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The authors regret to inform that there is a mistake in a reference given in the introduction. In page 935 end of line 17 “.See also [43], [18], [32]…” the reference [32] is NOT correct. The correct reference is [1]. Note that everywhere else [32] is cited, the reference to the paper “A. Maspero, D. Robert, J. Funct. Anal. 273 (2) (2017) 721–781” is correct. The authors would like to apologise for any inconvenience caused.

We prove the existence and the linear stability of small amplitude time {\it quasi-periodic} standing wave solutions (i.e. periodic and even in the space variable $ x $) of a $ 2 $-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure., Comment: Revised version to appear in Memoirs of the American Mathematical Society, MEMO 891

Different temporal forms can be used to describe one temporal relation (IT: camminava vs. camminò, walk-past- imperfective vs. walk-past-perfective) ; one form can be used to describe different temporal relations (D: life, walk-past- imperfective vs. walk-past-perfective) ; these relations concern objects not directly perceivable. This can make their acquisition a complicated task. We tested 355 children, 5 years-old, on tense comprehension in 17 languages. Participants watched movies where a protagonist walks along a road performing the same action at different locations (Wagner 2001). The test-questions were of the form: Where V-past? ; Where V-present? ; Where V-future? We found differences across languages (χ2(16)=100.14, p

We investigate whether Italian native speakers and Italian children can order the quantifiers of their language on a magnitude scale. We look at three indefinite low-magnitude quantifiers (alcuni, pochi, qualche,'a few') and three indefinite high-magnitude quantifiers (molti, parecchi, tanti,'many'). Dictionaries suggest that these quantifiers refer to different magnitudes, which would not only justify their co-existence in the language, but should also make scalar ordering possible. In two experiments, 96 adults and 16 five-year-old children took part in a magnitude comparison task. The results show a developmental difference in so called “synonymous” quantifiers, which we explain by suggesting that children apply the Principle of Contrast to quantifiers.

We consider a family of Schr\"odinger equations with unbounded Hamiltonian quadratic nonlinearities on a generic tori of dimension $d\geq1$. We study the behaviour of high Sobolev norms $H^{s}$, $s\gg1$, of solutions with initial conditions in $H^{s}$ whose $H^{\rho}$-Sobolev norm, $1\ll\rho\ll s$, is smaller than $\e\ll1$. We provide a control of the $H^{s}$-norm over a time interval of order $O(\e^{-2})$. %where $\e\ll1$ is the size of the initial condition in $H^{\rho}$. Due to the lack of conserved quantities controlling high Sobolev norms, the key ingredient of the proof is the construction of a modified energy equivalent to the "low norm" $H^{\rho}$ (when $\rho$ is sufficiently high) over a nontrivial time interval $O(\e^{-2})$. This is achieved by means of normal form techniques for quasi-linear equations involving para-differential calculus. The main difficulty is to control the possible loss of derivatives due to the small divisors arising form three waves interactions. By performing "tame" energy estimates we obtain upper bounds for higher Sobolev norms $H^{s}$.

We study the reducibility of a linear Schrodinger equation subject to a small unbounded almost periodic perturbation which is analytic in time and space. Under appropriate assumptions on the smallness, analyticity, and on the frequency of the almost periodic perturbation, we prove that such an equation is reducible to constant coefficients via an analytic almost periodic change of variables. This implies control of both Sobolev and analytic norms for the solution of the corresponding Schrödinger equation for all times.

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig–Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on $$\mathbb {T}^\infty $$ T ∞ .