Fuzzy sphere

Abstract: A model of Euclidean spacetime is presented in which at scales less than a certain length kappa the notion of a point does not exist. At scales larger then kappa the model resembles the 2-sphere S2. The algebra which determines the structure of the model, and which replaces the algebra of functions, is an algebra of matrices. The order of n of the matrices is connected with the length kappa and the radius r of the sphere by the relation kappa approximately r/n. The elements of differential calculus are sketched as well as the possible definitions of a metric and linear connection. A definition of the path integral is given and a few examples of field theory on a fuzzy sphere are finally referred to.

Abstract: 1 Preliminaries.- 2 Spherical Harmonics.- 3 Differentiation and Integration over the Sphere.- 4 Approximation Theory.- 5 Numerical Quadrature.- 6 Applications: Spectral Methods.

Abstract: Spherical microphone arrays have been recently studied for sound analysis and sound recordings, which have the advantage of spherical symmetry facilitating three-dimensional analysis. This paper complements the recent microphone array design studies by presenting a theoretical analysis of plane-wave decomposition given the sound pressure on a sphere. The analysis uses the spherical Fourier transform and the spherical convolution, where it is shown that the amplitudes of the incident plane waves can be calculated as a spherical convolution between the pressure on the sphere and another function which depends on frequency and the sphere radius. The spatial resolution of plane-wave decomposition given limited bandwidth in the spherical Fourier domain is formulated, and ways to improve the computation efficiency of plane-wave decomposition are introduced. The paper concludes with a simulation example of plane-wave decomposition.

Abstract: The spherical analog of the Milne problem for the half-plane is treated by an approximate method based on expanding the neutron distribution function in a finite number of spherical harmonics. The results are improved markedly in going from the first to the second approximation and more slowly in higher approximations. The neutron distribution is calculated in the first two approximations. Values of the "extrapolated endpoint"---as predicted by the first three approximations---are tabulated in Table I as a function of the radius of the sphere.