Ok, đây là tôi tự bổ túc văn hóa về complexity thôi.
Trong lý thuyết complexity, người ta nói đến các oracle, tức là các “hộp đen” để tính các hàm (hay trả lời các câu hỏi) nào đó, chỉ cần cho input (thuộc loại nào đó) vào đấy thì sẽ ra ngay output, . . . → Read More: Oracle phổ quát bậc 2 ?
These are the notes for an optional course on dynamical systems that I will give to 4th year mathematics students this semester.
The course is in French, but for convenience I will write the notes in English. (In order to become competitive, my French students will have to learn to use English anyway).
The . . . → Read More: Introduction to dynamical systems (1)
These are the notes that i’m trying to write up for the 5th lecture of my doctoral course “Basic principles of dynamical systems” in Toulouse. This lecture is about ergodicity.
I’m behind schedule in many things, so please be patient with me. The correct lecture notes will appear one day. I’m even planning to . . . → Read More: Dynamical Systems (5): Ergodicity
I know very little about complex (meaning over the field of complex numbers, and not in the sense of complexity) dynamical systems. But by chance I’m invited to give a talk in a seminar on complex dynamics in Paris this week, so I have to find a topic which is close to my work . . . → Read More: Complexification of real dynamical systems
Just learned about the following conjecture of Littlewood (1930), which looks very simple and which is apparently still open:
Let be two arbitrary real numbers. Denote by . Then
This conjecture is related to ergodic theory of It is not difficult to show that the set of pairs of number which donot satisfy . . . → Read More: Littlewood’s conjecture (1930)
I’m stumbling over the problem of inversion of a unbounded operator on a Hilbert space. My operator is a perturbation of a Casimir operator acting by double differentiation of a tensor space. It ooks like an elliptic operator, so the inverse is expected to be (at least) bounded, with a “very reasonabe” bound. But . . . → Read More: Inverse of a unbounded operator ?
Last updated 10/March/2012: This article is now finished. See
19/02/2012: We’re behind schedule now. Need to speed up !!!
This is the sketch of a research article in progress with a student of mine. I’m making the sketch of what we want to put in the article, and he takes care of writing . . . → Read More: Rn-actions on n-dimensional manifolds (article in preparation)
last updated: 18/jan/2012
This is a particular case of integrable non-Hamiltonian systems that my student Minh is working on with me for this thesis. We want to study such systems topologically. A real integrable system of type is nothing but a -action (generated by a family of commuting vector fields) on a -dimensional manifold.
. . . → Read More: Rn-actions on n-dimensional manifolds
While looking for journals to submit my new preprints (I prefer to submit to places where I have not published before, in order to “collect” the journals ), I came across the following list of A* journals in mathematics, according to the Australian Mathematical Society. I don’t know which are their selection criteria. The . . . → Read More: A* journals in mathematics, according to Aussies
I’m doing this project about 2nd-order pricing models with a PhD student of mine. The project is quite ambitious. It aims to to be better than known models (Black-Scholes, stochastic volatility, 1st-order jump models, equilibrium, etc.) and be able to explain things that can’t be explained by previous models. Potential applications include investing & . . . → Read More: Second-order models for asset prices
This is the research project of a PhD student of mine. Actually the general project is sufficiently large for a number of PhD theses. There are lots of open questions in the non-Hamiltonian case.
What my student is doing is to study simplest (mostly low-dimensional) cases, and with only nondegenerate singularities:
- Systems of . . . → Read More: Topology of integrable non-Hamiltonian systems
Whitehead’s lemma says that and where is a simple Lie algebra and is a linear representation of it. I know an algebraic proof which gives an explicit formula for the homotopy operator. I’m looking for a more geometric proof (using, for example, an averaging formula). Why ? Because the explicit homotopy operator in the . . . → Read More: Geometric proof of Whitehead’s lemma ?
(A work in progress of mine)
Recently there have been very interesting papers by Crainic, Fernandes and company about geometric approaches to normal forms problems in Poisson geometry. Using groupoid techniques (in particular integration & the path method & their integrability iteria) they recover Conn’s linearization theorem (for Poisson structures), and also my linearization . . . → Read More: Geometric approach to Levi decomposition