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
Lecture Notes in Mathematical Finance (links to the files)
Link đến các files bài giảng
Trường hè (Đại học kinh tế – luật & Trung tâm JVN):
Phạm Huyên, Stochastic Optimal Control in Finance
Luciano Campi, Pricing and Hedging in Energy Market
Mathieu Rosenbaum, High Frequency Trading: Part 1, Part 2, Part 3
Hội . . . → Read More: Toán tài chính: các bài giảng trường hè và hội nghị 2012
Last updated: 09/12/2011
This is a brief real-time report on the conference OSIF (Toulouse 1, 08-09/Dec/2011)
About 40+ participants. Some are my students :-) Lunchs and coffees are served at the conference free of charge for the participants :-). The organizers are A. Blanchet & S. Villeneuve (my colleagues at Toulouse 1)
9h30: . . . → Read More: Stochastic Control and Optimal Stopping in Finance (Toulouse 12/2011)
(Devoir Maison No. 3, Licence 2 en Mathématiques, Novembre 2011)
Exercice 1. (Somme de carrés des coefficients). Soit un espace euclidien de dimension , et soit .
1) Montrer que, si et sont deux bases orthonormales de , alors
En déduire que la quantité est indépendante de la base orthonormale choisie.
(Indication: utiliser . . . → Read More: Exercices d’algèbre linéaire (DM3/2011)
Finally, “Lectures on Poisson Geometry”, which is a collection of lectures on various aspects of Poisson geometry, edited by T. Ratiu, A. Weinstein, and myself, has appeared as Volume 17 of the series Geometry and Topology Monographs:
Poisson geometry is a rapidly growing subject, with many interactions and applications in areas of mathematics . . . → Read More: Lectures on Poisson Geometry (Geometry and Topology Monographs, Vol. 17)
Hou-Li solutions are regular, but exhibit a tremendous dynamic growth during a short period of time !
Dynamic Stability of the Three-Dimensional Axisymmetric Navier-Stokes Equations with Swirl, by HOU Thomas Y. (1) ; CONGMING LI (2), Communications on pure and applied mathematics, 2008, vol. 61, no5, pp. 661-697.
Abstract: In this paper, we study . . . → Read More: Notes on INS (18): Hou-Li axisymmetric swirl solutions
Last updated: 06/Novt/2010
The book by G. Lemarié-Rieusset: Recent developments in the Navier-Stokes problem (Research Notes in Mathematics Series, CRC Press, 2002) seems to be a very interesting recent book on the Navier-Stokes problem.
Part 1 of of the book (about 100 pages) contains tools from real harmonic analysis needed for the study of . . . → Read More: Notes on INS (17): Lemarié’s book on Navier-Stokes (2002)
Infinite speed is no big deal ?
There may be some very small regions in space-time where there are some points with infinite speed. The flow will sill be continuous. From the “physical” point of view, the flow map is more important than the velocity field ? The flow maps may have a few . . . → Read More: Notes on INS (16): Random thoughts
Last updated: 27/Oct/2010
CKN stands for Caffarelli-Kohn-Nirenberg. The theory is about partial regularity of solutions of INS. One should probably add the name of Scheffer, who introduced the concepts that CKN improved/generalized.
The main result is that the (parabolic) 1-dimensional Hausdorff measure of the (hypothetical) singular set in space-time is zero (which means that, . . . → Read More: Notes on INS (14): CKN theory
This part is about Yakov Sinai‘s papers on INS. Sinai wrote I-don’t-know-how-many papers on the Navier-Stokes equations. It seems that he has been trying for many years to study the properties of the N-S equation from many different points of view, and has a lot of interesting ideas, some of which may turn out . . . → Read More: Notes on INS (13): Sinai’s papers
Last updated: 20/Oct/2010
In this part we will prove the following theorem of Serrin (1962):
Serrin’s Theorem. If and is a weak solution such that
for some region of space-time, then for .
The exposition will mainly follow lecture notes by James Robinson (Campinas, 2010), Chapter II. (These lecture notes are much . . . → Read More: Notes on INS (11): Serrin’s regularity result
Last updated: 17/Oct/2010
Serrin-type regularity criteria based on pressure
Serrin was the first to write down a criterion for global smoothness of solutions of the INS equation in terms of some a-priori inequalities. In the 2D case, his inequalities are automatically satisfied, and so he proved regularity for 2D INS. In this part, we . . . → Read More: Notes on INS (10): Serrin-type regularity criteria based on pressure
Last updated: 19/Oct/2010
In this part: Ladyzhenskaya’s contributions to INS
Olga Ladyzhenskaya (1922-2004) is famous for her work on PDEs (regularity of elliptic and parabolic equations etc.), and especially the Navier-Stokes problem. In 2002 she was awarded the “Doctor Honoris Causa” by the University of Bonn. There is an interesting write-up by Michael Struwe . . . → Read More: Notes on INS (9): Ladyzhenskaya’s work on Navier-Stokes