Generalized Euler-Lagrange equations for variational problems with scale derivatives

We obtain several Euler-Lagrange equations for variational functionals defined on a set of H\"older curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale derivatives are considered.Comment: Submitted on 03-Aug-2009; accepted for publication 16-March-2010; in "Letters in Mathematical Physics"

On the positivity of solutions of systems of stochastic PDEs

We study the positivity of solutions of a class of semi-linear parabolic systems of stochastic partial differential equations by considering random approximations. For the family of random approximations we derive explicit necessary and sufficient conditions such that the solutions preserve positivity. These conditions imply the positivity of the solutions of the stochastic system for both Itô's and Stratonovich's interpretation of stochastic differential equations. We study the positivity of solutions of a class of semi-linear parabolic systems of stochastic partial differential equations by considering random approximations. For the family of random approximations we derive explicit necessary and sufficient conditions such that the solutions preserve positivity. These conditions imply the positivity of the solutions of the stochastic system for both Itô's and Stratonovich's interpretation of stochastic differential equations

Organic Farming Training and needs of stakeholders (Teachers and advisers); Status and proposals

The work carried out made it possible to better understand the devices of training, the knowledge of the training and development stakeholders, their expectations in the domainof organic farming. If the general principles and techniques of organic farming are fairly well known, their variation in practice is not easily normative. The courses, primarily based on knowledge transfer, are not very effective and the surveyed actors highlighted the value of networks and non-formal trainings. Proposals for changing devices are presented being based on such an innovative action in the Loire Valley with teachers and trainers. The issue of the integration of e-learning in these devices is presented with its potentialsand limits. The issue of educational resources concerning their format, their contents as the media used and their accessibility is addressed with proposals based on Swiss and the Loire Valley achievements

Scale calculus and the Schrodinger equation

We introduce the scale calculus, which generalizes the classical differential calculus to non differentiable functions. The new derivative is called the scale difference operator. We also introduce the notions of fractal functions, minimal resolution, and quantum representation of a non differentiable function. We then define a scale quantization procedure for classical Lagrangian systems inspired by the Scale relativity theory developped by Nottale. We prove that the scale quantization of Newtionian mechanics is a non linear Schrodinger equation. Under some specific assumptions, we obtain the classical linear Schrodinger equation.Comment: 49 page

A λ-lemma for Normally Hyperbolic Invariant Manifolds

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Séries hypergéométriques multiples et polyzêtas

International audienc

Scale relativity and fractal space-time: theory and applications

In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a generalization of possible physically relevant fractal laws, written as partial differential equation acting in the space of scales, and (ii) to a new geometric foundation of quantum mechanics and gauge field theories and their possible generalisations. In the second part, we discuss some examples of application of the theory to various sciences, in particular in cases when the theoretical predictions have been validated by new or updated observational and experimental data. This includes predictions in physics and cosmology (value of the QCD coupling and of the cosmological constant), to astrophysics and gravitational structure formation (distances of extrasolar planets to their stars, of Kuiper belt objects, value of solar and solar-like star cycles), to sciences of life (log-periodic law for species punctuated evolution, human development and society evolution), to Earth sciences (log-periodic deceleration of the rate of California earthquakes and of Sichuan earthquake replicas, critical law for the arctic sea ice extent) and tentative applications to system biology.Comment: 63 pages, 14 figures. In : First International Conference on the Evolution and Development of the Universe,8th - 9th October 2008, Paris, Franc

Arnold diffusion for a complete family of perturbations

In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, f, s) = p2/2+ cos q - 1 + I2/2 + h(q, f, s; e) — proving that for any small periodic perturbation of the form h(q, f, s; e) = e cos q (a00 + a10 cosf + a01 cos s) (a10a01 ¿ 0) there is global instability for the action. For the proof we apply a geometrical mechanism based on the so-called scattering map. This work has the following structure: In the first stage, for a more restricted case (I* ~ p/2µ, µ = a10/a01), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of instability for any µ). The bifurcations of the scattering map are also studied as a function of µ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.Peer ReviewedPostprint (published version

Canonical Melnikov theory for diffeomorphisms

We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection. We show how our definition reproduces the classical methods of Poincar\'{e} and Melnikov and specializes to methods previously used for exact symplectic and volume-preserving maps. We use the method to detect the transverse intersection of stable and unstable manifolds and relate this intersection to the set of zeros of the Melnikov displacement.Comment: laTeX, 31 pages, 3 figure

Fractional variational calculus of variable order

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted 13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck), Operator Theory: Advances and Applications, Birkh\"auser Verlag (http://www.springer.com/series/4850