Direct method of finding first integral of two-dimensional autonomous systems in polar coordinates

A direct method to find the first integral for two-dimensional autonomous system in polar coordinates is suggested. It is shown that if the equation of motion expressed by differential 1-forms for a given autonomous Hamiltonian system is multiplied by a set of multiplicative functions, then the general expression of the first integral can be obtained, An example is given to illustrate the application of the results.

ANALYSIS OF FDTD TO UPML FOR MAXWELL EQUATIONS IN POLAR COORDINATES

An FDTD system associated with uniaxial perfectly matched layer（UPML） for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic（TM） mode to Maxwell＇s equations is obtained by Yee＇s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis.

An optimized finite difference method based on a polar coordinate system for regional-scale irregular topography

The finite difference method(FDM)is an important numerical approach for simulating the propagation of seismic waves,and some FDMs can be used to study the impact of the Earth’s curvature and topography over large distances.To efficiently model the effects of the Earth’s irregular topography on the propagation of seismic waves,here we optimize a previously proposed grid mesh method and develop a novel two-dimensional boundary-conforming FDM based on a curvilinear polar coordinate system.This method efficiently simulates the propagation of seismic waves in an arc-shaped model with large variations in surface topography.Our method was benchmarked against other reported methods using several global-scale models.The consistency of the results confirms the validity of our proposed optimization strategy.Furthermore,our findings indicate that the proposed optimization strategy improves computational efficiency.

FURTHER APPLICATIONS OF POLAR COORDINATES IN C^n

Simplify the proof on the domain of convergence of multiple power series and consider the case where some of z1, …, zn are contained only in a finite number of terms of the series. Obtain some results on holomorphic functions in Cn.

Research of contour error compensation control for X-C non-circular grinding in polar coordinates

In the X-C linkage grinding of non-circular parts,the computation and control method of contour error in polar coordinates platform is different with that in the XY coordinates platform.To solve this problem,the analysis of the definition and computation methods for contour error and track error in polar coordinates platform will be made.Through the relative lead-lag relation of the linkage axes in the grinding process,the range of the estimation contour error is narrowed and a contour error calculation model is constructed.Then the contour compensation controllers along contour error direction and trajectory tracking error direction are designed respectively,and the error compensation decoupling matrix of the X-C linkage axes is given as well.In the end of this paper,we take the machining of the cylinder contour in Wankel rotary piston engine as an example.A simulation experiment of contour error compensation control based on relative lead-lag quantity is made.The result shows that the designed contour compensation controller can increase the contour machining accuracy effectively.

A novel particle tracking algorithm using polar coordinate system similarity

A new algorithm using polar coordinate system similarity （PCSS） for tracking particle in particle tracking velocimetry （PTV） is proposed. The essence of the algorithm is to consider simultaneously the changes of the distance and angle of surrounding particles relative to the object particle. Monte Carlo simulations of a solid body rotational flow and a parallel shearing flow are used to investigate flows measurable by PCSS and the influences of experimental parameters on the implementation of the new algorithm. The results indicate that the PCSS algorithm can be applied to flows subjected to strong rotation and is not sensitive to experimental parameters in comparison with the conventional binary image cross-correlation （BICC） algorithm. Finally, PCSS is applied to images of a real experiment.

Star Point Energy Center Correction of Star Simulator Based on Polar Coordinates

In this paper,various aberrations have been analyzed.Not only the effects of aberration on geometrical center position are taken into account,but also the deviation of displayed star position energy center caused by aberration is analyzed.These two aspects have been taken into comprehensive evaluation and star position correction.The correction method based on polar coordinates is proposed,and cumbersome partition correction and calculated quantity based on two-dimensional coordinates can be simplified.The experimental results show that the correction processing based on polar coordinates is simpler and easier compared with any other correction methods.In addition,the correction results are significantly more accurate.

Multiframe track-before-detect method based on velocity filtering in mixed coordinates

In this paper,a velocity filtering based track-before-detect algorithm in mixed coordinates is presented to address the problem of integration loss caused by inaccurate motion model in polar coordinate sensors.Since the motion of a con-stant velocity(CV)target is better modeled in Cartesian coordi-nates,the search of measurements for integration in polar sensor coordinates is carried out according to the CV model in Cartesian coordinates instead of an approximate model in polar sensor coordinates.The position of each cell is converted into Cartesian coordinates and predicted according to an assumed velocity.Then,the predicted Cartesian position is converted back to polar sensor coordinates for multiframe accumulation.The use of the correct model improves integration effectiveness and consequently improves algorithm performance.To handle the weak target with unknown velocity,a velocity filter bank in mixed coordinates is presented.The influence of velocity mis-match on the performance of filter bank is analyzed,and an effi-cient strategy for filter bank design is proposed.Numerical re-sults are presented to demonstrate the effectiveness of the pro-posed algorithm.

Stability analysis of the projectile based on random center manifold reduction

The center manifold method has been widely used in the field of stochastic dynamics as a dimensionality reduction method.This paper studied the angular motion stability of a projectile system under random disturbances.The random bifurcation of the projectile is studied using the idea of the Routh-Hurwitz stability criterion,the center manifold reduction,and the polar coordinates transformation.Then,an approximate analytical presentation for the stationary probability density function is found from the related Fokker–Planck equation.From the results,the random dynamical system of projectile generates three different dynamical behaviors with the changes of the bifurcation parameter and the noise strength,which can be a reference for projectile design.

SOME PROPERTIES OF MULTIPLE TAYLOR SERIES AND RANDOM TAYLOR SERIES

Some polar coordinates are used to determine the domain and the ball of convergence of a multiple Taylor series. In this domain and in this ball the series converges, converges absolutely and converges uniformly on any compact set properties of the series may also be studied. For some random multiple are some corresponding properties. Growth and other Taylor series there

SOME REMARKS ON HOLOMORPHIC FUNCTIONS AND TAYLOR SERIES IN C^n

Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.

FOURTH-ORDER COMPACT SCHEMES FOR HELMHOLTZ EQUATIONS WITH PIECEWISE WAVE NUMBERS IN THE POLAR COORDINATES

In this paper, fourth-order compact finite difference schemes are proposed for solving Helmholtz equation with piecewise wave numbers in polar coordinates with axis-symmetric and in some cases that the solution depends both of independent variables. The idea of the immersed interface method is applied to deal with the discontinuities in the wave number and certain derivatives of the solution. Numerical experiments are included to confirm the accuracy and efficiency of the proposed method.

Practical Method of Conical Cam Outline Expansion

Conical cam mechanism has been widely used in modern machinery and equipment.However,the commonly used planar expansion methods for the design of spatial cam contour produce significant errors,because these methods incorrectly use the distance from the axis of the follower to the main conical cam to replace the corresponding arc length on the conical cam.HSIEH,et al,used analytical methods to achieve higher accuracy,but these analytical methods have their own drawbacks since they are too complicated for practical use.Through the analysis of the errors created during the generation of conical cam contour using the existing expansion methods,this paper proposes to include diverge angle in the calculation of conical cam rotation angle in the equation of conical cam contour expansion.This correction eliminates the error generated by the commonly used methods.Based on the expression of the follower＇s 3D trajectory and the spatial geometry of conical cam,this paper has deduced the planar polar curve equation for determining polar coordinates for the curve of planar expansion outline.Furthermore,this paper provides an example of conical cam contour design based on sinusoidal acceleration variation.According to polar coordinates and the movement of curve equation function expression,this paper applies MATLAB software to solve coordinates for the cam expansion curve and uses AutoCAD software to generate conical cam expansion contour that meets the requirement of the law of motion.The proposed method provides a design process that is simple,intuitive and easy to master and implement.It also avoids the design error in the traditional methods for generating contour of conical cam with oscillating follower that requires high precision.

The Algebra of Projective Spheres on Plane, Sphere and Hemisphere

Numerous authors studied polarities in incidence structures or algebrization of projective geometry [1] [2]. The purpose of the present work is to establish an algebraic system based on elementary concepts of spherical geometry, extended to hyperbolic and plane geometry. The guiding principle is: “The point and the straight line are one and the same”. Points and straight lines are not treated as dual elements in two separate sets, but identical elements within a single set endowed with a binary operation and appropriate axioms. It consists of three sections. In Section 1 I build an algebraic system based on spherical constructions with two axioms: ab = ba and (ab)(ac) = a, providing finite and infinite models and proving classical theorems that are adapted to the new system. In Section Two I arrange hyperbolic points and straight lines into a model of a projective sphere, show the connection between the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. In Section Three I create another model of a projective sphere in the Cartesian coordinate system of the plane, and give methods and techniques for using the model in the theory of functions.

Establishment,simulation and verification of firepower safety control model

On a narrow warship platform,the coordinated use of shipborne weapon systems may cause firepower conflicts,which seriously endangers the ship safety.Meanwhile,with directed-energy weapons mounted on ships,firepower conflicts between weapons become a“high probability event”.Aiming at the problem of firepower safety control,based on the research about the collision probability model of air crafts and space targets and according to the cone of fire model of conventional weapons and directed-energy weapons,this paper solved the firepower conflict probabilities between conventional weapons as well as between conventional weapons and directed-energy weapons respectively using the methods of probability theory,and established the firepower safety control model.Then the calculation of firepower conflict probability was carried out using the dimensionality reduction method based on the equivalent conversion of polar coordinates and the power series method based on Laplace transform.The simulation results revealed that the proposed model and calculation methods are effective and reliable,which can provide theoretical basis and technical support for resolution of firepower conflicts between weapons.

Lagrange Interpolation on a Sphere

In this paper, we obtain a properly posed set of nodes for interpolation on a sphere. Moreover it is applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of total degree n.

Single-pixel wavefront sensing via vectorial polarization modulation[Invited]

The Shack–Hartmann wavefront sensor(SHWFS)is commonly used for its high speed and precision in adaptive optics.However,its performance is limited in low light conditions,particularly when observing faint objects in astronomical applications.Instead of a pixelated detector,we present a new approach for wavefront sensing using a single-pixel detector,which is able to code the spatial position of a light spot array into the polarization dimension and decode the polarization state in the polar coordinate.We propose validation experiments with simple and complex wavefront distortions to demonstrate our approach as a promising alternative to traditional SHWFS systems,with potential applications in a wide range of fields.

A Blind Watermarking Algorithm Resisting to Geometric Transforms Based on SVD

A new watermarking algorithm resisting to geometric transformation based on singular value decomposition （SVD） in logarithm polar coordinate is proposed. The log-polar mapping （LPM） is used to resist rotation and scaling attacks, and the odd-even quantization algorithm is used to embed watermark so it can be extracted without the original host image. The experiments show that the proposed algorithm not only resists various geomet- ric attacks but also is robust enough to the common signal processing.

Geometric Deformations Based on 3D Volume Morphing

This paper presents a new geometric deformation method based on 3D volume morphing by using a new concept called directional polar coordinate. The user specifies the source control object and the destination control object which act as the embedded spaces. The source and the destination control objects determine a 3D volume morphing which maps the space enclosed in the source control object to that of the destination control object. By embedding the object to be deformed into the source control object, the 3D volume morphing determines the deformed object automatically without the tiring moving of control points. Experiments show that this deformation model is efficient and intuitive, and it can achieve some deformation effects which are difficult to achieve for traditional methods.

A new modified group explicit iterative method for the numerical solution of time dependent viscous Burgers’equation

In this article,we discuss a new smart alternating group explicit method based on off-step discretization for the solution of time dependent viscous Burgers’equation in rectangu-lar coordinates.The convergence analysis for the new iteration method is discussed in details.We compared the results of Burgers’equation obtained by using the proposed iterative method with the results obtained by other iterative methods to demonstrate computationally the efficiency of the proposed method.