1. P. Zlatoš: Algebra a geometria, skriptá http://thales.doa.fmph.uniba.sk/zlatos/la/LAG_A4.pdf

G. Strang: Linear algebra and its applications, 1976 (ruský preklad 1980);

R. A. Horn, Ch. Johnson: Matrix Analysis, 1990.

2. T. Neubrunn, J. Vencko: Matematická analýza I, II. Skriptá UK Bratislava;

M. Barnovská, K. Smítalová: Matematická analýza III, IV. Skriptá UK Bratislava

Z. Kubáček, J. Valášek: Cvičenia z matematickej analýzy I, II. Skriptá UK Bratislava

(www.iam.fmph.uniba.sk/skripta/kubacek);

M. Barnovská, K. Smítalová: Matematická analýza III, IV. Skriptá UK Bratislava

M. Kollár, Ľ. Kossaczká, D. Ševčovič: Diferenciálny a integrálny počet funkcií viac premenných v

príkladoch, Knižničné a edičné centrum FMFI UK, 2012, tretie doplnené vydanie,

(www.iam.fmph.uniba.sk/institute/sevcovic/skripta/difint)

M. Barnovská a kol.: Cvičenia z matematickej analýzy III. Skriptá UK Bratislava

(www.iam.fmph.uniba.sk/skripta/maiii);

3. P. Brunovský: Diferenčné a diferenciálne rovnice, Skriptá UK Bratislava,

(www.iam.fmph.uniba.sk/skripta/brunovsky); S. H. Strogatz: Nonlinear Dynamics and

Chaos, Addison-Wesley/CRC Press, 2000;

J. Eliaš, J. Horváth, J. Kajan: Zbierka úloh z vyššej matematiky, 3.časť, Alfa, Bratislava, 1980;

4. K. Janková, A. Pázman: Pravdepodobnosť a štatistika, Univerzita Komenského Bratislava 2011;

R. Harman, E. Honschová, J. Somorčík: Zbierka úloh zo základov teórie pravdepodobnosti,

PACI, Bratislava 2009;

T. Amemiya: Introduction to Statistics and Econometrics. Harvard Univ. Press, 1994;

Examination: written and oral in front of the state commission

Slovak, English

1. Linear algebra and matrix calculus

Vector spaces: Linear dependence and independence of vectors, basis and dimension, linear envelope, subspaces, intersection, sum and direct sum of subspaces, projections, coordinate transformations.

Linear mappings: Matrix, kernel and image of linear mapping, coordinate transformations and similarity of linear representation matrices, eigenvalues ​​and eigenvectors, invariant subspaces, characteristic polynomial, criteria of diagonalizability of matrices, Cayley-Hamilton theorem, Jordan normal form.

Bilinear and quadratic forms: Matrix of bilinear form in various bases, symmetric and cososymmetric forms, canonical form of quadratic form, Lagrange's method, Sylvester's law of inertia, positive definite quadratic forms and their matrices, properties of positive definite matrices.

Vector spaces with scalar product: Euclidean vector space, Gram matrix, basic metric concepts, relation of norm and scalar product, Gram-Schmidt orthogonalization process, orthonormal basis.

Gaussian elimination method and LU matrix decomposition, Fredholm alternative, orthogonal matrices and QR-decomposition, orthogonal projections and least squares method, normal system of linear equations.

Complexification of real vector space, realization of complex vector space, complex extension of real linear representation, real matrix of complex linear representation. Hermitian vector space, special complex matrices, Schur decomposition, unitary equivalence of matrices and normal matrices, spectral decomposition of matrices.

2. Mathematical analysis

Functions of one and more variables: Limit, continuity, differentiability, derivation, directional derivation, total differential. Functions specified implicitly, implicit function theorem. Inverse function theorem. Higher order derivatives. Sufficient conditions for the existence of a local extreme of functions of one and more variables.

Numerical series and function series, convergence criteria of numerical series (D’Alembert, Cauchy, Raabe, integral), uniform convergence of function series, Weierstrass's majority criterion. Power series, Taylor evolution of a function of one or more variables. Analytical functions. Fourier series and the criterion of their point convergence. Bessel's inequality and Parseval's equality.

Multivariate function optimization: Free and bounded extremes of multivariate functions. Necessary and sufficient conditions. Lagrange's theorem and Lagrange multipliers

Convex functions. Convexity criteria of functions of one and more variables.

Integration theory: Riemann integral of functions of one and more variables. Integral as a function of the upper bound. Parametric integrals. Continuity and differentiability of parametric integrals. Gamma function and its basic properties. Multidimensional integrals and substitution theorem. Curve integrals. Independence of the curve integral from the integration path, vector field potential. Green's formula of per partes integration for multidimensional integrals.

Standard spaces, complete spaces, open, closed and compact sets. Banach's fixed point theorem and its applications. Properties of continuous functions on compact sets (Weierstrass theorem).

3. Differential and difference equations

Discrete dynamical systems: Equilibrium states and their stability. Trajectory calculation.

Linear differential equations: Solution of autonomous homogeneous equations. Solving nonhomogeneous equations by the method of indefinite coefficients. Stability. Classification of two-dimensional autonomous equations. Fixed point and its stability.

Nonlinear differential equations: Equilibrium states and their stability. Trajectories of autonomous equations. Phase portraits of two-dimensional autonomous equations.

4. Probability and mathematical statistics.

Classical and axiomatic definition of probability. Conditional probability, Bayesian formula.

Random variable: probability distribution, distribution function of a random variable and its properties.

Discrete probability distributions: binomial, hypergeometric, Poisson, geometric. Continuous random variables: density and basic types of distributions: uniform, exponential, normal, mean and dispersion of random variables. Independence of random variables and uncorrelation. Correlation coefficient and its basic properties.

Central limit theorem and the law of large numbers.

Random vectors, mean and covariance matrix of a random vector. Marginal and conditional distributions, convolution of densities. Distributions derived from the normal: chi square, Student's distribution.

Linear model, parameter estimation, least squares method, maximum likelihood method.

0/100

The result will be the completion of the final state examination in the subject of the Mathematical Basis. This means that the student demonstrates an understanding of the basics of each subject in the subject syllabus in their interdisciplinary contexts.

1. J. Plesník, J. Dupacova, M. Vlach: Lineárne programovanie. Alfa, Bratislava 1990

2. M. Hamala, M. Trnovská: Nelineárne programovanie, EPOS, Bratislava 2013.

3. P. A. Samuelson, W. D. Nordhaus: Ekonómia I, II. Bradlo, Bratislava 1992;

B. Felderer, S. Homburg: Makroekonomika a nová makroekonomika. Elita, Bratislava, 1995.

P. Brunovský: Mikroekonómia, učebný text (http://www.iam.fmph.uniba.sk/skripta/brunovsky2)

4. I. Melicherčík, L. Olšarová, V. Úradníček: Kapitoly z finančnej matematiky, EPOS, Bratislava 2005

5. G. Casella a R. L. Berger: Statistical inference (2nd edition). Cengane Learning, 2001.

J. Johnston, J. DiNardo: Econometric Methods. McGraw-Hill, 1997.

Examination: written and oral in front of the state commission

Slovak, English

1. Linear programming

Convex analysis of sets: convex sets and their properties, extreme points of convex sets, supporting hyperplane theorem, theorems on the separation of convex sets, Farkas' lemma.

Simplex method: Geometric idea. Simplex table and algorithm. Two-phase simplex method. Finite simplex method, anticyclic method. Dual simplex method.

Duality theory: General form of a dual problem. Weak theorem about duality and its consequences. Strong duality theorem. Complementarity theorem. optimality verification.

2. Nonlinear programming and free optimization methods

Optimization methods (Overview and basic principles): Minimization of the function of one variable (interval approximation methods and interpolation methods). Minimization of the function of n-variables (gradient method, cyclic coordinate reduction method, Newton's method, combined gradients method and quasi-Newtonian methods).

The concept of Lagrange function, the vector of Lagrange multipliers, the relation of the optimal solution and the associated vector of multipliers with the saddle point of the Lagrange function, nonlinear programming methods using the Lagrange function, generalizations of the Lagrange function, Lagrange's dual problem.

Necessary conditions of optimality: Lagrange's theorem and theorem on sensitivity for the classical problem to a bounded extreme. Kuhn-Tucker theorem for mathematical programming problem (with mixed boundary type), construction of Kuhn-Tucker conditions for general problems.

Overview of basic types of extremes and saddle points. Existence theorems for extreme and saddle point type "minmax". General principle of duality in extremal problems: General concept of dual problem. (Application of Roode's theorem and the "minmax" theorem.)

Convex analysis of functions: definition of convexity, conditions of the 1st and 2nd order of convexity, convexity criteria, operations preserving convexity, quasi-convex functions, strongly convex functions.

Convex programming: Kuhn-Tucker theorem for the convex programming problem. Weak and strong duality theorem. Wolfe's dual problem, Slater's condition and Slater's theorem.

3. Selected chapters of economic theory

Basic macroeconomic variables: gross domestic product, unemployment rate, inflation rate. Commodity market. Aggregate demand and its composition. Equilibrium output of the economy. Commodity market dynamics. Financial markets. Money and bonds. Demand for money, money supply and equilibrium interest rate. The role of the central bank and commercial banks. IS-LM model, balance of goods and financial markets. Fiscal and monetary policy. Expectations and macroeconomic policy. Commodity market in an open economy. Equilibrium output and trade balance. IS-LM model in the case of open economy. Exchange rates. The effectiveness of macroeconomic policy.

Labor market. Wage and pricing. Natural unemployment rate. Aggregate demand and aggregate supply. Inefficiency of monetary policy in the long run. The effectiveness of fiscal policy. Changes in the natural rate of unemployment. Philips curve. Inflation, expected inflation and unemployment. Economic growth. Saving, capital accumulation and economic output. Technological progress and growth.

Consumer: Preferences and utility function. Consumer balance. Marshall and Hicks demand function: Slutsky's equation.

Company: Technological set and production function. Company balance and cost function.

Equilibrium in a sub-market in perfect competition: Short-term equilibrium and equilibrium with free market entry. Impact of taxes and subsidies. Consumers and producers surplus.

Imperfect competition: The balance of the monopoly and its inefficiency. Cournot's equilibrium oligopoly. Cartel instability.

Equilibrium of the complete market: Walras 'law, Walras' equilibrium and its Pareto optimality. Externalities and property rights.

4. Financial mathematics

Interest rates: Coupon and zero-coupon bonds. Time structure of interest rates. Construction of the time structure of interest rates using bond market prices (bootstrapping). Net present value. Yield to maturity. Forward interest rates. Duration and change in the value of the bond portfolio with a parallel shift in interest rates.

Portfolio theory: Utility function, risk aversion, optimal portfolio selection by maximizing the mean value of the utility function. Markowitz model. Risk minimization with fixed return, optimization of the portfolio containing risk-free securities, market price of risk. Capital Asset Pricing Model as Equilibrium Model, Capital Market Line, Security Market Line.

Fundamentals of derivative valuation theory: Binomial tree model, calculation of risk-neutral probabilities and values ​​of derivatives based on them. Self-financed strategies and derivatives replication.

5. Statistical methods and econometrics

Properties of diameter and selective dispersion. Student's t-tests and F-test of dispersion equality. Confidence intervals for mean and dispersion. UMP-tests and the Neyman-Pearson lemma. Estimates of BUE and Cramér-Ra inequality. MLE estimation properties. Wald's test

0/100

The result will be the completion of the final state exam in the subject: Mathematical Methods. This means that the student demonstrates an understanding of the basics of each subject in the subject syllabus in their interdisciplinary contexts.