5,962 research outputs found
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 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 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 -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 -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 -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
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