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In this paper, a novel mixed spectral-Galerkin method is proposed and studied for a Maxwell transmission eigenvalue problem in a spherical domain. The method utilizes vector spherical harmonics to achieve dimension reduction. By introducing an auxiliary vector function, the original problem is rewritten as an equivalent fourth-order coupled linear eigensystem, which is further decomposed into a sequence of one-dimensional fourth-order decoupled transverse-electric (TE) and transverse-magnetic (TM) modes. Based on compact embedding theory and the spectral approximation property of compact operators, error estimates for both eigenvalue and eigenfunction approximations are established for the TE mode. For the TM mode, an efficient essential polar condition and a high-order polynomial approximation method are designed to cope with the singularity and complexity caused by the coupled boundary conditions. Numerical experiments are presented to demonstrate the efficiency and robustness of our algorithm. [ABSTRACT FROM AUTHOR]

We present a high‐accuracy numerical method based on a decoupled dimensionality reduction scheme for Maxwell eigenvalue problem in spherical domains. Using the orthogonality of vector spherical harmonics and the variable separation approach, we decompose the original problem into two classes of decoupled one‐dimensional TE mode and TM mode. For the TE mode, we establish a variational formulation and its discrete scheme and give the error estimations of the approximate eigenvalues and eigenfunctions. For the TM mode, it is different from TE mode which naturally meets the divergence‐free condition and will not generate some spurious eigenvalues. We design a numerical algorithm based on a parameterized method to filter out the spurious eigenvalues. Finally, some numerical results are presented to confirm the theoretical results and validate the algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an efficient spectral-Galerkin method based on a mixed formulation with dimension reduction for the Helmholtz transmission eigenvalue problem in spherical domains. By introducing an auxiliary function, we rewrite the original problem as an equivalent fourth-order coupled form in spherical coordinates. Using the properties of spherical harmonic and Laplace–Beltrami operator, we further decompose the original problem into a series of decoupled one-dimensional fourth-order linear eigenvalue problems, for which a new mixed variational formulation and its discretization is developed. For error estimates of numerical eigenvalues and eigenfunctions, we recall the spectral theory of compact operators. Towards this end, we derive the essential polar conditions, define a class of weighted Sobolev spaces, and most importantly, prove a sequence of two compact embedding properties for the weighted Sobolev spaces, based on which the spectral theory of compact operators for the variational formulation and discrete system can be established. Finally, some numerical examples are presented to confirm the theoretical error analysis and the efficiency of our algorithm. [ABSTRACT FROM AUTHOR]

The motion of the lymph has a very important role in the immune system, and it is influenced by the porosity of the lymph nodes: more than 90% takes the peripheral path without entering the lymphoid compartment. In this paper, we construct a mathematical model of a lymph node assumed to have a spherical geometry, where the subcapsular sinus is a thin spherical shell near the external wall of the lymph node and the core is a porous material describing the lymphoid compartment. For the mathematical formulation, we assume incompressibility and we use Stokes together with Darcy–Brinkman equation for the flow of the lymph. Thanks to the hypothesis of axisymmetric flow with respect to the azimuthal angle and the use of the stream function approach, we find an explicit solution for the fully developed pulsatile flow in terms of Gegenbauer polynomials. A selected set of plots is provided to show the trend of motion in the case of physiological parameters. Then, a finite element simulation is performed and it is compared with the explicit solution. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of determining a radially dependent coefficient a (r) in the equation Δ u − a (r) u = 0 , in the unit sphere Ω from the Dirichlet–Neumann data pair { u , ∂ u ∂ ν } | ∂ Ω . We discuss the uniqueness of this determination. [ABSTRACT FROM AUTHOR]

Time-fractional Klein–Gordon equation in a sphere is considered for the case of central symmetry under the time-variable Dirichlet condition. The time-fractional derivative with the power-law kernel is used. The Laplace transform and convenient transformations of the independent variable and unknown function are used to determine the general analytical solution of the problem in the Laplace domain. In order to obtain the solution in the real domain, the inverse Laplace transforms of two functions of exponential type whose expressions are new in the literature have been determined. The similar solution for ordinary Klein– Gordon equation is a limiting case of general solution but a simpler form for this solution is provided. The convergence of general solution to the ordinary solution and the effects of fractional parameter on this solution are graphically underlined. [ABSTRACT FROM AUTHOR]

We establish the sufficiency of Kalman's rank condition on the approximate boundary controllability at finite time for diagonalizable systems on a spherical domain in higher dimensional case. [ABSTRACT FROM AUTHOR]

The backward heat conduction problem with variable coefficients in a spherical domain was considered. This problem is ill-posed, i.e., the solution (if it exists) to this problem does not depend continuously on the measured data. A projected iteration regularization method was constructed to obtain the regularized approximate solution to this inverse problem, and the convergence error estimates between the exact solution and the corresponding regularized approximate solution were given under the a priori and a posteriori parameter choice rules. Numerical results verify the effectiveness of this method. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for dimension reduction of data observed on spherical surfaces. Several dimension reduction techniques have been developed in recent years for non-euclidean data analysis. As a pioneer work, (Hauberg 2016) attempted to implement principal curves on Riemannian manifolds. However, this approach uses approximations to process data on Riemannian manifolds, resulting in distorted results. This study proposes a new approach to project data onto a continuous curve to construct principal curves on spherical surfaces. Our approach lies in the same line of (Hastie and Stuetzle et al. 1989) that proposed principal curves for data on euclidean space. We further investigate the stationarity of the proposed principal curves that satisfy the self-consistency on spherical surfaces. The results on the real data analysis and simulation examples show promising empirical characteristics of the proposed approach. [ABSTRACT FROM AUTHOR]

Abstract In this study, a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on a spherical domain is developed. By employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, the error estimates of approximate eigenvalues and eigenfunctions are provided. By adopting orthogonal spherical base functions, the discrete model with sparse mass and stiff matrices is established so that it is very efficient for finding the numerical solutions of biharmonic eigenvalue equations on the spherical domain. Some numerical examples are provided to validate the theoretical results.

• Propose a new robust principal curve for data sets that deviate from the normality (Gaussian) assumption. • Investigate a theoretical property of the robust principal curve, termed 'stationarity,' which implies that the proposed method is canonical in the spherical domain. • Demonstrate the promising empirical performance of the proposed method through numerical experiments such as simulation study and real data analysis. Principal curves are a nonlinear generalization of principal components and go through the mean of data lying in Euclidean space. In this paper, we propose L 1 -type and Huber-type principal curves through the median of data to robustify the principal curves for a dataset that may contain outliers. We further investigate the stationarity of the proposed robust principal curves on S 2. Results from numerical experiments on S 2 and S 4 , including real data analysis, manifest promising empirical features of the proposed method. [ABSTRACT FROM AUTHOR]

We simulate LC director distribution and optical transmission of heterophase systems with different LC boundary conditions. The first type is a transparent isotropic material with spherical or cylindrical objects. The second type is an LC incorporating an isotropic transparent or non-transparent object. Systems parameters influence on LCD performances are discussed. [ABSTRACT FROM AUTHOR]

In this study, we develop a high accuracy spectral method based on the min/max principle for biharmonic eigenvalue problems in the spherical domain. By analyzing the orthogonal spherical harmonic and approximation and using the min/max principle, we first deduce the error estimates of approximate eigenvalues. Then we construct an appropriate set of orthogonal spherical functions contained in H 0 2 ( Ω ) and establish the matrix formulations for the discrete variational form, whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently. Finally, we provide some numerical experiments to validate that the theoretical results are correct. [ABSTRACT FROM AUTHOR]

We simulate LC director distribution and optical transmission of heterophase systems with different LC boundary conditions. The first type is a transparent isotropic material with spherical or cylindrical objects. The second type is an LC layer incorporating an isotropic transparent or non-transparent object. Systems parameters' influence on LCD performances is discussed. [ABSTRACT FROM AUTHOR]

In this study, a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on a spherical domain is developed. By employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, the error estimates of approximate eigenvalues and eigenfunctions are provided. By adopting orthogonal spherical base functions, the discrete model with sparse mass and stiff matrices is established so that it is very efficient for finding the numerical solutions of biharmonic eigenvalue equations on the spherical domain. Some numerical examples are provided to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Projection of omnidirectional videos in the spherical domain onto the planar domain is required to utilise existing video compression techniques. Due to its simplicity, the equirectangular projection (ERP) method has been widely adopted for sphere‐to‐plane mapping. However, the coding efficiency of the ERP method is inefficient because the ERP video includes visual redundancies in the high latitude areas of the sphere. To solve this problem, this Letter presents an efficient data compression scheme that employs latitude adaptive down‐sampling followed by pixel rearrangement. Experimental results show that the proposed method can achieve a bitrate reduction of up to 28.9%, as compared with the ERP. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the numerical approximation of the three-dimensional poroelastic wave equations in the spherical coordinate system. One difficulty in the design of an efficient numerical scheme is that the problem is singular in the center and the polar axes of the computational domain. Nevertheless, we develop a hybrid finite difference/control volume method for solving this problem. Our method is explicit and is second order accurate in both space and time. Numerical results are shown to confirm the convergence rate of our method and the effectiveness to simulate wave propagation in poroelastic media in the spherical coordinate system. [Copyright &y& Elsevier]

By using Kelvin's inversion, we construct analytical harmonic functions satisfying the zero-Neumann condition on the inner spherical wall everywhere except a single point where the velocity is infinite. This set of functions can be used to construct approximate velocity potentials associated with liquid sloshing in a spherical tank, and in other hydrodynamic problems. By using Kelvin's inversion, we construct analytical harmonic functions satisfying the zero-Neumann condition on the inner spherical wall everywhere except a single point where the velocity is infinite. This set of functions can be used to construct approximate velocity potentials associated with liquid sloshing in a spherical tank, and in other hydrodynamic problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the first two eigenvalues of the buckling problem on spherical domains. We obtain an estimate of the second eigenvalue in terms of the first eigenvalue, which improves on a recent result obtained by Wang and Xia (2007) . [Copyright &y& Elsevier]

We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere, and the three-dimensional ball. The grids are logically rectangular and the computational domain is a single Cartesian grid. Compared to alternative approaches based on a multiblock data structure or unstructured triangulations, this approach simplifies the implementation of numerical methods and the use of adaptive refinement. A more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain. Although these grids are highly nonorthogonal, we show that the high-resolution wave-propagation algorithm implemented in CLAWPACK can be used effectively to approximate hyperbolic problems on these grids. Since the ratio between the largest and smallest grids is below 2 for most of our grid mappings, explicit finite volume methods such as the wave-propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitude-longitude grids. Numerical test calculations illustrate the potential use of these grids for a variety of applications including Euler equations, shallow water equations, and acoustics in a heterogeneous medium. Pattern formation from a reaction-diffusion equation on the sphere is also considered. All examples are implemented in the CLAWPACK software package and full source code is available on the web, along with MATLAB routines for the various mappings. [ABSTRACT FROM AUTHOR]

This work is concerned with developing moving mesh strategies for solving problems defined on a sphere. To construct mappings between the physical domain and the logical domain, it has been demonstrated that harmonic mapping approaches are useful for a general class of solution domains. However, it is known that the curvature of the sphere is positive, which makes the harmonic mapping on a sphere not unique. To fix the uniqueness issue, we follow Sacks and Uhlenbeck [Ann. of Math. (2), 113 (1981), pp. 1-24] to use a perturbed harmonic mapping in mesh generation. A detailed moving mesh strategy including mesh redistribution and solution updating on a sphere will be presented. The moving mesh scheme based on the perturbed harmonic mapping is then applied to the moving steep front problem and the Fokker--Planck equations with high potential intensities on a sphere. The numerical experiments show that with a moderate number of grid points our proposed moving mesh algorithm can accurately resolve detailed features of singular problems on a sphere. [ABSTRACT FROM AUTHOR]

Reaction–diffusion equations on n -dimensional symmetric domains by boundary control are considered in this paper. By PDE symmetric backstepping, control laws and stability of closed loop are established. Results are applied to circular and spherical domains, obtaining boundary control laws which involve integrations on intervals, rather than the established results through ultra spherical coordinates. [ABSTRACT FROM AUTHOR]