ON THE CURRENT STATE OF COMPUTATIONAL MECHANICS

Mladen Berković

Faculty of Mathematics, Belgrade, Yugoslavia

In the paper the research methods and application areas in this contemporary and relatively new branch of science, connecting mathematics, mechanics, computer science and engineering sciences. Presently it is widely recognized that computational mechanics is of vital importance in the transition of the low technology industry into the high technology industry, i.e. for the economy and the security of the country.

In the first part of the paper the current state of the area worldwide will be presented. The second part is devoted to the state and results in computational mechanics in this country, while in the third part the work in the area performed at the Faculty of Mathematics Belgrade is described.

SYMMETRIES OF THE PSEUDO-RIEMANNIAN MANIFOLDS AND THE OSSERMAN CONDITION

Novica Blažić

Faculty of Mathematics, Belgrade, Yugoslavia

Studying symmetric type properties is very important topic in the pseudo-Riemann geometry: locally symmetric manifolds, Killing vector fields, holonomy groups, homogeneous manifolds,... Particulary are interesting Jacobi operator since Jacobi vector fields are usefull for estimating exponential map. Also, it is used in describing local dynamic of Lorentzian space-time (the second Newton law). In the 80’s, Sarnak and Osserman studied the entropy of geodesic flow of nonpositively curved manifolds and establish an important class of manifolds with algebraic structure of the Jacobi operator independent of direction and the point (known as Osserman manifolds). Osserman conjecture states that manifolds has to be locally symmetric of rang one spaces or flat.

In the communication will be presented joint results with Neda Bokan, Peter Gilkey and Zoran Rakić related to the Osserman conjecture. The conjecture is still open in the Riemannian setting, but in the Lorentzian setting this conjecture holds. For four-dimensional manifolds of neutral signature (− − ++), the conjecture doesn’t hold, i.e. there exist Osserman manifolds which are not even locally homogeneous manifolds. But, we establish that this manifolds posses some intrinsic symetries such as:

-isotropic derivative of the curvature tensor

-self-duality and integrability of the twistor space

-foliated by totally geodesic, flat, isotropic two dimensional submanifolds.

SELECTED TOPICS FROM THE THEORY OF GRAPH SPECTRA

Dragoš Cvetković

University of Belgrade, Faculty of Electrical Engineering

The spectrum of a graph is defined as the spectrum of any square matrix which is in a prescribed way associated to a graph. Two variants are most frequently used: the spectrum based on the adjacency matrix and the spectrum based on the Laplacian matrix of a graph. Graph eigenvalues can be successfully used in solving many graph theory problems. After a short introduction to the theory of graph spectra, we discuss a number of topics which are of interest in current research : enumeration of spanning trees, graph angles, cospectral graphs with the same angles, graph reconstruction based on eigenvalues and angles, star partitions, etc.

THE USE OF MATHEMATICAL MODELS FOR THE REPRESENTATION OF LINGUISTIC KNOWLEDGE

Maurice Gross

Université Paris VII Laboratoire d’automatique documentaire et linguistique

Various mathematical models of language have been proposed in the last half century, starting with the work of logicians such as Carnap, whose views led to the formalization of mathematical assertions in the form of logical formulas and systems. More specific models aimed at natural language descriptions followed, such as Lambeck’s categorial grammars, trees of various shapes (e.g. phrase structure and dependency trees) and finite state processes such as Hockett’s boxes. These models were reviewed by Chomsky and their defects emphasized with respect to transformational grammars.

We present a position closed to Z.S. Harris’ theory, which relies on elementary algebraic tools: Finite state automata and specific equivalence relations between syntactic structures. We show that these tools are empirically adequate from the strict linguistic point of view and that they are well-adapted to the computer analysis of texts.

ON SOME RECENT RESULTS IN THE THEORY OF ZETA FUNCTION

Aleksandar Ivić

Department of Electrical Engineering, University of Belgrade

This review lecture is devoted to the Reimann zeta fuction ζ(s), which plays a prominent rôle in Analytic Number Theory. Several topics will be discussed, and some of the recent results will be presented. These include the distribution of zeros of ζ(s) (with mention of the famous Riemann Hypothesis), zero density results, and power moments of ζ(s). Major advances have been recently obtained on error terms in the asymptotic formulas for ∫_{0}^{T} |ζ(1/2 + it)|^{2k}dt when k = 1 or k = 2. The latte is connected with powerful methods of spectral theory and sums over discrete spectrum of the non Euclidean Laplacian acting on SL(2, Z)- automorphic forms, and it is a very active area of research.

DIFFERENCE SCHEMES FOR PARTIAL DIFFERENTIAL EQUATIONS WITH GENERALIZED SOLUTIONS

Boško Jovanović

Faculty of Mathematics, Belgrade, Yugoslavia

A survey of results concerning convergence of difference schemes for partial differential equations of elliptic type satisfies convergence rate estimates of the type

∥u_{h} − u∥_{W_{2,h}^{k}} ≤ Ch^{s−k}∥u∥_{W_{2}^{s}}

where u and u_{h} are solutions of continuous and discretized problem, while h is discretization parameter. Such estimates can be obtained by using integral representation of residual or applying Brambe-Hilbert lemma. In the case of noniteger s and k Dupont-Scott lemma or interpolation of function spaces may be used.

Special attention is given to a priori estimates, averaging operators, problems with variable coefficients, nonstationary problems and finite difference schemes on nonuniform meshes.

ON EUCLIDEAN RINGS

Gojko Kalajdzić

Faculty of Mathematics, Belgrade, Yugoslavia

In this talk we describe the development of the idea of finding the greatest common divisor of two integers (Euclidean algorithm) and its role in the development of other algebraic notions (groups, rings). We give a historic overview of some important results related to euclidean rings, as well as a selection of open problems.

ASYMPTOTICAL CHARACTERISTICS OF THE SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS

Julka Knežević-Miljanović

Faculty of Mathematics, Belgrade, Yugoslavia

Asymptotic property of solutions has been considered for some nonlinear differential equations. The paper deals with investigation of bounded solutions, of prolongation of solutions, oscillatory solutions and another asymptotic property.

CHAOS IN THE ASTEROID MOTION

Zoran Knežević

Astronomical Observatory, Belgrade

The motions of all the bodies in the planetary system are basically chaotic, only the dynamical mechanisms giving rise to chaos, time scales involved and the extent of the resulting macroscopic instabilities are different. We briefly describe how to recognize and how to measure this chaos, and discuss the typical asteroidal ranges for the applied measures. We also introduce the “stable chaos” phenomena, and show how they can be used as a diagnostic tool for the study of the asteroid dynamics.

Among the examples of observed features of the asteroid main belt understood as due to slow chaotic diffusion, we thus first describe an attempt to estimate the upper limit to the age of an asteroid family, based on the “chaotic chronology” method. More than 500 “clones” of 5 chaotic real bodies, members of the Veritas family, are integrated for 100 Myr in the past, the output is analysed in terms of the metrics used to define the family, and the preferential time of escape from the region in the phase space occupied by the family is looked for by means of the straightforward statistical methods. It was found that the family is probably younger than 100 Myr, which would make it the youngest one originated from such a big parent body.

Next we present the most recent results regarding the mechanisms of transport of the meteoritic material from the main asteroid belt to the Earth. The “leakage” through the mean motion resonances and the strong ν_{6} secular resonance, and pumping up to the Mars-crossing values of the orbital eccentricity are demonstrated, and the chaos due to the subsequent close approaches to Mars indicated as responsible for the insertion of these bodies into the vicinity of our planet. We also discuss the possible chaotic evolution of the orbits of some larger asteroids (433 Eros), presently on Mars-crossing orbits, and the probability of such a body impacting the Earth in the future.

RECENT ADVANCES IN DEVELOPMENT OF METHODS AND SOFTWARE FOR NONLINEAR FINITE ELEMENT ANALYSIS AT UNIVERSITY OF KRAGUJEVAC

Miloš Kojić

Faculty of Mechanical Engineering, University of Kragujevac

An overview of research at Laboratory for Engineering software of Faculty of Mechanical Engineering, and at Center for Scientific Research of Serbian Academy of Sciences and Arts and University of Kragujevac, is presented in the paper. The following fields are included: Nonlinear Structural Analysis, especially inelastic deformation and some elements formulations; field problems, such as 3D seepage with free surface, coupled problems flow through porous deformable solids with heat transfer, and general solid-fluid interactions; biomechanics with emphasis on nonlinear tissue behaviour, blood flow with blood vessel deformation, lung and cartilage problems. A large number of solved examples, that illustrate theoretical developments and include complex engineering problems, is provided.

ON TANGENT SPHERE BUNDLES WITH ARBITRARY CONSTANT RADIUS

Oldrich Kowalski

Faculty of Mathematics and Physics, Charles University, Prague

Many papers have been devoted to the geometry of tangent bundles equipped with the Sasaki metric over a Riemannian manifold. Also the corresponding geometry of the unit tangent sphere bundles has been studied by many authors. The aim of the present research is to show how the geometry of a tangent sphere bundle /with an arbitrary constant radius/ depends on its radius. In particular, we present some rather surprising results for the cases when the radius is very small or, to the contrary, very large.

OPERATOR THEORY AND FUNCTIONAL ANALYSIS

Nebojša Lažetić

Faculty of Mathematics, Belgrade, Yugoslavia

In this lecture we present a short review of research accomplished in the last decade by people from Department for real and functional analysis of Faculty of Mathematics.

There is a variety of results in different areas such as asymptotic analysis, fixed point theory, geometry of Banach spaces, locally convex spaces, operator theory, spectral analysis of differential and integral operators etc.

Some of these results, published in most eminent mathematical journals, are described.

THE MODULE OF THE IMAGE ANNULI UNDER UNIVALENT HARMONIC MAPPINGS AND A CONJECTURE OF J. C. C. NITHSCHE

Abdallah Lyzzaik

Mathematics Department, AUB

The object of the talk is to show that if f is a univalent harmonic mapping of the annulus A(r, 1) = {z : r < |z| < 1} onto the annulus A(R, 1), and if s is the length of the segment of Grotsch’s ring domain associated with A(r, 1), then R < s. This gives the first quantitative upper bound of R which relates to a question of J. C. C. Nitsche that he raised in 1962. The question whether this bound is sharp remains open.

SOME GEOMETRIC QUESTIONS RELATED TO HARMONIC MAPS

Miodrag Mateljević

Faculty of Mathematics, Belgrade, Yugoslavia

We prove some versions of classical Ahlfors-Schwarz lemma for holomorphic and harmonic maps.

Some applications of the obtained results to the fixed point theory and complex dynamics are given.

Further, we study the images of harmonic diffeomorphisms of C into hyperbolic plane. Among other things, emphasize that our results give answers to a number of popular open problems in some cases.

Also, we study uniqueness of harmonic map in its homotopy class and get some new results using new version of Reich-Strebel inequality.

MÜNTZ ORTHOGONAL SYSTEMS AND APPLICATIONS

Gradimir V. Milovanović

University of Niš

Polynomial systems are very attractive in approximation theory as well as in many applications in mathematics, physics, and other computational and applied sciences. The orthogonal polynomial systems, especially classical orthogonal polynomials, play very important role in many problems in approximation theory and numerical analysis. In this survey we consider generalized polynomial systems and their applications.

Let Λ = {λ_{0}, λ_{1}, λ_{2}, …} be a complex sequence. A linear combination of the system {x^{λ_{0}}, x^{λ_{1}}, …, x^{λ_{n}}} is called a Müntz polynomial, or a Λ-polynomial. By M_{n}(Λ) we denote the set of all such polynomials, i.e.,

M_{n}(Λ) = span{x^{λ_{0}}, x^{λ_{1}}, …, x^{λ_{n}}},

where the linear span is over the real (or complex) numbers. Such generalized polynomials can be orthogonalized (see [?]–[?]) and applied to several approximation problems including quadrature problems.

We investigate two Müntz systems which are orthogonal with respect to some inner products. Beside the general properties including some representations and recurrence relations, we consider a few interesting special cases of generalized systems. In particular, the systems regarding to the real sequence Λ, as well as the case when some of λ’s are equal, are also considered. Zero distribution of such systems is investigated. Also, we connect orthogonal Müntz systems with Malmquist systems of orthogonal rational functions in the complex plane.

A direct evaluation of Müntz polynomials P_{n}(x) in the power form can be problematic in finite arithmetic, especially when n is a large number and x is close to 1. As a rule, such polynomials are ill-conditioned. We discuss such problems regarding to single, double, and Q-arithmetics. A new approach in numerical evaluation of these polynomials and their derivatives is given (see [?]).

A numerical algorithm for the construction of generalized Gaussian quadratures was originally introduced over three decades ago by Karlin and Studden, and recently investigated by Ma, Rokhlin and Wandzura [?]. Using theory of orthogonality for Müntz systems, we present an alternatively numerical method for constructing generalized Gaussian quadrature rules

∫_{0}^{1} f(x)dσ(x) = ∑_{k=1}^{n} A_{k}f(x_{k}) + R_{n}(f),

which are exact for each f ∈ M_{2n−1}(Λ). Here dσ(x) is a given nonnegative measure on [0, 1]. Our construction is quite different from the corresponding procedure for the classical Gaussian integration formulae with an algebraic degree of precision.

References

[1] G.V. Badalyan, Generalization of Legendre polynomials and some of their applications, Akad. Nauk. Armyan. SSR Izv. Ser. Fiz.-Mat. Estest. Tekhn. Nauk 8 (5) (1955) 1–28 and 9 (1) (1956) 3–22 (Russian).

[2] P. Borwein, T. Erdélyi, and J. Zhang, Müntz systems and orthogonal Müntz-Legendre polynomials, Trans. Amer. Math. Soc. 342 (1994) 523–542.

[3] B. Danković, G.V. Milovanović, and S. Lj. Rančić, Malmquist and Müntz orthogonal systems and applications, In: Inner Product Spaces and Applications (Th. M. Rassias, ed.), 22–41, Addison Wesley Longman, Harlow, 1997.

[4] G.V. Milovanović, Müntz orthogonal polynomials and their numerical evaluation, Oberwolfach Meeting “Applications and Computation of Orthogonal Polynomials” (March, 1998) (to appear).

[5] G.V. Milovanović, B. Danković, S. Lj. Rančić, Some Müntz orthogonal systems, J. Comput. Appl. Math. 99 (1998), 299–310.

[6] J. Ma, V. Rokhlin, and S. Wandzura, Generalized Gaussian quadrature rules for systems of arbitrary functions, SIAM J. Numer. Anal. 33 (1996), 971–996.

ON EXTREME VALUES OF RANDOM VARIABLES

Pavle Mladenović Faculty of Mathematics, Belgrade, Yugoslavia

The talk will be organized as follows: Introduction Classical extreme value theory Domains of attraction Examples Stationary random sequences Stationary Gaussian sequences Waiting time in a sequence of experiments Stationary random processes with continuous parameters Stationary Gaussian processes Non-stationary and non-Gaussian processes Maximal deviation of empirical distribution function Maximal deviation of spectral density estimate ON APPLIED ALGEBRAIC TOPOLOGY

Zoran Z. Petrović

Faculty of Mathematics, Belgrade, Yugoslavia

Ideas and methods of Algebraic Topology has pervaded many branches of today’s Mathematics and Theoretical Physics. And although Applied Algebraic Topology does not appear in Mathematics Subject Classification, there are many mathematicians in the world today who think of themselves as applied algebraic topologists. Here we present some of the numerous applications of Algebraic Topology to problems in Geometry, Topology and Algebra. Special emphasis is put on the work of the group of Belgrade topologists.

SOME ASPECTS OF LOCAL AND MICROLOCAL ANALYSIS

S. Pilipović Faculty of Sciences and Mathematics, Institute of Mathematics, University of Novi Sad,

The aim of this survey article is to present some parts of investigations of my collaborators and myself concerning local and microlocal analysis on extended real domains.

Local and microlocal properties of distributions, ultradistributions and hyperfunctions have been extensively studied in the last three decades. Applications were made in the analysis of linear problems while the non-linear ones required constructions of new algebras in which the product and some other non-linear operations are defined. This was the motivation for introducing the space of Colombeau generalized functions G and the developement of microlocal analysis in this space.

In the framwork of local analysis, we analyze the behavior of a generalized object at a point (quasiasymptotic and s-asymptotic behaviour); in the case of point ∞, this leads us to Wiener Tauberian type results.

Microlocal point of view is presented through the microlocal decomposition in some spaces of hyperfunctions. Also a new concept of a wave front in Colombeau space G is given.

SOFTWARE AGENTS: FORMALIZATION OF THE AGENT MODEL

Vojislav Stojković

Morgan State University, Baltimore, Maryland, USA

Artificial intelligence is the subfield of computer science that aims to design intelligent agents.

The concepts of ”agents,” ”agent-oriented programming languages (AOP languages),” ”agent-oriented programming systems (AOP systems),” ”agentoriented programming (AOP),” ”agent programs,” ”agents theory,” ”agent model,” etc. are central concepts of artificial intelligence. They are also one of the most important concepts in some other subfields of computer science, communications, robotics, and user interfaces.

Unfortunately some of these terms are still noise terms, subjects to abuse, misuse, and confusion even in the artificial intelligence agent community.

This article tries to define and explain the following important terms and topics of agent theory:

- Agents, rational agents, ideal agents, and information retrieval (network) agents - Agents versus non agents - Agent-oriented programming languages (AOP languages) AGENT0, APRIL, PLACA, Concurrent MetateM, TELE-SCRIPT, ABLE, etc. - Agent-oriented programming systems (AOP systems)

- Agent-oriented programming (AOP) - Agent-oriented programming (AOP) versus object-oriented programming (OOP) - Description of agents - Implementation of agents - Agent programs - Types of agents - Information retrieval (network) agent programs - Agents theory - Agent model

The paper includes original results on the formalization of the agent model.

THEORY OF AUTOMATON AND INTELLECTUAL SYSTEMS

Aleksandar Strogalov

Moscow State University

Many interesting results were obtained in this area by group of mathematicians from Moscow State University and other Russian universities during last years. These results concerning the problem of completness of automaton mappings, approximation of their behaviour and modelling of real processes on cellular automaton (homogenious structure). This results are the theoretical basis for development and synthesis of intellectual systems in many fields of human activty, in particular, computer learning systems. Some learning systems will be shown during this lecture.

ASTRONOMICAL REFERENCE FRAME, HIPPARCOS AND EARTH ORIENTATION

Jan Vondrak

Academy of Sciences of the Czech Republic

The International Astronomical Union recently adopted the new International Celestial Reference System (ICRS) and its realization in radio wavelength, International Celestial Reference Frame (ICRF). Its main realization in optical wavelength is the Hipparcos Catalogue that has recently been used to re-analyse the observations of latitude and universal time variations by the methods of optical astrometry since the beginning of the century. The resulting long series of Earth orientation parameters, covering the interval 1899.7-1992.0, is further combined with the series obtained by space geodetic methods (SPACE97), covering the interval 1976.75-1997.0, and the long-periodic behaviour of Earth orientation is analysed.

MODERN DEVELOPMENTS OF THE TEORY OF BERGMAN SPACES

Dragan Vukotić

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain dragan.vukoticuam.es

Over the last two decades we have witnessed some remarkable developments in the theory of Bergman spaces (of area-integrable analytic functions in the disk). This theory has a very diverse nature, which allows for combining techniques from different fields, ranging from Geometric Function Theory and Classical Real Analysis to Functional Analysis and Operator Theory. One of the sources of inspiration is the very rich knowledge of the (closely related) Hardy spaces which enables us to carry out a comparative study. However, the Bergman spaces exhibit a number of peculiarities and additional difficulties, and therefore often have to be dealt with by quite different methods.

The main objective of this lecture is to give a brief survey of the history of the subject, showing some of the main ideas and important achievements, obtained recently by numerous authors and groups of researchers. Some of the presenter’s own results will also be mentioned occasionally.

The talk will begin with a brief overview of the basic necessary facts about Bergman spaces in order to get the audience familiarized with the topic. The idea is to avoid communicating many tedious technical details, while indicating instead some directions of the current research and stating some of the challenging problems that stand open in the current theory.