ISAF (icosahedral symmetry-adapted functions) is a new approach used for 3D reconstruction of icosahedral macromolecules, which can achieve better resolutions compared with the commonly used Fourier-Bessel algorithm. However, ISAF is significantly slow. Among all reasons, the operation that rotating each image sixty times costs lots of running time. By analyzing the sixty symmetrical characteri...

in this paper, we establish some new results in ultra bessel sequences and ultra bessel sequences of subspaces. also, we investigate ultra bessel sequences in direct sums of hilbert spaces.specially, we show that {( fi, gi)}∞ i=1 is a an ultra bessel sequencefor hilbert space h ⊕ k if and only if { fi}∞ i=1 and {gi}∞ i=1 are ultrabessel sequences for hilbert spaces h and k, respectively.

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. The two identities are intimately connected with the classical circle and divisor problems, respectively. For each of Ramanujan’s identities, there are three possible interpretations for the double series. In two earlier papers, t...