Arithmetic theory of E-operators

In [S\'eries Gevrey de type arithm\'etique I Th\'eor\'emes de puret\'e et de dualit\'e, Annals of Math. 151 (2000), 705--740], Andr\'e has introduced E-operators, a class of differential operators intimately related to E-functions, and constructed local bases of solutions for these operators. In this paper we investigate the arithmetical nature of connexion constants of E-operators at finite distance, and of Stokes constants at infinity. We prove that they involve values at algebraic points of E-functions in the former case, and in the latter one, values of G-functions and of derivatives of the Gamma function at rational points in a very precise way. As an application, we define and study a class of numbers having certain algebraic approximations defined in terms of E-functions. These types of approximations are motivated by the convergents to the number e, as well as by recent constructions of approximations to Euler's constant and values of the Gamma function. Our results and methods are completely different from those in our paper [On the values of G-functions, Commentarii Math. Helv., to appear], where we have studied similar questions for G-functions

Multiple zeta values, Padé approximation and Vasilyev's conjecture

International audienceSorokin gave in 1996 a new proof that pi is transcendental. It is based on a simultaneous Padé approximation problem involving certain multiple polylogarithms, which evaluated at the point 1 are multiple zeta values equal to powers of pi. In this paper we construct a Padé approximation problem of the same flavour, and prove that it has a unique solution up to proportionality. At the point 1, this provides a rational linear combination of 1 and multiple zeta values in an extended sense that turn out to be values of the Riemann zeta function at odd integers. As an application, we obtain a new proof of Vasilyev's conjecture for any odd weight, concerning the explicit evaluation of certain hypergeometric multiple integrals; it was first proved by Zudilin in 2003

Arithmetic theory of E-operators

International audienceIn [Séries Gevrey de type arithmétique I Théorémes de pureté et de dualité, Annals of Math. 151 (2000), 705--740], André has introduced E-operators, a class of differential operators intimately related to E-functions, and constructed local bases of solutions for these operators. In this paper we investigate the arithmetical nature of connexion constants of E-operators at finite distance, and of Stokes constants at infinity. We prove that they involve values at algebraic points of E-functions in the former case, and in the latter one, values of G-functions and of derivatives of the Gamma function at rational points in a very precise way. As an application, we define and study a class of numbers having certain algebraic approximations defined in terms of E-functions. These types of approximations are motivated by the convergents to the number e, as well as by recent constructions of approximations to Euler's constant and values of the Gamma function. Our results and methods are completely different from those in our paper [On the values of G-functions, Commentarii Math. Helv., to appear], where we have studied similar questions for G-functions

The effects of digital predistortion in a CO-OFDM system – a stochastic approach

Digital predistortion is topic of significant interest in telecommunications – both in the wireless radio field and, more recently, in photonics. In the present letter, the authors undertake a sensitivity analysis of various digital predistortion algorithms. Using recent metamodeling techniques designed for efficient stochastic analysis, the authors show that using predistortion not only leads to a reduction of the error vector magnitude in general but can also make the system less sensitive to uncertainties

Resuming Training in High-Level Athletes After Mild COVID-19 Infection: A Multicenter Prospective Study (ASCCOVID-19)

BACKGROUND: There is a paucity of data on cardiovascular sequelae of asymptomatic/mildly symptomatic SARS-Cov-2 infections (COVID). OBJECTIVES: The aim of this prospective study was to characterize the cardiovascular sequelae of asymptomatic/mildly symptomatic COVID-19 among high/elite-level athletes. METHODS: 950 athletes (779 professional French National Rugby League (F-NRL) players; 171 student athletes) were included. SARS-Cov-2 testing was performed at inclusion, and F-NRL athletes were intensely followed-up for incident COVID-19. Athletes underwent ECG and biomarker profiling (D-Dimer, troponin, C-reactive protein). COVID(+) athletes underwent additional exercise testing, echocardiography and cardiac magnetic resonance imaging (CMR). RESULTS: 285/950 athletes (30.0%) had mild/asymptomatic COVID-19 [79 (8.3%) at inclusion (COVID(+)(prevalent)); 206 (28.3%) during follow-up (COVID(+)(incident))]. 2.6% COVID(+) athletes had abnormal ECGs, while 0.4% had an abnormal echocardiogram. During stress testing (following 7-day rest), COVID(+) athletes had a functional capacity of 12.8 ± 2.7 METS with only stress-induced premature ventricular ectopy in 10 (4.3%). Prevalence of CMR scar was comparable between COVID(+) athletes and controls [COVID(+) vs. COVID(-); 1/102 (1.0%) vs 1/28 (3.6%)]. During 289 ± 56 days follow-up, one athlete had ventricular tachycardia, with no obvious link with a SARS-CoV-2 infection. The proportion with troponin I and CRP values above the upper-limit threshold was comparable between pre- and post-infection (5.9% vs 5.9%, and 5.6% vs 8.7%, respectively). The proportion with D-Dimer values above the upper-limit threshold increased when comparing pre- and post-infection (7.9% vs 17.3%, P = 0.01). CONCLUSION: The absence of cardiac sequelae in pauci/asymptomatic COVID(+) athletes is reassuring and argues against the need for systematic cardiac assessment prior to resumption of training (clinicaltrials.gov; NCT04936503).L'Institut de Rythmologie et modélisation Cardiaqu

Measurement of the cosmic ray spectrum above 4×10184{\times}10^{18} eV using inclined events detected with the Pierre Auger Observatory

A measurement of the cosmic-ray spectrum for energies exceeding $4{\times}10^{18}$ eV is presented, which is based on the analysis of showers with zenith angles greater than $60^{\circ}$ detected with the Pierre Auger Observatory between 1 January 2004 and 31 December 2013. The measured spectrum confirms a flux suppression at the highest energies. Above $5.3{\times}10^{18}$ eV, the "ankle", the flux can be described by a power law $E^{-\gamma}$ with index $\gamma=2.70 \pm 0.02 \,\text{(stat)} \pm 0.1\,\text{(sys)}$ followed by a smooth suppression region. For the energy ($E_\text{s}$) at which the spectral flux has fallen to one-half of its extrapolated value in the absence of suppression, we find $E_\text{s}=(5.12\pm0.25\,\text{(stat)}^{+1.0}_{-1.2}\,\text{(sys)}){\times}10^{19}$ eV.Comment: Replaced with published version. Added journal reference and DO