Petak, 13. april 2007. u 14h, sala 2, SANU, BG:

Ljubica Velimirovic, Prorodno-matematicki fakultet, Nis
INFINITESIMAL BENDING

Abstract: The problem of infinitesimal bending of surfaces is a special part of Differential geometry. Under infinitesimal bending surface is included in continuous family of surfaces, so that the arc length is stationary. We are giving a brief historical overview of some results at the bending theory. Especialy, infinitesimal bending of the first, the second and the higher order for the toroidal rotational surfaces with polygonal meridian is given. Variation of the volume at the level of infinitesimal bending for rotational surfaces is examined.

Infinitesimal bending of a closed plane curve and variation of some geometric magnitudes (curvature, torsion, area and the volume) in relation with infinitesimal bending of a curve are considered.

We also consider graphical presentation of surface bending.

At the second part we consider infinitesimal deformations $$f:x^i\rightarrow x^i+\varepsilon z^i(x^j) $$ of a space $L_N$ with non-symmetric affine connection $L{}^{i}_{jk} $. Based on the non-symmetry of the connection, we use four kinds of covariant derivative to express the Lie derivative and the deformations. Rigidity of geometric objects (connection, tensors, curvature) is defined by virtue of Lie derivative.

Petak, 20. april 2007. sala 2 MI SANU:

Ljiljana Petrovic, Ekonomski fakultet, Beograd
STATISTICAL THEORY OF CAUSALITY

Abstract: The different concepts of causality between filtrations and between stochastic processes are considered. Then, some causality concepts are applied on stochastic filtering theory, control theory and on regular solutions of stochastic differential equations. The equivalence between some models of causality and weakly uniqueness of regular solutions is shown. Also, we show that the given concepts of causality is closely connected to the concept of extremality of measures and links Granger's causality with the concept of adapted distribution.

Petak, 27. april 2007. u 14h, sala 718, MF BGD:

Igor Dolinka, Departman za matematiku i informatiku, PMF, Novi Sad
PROBLEMI KONACNE BAZE IDENTITETA ZA KONACNE POLUPRSTENE

Sadrzaj: Neka Eq(A) oznacava skup svih identiteta koji vaze na algebri A. Kazemo da A ima/nema konacnu bazu ukoliko Eq(A) jeste/nije konacno aksiomatizabilno u okviru (standardne) jednakosne logike. Na primer, konacnu bazu identiteta imaju: sve konacne grupe (Oates, Powell, '64.), svi konacni prsteni (Ljvov, Kruse, '73.), sve konacne mreze (Mckenzie, '70.) i sve komutativne polugrupe (Perkins, '68.). S druge strane, Murskij je 1965. konstruisao 3-elementni grupoid koji nema konacnu bazu, a prvi primer polugrupe bez konacne baze dao je Perkins 1968. (u pitanju je cuveni 6-elementni Brandtov monoid B21). 1996. McKenzie je pokazao da ne postoji algoritam koji odlucuje da li data konacna algebra ima konacnu bazu - sto govori da je problem konacne baze prilicno netrivijalan...

Problemi konacne baze predstavljaju veoma zivo polje istrazivanja u teoriji polugrupa. U ovom predavanju, autor ce prezentovati svoju "inicijativu" u smislu istrazivanja analognog problema za poluprstene, kao i odgovarajuce novije rezultate. Oni ukljucuju:
- prvi primer konacnog poluprstena bez konacne baze identiteta (najmanji takav primer ima samo 7 elemenata);
- razmatranja u vezi neprebrojivih intervala u mrezi varijeteta poluprstena (a sto je u tesnoj vezi sa pitanjem konacne baze);
- prve primere konacnih poluprstena sa nasledno beskonacnom bazom. (Konacna algebra A ima nasledno beskonacnu bazu (eng. INFB) ako za svaki lokalno konacni varijetet V koji sadrzi A vazi da Eq(V) nema konacnu bazu. Potpunu karaterizaciju INFB polugrupa dao je M.V.Sapir 1987. sto i danas vazi za jedno od najznacajnijih dostignuca u teoriji polugrupa.) Izmedju ostalog, u okviru trece tacke autor daje dovoljan uslov da poluprsten bude INFB, sto povlaci da tu osobinu imaju poluprsteni binarnih relacija Reln za sve n takve da je 2 <= n< \omega, kao i sve algebre matrica Mn(\Sigma ) (n >= 2) nad svakim aditivno idempotentnim poluprstenom \Sigma koji ima nenula idempotent.

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