On bare masses in time-symmetric initial-value solutions for two black holes

The Brill-Lindquist time-symmetric initial-value solution for two uncharged black holes is rederived using the Hamiltonian constraint equation with Dirac delta distributions as a source for the binary black-hole field. The bare masses of the Brill-Lindquist black holes are introduced in a way which is applied, after straightforward modification, to the Misner-Linquist binary black-hole solution.Comment: LaTeX, 4 page

Correction to ‘Shifted convolution and the Titchmarsh divisor problem over Fq[t]

PublishedCorrection to original article: Phil. Trans. R. Soc. A 373, 20140308 (28 April 2015; Published online 23 March 2015) (doi:10.1098/rsta.2014.0308). Two of the equations in the original article contained a typographical error. The author's accepted manuscript of the original article is available in this repository via: http://hdl.handle.net/10871/2062

Walks on Apollonian networks

We carry out comparative studies of random walks on deterministic Apollonian networks (DANs) and random Apollonian networks (RANs). We perform computer simulations for the mean first passage time, the average return time, the mean-square displacement, and the network coverage for unrestricted random walk. The diffusions both on DANs and RANs are proved to be sublinear. The search efficiency for walks with various strategies and the influence of the topology of underlying networks on the dynamics of walks are discussed. Contrary to one's intuition, it is shown that the self-avoiding random walk, which has been verified as an optimal strategy for searching on scale-free and small-world networks, is not the best strategy for the DAN in the thermodynamic limit.Comment: 5 pages, 4 figure

Scanning Tunneling Microscopy of Proteins on Graphite Surfaces

We applied scanning tunneling microscopy (STM) to the observation of amino acids and proteins deposited and/or adsorbed on highly oriented pyrolytic graphite (HOPG). Although many questions remain, it is demonstrated that relatively high resolution images of uncoated proteins can often be obtained in air. We present images of five amino acids (glycine, leucine, lysine, methionine and tryptophan) and three proteins (lysozyme, albumin and fibrinogen) under various conditions of deposition and adsorption. We discuss the role of affinity of the amino acids and proteins to the substrate, their adsorbed states and distribution, and STM tip-induced deformation and/or destruction. STM studies of adsorbed proteins are expected to provide useful and even unique information on the conformation and packing of the proteins

The binary black-hole problem at the third post-Newtonian approximation in the orbital motion: Static part

Post-Newtonian expansions of the Brill-Lindquist and Misner-Lindquist solutions of the time-symmetric two-black-hole initial value problem are derived. The static Hamiltonians related to the expanded solutions, after identifying the bare masses in both solutions, are found to differ from each other at the third post-Newtonian approximation. By shifting the position variables of the black holes the post-Newtonian expansions of the three metrics can be made to coincide up to the fifth post-Newtonian order resulting in identical static Hamiltonians up the third post-Newtonian approximation. The calculations shed light on previously performed binary point-mass calculations at the third post-Newtonian approximation.Comment: LaTeX, 9 pages, to be submitted to Physical Review

Trapped and excited w modes of stars with a phase transition and R>=5M

The trapped $w$-modes of stars with a first order phase transition (a density discontinuity) are computed and the excitation of some of the modes of these stars by a perturbing shell is investigated. Attention is restricted to odd parity (``axial'') $w$-modes. With $R$ the radius of the star, $M$ its mass, $R_{i}$ the radius of the inner core and $M_{i}$ the mass of such core, it is shown that stars with $R/M\geq 5$ can have several trapped $w$-modes, as long as $R_{i}/M_{i}<2.6$. Excitation of the least damped $w$-mode is confirmed for a few models. All of these stars can only exist however, for values of the ratio between the densities of the two phases, greater than $\sim 46$. We also show that stars with a phase transition and a given value of $R/M$ can have far more trapped modes than a homogeneous single density star with the same value of $R/M$, provided both $R/M$ and $R_{i}/M_{i}$ are smaller than 3. If the phase transition is very fast, most of the stars with trapped modes are unstable to radial oscillations. We compute the time of instability, and find it to be comparable to the damping of the $w$-mode excited in most cases where $w$-mode excitation is likely. If on the other hand the phase transition is slow, all the stars are stable to radial oscillations.Comment: To appear in Physical Review

Extremal graphs for Estrada indices

Let $\mathcal{G}$ be a simple undirected connected graph. The signless Laplacian Estrada, Laplacian Estrada and Estrada indices of a graph $\mathcal{G}$ is the sum of the exponentials of the signless Laplacian eigenvalues, Laplacian eigenvalues and eigenvalues of $\mathcal{G}$, respectively. The present work derives an upper bound for the Estrada index of a graph as a function of its chromatic number, in the family of graphs whose color classes have order not less than a fixed positive integer. The graphs for which the upper bound is tight is obtained. Additionally, an upper bound for the Estrada Index of the complement of a graph in the previous family of graphs with two color classes is given. A Nordhaus-Gaddum type inequality for the Laplacian Estrada index when {$\mathcal{G}$ is a bipartite} graph with color classes of order not less than $2$, is presented. Moreover, a sharp upper bound for the Estrada index of the line graph and for the signless Laplacian index of a graph in terms of connectivity is obtained.publishe