4,904 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
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
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
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
Quantization of multidimensional cat maps
In this work we study cat maps with many degrees of freedom. Classical cat
maps are classified using the Cayley parametrization of symplectic matrices and
the closely associated center and chord generating functions. Particular
attention is dedicated to loxodromic behavior, which is a new feature of
two-dimensional maps. The maps are then quantized using a recently developed
Weyl representation on the torus and the general condition on the Floquet
angles is derived for a particular map to be quantizable. The semiclassical
approximation is exact, regardless of the dimensionality or of the nature of
the fixed points.Comment: 33 pages, latex, 6 figures, Submitted to Nonlinearit
Entanglement in Quantum Spin Chains, Symmetry Classes of Random Matrices, and Conformal Field Theory
We compute the entropy of entanglement between the first spins and the
rest of the system in the ground states of a general class of quantum
spin-chains. We show that under certain conditions the entropy can be expressed
in terms of averages over ensembles of random matrices. These averages can be
evaluated, allowing us to prove that at critical points the entropy grows like
as , where and are determined explicitly. In an important class of systems,
is equal to one-third of the central charge of an associated Virasoro algebra.
Our expression for therefore provides an explicit formula for the
central charge.Comment: 4 page
Introduction: Policymaking, Learning and Devolution
This collection explores the idea that devolution in European states can provide ‘laboratories of democracy' as states experiment in different ways to address social and economic problems and engage their citizens. While there is a substantial literature on policy diffusion and learning among US states, and on transfer and learning between countries, there is not much on learning among European federated and devolved governments. This collection fills that gap with a series of studies based primarily on experiences in Germany, Spain and the UK (including a particular focus on Northern Ireland). The introduction sets up the analytical framework, discussing the concepts of learning and transfer, the direction of transfer (for example, from the centre to the devolved territory), the mechanisms involved (from voluntary to coercive) and the degree of transfer
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