Evaluation of the capacitive force between an atomic force microscopy tip and a metallic surface

We propose a very simple method to determine the electrical tip-surface force in Atomic Force Microscopes used to study the electrical properties of metallic or insulating materials; the analysis of the measurements as well as determination of the appropriate experimental procedures requiring an analytical model of the tip-surface capacitance. The comparison of force expressions obtained by this method with those obtained by exact derivation in the case of the sphere-infinite plane system shows very good agreement. This method is then applied to determine the tip-surface force, the real shape of the tip being introduced in the derivation. The obtained expression is compared to experimental and numerical data. We emphasize that this method is very general and can be applied to any axially symmetric capacitor. Received: 17 June 1997 / Received: 16 September 1997 / Accepted: 7 November 1997     

Explicit solution for Lamé and other PDE systems

We provide a general series form solution for second-order linear PDE system with constant coefficients and prove a convergence theorem. The equations of three dimensional elastic equilibrium are solved as an example. Another convergence theorem is proved for this particular system. We also consider a possibility to represent solutions in a finite form as partial sums of the series with terms depending on several complex variables.     

过渡区泥沙沉速公式的比较

本文利用精确的试验资料对几个泥沙沉速公式进行了比较。结果表明窦国仁和沙玉清公式较符合实际,在理论上及因子处理上也较有根据,而我国水文测验规范所采用冈察洛夫公式则有系统偏差。     

冻结壁蠕变问题的位移反分析

本文以冻结壁蠕变模型试验获得的位移为已知条件,综合分析了各位移量之间的关系后,借助于有限元法,采用多目标函数及待分析参数分组的分步反分析法,实现了冻结壁蠕变这一多参数问题的位移反分析。     

On the convergence of certain sequences of rational approximants to meromorphic functions in several variables

In a previous paper, the author introduced a new class of multivariate rational interpolants, which are called Optimal Padé-type Approximants (OPTA). There, for this class of rational interpolants, which extends classical univariate Padé Approximants, a direct extension of the “de Montessus de Ballore's Theorem” for meromorphic functions in several variables is established. In the univariate case, this theorem ensures uniform convergence of a row of Pade Approximants when the denominator degree equals the number of poles (counting multiplicities) in a certain disc. When one overshoots the number of poles when fixing the denominator degree, convergence in measure or capacity has been proved and, under certain additional restrictions, the uniform convergence of a subsequence of the row. The author tackles the latter case and studies its generalization to functions in several variables by using OPTA.     

Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations

In this short paper the core of the direct method for proving stability of functional equations is described in a clear way and in a quite general form.     

The generalized Goursat-Darboux problem for a third order operator

It is proved that if a generalized Goursat-Darboux problem is -well posed then the operator cannot contain derivatives with respect to one of the variables.

     

Integration of the intertwining operator for -harmonic polynomials associated to reflection groups

Let be the intertwining operator with respect to the reflection invariant measure on the unit sphere in Dunkl's theory on spherical -harmonics associated with reflection groups. Although a closed form of is unknown in general, we prove that

where is the unit ball of and is a constant. The result is used to show that the expansion of a continuous function as Fourier series in -harmonics with respect to is uniformly Cesáro summable on the sphere if , provided that the intertwining operator is positive.

     

Extremal polynomials on arcs of the circle with zeros on these arcs

The paper gives a solution of an extremal problem of finding monic polynomial least deviating from zero on several arcs of the unit circle, under some restrictions on the location of zeros and additional conditions on mutual position of the arcs. The extremal polynomial is represented in the terms of density of harmonic measure. The work is done under the financial support of RFFI (Project 07-01-00167) and the President of RF Grant (Project NSh-2970.2008.1).