Discrete Signal Processing on Graphs: Frequency Analysis

Signals and datasets that arise in physical and engineering applications, as well as social, genetics, biomolecular, and many other domains, are becoming increasingly larger and more complex. In contrast to traditional time and image signals, data in these domains are supported by arbitrary graphs. Signal processing on graphs extends concepts and techniques from traditional signal processing to data indexed by generic graphs. This paper studies the concepts of low and high frequencies on graphs, and low-, high-, and band-pass graph filters. In traditional signal processing, there concepts are easily defined because of a natural frequency ordering that has a physical interpretation. For signals residing on graphs, in general, there is no obvious frequency ordering. We propose a definition of total variation for graph signals that naturally leads to a frequency ordering on graphs and defines low-, high-, and band-pass graph signals and filters. We study the design of graph filters with specified frequency response, and illustrate our approach with applications to sensor malfunction detection and data classification

Tearing of free-standing graphene

We examine the fracture mechanics of tearing graphene. We present a molecular dynamics simulation of the propagation of cracks in clamped, free-standing graphene as a function of the out-of-plane force. The geometry is motivated by experimental configurations that expose graphene sheets to out-of-plane forces, such as back-gate voltage. We establish the geometry and basic energetics of failure and obtain approximate analytical expressions for critical crack lengths and forces. We also propose a method to obtain graphene's toughness. We observe that the cracks' path and the edge structure produced are dependent on the initial crack length. This work may help avoid the tearing of graphene sheets and aid the production of samples with specific edge structures.CAPESNational Science Foundation DMR 1002428Physic

Vortex motion around a circular cylinder above a plane

The study of vortex flows around solid obstacles is of considerable interest from both a theoretical and practical perspective. One geometry that has attracted renewed attention recently is that of vortex flows past a circular cylinder placed above a plane wall, where a stationary recirculating eddy can form in front of the cylinder, in contradistinction to the usual case (without the plane boundary) for which a vortex pair appears behind the cylinder. Here we analyze the problem of vortex flows past a cylinder near a wall through the lenses of the point-vortex model. By conformally mapping the fluid domain onto an annular region in an auxiliary complex plane, we compute the vortex Hamiltonian analytically in terms of certain special functions related to elliptic theta functions. A detailed analysis of the equilibria of the model is then presented. The location of the equilibrium in front of the cylinder is shown to be in qualitative agreement with the experimental findings. We also show that a topological transition occurs in phase space as the parameters of the systems are variedComment: 17 pages, 8 figure

New Analyses of Double-Bang Events in the Atmosphere

We use CORSIKA+Herwig simulation code to produce ultra-high energy neutrino interactions in the atmosphere. Our aim is to reproduce extensive air showers originated by extragalactic tau-neutrinos. For charged current tau-neutrino interactions in the atmosphere, beside the air shower originated from the neutrino interaction, it is expected that a tau is created and may decay before reaching the ground. That phenomenon makes possible the generation of two related extensive air showers, the so called Double-Bang event. We make an analysis of the main characteristics of Double-Bang events in the atmosphere for mean values of the parameters involved in such phenomenon, like the inelasticity and tau decay length. We discuss what may happen for the ``out of the average'' cases and conclude that it may be possible to observe this kind of event in ultra-high energy cosmic ray observatories such as Pierre Auger or Telescope Array.Comment: 17 pages, 5 figures, final version to appear in BJ

Convergence Rate Analysis of Distributed Gossip (Linear Parameter) Estimation: Fundamental Limits and Tradeoffs

The paper considers gossip distributed estimation of a (static) distributed random field (a.k.a., large scale unknown parameter vector) observed by sparsely interconnected sensors, each of which only observes a small fraction of the field. We consider linear distributed estimators whose structure combines the information \emph{flow} among sensors (the \emph{consensus} term resulting from the local gossiping exchange among sensors when they are able to communicate) and the information \emph{gathering} measured by the sensors (the \emph{sensing} or \emph{innovations} term.) This leads to mixed time scale algorithms--one time scale associated with the consensus and the other with the innovations. The paper establishes a distributed observability condition (global observability plus mean connectedness) under which the distributed estimates are consistent and asymptotically normal. We introduce the distributed notion equivalent to the (centralized) Fisher information rate, which is a bound on the mean square error reduction rate of any distributed estimator; we show that under the appropriate modeling and structural network communication conditions (gossip protocol) the distributed gossip estimator attains this distributed Fisher information rate, asymptotically achieving the performance of the optimal centralized estimator. Finally, we study the behavior of the distributed gossip estimator when the measurements fade (noise variance grows) with time; in particular, we consider the maximum rate at which the noise variance can grow and still the distributed estimator being consistent, by showing that, as long as the centralized estimator is consistent, the distributed estimator remains consistent.Comment: Submitted for publication, 30 page

Consensus State Gram Matrix Estimation for Stochastic Switching Networks from Spectral Distribution Moments

Reaching distributed average consensus quickly and accurately over a network through iterative dynamics represents an important task in numerous distributed applications. Suitably designed filters applied to the state values can significantly improve the convergence rate. For constant networks, these filters can be viewed in terms of graph signal processing as polynomials in a single matrix, the consensus iteration matrix, with filter response evaluated at its eigenvalues. For random, time-varying networks, filter design becomes more complicated, involving eigendecompositions of sums and products of random, time-varying iteration matrices. This paper focuses on deriving an estimate for the Gram matrix of error in the state vectors over a filtering window for large-scale, stationary, switching random networks. The result depends on the moments of the empirical spectral distribution, which can be estimated through Monte-Carlo simulation. This work then defines a quadratic objective function to minimize the expected consensus estimate error norm. Simulation results provide support for the approximation.Comment: 52nd Asilomar Conference on Signals, Systems, and Computers (Asilomar 2017

One loop superstring effective actions and N=8 supergravity

In a previous article we have shown the existence of a new independent R^4 term, at one loop, in the type IIA and heterotic effective actions, after reduction to four dimensions, besides the usual square of the Bel-Robinson tensor. It had been shown that such a term could not be directly supersymmetrized, but we showed that was possible after coupling to a scalar chiral multiplet. In this article we study the extended (N=8) supersymmetrization of this term, where no other coupling can be taken. We show that such supersymmetrization cannot be achieved at the linearized level. This is in conflict with the theory one gets after toroidal compactification of type II superstrings being N=8 supersymmetric. We interpret this result in face of the recent claim that perturbative supergravity cannot be decoupled from string theory in d>=4, and N=8, d=4 supergravity is in the swampland.Comment: 28 pages, no figure