The Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs $G_1,\dots,G_k$ on the same set $V$ of $n$ vertices, the simultaneous geometric embedding (with mapping) problem, or simply $k$-SGE, is to find a set $P$ of $n$ points in the plane and a bijection $\phi: V \to P$ such that the induced straight-line drawings of $G_1,\dots,G_k$ under $\phi$ are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x) proved to be complete for $\exists\mathbb{R}$, the existential theory of the reals. Hence the problem $k$-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to $k$-SGE, with the property that both numbers $k$ and $n$ are linear in the number of pseudolines. This implies not only the $\exists\mathbb{R}$-hardness result, but also a $2^{2^{\Omega (n)}}$ lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is only $2^{2^{\Omega (\sqrt{n})}}$

Inhomogeneous extragalactic magnetic fields and the second knee in the cosmic ray spectrum

Various experiments indicate the existence of a second knee around energy E=3.10^{17} eV in the cosmic ray spectrum. This feature could be the signature of the end of the galactic component and of the emergence of the extragalactic one, provided that the latter cuts off at low energies. Recent analytical calculations have shown that this cut-off could be a consequence of the existence of extragalactic magnetic fields: low energy protons diffuse on extragalactic magnetic fields and cannot reach the observer within a given time. We study the influence of inhomogeneous magnetic fields on the magnetic horizon, using a new semi-analytical propagation code. Our results indicate that, at a fixed value of the volume averaged magnetic field , the amplitude of the low energy cut-off is mainly controled by the strength of magnetic fields in the voids of the large scale structure distribution.Comment: 15 pages, 10 figures. Version to appear in PRD (minor changes

Erratic Jet Wobbling in the BL Lacertae Object OJ287 Revealed by Sixteen Years of 7mm VLBA Observations

We present the results from an ultra-high-resolution 7mm Very Long Baseline Array (VLBA) study of the relativistic jet in the BL Lacertae object OJ287 from 1995 to 2011 containing 136 total intensity images. Analysis of the image sequence reveals a sharp jet-position-angle swing by >100 deg. during [2004,2006], as viewed in the plane of the sky, that we interpret as the crossing of the jet from one side of the line of sight to the other during a softer and longer term swing of the inner jet. Modulating such long term swing, our images also show for the first time a prominent erratic wobbling behavior of the innermost ~0.4mas of the jet with fluctuations in position angle of up to ~40 deg. over time scales ~2yr. This is accompanied by highly superluminal motions along non-radial trajectories, which reflect the remarkable non-ballistic nature of the jet plasma on these scales. The erratic nature and short time scales of the observed behavior rules out scenarios such as binary black hole systems, accretion disk precession, and interaction with the ambient medium as possible origins of the phenomenon on the scales probed by our observations, although such processes may cause longer-term modulation of the jet direction. We propose that variable asymmetric injection of the jet flow; perhaps related to turbulence in the accretion disk; coupled with hydrodynamic instabilities, leads to the non-ballistic dynamics that cause the observed non-periodic changes in the direction of the inner jet.Comment: Accepted for Publication in The Astrophysical Journal. 11 pages, 6 figures, 4 tables. High resolution images on figure 1 and complete tables 1 and 2 may be provided on reques

Locations of Auroral Kilometric Radiation Bursts Inferred From Multi-Spacecraft Wideband Cluster VLBI Observations i: Description of Technique and Initial Results

The Cluster Wideband Data instrument has been used to determine the locations of auroral kilometric radiation (AKR) using very long baseline interferometry. The technique involves cross-correlating individual AKR bursts from all six Cluster baselines using time and frequency filtered waveforms. We report the locations of over 1,700 individual AKR bursts during six observing epochs between 10 July 2002 and 22 January 2003 when the Cluster constellation was high above the southern or northern hemisphere. In general we find that the AKR burst locations lie along magnetic field lines which map onto the nighttime auroral zone as expected from previous AKR studies. The distribution of AKR auroral footprint locations at each epoch had a overall spatial scale between 1000 - 2000 km, much larger than the positional uncertainty of an individual AKR burst location magnetic footprint, but a small fraction of the auroral oval. For two of the six epochs, there was a significant drift in the mean location of AKR activity over a period of 1-2 hours. The drift was predominantly in latitude at one epoch and in longitude at the other, with average drift speed V ~ 80-90 m s-1 at the AKR emission location.Comment: 31 pages, 9 figures, accepted for publication 19 June 2003 in JGR Space Physics. accepted for publicatio

Splitting neutrino masses and showering into Sky

Neutrino masses might be as light as a few time the atmospheric neutrino mass splitting. High Energy ZeV cosmic neutrinos (in Z-Showering model) might hit relic ones at each mass in different resonance energies in our nearby Universe. This non-degenerated density and energy must split UHE Z-boson secondaries (in Z-Burst model) leading to multi injection of UHECR nucleons within future extreme AUGER energy. Secondaries of Z-Burst as neutral gamma, below a few tens EeV are better surviving local GZK cut-off and they might explain recent Hires BL-Lac UHECR correlations at small angles. A different high energy resonance must lead to Glashow's anti-neutrino showers while hitting electrons in matter. In air, Glashow's anti-neutrino showers lead to collimated and directional air-showers offering a new Neutrino Astronomy. At greater energy around PeV, Tau escaping mountains and Earth and decaying in flight are effectively showering in air sky. These Horizontal showering is splitting by geomagnetic field in forked shapes. Such air-showers secondaries release amplified and beamed gamma bursts (like observed TGF), made also by muon and electron pair bundles, with their accompanying rich Cherenkov flashes. Also planet' s largest (Saturn, Jupiter) atmosphere limbs offer an ideal screen for UHE GZK and Z-burst tau neutrino, because their largest sizes. Titan thick atmosphere and small radius are optimal for discovering up-going resonant Glashow resonant showers. Earth detection of Neutrino showering by twin Magic Telescopes on top mountains, or by balloons and satellites arrays facing the limbs are the simplest and cheapest way toward UHE Neutrino Astronomy .Comment: 4 pages, 7 figures; an author's name correction and Journal Referenc

Ultra-High Energy Cosmic Ray Propagation in the Local Supercluster

We present detailed numerical simulations and analytical approximations of the propagation of nucleons above 10**(19) eV in the Local Supercluster, assuming that the ambient magnetic field is turbulent, and its strength 0.01 < B_rms < 1 micro-Gauss. In such strong magnetic fields, protons in the low energy part of the spectrum, 10**(19) eV < E < E_C diffuse, while the higher energy particles, with E > E_C propagate along nearly straight lines. The magnitude of the transition energy E_C depends mainly on the strength of the magnetic field, the coherence length, and the distance to the source; for B_rms=0.1 micro-Gauss, a largest eddy of length 10 Mpc, and a distance to the source of 10 Mpc, E_C=100 EeV. Our numerical treatment substantially improves on previous analytical approximations, as it allows to treat carefully the transition between the two propagation regimes, as well as the effects due to inhomogeneities expected on scales of a few Mpc. We show that a turbulent magnetic field B_rms=0.1 micro-Gauss, close to equipartition, would allow to reproduce exactly the observed spectrum of ultra high energy cosmic rays, up to the highest energy observed, for a distance to the source below 10 Mpc, for the geometry of the Local Supercluster, i.e. a sheet of thickness 10 Mpc. Diffusion, in this case, allows to reproduce the high flux beyond the Greisen Zatsepin Kuzmin cut-off, with a soft injection spectrum proportional to E**(-2.4). Moreover, the large deflection angles at the highest energies observed, typically 10 degrees for the above values, would explain why no close-by astrophysical counterpart could be associated with these events.Comment: 17 latex pages (tightened format), 9 updated postscript figures, uses revtex.sty and epsf.sty, extended discussion of numerical results, to appear in Astroparticle Physic

Maximizing the Total Resolution of Graphs

A major factor affecting the readability of a graph drawing is its resolution. In the graph drawing literature, the resolution of a drawing is either measured based on the angles formed by consecutive edges incident to a common node (angular resolution) or by the angles formed at edge crossings (crossing resolution). In this paper, we evaluate both by introducing the notion of "total resolution", that is, the minimum of the angular and crossing resolution. To the best of our knowledge, this is the first time where the problem of maximizing the total resolution of a drawing is studied. The main contribution of the paper consists of drawings of asymptotically optimal total resolution for complete graphs (circular drawings) and for complete bipartite graphs (2-layered drawings). In addition, we present and experimentally evaluate a force-directed based algorithm that constructs drawings of large total resolution

Reconstruction of cosmological density and velocity fields in the Lagrangian Zel'dovich Approximation

We present a method for reconstructing cosmological densityn for and velocity fields using the Lagrangian Zel'dovich formalism. . The method involves finding the least action solution for straight line particle paths in an evolving density field. Our starting point is the final, evolved density , so that we are in effect carrying out the standard Zel'dovich Approximation based process in reverse. Using a simple numerical algorithm we are able to minimise the action for the trajectories of several million particles. We apply our method to the evolved density taken from N-body simulations of different cold dark matter dominated universes, testing both the prediction for the present day velocity field and for the initial density field. The method is easy to apply, reproduces the accuracy of the forward Zel'dovich Approximation, and also works directly in redshift space with minimal modification.Comment: 13 pages with only 2 (out 9) figures. MNRAS in press. New Appendix shows the relation between shell crossing and PIZA. A completed version with all 9 figures available by anonymous ftp at ftp://bessel.mps.ohio-state.edu/pub/racc/piza.ps.gz (USA) or ftp://ftp-astro.physics.ox.ac.uk/pub/eg/piza3.ps.gz (UK