### Departments

# Department of Mathematical Sciences

The teachers and associates of the department were involved in research in mathematical calculus (the theory of summability), functional analysis (linear operators), algebra (group theory), geometrical theory of complex function unknowns,mathematical programming - non-linear programming, mathematical modeling, mathematical cybernetics, technical cybernetics, discrete mathematics (theory of automation, theory of functional systems, theory of Boole functions, combinatorial analysis), informatics and the mathematical theory of intelligent systems.

The teachers of the department, in cooperation with the Mathematical Institute of the Serbian Academy of Sciences and Arts, have been organizing a seminar on the theory of automation for several years, and a seminar on the theory of automation and image recognition.

The department has developed institutional scientific and research cooperation with the Department of Discrete Mathematics, and later with the Department of Mathematical Theory of Intelligent Systems at the Faculty of Mechanics and Mathematics, Moscow State University "M. V. Lomonosov".

**Subjects**

**Mathematics 3**

*Ordinary differential equations – *the first-order differential equations, differential equations of the second and higher orders, some applications of ordinary differential equations;

*Systems of ordinary sifferential equations-*definition, methods of solution; some applications of systems of ordinary differential equations;

*Series-numerical series-* definition, properties, converging criteria, *power series*- definition, properties, the domain of convergence, converging criteria, expansion of functions into power series, Taylor’s and Maclaurin’s series.

*Probability *– events,definition of probability, characteristics of probability, conditional probability,total probability theorem, Bayes’ theorem.

**Elements of Probability and Statistics**

*Series-numerical series-* definition, properties, converging criteria, *power series*- definition, properties, the domain of convergence, converging criteria, expansion of functions into power series, Taylor’s and Maclaurin’s series.

*Probability – *definition, characteristics, total probability theorem, Bayes’ theorem, random variable, the most important discrete and continuous probability distributions, numerical characteristics of distributions, the central limit theorem of the calculus of probabilities;

*Statistics *– random sample, examples of the most important statistics, tabular and graphical representation of statistical data, point estimation of distribution parameters, methods of obtaining point estimations , confidence intervals for parameters of normal distribution , parametric hypothesis testing, chi-squarse tests, regression (linear, non-linear).

**Differential Equations**

*Ordinary differential equations – *the first-order differential equations, differential equations of the second and higher orders, some applications of ordinary differential equations;

*Systems of ordinary sifferential equations-*definition, methods of solution; some applications of systems of ordinary differential equations;

*Partial differential equations- *the first order partial differential equations, the second order partial differential equations, numerical solution of partial differential equations, some applications of partial differential equations;

*Laplace transformations-* definition, properties, inverse Laplace transformations, application of Laplace transformations to solving differential equations and systems of differential equations.

**Mathematical Processing of Experimental Data**

*Probability* – definition, characteristics, total probability theorem, Bayes’ theorem, random variable, the most important discrete and continuous probability distributions, multidimensional random variables, the most important multidimensional distributions, numerical characteristics of distributions, numerical characteristics of multidimensional distributions, law of large numbers and central limit theorem of the calculus of probabilities; *Statistics* – random sample, examples of the most important statistics, tabular and graphical representation of statistical data, point estimation of distribution parameters, methods of obtaining point estimations , confidence intervals for parameters of normal distribution , parametric hypothesis testing, non-parametric tests, regression (linear, non-linear, multidimensional).

**Mathematics 1**

Basic elements of modern mathematics-mathematical logic, set theory, real numbers, complex numbers.

Real functions of one real variable-binar relations, elementary functions, polynomial functions, sequences of real numbers, limiting value of sequences, limiting value of functions, continuity of functions.

Derivatives, diferentiation, high order derivatives, fundamental theorems from diferentiation calculus, Taylor's theorem.

Elements of linear algebra and analytic geometry-determinants, matrices, systems of linear equations, vectors, equations of straight lines and plains.

**Mathematics 2**

Integral calaculus—primitive function, the definite integral, inproper integrals, applications of integrals.

Real functions of several real variables-definition, limit and continuity, partial derivatives and diferentiation.

Complex functions of complex variables-definition, differential calculus, elementary functions.

Line and multiple integrals-lines and surfaces, line integrals, double integrals, triple integrals, surface integrals, Green-Stocks and Gauss-Ostrogradski theorems.

Scalar and vector fields-definition, divergence and curl of a vector field, Hamilton's operator.

**Selected topics of mathematical analysis**

Complex functions of complex variable-definition, complex sequences,limit and conitnuity, derivative and differentiability, Cauchy-Riemann equations, integration, Cauchy’s integral formulas, Taylor's and Loran's sereies, residues and residue theorem.

Calculus of variations-unconstrained and constarined minimum of functions of several variables, basic problem of the calculus of variations, problems with high order derivatives.

Series Fourier-ortogonality of trigonometric functions, Dirichle theorem, seriees Fourier of some functions.

**Selected topics of numerical analysis**

Approximate numbers and errors-sources of error, absolute and relative error, aproximate calculations of function values.

Numerical solution of nonlinear equations-bisection method, Newton method, iterative methods, iterative methods for systems of the equations.

Interpolation-the interpolation formula of Lagrange, Newton's interpolation formula, Hermite interpolation, Stirling interpolation Bessel interpolation.

Numerical differentiation and integration-trapeziodal rule, Simpson's rule.

Numerical solution of ordinary differential equations-Euler and Newton method, Runge-Kutta methods, Adam-Moulton methods. Numerical solution of partial differential equations-the finite difference method.