JAIRO CHARRIS SEMINAR IN ALGEBRA AND
ANALYSIS
Bogotá – Colombia, August 1 – 3, 2008
Scientific program
Download here the pdf
version
Schedule
|
|
Friday – Chairman V. Albis |
Saturday – Chairman J. Lesmes |
Sunday – Chairman F. Marcellan |
|
Speaker |
Mourad Ismail |
Francisco Marcellán Español |
David Mond |
|
|
16:30 – 18:00 |
8:00 - 9:00 |
8:00 – 9:00 |
|
|
|
|
|
|
Speaker |
Primitivo Acosta-Humanez |
Mohammed El Aïdi |
Guillermo Rodríguez |
|
Time |
18:00 – 19:00 |
9:00 – 10:00 |
9:00 – 10:00 |
|
|
|
|
|
|
|
Luis Manuel Tovar |
David Blazquez-Sanz |
Felix Soriano |
|
|
19:30 – 20:30 |
10:30 – 11:30 |
10:30 – 11: 30 |
|
|
|
|
|
|
Speaker |
----------------------------- |
Marlio Paredes |
Walter Van Assche |
|
Time |
---------------------------- |
11:30 – 12: 30 |
11:30 – 12:30 |
|
|
|
|
|
|
|
Poster Session |
Poster Session |
Poster Session |
|
Time |
19:00 – 19:30 |
10:00 – 10:30 |
10:00 – 10:30 |
Abstracts
Conferences
Primitivo B. ACOSTA HUMÁNEZ (U. Pol. Catalunya
Talk: DIFFERENTIAL GALOIS THEORY AND ORTHOGONAL
POLYNOMIALS
Abstract. Differential Galois Theory is the Galois theory in
the context of differential equations (also known as Picard-Vessiot
theory in the linear case). In this talk we show examples that relates the
Picard-Vessiot theory with the theory of orthogonal
polynomials and special functions. We compute the Galois group of differential
equations in which their solutions are given in terms of orthogonal polynomials
(Chebyshev, Hermite, Laguerre, etc.). In the same way to differential equations
that arises from the recurrence relations of some special kind of orthogonal
polynomials. The main tools used here to solve the di®erential
equations and compute the Galois groups are two powerful algorithms: Kovacic algorithm and algebrization
procedure (see [AB.]).
References
[AC1.] P. ACOSTA HUMÁNEZ, Sobre las ecuaciones diferenciales lineales
de segundo orden y el algoritmo de Kovacic,
Civilizar, (2004), 209 - 220.
[AC2.] P. ACOSTA HUMÁNEZ, La teoría de Morales - Ramis y el algoritmo
de Kovacic, Lecturas matemáticas Volumen Especial,
(2006), 21--56.
[AB.] P. ACOSTA HUMÁNEZ \& D. BLÁZQUEZ SANZ, Non-Integrability of some hamiltonian
systems with rational potential. Preprint:
www.arxiv.org/06010010
[AP1.] P. ACOSTA HUMÁNEZ
\& J.H. PÉREZ, Teoría de Galois Diferencial: Una aproximación, Matemáticas
Enseñanza Universitaria, Vol XV 2, (2007), 91--102.
[AP2.] P. ACOSTA HUMÁNEZ
& J.H. PÉREZ, Una
Introducción a
[KO.] J. KOVACIC, An Algorithm for Solving Second Order Linear Homogeneous Differential
Equations, J. Symbolic Computation, 2,
(1986), 3-43.
[MO.] J.J. MORALES-RUIZ, “Differential Galois Theory and Non-Integrability of Hamiltonian Systems”, BirkhÄauser,
[MR1.] J.J. MORALES-RUIZ & J. P. RAMIS, Galoisian obstructions to integrability of Hamiltonian
systems I, Methods and Applications of Analysis 8 (2001), 33-95.
[MR2.] J.J. MORALES-RUIZ & J. P. RAMIS, Galoisian obstructions to integrability of Hamiltonian
systems II, Methods and Applications of Analysis 8 (2001).
[MRS.] J. J. MORALES-RUIZ, J. P. RAMIS & C. SIMÓ, Integrability of Hamiltonian Systems and Differential
Galois Groups of Higher Variational Equations,
Preprint.
[MS.] J.J. MORALES-RUIZ & C. SIMÓ, Non-integrability criteria for Hamiltonians
in the case of Lamé Normal Variational
Equations, J. Diff. Eq. 129
(1996) 111-135.
[VS.] M. VAN DER PUT & M. SINGER, “Galois Theory in Linear
Differential Equations”, Springer Verlag,
David BLÁZQUEZ SANZ (U. Sergio Arboleda
Talk: GLOBAL CONDITIONS FOR SUPERPOSITION OF
SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
Abstract. Differential equations that admit superposition laws
of Solutions were introduced by S. Lie in 1885 [1]. They are a natural
generalization of the linear systems of ordinary differential equations. They
form a special class of differential equations. The local characterization of
such a class was done by S. Lie and G. Scheffers in
1893, and further development is due to E. Vessiot
[2]. This is the well known Lie-Scheffers theorem.
These conditions are necessary for the existence of a superposition law, but
they are not sufficient. There are some additional global conditions. Here, we
give necessary and sufficient conditions for the existence of a superposition
law for a differential equation. This result is included in [3].
References
[1.] S. LIE, Allgemeine Untersuchungen über continuerliche endliche Gruppe gestatten, Math. Ann., Bd. 25 (1885), 71-151
[2.] E. VESSIOT, L Sur les
systèmes d'équations différentielles du premier ordre qui ont des systèmes fundamentaux d'integrales, Ann. Sci. Fac. Sci. Tou., Sr. 1, 8, No 3
(1894) 1-33.
[3.] D. BLÁZQUEZ SANZ, ”Differential Galois
Theory and Lie-Vessiot Systems", PhD. Dissertation, Universitat Politécnica
de Catalunya, 2008.
Héctor José CABARCAS URRIOLA (U. Distrital
Talk: WELL-POSEDNESS FOR A NONLINEAR PERTURBATION OF THE
KURAMOTO-SIVASHINSKY EQUATION
Abstract. In this lecture
we are interested in the well-posedness in the
periodical case for a nonlinear perturbation of the Kuramoto-Sivashinsky
equation, which describes near planar interfaces which are marginally long-wave
unstable. We will show that this equation is locally well-posed in $H^s$, for $s\geq 2$ improving the
result obtained by Bernoff-Sarocka. Also, we are
going to present results related with the persistence at all time
of the solutions, as a result of the blow up in finite time.
Mohammed EL AIDI
Talk: EXPLICIT LOWER BOUND FOR A
SCHR\"ODINGER OPERATOR WITH A REAL POTENTIAL IN $L^1_{loc}(\mathbb{R}^d), d\geq 3$
Abstract. The goal of this talk, is to give the proof in
detail the note “Lower bound for an elliptic operator perturbed by a
potential”, published on C.R.Acad.Paris (see [El]).
Especially we give, in an explicit way, the lower bound for the Schr\”odinger’s operator $L = −\triangle + V, with V
semi-bounded from bellow, $V\in L^1_{loc}(\mathbb{R}^d),$
and $d\geq 3$.
In this case the bottom of the Schr\”odinger’s
operator $L$ has an explicit lower bound. The bottom of $L$ is either the first
real eigenvalue of $L$ or the bottom of the essential
spectrum of $L$. So to look for in an explicit away the lower or/and the upper
bound of the bottom of $L$ is importantly in the field of quantum physics.
Bibliography
[El] M. EL AIDI, Borne inférieure d’un opérateur elliptique
perturbé par un potentiel, C. R. Acad. Sci.
Paris, t.325, Série I, (1997).
273--276.
Mourad ISMAIL (U. C. F - USA
Talk: ADDITION
THEOREMS VIA CONTINUED FRACTIONS
Abstract. We show connections between a special type of addition
formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We
apply our approach to derive special addition theorems for Bessel functions and
confluent hypergeometric functions. We also derive
several additions theorems for basic hypergeometric
functions. Applications to the
evaluation of Hankel determinants are also given.
Francisco MARCELLÁN (U. Carlos III.
Talk: RECENT TRENDS ON TWO VARIABLE ORTHOGONAL
POLYNOMIALS.
Abstract. For a measure supported on a subset of the plane we
introduce two variable orthogonal polynomial sequences taking into account the
Gram-Schmidt orthogonalization process for several
choices in the ordering of the canonical basis of monomials. From them, we
deduce three-term recurrence relations with matrix coefficients that such
polynomial sequences satisfy. A connection with matrix orthogonal polynomials
is established according to [1].
Following an historical approach (see [5], [6], and [7]) based on the
extension of the Routh-Bochner characterization of
classical orthogonal polynomials in one variable (Hermite,
Laguerre, Jacobi, and Bessel) , we give a
constructive approach of some families of two variable orthogonal polynomials
which are eigenfunctions of second order partial
differential operators with polynomial coefficients. Then, using standard
techniques for the symmetrisation of partial
differential operators, we can deduce the weight function as well as the
corresponding domain of orthogonality .
In the more general framework of the orthogonality
associated with moment functionals on the linear
space of polynomials in two variables with real coefficients, classical
orthogonal polynomials are defined in terms of a matrix analogue of the Pearson
differential equation that such a functional satisfies. They can also be
characterized as the polynomial solutions of a matrix second order partial
differential equation (see [3]).
On the other hand, the discretization of the
second order partial linear differential operator in a uniform lattice in the
plane yields a difference operator in two variables. We present some recent
results by Iliev and Xu (
see, for instance, [4]) about such operators assuming that they have polynomial
solutions. We also show that their eigenvalues are
either quadratic or linear polynomials of the index. Furthermore, the
polynomial coefficients of the difference operator need to have certain simpler
forms. The complete solution of this problem is given.
Finally, following [2] we will present some applications of orthogonal
polynomials in two variables in different frameworks.
Bibliography
[1.] A. M. DELGADO, J. S.
GERONIMO, P. ILIEV, AND F. MARCELLAN, Two
Variable Orthogonal Polynomials and Structured Matrices,
[2.] C. F. DUNKL AND Y. XU, Orthogonal Polynomials of Several Variables,
Encyclopedia of Math. and its Appl., vol. 81
[3.] L. FERNANDEZ, T. E. PEREZ,
AND M. A. PIÑAR, Classical Orthogonal
Polynomials in Two Variables. A Matrix Approach, Numer.
Algor. 39
(2005), 131--142.
[4.] P. ILIEV AND Y. XU, Discrete Orthogonal polynomials and
Difference Equations of Several Variables, Adv. in Math., 212
(2007), 1--36.
[5.] Y. J. KIM, K. H. KWON, AND
J. K. LEE, Orthogonal Polynomials in Two
Variables and Second-Order Partial Differential Equations, J. Comput. Appl. Math., 82
(1997), 239--260.
[6.] H. L. KRALL AND I. M. SHEFFER,
Orthogonal polynomials in two variables,
Ann. Mat. Pura.Appl, 76 (1967), 325--376.
[7.] P. K. SUETIN, Orthogonal Polynomials in Two Variables.
Gordon and Breach,
David MOND (
Talk: MONODROMY OF SOLUTIONS OF COMPLEX LINEAR
DIFFERENTIAL EQUATIONS
Abstract. My Master's thesis with Jairo
Charris is concerned with the construction of a long exact
sequence associated to a linear ordinary differential operator in the complex
plane:
$$0\to \ker(P)\to \s O(\Omega)\stackrel{P}{\too}\s O(\Omega)\stackrel{\delta}{\too}
\Hom_{\CC}(H_1(\Omega;\ZZ),\CC)\to 0.$$
The morphism $\delta$ is defined using the
method of variation of parameters. It later became clear that a long exact sequnce of sheaf cohomology,
formally identical to this sequence, results from applying the global section functor to the short exact sequence of sheaves
$$0\to{\s Ker}(P)\to\s O\stackrel{P}{\too} \s
O\to 0.$$
In the talk I will explain the constructions, comment on the relation
between the different cohomology theories involved,
and briefly touch on the Riemann Hilbert correspondence.
Marlio PAREDES
(U. Turabo
Talk: PARABOLIC ALMOST COMPLEX STRUCTURES ON
MAXIMAL FLAG MANIFOLDS
Abstract. In this talk we
present some results about parabolic almost complex structures on the classical
maximal flag manifold $\mathbb{F}(n) =U(n)/((U(1)x…xU(1))$. Burstall and Salamon showed the existence of a one to one correspondence
between al- most complex structures on a flag manifold and tournaments. This
correspondence has been used in several papers in order to prove geometric
facts using properties of the tournaments. We give a necessary condition for an
almost complex structure to be parabolic. In addition, we give a new proof of
the theorem, due to Mo and Negreiros, which shows
that all the parabolic almost complex structures admit (1;2)-symplectic metrics.
Felix SORIANO
(U. Nacional – Colombia
Talk: EXISTENCE AND STABILITY OF SOLITARY WAVES
FOR A DISPERSIVE EQUATION
Abstract. In this conference we talk about of the existence
and stability of solitary waves for a fifth order modification of the Camassa – Holm equation. For this we use the compactness –
concentration principle introduced by P. L. Lions. This work was done in conjunction
with Ruth Milena Cortés.
Luis Manuel TOVAR (Instituto Politécnico Nacional – México
Talk: WEIGHTED FUNCTION SPACES
Abstract. In this talk I present a survey about the main
results and relationships among the classical and recent weighted function
spaces of analytic functions defined on the complex open unit disk: Dirichlet, Dp, BMOA, Bloch, Bezov, Hardy, Bergman, Qp and F(p,q,s).
Walter VAN ASSCHE (Katholieke Universiteit
Leuven
Talk: POLYNOMIAL TRANSFORMATIONS FOR ORTHOGONAL
POLYNOMIALS AND SIEVED ORTHOGONAL POLYNOMIALS
Abstract. Jairo Charris wrote some interesting papers on sieved orthogonal polynomials.
I will describe how one can obtain a new sequence of orthogonal polynomials
from a given sequence of orthogonal polynomials by a polynomial transformation.
If the polynomial in this transformation is a Chebyshev
polynomial of the first kind, then one finds sieved orthogonal polynomials. I
will give an application which consists of the construction of a sequence of orthogonal
polynomials with a discrete measure (on a dense set of points) for which the
recurrence coefficients converge.
Posters
Title: MODELLING THE NONLINEAR DYNAMICS IN THE
EARTH’S MAGNETOSPHERE
Author:
Marlon NÚÑEZ PAZ – Universidad de Málaga (Spain)
Abstract. Download here
the pdf file.