The Problem of Topologies of Grothendieck and the Class of Fréchet T-Spaces

We examine the interpolation with periodic polynomial splines of degree d and defect r (d ≦ r) on equidistant partitions of the real axis and generalize known results for r = 0. We prove necessary and sufficient conditions for the existence and a certain L²-stability of the interpolants as well as their approximation properties in the scale of the periodic SOBOLEV spaces.     

Validity of the NLS approximation for periodic quantum graphs

We consider a nonlinear Schrödinger (NLS) equation on a spatially extended periodic quantum graph. With a multiple scaling expansion, an effective amplitude equation can be derived in order to describe slow modulations in time and space of an oscillating wave packet. Using Bloch wave analysis and Gronwall’s inequality, we estimate the distance between the macroscopic approximation which is obtained via the amplitude equation and true solutions of the NLS equation on the periodic quantum graph. Moreover, we prove an approximation result for the amplitude equations which occur at the Dirac points of the system.     

Approximation and existence of periodic solutions for controlled diffusion equations

This paper concerns the existence of control functions such that a system of controlled stochastic differential equations (SDE) with periodic coefficients has a solution which is periodic in distribution. We show that bounded periodic controls acting in the same direction as the Driving Wiener process will achieve this under some nondegeneracy condition on the diffusion part. The main tool is an approximation theorem for solutions of SDEs which enables one to check certain stability conditions on a more suitable differential equation     

Hermite–Padé approximation and simultaneous quadrature formulas

We study Hermite–Padé approximation of the so-called Nikishin systems of functions. In particular, the set of multi-indices for which normality is known to take place is considerably enlarged as well as the sequences of multi-indices for which convergence of the corresponding simultaneous rational approximants takes place. These results are applied to the study of the convergence properties of simultaneous quadrature rules of a given function with respect to different weights.     

An Analog of the Jackson Inequality for Coconvex Approximation of Periodic Functions

We prove an analog of the Jackson inequality for a coconvex approximation of continuous periodic functions with the second modulus of continuity and a constant that depends on the location of the points at which a function changes its convexity.     

Hermite–Fejér interpolation operator and characterization of functions

In this paper we investigate the approximation behaviour of the so‐called Hermite–Fejér interpolation operator based on the zeros of Jacobi polynomials. As a result we obtain the asymptotic formula of approximation rate for these operators. Moreover, such a formula is valid for any individual continuous function. We will also study the K ‐functional deduced by this operator. Consequently the asymptotic term of this K ‐functional is established. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An algorithm for determining the approximation orders of multivariate periodic spline spaces

We give an algorithm which computes the approximation order of spaces of periodic piece-wise polynomial functions, given the degree, the smoothness and tesselation. The algorithm consists of two steps. The first gives an upper bound and the second a lower bound on the approximation order. In all known cases the two bounds coincide.     

Numerical solution of the Monge–Ampère equation by a Newton's algorithm

We solve numerically the Monge–Ampère equation with periodic boundary condition using a Newton's algorithm. We prove convergence of the algorithm, and present some numerical examples, for which a good approximation is obtained in 10 iterations. To cite this article: G. Loeper, F. Rapetti, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Approximation of semigroups and cosine functions in spaces of periodic functions

In this article we furnish a representation of the solutions of some classes of first-order and second-order evolution problems as limit of iterates of classical sequences of approximating operators. The method is based on Trotter's theorem on the approximation of semigroups which is applied here also for the approximation of groups and cosine functions. We apply this method in spaces of continuous periodic functions and using some classical sequences of trigonometric polynomials.     

Hyperbolic Wavelet Approximation

We study the multivariate approximation by certain partial sums (hyperbolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets. We characterize spaces of functions which have a prescribed approximation error by hyperbolic wavelet sums in terms of a K -functional and interpolation spaces. The results parallel those for hyperbolic trigonometric cross approximation of periodic functions [DPT]. October 16, 1995. Date revised: August 28, 1996.     

Semidiscretisation Methods for Quasi‐periodic Solutions

The aim of our approach is a reliable numerical approximation of quasi‐periodic solutions of periodically forced ODEs without using a‐priori transformations into new coordinates [1]. The invariant torus is computed as a solution of a special invariance equation. In the case of two basic frequencies this system can be solved by semidiscretisation, which transforms the system into a higher dimensional autonomous ODE system with periodic solutions.     

Bounds and approximation for clocked interprocessor communication

We consider a clocked message transfer scheme in which message transfer is implemented as a periodic task with fixed overhead. Messages are then processed according to a priority discipline with preemption. Simple closed-form lower and upper bounds, and an approximation based on these bounds are found for the total system mean response time. Both bounds are tight for high processor occupancies, and simulation shows that the approximation is excellent for medium to high occupancies. Application to two different processors is given; the bounds and the approximation are found to be sufficiently tight, demonstrating the usefulness of this bounding and approximation technique for performance modelling of systems early in the design cycle.     

球型平移网络逼近周期函数

研究了球型平移网络对周期函数的逼近问题．文章首先将基函数eimx分别表示成为两种球型平移网络．进一步,将有关多重Fourier级数的Bochner-Riesz平均表示成为球型平移网络的形式．在此基础上构造出了两类球型平移网络序列,并借助于有关Bochner-Riesz平均对Lp空间中函数的逼近结果给出了这两类球型平移网络序列在Lp空间中的逼近阶．     

Validity of the Weakly Nonlinear Solution of the Cauchy Problem for the Boussinesq‐Type Equation

We consider the initial‐value problem for the regularized Boussinesq‐type equation in the class of periodic functions. Validity of the weakly nonlinear solution, given in terms of two counterpropagating waves satisfying the uncoupled Ostrovsky equations, is examined. We prove analytically and illustrate numerically that the improved accuracy of the solution can be achieved at the timescales of the Ostrovsky equation if solutions of the linearized Ostrovsky equations are incorporated into the asymptotic solution. Compared to the previous literature, we show that the approximation error can be controlled in the energy space of periodic functions and the nonzero mean values of the periodic functions can be naturally incorporated in the justification analysis.     

A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations

In this Note, we propose a new method, based on perturbation theory, to post-process the planewave approximation of the eigenmodes of periodic Schrödinger operators. We then use this post-processing to construct an accurate a posteriori estimator for the approximations of the (nonlinear) Gross–Pitaevskii equation, valid at each step of a self-consistent procedure. This allows us to design an adaptive algorithm for solving the Gross–Pitaevskii equation, which automatically refines the discretization along the convergence of the iterative process, by means of adaptive stopping criteria.     

Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions and additive noise in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in a linear stochastic parabolic equation for vorticity. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot-Savart-type law. We prove the first mean-square convergence order of the vorticity approximation.     

Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg–de Vries Equation

In this paper, we consider the spectral stability of spatially periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long‐wavelength perturbations. Specifically, we extend the work of Bronski and Johnson by demonstrating that the homogenized system describing the mean behavior of a slow modulation (WKB) approximation of the solution correctly describes the linearized dispersion relation near zero frequency of the linearized equations about the background periodic wave. The latter has been shown by rigorous Evans function techniques to control the spectral stability near the origin, that is, stability to slow modulations of the underlying solution. In particular, through our derivation of the WKB approximation we generalize the modulation expansion of Whitham for the KdV to a more general class of equations which admit periodic waves with nonzero mean. As a consequence, we will show that, assuming a particular nondegeneracy condition, spectral stability near the origin is equivalent with the local well‐posedness of the Whitham system.     

Fonctions de partitions à parité périodique

Let be the set of positive integers and a subset of . For , let denote the number of partitions of n with parts in . In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any and , there is a unique set , such that is even for n>N. Soon after, Ben Saïd and Nicolas (Acta Arith. 106 (2003) 183) considered , and proved that for all k≥0, the sequence is periodic on n. In this paper, we generalise the above works for any formal power series f in with f(0)=1, by constructing a set such that the generating function of is congruent to f modulo 2, and by showing that if f=P/Q, where P and Q are in with P(0)=Q(0)=1, then for all k≥0 the sequence is periodic on n.     

Lower part of the spectrum for the two-dimensional Schrödinger operator periodic in one variable and application to quantum dimers

We study the semiclassical asymptotic approximation of the spectrum of the two-dimensional Schrödinger operator with a potential periodic in x and increasing at infinity in y. We show that the lower part of the spectrum has a band structure (where bands can overlap) and calculate their widths and dispersion relations between energy and quasimomenta. The key role in the obtained asymptotic approximation is played by librations, i.e., unstable periodic trajectories of the Hamiltonian system with an inverted potential. We also present an effective numerical algorithm for computing the widths of bands and discuss applications to quantum dimers.     

An extension of the ‘ 1/9 ’-problem

We consider the rational approximation of a perturbed exponential function

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