The Statistics of the Points Where Nodal Lines Intersect a Reference Curve

We study the intersection points of a fixed planar curve $\Gamma$ with the nodal set of a translationally invariant and isotropic Gaussian random field \Psi(\bi{r}) and the zeros of its normal derivative across the curve. The intersection points form a discrete random process which is the object of this study. The field probability distribution function is completely specified by the correlation G(|\bi{r}-\bi{r}'|) = . Given an arbitrary G(|\bi{r}-\bi{r}'|), we compute the two point correlation function of the point process on the line, and derive other statistical measures (repulsion, rigidity) which characterize the short and long range correlations of the intersection points. We use these statistical measures to quantitatively characterize the complex patterns displayed by various kinds of nodal networks. We apply these statistics in particular to nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of special interest is the observation that for monochromatic random waves, the number variance of the intersections with long straight segments grows like $L \ln L$, as opposed to the linear growth predicted by the percolation model, which was successfully used to predict other long range nodal properties of that field.Comment: 33 pages, 13 figures, 1 tabl

Scalable Byzantine Reliable Broadcast

Byzantine reliable broadcast is a powerful primitive that allows a set of processes to agree on a message from a designated sender, even if some processes (including the sender) are Byzantine. Existing broadcast protocols for this setting scale poorly, as they typically build on quorum systems with strong intersection guarantees, which results in linear per-process communication and computation complexity. We generalize the Byzantine reliable broadcast abstraction to the probabilistic setting, allowing each of its properties to be violated with a fixed, arbitrarily small probability. We leverage these relaxed guarantees in a protocol where we replace quorums with stochastic samples. Compared to quorums, samples are significantly smaller in size, leading to a more scalable design. We obtain the first Byzantine reliable broadcast protocol with logarithmic per-process communication and computation complexity. We conduct a complete and thorough analysis of our protocol, deriving bounds on the probability of each of its properties being compromised. During our analysis, we introduce a novel general technique that we call adversary decorators. Adversary decorators allow us to make claims about the optimal strategy of the Byzantine adversary without imposing any additional assumptions. We also introduce Threshold Contagion, a model of message propagation through a system with Byzantine processes. To the best of our knowledge, this is the first formal analysis of a probabilistic broadcast protocol in the Byzantine fault model. We show numerically that practically negligible failure probabilities can be achieved with realistic security parameters

Intuitive Beliefs

Beliefs are intuitive if they rely on associative memory, which can be described as a network of associations between events. A belief-theoretic characterization of the model is provided, its uniqueness properties are established, and the intersection with the Bayesian model is characterized. The formation of intuitive beliefs is modelled after machine learning, whereby the network is shaped by past experience via minimization of the difference from an objective probability distribution. The model is shown to accommodate correlation misperception, the conjunction fallacy, base-rate neglect/conservatism, etc

Reverse Shock Emission Revealed in Early Photometry in the Candidate Short GRB 180418A

We present observations of the possible short GRB 180418A in $\gamma$-rays, X-rays, and in the optical. Early optical photometry with the TAROT and RATIR instruments show a bright peak ($\approx$ 14.2 AB mag) between $T+28$ and $T+90$ seconds that we interpret as the signature of a reversal shock. Later observations can be modeled by a standard forward shock model and show no evidence of jet break, allowing us to constrain the jet collimation to $\theta_j> 7^\circ$. Using deep late-time optical observations we place an upper limit of $r>24$ AB mag on any underlying host galaxy. The detection of the afterglow in the \textit{Swift} UV filters constrains the GRB redshift to $z<1.3$ and places an upper bound on the $\gamma$-ray isotropic equivalent energy $E_{\rm{\gamma,iso}} < 3 \times 10^{51}$ erg. The properties of this GRB (e.g. duration, hardness ratio, energetic, and environment) lie at the intersection between short and long bursts, and we can not conclusively identify its type. We estimate that the probability that it is drawn from the population of short GRBs is 10\%-30\%.Comment: Accepted por publication in Ap

On Topological Properties of Wireless Sensor Networks under the q-Composite Key Predistribution Scheme with On/Off Channels

The q-composite key predistribution scheme [1] is used prevalently for secure communications in large-scale wireless sensor networks (WSNs). Prior work [2]-[4] explores topological properties of WSNs employing the q-composite scheme for q = 1 with unreliable communication links modeled as independent on/off channels. In this paper, we investigate topological properties related to the node degree in WSNs operating under the q-composite scheme and the on/off channel model. Our results apply to general q and are stronger than those reported for the node degree in prior work even for the case of q being 1. Specifically, we show that the number of nodes with certain degree asymptotically converges in distribution to a Poisson random variable, present the asymptotic probability distribution for the minimum degree of the network, and establish the asymptotically exact probability for the property that the minimum degree is at least an arbitrary value. Numerical experiments confirm the validity of our analytical findings.Comment: Best Student Paper Finalist in IEEE International Symposium on Information Theory (ISIT) 201

GRB 070201: A possible Soft Gamma Ray Repeater in M31

The gamma-ray burst (GRB) 070201 was a bright short-duration hard-spectrum GRB detected by the Inter-Planetary Network (IPN). Its error quadrilateral, which has an area of 0.124 sq. deg, intersects some prominent spiral arms of the nearby M31 (Andromeda) galaxy. Given the properties of this GRB, along with the fact that LIGO data argues against a compact binary merger origin in M31, this GRB is an excellent candidate for an extragalactic Soft Gamma-ray Repeater (SGR) giant flare, with energy of 1.4x10^45 erg. Analysis of ROTSE-IIIb visible light observations of M31, taken 10.6 hours after the burst and covering 42% of the GRB error region, did not reveal any optical transient down to a limiting magnitude of 17.1. We inspected archival and proprietary XMM-Newton X-ray observations of the intersection of the GRB error quadrilateral and M31, obtained about four weeks prior to the outburst, in order to look for periodic variable X-ray sources. No SGR or Anomalous X-ray Pulsar (AXP) candidates (periods in range 1 to 20 s) were detected. We discuss the possibility of detecting extragalactic SGRs/AXPs by identifying their periodic X-ray light curves. Our simulations suggest that the probability of detecting the periodic X-ray signal of one of the known Galactic SGRs/AXPs, if placed in M31, is about 10% (50%), using 50 ks (2 Ms) XMM-Newton exposures.Comment: 7 pages, submitted to ApJ (Fig. 2 resolution reduced

Sets and Probability

In this article the idea of random variables over the set theoretic universe is investigated. We explore what it can mean for a random set to have a specific probability of belonging to an antecedently given class of sets