This paper provides a rigorous proof of the existence of an infinite number of black hole solutions to the Einstein-Yang/Mills equations with gauge group<i>SU</i>(2), for any event horizon. It is also demonstrated that the ADM mass of each solutions is finite, and that the corresponding Einstein metric tends to the associated Schwarzschild metric at a rate 1/<i>r</i> ², as<i>r</i> tends to infinity.

We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the gap has a lower bound independent of the dimension. We also estimate the gap when the potential is not necessarily convex.

We study deformations of maximally supersymmetric gauge theories by higher dimensional operators in various spacetime dimensions. We classify infinitesimal deformations that preserve all 16 supersymmetries, while allowing the possibility of breaking either Lorentz or R-symmetry, using an on-shell algebraic method developed by Movshev and Schwarz. We also consider the problem of extending the deformation beyond the first order.

We analyze spectrum of Laplacian supported by a periodic honeycomb lattice with generally unequal edge lengths and a <i></i> type coupling in the vertices. Such a quantum graph has nonempty point spectrum with compactly supported eigenfunctions provided all the edge lengths are commensurate. We derive conditions determining the continuous spectral component and show that existence of gaps may depend on number-theoretic properties of edge lengths ratios. The case when two of the three lengths coincide is discussed in detail.

We give a topological simulation for tensor networks that we call the two-string model. In this approach we give a new way to design protocols, and we discover a new multipartite quantum communication protocol. We introduce the notion of topologically compressed transformations. Our new protocol can implement multiple, non-local compressed transformations among multi-parties using one multipartite resource state.