The plancherel formula for the line bundles on SL(n + 1, ℝ)/S(GL(1,ℝ) × GL(n, ℝ))

In this paper we obtain the Plancherel formula for the spaces of L²-sections of the line bundles over the pseudo-Riemannian symmetric space G/H where G = SL(n + 1, ?) and H = S(GL(1, ?) × GL(n 1, ?)). The Plancherel formula is given in an explicit form by means of spherical distributions associated with the character χ_λ of the subgroup H. We follow the method of Faraut, Kosters and van Dijk.     

Satellites in the gravitational field of the moon

Expressions are derived correct to the first order in the ellipticities of the Moon for the radial shift and the rotation of the unperturbed orbit of a close satellite in the plane of its instantaneous motion. Criteria for the rotation to be a libration lead to the existence of a critical range of values of orbital inclination as against a single value cos^?1 (·2)^1/2=63·4° quoted for the Earth satellites. It appears that mean radial shift and the perturbation in the period of the satellite, also possess similar ranges. The perturbation in the velocity vector is then analysed and it is shown that the angle between the unperturbed and real trajectories vanishes only at points where the radial distance of the satellite from the selenocentre attains extreme values. The interesting case of nearly circular trajectories is considered next in some detail. It is deduced that the critical inclination for the orbital libration is given byα=60° for an Earth satellite in this case. The paper concludes with numerical results based on Jeffreys’ estimates.     

A generalized Harish-Chandra method of descent for reductive Lie algebras

Let G be a connected real reductive group and M a connected reductive subgroup of G with Lie algebras g and m, respectively. We assume that g and m have the same rank. We define a map from the space of orbital integrals of m into the space of orbital integrals of g which we call a transfer. We then consider the transpose of the transfer. This can be viewed as a map from the space of G-invariant distributions of g to the space of M-invariant distributions of m and can be considered as a restriction map from g to m. We prove that this map extends Harish-Chandra method of descent and we obtain a generalization of the radial component theorem. We give an application.     

Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration x_n+1=g(x_n) can be determined through sublevel sets of a Lyapunov function. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13(6) (2007) 523–546] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative discrete orbital derivative in a neighborhood of the fixed point. In this paper we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative discrete orbital derivative both locally and globally.     

Statistical analysis of Fe abundances gradients in the Galaxy

Using the high precision data of the proper motions and parallaxes from Hipparcos catalogue, we obtained the orbital parameters of 1302 stars in the Galaxy based on the mass distribution model provided by Allen and Santillón. Fe abundances of 1295 stars among our samples were analyzed. With the correlation analyses between [Fe/H] and orbital parameters, we obtained that the Fe gradient is - 0.057 ± 0.007 dex/kpc along the direction of the maximum galactocentric distance (hereafter DG_max) in the range of 8.5 kpc < DG_max < 17 kpc. We also got the result that the vertical gradient is steeper than the radial gradient. Furthermore, we divided the samples into two subgroups: giants and dwarfs; F, G and K stars; and then analyzed them respectively. Our results show that the gradient becomes flatter and flatter from giants to dwarfs, from F type to G and K type stars. We also divided the samples into disk and halo stars using maximum vertical distance Z_max = 1 kpc as the criterion and got the result that the abundances of the disk stars are much higher than that of the halo stars. Our work suggests the existence of the galactic gradient and supports those chemlcal evolution medels which show that the halo was formed before the disk at the early stage of the Galaxy.     

Orbital stability of bound states of nonlinear Schrödinger equations with linear and nonlinear optical lattices

We study the orbital stability and instability of single-spike bound states of semi-classical nonlinear Schrödinger (NLS) equations with critical exponent, linear and nonlinear optical lattices (OLs). These equations may model two-dimensional Bose-Einstein condensates in linear and nonlinear OLs. When linear OLs are switched off, we derive the asymptotic expansion formulas and obtain necessary conditions for the orbital stability and instability of single-spike bound states, respectively. When linear OLs are turned on, we consider three different conditions of linear and nonlinear OLs to develop mathematical theorems which are most general on the orbital stability problem.     

Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations

We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain the full radial Strichartz estimates up to some endpoints for the Schrödinger equation. Using these estimates, we obtain some new results related to nonlinear problems, including small data scattering and large data LWP for the nonlinear Schrödinger and wave equations with radial critical initial data and the well-posedness theory for the fractional order Schrödinger equation in the radial case.     

Mathematical modelling and stabilization of large flexible spacecraft subject to orbital perturbation

In this paper the problem of modelling of large flexible spacecraft and their stabilization under the influence of orbital (radial) perturbation is considered. A complete dynamics of the spacecraft consisting of a rigid bus and a flexible beam is derived using Hamilton's principle. The equations of motion consist of a coupled system of partial differential equations governing the vibration of the flexible beam and ordinary differential equations describing the translational and rotational motions of the rigid bus. The asymptotic stability of the system is proved using Lyapunov's approach. Simple feedback controls are suggested for the stabilization of the system. For illustration, numerical simulations are carried out, giving interesting results.     

The Analytic and Formal Normal Form for the Nilpotent Singularity

We study orbital normal forms for analytic planar vector fields with nilpotent singularity. We show that the Takens normal form is analytic. In the case of generalized cusp we present the complete formal orbital normal form; it contains functional moduli. We interprete the coefficients of these moduli in terms of the hidden holonomy group.     

用径向基函数插值解自共轭椭圆型方程

本文讨论用MQ作为插值的径向基函数,对自共轭椭圆型方程进行插值,证明了插值系数的唯一性,并用投影法证明了用径向基函数解自共轭椭圆型方程的收敛性.     

A targeting strategy for the deployment of a tethered satellite system

^** Email: hans.troger{at}tuwien.ac.at One of the most important operations during a tethered satellitesystem mission is the deployment of a subsatellite from a spaceship.We restrict to the simple, but practically important case thatthe system is moving on a circular Keplerian orbit around theEarth. The main problem during deployment due to the local gravitygradient vector is that due to the Coriolis acceleration thetwo satellites do not move along the straight radial relativeequilibrium position in the orbital frame. Instead, a strongdeviation from the local vertical direction occurs, which afterthe deployment process is finished results in weakly dampedlarge-amplitude oscillations, which in some cases are even transientchaotic. This chaotic dynamics will be used to steer the satellitewith small control actions into the final radial relative equilibriumposition far away from the spaceship. Both deployment time andenergy input are computed and compared to other deployment strategies.     

Lp径向Blaschke-Minkowski同态的Lp对偶几何表面积

2006年，Schuster提出了径向Blaschke-Minkowski同态的概念.随后，汪卫等人将其推广到L_p径向Blaschke-Minkowski同态.本文结合L_p对偶几何表面积，建立了L_p径向Blaschke-Minkowski同态的若干不等式，包括Brunn-Minkowski型不等式和单调不等式.并给出了L_p径向Blaschke-Minkowski同态的Busemann-Petty问题的肯定和否定形式.     

关于p－拉普拉斯方程径向解的一点注记

本文研究方程ｄｉｖ（｜ｕ｜ｐ－２ｕ）＋｜ｘ｜ｌ｜ｕ｜τ－１ｕ＝０，ｘ∈Ｂ的Ｄｉｒｉｃｈｌｅｔ边值问题ｕ｜Ｂ＝０的径向解。应用山路引理，我们将文［１］的结果（ｐ＝２时）推广到一般情形，证明上述问题有一非平凡的径向解。     

Orbital stability of solitary waves for Kundu equation

In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c₃+s₂υ<0, while Guo and Wu (1995) only considered the case 2c₃+s₂υ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.     

The diagram theory for the degenerate two-orbital hubbard model

We investigate the minimal model that takes orbital degrees of freedom into account: the degenerate two-orbital Hubbard model. Our consideration includes the intraatomic Coulomb interaction of two electrons with opposite spins on the same orbital and on different orbitals and interorbital hopping of tunneling electrons. We take the influence of states caused by Hund’s rule coupling on the metal-insulator phase transition into account. We generalize the diagram theory developed for strongly correlated orbitally nondegenerate systems to the case of orbital degeneration. For the one-particle renormalized Green’s function, we establish an equation of Dyson type for calculating the system spectral function using a simple approximation based on summing chain diagrams.     

Radial symmetry of standing waves for nonlinear fractional Hardy–Schrödinger equation

In this paper, by applying the method of moving planes, we conclude the conclusions for the radial symmetry of standing waves for a nonlinear Schrödinger equation involving the fractional Laplacian and Hardy potential. First, we prove the radial symmetry of solution under the condition of decay near infinity. Based upon that, under the condition of no decay, by the Kelvin transform, we establish the results for the non-existence and radial symmetry of solution.     

A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

In this article, we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation technique. Our results complete and improve a work published recently by Zhang and Zhou(existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Applied Mathematics Letters,Volume 50, December 2015, Pages 48–55).     

Existence,orbital stability and instability of solitary waves for coupled BBM equations

This paper is concerned with the orbital stability/instability of solitary waves for coupled BBM equations which have Hamiltonian form. The explicit solitary wave solutions will be worked out first. Then by detailed spectral analysis and decaying estimates of solutions for the initial value problem, we obtain the orbital stability/instability of solitary waves.     

球面混合插值的逼近性质

球面径向基函数(SBF)和多项式样条函数均为处理球面散乱数据的有效工具. 本文考虑由球面径向基函数与球面多项式函数组成的混合插值模型, 并利用最小二乘法求解该模型. 对于该插值模型, 首先, 给出带Bessel势的Sobolev空间中的Bernstein不等式, 然后利用该不等式建立逼近正定理,并进一步给出该插值工具的误差估计. 最后, 研究该插值方式(即利用最小二乘法求解混合插值模型)的稳定性.