Fast Digital Convolutions using Bit-Shifts

An exact, one-to-one transform is presented that not only allows digital circular convolutions, but is free from multiplications and quantisation errors for transform lengths of arbitrary powers of two. The transform is analogous to the Discrete Fourier Transform, with the canonical harmonics replaced by a set of cyclic integers computed using only bit-shifts and additions modulo a prime number. The prime number may be selected to occupy contemporary word sizes or to be very large for cryptographic or data hiding applications. The transform is an extension of the Rader Transforms via Carmichael's Theorem. These properties allow for exact convolutions that are impervious to numerical overflow and to utilise Fast Fourier Transform algorithms.Comment: 4 pages, 2 figures, submitted to IEEE Signal Processing Letter

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

The Construction of Mirror Symmetry

The construction of mirror symmetry in the heterotic string is reviewed in the context of Calabi-Yau and Landau-Ginzburg compactifications. This framework has the virtue of providing a large subspace of the configuration space of the heterotic string, probing its structure far beyond the present reaches of solvable models. The construction proceeds in two stages: First all singularities/catastrophes which lead to ground states of the heterotic string are found. It is then shown that not all ground states described in this way are independent but that certain classes of these LG/CY string vacua can be related to other, simpler, theories via a process involving fractional transformations of the order parameters as well as orbifolding. This construction has far reaching consequences. Firstly it allows for a systematic identification of mirror pairs that appear abundantly in this class of string vacua, thereby showing that the emerging mirror symmetry is not accidental. This is important because models with mirror flipped spectra are a priori independent theories, described by distinct CY/LG models. It also shows that mirror symmetry is not restricted to the space of string vacua described by theories based on Fermat potentials (corresponding to minimal tensor models). Furthermore it shows the need for a better set of coordinates of the configuration space or else the structure of this space will remain obscure. While the space of LG vacua is {\it not} completely mirror symmetric, results described in the last part suggest that the space of Landau--Ginburg {\it orbifolds} possesses this symmetry.Comment: 58 pages, Latex file, HD-THEP-92-1