Planetary theories with the aid of the expansions of elliptic functions

For coplanar circular orbits, the mutual perturbations between two bodies can be expressed in term of the argument of Jacobian elliptic functions instead of the difference of the mean longitudes. For a given pair of planets, such a change of time variable improves the convergence of the developments. At the first order of planetary masses an integration of Lagrange's equations for the osculating elements is performed. When compared to classical developments the results are reduced by an important factor. The method is then extended to the mutual perturbations of Jupiter and Saturn, at any order of planetary masses, either with Fourier series with two arguments, or with one argument solely, taking advantage of the close commensurability of the mean motions.     

On the Distance Function Between Two Keplerian Elliptic Orbits

The problem of finding critical points of the distance function between two Keplerian elliptic orbits is reduced to the determination of all real roots of a trigonometric polynomial of degree 8. The coefficients of the polynomial are rational functions of orbital parameters. Using computer algebra methods we show that a polynomial of a smaller degree with such properties does not exist. This fact shows that our result cannot be improved and it allows us to construct an optimal algorithm to find the minimal distance between two Keplerian orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

北部湾金线鱼、深水金线鱼与日本金线鱼矢耳石形态识别的初步研究

采用2010年1～9月采自北部湾及邻近海域流刺网渔获的54尾金线鱼(Nemipterus virgatus( Houttuyn,1782))、71尾深水金线鱼(Nemi pterus bath ybius Snyder,1911)和65尾日本金线鱼(Nemi pterus ja ponicus (Bloch,1791))耳石样本,进行了椭圆傅立叶分析.耳石轮廓经图像处理后进行灰阶转换和二值化处理,消除噪点,然后解析为椭圆傅立叶标码( EFDs).采用77个EFD系数进行主成分分析和判别分析来区分3种金线鱼的耳石.结果表明,日本金线鱼幼鱼和成鱼耳石在性成熟前后存在显著的体长效应(P＜0.05),金线鱼和深水金线鱼则体长效应不显著(P＞0.05).因此在耳石椭圆傅立叶分析中,日本金线鱼采用性成熟个体样本,金线鱼和深水金线鱼采用全部样本.主成分分析中,第1、第2和第3主成分,分别解释了总变异的65.1％,15.3％和5.3％,可以将3种金线鱼耳石区分开来.EFD系数的逐步判别分析中,3种鱼的平均正判率达98.6％;交互验证结果显示,3种鱼的平均正判率达97.9％,这表明耳石椭圆傅立叶分析可以作为鱼种识别的有效工具.     

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具梯度项的次线性椭圆型方程解的分析

本文研究了1类具梯度项的次线性椭圆型方程大解和完全有界解的存在性问题。运用上下解方法和极值原理分别得到了Rn上方程存在完全大解的充分必要条件和存在完全有界解的充分条件,并且证明了该方程在Rn中光滑有界区域Ω上不存在大解。     

IMPLICIT NONLINEAR NORMAL MODE INITIALIZATION:A MULTIGRID APPROACH

The implicit nonlinear normal mode initialization (INMI) is applied to a tropical limited area shallow water modelin spherical coordinates.The boundary condition for the INMI scheme is based on the boundary formulation of themodel.The INMI scheme is found to be very efficient in suppressing spurious gravity wave oscillation and providing awell balanced initial data set for the model.The INMI scheme involves solving a number of elliptic type equations withvarying complexity.and hence an efficient numerical technique is required for solving such equations.In order to makeINMI computationally more attractive,we are employing the multigrid method for solving all the elliptic type equationsin the INMI scheme.The numerical procedures for the development of such multigrid solvers are briefly described.     

Distance Between Two Arbitrary Unperturbed Orbits

In the paper by Kholshevnikov and Vassilie, 1999, (see also references therein) the problem of finding critical points of the distance function between two confocal Keplerian elliptic orbits (hence finding the distance between them in the sense of set theory) is reduced to the determination of all real roots of a trigonometric polynomial of degree eight. In non-degenerate cases a polynomial of lower degree with such properties does not exist. Here we extend the results to all possible cases of ordered pairs of orbits in the Two–Body–Problem. There are nine main cases corresponding to three main types of orbits: ellipse, hyperbola, and parabola. Note that the ellipse–hyperbola and hyperbola–ellipse cases are not equivalent as we exclude the variable marking the position on the second curve. For our purposes rectilinear trajectories can be treated as particular (not limiting) cases of elliptic or hyperbolic orbits.     

Expansions of (r/a)m cos jv and (r/a)m sin jv to high eccentricities

Expansions of the functions (r/a)^cos jv and (r/a)m ^sin jv of the elliptic motion are extended to highly eccentric orbits, 0.6627 ... <e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of these expansions are expressed in terms of the derivatives of Hansen's coefficients with respect to the eccentricity. The new expansions are convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is the same of the extended solution of Kepler's equation. The new expansions are intrinsically related to Lagrange's series.     

直立椭圆柱体重力场的谱分析及其应用

利用傅里叶积分变换法导出了埋藏的直立椭圆柱体的重力场在频率域中的解析公式，依此得了确定此直立椭圆柱体尺寸的公式，提出依据这些重力公式来计算侵入大洋地壳中的地幔柱的尺寸和洋壳厚度的方法。     

Analytical Solutions for Pore-Fluid Flow Focusing Within Inclined Elliptic Inclusions in Pore-Fluid-Saturated Porous Rocks: Solutions Derived in an Elliptical Coordinate System

Exact analytical solutions have been derived rigorously for the pore-fluid velocity, pore-fluid-flow focusing factor, stream function and excess pore-fluid pressure around and within a buried inclined elliptic inclusion in pore-fluid-saturated porous rocks. The geometric characteristics of the buried inclined elliptic inclusion are represented by the aspect ratio and dip angle of the inclusion, while the hydrodynamic characteristic is represented by the permeability ratio of the elliptic inclusion to its surrounding rock. Since an elliptic inclusion of any aspect ratio can be used to approximately represent geological faults and cracks, the present analytical solutions can be used to investigate the pore-fluid-flow patterns around buried faults and cracks within the crust of the Earth. Therefore, the present analytical solution not only provides a better understanding of the physics behind the pore-fluid-flow focusing problem around and within buried faults and cracks, but also provides a valuable benchmark solution for validating any numerical method in dealing with this kind of pore-fluid-flow focusing problem. The pore-fluid-flow focusing factor of a buried elliptic inclusion is demonstrated to be dependent on the aspect ratio, the permeability ratio and the dip angle.