Congruences between modular forms and related modules

We fix $\ell$ a prime and let $M$ be an integer such that $\ell\not|M$; let $f\in S_2(\Gamma_1(M\ell^2))$ be a newform supercuspidal of fixed type related to the nebentypus, at $\ell$ and special at a finite set of primes. Let \TT^\psi be the local quaternionic Hecke algebra associated to $f$. The algebra \TT^\psi acts on a module $\mathcal M^\psi_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, \TT^\psi is the universal deformation ring of a global Galois deformation problem associated to \orho_f. Moreover $\mathcal M^\psi_f$ is free of rank 2 over \TT^\psi. If $f$ occurs at minimal level, by a generalization of a Conrad, Diamond and Taylor's result and by the classical Ihara's lemma, we prove a theorem of raising the level and a result about congruence ideals. The extension of this results to the non minimal case is an open problem

A crack-like notch analogue for a safe-life fretting fatigue design methodology

Various analogies have recently been proposed for comparing the stress fields induced in fretting fatigue contact situations, with those of a crack and a sharp or a rounded notch, resulting in a degree of uncertainty over which model is most appropriate in a given situation. However, a simple recent approach of Atzori–Lazzarin for infinite-life fatigue design in the presence of a geometrical notch suggests a corresponding unified model also for fretting fatigue (called Crack-Like Notch Analogue model) considering only two possible behaviours: either 'crack-like' or 'large blunt notch.' In a general fretting fatigue situation, the former condition is treated with a single contact problem corresponding to a Crack Analogue model; the latter, with a simple peak stress condition (as in previous Notch Analogue models), simply stating that below the fatigue limit, infinite life is predicted for any size of contact. In the typical situation of constant normal load and in phase oscillating tangential and bulk loads, both limiting conditions can be readily stated. Not only is the model asymptotically correct if friction is infinitely high or the contact area is very small, but also remarkably accurate in realistic conditions, as shown by excellent agreement with Hertzian experimental results on Al and Ti alloys. The model is useful for preliminary design or planning of experiments reducing spurious dependences on an otherwise too large number of parameters. In fact, for not too large contact areas ('crack-like' contact) no dependence at all on geometry is predicted, but only on three load factors (bulk stress, tangential load and average pressure) and size of the contact. Only in the 'large blunt notch' region occurring typically only at very large sizes of contact, does the size-effect disappear, but the dependence is on all other factors including geometry