A new summation method for power series with rational coefficients

We show that an asymptotic summation method, recently proposed by the authors, can be conveniently applied to slowly convergent power series whose coefficients are rational functions of the summation index. Several numerical examples are presented.

     

The Sampling Theorem and Several Equivalent Results in Analysis

First we show that several fundamental results on functions from theBernstein spaces (such as Bernstein's inequality andthe reproducing formula) can be deduced from a weak form of the classicalsampling theorem. In §3 we discuss the mutual equivalence of thesampling theorem, the derivative sampling theorem and a harmonic functionsampling theorem. In §§4–6 we discuss connections between thesampling theorem and various important results in complex analysis andFourier analysis. Our considerations include Cauchy's integral formula,Poisson's summation formula, a Gaussian integral, certain properties ofweighted Hermite polynomials, Plancherel's theorem, the maximum modulusprinciple, and the Phragmén–Lindelöf principle.     

Hardy-Hilbert重级数不等式的推广与改进

本文将著名的 Hardy-Hilbert重级数不等式∑∞m=1 ∑∞n=1ambnm + n≤ πsin(π/p) ∑∞n=1apn1p ∑∞n=1bqn1q∑∞m=1 ∑∞n=1anm + np ≤ πsin(π/p)p∑∞n=1apn进行了带参数形式的推广 ,同时改进了这些不等式     

Steady-state conditions of a nonisothermal film with a heat-insulated free boundary

The equilibrium shapes of a nonisothermal liquid film with a heat-insulated free surface for large Marangoni numbers are investigated in the long-wave approximation using a combination of analytical and numerical methods. It is proved that the two-dimensional problem of the equilibrium of a strip-shaped film has a steady-state solution for an arbitrary large temperature gradient on the boundaries of the strip. An increase in this gradient leads to an abrupt thinning of the film near the heated boundary, which can result in instability and rupture of the film. In the equilibrium problem for a film fixed on a circular contour, the nonuniform distribution of the heat flux on the contour was found to have a significant influence on the free-surface shape. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 59–73, July–August, 2008.     

Numerical computation of a three-dimensional vortex sheet in a swirl flow

We consider an incompressible and inviscid fluid flow, called “swirl flow” that rotates around a certain axis in three-dimensional space. We investigate numerically the dynamics of a three-dimensional vortex sheet which is defined as a surface across which the velocity field of the swirl flow changes discontinuously. The vortex method and a fast summation method are implemented on a parallel computer. These numerical methods make it possible to compute the evolution of the vortex sheet for a long time and to describe the complex dynamics of the sheet.     

On the Extrapolation of Perturbation Series

We discuss certain special cases of algebraic approximants that are given as zeroes of so-called effective characteristic polynomials and their generalization to a multiseries setting. These approximants are useful for the convergence acceleration or summation of quantum mechanical perturbation series. Examples are given and some properties discussed.     

A kinetic energy preserving nodal discontinuous Galerkin spectral element method

In this work, we discuss the construction of a skew‐symmetric discontinuous Galerkin (DG) collocation spectral element approximation for the compressible Euler equations. Starting from the skew‐symmetric formulation of Morinishi, we mimic the continuous derivations on a discrete level to find a formulation for the conserved variables. In contrast to finite difference methods, DG formulations naturally have inter‐domain surface flux contributions due to the discontinuous nature of the approximation space. Thus, throughout the derivations we accurately track the influence of the surface fluxes to arrive at a consistent formulation also for the surface terms. The resulting novel skew‐symmetric method differs from the standard DG scheme by additional volume terms. Those volume terms have a special structure and basically represent the discretization error of the different product rules. We use the summation‐by‐parts (SBP) property of the Gauss–Lobatto‐based DG operator and show that the novel formulation is exactly conservative for the mass, momentum, and energy. Finally, an analysis of the kinetic energy balance of the standard DG discretization shows that because of aliasing errors, a nonzero transport source term in the evolution of the discrete kinetic energy mean value may lead to an inconsistent increase or decrease in contrast to the skew‐symmetric formulation. Furthermore, we derive a suitable interface flux that guarantees kinetic energy preservation in combination with the skew‐symmetric DG formulation. As all derivations require only the SBP property of the Gauss–Lobatto‐based DG collocation spectral element method operator and that the mass matrix is diagonal, all results for the surface terms can be directly applied in the context of multi‐domain diagonal norm SBP finite difference methods. Numerical experiments are conducted to demonstrate the theoretical findings. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimates of three classical summations on the spaces F_p~(α,q)(R~n)(0 < p ≤ 1)

We obtain the boundedness on ˙Fα,qp(Rn) for the Poisson summation and Gauss summation. Their maximal operators are proved to be bounded from˙Fα,qp(Rn) to L∞(Rn).For the maximal operator of the Bochner-Riesz summation, we prove that it is bounded from˙Fα,qp(Rn) to Lpnn-pα,∞(Rn).