Secret Sharing Scheme Based on the Differential Manifold

In this paper, the concepts of topological space and differential manifold are introduced, and it is proved that the surface determined by function F (x2, x2, …, xt) of class Cr in Euelidean Rt is a differential manifold. Using the intersection of the tangent plane and the hypernormal of the differential manifold to construct the shared master key of participants, an intuitive, secure and complete (t,n)-threshold secret sharing scheme is designed. The paper is proved to be safe, and the probability of successful attack of attackers is only 1/pt-1. When the prime number p is sufficiently large, the probability is almost 0. The results show that this scheme has the characteristics of single-parameter representation of the master key in the geometric method, and is more practical and easy to implement than the Blakley threshold secret sharing scheme.

NEWTONIAN MECHANICS ON KAHLER MANIFOLD

In this paper we discuss Newtonian Mechanics on Kahler Manifold, and also givefoe complex mathematical aspects of Newton's law, the law of kinetic energy, the lawof kinetic quantity,the equation of motion and the 'general equation of dynamics',and so on.

A Topological Transformation of Quantum Dynamics

In this work, we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces. This is not the view from different perspectives of an observer who simply uses different coordinate systems to describe the same physical phenomenon but rather possible geometric and topological structures that quantum particles are endowed with when they are identified with differentiable manifolds that are embedded or immersed in Euclidean spaces of higher dimension. We present our discussions in the form of Bohr model in one, two and three dimensions using linear wave equations. In one dimension, the fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom. In two dimensions, the fundamental polygon is a square and the universal covering space is the plane and in this case, the standing wave on the square is transformed into the standing wave on different surfaces that can be formed by gluing opposite sides of the square, which include a 2-sphere, a 2-torus, a Klein bottle and a projective plane. In three dimensions, the fundamental polygon is a cube and the universal covering space is the three-dimensional Euclidean space. It is shown that a 3-torus and the manifold K?× S1?defined as the product of a Klein bottle and a circle can be constructed by gluing opposite faces of a cube. Therefore, in three-dimensions, the standing wave on a cube is transformed into the standing wave on a 3-torus or on the manifold K?× S1. We also suggest that the mathematical degeneracy may play an important role in quantum dynamics and be associated with the concept of wavefunction collapse in quantum mechanics.

Theory of constructing closed parametric curves based on manifolds

A parametric curve based on a manifold is de-signed for constructing an accurate closed curve.A circle was defined as the parametric space and a non-uniform B-splines defined on the unit circle were used as base func-tions.A method to construct knot vectors,control points and corresponding parameters were proposed.A method to de-termine the coordinates for any point on a curve was also proposed.Some non-uniform rational B-splines(NURBS)control techniques,such as curves with an embedded line,a sharp angle,and so on,were used to verify the proposed method’s compatibility with NURBS.Some examples were used to compare the arithmetic with that of NURBS.The results show that the method is not only simple,feasible and reliable but also compatible with a CAD system using NURBS.

Envelope protection for aircraft encountering upset condition based on dynamic envelope enlargement

Upset condition encountered by an aircraft in flight could pose great threat to flight safety, which is of chief importance in civil aviation. The causal factors have the nonlinear and multiple characteristics, and as a result the conventional envelope protection system cannot successfully do with the condition. So dynamic envelope based on differential manifold theory, which can take more coupling factors into account, is proposed as a basis to design a novel envelope protection system. Then the relationship between the dynamic envelope and the control coefficient or pilot command is obtained, and the result shows that the dynamic envelope can be enlarged with the change of control coefficient. Furthermore, quantification of flight security is realized via defining relative distance between flight state and dynamic envelope, which can detect whether there is a risk for an aircraft in flight. Finally, an envelope protection system based on dynamic envelope enlargement is proposed on the basis. NASA＇s Generic Transport Model encountering hazard gust wind in climbing phase is taken as an example to verify the system＇s feasibility. The result shows that the system can give a better operation encountering upset condition and to a certain extent reduce the number of accidents or incidents.

IMPASSE POINTS AND REDUCTION OF QUASILINEAR DAES WITH INDEX-K

This paper aims to analyze the reduction of quasilinear differential equations,and the existence and uniqueness of its solution on its matrice coefficients. Anda mathematical charactorization of its impasse point is provided. One examplefrom practicc electrical network problem is used to test all these theory.