Calculus 1. Semester
Limit of functions. Theorems on limits. Continuous functions. The derivative of a function. Rules for finding derivatives. The chain rule. Increments and differentials. Implicit differentiation. Newton's method. Local extrema of functions. Rolle's theorem and the mean value theorem. The first derivative test. Convexity and the second derivative test.
Calculus 2. Semester
Antiderivatives. Definite integral. The mean value theorem. The Newton-Leibniz theorem. Methods of integration (by parts, by substitution etc.) Integration of certain types of functions. Area. Solids of revolution. Arc length and other applications. Exponential and logarithmic functions. Trigonometric functions and their inverses. Hyperbolic functions.
Calculus 3. Semester
Infinite Sequences. Absolute convergence. Power series, Taylor and MacLaurin series. The binomial series. Conic sections. Plain curves. Polar equations of curves. Areas in polar coordinates. Length of curves. Surfaces of revolution. Vector valued functions. Limits, derivatives, integrals. Motion. Curvatures.
Calculus 4. Semester
Multiple integrals, applications, vector calculus, line integrals, surface integrals, Green’s Theorem, Stokes’ Theorem.
Linear Algebra 1. Semester
Introduction to Systems of Linear Equations, Gaussian Elimination, Homogeneous Systems, Matrices and Matrix Operations, Rules of Matrix Arithmetic, Different Methods of Finding the Inverse, Determinant, Properties of Determinant Function, Cofactor Expansion, Cramer’s Rule, Vectors in 2D and 3D, Norm, Dot Product, Projection, Cross Product, Lines and Planes, Vector Spaces, Subspaces, Linear Independence.
Linear Algebra 2. Semester
Basis, Dimension, Orthonormal Basis, Gram-Schmidt Process, Change of Basis, Linear Transformations, Kernel, Range, Matrices of Linear Transformations, Similarity, Eigenvalues, Eigenvectors, Diagonalization of Matrices, Symmetric Matrices.
Numerical Analysis 2. Semester
Floating Point Representation of Numbers, Absolute Error, Relative Error, Rounding Error, Instability, Gaussian Elimination, Pivoting, LU Decomposition.
Maple - first part.
Numerical Analysis 3. Semester
Iterative Methods for Solving Systems of Linear Equations: Jacobi Iteration, Gauss-Seidel Iteration, Nonlinear Equations: Bracketing Methods, Fixed Point Iteration, Newton Method, Eigenvalues, Eigenvectors: The Power Method, Deflation, Jacobi’s Method.
Maple - second part.
Numerical Analysis 4. Semester
Approximation. Polynomial Interpolation: Lagrange Form, Divided Difference Form, Error of the Interpolation, Hermite Interpolation, Cubic Spline Interpolation, Least Squares Approximation to Discrete Data. Numerical Integration: Newton Cotes Formulas, Composite Forms. Ordinary Differential Equations, Initial Value Problems: Euler Method, Trapesoidal Formula, Convergence, Stability.
Maple - third part.
Probability theory and Mathematical statistics 4. Sem.
The notion of probability, elementary properties. Kolmogorov probability field. Combinatorial calculation of probabilities. Conditional probability, properties, calculation. Bayes' theorem. Independency. Random (vector)variable and its distribution, joint distributions. Independent random variables. Random walk and ruin probabilities. Particular discrete distributions. Mean and variance, properties, calculation, inequalities. Median, moments. Covariance and the coefficient of correlation. Distribution and density functions. The distribution of sums of independent random variables (convolution). Particular absolute continuous distributions and their properties. Weak law of large numbers. Central limit theorem. Normal and multivariate normal distribution.
Probability theory and Mathematical statistics 5. Sem.
Statistical space, samples, statistics. Ordered statistics, empirical distribution functions. Unbiased, efficient and consistent estimators. Complete and sufficient statistics. Neyman factorization theorem. Fisher information, Cramer-Rao inequality. Rao-Blackwell-Kolmogorov theorem. Confidence intervals. Maximum likelihood estimators, properties. The method of moments. Hypothesis testing. Comparison of tests. Randomized and sequential tests. The Neyman-Pearson lemma. U-, Student t-, and F-tests, C2-test, and its applications. Linear regression and the method of least squares. The simplest cases of variance analysis.