7,949 research outputs found
Order Statistics and Benford's Law
Fix a base B and let zeta have the standard exponential distribution; the
distribution of digits of zeta base B is known to be very close to Benford's
Law. If there exists a C such that the distribution of digits of C times the
elements of some set is the same as that of zeta, we say that set exhibits
shifted exponential behavior base B (with a shift of log_B C \bmod 1). Let X_1,
>..., X_N be independent identically distributed random variables. If the X_i's
are drawn from the uniform distribution on [0,L], then as N\to\infty the
distribution of the digits of the differences between adjacent order statistics
converges to shifted exponential behavior (with a shift of \log_B L/N \bmod 1).
By differentiating the cumulative distribution function of the logarithms
modulo 1, applying Poisson Summation and then integrating the resulting
expression, we derive rapidly converging explicit formulas measuring the
deviations from Benford's Law. Fix a delta in (0,1) and choose N independent
random variables from any compactly supported distribution with uniformly
bounded first and second derivatives and a second order Taylor series expansion
at each point. The distribution of digits of any N^\delta consecutive
differences \emph{and} all N-1 normalized differences of the order statistics
exhibit shifted exponential behavior. We derive conditions on the probability
density which determine whether or not the distribution of the digits of all
the un-normalized differences converges to Benford's Law, shifted exponential
behavior, or oscillates between the two, and show that the Pareto distribution
leads to oscillating behavior.Comment: 14 pages, 2 figures, version 4: Version 3: most of the numerical
simulations on shifted exponential behavior have been suppressed (though are
available from the authors upon request). Version 4: a referee pointed out
that we need epsilon > 1/3 - delta/2 in the proof of Theorem 1.5; this has
now been adde
On the relation between the Deuteron Form Factor at High Momentum Transfer and the High Energy Neutron-Proton Scattering Amplitude
A non-relativistic potential-model version of the factorization assumption,
used in perturbative QCD calculations of hadronic form factors, is used, along
with the Born approximation valid at high energies, to derive a remarkably
simple relationship between the impulse approximation contribution to the
deuteron form factor at high momentum transfer and the high energy
neutron-proton scattering amplitude. The relation states that the form factor
at a given value of is proportional to the scattering amplitude at a
specific energy and scattering angle. This suggests that an accurate
computation of the form factors at large requires a simultaneous
description of the phase-shifts at a related energy, a statement that seems
reasonable regardless of any derivation. Our form factor-scattering amplitude
relation is shown to be accurate for some examples. However, if the potential
consists of a strong short distance repulsive term and a strong longer ranged
attractive term, as typically occurs in many realistic potentials, the relation
is found to be accurate only for ridiculously large values of . More general
arguments, using only the Schroedinger equation, suggest a strong, but
complicated, relationship between the form factor and scattering amplitude.
Furthermore, the use of recently obtained soft potentials, along with an
appropriate current operator, may allow calculations of form factors that are
consistent with the necessary phase shifts.Comment: 14 pages, 4 figures, The discussion has been extended by including
numerical examples and general argument
A Rapidly Spinning Black Hole Powers the Einstein Cross
Observations over the past 20 years have revealed a strong relationship
between the properties of the supermassive black hole (SMBH) lying at the
center of a galaxy and the host galaxy itself. The magnitude of the spin of the
black hole will play a key role in determining the nature of this relationship.
To date, direct estimates of black hole spin have been restricted to the local
Universe. Herein, we present the results of an analysis of 0.5 Ms of
archival Chandra observations of the gravitationally lensed quasar Q 2237+305
(aka the "Einstein-cross"), lying at a redshift of z = 1.695. The boost in flux
provided by the gravitational lens allows constraints to be placed on the spin
of a black hole at such high redshift for the first time. Utilizing state of
the art relativistic disk reflection models, the black hole is found to have a
spin of at the 90% confidence level. Placing a
lower limit on the spin, we find (4). The high value of
the spin for the black hole in Q 2237+305 lends
further support to the coherent accretion scenario for black hole growth. This
is the most distant black hole for which the spin has been directly constrained
to date.Comment: 5 pages, 3 figures, 1 table, formatted using emulateapj.cls. Accepted
for publication in ApJ
Thermodynamics and the Global Optimization of Lennard-Jones clusters
Theoretical design of global optimization algorithms can profitably utilize
recent statistical mechanical treatments of potential energy surfaces (PES's).
Here we analyze the basin-hopping algorithm to explain its success in locating
the global minima of Lennard-Jones (LJ) clusters, even those such as \LJ{38}
for which the PES has a multiple-funnel topography, where trapping in local
minima with different morphologies is expected. We find that a key factor in
overcoming trapping is the transformation applied to the PES which broadens the
thermodynamic transitions. The global minimum then has a significant
probability of occupation at temperatures where the free energy barriers
between funnels are surmountable.Comment: 13 pages, 13 figures, revte
The double-funnel energy landscape of the 38-atom Lennard-Jones cluster
The 38-atom Lennard-Jones cluster has a paradigmatic double-funnel energy
landscape. One funnel ends in the global minimum, a face-centred-cubic (fcc)
truncated octahedron. At the bottom of the other funnel is the second lowest
energy minimum which is an incomplete Mackay icosahedron. We characterize the
energy landscape in two ways. Firstly, from a large sample of minima and
transition states we construct a disconnectivity tree showing which minima are
connected below certain energy thresholds. Secondly we compute the free energy
as a function of a bond-order parameter. The free energy profile has two
minima, one which corresponds to the fcc funnel and the other which at low
temperature corresponds to the icosahedral funnel and at higher temperatures to
the liquid-like state. These two approaches show that the greater width of the
icosahedral funnel, and the greater structural similarity between the
icosahedral structures and those associated with the liquid-like state, are the
cause of the smaller free energy barrier for entering the icosahedral funnel
from the liquid-like state and therefore of the cluster's preferential entry
into this funnel on relaxation down the energy landscape. Furthermore, the
large free energy barrier between the fcc and icosahedral funnels, which is
energetic in origin, causes the cluster to be trapped in one of the funnels at
low temperature. These results explain in detail the link between the
double-funnel energy landscape and the difficulty of global optimization for
this cluster.Comment: 12 pages, 11 figures, revte
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