在1954年，科学界庆祝了Poincare的诞辰100周年纪念日，在那个时候，Poincare在数学家们当中的名声并没有达到最高峰，而希尔伯特的精神则主导了很多数学家的思想，Poincare在物理学家们当中的名声也没有达到最高峰，因为在那时候，物理学在本质上是和量子理论相关的。

Henri Poincare，为科学服务的一生

This paper presents an opportunity for the uncertainty that has plagued the novel's criticism to appear as absences in the body of historical knowledge, particularly regarding the notion of life after death. Taking appearance (eg. proof of existence), as opposed to disappearance, as a universally accepted value allows this analysis to interrogate the novel's logic in relation to a variety of conventional systems whose very existence depends on the reproduction of their systems. The ineffectuality of Foucauldian disciplinary institutions in the novel establishes the threat of nonexistence. A significant relationship to Dante's Inferno is rendered, lending the appearance of language an 'enchanted' value through allusions to Dante's intentional invocation of Augustinian corporeal vision. The novel's metalanguage appears enchanted by the body of historical knowledge, particularly as the product of capitalism, discipline and Judeo-Christianity, and programmed by literary precursors William S. Burroughs, Gertrude Stein and Ernest Hemingway. Foregrounded by this complex network, an analysis of the novel’s first chapter demonstrates how an attention to appearance brings the language to life and draws the narrator, equally invested in appearance, into its realm of representation.

Will To Appear

We prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. At this point we implement a Nash–Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients.

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KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations

A KAM algorithm for the resonant non-linear Schrödinger equation

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KAM for autonomous quasi-linear perturbations of KdV

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions—namely periodic and even in the space variable x—of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the fully nonlinear nature of the gravity water waves equations—the highest order x-derivative appears in the nonlinear term but not in the linearization at the origin—and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators, obtained at each approximate quasi-periodic solution along a Nash–Moser iterative scheme, to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory.

Time quasi-periodic gravity water waves in finite depth

We prove the existence and the linear stability of Cantor families of small amplitude time 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.

Quasi-Periodic Standing Wave Solutions of Gravity-Capillary Water Waves

We prove a reducibility result for a class of quasi-periodically forced linear wave equations on the $d$-dimensional torus $\mathbb{T}^d$ of the form $$ \partial_{tt} v - \Delta v + \varepsilon {\cal P}(\omega t)[v] = 0 $$ where the perturbation ${\cal P}(\omega t)$ is a second order operator of the form ${\cal P}(\omega t) = - a(\omega t) \Delta - {\cal R}(\omega t)$, the frequency $\omega \in {\cal R}^\nu$ is in some Borel set of large Lebesgue measure, the function $a : \mathbb{T}^\nu \to {\cal R}$ (independent of the space variable) is sufficiently smooth and ${\cal R}(\omega t)$ is a time-dependent finite rank operator. This is the first reducibility result for linear wave equations with unbounded perturbations on the higher dimensional torus $\mathbb{T}^d$. As a corollary, we get that the linearized Kirchhoff equation at a smooth and sufficiently small quasi-periodic function is reducible.

Riccardo Montalto

Papers

KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

KAM for autonomous quasi-linear perturbations of KdV

Time quasi-periodic gravity water waves in finite depth

Quasi-Periodic Standing Wave Solutions of Gravity-Capillary Water Waves

A Reducibility Result for a Class of Linear Wave Equations on ${\mathbb T}^d$