Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable oscillations over the entire sphere.
Recently, methods for removing the Gibbs phenomenon for one-dimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function is enough to recover an exponentially convergent approximation to the point values of in any subinterval in which the function is analytic.
Here we take a similar approach, proving that knowledge of the first spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function in any subinterval , where the function is analytic. Thus we entirely overcome the Gibbs phenomenon.
In this paper we will investigate an isoperimetric type problem in lattices. If K is a bounded O-symmetric (centrally symmetric with respect to the origin) convex body in En of volume v(K) = 2n det L which does not contain non-zero lattice points in its interior, we say that K is extremal with respect to the given lattice L. There are two variations of the isoperimetric problem for this class of polyhedra. The first one is: Which bodies have minimal surface area in the class of extremal bodies for a fixed n-dimensional lattice? And the second one is: Which bodies have minimal surface area in the class of extremal bodies with volume 1 of dimension n? We characterize the solutions of these two problems in the plane. There is a consequence of these results, the solutions of the above problems in the plane give the solution of the lattice-like covering problem: Determine those centrally symmetric convex bodies whose translated copies (with respect to a fixed lattice L) cover the space and have minimal surface area.
The well-balanced distribution of points over the surface of a sphere is of significant interest in various fields of science. The quality of point configurations is typically expressed by criterion functions that have many local optima. A general global optimization framework is suggested to solve such problems. To illustrate the viability of this approach, the model development and solver system LGO is applied to four different model versions. Numerical results – including the visual representation of criterion functions in these models – are presented. The global optimization approach can be tailored to specific problem settings, and it is also applicable to a large variety of other model forms. 相似文献
Based on the hypothesis of similarity of transverse displacements in thin-walled sandwich shells with a transversely soft core under dynamic and static loads, refined geometrically nonlinear dynamic equations of motion are constructed in the case of large variations in the parameters of the stress-strain state (SSS) in the tangential directions. For shells structurally symmetric across the thickness and loaded with initial static loads, linearized dynamic equations are derived, which, upon introducing the synphasic and antiphasic functions of displacements and forces, can be used to describe the synphasic and antiphasic buckling forms in the transverse and tangential directions. For nonshallow cylindrical and shallow spherical shells, the nonclassical problems on all possible vibration forms realized at zero indices of variability of the SSS parameters in the tangential directions are formulated and solved. For shallow shells of symmetric structure, the resolving equations are obtained by introducing, instead of tangential displacements and transverse tangential stresses in the core, the corresponding potential and vortex functions. 相似文献