In the present paper we prove a multiplicity theorem for a quasilinear elliptic problem with dependence on the gradient ensuring the existence of a positive solution and of a negative solution. In addition, we show the existence of the extremal constant-sign solutions: the smallest positive solution and the biggest negative solution. Our approach relies on extremal solutions for an auxiliary pa...

We explore a general-purpose heuristic algorithm for finding high-quality solutions to continuous optimization problems. The method, called continuous extremal optimization (CEO), can be considered as an extension of extremal optimization and consists of two components, one which is responsible for global searching and the other which is responsible for local searching. The CEO's performance pr...

Any spacetime containing a degenerate Killing horizon, such as an extremal black hole, possesses a well-defined notion of a near-horizon geometry. We review such near-horizon geometry solutions in a variety of dimensions and theories in a unified manner. We discuss various general results including horizon topology and near-horizon symmetry enhancement. We also discuss the status of the classif...

We construct globally regular gravitating solutions, which possess only discrete symmetries. These solutions of Yang-Mills-dilaton theory may be viewed as exact (numerical) solutions of scalar gravity, by considering the dilaton as a kind of scalar graviton, or as approximate solutions of Einstein-Yang-Mills theory. We focus on platonic solutions with cubic symmetry, related to a rational map o...

We study exponentially suppressed contributions to the degeneracies of extremal black holes. Within Sen’s quantum entropy function framework and focusing on extremal black holes with an intermediate AdS3 region, we identify an infinite family of semi-classical AdS2 geometries which can contribute effects of order exp(S0/c), where S0 is the Bekenstein-Hawking-Wald entropy and c is an integer gre...

We present new analytic and numerical results for self-gravitating SU(2)Higgs magnetic monopoles approaching the black hole threshold. Our investigation extends to large Higgs self-coupling, λ, a regime heretofore unexplored. When λ is small, the critical solution where a horizon first appears is extremal Reissner-Nordstrom outside the horizon but has a nonsingular interior. When λ is large, th...

We present a new method for the numerical solution of the Hamilton Jacobi Bellman PDE that arises in an infinite time optimal control problem. The method can be of higher order to reduce ”the curse of dimensionality”. It proceeds in two stages. First the HJB PDE is solved in a neighborhod of the origin using the power series method of Al’brecht. From a boundary point of this neighborhood, an ex...

Let (M, J, Ω) be a polarized complex manifold of Kähler type. Let G be the maximal compact subgroup of the automorphism group of (M, J). On the space of Kähler metrics that are invariant under G and represent the cohomology class Ω, we define a flow equation whose critical points are extremal metrics, those that minimize the square of the L 2-norm of the scalar curvature. We prove that the dyna...