Quasi-periodic solutions of completely resonant forced wave equations

We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.Comment: 25 pages, 1 figur

The LHCf experiment: present status and physics results

The main aim of the LHCf experiment is to provide precise measurements of the production spectra relative to neutral particle produced by high energy proton-ion collisions in the very forward region. This information is necessary in order to test and tune hadronic interaction models used by ground-based cosmic rays experiments. In order to reach this goal, LHCf makes use of two small sampling calorimeters installed in the LHC tunnel at $\pm 140$ m from IP1, able to detect neutral particles having pseudo-rapidity $\eta > 8.4$. In this paper we will present the current status of the LHCf experiment, regarding in particular the first analysis results from data taking relative to p-p collisions at $\sqrt{s} =$ 13 TeV

Periodic solutions of nonlinear wave equations with general nonlinearities

We prove the existence of small amplitude periodic solutions, with strongly irrational frequency \om close to one, for completely resonant nonlinear wave equations. We provide multiplicity results for both monotone and nonmonotone nonlinearities. For \om close to one we prove the existence of a large number N_\om of 2 \pi \slash \om -periodic in time solutions $u_1, ..., u_n, ..., u_N$: N_\om \to + \infty as \om \to 1 . The minimal period of the $n$-th solution $u_n$ is proved to be 2 \pi \slash n \om . The proofs are based on a Lyapunov-Schmidt reduction and variational arguments.Comment: 29 page

Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential

We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T^d, d \geq 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash-Moser iterative scheme as in [5]. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove the "separation properties" of the small divisors assuming weaker non-resonance conditions than in [11]

Temperature estimation and slip-line force analytical models for the estimation of the radial forming force in the RARR process of flat rings

open2noIn this study, a mathematical model for the prediction of the temperature evolution in the ring during the radial-axial ring rolling process is developed and used, together with the authors’ previous results, to determine analytically the flow stress of the material during process. These results, combined with Hill's slip-line field solution adapted to the RARR process, allow a fast and reasonably precise calculation of the radial forming force, a key parameter at the preliminary stage of the process design. The approach is validated by applying the proposed model to a case available in the literature and comparing the analytical results with those of the laboratory experiment and FEM simulation. Following the successful comparison, the models were applied to a large variety of flat rings, comparing analytical predictions with the results of FEM simulations. The accuracy of the analytical calculation and the reliability of the proposed models, for different ring configuration and process parameters, are presented and discussed.embargoed_20190501Quagliato, Luca; Berti, GuidoQuagliato, Luca; Berti, Guid