On a conjecture of Bennewitz, and the behaviour of the Titchmarsh-Weyl matrix near a pole

For any real limit-$n$ $2n$th-order selfadjoint linear differential expression on $[0,\infty)$, Titchmarsh- Weyl matrices $M(\lambda)$ can be defined. Two matrices of particu lar interest are the matrices $M_D(\lambda)$ and $M_N(\lambda)$ assoc iated respectively with Dirichlet and Neumann boundary conditions at $x=0$. These satisfy $M_D(\lambda) = -M_{N}(\lambda)^{-1}$. It is known that when these matrices have poles (which can only lie on the real axis) the existence of valid HELP inequalities depends on their behaviour in the neighbourhood of these poles. We prove a conjecture of Bennewitz and use it, together with a new algorithm for computing the Laurent expansion of a Titchmarsh-Weyl matrix in the neighbourhood of a pole, to investigate the existence of HELP inequalities for a number of differential equations which have so far proved awkward to analys

Fast algorithm for detecting community structure in networks

It has been found that many networks display community structure -- groups of vertices within which connections are dense but between which they are sparser -- and highly sensitive computer algorithms have in recent years been developed for detecting such structure. These algorithms however are computationally demanding, which limits their application to small networks. Here we describe a new algorithm which gives excellent results when tested on both computer-generated and real-world networks and is much faster, typically thousands of times faster than previous algorithms. We give several example applications, including one to a collaboration network of more than 50000 physicists.Comment: 5 pages, 4 figure

Quantum Singularities in Horava-Lifshitz Cosmology

The recently proposed Horava-Lifshitz (HL) theory of gravity is analyzed from the quantum cosmology point of view. By employing usual quantum cosmology techniques, we study the quantum Friedmann-Lemaitre-Robertson-Walker (FLRW) universe filled with radiation in the context of HL gravity. We find that this universe is quantum mechanically nonsingular in two different ways: the expectation value of the scale factor $(t)$ never vanishes and, if we abandon the detailed balance condition suggested by Horava, the quantum dynamics of the universe is uniquely determined by the initial wave packet and no boundary condition at $a=0$ is indeed necessary.Comment: 13 pages, revtex, 1 figure. Final version to appear in PR

Analytic Kramer kernels, Lagrange-type interpolation series and de Branges spaces

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. In particular, when the involved kernel is analytic in the sampling parameter it can be stated in an abstract setting of reproducing kernel Hilbert spaces of entire functions which includes as a particular case the classical Shannon sampling theory. This abstract setting allows us to obtain a sort of converse result and to characterize when the sampling formula associated with an analytic Kramer kernel can be expressed as a Lagrange-type interpolation series. On the other hand, the de Branges spaces of entire functions satisfy orthogonal sampling formulas which can be written as Lagrange-type interpolation series. In this work some links between all these ideas are established

On the time delay in binary systems

The aim of this paper is to study the time delay on electromagnetic signals propagating across a binary stellar system. We focus on the antisymmetric gravitomagnetic contribution due to the angular momentum of one of the stars of the pair. Considering a pulsar as the source of the signals, the effect would be manifest both in the arrival times of the pulses and in the frequency shift of their Fourier spectra. We derive the appropriate formulas and we discuss the influence of different configurations on the observability of gravitomagnetic effects. We argue that the recently discovered PSR J0737-3039 binary system does not permit the detection of the effects because of the large size of the eclipsed region.Comment: 7 pages, 2 eps figures, RevTex, to appear in Physical Review