We derive all the axi-symmetric, vacuum and electrovac extremal isolated horizons. It turns out that for every horizon in this class, the induced metric tensor, the rotation 1-form potential and the pullback of the electromagnetic field necessarily coincide with those induced by the monopolar, extremal Kerr– Newman solution on the event horizon. We also discuss the general case of a symmetric, ...

We study the properties of anti–de Sitter black holes with a Gauss-Bonnet term for various horizon topologies (k = 0, ±1) and for various dimensions, with emphasis on the less well understood k = −1 solution. We find that the zero temperature (and zero energy density) extremal states are the local minima of the energy for AdS black holes with hyperbolic event horizons. The hyperbolic AdS black ...

In this article, we prove the existence of extremal positive solution for the distributed order fractional hybrid differential equation$$int_{0}^{1}b(q)D^{q}[frac{x(t)}{f(t,x(t))}]dq=g(t,x(t)),$$using a fixed point theorem in the Banach algebras. This proof is given in two cases of the continuous and discontinuous function $g$, under the generalized Lipschitz and Caratheodory conditions.

We construct a four-dimensional BPS saturated heterotic string solution from the Taub-NUT solution. It is a non-extremal black hole solution since its Euler number is non-zero. We evaluate its black hole entropy semiclassically. We discuss the relation between the black hole entropy and the degeneracy of string states. The entropy of our string solution can be understood as the microscopic entr...

We show how a recently discovered black ring solution with a rotating 2-sphere can be turned into two new solutions of Einstein–Maxwell–dilaton theory. The first is a four-dimensional solution describing a pair of oppositely charged, extremal black holes—known as a black dihole—undergoing uniform acceleration. The second is a five-dimensional solution describing a pair of concentric, static ext...

We study the regularity of the extremal solution of the semilinear biharmonic equation β∆u−τ∆u = λ (1−u)2 on a ball B ⊂ R , under Navier boundary conditions u = ∆u = 0 on ∂B, where λ > 0 is a parameter, while τ > 0, β > 0 are fixed constants. It is known that there exists a λ∗ such that for λ > λ∗ there is no solution while for λ < λ∗ there is a branch of minimal solutions. Our main result asse...

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...

For a given family JC of digraphs, we study the "extremal" digraphs on n vertices containing no member of JC, and having the maximum number of arcs, e\(n,^f). We resolve conjectures concerning the set {lim,, ^x (ex(n,JC )/n2)) as JC ranges over all possible families, and describe a "finite" algorithm that can determine, for any JC, all matrices A for which a sequence {A(n)} of "matrix digraphs"...

Classical solutions describing charged dilaton black holes accelerating in a background magnetic field have recently been found. They include the Ernst metric of the EinsteinMaxwell theory as a special case. We study the extremal limit of these solutions in detail, both at the classical and quantum levels. It is shown that near the event horizon, the extremal solutions reduce precisely to the s...