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A bstract An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics. A variety of examples are constructed, having different combinations of ring S 1 × S 2 and sphere S 3 cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number.

Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by S = ln χ ( x ) with χ ( x ) being the distance between two nearby geodesics. We derive an equation for the entropy, which by transformation to a Riccati-type equation becomes similar to the Jacobi equation. We further show that the geodesic equation for a null geodesic in a double-warped spacetime leads to the same entropy equation. By applying a Robertson–Walker metric for a flat three-dimensional Euclidean space expanding as a function of time, we again reach the entropy equation stressing the connection between the chosen entropy measure and time. We finally turn to the Raychaudhuri equation for expansion, which also is a Riccati equation similar to the transformed entropy equation. Those Riccati-type equations have solutions of the same form as the Jacobi equation. The Raychaudhuri equation can be transformed to a harmonic oscillator equation, and it has been shown that the geodesic deviation equation of Jacobi is essentially equivalent to that of a harmonic oscillator. The Raychaudhuri equations are strong geometrical tools in the study of general relativity and cosmology. We suggest a refined entropy measure applicable in cosmology and defined by the average deviation of the geodesics in a congruence.

ApoA-I is a uniquely flexible lipid-scavenging protein capable of incorporating phospholipids into stable particles. Here we report molecular dynamics simulations on a series of progressively smaller discoidal high density lipoprotein particles produced by incremental removal of palmitoyloleoylphosphatidylcholine via four different pathways. The starting model contained 160 palmitoyloleoylphosphatidylcholines and a belt of two antiparallel amphipathic helical lipid-associating domains of apolipoprotein (apo) A-I. The results are particularly compelling. After a few nanoseconds of molecular dynamics simulation, independent of the starting particle and method of size reduction, all simulated double belts of the four lipidated apoA-I particles have helical domains that impressively approximate the x-ray crystal structure of lipid-free apoA-I, particularly between residues 88 and 186. These results provide atomic resolution models for two of the particles produced by in vitro reconstitution of nascent high density lipoprotein particles. These particles, measuring 95 Å and 78 Å by nondenaturing gradient gel electrophoresis, correspond in composition and in size/shape (by negative stain electron microscopy) to the simulated particles with molar ratios of 100:2 and 50:2, respectively. The lipids of the 100:2 particle family form minimal surfaces at their monolayer-monolayer interface, whereas the 50:2 particle family displays a lipid pocket capable of binding a dynamic range of phospholipid molecules.

The topology of the domain of outer communication for 5-dimensional stationary bi-axisymmetric black holes is classified in terms of disc bundles over the 2-sphere and plumbing constructions. In particular we find an algorithmic bijective correspondence between the plumbing of disc bundles and the rod structure formalism for such spacetimes. Furthermore, we describe a canonical fill-in for the black hole region and cap for the asymptotic region. The resulting compactified domain of outer communication is then shown to be homeomorphic to S4S^4, a connected sum of S2×S2S^2\\times S^2’s, or a connected sum of complex projective planes CP2\\mathbb {CP}^2. Combined with recent existence results, it is shown that all such topological types are realized by vacuum solutions. In addition, our methods treat all possible types of asymptotic ends, including spacetimes which are asymptotically flat, asymptotically Kaluza-Klein, or asymptotically locally Euclidean.

Abstract We produce new examples, both explicit and analytical, of bi-axisymmetric stationary vacuum black holes in five dimensions. A novel feature of these solutions is that they are asymptotically locally Euclidean, in which spatial cross-sections at infinity have lens space $L(p,q)$ topology, or asymptotically Kaluza–Klein so that spatial cross-sections at infinity are topologically $S^1\\times S^2$. These are nondegenerate black holes of cohomogeneity 2, with any number of horizon components, where the horizon cross-section topology is any one of the three admissible types: $S^3$, $S^1\\times S^2$, or $L(p,q)$. Uniqueness of these solutions is also established. Our method is to solve the relevant harmonic map problem with prescribed singularities, having target symmetric space $SL(3,\\mathbb{R})/SO(3)$. In addition, we analyze the possibility of conical singularities and find a large family for which geometric regularity is guaranteed.

This volume contains the proceedings of the Fifth International Conference on Complex Analysis and Dynamical Systems, held from May 22-27, 2011, in Akko (Acre), Israel. The papers cover a wide variety of topics in complex analysis and partial differential equations, including meromorphic functions, one-parameter semigroups, subordination chains, quasilinear hyperbolic systems, and the Euler-Poisson-Darboux equation. In addition, there are several articles dealing with various aspects of fixed point theory, hyperbolic geometry, and optimal control.

We study the force between rotating coaxial black holes, as it was defined in [9 and 10]. We show that under a certain limit, the force is attractive, and in fact tends to infinity. This lends support to the conjecture that the force is always positive.

We study the force between rotating coaxial black holes, as it was defined in [9 and 10]. We show that under a certain limit, the force is attractive, and in fact tends to infinity. This lends support to the conjecture that the force is always positive.

We present the outline of a proof of the Riemannian Penrose inequality with charge

An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics. A variety of examples are constructed, having different combinations of ring \\(S^1\\times S^2\\) and sphere \\(S^3\\) cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number.