The validity of the KdV approximation in case of resonances arising from periodic media

It is the purpose of this short note to discuss some aspects of the validity question concerning the Korteweg-de Vries (KdV) approximation for periodic media. For a homogeneous model possessing the same resonance structure as it arises in periodic media we prove the validity of the KdV approximation with the help of energy estimates.     

Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate

We consider a prototype reaction-diffusion system which models a network of two consecutive reactions in which chemical components A and B form an intermediate C which decays into two products P and Q. Such a situation often occurs in applications and in the typical case when the intermediate is highly reactive, the species C is eliminated from the system by means of a quasi-steady-state approximation. In this paper, we prove the convergence of the solutions in L², as the decay rate of the intermediate tends to infinity, for all bounded initial data, even in the case of initial boundary layers. The limiting system is indeed the one which results from formal application of the QSSA. The proof combines the recent L²-approach to reaction-diffusion systems having at most quadratic reaction terms, with local L^∞-bounds which are independent of the decay rate of the intermediate. We also prove existence of global classical solutions to the initial system.     

The strong approximation property and the weak bounded approximation property

We show that the strong approximation property (strong AP) (respectively, strong CAP) and the weak bounded approximation property (respectively, weak BCAP) are equivalent for every Banach space. This gives a negative answer to Oja's conjecture. As a consequence, we show that each of the spaces _c0$c_0$ and _?1${?}_1$ has a subspace which has the AP but fails to have the strong AP.     

Guaranteed-accuracy approximation of reachable sets for a linear dynamic system subject to impulse actions

A method is proposed for approximating the reachable set of a dynamic system with a state space dimension no higher than six-eight considered on a finite time interval. The system is governed by linear differential equations with piecewise constant coefficients and impulse actions specified at prescribed times. The method is based on guaranteed-accuracy polyhedral approximations of reachable sets at researcher-specified times. Every approximation is constructed using the preceding one. A procedure is described for choosing parameters of the method that ensure the required accuracy with close-to-minimal time costs.     

Rational approximation with sign-sensitive weight

For metrics of sign-sensitive weight, a generalization of the Jackson theorem and an analog of the Dolzhenko theorem on the estimate of the variation of a function in terms of its least rational deviations are obtained. In the same metric, two-sided estimates for rational deviations of the function signx are given.Translated fromMatematickeskie Zametki, Vol. 60, No. 5, pp. 715–725, November, 1996.     

Estimation of a class of quasi‐resonances generated by multiple small particles with high surface impedances

We deal with the stationary acoustic waves propagating in a cluster of small particles enjoying high contrasts. Such contrasts allow the appearance of (complex valued) resonances that are close to the real line as the size of the particles becomes small. For single (but not necessarily small) particles, we derive the characteristic equation that generates a class of these resonances (the ones for which the corresponding eigenfunctions are uniformly constant). For multiple and small particles, we provide sufficient conditions on the contrasts that generates quasi‐resonances for which the corresponding eigenfunctions are uniformly constant. Precisely, we show that, if we distribute the particles on a uniform line, then the existence of such quasi‐resonances is related to the eigenvalues of the Harary matrix. To show these results, we take, as the small contrasted particles, small obstacles with high surface impedances λ of the form λ: = βa^?1 ? αβa^{?1 + h} where a is the maximum radi of the particles, with a < <1, and β is a universal and positive constant depending only on the shape of the particles (but not on their size). In this case, if the relative constant α is an eigenvalue of the Harary matrix, then the used frequency is a quasi resonance of the cluster of the small particles where the error of approximation is of the order for h ∈ (0,1) as a < <1.     

Solving a large dense linear system by adaptive cross approximation

An efficient algorithm for the direct solution of a linear system associated with the discretization of boundary integral equations (in two dimensions) is described without having to compute the complete matrix of the linear system. This algorithm is based on the unitary-weight representation, for which a new construction based on adaptive cross approximation is proposed. This low rank approximation uses only a small part of the entries to construct the adaptive cross representation, and therefore the linear system can be solved efficiently.     

Clustering estimates for spatial point distributions with unstable potentials

Summary Several authors have tried to model highly clustered point patterns by using Gibbs distributions with attractive potentials. Some of these potentials violate a stability condition well known in statistical mechanics. We show that such potentials produce patterns which are much more tightly clustered than those considered by the authors. More generally, our estimates provide a useful test for rejecting unsuitable potentials in models for given patterns. We also use instability arguments to reject related approximations and simulations. Csiro     

The estimate for approximation error of spherical neural networks

Compared with planar hyperplane, fitting data on the sphere has been an important and an active issue in geoscience, metrology, brain imaging, and so on. In this paper, with the help of the Jackson‐type theorem of polynomial approximation on the sphere, we construct spherical feed‐forward neural networks to approximate the continuous function defined on the sphere. As a metric, the modulus of smoothness of spherical function is used to measure the error of the approximation, and a Jackson‐type theorem on the approximation is established. Copyright © 2011 John Wiley & Sons, Ltd.     

An approximation analysis for an assembly-like queueing system with time-constraint items

We study an assembly-like queueing system one of whose queues has items with generally distributed time-constraints, where this system has a single server providing services using each item individually. It is well-known that analysis of a queueing system which has items with time-constraint (i.e., impatient items) is difficult since the analytical model must involve all the departure times of these impatient items. We therefore propose to employ the techniques of Whitt’s approximation and show the method for obtaining the stationary distribution of the model. Through some simulation experiments, we discuss the validation of our approximation model, and show that the approximation is accurate in various kinds of situations (e.g., service time distribution and the number of queues).     

KdV方程的时间谱离散方法

本文提出了解KdV方程周期边值问题的安全港离散方法:在时间方向上采用Chebyshev拟谱逼近,在空间方向上采用Fourier Galerkin逼近。谱展开的系数由目标泛函的极小值来确定。同时证明了该方法的收敛性。     

Global stable and unstable manifolds for a class of semilinear equations with sectorially dichotomous operator

In this paper, the existence and smoothness of global stable and unstable manifolds at an equilibrium are established for a class of semilinear equations with sectorially dichotomous operator. As an application, an elliptic PDE in infinite cylindrical domain is discussed.     

The limits of Riemann solutions to the simplified pressureless Euler system with flux approximation

The formation of vacuum state and delta shock wave in the solutions to the Riemann problem for the simplified pressureless Euler system is considered under the linear approximations of flux functions. The method is to perturb the non‐strictly hyperbolic system into a nearby strictly hyperbolic system by introducing appropriately the linear approximations of flux functions. The solutions to the Riemann problem for the approximated system can be constructed explicitly and then the formation of vacuum state and delta shock wave can be observed by taking the perturbation parameter tend to zero in the solutions.     

Convergence analysis of a space-time approximation to a two-dimensional system of shallow water equations

A characteristic-Galerkin approximation of a new two-dimensional Shallow Water viscous parabolic model is presented and a semi-implicit-Lagrangian finite element scheme is used. A convergence result is proved and numerical experiments on academic tests are shown.     

Global well-posedness for a coupled modified KdV system

We prove the sharp global well-posedness result for the initial value problem (IVP) associated to the system of the modified Korteweg-de Vries (mKdV) equation. For the single mKdV equation such result has been obtained by using Mirura’s Transform that takes the KdV equation to the mKdV equation [8]. We do not know the existence of Miura’s Transform that takes a KdV system to the system we are considering. To overcome this difficulty we developed a new proof of the sharp global well-posedness result for the single mKdV equation without using Miura’s Transform. We could successfully apply this technique in the case of the mKdV system to obtain the desired result.     

Convergence of solutions for a network approximation of the two-dimensional Laplace equation in a domain with a system of absolutely conducting disks

A boundary value problem for the Laplace equation describing the (electric, thermal, etc.) field of a system of ideally conducting disks of radius R is considered. The solution to the problem is analyzed under the condition that the characteristic distance δ between the disks is small. It was previously proved that the original continuous problem can be approximated as δ → 0 by a finite-dimensional network problem in the sense that the effective conductivities (energies) of the continuous problem are close to those of its network model. It is shown that the potentials of the ideally conducting disks determined from the continuous problem and the network model are also close to each other as δ → 0, and the difference between the potentials is O(ε^1/4), where ε = δ/R is the characteristic relative distance between the disks.     

Copositive approximation by rational functions with prescribed numerator degree

The paper proves that, if f（x） ∈ L^p[-1,1],1≤p〈∞ ,changes sign I times in （-1, 1）,then there exists a real rational function r（x） ∈ Rn^（2μ-1）l which is eopositive with f（x）, such that the following Jackson type estimate ｜｜f-r｜｜p≤Cδl^2μωφ（f,1/n）p holds, where μ is a natural number ≥3/2＋1/p, and Cδ is a positive constant depending only on δ.     

The similarity between the best approximation and the best copositive approximation

In this paper the author studies the copositive approximation in C(?) by elements of finite dimensional Chebyshev subspaces in the general case when ? is any totally ordered compact space. He studies the similarity between me behavior of the ordinary best approximation and the behavior pf the copositive best approximation. At the end of this paper, the author isolates many cases at which the two behaviors are the same.     

The sample average approximation method for empty container repositioning with uncertainties

One of the challenges faced by liner operators today is to effectively operate empty containers in order to meet demand and to reduce inefficiency in an uncertain environment. To incorporate uncertainties in the operations model, we formulate a two-stage stochastic programming model with random demand, supply, ship weight capacity, and ship space capacity. The objective of this model is to minimize the expected operational cost for Empty Container Repositioning (ECR). To solve the stochastic programs with a prohibitively large number of scenarios, the Sample Average Approximation (SAA) method is applied to approximate the expected cost function. To solve the SAA problem, we consider applying the scenario aggregation by combining the approximate solution of the individual scenario problem. Two heuristic algorithms based on the progressive hedging strategy are applied to solve the SAA problem. Numerical experiments are provided to show the good performance of the scenario-based method for the ECR problem with uncertainties.