We review what has been learnt and what remains unknown about the physics of hot enhancons following studies in supergravity. We recall a rather general family of static, spherically symmetric, non-extremal enhancon solutions describing D4 branes wrapped on K3 and discuss physical aspects of the solutions. We embed these solutions in the six dimensional supergravity describing type IIA strings on K3 and generalize them to have arbitrary charge vector. This allows us to demonstrate the equivalence with a known family of hot fractional D0 brane solutions, to widen the class of solutions of this second type and to carry much of the discussion across from the D4 brane analysis. In particular we argue for the existence of a horizon branch for these branes.
Here, the story is particularly simple. We find that, at some radius greater than @math , the volume of the K3 always shrinks to zero, indicating that somewhere outside this radius, the K3 has reached its stringy volume. Note that the old ( @math ) shell solution @cite_0 falls into this category.
@cite_0
2064943293 2038932251 2056986231 2169164634
Abstract The problem of finding the maximum diameter of n equal mutually disjoint circles inside a unit square is addressed in this paper. Exact solutions exist for only n = 1, …, 9,10,16,25,36 while for other n only conjectural solutions have been reported. In this work a max-min optimization approach is introduced which matches the best reported solutions in the literature for all n ⩽ 30, yields a better configuration for n = 15, and provides new results for n = 28 and 29. It is known that random k-CNF formulas have a so-called satisfiability threshold at a density (namely, clause-variable ratio) of roughly 2kln2: at densities slightly below this threshold almost all k-CNF formulas are satisfiable, whereas slightly above this threshold almost no k-CNF formula is satisfiable. In the current work we consider satisfiable random formulas and inspect another parameter—the diameter of the solution space (that is, the maximal Hamming distance between a pair of satisfying assignments). It was previously shown that for all densities up to a density slightly below the satisfiability threshold the diameter is almost surely at least roughly n 2 (and n at much lower densities). At densities very much higher than the satisfiability threshold, the diameter is almost surely zero (a very dense satisfiable formula is expected to have only one satisfying assignment). In this paper we show that for all densities above a density that is slightly above the satisfiability threshold (more precisel... Let P be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n , k by joining each point of P to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n , k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n , k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n , k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k > 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞. The kissing number k(3) is the maximal number of equal size nonoverlapping spheres in three dimensions that can touch another sphere of the same size. This number was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. The first proof that k(3) = 12 was given by Schutte and van der Waerden only in 1953. In this paper we present a new solution of the Newton--Gregory problem that uses our extension of the Delsarte method. This proof relies on basic calculus and simple spherical geometry.