30 research outputs found
On the ratio of consecutive gaps between primes
In the present work we prove a common generalization of Maynard-Tao's recent
result about consecutive bounded gaps between primes and on the
Erd\H{o}s-Rankin bound about large gaps between consecutive primes. The work
answers in a strong form a 60 years old problem of Erd\"os, which asked whether
the ratio of two consecutive primegaps can be infinitely often arbitrarily
small, and arbitrarily large, respectively
Primes in short segments of arithmetic progressions
Consider the variance for the number of primes that are both in the interval [y, y + h] for y ∈ [x, 2x] and in an arithmetic progression of modulus q. We study the total variance obtained by adding these variances over all the reduced residue classes modulo q. Assuming a strong form of the twin prime conjecture and the Riemann Hypothesis one can obtain an asymptotic formula for the total variance in the range when 1 ≤ h / q ≤ x1/2-∈, for any ∈ > 0. We show that one can still obtain some weaker asymptotic results assuming the Generalized Riemann Hypothesis (GRH) in place of the twin prime conjecture. In their simplest form, our results are that on GRH the same asymptotic formula obtained with the twin prime conjecture is true for "almost all" q in the range 1 ≤ h / q ≤ h1/4-∈, that on averaging over q one obtains an asymptotic formula in the extended range 1 ≤ h / q ≤ h1/2-∈, and that there are lower bounds with the correct order of magnitude for all q in the range 1 ≤ h/q ≤ x1/3-∈
Patterns of primes in arithmetic progressions
After the proof of Zhang about the existence of infinitely many bounded gaps between consecutive primes the author showed the existence of a bounded d such that there are arbitrarily long arithmetic progressions of primes with the property that p′ = p + d is the prime following p for each element of the progression. This was a common generalization of the results of Zhang and Green-Tao. In the present work it is shown that for every m we have a bounded m-tuple of primes such that this configuration (i.e. the integer translates of this m-tuple) appear as arbitrarily long arithmetic progressions in the sequence of all primes. In fact we show that this is true for a positive proportion of all m-tuples. This is a common generalization of the celebrated works of Green-Tao and Maynard/Tao.
Dedicated to the 60th birthday of Robert F. Tich
Bandlimited approximations to the truncated Gaussian and applications
In this paper we extend the theory of optimal approximations of functions in the -metric by entire functions of prescribed
exponential type (bandlimited functions). We solve this problem for the
truncated and the odd Gaussians using explicit integral representations and
fine properties of truncated theta functions obtained via the maximum principle
for the heat operator. As applications, we recover most of the previously known
examples in the literature and further extend the class of truncated and odd
functions for which this extremal problem can be solved, by integration on the
free parameter and the use of tempered distribution arguments. This is the
counterpart of the work \cite{CLV}, where the case of even functions is
treated.Comment: to appear in Const. Appro
Exploration of the equilibrium operating space for NSTX-Upgrade
This paper explores a range of high-performance equilibrium scenarios available in the NSTX-Upgrade device [J.E. Menard, submitted for publication to Nuclear Fusion]. NSTX-Upgrade is a substantial upgrade to the existing NSTX device [M. Ono, et al., Nuclear Fusion 40, 557 (2000)], with significantly higher toroidal field and solenoid capabilities, and three additional neutral beam sources with significantly larger current drive efficiency. Equilibria are computed with freeboundary TRANSP, allowing a self consistent calculation of the non-inductive current drive sources, the plasma equilibrium, and poloidal field coil current, using the realistic device geometry. The thermal profiles are taken from a variety of existing NSTX discharges, and different assumptions for the thermal confinement scalings are utilized. The no-wall and idealwall n=1 stability limits are computed with the DCON code. The central and minimum safety factors are quite sensitive to many parameters: they generally increases with large outer plasmawall gaps and higher density, but can have either trend with the confinement enhancement factor. In scenarios with strong central beam current drive, the inclusion of non-classical fast ion diffusion raises qmin, decreases the pressure peaking, and generally improves the global stability, at the expense of a reduction in the non-inductive current drive fraction; cases with less beam current drive are largely insensitive to additional fast ion diffusion. The non-inductive current level is quite sensitive to the underlying confinement and profile assumptions. For instance, for BT=1.0 T and Pinj=12.6 MW, the non-inductive current level varies from 875 kA with ITER-98y,2 thermal confinement scaling and narrow thermal profiles to 1325 kA for an ST specific scaling expression and broad profiles. This sensitivity should facilitate the determination of the correct scaling of transport with current and field to use for future fully non-inductive ST devices. Scenarios are presented which can be sustained for 8-10 seconds, or (20-30)τCR, at βN=3.8-4.5, facilitating, for instance, the study of disruption avoidance for very long pulse. Scenarios have been documented which can operate with βT~25% and equilibrated qmin>1. The value of qmin can be controlled at either fixed non-inductive fraction of 100% or fixed plasma current, by varying which beam sources are used, opening the possibility for feedback qmin control. In terms of quantities like collisionality, neutron emission, non-inductive fraction, or stored energy, these scenarios represent a significant performance extension compared to NSTX and other present spherical torii