**Mathematical Colloquim
**

**PROGRAM**

ODELJENJE ZA MATEMATIKUMATEMATICKOG INSTITUTA SANU |
OPSTI MATEMATICKI SEMINAR
NA MATEMATICKOM FAKULTETU U BEOGRADU |

**
-- PROGRAM ZA MAJ 2007 --
**

__Petak, 04. maj 2007. u 14h, sala 2 MI
SANU:__

*
Marko Stosic, Instituto Superior Tecnico, Lisbon, Portugal
*

LINK HOMOLOGIES

Abstract: In this talk, we shall give basic ideas of the "categorification
of the Jones polynomial", as well as the other link polynomial invariants.
This theory, initiated by M. Khovanov in 1999. is passing huge development
and it is one of the most perspective ideas in topology. We shall give
overview of the most important results and open problems.

__Petak, 11. maj 2007. u 14h, sala 718, MF, BG:__

!!!OBRATITE PA~NNJU NA MESTO!!!

*
David Kaljaj, Prirodno-matemati?ki fakultet Podgorica
*

QUASICONFORMAL HARMONIC MAPS, JORDAN DOMAINS, RIEMANN
SURFACE, SPHERICAL METRIC, POINCARÉ METRIC, LIPSCHITZ CONDITION

Abstract: The object of this lecture is to announce the following results:

1. Let ? and ?' be Jordan domains, let (0,1], and let f: ? ? ?' be a
harmonic homeomorphism.
(a) If M ?, M ?' C1, , then f is Lipschitz;
(b) if M ?, M ?' C1, and ?' is convex, then f is bi-Lipschitz; and
(c) if ? is the unit disk, ?' is convex, and M ?, M ? ' C1, , then f is
quasiconformal if and only if its boundary function is bi-Lipschitz and the
Hilbert transform of its derivative is in L .

2. Another approach is given in some other direction, in join work with
professor M. Mateljevic: i.e. it is proved the following theorem (D. Kalaj,
M. Mateljevic, Inner estimate and quasiconformal harmonic maps between
smooth domains}, Journal d'Analise Math. 100. (2006).):

(a) Let f be a quasiconformal C2 diffeomorphism from the C1, Jordan domain
? onto the C2, Jordan domain D. If there exists a constant M such that
|? f| M|fzA fû|, z ?, then f has bounded partial derivatives. In
particular, it is a Lipschitz mapping. One of important consequence is the
following theorem:

(b) If f is quasiconformal harmonic mapping between two smooth surfaces with
smooth boundary, then it is a lipshitz mapping.

__Petak, 18. maj 2007. sala 2 MI SANU:__

*
Aleksandar Cvetkovic, Univerzitet u Nisu
*

EXTREMAL PROBLEMS FOR THE SEMI-CLASSICAL WEIGHT FUNCTIONS IN THE WEIGHTED L2
NORM

Abstract. The general form of an extremal problem can be stated as follows:
find the constant Mn and the polynomial Qn Pn, such that
???
where Pn is a linear space of algebraic polynomials of degree less than or
equal to n, ||A|| is a norm on the linear space P and k N.
The beginning of a study of extremal problems goes back to the early results
of brothers Markov. In 1889, A.A. Markov solved the extremal problem in the
uniform norm for first derivative. V.A. Markov, in 1916, solved the
corresponding problem the uniform norm for higher derivatives. After this
initial results many other results appeared for various norms ||A||.
Especially, there appeared results for Lr norm (Hille, Tamarkin, Szegö,
1937). In the case of weighted L2 norm, with Hermite and generalized
Laguerre weight functions, results have been obtained by Schmidt (1944) and
Turan (1960), respectively. In 1983, Mirsky obtained an upper bound for the
best constant Mn in the weighted L2 norm. After that Dörfler (1987)
computed best constants Mn for various weighted L2 norms, using singular
values decomposition of certain matrices. Milovanovi? (1987) considered a
special case of the weighted L2 norm with even weight functions, where
special place takes the Gegenbauer weight function.
In this lecture we give an overview of these results of the extremal
problems in the various norms. We present a systematic way of treating
extremal problems in the weighted L2 norms with the semi-classical weight
functions. We focus our attention on the construction of an algebraic
equation satisfied by the best constant Mn and a construction of the
extremal polynomial Qn. We also present some new results obtained for the
generalized Hermite and Gegenbauer weight function.

__Petak, 25. maj 2007. u 14h, sala 2, MI SANU, BGD:__

*
Zoran Pop-Stojanovic, Department of Mathematics, University of Florida, USA
*

DIRICHLET PROBLEM: A PROBABILISTIC TREATMENT

Abstract. It is well-known that connection between Brownian motion process
and harmonic functions is profound, although simply explained. To illustrate
this connection, we shall present a probabilistic treatment of Dirichlet
problem. Recall that a function u mapping an open subset D of Rd into R is
called harmonic in D if u is of class C2 and in D. The classical Dirichlet
problem (D,f) states: given an open subset D of R and a continuous function
f: MD?R. Find a continuous function u: ?R such that u is harmonic in D and
takes on boundary values given by f, that is, u is of class C2(D), ?u=0 in D
and u=f on MD. It is well-known that this problem may have no a solution.
Using Brownian motion process, one gets necessary and sufficient conditions
for the existence of a solution of (D,f).

U maju ce Odeljenje za matematiku ucestvovati u organizovanju jos dva predavanja i to:

Cetvrtak, 10.05. 18 casova, sala 2.

MATEMATICKI INSTITUT SANU

(zajednicki sastanak sa Odeljenjem za mehaniku)

Jan Vondrak, Astronomical Institute, Academy of Sciences of the Czech Republic, Prague

MODERN SPACE OBSERVATIONS OF EARTH ORIENTATION PARAMETERS AND THEIR EXCITATION BY GEOPHYSICAL FLUIDS

The motion of Earth's spin axis in space has been monitored for more than 25 years by VLBI, and since 1994 also its rate has been measured by GPS. The method of "combined smoothing", developed recently at the Astronomical Institute in Prague enables to combine both series. The analysis of the combined solution revealed statistically significant deviations from the model. These differences can be identified with the retrograde Free Core Nutation (FCN) and several forced nutation terms. From the direct analysis of celestial pole offsets follows that the period of FCN apparently grew from original 435 days to 460 days during the past ten years. A study of indirect determination of FCN period, based on the observed nutation terms through the resonance effects however shows that this period is in fact close to 430 days and very stable. We also found that a small additional excitation by geophysical fluids (atmosphere, ocean) should exist, near the FCN frequency, to account for the observed celestial pole offsets in this frequency range.

In order to estimate how well the observed geophysical excitation agrees with the celestial pole offsets, we use an alternative 'integration' approach. We integrate numerically Brzezinski's broad-band Liouville equations in celestial reference frame, using appropriately chosen initial conditions. Only the long-periodic part of geophysical excitation (in celestial frame) is used, and external torques exerted by the Moon, Sun and planets are neglected. The results are then compared with the observed celestial pole offsets measured by VLBI with respect to the IAU2000 model of precession-nutation.

It is demonstrated that these small excitations are capable, thanks to a large amplification due to the resonance, to excite the Earth's nutation significantly. The amplitudes of the excited motion are comparable to the celestial pole offsets observed by VLBI. Both amplitudes and phases of individual nutation terms are sensitive to parameters characterizing the internal structure of the Earth. Among these, the flattening of the outer fluid core is dominant since it gives rise to large resonance effects. All nutation terms are affected, but the most sensitive ones are those with periods close to the period of FCN which, in turn, heavily depends on the dynamical flattening of the core. The nutation terms observed by VLBI, with atmospheric and oceanic excitations removed, can thus serve to determine the flattening, and also the quality factor Q of the Earth at the FCN frequency.

(Profesor Vondrak ce odrzati jos jedno predavanje pod naslovom "An improved astrometric catalogue EOC-3 -step towards a better reference frame for long-term Earth orientation studies"u ponedeljak, 14.05.2007. u 13h, Biblioteka Astronomske opservatorije u Beogradu (ul.Volgina 7).)

Utorak, 22.05. 14 casova, sala 2. MATEMATICKI INSTITUT SANU

(zajednicki sastanak sa Seminarom za primenjenu i industrijsku matematiku)

Robert Kooij, Delft University of Technology, Holandija

NASLOV I SADRZAJ CE BITI DOSTAVLJENI NAKNADNO

Rukovodioci Odeljenja za matematiku Matematickog instituta SANU i Opsteg matematickog seminara na Matematickom fakultetu u Beogradu, Stevan Pilipovic i Sinisa Vrecica predlazu zajednicki program rada naucnih sastanaka.

Predavanja ce se odrzavati na Matematickom Institutu (sala 2), petkom sa pocetkom u 14 casova. Odeljenje za matematiku je opsti seminar sa najduzom tradicijom u Institutu.

Svakog meseca, jedno predavanje ce biti odrzano na Matematickom Fakultetu u terminu koji ce biti posebno odredjen.

Molimo sve zainteresovane ucesnike u radu naucnih sastanaka da posebno obrate paznju na vreme odrzavanja svakog sastanka. Na Matematickom fakultetu su moguce izmene termina.

Obavestenje o programu naucnih sastanaka ce biti objavljeno na oglasnim tablama MI (Beograd), MF (Beograd), PMF (Novi Sad), PMF (Nis) i PMF (Kragujevac).

Odeljenje za matematiku Matematickog instituta SANU

Stevan Pilipovic

Opsti matematicki seminar na Matematickom fakultetu u Beogradu,

Sinisa Vrecica

Ako zelite da se obavestenja o Vasim naucnim
skupovima pojave u
Newsletter of EMS (European Mathematical Society)
i na Internetu na lokaciji EMS, onda se obratite na
**emsvesti@mi.sanu.ac.yu**
gde cete dobiti format obavestenja.