Search results for " Mathematics"
showing 10 items of 10797 documents
Lie Algebras Generated by Extremal Elements
1999
We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals of L) over a field of characteristic distinct from 2. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the Lie algebras of type An, Bn (n>2), Cn (n>1), Dn (n>3), En (n=6,7,8), F4 and G2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.
Pietro Mengoli and the six-square problem
1994
The aim of this paper is to analyze a little known aspect of Pietro Mengoli's (1625-1686) mathematical activity: the difficulties he faced in trying to solve some problems in Diophantine analysis suggested by J. Ozanam. Mengoli's recently published correspondence reveals how he cherished his prestige as a scholar. At the same time, however, it also shows that his insufficient familiarity with algebraic methods prevented him, as well as other Italian mathematicians of his time, from solving the so-called “French” problems. Quite different was the approach used for the same problems by Leibniz, who, although likewise partially unsuccessful, demonstrated a deeper mathematical insight which led…
Introduction to Gestural Similarity in Music. An Application of Category Theory to the Orchestra
2019
Mathematics, and more generally computational sciences, intervene in several aspects of music. Mathematics describes the acoustics of the sounds giving formal tools to physics, and the matter of music itself in terms of compositional structures and strategies. Mathematics can also be applied to the entire making of music, from the score to the performance, connecting compositional structures to acoustical reality of sounds. Moreover, the precise concept of gesture has a decisive role in understanding musical performance. In this paper, we apply some concepts of category theory to compare gestures of orchestral musicians, and to investigate the relationship between orchestra and conductor, a…
A Study of the Direct Spectral Transform for the Defocusing Davey‐Stewartson II Equation the Semiclassical Limit
2019
International audience; The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrodinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a sing…
Some Applications of the Poincaré-Bendixson Theorem
2021
We consider a C 1 vector field X defined on an open subset U of the plane, with compact closure. If X has no singular points and if U is simply connected, a weak version of the Poincaré-Bendixson Theorem says that the limit sets of X in U are empty but that one can defined non empty extended limit sets contained into the boundary of U. We give an elementary proof of this result, independent of the classical Poincaré-Bendixson Theorem. A trapping triangle T based at p, for a C 1 vector field X defined on an open subset U of the plane, is a topological triangle with a corner at a point p located on the boundary ∂U and a good control of the tranversality of X along the sides. The principal app…
On presentations for mapping class groups of orientable surfaces via Poincaré's Polyhedron theorem and graphs of groups
2021
The mapping class group of an orientable surface with one boundary component, S, is isomorphic to a subgroup of the automorphism group of the fundamental group of S. We call these subgroups algebraic mapping class groups. An algebraic mapping class group acts on a space called ordered Auter space. We apply Poincaré's Polyhedron theorem to this action. We describe a decomposition of ordered Auter space. From these results, we deduce that the algebraic mapping class group of S is a quotient of the fundamental group of a graph of groups with, at most, two vertices and, at most, six edges. Vertex and edge groups of our graph of groups are mapping class groups of orientable surfaces with one, tw…
Spatio-Temporal Spread Pattern of COVID-19 in Italy
2021
This paper investigates the spatio-temporal spread pattern of COVID-19 in Italy, during the first wave of infections, from February to October 2020. Disease mappings of the virus infections by using the Besag–York–Mollié model and some spatio-temporal extensions are provided. This modeling framework, which includes a temporal component, allows the studying of the time evolution of the spread pattern among the 107 Italian provinces. The focus is on the effect of citizens’ mobility patterns, represented here by the three distinct phases of the Italian virus first wave, identified by the Italian government, also characterized by the lockdown period. Results show the effectiveness of the lockdo…
Building a statistical surveillance dashboard for COVID-19 infection worldwide
2020
When a pandemic like the current novel coronavirus (COVID-19) breaks out, it is important that authorities, healthcare organizations and official decision makers, have in place an effective monitoring system to promptly analyze data, create new insights into problematic areas and generate actionable knowledge for fact-based decision making. The aim of this article is to describe an initial work focused on building a comprehensive statistical surveillance dashboard for the epidemic of COVID-19, which can be exploited also for future needs. We propose novel ways of exploring, analyzing and presenting data, using metrics that have not been used previously. We also show the steps necessary to b…
Random resampling numerical simulations applied to a SEIR compartmental model
2021
AbstractIn this paper, we apply resampling techniques to a modified compartmental SEIR model which takes into account the existence of undetected infected people in an epidemic. In particular, we implement numerical simulations for the evolution of the first wave of the COVID-19 pandemic in Spain in 2020. We show, by using suitable measures of goodness, that the point estimates obtained by the bootstrap samples improve the ones of the original data. For example, the relative error of detected currently infected people is equal to 0.061 for the initial estimates, while it is reduced to 0.0538 for the mean over all bootstrap estimated series.
Comparison of Reported Deaths From COVID-19 and Increase in Total Mortality in Italy
2020
This analysis compares reported deaths from COVID-19, February 23 to April 4, 2020, and total mortality in Italy from January 12 through April 4 in the years 2015 through 2020.