Toward the Universal Rigidity of General Frameworks

Let (G,P) be a bar framework of n vertices in general position in R^d, d <= n-1, where G is a (d+1)-lateration graph. In this paper, we present a constructive proof that (G,P) admits a positive semi-definite stress matrix with rank n-d-1. We also prove a similar result for a sensor network where the graph consists of m(>= d+1) anchors.Comment: v2, a revised version of an earlier submission (v1

Cosmic rays in the surroundings of SNR G35.6-0.4

HESS J1858+020 is a TeV gamma-ray source that was reported not to have any clear cataloged counterpart at any wavelength. However, it has been recently proposed that this source is indirectly associated with the radio source, re-identified as a supernova remnant (SNR), G35.6-0.4. The latter is found to be middle-aged ($\sim 30$ kyr) and to have nearby molecular clouds (MCs). HESS J1858+020 was proposed to be the result of the interaction of protons accelerated in the SNR shell with target ions residing in the clouds. The Fermi Large Area Telescope (LAT) First Source Catalog does not list any source coincident with the position of HESS J1858+020, but some lie close. Here, we analyse more than 2 years of data obtained with the Fermi-LAT for the region of interest, and consider whether it is indeed possible that the closest LAT source, 1FGL J1857.1+0212c, is related to HESS J1858+020. We conclude it is not, and we impose upper limits on the GeV emission originating from HESS J1858+020. Using a simplified 3D model for the cosmic-ray propagation out from the shell of the SNR, we consider whether the interaction between SNR G35.6-0.4 and the MCs nearby could give rise to the TeV emission of HESS J1858+020 without producing a GeV counterpart. If so, the pair of SNR/TeV source with no GeV detection would be reminiscent of other similarly-aged SNRs, such as some of the TeV hotspots near W28, for which cosmic-ray diffusion may be used to explain their multi-frequency phenomenology. However, for HESS J1858+020, we found that although the phase space in principle allows for such GeV--TeV non-correlation to appear, usual and/or observationally constrained values of the parameters (e.g., diffusion coefficients and cloud-SNR likely distances) would disfavor it.Comment: In press in MNRA

On dimensional rigidity of bar-and-joint frameworks

AbstractLet V={1,2,…,n}. A mapping p:V→Rr, where p1,…,pn are not contained in a proper hyper-plane is called an r-configuration. Let G=(V,E) be a simple connected graph on n vertices. Then an r-configuration p together with graph G, where adjacent vertices of G are constrained to stay the same distance apart, is called a bar-and-joint framework (or a framework) in Rr, and is denoted by G(p). In this paper we introduce the notion of dimensional rigidity of frameworks, and we study the problem of determining whether or not a given G(p) is dimensionally rigid. A given framework G(p) in Rr is said to be dimensionally rigid iff there does not exist a framework G(q) in Rs for s⩾r+1, such that ∥qi-qj∥2=∥pi-pj∥2 for all (i,j)∈E. We present necessary and sufficient conditions for G(p) to be dimensionally rigid, and we formulate the problem of checking the validity of these conditions as a semidefinite programming (SDP) problem. The case where the points p1,…,pn of the given r-configuration are in general position, is also investigated

On rigidity and realizability of weighted graphs

AbstractRecently, a new characterization of rigid graphs was introduced using Euclidean distance matrices (EDMs) [A.Y. Alfakih, Linear Algebra Appl. 310 (2000) 149]. In this paper, we address the computational aspects of this characterization. Also we present a characterization of graphs which are realizable in Rr for some 1⩽r⩽n−2 but not realizable in R(n−1), where n is the number of nodes

On the uniqueness of Euclidean distance matrix completions

AbstractThe Euclidean distance matrix completion problem (EDMCP) is the problem of determining whether or not a given partial matrix can be completed into a Euclidean distance matrix. In this paper, we present necessary and sufficient conditions for a given solution of the EDMCP to be unique

On the universal rigidity of generic bar frameworks

Let $V=\{1,\ldots,n\}$ be a finite set. An $r$-configuration is a mapping $p:V \rightarrow R^r$, where $p^1,\ldots,p^n$ are not contained in a proper hyper-plane. A framework $G(p)$ in $R^r$ is an $r$-configuration together with a graph $G=(V,E)$ such that every two points corresponding to adjacent vertices of $G$ are constrained to stay the same distance apart. A framework $G(p)$ is said to be generic if all the coordinates of $p^1,\ldots, p^n$ are algebraically independent over the integers. A framework $G(p)$ in $R^r$ is said to be unique if there does not exist a framework $G(q)$ in $R^s$, for some $s$, $1 \leq s \leq n-1$, such that $||q^i-q^j||=||p^i-p^j||$ for all $(i,j) \in E$. In this paper we present a sufficient condition for a generic framework $G(p)$ to be unique, and we conjecture that this condition is also necessary. Connections with the closely related problems of global rigidity and dimensional rigidity are also discussed

Fermi Large Area Telescope Observations of the Cygnus Loop Supernova Remnant

We present an analysis of the gamma-ray measurements by the Large Area Telescope(LAT) onboard the \textit{Fermi Gamma-ray Space Telescope} in the region of the supernova remnant(SNR) Cygnus Loop(G74.0$-$8.5). We detect significant gamma-ray emission associated with the SNR in the energy band 0.2--100 GeV. The gamma-ray spectrum shows a break in the range 2--3 GeV. The gamma-ray luminosity is $\sim$ $1 \times 10^{33}$erg s$^{-1}$ between 1--100 GeV, much lower than those of other GeV-emitting SNRs. The morphology is best represented by a ring shape, with inner/outer radii 0$^\circ$.7 $\pm$ 0$^\circ$.1 and 1$^\circ$.6 $\pm$ 0$^\circ$.1. Given the association among X-ray rims, \halpha filaments and gamma-ray emission, we argue that gamma rays originate in interactions between particles accelerated in the SNR and interstellar gas or radiation fields adjacent to the shock regions. The decay of neutral pions produced in nucleon-nucleon interactions between accelerated hadrons and interstellar gas provides a reasonable explanation for the gamma-ray spectrum.Comment: accepted by ApJ, 34 pages, 6 figure

Suzaku X-ray Imaging of the Extended Lobe in the Giant Radio Galaxy NGC6251 Associated with the Fermi-LAT Source 2FGLJ1629.4+8236

We report the results of a Suzaku X-ray imaging study of NGC6251, a nearby giant radio galaxy with intermediate FR I/II radio properties. Our pointing direction was centered on the gamma -ray emission peak recently discovered with Fermi-LAT around the position of the north-west radio lobe 15 arcmin offset from the nucleus. After subtracting two "off-source" pointings adjacent to the radio lobe, and removing possible contaminants in the XIS field of view, we found significant residual X-ray emission most likely diffuse in nature. The spectrum of the excess X-ray emission is well fit by a power law with photon index \Gamma = 1.90 +- 0.15 and a 0.5 - 8 keV flux of 4 x 10^{-13} erg cm^{-2} s^{-1}. We interpret this diffuse X-ray emission component as being due to inverse-Compton up-scattering of the cosmic microwave background photons by ultrarelativistic electrons within the lobe, with only a minor contribution from the beamed emission of the large-scale jet. Utilizing archival radio data for the source, we demonstrate by means of broad-band spectral modeling that the -ray flux of the Fermi-LAT source 2FGL J1629.4+8236 may well be accounted for by the high-energy tail of the inverse-Compton continuum of the lobe. Thus, this claimed association of gamma-rays from the north-west lobe of NGC6251, together with the recent Fermi-LAT imaging of the extended lobes of Centaurus A, indicates that particles may be efficiently (re-)accelerated up to ultrarelativistic energies within extended radio lobes of nearby radio galaxies in general.Comment: 19 pages, 4 figures, accepted for publication in the Ap

Outer gap accelerator closed by magnetic pair-creation process

We discuss outer gap closure mechanism in the trans-field direction with the magnetic pair-creation process near the stellar surface. The gap closure by the magnetic pair-creation is possible if some fraction of the pairs are produced with an outgoing momentum. By assuming that multiple magnetic field will affect the local field near the stellar surface, we show a specific magnetic field geometry near the stellar surface resulting in the outflow of the pairs. Together with the fact that the electric field is weak below null charge surface, the characteristic curvature photon energy emitted by incoming particles, which were accelerated in the outer gap, decreases drastically to $\sim 100$MeV near the stellar surface. We estimate the height measured from the last-open field line, above which 100MeV photons is converted into pairs by the magnetic pair-creation. We also show the resultant multiplicity due to the magnetic pair-creation process could acquire $M_{e^{\pm}}\sim 10^4-10^5$. In this model the fractional outer gap size is proportional to $P^{-1/2}$. The predicted gamma-ray luminosity ($L_{\gamma}$) and the characteristic curvature photon energy ($E_c$) emitted from the outer gap are proportional to $B^2P^{-5/2}$ and $B^{3/4}P^{-1}$ respectively. This model also predicts that $L_{\gamma}$ and $E_c$ are related to the spin down power ($L_{sd}$) or the spin down age of pulsars ($\tau$) as $L_{\gamma} \propto L_{sd}^{5/8}$ or $L_{\gamma} \propto \tau^{-5/4}$, and $E_c \propto L_{sd}^{1/4}$ or $E_c \propto \tau^{-1/2}$ respectively.Comment: 33 pages, 9 figures, Accepted for publication in ApJ

Adjacency on the constrained assignment problem

AbstractLet Qc,r be the integer hull of the intersection of the assignment polytope with a given hyper-plane H = {x = (xij) ϵ Rn × n: ∑ni = 1 ∑nj = 1 cijxij = r}. We show that the problem of checking whether two given extreme points of Qc,r are nonadjacent on Qc,r is solvable in O(n5) time if c = (cij) is a 0–1 matrix, and that it is NP-Complete if c is a general integer matrix