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.
Studying symmetric type properties is very important topic in the pseudoRiemann 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 spacetime (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 Raki\'c related to the Osserman conjecture. The conjecture is still open in the Riemannian setting, but in the Lorentzian setting this conjecture holds. For fourdimensional 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:
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.
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 welladapted to the computer analysis of texts.
This review lecture is devoted to the Reimann zeta fuction z(s), which plays a prominent r^ ole 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 z(s) (with mention of the famous Riemann Hypothesis), zero density results, and power moments of z(s). Major advances have been recently obtained on error terms in the asymptotic formulas for ò_{0}^{T} z(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.
A survey of results concerning convergence of difference schemes for partial differential equations of elliptic type satisfies convergence rate estimates of the type

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 BrambeHilbert lemma. In the case of noniteger s and k DupontScott 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.
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.
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.
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 n_{6} secular resonance, and pumping up to the Marscrossing 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 Marscrossing orbits, and the probability of such a body impacting the Earth in the future.
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 solidfluid 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.
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.
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 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.
We prove some versions of classical AhlforsSchwarz 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 ReichStrebel inequality.
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 L = {l_{0},l_{1},l_{2},¼} be a complex sequence. A linear combination of the system {x^{l0},x^{l1},¼,x^{ln}} is called a Müntz polynomial, or a Lpolynomial. By M_{n}(L) we denote the set of all such polynomials, i.e.,

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 L, as well as the case when some of l'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 illconditioned. We discuss such problems regarding to single, double, and Qarithmetics. 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

The talk will be organized as follows:
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.
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 nonlinear ones required constructions of new algebras in which the product and some other nonlinear operations are defined. This was the motivation for introducing the space of Colombeau generalized functions \cal 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 sasymptotic 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 \cal G is given.
Artificial intelligence is the subfield of computer science that aims to design intelligent agents.
The concepts of ägents," ägentoriented programming languages (AOP languages)," ägentoriented programming systems (AOP systems)," ägentoriented programming (AOP)," ägent programs," ägents theory," ägent 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  Agentoriented programming languages (AOP languages) AGENT0, APRIL, PLACA, Concurrent MetateM, TELESCRIPT, ABLE, etc.  Agentoriented programming systems (AOP systems)  Agentoriented programming (AOP)  Agentoriented programming (AOP) versus objectoriented 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.
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.
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 reanalyse 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 1899.71992.0, is further combined with the series obtained by space geodetic methods (SPACE97), covering the interval 1976.751997.0, and the longperiodic behaviour of Earth orientation is analysed.
Over the last two decades we have witnessed some remarkable developments in the theory of Bergman spaces (of areaintegrable 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.