19,189 research outputs found
Linear correlations amongst numbers represented by positive definite binary quadratic forms
Given a positive definite binary quadratic form f, let r(n) = |{(x,y):
f(x,y)=n}| denote its representation function. In this paper we study linear
correlations of these functions. For example, if r_1, ..., r_k are
representation functions, we obtain an asymptotic for sum_{n,d} r_1(n) r_2(n+d)
... r_k(n+ (k-1)d).Comment: 60 pages. Small correction
Lower Bounds on Mutual Information
We correct claims about lower bounds on mutual information (MI) between
real-valued random variables made in A. Kraskov {\it et al.}, Phys. Rev. E {\bf
69}, 066138 (2004). We show that non-trivial lower bounds on MI in terms of
linear correlations depend on the marginal (single variable) distributions.
This is so in spite of the invariance of MI under reparametrizations, because
linear correlations are not invariant under them. The simplest bounds are
obtained for Gaussians, but the most interesting ones for practical purposes
are obtained for uniform marginal distributions. The latter can be enforced in
general by using the ranks of the individual variables instead of their actual
values, in which case one obtains bounds on MI in terms of Spearman correlation
coefficients. We show with gene expression data that these bounds are in
general non-trivial, and the degree of their (non-)saturation yields valuable
insight.Comment: 4 page
Exploring Minimal Scenarios to Produce Transversely Bright Electron Beams Using the Eigen-Emittance Concept
Next generation hard X-ray free electron lasers require electron beams with
low transverse emittance. One proposal to achieve these low emittances is to
exploit the eigen-emittance values of the beam. The eigen-emittances are
invariant under linear beam transport and equivalent to the emittances in an
uncorrelated beam. If a correlated beam with two small eigen-emittances can be
produced, removal of the correlations via appropriate optics will lead to two
small emittance values, provided non-linear effects are not too large. We study
how such a beam may be produced using minimal linear correlations. We find it
is theoretically possible to produce such a beam, however it may be more
difficult to realize in practice. We identify linear correlations that may lead
to physically realizable emittance schemes and discuss promising future
avenues.Comment: 7 pages, 2 figures, to appear in NIM
Possibilities for reduction of transverse projected emittances by partial removal of transverse to longitudinal beam correlations
We show that if in the particle beam there are linear correlations between
energy of particles and their transverse positions and momenta (linear beam
dispersions), then the transverse projected emittances always can be reduced by
letting the beam to pass through magnetostatic system with specially chosen
nonzero lattice dispersions. The maximum possible reduction of the transverse
projected emittances occurs when all beam dispersions are zeroed, and the
values of the lattice dispersions required for that are completely defined by
the values of the beam dispersions and the beam rms energy spread and are
independent from any other second-order central beam moments. Besides that, we
prove that, alternatively, one can also use the lattice dispersions to remove
linear correlations between longitudinal positions of particles and their
transverse coordinates (linear beam tilts), but in this situation solution for
the lattice dispersions is nonunique and the reduction of the transverse
projected emittances is not guaranteed.Comment: 13 pages, 2 figure
Correlations of the divisor function
In this paper we study linear correlations of the divisor function tau(n) =
sum_{d|n} 1 using methods developed by Green and Tao. For example, we obtain an
asymptotic for sum_{n,d} tau(n) tau(n+d) ... tau(n+ (k-1)d).Comment: 33 pages. Corrections and journal referenc
Linear correlations between 4He trimer and tetramer energies calculated with various realistic 4He potentials
In a previous work [Phys. Rev. A 85, 022502 (2012)] we calculated, with the
use of our Gaussian expansion method for few-body systems, the energy levels
and spatial structure of the 4He trimer and tetramer ground and excited states
using the LM2M2 potential, which has a very strong short-range repulsion. In
this work, we calculate the same quantities using the presently most accurate
4He-4He potential [M. Przybytek et al., Phys. Rev. Lett. 104, 183003 (2010)]
that includes the adiabatic, relativistic, QED and residual retardation
corrections. Contributions of the corrections to the tetramer
ground-(excited-)state energy, -573.90 (-132.70) mK, are found to be,
respectively, -4.13 (-1.52) mK, +9.37 (+3.48) mK, -1.20 (-0.46) mK and +0.16
(+0.07) mK. Further including other realistic 4He potentials, we calculated the
binding energies of the trimer and tetramer ground and excited states, B_3^(0),
B_3^(1), B_4^(0) and B_4^(1), respectively. We found that the four kinds of the
energies for the different potentials exhibit perfect linear correlations
between any two of them over the range of binding energies relevant for 4He
atoms (namely, six types of the generalized Tjon lines are given). The
dimerlike-pair model for 4He clusters, proposed in the previous work, predicts
a simple universal relation B_4^(1)/B_2 =B_3^(0)/B_2 + 2/3, which precisely
explains the correlation between the tetramer excited-state energy and the
trimer ground-state energy, with B_2 being the dimer binding energy.Comment: 10 pages, 3 figures, published version in Phys. Rev. A85, 062505
(2012), Figs. 2, 5, and 6 added, minor changes in the description of the
dimerlike-pair mode
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