Asymptotic Behavior of Global Classical Solutions of Quasilinear Non-strictly Hyperbolic Systems with Weakly Linear Degeneracy

In this paper, we study the asymptotic behavior of global classical solutions of the Cauchy problem for general quasilinear hyperbolic systems with constant multiple and weakly linearly degenerate characteristic fields. Based on the existence of global classical solution proved by Zhou Yi et al., we show that, when t tends to infinity, the solution approaches a combination of C1 travelling wave solutions, provided that the total variation and the L1 norm of initial data are sufficiently small.     

Veiled assonance between geodesics on ellipsoid and billiard in its focal ellipse

We introduce new special ellipsoidal confocal coordinates in ⁿ (n ≥ 3) and apply them to the geodesic problem on a triaxial ellipsoid in ³ as well as the billiard problem in its focal ellipse.

Using such appropriate coordinates we show that these different dynamical systems have the same common analytic first integral. This fact is not evident because there exists a geometrical spatial gap between the geodesic and billiard flows under consideration, and this separating gap just “veils” the resemblance of the two systems.

In short, a geodesic on the ellipsoid and a billiard trajectory inside its focal ellipse are in a “veiled assonance”—under the same initial data they will be tangent to the same confocal hyperboloid. But this assonance is rather incomplete: the dynamical systems in question differ by their intrinsic action angle-variables, thereby the different dynamics arise on the same phase space (i.e. the same phase curves in the same phase space bear quite different rotation numbers).

Some results of this work have been published before in Russian (Tabanov, 1993) and presented to the International Geometrical Colloquium (Moscow, May 10–14, 1993) and the International Symposium on Classical and Quantum Billiards (Ascona, Switzerland, July 25–30, 1994).     

Separation coordinates in Hénon-Heiles systems

In this paper we apply the method of the Kowalewski's Conditions to separate the seven Hénon-Heiles integrable systems. For each of them we provide explicitly the separation coordinates in the form of eigenvalues of a matrix M called Control Matrix. A couple of systems (HH3 KK and HH4 1:12:16) are presented and discussed in a more general form than usually in the literature. We show that the process of separation of coordinates can be reduced, at the end, to the choice of a single function and, eventually, a vector field transversal to the Lagrangian foliation in an extended phase space.     

Analysis and algorithms for a regularized cauchy problem arising from a non-linear elliptic PDE for seismic velocity estimation

In the present work we derive and study a non-linear elliptic PDE coming from the problem of estimation of sound speed inside the Earth. The physical setting of the PDE allows us to pose only a Cauchy problem, and hence is ill-posed. However, we are still able to solve it numerically on a long enough time interval to be of practical use. We used two approaches. The first approach is a finite difference time-marching numerical scheme inspired by the Lax–Friedrichs method. The key features of this scheme is the Lax–Friedrichs averaging and the wide stencil in space. The second approach is a spectral Chebyshev method with truncated series. We show that our schemes work because of (i) the special input corresponding to a positive finite seismic velocity, (ii) special initial conditions corresponding to the image rays, (iii) the fact that our finite-difference scheme contains small error terms which damp the high harmonics; truncation of the Chebyshev series, and (iv) the need to compute the solution only for a short interval of time. We test our numerical schemes on a collection of analytic examples and demonstrate a dramatic improvement in accuracy in the estimation of the sound speed inside the Earth in comparison with the conventional Dix inversion. Our test on the Marmousi example confirms the effectiveness of the proposed approach.     

电介质椭球内极化场强方向的研究

导出了椭球体内极化场强方向与外电场方向之间夹角大小的表达式,并对其进行了数值计算,分析认为对于椭球体而言,其体内的极化场强方向并不一定严格和外电场方向相反。     

Parametric splines on a hyperbolic paraboloid

A hyperbolic paraboloid over a tetrahedron, constructed in B–B algebraic reduced form with its barycentric coordinate system, can be conveniently represented by two parameters. An arc on the surface, obtained by determining a type of function relation about the two parameters, has multiformity and consistent endpoint properties. We analyze the equivalence and boundedness of an arc’s curvature, and give a process of the proof. These arcs can be connected into an approximate G²$G^2$-continuity space curve for fitting to a sequence of points with their advantages, and the curves, connected by this type arcs, are quite different from other algebraic and parametric splines.     

(Semi-)Riemannian geometry of (para-)octonionic projective planes

We use reduced homogeneous coordinates to construct and study the (semi-)Riemannian geometry of the octonionic (or Cayley) projective plane , the octonionic projective plane of indefinite signature , the para-octonionic (or split octonionic) projective plane and the hyperbolic dual of the octonionic projective plane . We also show that our manifolds are isometric to the (para-)octonionic projective planes defined classically by quotients of Lie groups.     

测地极坐标下空间形式的度量特征

将R3中Gauss曲率为常数k0的常曲率曲面S的度量特征推广到了一般的空间形式,主要工作是利用活动标架法和测地极坐标的理论,通过对结构方程的微分,找到了关于度量的一个内在方程,由此给出了定理的一个证明。     

极坐标系统下进行直角坐标扫描的离子束抛光

在离子束抛光设备研制过程中,离子源扫描运动方式的选择是很关键的,一般分为直角坐标方式扫描和极坐标方式扫描两种。根据两种扫描方式的特点,在极坐标系统下进行直角坐标扫描方式加工。该种方法采用直角坐标扫描方式下的驻留时间计算,算法相对简单。该种方法在极坐标系统下进行加工,同等情况下可加工圆形镜面的口径比直角坐标系统下更大些;而且离子源的可移动区域是一条直线,其余地方可以摆放其他设备,空间利用率较高。对这种新思路进行仿真分析,证实了其具有可行性。     

梯形模糊数的有序表示及中心平均排序方法

模糊数的排序在决策分析和优化问题中占有十分重要的地位,而一般模糊数均可近似分解为若干分片小梯形的叠加形式,故梯形模糊数的排序问题至关重要!本文首先引入等距分片方法对梯形模糊数实施纵向分割,进而获得梯形模糊数的有序表示。其次,依中心平均加权准则改进梯形模糊数的横向和纵向中心坐标公式,并提出新的指标排序准则。最后,通过实例分析考证了新的排序方法的有效性。