Related papers: A lecture on the classical KAM theorem
This is a series of lecture notes explaining topos theory and its application in physics.
In this paper, we give a new proof of the classical KAM theorem which avoids small divisors and relies on two basic principles of Diophantine approximation: Dirichlet's box and Khintchine transference principles.
There are many falsely intuitive introductions to quantum theory and quantum computation in a handwave. There are also numerous documents which teach those subjects in a mathematically sound manner. To my knowledge this paper is the…
The quantum nature of a black hole is revealed using the simplest terms that one learns in undergraduate and beginning graduate courses. The exposition demonstrates -- vividly -- the importance and power of the quantum oscillator in…
This lecture addresses some general ideas behind numerical computations ranging from representation of numbers in computers to stability and accuracy of standard algorithms for some simple mathematical problems.
These lectures provide an introduction to effective theories concentrating on the basic ideas and providing some simple applications
This is part II of our book on KAM theory. We start by defining functorial analysis and then switch to the particular case of Kolmogorov spaces. We develop functional calculus based on the notion of local operators. This allows to define…
Introduce several KAM theorems for infinite dimensional Hamiltonian with short range and discuss the relationship between spectra of linearized operator and invariant tori. Especially, introduce a KAM theorem in the paper(Cummun. Math.…
An introduction to loop quantum gravity is given, focussing on the fundamental aspects of the theory, different approaches to the dynamics, as well as possible future directions. It is structured in five lectures, including exercises, and…
The goal of this lecture is to introduce the student to the theory of Special Relativity. Not to overload the content with mathematics, the author will stick to the simplest cases; in particular only reference frames using Cartesian…
The term quantum logic has different connotations for different people, having been considered as everything from a metaphysical attack on classical reasoning to an exercise in abstract algebra. Our aim here is to give a uniform…
The structure of covariant instruments is studied and a general structure theorem is derived. A detailed characterization is given to covariant instruments in the case of an irreducible representation of a locally compact group.
This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we…
We prove an abstract KAM theorem adapted to space-multidimensional hamiltonian PDEs with regularizing nonlinearities. It applies in particular to the singular perturbation problem studied in the first part of this work.
This is an introduction to some recent developments in string theory and M theory. We try to concentrate on the main physical aspects, and often leave more technical details to the original literature.
Written with respect to an appropriate Poisson structure, a partially integrable Hamiltonian system is viewed as a completely integrable system with parameters. Then, the theorem on quasi-periodic stability in Ref. [1] (the KAM theorem) can…
A selfcontained proof of the KAM theorem in the Thirring model is discussed, completely relaxing the ``strong diophantine property'' hypothesis used in previous papers. Keywords: \it KAM, invariant tori, classical mechanics, perturbation…
Constructor theory seeks to express all fundamental scientific theories in terms of a dichotomy between possible and impossible physical transformations - those that can be caused to happen and those that cannot. This is a departure from…
In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-R\"ussmann condition, in real-analytic non-degenerate Hamiltonian systems…
We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis-Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash-Moser…