Some New Results on Indices of Singular Points of a Polynomial Vector Field

Several new results on the indices of singular points in a polynomial vector field areproved in this paper(see Theorems 1-5).

Necessary and sufficient condition of C^0 flows on closed surfaces with isolated singular points having the pseudo-orbit tracing property

IN this letter we discuss the necessary and sufficient condition of C^0 flows on closed surfaces with isolated singular points having the pseudo-orbit tracing property. According to ref. [1], on a closed surface, every minimal set of a C^r(r≥2) flow is trivial, but it is possible for a C^0 flow to contain non-trivial minimal sets. Thus C^0 flows on closed surfaces are more complicated than C^r(r≥2) flows.

A Note on Parameterized Preconditioned Method for Singular Saddle Point Problems

Recently, some authors (Li, Yang and Wu, 2014) studied the parameterized preconditioned HSS (PPHSS) method for solving saddle point problems. In this short note, we further discuss the PPHSS method for solving singular saddle point problems. We prove the semi-convergence of the PPHSS method under some conditions. Numerical experiments are given to illustrate the efficiency of the method with appropriate parameters.

Hydraulics Management Model of Response Function Solved by Laplace Transform Boundary Element Method

In solving a response function by the boundary element method, the use of the singular valued method and the Laplace transform in a time domain makes the solving process be simplified and the result be accurate. The restricted condition matrix formed by the response matrix method is much smaller than that by embedding method. In addition, the response function may realize directly the management decision making. So it is efficient for establishing and solving hydraulics management models.

BIFURCATIONS OF A CUBIC SYSTEM PERTURBED BY DEGREE FIVE

In this article, using multi-parameter perturbation theory and qualitative analysis, the authors studied a kind of cubic system perturbed by degree five and ob-tained the system that can have 17 limit cycles giving their two kinds of distributions (see Fig.5).

Discrete Differential Geometry of Triangles and Escher-Style Trick Art

This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.

Study on Stable Scanning of Terminal Sensing Ammunition Based on Quaternion Transformation

Euler angles and Euler kinematics equation of terminal sensing ammunition are expressed and rewritten by using quaternion to solve the singular problem in using Euler angles to describe the motion.The contrastive simulations are performed in order to validate the correctness and advantage of the quaternion description.The simulation results show that the dynamic model with quaternion have stable solution,there is not singular point in the calculation,and the ballistic model rewritten by using the quaternion is suitable for describing the terminal sensing ammunition's scanning motion than the common Euler equation.

奇异点附近的复微分方程解的增长性与系数下级的关系

We investigate the growth of solutions of the following complex linear di er-ential equation f″+A(z)f′+B(z)f=0,where A(z)and B(z)are analytic functions in -C-{z0},z0∈C.Some estimations of lower bounded of growth of solutions of the di erential equation are obtained by using the concept of lower order.

Frobenius Method for Solving Second-Order Ordinary Differential Equations

As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point t = 0 and determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.

Geometric and analytic problems for a real submanifold in C^n with CR singularities

In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems that the author and his coauthors have recently worked on. The article also contains an example of a Bishop submanifold in C^3 of real codimension two, which cannot be quadratically flattened at a CR singular point but is CR non-minimal at any CR point. This provides a counter-example to a question asked in a private communication by Zaistev(2013).

Singular PT-symmetry broken point with infinite transmittance and reflectance--a classical analytical demonstration

To demonstrate the existence of singularparity-time symmetry(PT-symmetry)broken point inoptics system,we designed a one-dimensional PT symmetricstructure including N unit-cell with loss and gainmaterials in half.We performed an analytical deduction toobtain the transmittance and reflectance of the structurebasing on Maxwell’s equations.We found that with theexact structure unit-cell number and the imaginary part ofrefraction index,the transmittance and reflectance are bothclose to infinite.Such strict condition is called the singularpoint in this study.At the singular point position,both thetransmission and reflection are direction-independent.Away from the singular point,the transmittance andreflectance become finite.In light of classical wave optics,the single unit and total structure both become theresonance units.The infinite transmittance and reflectanceresult from the resonance matching of single unit and totalstructure.In light of quantum theory,the singular pointcorresponds to the single eigenvalue of electromagneticscattering matrix.The infinite transmittance and reflectancemean a huge energy transformation from pumpingsource to light waves.Numerical calculation and softwaresimulation both demonstrate the result.

Mathematical modeling of building structures and nonlinear differential equations

Nonlinear differential equations with moving singular points require emergence and development of new approximate methods of solution.In this paper,we give a solution to one of the problems of the analytical approximate method for solving nonlinear differential equations with moving singular points,and study the influence of the perturbation of the initial conditions on the analytical approximate solution in the analytic domain.Theoretical material was tested using a numerical experiment confirming its reliability.The theoretical material presented in this paper allows researchers to use nonlinear differential equations with moving singular points when designing mathematical models of building structures.

Planar Lyapunov Algebras

We study complex involutive algebras generated by a single nonselfadjoint idempotent and use them to construct a family of algebras,which we call planar Lyapunov algebras.As our main result,we prove that every 2-dimensional commutative real algebra whose homogeneous Riccati differential equation is stable at the origin must be isomorphic either to an algebra with zero multiplication or to some planar Lyapunov algebra.

Remark on Gauss curvature equations on punctured disk

We give a new argument on the classification of solutions of Gauss curvature equation on R2,which was first proved by W.Chen and C.Li[Duke Math.J.,1991,63(3):615-622].Our argument bases on the decomposition properties of the Gauss curvature equation on the punctured disk.