Showing papers on "Elliptic coordinate system published in 2014"
TL;DR: In this paper, it was shown that the Hamilton-Jacobi equation for massless geodesics can only separate in elliptic or spherical coordinates, and all known integrable backgrounds are covered by this separation.
Abstract: Motivated by the search for new backgrounds with integrable string theories, we classify the D-brane geometries leading to integrable geodesics. Our analysis demonstrates that the Hamilton-Jacobi equation for massless geodesics can only separate in elliptic or spherical coordinates, and all known integrable backgrounds are covered by this separation. In particular, we identify the standard parameterization of AdSp × Sq with elliptic coordinates on a flat base. We also find new geometries admitting separation of the Hamilton-Jacobi equation in the elliptic coordinates. Since separability of this equation is a necessary condition for integrability of strings, our analysis gives severe restrictions on the potential candidates for integrable string theories.
78 citations
TL;DR: In this article, the sufficient and necessary condition of boundary integral formulation for the uniqueness solution of 2D Laplace problem subject to the Dirichlet boundary condition is examined, and five regularization techniques to ensure the unique solution are analyzed and numerically implemented.
Abstract: The occurring mechanism of the ill-conditioned system due to degenerate scale in the direct boundary element method (BEM) and the indirect BEM is analytically examined by using degenerate kernels. Five regularization techniques to ensure the unique solution, namely hypersingular formulation, method of adding a rigid body mode, rank promotion by adding the boundary flux equilibrium (direct BEM), CHEEF method and the Fichera’s method (indirect BEM), are analytically studied and numerically implemented. In this paper, we examine the sufficient and necessary condition of boundary integral formulation for the uniqueness solution of 2D Laplace problem subject to the Dirichlet boundary condition. Both analytical study and BEM implementation are addressed. For the analytical study, we employ the degenerate kernel in the polar and elliptic coordinates to derive the unique solution by using five regularization techniques for any size of circle and ellipse, respectively. Full rank of the influence matrix in the BEM ...
29 citations
TL;DR: In this article, the authors considered the sub-Riemannian length minimization problem on the group of motions of pseudo-Euclidean plane that form the special hyperbolic group SH(2), which comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional.
Abstract: We consider the sub-Riemannian length minimization problem on the group of motions of pseudo-Euclidean plane that form the special hyperbolic group SH(2). The system comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional. We apply the Pontryagin maximum principle to obtain the extremal control input and the sub-Riemannian geodesics. A change of coordinates transforms the vertical subsystem of the normal Hamiltonian system into the mathematical pendulum. In suitable elliptic coordinates, the vertical and the horizontal subsystems are integrated such that the resulting extremal trajectories are parametrized by the Jacobi elliptic functions. Qualitative analysis reveals that the projections of normal extremal trajectories on the xy-plane have cusps and inflection points. The vertical subsystem being a generalized pendulum admits reflection symmetries that are used to obtain a characterization of the Maxwell strata.
25 citations
TL;DR: In this article, an elliptic-symmetry vector optical field was proposed, which is more flexible than the cylindrical vector fields for controlling the spatial structure of polarization and for engineering the focusing fields.
Abstract: We present in principle and demonstrate experimentally a new kind of vector fields: elliptic-symmetry vector optical fields. This is a significant development in vector fields, as this breaks the cylindrical symmetry and enriches the family of vector fields. Due to the presence of an additional degrees of freedom, which is the interval between the foci in the elliptic coordinate system, the elliptic-symmetry vector fields are more flexible than the cylindrical vector fields for controlling the spatial structure of polarization and for engineering the focusing fields. The elliptic-symmetry vector fields can find many specific applications from optical trapping to optical machining and so on.
21 citations
TL;DR: The spectral properties of boundary integral operators in acoustic scattering have important practical implications, such as for the analysis of the stability of boundary element discretisations or the convergence of iterative solvers as the wavenumber k grows as mentioned in this paper.
Abstract: Understanding the spectral properties of boundary integral operators in acoustic scattering has important practical implications, such as for the analysis of the stability of boundary element discretisations or the convergence of iterative solvers as the wavenumber k grows Yet little is known about spectral decompositions of the standard boundary integral operators in acoustic scattering Theoretical results are mainly available on the unit disk, where these operators diagonalise in a simple Fourier basis In this paper we investigate spectral decompositions for more general smooth domains Based on the decomposition of the acoustic Green’s function in elliptic coordinates we give spectral decompositions on ellipses For general smooth domains we show that approximate spectral decompositions can be given in terms of circle Fourier modes transplanted onto the boundary of the domain An important underlying question is whether or not the operators are normal Based on previous numerical investigations it appears that the standard boundary integral operators are normal only when the domain is a ball and here we prove that this is indeed the case for the acoustic single layer potential We show that the acoustic single, double and conjugate double layer potential are normal in a scaled inner product on the ellipse On more general smooth domains the operators can be split into a normal component plus a smooth perturbation Numerical computations of pseudospectra are presented to demonstrate the nonnonnormal behaviour on general domains
12 citations
TL;DR: The tetrahedron coordinate approach proposed in this paper does not need to induce any new variables when more points on the manipulator are considered, and has a prevailing advantage over the general natural coordinate method.
Abstract: This paper proposes a kinematics model with four noncoplanar points' Cartesian coordinates for a spatial parallel manipulator, which is called the tetrahedron coordinate method. The sufficient and necessary criteria of utilizing the Cartesian coordinates of the four noncoplanar points are proved. Because the constraint equations are either quadratic or linear, and the coordinates are complete Cartesian, the derivative matrix of the constraint equations only consists of linear or constant elements that are the advantages of the general natural coordinate method as well. However, the number of variables of the general natural coordinate method will increase with the increasing number of investigated points. The tetrahedron coordinate approach proposed in this paper does not need to induce any new variables when more points on the manipulator are considered. As a result, it has a prevailing advantage over the general natural coordinate method. This advantage is especially explicit when establishing the kinematics models for complex spatial parallel manipulators with three to six degrees of freedom, the virtues of which are demonstrated by a case study.
11 citations
TL;DR: In this article, an orthogonal curvilinear terrain-following coordinate (the OS coordinate) was designed to reduce the advection errors in the classic σ coordinate.
Abstract: . We have designed an orthogonal curvilinear terrain-following coordinate (the orthogonal σ coordinate, or the OS coordinate) to reduce the advection errors in the classic σ coordinate. First, we rotate the basis vectors of the z coordinate in a specific way in order to obtain the orthogonal, terrain-following basis vectors of the OS coordinate, and then add a rotation parameter b to each rotation angle to create the smoother vertical levels of the OS coordinate with increasing height. Second, we solve the corresponding definition of each OS coordinate through its basis vectors; and then solve the 3-D coordinate surfaces of the OS coordinate numerically, therefore the computational grids created by the OS coordinate are not exactly orthogonal and its orthogonality is dependent on the accuracy of a numerical method. Third, through choosing a proper b, we can significantly smooth the vertical levels of the OS coordinate over a steep terrain, and, more importantly, we can create the orthogonal, terrain-following computational grids in the vertical through the orthogonal basis vectors of the OS coordinate, which can reduce the advection errors better than the corresponding hybrid σ coordinate. However, the convergence of the grid lines in the OS coordinate over orography restricts the time step and increases the numerical errors. We demonstrate the advantages and the drawbacks of the OS coordinate relative to the hybrid σ coordinate using two sets of 2-D linear advection experiments.
11 citations
TL;DR: In this paper, a 3D semi-analytical formulation based on a multipole expansion, directional derivative, collocation technique and singular value decomposition (SVD) is proposed for 3D elliptical cylindrical cavities with multiple elliptical cylinders.
11 citations
TL;DR: In this article, the keypoint of the addition theorem in terms of the elliptical coordinates is utilized to expand the fundamental solution to the degenerate kernel and boundary densities are simulated by the eigenfunction expansion.
Abstract: Based on the successful experience of solving anti-plane problems containing multiple elliptical inclusions, we extend to deal with the piezoelectricity problems containing arbitrary elliptical inhomogeneities. In order to fully capture the elliptical geometry, the keypoint of the addition theorem in terms of the elliptical coordinates is utilized to expand the fundamental solution to the degenerate kernel and boundary densities are simulated by the eigenfunction expansion. Only boundary nodes are required instead of boundary elements. Therefore, the proposed approach belongs to one kind of meshless and semi-analytical methods. Besides, the error stems from the number of truncation terms of the eigenfunction expansion and the convergence rate of exponential order is better than the linear order of the conventional boundary element method. It is worth noting that there are Jacobian terms in the degenerate kernel, boundary density and contour integral. However, they would cancel each other out in the process of the boundary contour integral. As the result, the orthogonal property of eigenfunction is preserved and the boundary integral can be easily calculated. For verifying the validity of the present method, the problem of an elliptical inhomogeneity in an infinite piezoelectric material subject to anti-plane shear and in-plane electric field is considered to compare with the analytical solution in the literature. Besides, two circular inhomogenieties can be seen as a special case to compare with the available data by approximating the major and minor axes. Finally, the problem of two elliptical inhomogeneities in an infinite piezoelectric material is also provided in this paper.
4 citations
TL;DR: In this paper, the authors give a general 4-parameter expression for a model potential that is always integrable and is conformally superintegrable for some parameter choices.
Abstract: We analyze one particle, two-center quantum problems which admit separation of variables in prolate spheroidal coordinates, a natural restriction satisfied by the H$_2^+$ molecular ion. The symmetry operator is constructed explicitly. We give the details of the Hamiltonian reduction of the 3D system to a 2D system with modified potential that is separable in elliptic coordinates. The potentials for which there is double-periodicity of the Schrodinger operator in the space of prolate spheroidal coordinates, including one for the H$_2^+$ molecular ion, are indicated. We study possible potentials that admit exact-solvability is as well as all models known to us with the (quasi)-exact-solvability property for the separation equations. We find deep connections between second-order superintegrable and conformally superintegrable systems and these tractable problems. In particular we derive a general 4-parameter expression for a model potential that is always integrable and is conformally superintegrable for some parameter choices.
4 citations
Journal Article•
TL;DR: In this paper, the intersection angle between the direction along the third coordinate and the second coordinate corresponds to the parameter of the S-duality of the $\ensuremath{\beta}$-deformation.
Abstract: We discuss $\ensuremath{\beta}$-deformed geometries on two types of ${T}^{3}$'s where the direction along the third coordinate is not orthogonal to the direction along the second coordinate or the direction along the first coordinate. We show that the intersection angle between the direction along the third coordinate and the direction along the second coordinate corresponds to the parameter of the S-duality of the $\ensuremath{\beta}$-deformation, while the intersection angle between the direction along the third coordinate and the direction along the first coordinate generalizes the $\ensuremath{\beta}$-deformed geometry.
TL;DR: In this paper, the authors give a general four-parameter expression for a model potential that is always exactly solvable and integrable and is conformally superintegrable for some parameter choices.
Abstract: We analyze one particle, two-center quantum problems which admit separation of variables in prolate spheroidal coordinates, a natural restriction satisfied by the H molecular ion. The symmetry operator is constructed explicitly. We give the details of the Hamiltonian reduction of the 3D system to a 2D system with modified potential that is separable in elliptic coordinates. The potentials for which there is double-periodicity of the Schrodinger operator in the space of prolate spheroidal coordinates, including one for the H molecular ion, are indicated. We study possible potentials that admit exact-solvability is as well as all models known to us with the (quasi)-exact-solvability property for the separation equations. We find deep connections between second-order superintegrable and conformally superintegrable systems and these tractable problems. In particular we derive a general four-parameter expression for a model potential that is always exactly-solvable and integrable and is conformally superintegrable for some parameter choices.
TL;DR: In this article, the authors applied the region-point-matching technique to examine the scattering problem of truncated semi-elliptic canyons under plane SH-wave excitation.
Abstract: In this study, the region-point-matching technique (RPMT) is applied to examine the scattering problem of truncated semi-elliptic canyons under plane SH-wave excitation. The partition of the entire analyzed region into two subregions is carried out via an introduction of the elliptic-arc auxiliary boundary. Taking advantage of appropriate wavefunctions in elliptic coordinates, the expression of antiplane motions for each subregion can be obtained. To accomplish the indispensable coordinate shift, the coordinate-transformed relation, intended as a substitute for the addition theorem involving Mathieu functions, is well utilized. Integration of the coordinate-transformed relation into the RPMT brings about the rapid construction of simultaneous equations. Effects of pertinent parameters on steady-state and transient surface motions are demonstrated. Computed results show that, for horizontal incidence, the potential high level of ground shaking may occur near the illuminated upper corner of the canyon. In such a small localized region, due to the occurrence of constructive interference between the reflected waves from the horizontal ground surface and the scattered waves from the corners of the canyon, the peak amplifaction may be at least two times that of free-field response.
TL;DR: For a commutative ring A, a polynomial f ∈ A[x][n] is called a strongly residual coordinate if f becomes a coordinate (over A [x, x −1] ) upon going modulo x as discussed by the authors.
Abstract: For a commutative ring A, a polynomial f ∈ A[x][n] is called a strongly residual coordinate if f becomes a coordinate (over A) upon going modulo x, and f becomes a coordinate (over A[x, x −1]) upon inverting x. We study the question of when a strongly residual coordinate in A[x][n] is a coordinate, a question closely related to the Dolgachev–Weisfeiler conjecture. It is known that all strongly residual coordinates are coordinates for n = 2 over an integral domain of characteristic zero. We show that a large class of strongly residual coordinates that are generated by elementaries over A[x, x −1] are in fact coordinates for arbitrary n, with a stronger result in the n = 3 case. As an application, we show that all Venereau-type polynomials are 1-stable coordinates.
23 Dec 2014
TL;DR: A new circular effect generation approach is presented that transforms (i,j) coordinate system into (ρ,θ) coordinates system, which simplifies the handling of pixel location in an image.
Abstract: This paper presents a new circular effect generation approach. A pixel location in an image can be stated with i and j coordinate. To handle (i,j) coordinate system easier, we transform (i,j) coordinate system into (ρ,θ) coordinate system. Simulation results introduce performance comparison.
01 Jan 2014
TL;DR: In this article, the sufficiency and necessity of the four-point coordinate method for a type of spatial parallel mechanism with four non-coplanar points' Cartesian coordinates were proved.
Abstract: On the basis of the natural coordinate method, the sufficiency and necessity are firstly proved so that the kinematics model can be established for a type of spatial parallel mechanism only with four non-coplanar points’ Cartesian coordinates. This method has a significant advantage in dealing with kinematics issues of spatial parallel mechanisms with more than three degrees of freedom. As this model introduces only the Cartesian coordinate, and the elements of the derivative matrix contain only the linear term or even the constant term, the elements of the derivative matrix obtained through the modeling method of the rotation matrix transformation are often non-coplanar, or even contain transcendental functions. Therefore, these advantages of the four-point coordinate method have brought great convenience for the kinematic and dynamic analysis of the multi-degree-of-freedom spatial parallel robot mechanisms.
TL;DR: In this paper, it was shown that the Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Backlund transformation.
Abstract: The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Backlund transformation. We also prove that after similar Backlund transformations other curvilinear coordinates on the sphere and on the plane become variables of separations for the system with quartic potential, for the Henon-Heiles system and for the Kowalevski top. It allows us to say about some analog of the hetero Backlund transformations relating different Hamilton-Jacobi equations.
01 Feb 2014
TL;DR: A novel coordinate system that is invariant under isometries is defined that can easily be converted from the global vertex coordinates and can be efficiently recovered from the new coordinates by simply solving two sparse least-squares problems.
Abstract: Computing the natural-looking interpolation of different shapes is a fundamental problem of computer graphics. It is proved by some researchers that such an interpolation can be achieved by pursuing the isometry. In this paper, a novel coordinate system that is invariant under isometries is defined. The coordinate system can easily be converted from the global vertex coordinates. Furthermore, the global coordinates can be efficiently recovered from the new coordinates by simply solving two sparse least-squares problems. Since the proposed coordinate system is invariant under isometries, then transformations such as global rigid transformations, articulated posture deformations, or any other isometric deformations, do not change the coordinate values. Therefore, shape interpolation can be done in this framework without being affected by the distortions caused by the isometry.
Posted Content•
TL;DR: In this article, the convergence properties of new molecular auxiliary functions are investigated and the results obtained are compared with results found in the literature, and the comparison for two-center two-electron integrals is made with results obtained from one-center expansions by translation of wave function to same center with integer principal quantum number and results obtaining from the Cuba numerical integration algorithm, respectively.
Abstract: The two-center two-electron Coulomb and hybrid integrals arising in relativistic and nonrelativistic ab-initio calculations of molecules are evaluated over the non-integer Slater-type orbitals via ellipsoidal coordinates. These integrals are expressed through new molecular auxiliary functions and calculated with numerical Global-adaptive method according to parameters of non-integer Slatertype orbitals. The convergence properties of new molecular auxiliary functions are investigated and the results obtained are compared with results found in the literature. The comparison for two-center twoelectron integrals is made with results obtained from one-center expansions by translation of wavefunction to same center with integer principal quantum number and results obtained from the Cuba numerical integration algorithm, respectively. The procedures discussed in this work are capable of yielding highly accurate two-center two-electron integrals for all ranges of orbital parameters.
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TL;DR: In this article, several local elliptic coordinates are used to build a new polyelliptic coordinate system which is orthogonal and admits the separation of variables, which can give the exact solutions of some unsolved problems in quantum mechanics and diffraction theory.
Abstract: Several local elliptic coordinates are used to build a new polyelliptic coordinate system which is orthogonal and admits the separation of variables. Such coordinate systems can give the exact solutions of some unsolved problems in quantum mechanics and diffraction theory.
01 Jan 2014
TL;DR: In this paper, the authors examined the often overlooked problem of representing an optical field discretely using cylindrical coordinates, and they showed that the choice of coordinate system depends on the optical problem being examined.
Abstract: The Fresnel transform is a diffraction integral used in optics to calculate the propagation of a wave field in the paraxial domain. It is possible to express the Fresnel transform in Cartesian or cylindrical coordinates. Often the choice of coordinate system depends on the optical problem being examined. In this proceeding we examine the often overlooked problem of representing an optical field discretely using cylindrical coordinates.