We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernels. In contrast to the direct spectral method [L. Pareschi and G. Russo, SIAM J. Numer. Anal., 37 (2000), pp. 1217–1245; I. M. Gamba and S. H. Tharkabhushanam, J. Comput. Phys., 228 (2009), pp. 2012–2036], which requires O(N6) memory to store precomputed weights and has O(N6) numerical complexity, the new method has complexity O(MN4 log N), where N is the number of discretization points in each of the three velocity dimensions and M is the total number of discretization points on the sphere and M ≪ N2. Furthermore, it requires no precomputation for the variable hard sphere model and only O(MN4) memory to store precomputed functions for more general collision kernels. Although a faster spectral method is available [C. Mouhot and L. Pareschi, Math. Comp., 75 (2006), pp. 1833–1852] (with complexity O(MN3 logN)), it works only for hard sphere molecules, thus limiting its use for practical problems. Our new method, on the other hand, can apply to arbitrary collision kernels. A series of numerical tests is performed to illustrate the efficiency and accuracy of the proposed method.

What is the shape of a uniformly massive object that generates a gravitational potential equivalent to that of two equal point-masses? If the weight of each point-mass is sufficiently small compared to the distance between the points then the answer is a pair of balls of equal radius, one centered at each of the two points, but otherwise it is a certain domain of revolution about the axis passing through the two points. The existence and uniqueness of such a domain is known, but an explicit parameterization is known only in the plane where the region is referred to as a Neumann oval. We construct a four-dimensional “Neumann ovaloid”, solving explicitly this inverse potential problem.

We show that if X is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of X as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set EC coincides with the lower dimension of the condensation set C, while the Assouad dimension of EC is the maximum of the Assouad dimensions of the corresponding self-similar set E and the condensation set C. If the Assouad dimension of C is strictly smaller than the Assouad dimension of E, then the upper regularity dimension of any measure supported on EC is strictly larger than the Assouad dimension of EC. Surprisingly, the corresponding statement for the lower regularity dimension fails. 1

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We develop a method searching for quasi-conformal maps from point-cloud surfaces to Euclidean plane. The maps can be got by directly solving Beltrami equation with proper boundary condition or solving optimization problem of the variational formu- lation. The point integral methd (PIM) is used for discretization on point clouds. Numerical experiments suggest that the method converges as the density of points in- creases.