Normal product form of two-mode Wigner operator

In the context of normal product, we use the method of the integration within an ordered product (IWOP) of operators to derive three representations of the two-mode Wigner operator: SU (2) symmetric description, SU (1, 1) symmetric description and polar coordinate form. We find that two-mode Wigner operator has multiple potential degrees of freedom. As the physical meaning of the selected integral variable changes, Wigner operator shows different symmetries. In particular, in the case of polar coordinates, we reveal the natural connection between the two-mode Wigner operator and the entangled state representation.

Fixed Points Features in N-Point Gravitational Lenses

A set of fixed points in N-point gravitational lenses is studied in the paper. We use complex form of lens mapping to study fixed points. There are some merits of using a complex form over coordinate. In coordinate form gravitational lens is described by a system of two equations and in complex form is described by one equation. We transform complex equation of N-point gravitational lens into polynomial equation. It is convenient to study polynomial equation. Lens mapping presented as a linear combination of two mappings: complex analytical and identity. Analytical mapping is specified by deflection function. Fixed points are roots of deflection function. We show, that all fixed points of lens mapping appertain to the minimal convex polygon. Vertices of the polygon are points into which dimensionless point masses are. Method of construction of fixed points in N-point gravitational lens is shown. There are no fixed points in 1-point gravitational lens. We study properties of fixed points and their relation to the center of mass of the system. We obtained dependence of distribution of fixed points on center of mass. We analyzed different possibilities of distribution in N-point gravitational lens. Some cases, when fixed points merge with the center of mass are shown. We show a linear dependence of fixed point on center of mass in 2-point gravitational lens and we have built a model of this dependence. We obtained dependence of fixed point to center of mass in 3-point lens in case when masses form a triangle or line. In case of triangle, there are examples when fixed points merges. We study conditions, when there are no one-valued dependence of distribution of fixed points in case of 3-points gravitational lens and more complicated lens.

Physical deformation configuration of a spatial clamped cable based on Kirchhoff rods

Purpose In this study, a modeling method for a clamped deformable cable simulation based on Kirchhoff theory is proposed. This methodology can be used to describe the physical deformation configuration of any constrained flexible cable in a computer-aided design/manufacturing system. The modeling method, solution algorithm, simulation and experimental results are presented to prove the feasibility of the proposed methodology. The paper aims to discuss these issues. Design/methodology/approach First, Kirchhoff equations for deformable cables are proposed based on the nonlinear mechanics of thin elastic rods, and the general solution of the equations described by the Euler angles is given in the arc coordinate system. The parametric form solution of the Kirchhoff equations, which is easy to use, is then obtained in a cylindrical coordinate form based on Saint Venant’s theory. Finally, mathematical expressions that reflect the clamped cable configuration are given, and the deformable process is simulated based on an open source geometry kernel and is then tested by a 3D laser scanning technology. Findings The method presented in this paper can be adapted to any boundary condition for constrained cables as long as the external force and torque are known. The experimental results indicate that both the model and algorithm are efficient and accurate. Research limitations/implications A more comprehensive study must be executed for the physical simulation of more complicated constrained cables, such as the helical spring and asymmetric constraint. The influence of the material properties of the cable on the calculation efficiency must be considered in future analysis. Originality/value The semi-analytical algorithm of the cable simulation in cylindrical coordinates is a novel topic and is more accurate and efficient than the common numerical solution.

Research for Coupled van der Pol Systems with Parametric Excitation and Its Application

In this article, we study the primary resonances of van der Pol systems with parametric excitation using the multiple scales method (MSM) and the homotopy analysis method (HAM). First, we study the nonlinear dynamic response of a coupled system with parametric excitation when the ratio of internal resonances are different, and obtain the four-dimensional average equation of the rectangular coordinate form using the MSM, thereby periodic motions are found in the system. Second, using the HAM, we obtain the four periodic solutions, in which there are two sets of in-phase periodic solutions and two sets of out-of-phase periodic solutions. Finally, we obtain the frequency response curves using the MSM and the HAM, in which it is found that the differences could be ignored.

A Variation on Uncertainty Principle and Logarithmic Uncertainty Principle for Continuous Quaternion Wavelet Transforms

The continuous quaternion wavelet transform (CQWT) is a generalization of the classical continuous wavelet transform within the context of quaternion algebra. First of all, we show that the directional quaternion Fourier transform (QFT) uncertainty principle can be obtained using the component-wise QFT uncertainty principle. Based on this method, the directional QFT uncertainty principle using representation of polar coordinate form is easily derived. We derive a variation on uncertainty principle related to the QFT. We state that the CQWT of a quaternion function can be written in terms of the QFT and obtain a variation on uncertainty principle related to the CQWT. Finally, we apply the extended uncertainty principles and properties of the CQWT to establish logarithmic uncertainty principles related to generalized transform.

Dynamic Inhomogeneous Isoparametric Element with Coordinate Transformation

Based on the coordinate transformation method, the formula of the dynamic inhomogeneous isoparametric finite element method is presented for generating element stiffness, damping and mass matrices. First, the global coordinate form and simplified form of dynamic inhomogeneous finite element are given in this paper. Then, the discrete material parameter distributions under the isoparametric coordinate system are obtained by using the transformation relationship between the global coordinates and the isoparametric coordinates. The simplified form with the discrete material parameter distributions is obtained for generating the element stiffness and mass matrices of the dynamic inhomogeneous isoparametric element. The numerical examples show that the scheme proposed in present paper has high precision.

Optimization of Fractional Order Controller Parameters

A new training method is proposed, which could solve the problem of that parameters of fractional order controller are not easy to be selected. This method which based on the principle of gravity optimizes parameters. Random initial parameter based on step was set as coordinate form which in the midpoint of the multidimensional space. The error between the actual output and the target output was set as radius. This method had advantages which could not need to calculate the gradient and could randomly select initial. Through the simulation experiment, this method is successfully applied in the fractional order PID controller, which obtains the optimal parameters.

Bifurcation in calculus of variations with constraints

We describe a variational problem on a domain of a plane under a constraint of geometrical character. We provide sufficient and necessary conditions for the existence of bifurcation points. The problem in 2 coordinate form, corresponds to a quasilinear elliptic boundary value problem. The problem provides a physical model for several applications referring to continuum media and membranes.

Evaluation of Crowd Motion Direction Based on Wavelet Transform

Taking the wavelet decomposed approximate image as the main research object, a direction estimation method for moving object was proposed in this paper. Firstly, the approximate image for the frame of the video was obtained via wavelet decomposition; and furthermore, the motion estimation on the approximate image was achieved to obtain the motion vectors. Finally, the motion vectors were described as polar coordinate form to compute the number of motion vectors in specified angles and the information entropy of the motion directions. The experiment results show that the proposed method can remove the effect of noise and the results of direction estimation are consistent with the actual motion directions.