Fractional Calculus as a Macroscopic Manifestation of Randomness

We generalize the method of Van Hove so as to deal with the case of non-ordinary statistical mechanics, that being phenomena with no time-scale separation. We show that in the case of ordinary statistical mechanics, even if the adoption of the Van Hove method imposes randomness upon Hamiltonian dynamics, the resulting statistical process is described using normal calculus techniques. On the other hand, in the case where there is no time-scale separation, this generalized version of Van Hove's method not only imposes randomness upon the microscopic dynamics, but it also transmits randomness to the macroscopic level. As a result, the correct description of macroscopic dynamics has to be expressed in terms of the fractional calculus.Comment: 20 pages, 1 figur

Galaxy Orientations in the Coma Cluster

We have examined the orientations of early-type galaxies in the Coma cluster to see whether the well-established tendency for brightest cluster galaxies to share the same major axis orientation as their host cluster also extends to the rest of the galaxy population. We find no evidence of any preferential orientations of galaxies within Coma or its surroundings. The implications of this result for theories of the formation of clusters and galaxies (particularly the first-ranked members) are discussed.Comment: Accepted for publication in the Astrophysical Journal Letters. 4 pages, 4 figure

Development of optical modulators for measurements of solar magnetic fields

The measurement of polarized light allows solar astronomers to infer the magnetic field on the Sun. The accuracy of these measurements is dependent on the stable retardation characteristics of the polarization modulators used to minimize the atmospheric effects seen in ground-based observations. This report describes the work by the Space Science Laboratory at Marshall Space Flight Center to improve two types of polarization modulators. As a result, the timing characteristics for both electrooptic crystals (KD*Ps) and liquid crystal devices (LCDs) have been studied and will be used to enhance the capabilities of the MSFC Vector Magnetograph

An intensity-expansion method to treat non-stationary time series: an application to the distance between prime numbers

We study the fractal properties of the distances between consecutive primes. The distance sequence is found to be well described by a non-stationary exponential probability distribution. We propose an intensity-expansion method to treat this non-stationarity and we find that the statistics underlying the distance between consecutive primes is Gaussian and that, by transforming the distance sequence into a stationary one, the range of Gaussian randomness of the sequence increases.Comment: 11 pages, 7 figures, in press on 'Chaos, Solitons & Fractals

Invisible Pixels Are Dead, Long Live Invisible Pixels!

Privacy has deteriorated in the world wide web ever since the 1990s. The tracking of browsing habits by different third-parties has been at the center of this deterioration. Web cookies and so-called web beacons have been the classical ways to implement third-party tracking. Due to the introduction of more sophisticated technical tracking solutions and other fundamental transformations, the use of classical image-based web beacons might be expected to have lost their appeal. According to a sample of over thirty thousand images collected from popular websites, this paper shows that such an assumption is a fallacy: classical 1 x 1 images are still commonly used for third-party tracking in the contemporary world wide web. While it seems that ad-blockers are unable to fully block these classical image-based tracking beacons, the paper further demonstrates that even limited information can be used to accurately classify the third-party 1 x 1 images from other images. An average classification accuracy of 0.956 is reached in the empirical experiment. With these results the paper contributes to the ongoing attempts to better understand the lack of privacy in the world wide web, and the means by which the situation might be eventually improved.Comment: Forthcoming in the 17th Workshop on Privacy in the Electronic Society (WPES 2018), Toronto, AC

Standard and Non-standard Extensions of Lie algebras

We study the problem of quadruple extensions of simple Lie algebras. We find that, adding a new simple root $\alpha_{+4}$, it is not possible to have an extended Kac-Moody algebra described by a Dynkin-Kac diagram with simple links and no loops between the dots, while it is possible if $\alpha_{+4}$ is a Borcherds imaginary simple root. We also comment on the root lattices of these new algebras. The folding procedure is applied to the simply-laced triple extended Lie algebras, obtaining all the non-simply laced ones. Non- standard extension procedures for a class of Lie algebras are proposed. It is shown that the 2-extensions of $E_{8}$, with a dot simply linked to the Dynkin-Kac diagram of $E_{9}$, are rank 10 subalgebras of $E_{10}$. Finally the simple root systems of a set of rank 11 subalgebras of $E_{11}$, containing as sub-algebra $E_{10}$, are explicitly written.Comment: Revised version. Inaccurate statements corrected. Expanded version with added reference

Asynchronous Variational Integrators

We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton’s variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly. To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system.A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorith