Autocorrelation of Random Matrix Polynomials

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions

Determination of mean atmospheric densities from the explorer ix satellite

Mean atmospheric densities from changes in orbital elements of Explorer IX satellit

Diagnosing Disaster Resilience of Communities as Complex Socioecological Systems

Global environmental change, growing anthropogenic influence, and increasing globalization of society have made it clear that disaster vulnerability and resilience of communities cannot be understood without knowledge of the broader social-ecological system in which they are embedded. Inspired by iterative multiscale analysis employed by the Resilience Alliance, the related Social-Ecological Systems Framework initially designed by Elinor Ostrom, and the Sustainable Livelihood Framework, we developed a multi-tier framework for conceptualizing communities as multiscale social-ecological systems. We use the framework to diagnose and analyze community resilience to disasters, as a form of disturbance to social-ecological systems, with feedbacks from the local to the global scale. We highlight the cross-scale influences and feedback on communities that exist from lower (e.g., household) to higher (e.g., regional, national) scales. The framework is then applied to real-world community resilience assessment in Nepal and China, to illustrate how key components of socio-ecological systems, including natural hazards, natural and man-made environment, and community capacities can be delineated and analyzed

Revealing Cosmic Rotation

Cosmological Birefringence (CB), a rotation of the polarization plane of radiation coming to us from distant astrophysical sources, may reveal parity violation in either the electromagnetic or gravitational sectors of the fundamental interactions in nature. Until only recently this phenomenon could be probed with only radio observations or observations at UV wavelengths. Recently, there is a substantial effort to constrain such non-standard models using observations of the rotation of the polarization plane of cosmic microwave background (CMB) radiation. This can be done via measurements of the $B$-modes of the CMB or by measuring its TB and EB correlations which vanish in the standard model. In this paper we show that $EB$ correlations-based estimator is the best for upcoming polarization experiments. The $EB$ based estimator surpasses other estimators because it has the smallest noise and of all the estimators is least affected by systematics. Current polarimeters are optimized for the detection of $B$-mode polarization from either primordial gravitational waves or by large scale structure via gravitational lensing. In the paper we also study optimization of CMB experiments for the detection of cosmological birefringence, in the presence of instrumental systematics, which by themselves are capable of producing $EB$ correlations; potentially mimicking CB.Comment: 10 pages, 3 figures, 2 table

Properties of a continuous-random-network model for amorphous systems

We use a Monte Carlo bond-switching method to study systematically the thermodynamic properties of a "continuous random network" model, the canonical model for such amorphous systems as a-Si and a-SiO$_2$. Simulations show first-order "melting" into an amorphous state, and clear evidence for a glass transition in the supercooled liquid. The random-network model is also extended to study heterogeneous structures, such as the interface between amorphous and crystalline Si.Comment: Revtex file with 4 figure

Zeta Function Zeros, Powers of Primes, and Quantum Chaos

We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised of an infinite series of oscillatory terms, one for each zero of the zeta function on the critical line and was derived by Riemann in his paper on primes assuming the Riemann hypothesis. We show that high resolution spectral lines can be generated by the truncated series at all powers of primes and demonstrate explicitly that the relative line intensities are correct. We then derive a Gaussian sum rule for Riemann's formula. This is used to analyze the numerical convergence of the truncated series. The connections to quantum chaos and semiclassical physics are discussed

Observational constraints on Cosmic Reionization

Recent observations have set the first constraints on the epoch of reionization (EoR), corresponding to the formation epoch of the first luminous objects. Studies of Gunn-Peterson (GP) absorption, and related phenomena, suggest a qualitative change in the state of the intergalactic medium (IGM) at $z \sim 6$, indicating a rapid increase in the neutral fraction of the IGM, from $x_{HI} 10^{-3}$, perhaps up to 0.1, at $z \ge 6$. Conversely, transmission spikes in the GP trough, and the evolution of the \lya galaxy luminosity function indicate $x_{HI} < 0.5$ at $z\sim 6.5$, while the large scale polarization of the cosmic microwave background (CMB) implies a significant ionization fraction extending to higher redshifts, $z \sim 11 \pm 3$. The results suggest that reionization is less an event than a process, with the process beginning as early as $z \sim 14$, and with the 'percolation', or 'overlap' phase ending at $z \sim 6$. The data are consistent with low luminosity star forming galaxies as being the dominant sources of reionizing photons. Low frequency radio telescopes currently under construction should be able to make the first direct measurements of HI 21cm emission from the neutral IGM during the EoR, and upcoming measurements of secondary CMB temperature anisotropy will provide fine details of the dynamics of the reionized IGM.Comment: to appear in ARAA 2006, vol 44, page 415-462; latex. 84 pages. 15 fi

Stochastic stabilization of cosmological photons

The stability of photon trajectories in models of the Universe that have constant spatial curvature is determined by the sign of the curvature: they are exponentially unstable if the curvature is negative and stable if it is positive or zero. We demonstrate that random fluctuations in the curvature provide an additional stabilizing mechanism. This mechanism is analogous to the one responsible for stabilizing the stochastic Kapitsa pendulum. When the mean curvature is negative it is capable of stabilizing the photon trajectories; when the mean curvature is zero or positive it determines the characteristic frequency with which neighbouring trajectories oscillate about each other. In constant negative curvature models of the Universe that have compact topology, exponential instability implies chaos (e.g. mixing) in the photon dynamics. We discuss some consequences of stochastic stabilization in this context.Comment: 4 pages, 3 postscript figures in color which are also appropriate for black and white printers; v2 emphasizes relevance to flat as well as negatively curved cosmologies; to appear in J. Phys.

Random polynomials, random matrices, and LL-functions

We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function.Comment: Added background material. Final version. To appear in Nonlinearit

Random matrix theory, the exceptional Lie groups, and L-functions

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are here extended to non-classical groups. We focus on an explicit example: the exceptional Lie group G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G_2, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the 7-dimensional representation of G_2. The random matrix calculations extend to all exceptional Lie groupsComment: 14 page