Zeta and L-functions of elliptic curves

In questa tesi si studiano alcune proprietà fondamentali delle funzioni Zeta e L associate ad una curva ellittica. In particolare, si dimostra la razionalità della funzione Zeta e l'ipotesi di Riemann per due famiglie specifiche di curve ellittiche. Si studia poi il problema dell'esistenza di un prolungamento analitico al piano complesso della funzione L di una curva ellittica con moltiplicazione complessa, attraverso l'analisi diretta di due casi particolari.

Prokaryotic community structure in ultra-slow spreading Southwest Indian Ridge.

The Southwest Indian Ridge segment that extends between 10° and 16° E has the slowest spreading rate of any other oceanic ridge (about 8.4 mm/year). In 2013 during the expedition ANTXXIX/8 seismology, geology, microbiology, heat flow analyses were carried out. Here, no hydrothermal plumes or black smoker systems were found but the results of the survey allowed to identify areas with peculiar characteristics: Area 1 with higher heat flux bsf; Area 2 where in 2002 the presence of hydrothermal emissions was hypothesized (Bach et al., 2002); Area 3 with anomalies of methane, ammonium, sulphide and dissolved inorganic carbon in pore water sediment profiles, and recovery of fauna vents. All these aspects suggest the presence of a hydrothermal circulation. Using Illumina 16S gene tag, statistical tools and phylogenetic trees, I provided a biological proof of the presence of hydrothermal circulation in this ridge segment. At Area 3, alpha and beta diversity indexes showed similarities with those described for venting microbial communities and about 40-70% of the dominant microbial community was found phylogenetically related to clones isolated hydrothermal-driven environments. Although the majority of chemosynthetic environment related taxa were not classified like autotrophic prokaryotes, some of them are key taxa in support of the presence of hydrothermal circulation, since they are partners of consortia or mediate specific reaction typically described for hydrothermal and seep environments, or are specialized organisms in exploiting labile organic substrates. Concluding, these results are remarkable because support the importance of ultra slow spreading ridge systems in contributing to global geochemical cycles and larval dispersion of vent fauna.

Applications of elliptic functions to solve differential equations

In questa tesi si mostrano alcune applicazioni degli integrali ellittici nella meccanica Hamiltoniana, allo scopo di risolvere i sistemi integrabili. Vengono descritte le funzioni ellittiche, in particolare la funzione ellittica di Weierstrass, ed elenchiamo i tipi di integrali ellittici costruendoli dalle funzioni di Weierstrass. Dopo aver considerato le basi della meccanica Hamiltoniana ed il teorema di Arnold Liouville, studiamo un esempio preso dal libro di Moser-Integrable Hamiltonian Systems and Spectral Theory, dove si prendono in considerazione i sistemi integrabili lungo la geodetica di un'ellissoide, e il sistema di Von Neumann. In particolare vediamo che nel caso n=2 abbiamo un integrale ellittico.