Formal series and numerical integrators: some history and some new techniques

This paper provides a brief history of B-series and the associated Butcher group and presents the new theory of word series and extended word series. B-series (Hairer and Wanner 1976) are formal series of functions parameterized by rooted trees. They greatly simplify the study of Runge-Kutta schemes and other numerical integrators. We examine the problems that led to the introduction of B-series and survey a number of more recent developments, including applications outside numerical mathematics. Word series (series of functions parameterized by words from an alphabet) provide in some cases a very convenient alternative to B-series. Associated with word series is a group G of coefficients with a composition rule simpler than the corresponding rule in the Butcher group. From a more mathematical point of view, integrators, like Runge-Kutta schemes, that are affine equivariant are represented by elements of the Butcher group, integrators that are equivariant with respect to arbitrary changes of variables are represented by elements of the word group G.Comment: arXiv admin note: text overlap with arXiv:1502.0552

A Stroboscopic Numerical Method for Highly Oscillatory Problems

International audienceWe suggest a method for the integration of highly oscillatory systems with a single high frequency. The new method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging. The technique may be easily implemented in combination with standard software and may be applied with variable step sizes. Numerical experiments show that the suggested algorithms may be substantially more efficient than standard numerical integrators

Higher-order averaging, formal series and numerical integration II: the quasi-periodic case

International audienceThe paper considers non-autonomous oscillatory systems of ordinary differential equations with d>1 non-resonant constant frequencies. Formal series like those used nowadays to analyze the properties of numerical integrators are employed to construct higher-order averaged systems and the required changes of variables. With the new approach, the averaged system and the change of variables consist of vector-valued functions that may be written down immediately and scalar coefficients that are universal in the sense that they do not depend on the specific system being averaged and may therefore be computed once and for all. The new method may be applied to obtain a variety of averaged systems. In particular we study the quasi-stroboscopic averaged system characterized by the property that the true oscillatory solution and the averaged solution coincide at the initial time. We show that quasi-stroboscopic averaging is a geometric procedure because it is independent of the particular choice of co-ordinates used to write the given system. As a consequence, quasi-stroboscopic averaging of a canonical Hamiltonian (resp. of a divergence-free) system results in a canonical (resp. in a divergence-free) averaged system. We also study the averaging of a family of near-integrable systems where our approach may be used to construct explicitly d formal first integrals for both the given system and its quasi-stroboscopic averaged version. As an application we construct three first integrals of a system that arises as a nonlinear perturbation of five coupled harmonic oscillators with one slow frequency and four resonant fast frequencies

A formal series approach to averaging: exponentially small error estimates

International audienceThe techniques, based on formal series and combinatorics, used nowadays to analyze numerical integrators may be applied to perform high-order averaging in oscillatory periodic or quasi-periodic dynamical systems. When this approach is employed, the averaged system may be written in terms of (i) scalar coefficients that are universal, i.e. independent of the system under consideration and (ii) basis functions that may be written in an explicit, systematic way in terms of the derivatives of the Fourier coefficients of the vector field being averaged. The coefficients may be recursively computed in a simple fashion. We show that this approach may be used to obtain exponentially small error estimates, as those first derived by Neishtadt. All the constants that feature in the estimates have a simple explicit expression

Bleeding Follicular Conjunctivitis due to Influenza H1N1 Virus

Influenza H1N1 or A virus is a new virus serotype capable of human-to-human transmission. This infection causes a flu syndrome similar to that of seasonal influenza, with only one case of conjunctivitis described and no clinical details or microbiological confirmation. Its diagnosis is performed by PCR of pharyngeal smear of the patients affected. We report the first well-documented case in the medical literature of conjunctivitis by H1N1 virus

Numerical stroboscopic averaging for ODEs and DAEs

International audienceThe stroboscopic averaging method (SAM) is a technique for the integration of highly oscillatory differential systems dy/dt = f(y; t) with a single high frequency. The method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging which constructs an averaged differential system dY/dt = F(Y ) whose solutions Y interpolate the sought highly oscillatory solutions y. SAM integrates numerically the averaged system without using the analytic expression of F; all information on F required by the algorithm is gathered on the fly by numerically integrating the originally given system in small time windows. SAM may be easily implemented in combination with standard software and may be applied with variable step sizes. Furthermore it may also be used successfully to integrate oscillatory DAEs. The paper provides an analytic and experimental study of SAM and two related techniques: the LISP algorithms of Kirchgraber and multirevolution methods

Assessing population-sampling strategies for reducing the COVID-19 incidence

As long as critical levels of vaccination have not been reached to ensure heard immunity, and new SARS-CoV-2 strains are developing, the only realistic way to reduce the infection speed in a population is to track the infected individuals before they pass on the virus. Testing the population via sampling has shown good results in slowing the epidemic spread. Sampling can be implemented at different times during the epidemic and may be done either per individual or for combined groups of people at a time. The work we present here makes two main contributions. We first extend and refine our scalable agent-based COVID-19 simulator to incorporate an improved socio-demographic model which considers professions, as well as a more realistic population mixing model based on contact matrices per country. These extensions are necessary to develop and test various sampling strategies in a scenario including the 62 largest cities in Spain; this is our second contribution. As part of the evaluation, we also analyze the impact of different parameters, such as testing frequency, quarantine time, percentage of quarantine breakers, or group testing, on sampling efficacy. Our results show that the most effective strategies are pooling, rapid antigen test campaigns, and requiring negative testing for access to public areas. The effectiveness of all these strategies can be greatly increased by reducing the number of contacts for infected individual.This work has been supported by the Carlos III Institute of Health under the project grant 2020/00183/001, the project grant BCV-2021-1-0011, of the Spanish Supercomputing Network (RES) and the European Union's Horizon 2020 JTI-EuroHPC research and innovation program under grant agreement No 956748. The role of all study sponsors was limited to financial support and did not imply participation of any kind in the study and collection, analysis, and interpretation of data, nor in the writing of the manuscript.S

Potassium fertilisation and the thermal behaviour of Cynara cardunculus L.

Herbaceous biomass like Cynara is commonly high in potassium, chlorine and ash, which has been reported as a source of problems for combustion applications. An appropriate management of the potassium fertilisation is suggested as a way of improving the quality of the Cynara biomass for solid fuel applications. In this work a factorial experiment was designed involving two types of fertilisers, KCl and K2SO4, and two K rates, in order to study the effect of potassium fertilisation on the composition and thermal behaviour of Cynara biomass. The results proved that the potassium content of Cynara biomass increases with the potassium fertilisation. The thermogravimetric study showed that sintering phenomena can be expected at temperatures higher than 900 °C when the crop has been highly K-fertilised, irrespective of the type of fertiliser used, KCl or K2SO4. However, the SEM images taken of samples of the four K treatments of this experiment did not reveal signs of ash melting, although some particles with crystalline appearance appeared in the samples from highly K-fertilised treatments

Structure of stratlingite and effect of hydration methodology on microstructure

Stratlingite, Ca4Al2(OH)12[AlSi(OH)8]2•2H2O, is an AFm phase which appears as hydration product of aluminum-rich cements. These binders may be calcium aluminate cements, calcium sulfoaluminate cements and also Belite Calcium Sulfo-Aluminate (BCSA) cements. The structure of stratlingite is known from single crystal studies of tiny minerals but their bulk formation, crystal structure and microstructure of powders is poorly understood. Here, we report the synthesis of stratlingite and a complete structural and microstructural characterization by synchrotron X-ray powder diffraction, nuclear magnetic resonance, scanning electron microscopy and thermal analyses. The structural and microstructural models have important implications for a correct quantitative phase analysis of stratlingite in cement pastes (for instance, in pastes of BCSA cements). The microstructure of stratlingite formed in cement pastes is highly dependent on the hydration conditions. In BCSA pastes, the (003) line position of stratlingite appears slightly shifted towards higher diffracting angles (lower inter-layered distance) after stopping hydration compared to that of a similar phase present in a paste analyzed without stopping hydration. This is related to dehydration and disorder. This shift and peak broadening is even larger when the paste has suffered partial dehydration during curing (apart from stopping hydration). A microstructural study is reportedThis work has been supported by Junta de Andalucía through P11-FQM-07517 research project. I. Santacruz thanks a Ramón y Cajal fellowship, RYC-2008-03523. Synchrotron experiments were performed in MSPD beamline at ALBA Synchrotron Light Facility with the collaboration of François Fauth