Arithmetic of Plane Cremona Transformations and the Dimensions of Transfinite Heterotic String Space-Time

It is shown that the two sequences of characteristic dimensions of transfinite heterotic string space-time found by El Naschie can be remarkably well accounted for in terms of the arithmetic of self-conjugate homaloidal nets of plane algebraic curves of orders 3 to 20. A firm algebraic geometrical justification is thus given not only for all the relevant dimensions of the classical theory, but also for other two dimensions proposed by El Naschie, viz. the inverse of quantum gravity coupling constant (~42.36067977) and that of (one half of) fine structure constant (~68.54101967). A non-trivial coupling between the two El Naschie sequences is also revealed.Comment: 4 pages, no figures, accepted in Chaos, Solitons & Fractal

On Veech's proof of Sarnak's theorem on the M\"{o}bius flow

We present Veech's proof of Sarnak's theorem on the M\"{o}bius flow which say that there is a unique admissible measure on the M\"{o}bius flow. As a consequence, we obtain that Sarnak's conjecture is equivalent to Chowla conjecture with the help of Tao's logarithmic Theorem which assert that the logarithmic Sarnak conjecture is equivalent to logaritmic Chowla conjecture, furthermore, if the even logarithmic Sarnak's conjecture is true then there is a subsequence with logarithmic density one along which Chowla conjecture holds, that is, the M\"{o}bius function is quasi-generic.Comment: 11 pages. Some misprints are corrected, and some details about the proof of the main result is added. We further add :"W. Veech in his letter indicated to me that there is only four persons in the world who has a copy of his notes including me.

A Further Note on a Formal Relationship Between the Arithmetic of Homaloidal Nets and the Dimensions of Transfinite Space-Time

A sequence of integers generated by the number of conjugated pairs of homaloidal nets of plane algebraic curves of even order is found to provide an >exact< integer-valued match for El Naschie's primordial set of fractal dimensions characterizing transfinite heterotic string space-time.Comment: 3 pages, no figures, submitted to Chaos, Solitons & Fractal

On the spectral type of some class of rank one flows

It is shown that a certain class of Riesz product type measure on $\mathbb{R}$ is singular. This proves the singularity of the spectral types of some class of rank one flows. Our method is based on the extension of the Central Limit Theorem approach to the real line which gives a new extension of Salem-Zygmund Central Limit Theorem.Comment: revised version with 19 pages. Submitted for publicatio

A new class of rank one transformations with singular spectrum

We introduce a new tool to study the spectral type of rank one transformations using the method of central limit theorem for trigonometric sums. We get some new applications.Comment: 16 pages et 27 r\'ef\'erence

On the spectrum of α\alpha-rigid maps

It is shown that there exists an $\alpha$-rigid transformation with $\alpha$ less or equal to $\frac12$ whose spectrum has Lebesgue component. This answers the question raised by Klemes and Reinhold in \cite{Klemes-Reinhold}. We exhibit also a large class of $\alpha$-rigid transformations with singular spectrum.Comment: 15 pages. Accepted for publication in Journal of Dynamical and Control System