Chirikov and Nekhoroshev diffusion estimates: bridging the two sides of the river

We present theoretical and numerical results pointing towards a strong connection between the estimates for the diffusion rate along simple resonances in multidimensional nonlinear Hamiltonian systems that can be obtained using the heuristic theory of Chirikov and a more formal one due to Nekhoroshev. We show that, despite a wide-spread impression, the two theories are complementary rather than antagonist. Indeed, although Chirikov's 1979 review has thousands of citations, almost all of them refer to topics such as the resonance overlap criterion, fast diffusion, the Standard or Whisker Map, and not to the constructive theory providing a formula to measure diffusion along a single resonance. However, as will be demonstrated explicitly below, Chirikov's formula provides values of the diffusion coefficient which are quite well comparable to the numerically computed ones, provided that it is implemented on the so-called optimal normal form derived as in the analytic part of Nekhoroshev's theorem. On the other hand, Chirikov's formula yields unrealistic values of the diffusion coefficient, in particular for very small values of the perturbation, when used in the original Hamiltonian instead of the optimal normal form. In the present paper, we take advantage of this complementarity in order to obtain accurate theoretical predictions for the local value of the diffusion coefficient along a resonance in a specific 3DoF nearly integrable Hamiltonian system. Besides, we compute numerically the diffusion coefficient and a full comparison of all estimates is made for ten values of the perturbation parameter, showing a very satisfactory agreement.Comment: 25 pages, 9 figures. NOTICE: this is the author's version of a work that was accepted for publication in Physica D. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publicatio

Stochastic approach to diffusion inside the chaotic layer of a resonance

We model chaotic diffusion, in a symplectic 4D map by using the result of a theorem that was developed for stochastically perturbed integrable Hamiltonian systems. We explicitly consider a map defined by a free rotator (FR) coupled to a standard map (SM). We focus in the diffusion process in the action, $I$, of the FR, obtaining a semi--numerical method to compute the diffusion coefficient. We study two cases corresponding to a thick and a thin chaotic layer in the SM phase space and we discuss a related conjecture stated in the past. In the first case the numerically computed probability density function for the action $I$ is well interpolated by the solution of a Fokker-Planck (F-P) equation, whereas it presents a non--constant time delay respect to the concomitant F-P solution in the second case suggesting the presence of an anomalous diffusion time scale. The explicit calculation of a diffusion coefficient for a 4D symplectic map can be useful to understand the slow diffusion observed in Celestial Mechanics and Accelerator Physics.Comment: This is the author's version of a work that was submitted to Physical Review E (http://pre.aps.org

Elliptic curves of large rank and small conductor

For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search methods, and tabulate, for each r=5,6,...,11, the five curves of lowest conductor, and (except for r=11) also the five of lowest absolute discriminant, that we found.Comment: 16 pages, including tables and one .eps figure; to appear in the Proceedings of ANTS-6 (June 2004, Burlington, VT). Revised somewhat after comments by J.Silverman on the previous draft, and again to get the correct page break

CANreduce-SP—adding psychological support to web-based adherence-focused guided self-help for cannabis users: study protocol for a three-arm randomized control trial

Background: Cannabis is the most-frequently used illicit drug in Europe. Over the last few years in Spain, treatment demand has increased, yet most cannabis users do not seek treatment despite the related problems. A web-based self-help tool, like CANreduce 2.0, could help these users to control their consumption. Methods: This study protocol describes a three-arm randomized controlled trial (RCT) comparing the effectiveness of three approaches, in terms of reducing cannabis use among problematic cannabis users, the first two treatment arms including the Spanish version of CANreduce 2.0 (an adherence-focused, guidance-enhanced, web-based self-help tool) (1) with and (2) without psychological support; and the third group (3) treatment as usual (TAU). Study hypotheses will be tested concerning the primary outcome: change in the number of days of cannabis use over the previous week, comparing assessments at 6 weeks and 3 and 6 months follow-up between groups and against baseline. Secondary outcomes related to cannabis use will be tested similarly. Mental disorders will be explored as predictors of adherence and outcomes. Analyses will be performed on an intention-to-treat basis, then verified by complete case analyses. Discussion: This study will test how effective the Spanish version of CANreduce 2.0 (CANreduce-SP) is at reducing both the frequency and quantity of cannabis use in problematic users and whether adding psychological support increases its effectiveness. Trial registration: This trial is registered with the Clinical Trials Protocol Registration and Results System (PRS) number: NCT04517474 . Registered 18 August 2020, (Archived by archive.is https://archive.is/N1Y64 ). The project commenced in November 2020 and recruitment is anticipated to end by November 2022. Keywords: Adherence; CANreduce; Cannabis use disorder; Cognitive behavioural therapy; Guidance; Psychological support; Randomized controlled trial; Reducing cannabis; Self-help too

Testing Hardy nonlocality proof with genuine energy-time entanglement

We show two experimental realizations of Hardy ladder test of quantum nonlocality using energy-time correlated photons, following the scheme proposed by A. Cabello \emph{et al.} [Phys. Rev. Lett. \textbf{102}, 040401 (2009)]. Unlike, previous energy-time Bell experiments, these tests require precise tailored nonmaximally entangled states. One of them is equivalent to the two-setting two-outcome Bell test requiring a minimum detection efficiency. The reported experiments are still affected by the locality and detection loopholes, but are free of the post-selection loophole of previous energy-time and time-bin Bell tests.Comment: 5 pages, revtex4, 6 figure