Polynomial In Cartesian Coordinates Research Articles

Integrable systems in magnetic fields: the generalized parabolic cylindrical case

This article is a contribution to the classification of quadratically integrable systems with vector potentials whose integrals are of the nonstandard, nonseparable type. We focus on generalized parabolic cylindrical case, related to non-subgroup-type coordinates. We find three new systems, two with magnetic fields polynomial in Cartesian coordinates and one with unbounded exponential terms. The limit in the parameters of the integrals yields a new parabolic cylindrical system; the limit of vanishing magnetic fields leads to the free motion. This confirms the conjecture that non-subgroup type integrals can be related to separable systems only in a trivial manner.

Spherical radial basis function approximation of some physical quantities measured

The article treats the spherical interpolation and approximation, i.e., a way of measured data processing in case when the scalar data sampling is performed at nodes on the unit 2D sphere surface in the 3D Euclidean space (in general, on the (d−1)-dimensional sphere surface in the d-dimensional space). We use the spherical radial basis function interpolation with an inverse multiquadric basis function and a second degree polynomial in Cartesian coordinates as a trend.The formulae of this type can be useful in the interpretation of various physical measurements and have wide applications in geosciences, e.g., in the treatment of anisotropy of magnetic susceptibility data measured. We present the advantages of the formula chosen on numerical examples. However, in practical computation with high number of sampling nodes, the matrix of the system for determining interpolation coefficients may be ill-conditioned. Then the standard double precision LU factorization does not give a reliable solution due to the effect of round-off error accumulation and some more sophisticated means of solving the system are to be used.

Improving the Targets’ Trajectories Estimated by an Automotive RADAR Sensor Using Polynomial Fitting

A way to reduce the uncertainty at the output of a Kalman filter embedded into a tracker connected to an automotive RADAR sensor consists of the adaptive selection of parameters during the tracking process. Different informed strategies for automatically tuning the tracker’s parameters and to jointly learn the parameters and state/output sequence using: expectation maximization; optimization approaches, including the simplex algorithm; coordinate descent; genetic algorithms; nonlinear programming using finite differencing to estimate the gradient; Bayesian optimization and reinforcement learning; automatically tuning hyper-parameters in the least squares, were already proposed. We develop here a different semi-blind post-processing approach, which is faster and more robust. Starting from the conjecture that the trajectory is polynomial in Cartesian coordinates, our method supposes to fit the data obtained at the output of the tracker to a polynomial. We highlight, by simulations, the improvement of the estimated trajectory’s accuracy using the polynomial fitting for single and multiple targets. We propose a new polynomial fitting method based on wavelets in two steps: denoising and polynomial part extraction, which compares favorably with the classical polynomial fitting method. The effect of the proposed post-processing methods is visible, the accuracy of targets’ trajectories estimations being hardly increased.

Adaptation and performance of the Cartesian coordinates fast multipole method for nanomagnetic simulations

An implementation of the fast multiple method (FMM) is performed for magnetic systems with long-ranged dipolar interactions. Expansion in spherical harmonics of the original FMM is replaced by expansion of polynomials in Cartesian coordinates, which is considerably simpler. Under open boundary conditions, an expression for multipole moments of point dipoles in a cell is derived. These make the program appropriate for nanomagnetic simulations, including magnetic nanoparticles and ferrofluids. The performance is optimized in terms of cell size and parameter set (expansion order and opening angle) and the trade off between computing time and accuracy is quantitatively studied. A rule of thumb is proposed to decide the appropriate average number of dipoles in the smallest cells, and an optimal choice of parameter set is suggested. Finally, the superiority of Cartesian coordinate FMM is demonstrated by comparison to spherical harmonics FMM and FFT.

The classification of anisotropic stressed states in spherical bodies

Tensor fields (for example, of pressure) with desired properties of symmetry under rotation were constructed in an explicit form. Tensor components represented as polynomials in Cartesian coordinates satisfy a biharmonic equation. Possible applications of the obtained results in various domains of technical physics are discussed.

Fractal escapes in Newtonian and relativistic multipole gravitational fields

We study the planar motion of test particles in gravitational fields produced by an external material halo. Both the Newtonian and the general-relativistic dynamics are examined, and in the relativistic case the dynamics of both massive and massless particles are investigated. The halo field is given in general by a multipole expansion; we restrict ourselves to multipole fields of pure order, whose Newtonian potentials are homogeneous polynomials in cartesian coordinates. A pure n-pole field has n different escapes, one of which is chosen by the particle according to its initial conditions. We find that the escape has a fractal dependency on the initial conditions for n>2 both in the Newtonian and the relativistic cases for massive test particles, but with important differences between them. The relativistic motion of massless particles, however, was found to be regular for all the fields we could study. The box-counting dimension was used in each case to quantify the sensitivity to initial conditions which arises from the fractality of the escape route.

Closed cartesian representation of the Zernike polynomials

A general closed representation is presented of Zernike polynomials in cartesian coordinates. This form substantially increases the precision of numerical calculations when x-y coordinates are used to sample an interferogram, by avoiding any numerical conversions from cartesian to polar coordinates.

Transition Plate‐Bending Elements for Compatible Mesh Gradation

In this paper, a new approach to the development of the transition plate‐bending elements with variable midside nodes is presented. These elements will facilitate abrupt changes in mesh refinement between domains consisting of quadrilateral elements. For compatible mesh gradation, special shape functions with slope discontinuity at midside nodes are used. The transition elements are formulated based on the Mindlin‐Reissner plate theory to take shear deformation into account. To evaluate the proper element stiffness pertinent to shear, the substitute shear strain polynomials in Cartesian coordinates are constructed. A modified Gaussian integration is adopted to eliminate the slope discontinuity problem in the element domain. Some numerical examples are presented that illustrate the validity and effectiveness of these transition elements for the efficient analysis of plate‐bending problems.