Twin prime correlations from the pair correlation of Riemann zeros

We establish, via a formal/heuristic Fourier inversion calculation, that the Hardy-Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height $E$ on the critical line. Previously it was known that the Hardy-Littlewood conjecture implies the pair correlation formula, and we show that the reverse implication also holds. A smooth form of the Hardy-Littlewood conjecture is obtained by inverting the $E \rightarrow \infty$ limit of the two-point correlation function and the precise form of the conjecture is found by including asymptotically lower order terms in the two-point correlation function formula.Comment: 11 page

On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class

We establish the equivalence of conjectures concerning the pair correlation of zeros of $L$-functions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the results of Goldston & Montgomery [7] and Montgomery & Soundararajan [11] for the Riemann zeta-function to other $L$-functions in the Selberg class. Our approach is based on the statistics of the zeros because the analogue of the Hardy-Littlewood conjecture for the auto-correlation of the arithmetic functions we consider is not available in general. One of our main findings is that the variances of sums of these arithmetic functions over primes in short intervals have a different form when the degree of the associated $L$-functions is 2 or higher to that which holds when the degree is 1 (e.g. the Riemann zeta-function). Specifically, when the degree is 2 or higher there are two regimes in which the variances take qualitatively different forms, whilst in the degree-1 case there is a single regime

Quantum chaotic resonances from short periodic orbits

We present an approach to calculating the quantum resonances and resonance wave functions of chaotic scattering systems, based on the construction of states localized on classical periodic orbits and adapted to the dynamics. Typically only a few of such states are necessary for constructing a resonance. Using only short orbits (with periods up to the Ehrenfest time), we obtain approximations to the longest living states, avoiding computation of the background of short living states. This makes our approach considerably more efficient than previous ones. The number of long lived states produced within our formulation is in agreement with the fractal Weyl law conjectured recently in this setting. We confirm the accuracy of the approximations using the open quantum baker map as an example.Comment: 4 pages, 4 figure

Attitude determination of the spin-stabilized Project Scanner spacecraft

Attitude determination of spin-stabilized spacecraft using star mapping techniqu

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

Periodic orbit bifurcations and scattering time delay fluctuations

We study fluctuations of the Wigner time delay for open (scattering) systems which exhibit mixed dynamics in the classical limit. It is shown that in the semiclassical limit the time delay fluctuations have a distribution that differs markedly from those which describe fully chaotic (or strongly disordered) systems: their moments have a power law dependence on a semiclassical parameter, with exponents that are rational fractions. These exponents are obtained from bifurcating periodic orbits trapped in the system. They are universal in situations where sufficiently long orbits contribute. We illustrate the influence of bifurcations on the time delay numerically using an open quantum map.Comment: 9 pages, 3 figures, contribution to QMC200

Quantum statistics on graphs

Quantum graphs are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter. We consider quantum statistics for indistinguishable spinless particles on a graph, concentrating on the simplest case of abelian statistics for two particles. In spite of the fact that graphs are locally one-dimensional, anyon statistics emerge in a generalized form. A given graph may support a family of independent anyon phases associated with topologically inequivalent exchange processes. In addition, for sufficiently complex graphs, there appear new discrete-valued phases. Our analysis is simplified by considering combinatorial rather than metric graphs -- equivalently, a many-particle tight-binding model. The results demonstrate that graphs provide an arena in which to study new manifestations of quantum statistics. Possible applications include topological quantum computing, topological insulators, the fractional quantum Hall effect, superconductivity and molecular physics.Comment: 21 pages, 6 figure

Determination of mean atmospheric densities from the explorer ix satellite

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

Number fields and function fields:Coalescences, contrasts and emerging applications

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained

A CubeSat for Calibrating Ground-Based and Sub-Orbital Millimeter-Wave Polarimeters (CalSat)

We describe a low-cost, open-access, CubeSat-based calibration instrument that is designed to support ground-based and sub-orbital experiments searching for various polarization signals in the cosmic microwave background (CMB). All modern CMB polarization experiments require a robust calibration program that will allow the effects of instrument-induced signals to be mitigated during data analysis. A bright, compact, and linearly polarized astrophysical source with polarization properties known to adequate precision does not exist. Therefore, we designed a space-based millimeter-wave calibration instrument, called CalSat, to serve as an open-access calibrator, and this paper describes the results of our design study. The calibration source on board CalSat is composed of five "tones" with one each at 47.1, 80.0, 140, 249 and 309 GHz. The five tones we chose are well matched to (i) the observation windows in the atmospheric transmittance spectra, (ii) the spectral bands commonly used in polarimeters by the CMB community, and (iii) The Amateur Satellite Service bands in the Table of Frequency Allocations used by the Federal Communications Commission. CalSat would be placed in a polar orbit allowing visibility from observatories in the Northern Hemisphere, such as Mauna Kea in Hawaii and Summit Station in Greenland, and the Southern Hemisphere, such as the Atacama Desert in Chile and the South Pole. CalSat also would be observable by balloon-borne instruments launched from a range of locations around the world. This global visibility makes CalSat the only source that can be observed by all terrestrial and sub-orbital observatories, thereby providing a universal standard that permits comparison between experiments using appreciably different measurement approaches