On Rotation About Lightlike Axis in Three-Dimensional Minkowski Space

We obtain matrix of the rotation about arbitrary lightlike axis in three-dimensional Minkowski space by deriving the Rodrigues’ rotation formula and using the corresponding Cayley map. We prove that a unit timelike split quaternion q with a lightlike vector part determines rotation R _q about lightlike axis and show that a split quaternion product of two unit timelike split quaternions with null vector parts determines the rotation about a spacelike, a timelike or a lightlike axis. Finally, we give some examples.     

On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions

In this paper, we examine eigenvalue problem of a rotation matrix in Minkowski 3 space by using split quaternions. We express the eigenvalues and the eigenvectors of a rotation matrix in term of the coefficients of the corresponding unit timelike split quaternion. We give the characterizations of eigenvalues (complex or real) of a rotation matrix in Minkowski 3 space according to only first component of the corresponding quaternion. Moreover, we find that the casual characters of rotation axis depend only on first component of the corresponding quaternion. Finally, we give the way to generate an orthogonal basis for ${\mathbb{E}^{3}_{1}}$ by using eigenvectors of a rotation matrix.     

Comparative study of Mixed product and quaternion product

Mixed number is the sum of a scalar and a vector. The quaternion can also be written as the sum of a scalar and a vector but the product of mixed numbers and the product of quaternions are different. Here we studied the Mixed product which is derived from the product of mixed numbers and the quaternion product which is derived from the product of quaternions. It was observed that Mixed product is more consistent with Physics than that of quaternion product.     

Commutative Quaternion Matrices

In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some properties of commutative quaternions and their fundamental matrices. After that we investigate commutative quaternion matrices using properties of complex matrices. Then we define the complex adjoint matrix of commutative quaternion matrices and give some of their properties.     

A fully diagonalized spectral method using generalized Laguerre functions on the half line

A fully diagonalized spectral method using generalized Laguerre functions is proposed and analyzed for solving elliptic equations on the half line. We first define the generalized Laguerre functions which are complete and mutually orthogonal with respect to an equivalent Sobolev inner product. Then the Fourier-like Sobolev orthogonal basis functions are constructed for the diagonalized Laguerre spectral method of elliptic equations. Besides, a unified orthogonal Laguerre projection is established for various elliptic equations. On the basis of this orthogonal Laguerre projection, we obtain optimal error estimates of the fully diagonalized Laguerre spectral method for both Dirichlet and Robin boundary value problems. Finally, numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized method.     

On the product of rotations

Using the elementary tools of matrix theory, we show that the product of two rotations in the three-dimensional Euclidean space is a rotation again. For this purpose, three types of rotation matrices are identified which are of simple structure. One of them is the identity matrix, and each of the other two types can be uniquely characterized by exactly one vector. The resulting products are investigated by using the basic properties of the vector cross product.     

Combinatorics of rooted trees and Hopf algebras

We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities.

Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.

     

On an integrable case of perturbed keplerian motion

A general solution of a differential vector equation of perturbed Keplerian motion is derived for the case when the position vector and perturbing acceleration vector are collinear. A variable change is employed, in which the new independent variable is expressed in terms of the initial values of the phase variables and time, using the elliptical Jacobi function. The two-point boundary value problem for the initial equation is reduced to the Cauchy problem, A parametric representation is obtained for the regularized trajectory of motion of a material point under the action of a central force.     

Conformal cochains

In this paper we define holomorphic cochains and an associated period matrix for triangulated closed topological surfaces. We use the combinatorial Hodge star operator introduced in the author's paper of 2007, which depends on the choice of an inner product on the simplicial 1-cochains.

We prove that for a triangulated Riemannian 2-manifold (or a Riemann surface), and a particularly nice choice of inner product, the combinatorial period matrix converges to the (conformal) Riemann period matrix as the mesh of the triangulation tends to zero.

     

Justification asymptotique des hypothèses de Kirchhoff-Love pour une coque encastrée linéairement élastique

The displacement vector of a linearly elastic shell can be computed by using the twodimensional Koiter's model, based on the a priori Kirchhoff-Love assumptions. These hypotheses imply that the displacement of any point of the shell is an affine function of the transverse variable x₃. The term independent of x₃ of this approximation is equal to the displacement vector of the two-dimensional Koiter's model. The term linear in x₃ depends on the rotation vector of the normal. After an appropriate scaling, we here estimate the difference between the three-dimensional displacement and the affine function in the case of shells clamped along their entire lateral face. Besides, in the case of shells with uniformly elliptic middle surface, taking into account the term depending of the rotation of the normal, allows to improve the asymptotic estimate between the three-dimensionnal displacement and Koiter's bidimensional displacement.     

A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance

We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.     

Dualité dans l’espace des caractères virtuels tempérés d’un groupe réductif <Emphasis Type="Italic">p</Emphasis>-adique

Résumé We define an involution on the set of tempered virtual characters of a connected reductive p-adic group. This involution commutes with character of parabolic induction and with truncation. It also preserves the irreducible characters up to sign and the elliptic inner product.     

基于对数极坐标及对称循环矩阵的形状特征描述

在形状检索算法中,满足尺度和旋转不变是基本要求.本文将形状的边界用对数极坐标表示,使得形状的放缩和旋转化为简单的平移.由于计算机读取形状边界信息时与起点有关,当形状旋转时会带来边界点列的循环,影响旋转不变性.为消除边界点列循环带来的影响,本文首先证明"奇数阶对称循环矩阵,当生成元循环时,所得循环矩阵的特征值不变",在这个数学理论基础上,把形状边界点数插值到奇数,构造相应的对称循环矩阵,通过这个循环矩阵的特征值来描述形状特征,由此得到一种具有放缩旋转不变的形状检索新算法.实验表明,本文算法对运动目标和非刚性形变的形状检索具有良好的鲁棒性和快捷的运行速度,这在目标跟踪方面将发挥作用.     

Canal Surfaces with Quaternions

Quaternions are more usable than three Euler angles in the three dimensional Euclidean space. Thus, many laws in different fields can be given by the quaternions. In this study, we show that canal surfaces and tube surfaces can be obtained by the quaternion product and by the matrix representation. Also, we show that the equation of canal surface given by the different frames of its spine curve can be obtained by the same unit quaternion. In addition, these surfaces are obtained by the homothetic motion. Then, we give some results.     

复Finsler流形上的Hodge-Laplace算子

本文定义了强拟凸复Finsler流形上的Hodge-Laplace算子,并给出其水平部分的局部坐标表示．     

Canonical forms for mixed symmetric-skewsymmetric quaternion matrix pencils

Canonical forms are described for pairs of quaternionic matrices, or equivalently matrix pencils, where one matrix is symmetric and the other matrix is skewsymmetric, under strict equivalence and symmetry respecting congruence. The symmetry is understood in the sense of a fixed involutory antiautomorphism of the skew field of the real quaternions; the involutory antiautomorphism is assumed to be nonstandard, i.e., other than the quaternionic conjugation. Some applications are developed, such as canonical forms for quaternionic matrices under symmetry respecting congruence, and canonical forms for matrices that are skewsymmetric with respect to a nondegenerate symmetric or skewsymmetric quaternion valued inner product.     

Petri net approach to disassembly process planning for products with complex AND/OR precedence relationships

We present a Petri net (PN)-based approach to automatically generate disassembly process plans (DPPs) for product recycling or remanufacturing. We define an algorithm to generate a geometrically-based disassembly precedence matrix (DPM) from a CAD drawing of the product. We then define an algorithm to automatically generate a disassembly Petri net (DPN) from the DPM; the DPN is live, bounded, and reversible. The resulting DPN can be analyzed using the reachability tree method to generate feasible DPPs, and cost functions can be used to determine the optimal DPP. Since reachability tree generation is NP-complete, we develop a heuristic to dynamically explore the v likeliest lowest cost branches of the tree, to identify optimal or near-optimal DPPs. The cost function incorporates tool changes, changes in direction of movement, and individual part characteristics (e.g., hazardous). An example is used to illustrate the procedure. This approach can be used for products containing AND, OR, and complex AND/OR disassembly precedence relationships.     

The problem of the time-optimal control of spacecraft reorientation

The use of Pontryagin's maximum principle to solve spacecraft motion control problems is demonstrated. The problem of the optimal control of the spatial reorientation of a spacecraft (as a rigid body) from an arbitrary initial angular position to an assigned final angular position in the minimum rotation time is investigated in detail. The case in which velocity parameters of the motion are constrained is considered. An analytical solution of the problem is obtained in closed form using the method of quaternions, and mathematical expressions for synthesizing the optimal control programme are given. The kinematic problem of spacecraft reorientation is solved completely. A design scheme for solving the maximum principle boundary-value problem for arbitrary turning conditions and inertial characteristics of the spacecraft is given. A solution of the problem of the optimal control of spatial reorientation for a dynamically symmetrical spacecraft is presented in analytical form (to expressions in elementary functions). The results of mathematical modelling of the motion of a spacecraft under optimal control, which confirm the practical feasibility of the control algorithm developed, are given. Estimates have shown that the turn time of modern spacecraft with a constrained magnitude of the angular momentum can be reduced by 15–25% compared with conventional reorientation methods. The greatest effect is achieved for turns through large angles (90° or more) when the final rotation vector is equidistant from the longitudinal axis and the transverse plane of the spacecraft.     

Cauchy’s Formula on Surfaces Embedded in the Quaternions

Cauchy’s theorem for analytic functions on complex numbers is extended to analytic functions on the quaternions. For this purpose, we carefully define the notions of differentiation and integration on two or three dimensional manifolds embedded in the quaternions.