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