Basic requirements for specialty 05.13.18
Basic requirements
at qualifying exams for admission to postgraduate study
05.13.18 “Mathematical modelling,
numerical methods and programs’ complexes”
in Physics, Mathematics and Engineering sciences
Introduction
The requirements are based on the following courses’ material: Functional analysis, Mathematical Physics, Theory of probability, Mathematical statistics and Numerical methods.
1. Mathematical basis
Elements of theory of function and functional analysis. Concept of measure and Lebesgue integral. Metric and normalized space. Integrated functions spaces. Sobolev space. Linear continuous functional. Banach-Hahn theorem. Linear operators. Spectral theory elements. Differential and integral operators.
Extremum problems. Convex analysis. Extremum problems in Euclidean space. Convex minimum problems. Mathematical programming, linear programming, convex programming. Minimax problems. Variational calculus basics. Optimal management problems. Maximum principle. Dynamical programming principle.
Theory of probability. Mathematical statistics. Probability theory axiomatics. Probability, transitional probability. Independency. Random variables and vectors. Elements of correlation theory for random vectors. Elements of random processes theory. Point and interval estimation of distribution parameters. Elements of statistical hypothesis check theory. Elements of polydimensional statistical analysis. Basic concepts of statistical decisions theory. Information theory basics.
2. Information technologies
Decision taking. General decision problem. Regret function. Bayesian and minimax approaches. Sequential decision method.
Study of artificial intelligence operations and tasks. Expertise and nonformal procedures. Design automation. Artificial intelligence. Pattern recognition.
3. Computer technologies
Numerical methods. Interpolation and approximation of functional dependences. Numerical differentiation and integration. Numerical methods in extremum detection. Computational method of linear algebra. Numerical methods for solution of differential equations systems. Spline-approximation, interpolation, finite element method. Transformation of Fourier, Laplace and Haar, etc. Numerical methods of wavelet analysis.
Computational experiment. Computational experiment principles. Model, algorithm, program.
Algorithmical languages. Concept of high-level programming languages. Application programs packages.
4. Mathematical modelling methods
Basic principles of mathematical modelling. Elementary mathematical models in mechanics, hydrodynamics and electrodynamics. Mathematical models flexibility. Methods of mathematical models development basing on fundamental natural laws. Variational methods of mathematical models building.
Methods of mathematical models studying. Sustainability. Check of mathematical models adequacy.
Mathematical models in scientific studies. Mathematical models in statistical mechanics, economics and biology. Mathematical modelling methods for measuring and computing systems.
Reduction tasks in ideal device. Synthesis of output signal from ideal device. Checking of measuring model adequacy and reduction outcome adequacy.
Dynamic systems models. Crucial points. Bifurcation. Dynamic chaos. Ergodism and hashing. Concept of self-organization. Dissipative structures. Blow-up regimes.
Basic references
Kolmogorov A.N., Fomin S.V. Functional analysis. M.: Nauka, 1984.
Vasiljev F.P. Numerical methods for extremum problems solving. M.: Nauka, 1981.
Borovkov A.A. Theory of probability. M.: Nauka, 1984.
Borovkov A.A. Mathematical statistics. M.: Nauka, 1984.
Kalitkin N.N. Numerical methods. M.:Nauka, 1978.
Samarsky A.A., Mikhailov A.P. Mathematical modelling. M.: Phizmatlit, 1997.
Mathematical modelling / edited by A.N. Tikhonov, V.A. Sadovnichy and others. M.:P.H. MSU, 1993.
Lebedev V.V. Mathematical modelling of socio-economic processes. M.:IZOGRAPH, 1997.
Petrov A.A., Pospelov I.G., Shananin A.A. Experience in mathematical modelling of economy. M.: Energoatomizdat, 1996.
Pytjev Yu.P. Mathematical modelling methods for measuring and computing systems. M.: Phizmatlit, 2002.
Related references
Tikhonov A.N., Arsenin V.Ya. Solving methods of ill-defined problems.M.: Phizmatlit, 2002.
Pytjev A.N. Mathematical methods for experiment analysis. M.: Vyssh.shkola, 1989.
Chulichkov A.I. Mathematical models for nonlinear dynamics. M.: Phizmatlit, 2000.
Demyanov V.F., Malozemov V.N. Introduction into minimax. M.: Nauka, 1972.
Krasnoschekov P.S., Petrov A.A. Models building principles. M.: P.H. MSU, 1984.
Ventsel Ye.S. Operations study. M.: Sov.Radio, 1972.
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