Petak, 06. maj 2005. u 14h, sala 2 MI SANU BG:

Prof. Mirko Lepovic, Prirodno-matematicki fakultet, Kragujevac
ON INEGRAL GRAPHS WHICH BELONG TO THE CLASS \overline{\alpha K_{a,a,...,a,b,b,...,b}}

Abstract. Let $G$ be a simple graph and let $\bar{G}$ denote its complement. We say that $G4 is integral if its spectrum consists of integral values. Let $K_{xa,yb}$ = $K_{a,a,...,a,b,b...,b}$ be the complete $m$-partite graph with $xa + yb$ vertices, where $x$ and $y$ are positive integers and $m = x + y$. In this work we establish a characterization of integral graphs which belong to the class $\alpha K_{xa,yb}$ , for any $\alpha > 1$ and $a > b$, where $mG$ denotes the $m$-fold union of the graph $G$.

Petak, 13. maj 2005. u 14h, sala 2 MI SANU:

Prof. Radomir Stankovic, Elektronski fakultet, Nis
IZRACUNAVANJE SPEKTRALNIH TRANSFORMACIJA NA KONACNIM GRUPAMA

Sadrzaj. Razmatrace se istorijski razvoj brzih algoritama za izracunavanje diskretnih spektralnih transformacija, sa posebnim osvrtom na problem izracunavnja spektara funkcija velikog broja promenljivih. U predavanju ce najpre biti izlozen istorijski pregled razvoja brze Fourierove transformacije (FFT), razliciti nacini faktorizacije transformacione matrice Diskretne Fourierove transformacije (DFT), a zatim prosirenja na slucaj drugih spektralnih transformacija, na primeru Walshove, Haarove i Reed-Muller transformacije. Diskutovace se veza izmedju strukture dijagrama odlucivanja za predstavljanje diskretnih funkcija i faktorizacije transformacionih matrica, i nacin izracunavanja spektara diskretnih transformacija preko dijagrama odlucivanja.

Petak, 20. maj 2005. u 14h, sala 2 MI SANU:

Professor Jean Francois Pommaret, Ecole Nationale des Points et Chaussees, Francuska
ALGEBRAIC ANALYSIS OF MULTIDIMENSIONAL CONTROL SYSTEMS

Abstract: The purpose of this self-contained lecture will be to provide an introduction to "algebraic analysis" and its application to control theory in order to study control systems defined by partial differential equations, by means of new methods from module theory and homological algebra.We shall revisit a few basic concepts and prove, in particular, that controllability, contrary to a well established engineering tradition or intuition, is an intrinsic structural property of a control system, not depending on the choice of inputs and outputs among the control variables. The link with partial differential algebra will also be presented and illustrated.

Petak, 27. maj 2005. u 14h, sala 718, MF BG:

Justus Diller Institut für Mathematische Logik und Grundlagenforschung der Westfälischen, Wilhelms Universität in Münster, Nemacka
FUNCTIONAL INTERPRETATIONS OF ARITHMETIC IN ALL FINITE TYPES

Abstract: In his 1958 Dialectica interpretation of Heyting arithmetic HA in a quantifier free theory T of primitive recursive functionals of finite type, Gödel reduced the consistency problem of HA to the computability problem of these primitive recursive functionals. We develop functional interpretations that - in contrast to the Dialectica interpretation - extend to Heyting arithmetic in all finite types (HA ù), the natural union or "span" of HA and Gödel's theory T. We thereby prove closure of HAù under a new form of Markov's rule. Following Shoenfield, we also give an interpretation of Peano arithmetic in all finite types (PAù). Finally, we consider hybrids of functional interpretations of HA ù which yield closure of this theory under a rule of choice.

OBAVESTENJE 1

Profesor Pommaret ce pre svog predavanja u okviru Odeljenja za matematiku, odrzati seriju od tri predavanja na temu diferencijalne teorije Galois od ponedeljka 16.5 do 19.5., u tri termina sa trajanjem od po sat vremena.

AN INTRODUCTION TO THE GALOIS THEORY FOR PARTIAL DIFFERENTIAL EQUATIONS

The purpose of this series of lectures is to prove the "equality": DIFFERENTIAL GALOIS THEORY = PHS + PDE along the following schedule:

1) FROM PURE ALGEBRA TO PARTIAL DIFFERENTIAL ALGEBRA. After a brief exposition of the main motivations, this first lecture will sketch the main trends of the formal theory of partial differential equations (PDE)and its application to differential algebra.

2) FROM LIE GROUPS TO LIE PSEUDOGROUPS. Exactly as Lie (algebraic) groups in the Picard-Vessiot theory must be used in place of permutation groups in the classical Galois theory, Lie (algebraic) pseudogroups must be used in the differential Galois theory.

3) FROM CLASSICAL GALOIS THEORY TO DIFFERENTIAL GALOIS THEORY. Our aim will be to prove that differential Galois theory is nothing else than a theory of principal homogeneous spaces (PHS)for Lie pseudogroups, both being defined by systems of algebraic partial differential equations. Roughly speaking, classical Galois theory works for extensions of zero transcendence degree, Picard-vessiot theory works for differential extensions of finite transcendence degree but zero differential transcendence degree while differential Galois theory works for differential extensions with finite differential transcendence degree.

OBAVESTENJE 2

Profesor Diller ce odrzati jos jedno predavanje u Matematickom institutu, u okviru Seminara za logiku, u petak, 27. maja u 16 casova.

FUNCTIONAL INTERPRETATIONS OF SET THEORY

Abstract. We start by developing a theory T* of constructive set functionals of finite type which employs bounded quantifiers only. In analogy to the interpretations of PAù in the theory T, we give a functional interpretation of Kripke-Platek set theory KP in the theory T*. We then present the Burr translation for the interpretation of constructive set theory CZF in all finite types in the theory T* and characterize the strength of this translation. Finally, we study the Schulte hybrid of the Burr interpretation which yields an existence property for CZF which diverges typically from the well known existential definability for HA.

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