Analytical method for perturbed frozen orbit around an Asteroid in highly inhomogeneous gravitational fields : A first approach

This article provides a method for nding initial conditions for perturbed frozen orbits around inhomogeneous fast rotating asteroids. These orbits can be used as reference trajectories in missions that require close inspection of any rigid body. The generalized perturbative procedure followed exploits the analytical methods of relegation of the argument of node and Delaunay normalisation to arbitrary order. These analytical methods are extremely powerful but highly computational. The gravitational potential of the heterogeneous body is rstly stated, in polar-nodal coordinates, which takes into account the coecients of the spherical harmonics up to an arbitrary order. Through the relegation of the argument of node and the Delaunay normalization, a series of canonical transformations of coordinates is found, which reduces the Hamiltonian describing the system to a integrable, two degrees of freedom Hamiltonian plus a truncated reminder of higher order. Setting eccentricity, argument of pericenter and inclination of the orbit of the truncated system to be constant, initial conditions are found, which evolve into frozen orbits for the truncated system. Using the same initial conditions yields perturbed frozen orbits for the full system, whose perturbation decreases with the consideration of arbitrary homologic equations in the relegation and normalization procedures. Such procedure can be automated for the first homologic equation up to the consideration of any arbitrary number of spherical harmonics coefficients. The project has been developed in collaboration with the European Space Agency (ESA)

Kustaanheimo-Stiefel Regularization and the Quadrupolar Conjugacy

In this note, we present the Kustaanheimo-Stiefel regularization in a symplectic and quaternionic fashion. The bilinear relation is associated with the moment map of the $S^{1}$- action of the Kustaanheimo-Stiefel transformation, which yields a concise proof of the symplecticity of the Kustaanheimo-Stiefel transformation symplectically reduced by this circle action. The relation between the Kustaanheimo-Stiefel regularization and the Levi-Civita regularization is established via the investigation of the Levi-Civita planes. A set of Darboux coordinates (which we call Chenciner-F\'ejoz coordinates) is generalized from the planar case to the spatial case. Finally, we obtain a conjugacy relation between the integrable approximating dynamics of the lunar spatial three-body problem and its regularized counterpart, similar to the conjugacy relation between the extended averaged system and the averaged regularized system in the planar case.Comment: 19 pages, corrected versio

Volume-preserving normal forms of Hopf-zero singularity

A practical method is described for computing the unique generator of the algebra of first integrals associated with a large class of Hopf-zero singularity. The set of all volume-preserving classical normal forms of this singularity is introduced via a Lie algebra description. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a non-zero condition on their quadratic parts are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any non-degenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are derived. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and applied to appropriately modified R\"{o}ssler and generalized Kuramoto--Sivashinsky equations to demonstrate the applicability of our theoretical results. An approach (introduced by Iooss and Lombardi) is applied to find an optimal truncation for the first level normal forms of these examples with exponentially small remainders. The numerically suggested radius of convergence (for the first integral) associated with a hypernormalization step is discussed for the truncated first level normal forms of the examples. This is achieved by an efficient implementation of the results using Maple

LÍmite infinito de sucesiones en libros de texto espaÑoles: Desde 1936 hasta 2019

En este trabajo se realiza la búsqueda y el análisis de fenómenos organizados por el límite infinito de una sucesión en 35 libros de texto españoles de matemáticas editados desde el año 1936 hasta el año 2019. A partir de la construcción de un instrumento basado en los fenómenos caracterizados para el límite infinito de una sucesión y el análisis de la información recogida por este, se establece la identificación de estos fenómenos en los diferentes sistemas de representación y formatos. La evolución histórica de estos fenómenos y la comparación entre algunos periodos legislativos muestra que dichos fenómenos se presentan de manera aislada sin utilizar diferentes sistemas de representación y formatos, dificultando con ello su proceso de enseñanza-aprendizaje. In this paper, we carry out the research and the analysis of phenomena organized by the infinite limit of a sequence in 35 Spanish textbooks of Mathematics published from 1936 to 2019. From a construction of an instrument based on the phenomena characterized by an infinite limit of a sequence and the analysis of the information collected by it, we establish the identification of these phenomena in a different way of representation and formats. The historical evolution of these phenomena and the comparison between some of legislative periods shows these phenomena are presented in an isolated way without using different systems of representation and formats, making difficult with it their teaching-learning process

Aspects of the planetary Birkhoff normal form

The discovery in [G. Pinzari. PhD thesis. Univ. Roma Tre. 2009], [L. Chierchia and G. Pinzari, Invent. Math. 2011] of the Birkhoff normal form for the planetary many--body problem opened new insights and hopes for the comprehension of the dynamics of this problem. Remarkably, it allowed to give a {\sl direct} proof of the celebrated Arnold's Theorem [V. I. Arnold. Uspehi Math. Nauk. 1963] on the stability of planetary motions. In this paper, using a "ad hoc" set of symplectic variables, we develop an asymptotic formula for this normal form that may turn to be useful in applications. As an example, we provide two very simple applications to the three-body problem: we prove a conjecture by [V. I. Arnold. cit] on the "Kolmogorov set"of this problem and, using Nehoro{\v{s}}ev Theory [Nehoro{\v{s}}ev. Uspehi Math. Nauk. 1977], we prove, in the planar case, stability of all planetary actions over exponentially-long times, provided mean--motion resonances are excluded. We also briefly discuss perspectives and problems for full generalization of the results in the paper.Comment: 44 pages. Keywords: Averaging Theory, Birkhoff normal form, Nehoro{\v{s}}ev Theory, Planetary many--body problem, Arnold's Theorem on the stability of planetary motions, Properly--degenerate kam Theory, steepness. Revised version, including Reviewer's comments. Typos correcte

Penicillin susceptibility among invasive MSSA infections: a multicentre study in 16 Spanish hospitals

Objectives: To determine the prevalence of penicillin susceptibility among MSSA causing bloodstream infections (BSIs) in 16 Spanish hospitals and to characterize the penicillin-susceptible MSSA (MSSA-PENS) isolates. Methods: A total of 1011 Staphylococcus aureus isolates were collected from blood cultures in 16 Spanish hospitals during 2018–19 (6–12 months) and their susceptibility to 18 antimicrobials was determined. The MSSA-PENS isolates were selected and examined by PCR to determine the presence of the blaZ gene, other resistance genes and the genes lukF/lukS-PV, eta, etb and tst. The immune evasion cluster (IEC) type was also analysed. All the MSSA-PENS isolates were submitted to S. aureus protein A (spa) typing and the clonal complexes (CCs) were assigned according to their spa type. Results: The prevalence of MSSA was 74.6% (754/1011) and 14.9% (151/1011) were MSSA-PENS-blaZnegative. MSSA-PENS-blaZnegative isolates (n = 151) were ascribed to 88 spa types and 11 CCs. The most frequent CCs were CC5 (35/151) and CC398 (25/151), with t002-CC5 and t571-CC398 being the most common lineages. Pan-susceptibility was identified in 117 of the 151 MSSA-PENS-blaZnegative isolates (77.5%). In the remaining isolates, erythromycin and clindamycin resistance was the most frequent resistance found, although tobramycin, ciprofloxacin, fusidic acid, mupirocin and/or tetracycline resistance was also detected. Thirty-eight MSSA-PENS-blaZnegative isolates were IEC negative and four isolates were Panton–Valentine leucocidin (‘PVL’) positive. Conclusions: A high penicillin susceptibility rate was detected among MSSA, opening therapeutic opportunities for BSIs. The emergence of new successful MSSA-PENS clones could be responsible for these data. The detection among MSSA-PENS-blaZnegative isolates of the clonal lineage CC398 or the absence of an IEC raises questions about their possible animal origin, requiring further analysis

Role of age and comorbidities in mortality of patients with infective endocarditis

[Purpose]: The aim of this study was to analyse the characteristics of patients with IE in three groups of age and to assess the ability of age and the Charlson Comorbidity Index (CCI) to predict mortality. [Methods]: Prospective cohort study of all patients with IE included in the GAMES Spanish database between 2008 and 2015.Patients were stratified into three age groups:<65 years,65 to 80 years,and ≥ 80 years.The area under the receiver-operating characteristic (AUROC) curve was calculated to quantify the diagnostic accuracy of the CCI to predict mortality risk. [Results]: A total of 3120 patients with IE (1327 < 65 years;1291 65-80 years;502 ≥ 80 years) were enrolled.Fever and heart failure were the most common presentations of IE, with no differences among age groups.Patients ≥80 years who underwent surgery were significantly lower compared with other age groups (14.3%,65 years; 20.5%,65-79 years; 31.3%,≥80 years). In-hospital mortality was lower in the <65-year group (20.3%,<65 years;30.1%,65-79 years;34.7%,≥80 years;p < 0.001) as well as 1-year mortality (3.2%, <65 years; 5.5%, 65-80 years;7.6%,≥80 years; p = 0.003).Independent predictors of mortality were age ≥ 80 years (hazard ratio [HR]:2.78;95% confidence interval [CI]:2.32–3.34), CCI ≥ 3 (HR:1.62; 95% CI:1.39–1.88),and non-performed surgery (HR:1.64;95% CI:11.16–1.58).When the three age groups were compared,the AUROC curve for CCI was significantly larger for patients aged <65 years(p < 0.001) for both in-hospital and 1-year mortality. [Conclusion]: There were no differences in the clinical presentation of IE between the groups. Age ≥ 80 years, high comorbidity (measured by CCI),and non-performance of surgery were independent predictors of mortality in patients with IE.CCI could help to identify those patients with IE and surgical indication who present a lower risk of in-hospital and 1-year mortality after surgery, especially in the <65-year group