Extreme deviations and applications

Stretched exponential probability density functions (pdf), having the form of the exponential of minus a fractional power of the argument, are commonly found in turbulence and other areas. They can arise because of an underlying random multiplicative process. For this, a theory of extreme deviations is developed, devoted to the far tail of the pdf of the sum $X$ of a finite number $n$ of independent random variables with a common pdf $e^{-f(x)}$. The function $f(x)$ is chosen (i) such that the pdf is normalized and (ii) with a strong convexity condition that $f''(x)>0$ and that $x^2f''(x)\to +\infty$ for $|x|\to\infty$. Additional technical conditions ensure the control of the variations of $f''(x)$. The tail behavior of the sum comes then mostly from individual variables in the sum all close to $X/n$ and the tail of the pdf is $\sim e^{-nf(X/n)}$. This theory is then applied to products of independent random variables, such that their logarithms are in the above class, yielding usually stretched exponential tails. An application to fragmentation is developed and compared to data from fault gouges. The pdf by mass is obtained as a weighted superposition of stretched exponentials, reflecting the coexistence of different fragmentation generations. For sizes near and above the peak size, the pdf is approximately log-normal, while it is a power law for the smaller fragments, with an exponent which is a decreasing function of the peak fragment size. The anomalous relaxation of glasses can also be rationalized using our result together with a simple multiplicative model of local atom configurations. Finally, we indicate the possible relevance to the distribution of small-scale velocity increments in turbulent flow.Comment: 26 pages, 1 figure ps (now available), addition and discussion of mathematical references; appeared in J. Phys. I France 7, 1155-1171 (1997

Non-unique factorization of polynomials over residue class rings of the integers

We investigate non-unique factorization of polynomials in Z_{p^n}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring arbitrary non-zero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of Z_{p^n}[x] is a direct sum of monoids corresponding to irreducible polynomials in Z_p[x], and we show that each of these monoids has infinite elasticity. Moreover, for every positive integer m, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.Comment: 11 page

High accuracy precession measurement with an autometric gyro

High accuracy precession measurement with autometric gyroscope

The Sun's Journey Through the Local Interstellar Medium: The PaleoLISM and Paleoheliosphere

Over the recent past, the galactic environment of the Sun has differed substantially from today. Sometime within the past ~130,000 years, and possibly as recent as ~56,000 years ago, the Sun entered the tenuous tepid partially ionized interstellar material now flowing past the Sun. Prior to that, the Sun was in the low density interior of the Local Bubble. As the Sun entered the local ISM flow, we passed briefly through an interface region of some type. The low column densities of the cloud now surrounding the solar system indicate that heliosphere boundary conditions will vary from opacity considerations alone as the Sun moves through the cloud. These variations in the interstellar material surrounding the Sun affected the paleoheliosphere.Comment: To be published in Astrophysics and Space Sciences Transactions (ASTRA), for the proceedings of the workshop "Future Perspectives in Heliospheric Research: Unsolved Problems, New Missions - New Sciences" Bad Honnef, Germany, April 6-8, 2005, held in honor of Prof. Hans Fahr's 65th birthda

Electromagnetic Vacuum of Complex Media: Dipole Emission vs. Light Propagation, Vacuum Energy, and Local Field Factors

We offer a unified approach to several phenomena related to the electromagnetic vacuum of a complex medium made of point electric dipoles. To this aim, we apply the linear response theory to the computation of the polarization field propagator and study the spectrum of vacuum fluctuations. The physical distinction among the local density of states which enter the spectra of light propagation, total dipole emission, coherent emission, total vacuum energy and Schwinger-bulk energy is made clear. Analytical expressions for the spectrum of dipole emission and for the vacuum energy are derived. Their respective relations with the spectrum of external light and with the Schwinger-bulk energy are found. The light spectrum and the Schwinger-bulk energy are determined by the Dyson propagator. The emission spectrum and the total vacuum energy are determined by the polarization propagator. An exact relationship of proportionality between both propagators is found in terms of local field factors. A study of the nature of stimulated emission from a single dipole is carried out. Regarding coherent emission, it contains two components. A direct one which is transferred radiatively and directly from the emitter into the medium and whose spectrum is that of external light. And an indirect one which is radiated by induced dipoles. The induction is mediated by one (and only one) local field factor. Regarding the vacuum energy, we find that in addition to the Schwinger-bulk energy the vacuum energy of an effective medium contains local field contributions proportional to the resonant frequency and to the spectral line-width.Comment: Typos fixed, journal ref. adde

Experimental evidence of accelerated energy transfer in turbulence

We investigate the vorticity dynamics in a turbulent vortex using scattering of acoustic waves. Two ultrasonic beams are adjusted to probe simultaneously two spatial scales in a given volume of the flow, thus allowing a dual channel recording of the dynamics of coherent vorticity structures. Our results show that this allows to measure the average energy transfer time between different spatial length scales, and that such transfer goes faster at smaller scales.Comment: 5 pages, 5 figure

Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem

Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to $4096^3$. The results are analyzed in terms of the classical analyticity strip method and Beale, Kato and Majda (BKM) theorem. A well-resolved acceleration of the time-decay of the width of the analyticity strip $\delta(t)$ is observed at the highest resolution for $3.7<t<3.85$ while preliminary 3D visualizations show the collision of vortex sheets. The BKM criterium on the power-law growth of supremum of the vorticity, applied on the same time-interval, is not inconsistent with the occurrence of a singularity around $t \simeq 4$. These new findings lead us to investigate how fast the analyticity strip width needs to decrease to zero in order to sustain a finite-time singularity consistent with the BKM theorem. A new simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the BKM theorem with the analyticity-strip method. It is shown that a finite-time blowup can exist only if $\delta(t)$ vanishes sufficiently fast at the singularity time. In particular, if a power law is assumed for $\delta(t)$ then its exponent must be greater than some critical value, thus providing a new test that is applied to our $4096^3$ Taylor-Green numerical simulation. Our main conclusion is that the numerical results are not inconsistent with a singularity but that higher-resolution studies are needed to extend the time-interval on which a well-resolved power-law behavior of $\delta(t)$ takes place, and check whether the new regime is genuine and not simply a crossover to a faster exponential decay

Fluctuations of the vortex line density in turbulent flows of quantum fluids

We present an analytical study of fluctuations of the Vortex Line Density (VLD) $$ in turbulent flows of quantum fluids. Two cases are considered. The first one is the counterflowing (Vinen) turbulence, where the vortex lines are disordered, and the evolution of quantity $\mathcal{L}(t)$ obeys the Vinen equation. The second case is the quasi-classic turbulence, where vortex lines are believed to form the so called vortex bundles, and their dynamics is described by the HVBK equations. The latter case, is of a special interest, since a number of recent experiments demonstrate the $\omega ^{-5/3}$ dependence for spectrum VLD, instead of $\omega ^{1/3}$ law, typical for spectrum of vorticity. In nonstationary situation, in particular, in the fluctuating turbulent flow there is a retardation between the instantaneous value of the normal velocity and the quantity $\mathcal{L}$. This retardation tends to decrease in the accordance with the inner dynamics, which has a relaxation character. In both cases the relaxation dynamics of VLD is related to fluctuations of the relative velocity, however if for the Vinen case the rate of temporal change for $\mathcal{L}(t)$ is directly depends on $\delta \mathbf{v}_{ns}$, for the HVBK dynamics it depends on $\nabla \times \delta \mathbf{v}_{ns}$. As a result, for the disordered case the spectrum $<\delta \mathcal{L}(\omega) \delta \mathcal{L}(-\omega)>$ coincides with the spectrum $\omega ^{-5/3}$. In the case of the bundle arrangement, the spectrum of the VLD varies (at different temperatures) from $\omega ^{1/3}$ to $\omega ^{-5/3}$ dependencies. This conclusion may serve as a basis for the experimental determination of what kind of the turbulence is implemented in different types of generation.Comment: 8 pages, 29 reference

An order (n) algorithm for the dynamics simulation of robotic systems

The formulation of an Order (n) algorithm for DISCOS (Dynamics Interaction Simulation of Controls and Structures), which is an industry-standard software package for simulation and analysis of flexible multibody systems is presented. For systems involving many bodies, the new Order (n) version of DISCOS is much faster than the current version. Results of the experimental validation of the dynamics software are also presented. The experiment is carried out on a seven-joint robot arm at NASA's Goddard Space Flight Center. The algorithm used in the current version of DISCOS requires the inverse of a matrix whose dimension is equal to the number of constraints in the system. Generally, the number of constraints in a system is roughly proportional to the number of bodies in the system, and matrix inversion requires O(p exp 3) operations, where p is the dimension of the matrix. The current version of DISCOS is therefore considered an Order (n exp 3) algorithm. In contrast, the Order (n) algorithm requires inversion of matrices which are small, and the number of matrices to be inverted increases only linearly with the number of bodies. The newly-developed Order (n) DISCOS is currently capable of handling chain and tree topologies as well as multiple closed loops. Continuing development will extend the capability of the software to deal with typical robotics applications such as put-and-place, multi-arm hand-off and surface sliding

Wallace v. City of Chicago and Accrual of 1983 Claims

This comment will analyze the recent 7th circuit case, Wallace v. City of Chicago. By ruling that claims under 1983 accrue from the moment of the injury, Wallace basically prevents convicts from recovering under 1983. I will examine the case and suggest resolutions for when the Supreme Court hears the case this term. See 440 F.3d 42