It is based on a variational framework recently introduced in S. Amat and P. Pedregal [A variational approach to implicit ODEs and differential inclusions, ESAIM: COCV 15 (2009), 149–172] where the solution is sought as the minimizer of an error functional tailored after the ODE in a rather straightforward way. A suitable discretization of this error functional is pursued, and it is performed using Hermite's interpolation and quadrature formulae. Notice that only Hermite's interpolation is necessary when polynomial systems of ODEs are considered (many models in practice use these types of equations). A comparison with implicit Runge–Kutta methods is analysed. With this variational strategy not only some classical collocation methods, but also new schemes that seem to have better numerical behaviour can be recovered. Although the driving idea is very simple, the strategy turns out to be very general and flexible. At the same time, it can be implemented efficiently.     

High-order generalized Gauss–Radau and Gauss–Lobatto formulae for Jacobi and Laguerre weight functions

The generation of generalized Gauss–Radau and Gauss–Lobatto quadrature formulae by methods developed by us earlier breaks down in the case of Jacobi and Laguerre measures when the order of the quadrature rules becomes very large. The reason for this is underflow resp. overflow of the respective monic orthogonal polynomials. By rescaling of the polynomials, and other corrective measures, the problem can be circumvented, and formulae can be generated of orders as high as 1,000. In memoriam Gene H. Golub.     

Cumulants on the Wiener space

We combine infinite-dimensional integration by parts procedures with a recursive relation on moments (reminiscent of a formula by Barbour (1986)), and deduce explicit expressions for cumulants of functionals of a general Gaussian field. These findings yield a compact formula for cumulants on a fixed Wiener chaos, virtually replacing the usual “graph/diagram computations” adopted in most of the probabilistic literature.     

光波在左手材料中的菲涅尔公式和布儒斯特定律

根据Maxwell的电磁场理论,推导出单色平面光波入射到介电常量和磁导率分别为ε1、μ1和ε2、μ2的两种介质时的菲涅耳公式和反射率及透射率的普适表达形式.对于ε2<0 和,μ2<0的左手材料,普适菲涅耳公式同样适用.研究了光从右手材料入射到左手材料时反射和折射的特性,给出了反射光为全偏光时布儒斯特角应满足的条件,此条件说明,在左手材料中布儒斯特角不但与折射率有关而且与磁导率有关.     

ζ-function and heat kernel formulae

We present a systematic study of asymptotic behaviour of (generalised) ζ-functions and heat kernels used in noncommutative geometry and clarify their connections with Dixmier traces. We strengthen and complete a number of results from the recent literature and answer (in the affirmative) the question raised by M. Benameur and T. Fack (2006) [1].     

Linear variance bounds for particle approximations of time-homogeneous Feynman–Kac formulae

This article establishes sufficient conditions for a linear-in-time bound on the non-asymptotic variance for particle approximations of time-homogeneous Feynman–Kac formulae. These formulae appear in a wide variety of applications including option pricing in finance and risk sensitive control in engineering. In direct Monte Carlo approximation of these formulae, the non-asymptotic variance typically increases at an exponential rate in the time parameter. It is shown that a linear bound holds when a non-negative kernel, defined by the logarithmic potential function and Markov kernel which specify the Feynman–Kac model, satisfies a type of multiplicative drift condition and other regularity assumptions. Examples illustrate that these conditions are general and flexible enough to accommodate two rather extreme cases, which can occur in the context of a non-compact state space: (1) when the potential function is bounded above, not bounded below and the Markov kernel is not ergodic; and (2) when the potential function is not bounded above, but the Markov kernel itself satisfies a multiplicative drift condition.     

Numerical performance of approximate queuing formulae with application to flexible manufacturing systems

This paper calls attention to two of the more successful queuing approximation formulae — one by Kramer and one by Marchal. The analytic solution of a range of single server Erlang cases is compared to the two approximation formulae. Then a family of H₂/M/1 cases is similarly considered. Maximum errors are seen to be about three percent. The Kramer formula seems to be better when the interarrival coefficient of variation is less than 0.66 and the Marchal formula is better for larger interarrival coefficients of variation. Finally, a multiserver refinement function (the ratio of G/G/1 results to M/M/1 results) is proposed to scale M/M/s as an approximation for G/G/s. In most of these multiple channel cases, the maximum error is less than six percent. The last section of this paper presents a simple, representative FMS. It is modelled as an open queuing network. Then the approximation procedure is applied node by node to illustrate the estimation of system performance measures such as machine utilizations and throughput.