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Showing papers on "Elliptic coordinate system published in 2003"


Journal Article
TL;DR: In this article, a new nonlinear function for independent component analysis to process complex-valued signals, which is used in frequency-domain blind source separation, is presented. But the difference between the two types of functions is in the assumed densities of independent components.
Abstract: This paper presents a new type of nonlinear function for independent component analysis to process complex-valued signals, which is used in frequency-domain blind source separation. The new function is based on the polar coordinates of a complex number, whereas the conventional one is based on the Cartesian coordinates. The new function is derived from the probability density function of frequency-domain signals that are assumed to be independent of the phase. We show that the difference between the two types of functions is in the assumed densities of independent components. Experimental results for separating speech signals show that the new nonlinear function behaves better than the conventional one.

148 citations


Journal ArticleDOI
TL;DR: In this article, the (2+1)-dimensional Toda lattice is decomposed into solvable ordinary differential equations and the straightening out of the continuous flow and the discrete flow is exactly given through the Abel-Jacobi coordinates.
Abstract: Resorting to the finite-order expansion of the Lax matrix, the elliptic coordinates are introduced, from which the discrete Ablowitz–Ladik equations and the (2+1)-dimensional Toda lattice are decomposed into solvable ordinary differential equations. The straightening out of the continuous flow and the discrete flow is exactly given through the Abel–Jacobi coordinates. As an application, explicit quasiperiodic solutions for the (2+1)-dimensional Toda lattice are obtained.

62 citations


Journal ArticleDOI
TL;DR: In this article, the absolute nodal coordinate formulation is used to describe the motion of flexible and rigid bodies and natural coordinates are used to represent the motions of the rigid and flexible bodies.
Abstract: This paper deals with the dynamic description of interconnected rigid and flexible bodies. The absolute nodal coordinate formulation is used to describe the motion of flexible bodies and natural coordinates are used to describe the motion of the rigid bodies. The absolute nodal coordinate formulation is a nonincremental finite element procedure, especially suitable for the dynamic analysis of flexible bodies exhibiting rigid body motion and large deformations. Nodal coordinates, which include global position vectors and global slopes, are all defined in a global inertial coordinate system. The advantages of using the absolute nodal coordinate formulation include constancy in the mass matrix and the need for only a minimal set of nonlinear constraint equations when connecting different flexible bodies with kinematic joints. When bodies within the system can be considered rigid, the above-mentioned advantages of the equations of motion can be preserved, provided natural coordinates are used. In the natural coordinate method, the coordinates used to describe rigid bodies include global position vectors of basic points and global unit vectors. As occurs in absolute nodal coordinate formulation, rotational coordinates are avoided and the mass matrix is also constant. This paper provides computer implementation of this formulation that uses absolute coordinates for general two-dimensional multibody systems. The constraint equations needed to define kinematic joints between different bodies can be linear or nonlinear. The linear constraint equations, which include those needed to define rigid connections and revolute joints, are used to define constant connectivity matrices that reduce the size of the system coordinates. These constant connectivity matrices are also used to obtain the mass matrix and generalized forces of the system. However, the nonlinear constraint equations that account for sliding joints require the use of the Lagrange multipliers technique. Numerical examples are provided and compared to the results of other existing formulations.

61 citations


Journal ArticleDOI
TL;DR: The Lame equations of triply orthogonal coordinate systems were studied in detail by Bianchi [Bi2] and Darboux [Da] as mentioned in this paper, and a variety of explicit solutions with the help of dressing method were constructed by Krichever [K].
Abstract: Triply orthogonal coordinate systems have attracted the attention of mathematicians and physicists for almost two hundred years now. Particular examples of them were already used by Leibniz and Euler to evaluate multiple integrals in canonical coordinates, and later Lame and Jacobi carried out calculations in analytical mechanics with the help of the famous elliptic coordinates. The first general results on the geometry of triply orthogonal systems date back to the 19th century – like the famous theorem of Dupin, saying that coordinate surfaces intersect along curvature lines. The Lame equations which analytically describe the triply orthogonal systems were studied in detail by Bianchi [Bi2] and Darboux [Da]. Recently orthogonal systems came back into the focus of interest in mathematical physics as an example of an integrable system. Zakharov [Z] has shown how the Lame equations can be solved by the ∂-method and constructed a variety of explicit solutions with the help of the dressing method. Algebro-geometric solutions of the Lame equations were constructed by Krichever [K]. The recent interest to the orthogonal coordinate systems is in particular motivated by their applications to the theory of the associativity equations [Du].

33 citations


Journal ArticleDOI
TL;DR: In this article, a general analytic formula for the two-center Coulomb integrals over Slater-type orbitals in elliptical coordinates is obtained, expressed in terms of a product of the well-known auxiliary functions Ak(p) and Bk(p), and incomplete gamma functions.
Abstract: A general analytic formula is obtained for the two-center Coulomb integrals over Slater-type orbitals in elliptical coordinates. Finite series expansions are used in the evaluation of the radial part of the integrals. The analytic formula is expressed in terms of a product of the well-known auxiliary functions Ak(p) and Bk(p) and incomplete gamma functions. Recursive relations for the computer evaluation of these functions are given as well. The recursive relations are stable and our computer results are in good agreement with the benchmark values given in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

23 citations


Journal ArticleDOI
TL;DR: In this paper, an analytic solution to the problem of a plane electromagnetic wave scattering by two parallel elliptic conducting cylinders is presented using an iterative procedure to account for the multiple scattered field between the cylinders.
Abstract: An analytic solution to the problem of a plane electromagnetic wave scattering by two parallel elliptic conducting cylinders is presented using an iterative procedure to account for the multiple scattered field between the cylinders. To compute the higher order terms of the scattered fields, the translation addition theorem for Mathieu functions is implemented to express the field scattered by one cylinder in terms of the elliptic coordinate system of the other cylinder in order to impose the boundary conditions. Scattered field coefficients of various orders are obtained and written in matrix form. Numerical results are obtained for the scattered field in the far zone for different axial ratios, electrical separations and angles of incidence.

16 citations


Journal ArticleDOI
TL;DR: In this paper, a brief survey of fractional calculus and fractional differential forms was given, and the fractional exterior transition to curvilinear coordinate at the origin was discussed.
Abstract: A brief survey of fractional calculus and fractional differential forms was firstly given The fractional exterior transition to curvilinear coordinate at the origin were discussed and the two coordinate transformations for the fractional differentials for three-dimensional Cartesian coordinates to spherical and cylindrical coordinates are obtained, respectively In particular, for v=m=1, the usual exterior transformations, between the spherical coordinate and Cartesian coordinate, as well as the cylindrical coordinate and Cartesian coordinate, are found respectively, from fractional exterior transformation

15 citations


Proceedings ArticleDOI
16 May 2003
TL;DR: In this paper, an analytic solution to the problem of a plane electromagnetic wave scattering by two parallel elliptic conducting cylinders is presented using an iterative procedure to account for the multiple scattered field between the cylinders.
Abstract: An analytic solution to the problem of a plane electromagnetic wave scattering by two parallel elliptic conducting cylinders is presented using an iterative procedure to account for the multiple scattered field between the cylinders. To compute the higher order terms of the scattered fields, the translation addition theorem for Mathieu functions is implemented to express the field scattered by one cylinder in terms of the elliptic coordinate system of the other cylinder to impose the boundary conditions. Scattered field coefficients of various orders are obtained and written in matrix form. Numerical results are obtained for the scattered field in the far zone for different axial ratios, electrical separations and angles of incidence

8 citations


Journal ArticleDOI
TL;DR: In this paper, the Siegert pseudostate method was applied to the case of hyperspherical elliptic coordinates using the S-wave resonances and antiresonances, and the resonance positions and widths were compared to other accurate calculations.
Abstract: The recently developed two-channel Siegert pseudostate method of Sitnikov and Tolstikhin [Phys. Rev. A 67, 032714 (2003)] is here applied to the case of ${\mathrm{Ps}}^{\ensuremath{-}}$ ${(e}^{+}ee)$ exploiting the hyperspherical elliptic coordinates method. S-wave resonances and antiresonances are calculated and classified according to their locations in the complex plane of the uniformization parameter. The resonance positions and widths are compared to other accurate calculations.

8 citations


Posted Content
TL;DR: In this article, the authors studied the separability of Newton systems that admit n quadratic first integrals and showed that a related Newton system with the same integrals can be trans-formed into a Stackel separa-ble Hamiltonian system and solved by qudratures.
Abstract: A conservative Newton system ¨ q = −∇V (q) in R n is called separa- ble when the Hamilton-Jacobi equation for the natural Hamiltonian H = 1 p 2 +V (q) can be solved through separation of variables in some curvilinear coordinates. If these coordinates are orhogonal, the New- ton system admits n first integrals, which all have separable Stackel form with quadratic dependence on p. We study here separability of the more general class of Newton systems ¨ q = −(cof G) −1 ∇W(q) that admit n quadratic first integrals. We prove that a related system with the same integrals can be trans- formed through a non-canonical transformation into a Stackel separa- ble Hamiltonian system and solved by qudratures, providing a solution to the original system. The separation coordinates, which are defined as characteristic roots of a linear pencil G −µ ˜ G of elliptic coordinates matrices, gener- alize the well known elliptic and parabolic coordinates. Examples of such new coordinates in two and three dimensions are given. These results extend, in a new direction, the classical separability theory for natural Hamiltonians developed in the works of Jacobi, Liouville, Stackel, Levi-Civita, Eisenhart, Benenti, Kalnins and Miller.

8 citations


Book ChapterDOI
01 Jan 2003
TL;DR: In this article, a unified coordinate system was proposed to resolve contact discontinuities for unsteady viscous flow, as well as the special case of steady flow when h = k. The functions h and k are determined by requiring that fluid pathlines coincide with coordinate lines and the grid angles are preserved.
Abstract: Publisher Summary This chapter introduces a coordinate system, which moves with velocity Q = (hu,kv), where q = (u,v) is fluid velocity. It includes the Eulerian coordinates as a special case when h = k = 0 and the Lagrangian when h = k = 1. The functions h and k are determined by requiring that fluid pathlines coincide with coordinate lines and the grid angles are preserved. This unified coordinate system is shown to be superior to the Eulerian one in resolving contact discontinuities for unsteady viscous flow, as well as the special case of steady flow when h = k. It also avoids computation breakdown associated with Lagrangian coordinates.

Journal ArticleDOI
TL;DR: In this article, the authors presented a quantum model for muon transfer between muonic hydrogen and an oxygen nuclei for s waves and collision energies in the range 10{sup -3}-10{sup 3} eV.
Abstract: Quantum-mechanical calculations of muon transfer between muonic hydrogen and an oxygen nuclei for s waves and collision energies in the range 10{sup -3}-10{sup 3} eV are presented. Close-coupling time-independent Schroedinger equations, written in terms of hyperspherical elliptic coordinates, were integrated along the hyperradius to obtain the partial and total muon-transfer probabilities. The results show the expected Wigner-Bethe threshold behavior up to collision energies of the order of 10{sup -2} eV and pronounced maxima at 10{sup 2} eV which can be interpreted in terms of crossings between potential energy curves corresponding to the entrance channel state ({mu}p){sub 1s}+O and two product channels which asymptotically correlate to p+(O{mu}){sub n=5,6}. The population of the final states with different orbital angular momenta is found to be essentially independent of energy in the range considered in this work. This can be attributed to a strong selection rule for the conservation of the quantum number associated with one of the elliptic hyperangles.

Journal ArticleDOI
TL;DR: In this article, the quasi-TEM modes on a planar multiconductor transmission line embedded in an elliptically stratified cross section are considered, and asymptotic solutions for the radial dependences of the terms of the series can be used under certain conditions on the profiles of stratification.
Abstract: The quasi-TEM modes on a planar multiconductor transmission line embedded in an elliptically stratified cross section are considered. Electro- and magneto-static problems are solved using separation of variables in elliptical coordinates. It is shown that asymptotic solutions for the radial dependences of the terms of the series can be used, under certain conditions, on the profiles of stratification. Results about the convergence and the usefulness of the asymptotic solution are presented.

Posted Content
TL;DR: In this paper, the integrable Camassa-Holm equation on the line with positive initial data rapidly decaying at infinity was considered and a one parameter family of integrably hierarchies which preserves the mixed spectrum of the associated string spectral problem was constructed.
Abstract: We consider the integrable Camassa--Holm equation on the line with positive initial data rapidly decaying at infinity. On such phase space we construct a one parameter family of integrable hierarchies which preserves the mixed spectrum of the associated string spectral problem. This family includes the CH hierarchy. We demonstrate that the constructed flows can be interpreted as Hamiltonian flows on the space of Weyl functions of the associated string spectral problem. The corresponding Poisson bracket is the Atiyah--Hitchin bracket. Using an infinite dimensional version of the Jacobi ellipsoidal coordinates we obtain a one parameter family of canonical coordinates linearizing the flows.

Posted Content
26 Mar 2003
TL;DR: In this article, the authors considered integrability of the Camassa-Holm equation on the line with positive initial data rapidly decaying at infinity and constructed one parameter family of integrable hierarchies which preserves mixed spectrum of the associated string spectral problem.
Abstract: We consider integrable the Camassa–Holm equation on the line with positive initial data rapidly decaying at infinity. On such phase space we construct one parameter family of integrable hierarchies which preserves mixed spectrum of the associated string spectral problem. This family includes the CH hierarchy. We demonstrate that the constructed flows can be interpreted as Hamiltonian flows on the space of Weyl functions of the associated string spectral problem. The corresponding Poisson bracket is the Atiyah–Hitchin bracket. Using an infinite dimensional version of the Jacobi elliptic coordinates we obtain one parameter family of canonical coordinates linearizing the flows.

01 Jan 2003
TL;DR: In this paper, a ray theoretic formulation is developed that allows rays to be traced directly from existing solutions to the Helmholtz equation using a generalized coordinate system (GCS) approach.
Abstract: A ray theoretic formulation is developed that allows rays to be traced directly from existing solutions to the Helmholtz equation. These rays, termed phase-rays, are dened by the direction normal to surfaces of constant waveeld phase. Phase-rays exhibit a number of attractive characteristics, including triplication-free ray-elds, an ability to shoot rays forward or backward, and an ability to shoot inll rays for ensuring adequate ray density. Because of these traits, we use phase-rays as a coordinate basis on which to extrapolate waveelds using the generalized coordinate system approach. Examples of waveelds successfully extrapolated in phase-ray coordinates are presented, and the merits and drawbacks of this approach, relative to conventionally traced ray coordinates, are discussed.

Proceedings ArticleDOI
A.C. Kabel1
12 May 2003
TL;DR: In this article, the authors considered the trajectory of a charged particle in an arbitrary external magnetic field and derived partial differential equations for the magnetic fields fix conditions among its local multipole components, which can be viewed as a generalization of the usual multipole expansion of the magnetic elements.
Abstract: We consider the trajectory of a charged particle in an arbitrary external magnetic field A local orthogonal coordinate system is given by the tangential, curvature, and torsion vectors We write down Maxwell's equations in this coordinate system The resulting partial differential equations for the magnetic fields fix conditions among its local multipole components, which can be viewed as a generalization of the usual multipole expansion of the fields of magnetic elements

Journal Article
TL;DR: According to the formula of rectangular coordinate in the plane of Gauss-Kruger projection, the authors designed a calculation program suitable for the CASIO fx 4800P calculator, put forward the steps and method of inversing of geographic coordinate by using rational approximation.
Abstract: According to the formula of rectangular coordinate in the plane of Gauss-Kruger projection, the authors designed a calculation program suitable for the CASIO fx 4800P calculator, put forward the steps and method of inversing of geographic coordinate by using rational approximation

Journal ArticleDOI
TL;DR: In this article, the problem of the propagation of optical waves in homogeneous and inhomogeneous media is reduced to the solution of the Helmholtz equation or the parabolic approximation to this equation in a coordinate system in which the inhomogeneities of the medium are arranged inthe directions of coordinate axes.
Abstract: INTRODUCTIONAs a rule, problems on the propagation of optical waves in homogeneous and inhomogeneousmedia, including the computation of eigenmodes of dielectric waveguides and cavity resonators,can be reduced to the solution of the Helmholtz equation or the parabolic approximation to thisequation [1, 2]. Both the statement of the problem and its subsequent analysis are simpli eddramatically in a coordinate system in which the inhomogeneities of the medium are arranged inthe directions of coordinate axes [3, 4]. In a number of practically interesting cases, for example,in simulation wedge- and cone-shaped waveguides, the adaptation of the coordinate system tothe inhomogeneities can be implemented with the use of the corresponding standard orthogonalcurvilinear coordinates.The statement of the problem for the Helmholtz equation in curvilinear coordinates does notcause any diculties [5, 6]; however, the speci c features of the passage to the paraxial approxima-tion to the wave equation were considered only for Cartesian and cylindrical coordinates with thepropagation direction along the

01 Jan 2003
TL;DR: In view of morbid matrix of GPS coordinate transforming into local coordinate, the paper puts forward the separation regression method and introduces the principle, the transformation process and the transformation example of the separation regressors.
Abstract: In view of morbid matrix of GPS coordinate transforming into local coordinate, the paper puts forward the separation regression method and introduces the principle, the transformation process and the transformation example of the separation regression method.

Book ChapterDOI
Zi-Niu Wu1, Jing Shi1
01 Jan 2003
TL;DR: In this article, the equivalence of solutions (such as simple waves, weak solutions, and hyperbolicity) in different coordinate systems is discussed, including given, flow driven and space-time transformations.
Abstract: In this paper we consider rather general coordinate transformation including given, flow driven (such as the unified coordinate and the generalized characteristic coordinates proposed here), and space-time transformation. We discuss the equivalence of solutions (such as simple waves, weak solutions, and hyperbolicity) in different coordinate systems.

Proceedings ArticleDOI
TL;DR: In this article, a new class of invariant optical fields which are exact solutions of the Helmholtz equation in parabolic coordinates is introduced, and the spatial distribution of these PIOFs is described in terms of the even and odd Parabolic cylinder functions Pe and Po.
Abstract: From group theory, it is known that the three-dimensional Helmholtz equation is separable in four orthogonal cylindrical coordinate systems: rectangular (i.e. Cartesian), circular, elliptic, and parabolic systems [1]. Invariant Optical Fields (often referred to as nondiffracting beams) have been demonstrated theoretical and experimentally for plane waves in Cartesian coordinates, for Bessel beams in circular cylindrical coordinates [2,3], and for Mathieu beams in elliptic coordinates [4,5]. We introduce in this work a new class of invariant optical fields which are exact solutions of the Helmholtz equation in parabolic coordinates. The spatial distribution of these PIOFs is described in terms of the even and odd Parabolic cylinder functions Pe and Po, namely