On high brightness temperature of pulsar giant pulses

A wide range of events observed at the giant pulses (high energy density, observed localization of giant pulses (GPs) relative to the average pulse, fine structure of GPs with duration up to some nanoseconds, observed circular polarization of GPs, correlation between the GP phase and the phase of the hard pulsar emission X-ray and gamma) can be explained from the viewpoint that the internal polar gap is a cavity-resonator stimulated by discharges and radiating through the breaks in the magnetosphere. The new results in this field [the electromagnetic (em) waves generation in the gap in the process of longitudinal acceleration in the electric field vanishing on the star surface, high frequency break in the spectrum as a result of switching off this generation, formation in this process a power-low spectrum with a high frequency (hf) break, the possibility determination of pulsar magnetic field by the hf break position, the difference between main pulse and inter pulse mechanism generation, quantization of em tornado rotation in the gap and appearance of the bands in the inter pulse spectrum, influence the high energy density in the gap on pair generation and position of the dead line in pulsars] are added in the intermediate epilogue.Comment: 14 pages, 2 Postscript figures; added Appendix D (Intermediate Epilogue) with 20 references. The 8th International Conference on Physics of Neutron Stars in Saint-Petersburg,2008. Printed in Journal of Physical Science and Application 5 (2015) 48-6

The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves

We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through "congruence" subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f(c,d)=c^2+d^2, but restrict (c,d) to the orbit O = (0,1).Gamma, where Gamma is an infinite-index non-elementary finitely-generated subgroup of SL(2,Z). Assume that the Reimann surface Gamma\H^2 has a cusp at infinity. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R>4/(delta-theta), where theta>1/2 is the spectral gap and delta<1 is the Hausdorff dimension of the limit set of Gamma. If delta>149/150, then we can take theta=5/6, giving R=25. The limit of this method is R=9 for delta-theta>4/9. This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture.Comment: 33 pages, 1 figure, minor corrections. To appear, Duke Math

Magnetic properties of periodic nonuniform spin-1/2 XXXX chains in a random Lorentzian transverse field

Using continued fractions we examine the density of states, transverse magnetization and static transverse linear susceptibility of a few periodic nonuniform spin-1/2 $XX$ chains in a random Lorentzian transverse field.Comment: 3 figure

Benford's Law, Values of L-functions and the 3x+1 Problem

We show the leading digits of a variety of systems satisfying certain conditions follow Benford's Law. For each system proving this involves two main ingredients. One is a structure theorem of the limiting distribution, specific to the system. The other is a general technique of applying Poisson Summation to the limiting distribution. We show the distribution of values of L-functions near the central line and (in some sense) the iterates of the 3x+1 Problem are Benford.Comment: 25 pages, 1 figure; replacement of earlier draft (corrected some typos, added more exposition, added results for characteristic polynomials of unitary matrices