Schrodinger operators with strongly singular interactions, typically supported by a set of Lebesgue measure zero, eg, by a nite or countable set of points, appeared in quantum mechanics from the very beginning~cf., eg,[35, 28]. Their use had essentially two motivations. First of all, they were expected to provide a reasonable description in the physical situations when the real interaction has a very small range comparing to the other characteristic lengths of the problem, for instance, to the wavelength of the scattered particles. On the other hand, the fact that the interaction support is small and outside it a free solution of the corresponding Schrodinger equation may be used often makes such models explicitly solvable. Until the beginning of the sixties, however, the occasional use of these point interactions remained purely formal. Only after the work of Berezin and Faddeev  did it become clear they can be
Recent compressive sensing results show that it is possible to accurately reconstruct certain compressible signals from relatively few linear measurements via solving nonsmooth convex optimization problems. In this paper, we propose the use of the alternating direction method - a classic approach for optimization problems with separable variables (D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite-element approximations, Computer and Mathematics with Applications, vol. 2, pp. 17-40, 1976; R. Glowinski and A. Marrocco, Sur lapproximation par elements finis dordre un, et la resolution par penalisation-dualite dune classe de problemes de Dirichlet nonlineaires, Rev. Francaise dAut. Inf. Rech. Oper., vol. R-2, pp. 41-76, 1975) - for signal reconstruction from partial Fourier (i.e., incomplete frequency) measurements. Signals are reconstructed as minimizers of
In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author.