On singular moduli that are S-units

Recently Yu. Bilu, P. Habegger and L. K\"uhne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-units. Here we prove that when the set S contains only primes congruent to 1 modulo 3 then no singular modulus can be an S-unit. We then give some remarks on the general case and we study the norm factorizations of a special family of singular moduli.Comment: Version changed according to the referee's comments. The final version appears in Manuscripta Mathematica, https://doi.org/10.1007/s00229-020-01230-

Effective conditional bounds on singular S-units

We provide a proof that, for every prime $\ell \geq 5$, if $S_0$ is the set of all primes congruent to $1$ modulo $3$ and $S_\ell=S_0 \cup \{\ell \}$, then under certain assumptions on the $L$-functions attached to imaginary quadratic fields, the set of singular $S_\ell$-units is finite and its cardinality can be effectively bounded.Comment: 22 pages, comments are very welcom

Cyclic reduction densities for elliptic curves

For an elliptic curve E defined over a number field K, the heuristic density of the set of primes of K for which E has cyclic reduction is given by an inclusion–exclusion sum δE/K involving the degrees of the m-division fields Km of E over K. This density can be proved to be correct under assumption of GRH. For E without complex multiplication (CM), we show that δE/K is the product of an explicit non-negative rational number reflecting the finite entanglement of the division fields of E and a universal infinite Artin-type product. For E admitting CM over K by a quadratic order O , we show that δE/K admits a similar ‘factorization’ in which the Artin type product also depends on O . For E admitting CM over K¯ by an order O⊄ K , which occurs for K= Q , the entanglement of division fields over K is non-finite. In this case we write δE/K as the sum of two contributions coming from the primes of K that are split and inert in O . The split contribution can be dealt with by the previous methods, the inert contribution is of a different nature. We determine the ways in which the density can vanish, and provide numerical examples of the different kinds of densities

Algebraic relations over finite fields that preserve the endomorphism rings of CM jj-invariants

We characterise the integral affine plane curves over a finite field $k$ with the property that all but finitely many of their $\overline{k}$-points have coordinates that are $j$-invariants of elliptic curves with isomorphic endomorphism rings. This settles a finite field variant of the Andr\'e-Oort conjecture for $Y(1)^2_\mathbb{C}$, which is a theorem of Andr\'e. We use our result to solve the modular support problem for function fields of positive characteristic.Comment: 18 page

Unlocking Onboard SAR Processing: Focusing and Ship Detection on Sentinel-1 IW Data

This work demonstrates the possibility of enabling onboard processing of SAR data in real-time through the adoption of an innovative focusing technology coupled with object detection, using limited computational resources. Our approach aims to provide a coarse focused product onboard to unlock real-time monitoring capabilities, complementing the ground-based detailed focusing algorithms. The focusing algorithm transforms the Level-0 raw signal into Level-1 Single-Look-Complex (SLC) data. It consists of a two-layers hybrid architecture: a traditional Fast-Fourier Transform (FFT) algorithm for range processing and a Deep Neural Network (DNN), trained to solved the azimuth processing task, which provides scalability and modularity benefits. After focusing, an object detection network is trained to detect the presence of ships in the SLC data. The whole processing chain has been optimized and deployed on different embedded devices, including NVIDIA Jetson Nano, and NVIDIA Jetson Xavier, to demonstrate the feasibility of running the overall pipeline onboard the future generations of SAR missions

Automated vehicles and the rethinking of mobility and cities

The project CityMobil2 has carried out a forward-looking exercise to investigate a lternative cybermobility scenarios, including both niche and large-market innovations, and their impacts on European cities and their transport systems. The paper describes the current status of and main trends in automated vehicles, a preliminary vision of the future city with mobility supported mainly by automated vehicles, and freight distribution. The expected positive impacts derive from the development of car sharing, the reduction of space required for parking vehicles, the possibilities for older people or those with disabilities to use cars, the enhancement of safety, and the improvement of efficiency of the transport system

Labour-use Efficiency in the Italian Machinery Industry: a Non-parametric Stochastic Frontier Perspective

Firms’ efficiency is a mainstream in the study of economic growth. Within this broad research area, the present work, conducted as part of the research activities of SOSE S.p.A., analyses the labour use efficiency in the Italian machinery industry through the application of a non-parametric stochastic frontier model with the aim of suggesting new insights to better understand the recent dynamics of the Italian manufacturing system. An extended panel data of manufacturing Small and Medium Enterprises (SMEs) operating in the mechanical industry for the period 2002-2012 has been extracted (in anonymous form) from the Italian Ministry of Economy and Finance annual survey and used for the implementation of the proposed method. Results show the presence of a persistent level of labour-use inefficiency in the sample used for the analysis: this issue is particularly evident for the subset of firms using non standard jobs, while firms entitled to access to wage redundancy fund appear to have achieved higher levels of efficiency in labour input use on average. The analysis also shows that the inefficiency gap between the two subsets of firms tends to reduce in absolute terms over tim

A Semi-Automatic Numerical Algorithm for Turing Patterns Formation in a Reaction-Diffusion Model

The Turing pattern formation is modeled by reaction - diffusion (RD) type partial differential equations , and it plays a crucial role in ecological studies. Big data analytics and suitable frameworks to manage and predict structures and configurations are mandatory. The processing and resolution procedures of mathematical models relies upon numerical schemes, and concurrently upon the related automated algorithms. Starting from a RD model for vegetation patterns, we propose a semi-automatic algorithm based on a smart numerical criterion for observing ecological reliable results. Numerical experiments are carried out in the case of spot's formations