The lost proof of Fermat's last theorem

This work contains two papers: the first entitled "Euler's double equations equivalent to Fermat's Last Theorem" presents a marvellous "Eulerian" proof of Fermat's Last Theorem, which could have entered in a not very narrow margin, i.e. in only a few pages (less than 13). The second instead, entitled "The origin of the Frey elliptic curve in a too narrow margin" provides a proof, which is not elementary (25 pages): It is in various ways articulated and sometimes the author use facts with are proven later, but it is still addressed in an appropriate manner. This proof is however conditioned by presence of a right triangle (very often used by Fermat in his elusive digressions on natural numbers) or more precisely from a Pythagorean equation, which has a role decisive in the reconstruction of the lost proof. Regarding the first paper, following an analogous and almost identical approach to that of A. Wiles, I tried to translate the aforementioned bond into a possible proof of Fermat's Theorem. More precisely, through the aid of a Diophantine equation of second degree, homogeneous and ternary, solved at first not directly, but as a consequence of the resolution of the double Euler equations that originated it and finally in a direct I was able to obtain the following result: the intersection of the infinite solutions of Euler's double equations gives rise to an empty set and this only by exploiting a well known Legendre Theorem, which concerns the properties of all the Diophantine equations of the second degree, homogeneous and ternary. I report that the "Journal of Analysis and Number Theory" has made this paper in part (5 pages) available online at http://www.naturalspublishing.com/ContIss.asp?IssID=1779Comment: 39 pages, 1 figure- The double equations of Euler and the two fundamental theorems of this work are equivalent to the Fermat Last Theorem. The main goal is to rediscover what Fermat had in mind (no square number can be a congruent number). Also with the method of Induction, discovered by Fermat, we obtain a full proof of FLT. arXiv admin note: text overlap with arXiv:1604.0375

High Luminescence in Small Si/SiO2 Nanocrystals: A Theoretical Study

In recent years many experiments have demonstrated the possibility to achieve efficient photoluminescence from Si/SiO2 nanocrystals. While it is widely known that only a minor portions of the nanocrystals in the samples contribute to the observed photoluminescence, the high complexity of the Si/SiO2 interface and the dramatic sensitivity to the fabrication conditions make the identification of the most active structures at the experimental level not a trivial task. Focusing on this aspect we have addressed the problem theoretically, by calculating the radiative recombination rates for different classes of Si-nanocrystals in the diameter range of 0.2-1.5 nm, in order to identify the best conditions for optical emission. We show that the recombination rates of hydrogenated nanocrystals follow the quantum confinement feature in which the nanocrystal diameter is the principal quantity in determining the system response. Interestingly, a completely different behavior emerges from the OH-terminated or SiO2-embedded nanocrystals, where the number of oxygens at the interface seems intimately connected to the recombination rates, resulting the most important quantity for the characterization of the optical yield in such systems. Besides, additional conditions for the achievement of high rates are constituted by a high crystallinity of the nanocrystals and by high confinement energies (small diameters)

Auger recombination and carrier multiplication in embedded silicon and germanium nanocrystals

For Si and Ge nanocrystals (NCs) embedded in wide band-gap matrices, Auger recombination (AR) and carrier multiplication (CM) lifetimes are computed exactly in a three-dimensional real space grid using empirical pseudopotential wave functions. Our results in support of recent experimental data offer new predictions. We extract simple Auger constants valid for NCs. We show that both Si and Ge NCs can benefit from photovoltaic efficiency improvement via CM due to the fact that under an optical excitation exceeding twice the band gap energy, the electrons gain lion's share from the total excess energy and can cause a CM. We predict that CM becomes especially efficient for hot electrons with an excess energy of about 1 eV above the CM threshold.Comment: 4 pages, 6 figures (Published version

Carrier multiplication between interacting nanocrystals for fostering silicon-based photovoltaics

Being a source of clean and renewable energy, the possibility to convert solar radiation in electric current with high efficiency is one of the most important topics of modern scientific research. Currently the exploitation of interaction between nanocrystals seems to be a promising route to foster the establishment of third generation photovoltaics. Here we adopt a fully ab-initio scheme to estimate the role of nanoparticle interplay on the carrier multiplication dynamics of interacting silicon nanocrystals. Energy and charge transfer-based carrier multiplication events are studied as a function of nanocrystal separation showing benefits induced by the wavefunction sharing regime. We prove the relevance of these recombinative mechanisms for photovoltaic applications in the case of silicon nanocrystals arranged in dense arrays, quantifying at an atomistic scale which conditions maximize the outcome.Comment: Supplementary materials are freely available onlin

Gap opening in ultrathin Si layers: Role of confined and interface states

We present first principle calculations of ultrathin silicon (111) layers embedded in CaF2, a lattice matched insulator. Our all electron calculation allows a check of the quantum confinement hypothesis for the Si band gap opening as a function of thickness. We find that the gap opening is mostly due to the valence band while the lowest conduction band states shift very modestly due to their pronounced interface character. The latter states are very sensitive to the sample design. We suggest that a quasidirect band gap can be achieved by stacking Si layers of different thickness

An Alternative Form of the Functional Equation for Riemann's Zeta Function, II

This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for {\zeta}(s). We present here, after showing the first proof of Riemann, a new, simple and direct proof of the symmetric form of the functional equation for both the Eulerian Zeta function and the alternating Zeta function, connected with odd numbers. A proof that Euler himself could have arranged with a little step at the end of his paper "Remarques sur un beau rapport entre les s\'eries des puissances tant direct que r\'eciproches". This more general functional equation gives origin to a special function, here named {\cyr \E}(s), which we prove that it can be continued analytically to an entire function over the whole complex plane using techniques similar to those of the second proof of Riemann. Moreover we are able to obtain a connection between Jacobi's imaginary transformation and an infinite series identity of Ramanujan. Finally, after studying the analytical properties of the function {\cyr \E}(s), we complete and extend the proof of a Fundamental Theorem, both on the zeros of Riemann Zeta function and on the zeros of Dirichlet Beta function, using also the Euler-Boole summation formula.Comment: 26 pages, 2 figure

Marie Curie, Hertha Ayrton e le altre. Donne e Scienziate

Il ruolo di Marie Curie e Hertha Ayrton come pioniere della scienza