BSC IN MATHEMATICS

During the six-semester Mathematics BSc program students acquire skills in pure and applied mathematics which enable them to pursue successful Master’s studies at inland or foreign leading universities or get employed in different areas of technology, economics, statistics and informatics. Profiting of the environment given by the University of Technology and Economics we train experts who are interested in practical problems and are able to use their knowledge creatively. In addition to being familiar with abstract fields of mathematics, they are able to communicate and collaborate with representatives of other professions. Through extensive relationships of our Institute our students can gain an insight into various fields of applications of mathematics and mathematical modelling of real life problems.
Students getting a BSc degree in Mathematics at our university can quickly and easily find a decent high-paying job either in Hungary or abroad. Several banks, investment funds, insurance, business consulting companies as well as those engaged in data mining or optimization employ mathematicians in senior positions. After receiving the BSc degree students can be admitted to the Applied Mathematics or Mathematics master program or other MSc programs subject to special conditions.
Special characteristics of our BSc program are the tutorial system, a large number of homework problems, opportunities to take part in project works and to get involved in high-level scientific research and a significantly higher than average personal attention paid to students thanks to their relatively small number.

CURRICULUM from 2024

Code

Title

Parameters*

ECTS credits
per semester

III

Obligatory courses (140 ECTS credits)

BMETEAGBsMMMOD-00

Mathematical Methods

BMETE92AP61

Calculus

BMETEAGBsMVMAL-00

Vector and Matrix Algebra for Mathematicians

BMETEAGBsMGE1E-00

Geometry 1e

BMETE91AM42

Informatics 1

BMETE92AP62

Multivariable Calculus

BMETEAGBsMBALG-00

Introduction to Algebra

BMEVISZA025

Combinatorics and Graph Theory 1

BMETEAGBsMGE2E-00

Geometry 2e

BMETE91AM43

Informatics 2

BMEGT35A410

Accounting

BMETEAOBsMAN1E-00

Analysis 1e

BMETE91AM38

Algebra 1

BMETE95AM29

Probability Theory 1

BMETE91AM56

Programming Exercises for Probability Theory

BMETE93AM15

Differential Equations 1

BMETE91AM44

Informatics 3

BMETE95AM31

Mathematical Statistics 1

BMETEAOBsMAN2E-00

Analysis 2e

BMETE94AM26

Differential Geometry 1

BMETE93AM19

Operations Research

BMEVISZAA08

Theory of Algorithms

BMETE91AM57

Programming Exercises for Theory of Algorithms

BMETE95AM12

Creating Mathematical Models

BMEGT30A410

Micro- and Macroeconomics

BMEGT35A411

Finance

BMETE90AM47

BSc Thesis Project

Specialization courses (30 ECTS credits)

BMETE91AM39

Algebra 2

BMETE92AM57

Functional analysis 1

BMETE95AM41

Stochastic Processes

BMEVISZA026

Combinatorics and Graph Theory 2

BMETE90AM48

Individual Research Project 1

BMETE13AM16

Physics 1 for Mathematicians

BMETE95AM33

Tools of Modern Probability Theory

BMETEAOBsMMATR-00

Matrix Analysis

BMETEAOBsMBKIE-00

Introduction to Quantum Information Theory

BMETE91AM40

Set Theory

BMETE91AM52

Mathematical Logic

BMETE95AM36

Introduction to Data Science 1

BMETE95AM43

Programming Exercises for Data Science

BMETE92AM54

Applied Numerical Methods with Matlab

BMETE91AM59

Number Theory

BMETE95AM30

Probability Theory 2

BMETE92AM45

Partial Differential Equations

BMETE94AM22

Convex Geometry

Professional subject completed in the framework of international mobility

n**

Optional courses (9 ECTS credits must be completed)

All courses (180 ECTS credits)

*Parameters:

Lc = lecture, Pr = practice, Lb = laboratory (hours per week);

Rq = requirement or exam type (v = examination, f = midterm exam);

Cr = ECTS credits.

**The credit number corresponding to the credit number of the completed subject will be taken
into account in case of a positive decision.

The 6th semester is the semester of mobility.

Preliminary requirements: see below.

Previous Curriculums:

DESCRIPTION OF SUBJECTS

Code

Title

III

BMETEAGBsMMMOD-00

Mathematical Methods

Course coordinator: Dr. Pál Hegedüs

Descripton: Elementary problems in combinatorics: counting and graphs. Natural language logic. Propositions, negations, reversing, logical operations. Single quantifier expressions (syllogisms), sets, their Boolean algebra. Proof methods. Case separation. Conditional statements. Provablity. Proofs by contradiction. Constructive proofs. Existence proofs. Pigeonhole principle. Invariants and algorithmic proofs. Isomorphism. Ordering and relations. Equivalence relations. Well ordering, principle of induction, infinite descent, recursion. Descartes product of sets. Equivalence of sets, cardinality. Countable and uncountable sets and their existence. Cantor's diagonal method. Russell's paradox and others.

Literature:
– G. Chartrand, A. Polimeni, P. Zhang: Mathematical Proofs – A Transition to Advanced Mathematics. Pearson 2018.

Code

Title

III

BMETE92AP61

Calculus

Course coordinator: Dr. Máté Matolcsi

Description: Set theoretical basis: logical symbols, truth tables, negation of statements, proof by contradiction, set theoretical operations. Real numbers, complex numbers: basic arithmetical operations, ordering, fractional parts, Bernoulli inequality, binomial theorem, absolute value, triangle inequality, mathematical induction, arithmetic of complex numbers, arithmetic-geometric mean inequality. Topology of the real line: open sets, closed sets, bounded sets, interior, exterior, boundary, closure of a set, dense sets, compact sets, Cantor intersection theorem, Borel-Lebesgue theorem (possibly without proof). Sequences: the notion of limit. Monotone sequences, subsequences, accumulation points, Bolzano-Weierstrass theorems. Liminf, limsup. Cauchy criterion. Limit of specific well-known sequences. Numeric series: convergence of a series, partial sums, Cauchy criterion. Majorant criterion, ratio criterion, root criterion. Leibniz-type series. Absolute and conditional convergence. Cauchy product. Mertens theorem, Abel rearrangement. Elementary functions (exp, log, sin, cos, sh, ch) and their identities.
Real functions: Notion of even, odd monotone, periodic functions. Convex, concave functions, Jensen-inequality. Limits, one-sided limits, continuity, transference principle. Properties of continuous functi-ons: topological characterization, Bolzano theorem. Continuous image of compact set is compact, Weierstrass min-max principle, uniform continuity, Heine theorem. Differentiation: notion of the derivative, its relation to continuity. Derivative of sums, products, quotients, chain rule. Local maxima and minima, and their connection to derivatives. Mean value theorems: Rolle, Cauchy. L’Hospital rule. Darboux property of the derivative. Higher order derivatives, Taylor polynomials, Taylor series. Specific Taylor series of well-known functions. Convex and concave functions and their connection to second derivatives. Derivative of a convex differentiable function is continuous. Jensen inequality, inequality of various means, Cauchy-Schwarz, Holder inequalities. Plotting functions by analysis of derivatives.
Indefinite integrals: definition, and elementary integrals. Integration by parts, and by substitution. Partial fraction decomposition, integration of rational functions. Integration of trigonometric, hyper-bolic functions. Definite (Riemann) integrals: Riemann approximation sums, oscillation sums, upper and lower integral. Riemann integrability of a function, sum and products of integrable functions. Newton-Leibniz formula. The integral function. Continuous or monotonic functions are integrable. Applications of the integral, improper integrals: arc-length, area. Volume and surface of a body of rotation. Center of gravity. Improper integrals, majorant and minorant criteria.

Literature:
– F.R. Giordano, J. Hass, G.B. Thomas, M.D. Weir: Thomas-féle kalkulus 1, Typotex kiadó, 2011, ISBN: 9789632798332
– G.B. Thomas: Thomas' calculus, Addison Wesley (2004)
– Laczkovich Miklós, T. Sós Vera: Valós analízis 1, Typotex kiadó, 2012, ISBN: 978-963-2797-32-8.

Code

Title

III

BMETEAGBsMVMAL-00

Vector and Matrix Algebra for Mathematicians

Course coordinator: Dr. Pál Hegedüs

Description: Elementary Real Analysis: Complex Numbers and Their Arithmetics. Algebraic, Trigonometric and Exponential Representations. Euler's Formula. The complex plane. Roots and primitive roots of unity. Elementary Functions. Algebra of polynomials. The Fundamental Theorem of Algebra.
Vector Spaces: Motivation. Linear Independence and Bases. Direct Sums. Inner Product Spaces. Orthogonal Sets.
Linear Equations and Matrices: Systems of Linear Equations. Elementary Row Operations. Row and Column Spaces. Solutions to Systems of Linear Equations. Matrix Algebra. Invertible Matrices. Elementary Matrices.
Determinants: Permutations. The Levi-Civita Symbol. Definitions and Elementary Properties. Additional Properties of Determinants. Determinants and Linear Equations. Expansion by Cofactors.
Linear Transformations and Matrices: Linear Transformations and Properties. Matrix Representations.Change of Basis. Orthogonal Transformations. Reflections, Rotations and Projections.
Eigenvalues and Eigenvectors: Eigenvalues and Eigenvectors. Characteristic Polynomials. Block Matrices. Invariant Subspaces. More on Diagonalization. Spectral theorem. Diagonalizing Normal Matrices. The Singular Value Decomposition.
Numerical and Algorithmic Approach: The LU and QR Factorizations. The Least Squares Method. The Jacobi Eigenvalue Algorithm for Symmetric Matrices.
Operators and Diagonalization: The Adjoint Operator. Normal Operators. More on Orthogonal Transformations. Projections. The Spectral Theorem. Positive Operators. The Matrix Exponential Series.

Literature:
– G. Strang: Introduction to Linear Algebra. (Fifth Edition) Wellesley-Cambridge 2016.
– R. Irving: Integers, Polynomials, and Rings - A Course in Algebra. Springer 2004.

Code

Title

III

BMETEAGBsMGE1E-00

Geometry 1e

Course coordinator: Dr. Géza Csima

Description: Axioms of geometry; Absolute geometry: absolut theorems; Spherical geometry: spheric axioms, spherical trygonometry, sine a cosine theorems, spherical area, stereographic projection; Hyperbolic geometry: models and their connection, cross ratio, angle an distance in hyperbolic geometry, hyperbolic area; Analytic geometry: vectors and coordinates, vector products, Lagrange and Jacobi theorems, spatial geometry, introduction to higher dimensional analytic geometry.

Literature:
– Á.G.Horváth: Wonderful Geometry, Typotex 2015.
– H.S.M. Coxeter, H.S.M: Non-Euclidean Geometry, The Univ. of Toronto Press, 1947.
– J.T. Smith: Methods of Geometry , Wiley and Sons, Inc. 2000

Code

Title

III

BMETE91AM42

Informatics 1

Course coordinator: Dr. András Simon

Descripton: The aim of the course is to study the basic notions of information technology. Basics of hardware (CPU, memory, mass storage,...), the hardware environment of the Institute. Basics of operating systems: program, process, file, folder, file system of Linux and Windows (bash, mc, Windows Total Commander). Graphic user interface, terminal user interface, bash language. Internet, network, IP address, wifi, Internet security. Data on machine: number representation, character encodings. Computer algebra, symbolic calculation (Sage, Mathematica,...), variable, recursion instead of iterative programming, deepening the secondary school function concept (factorial, Fibonacci sequence, Euclidean algorithm, exponentiation, quick exponentiation...). Programming paradigms in computer algebra languages. HTML, the markup language concept, homepage. CSS, separation of the content and presentation. Editing mathematical text: TeX, LaTeX, mathematics on the web. Presentation of math (beamer). Basic concepts of graphic file formats, graphics in mathematical text (TikZ).

Code

Title

III

BMETE92AP62

Multivariable Calculus

Course coordinator: Dr. Máté Matolcsi

Preliminary requirement: Calculus

Description: The n-dimensional Euclidean space, functions of several variable : Scalar product and the induced Euclidean norm on R^n. Cauchy-Schwarz inequality. The basic properties of the norm. Examples: p-norms on R^n. Topology of R^n: open, closed, compact sets, interior, boundary. Cauchy sequences, completeness of R^n. Borel-Lebesgue theorem for compact sets (possibly without proof). Limits and continuity of functions of several variable. The topological characterization of continuity, the level sets of continuous functions are open or closed. Convex sets, separation of convex sets and an exterior point, separation of two disjoint convex bodies (possibly without proof).
Differentiation of functions of several variable: Differentiable functions. Partial derivatives, gradient, tangent hyperplane, Jacobi matrix, Jacobi determinant, chain rule. Continuously differentiable functions, higher order derivatives, Young theorem, Description: Taylor formula (specifically of order two). Multilinear mappings, positive and negative definite mapping. Convex functions and the second derivative. Local maximum, minimum and connection to the derivatives. Conditional maxima and minima, Lagrange multiplier. Banach fixpoint theorem, inverse function theorem, implicit function theorem. Rotation, divergence, nabla symbol, Laplace operator. The existence of scalar potential.
Integration of functions of several variable: Definition and properties of the Jordan measure (without proofs). Integration of a continuous function of several variable on an n-dimensional domain. Integration by substitution: polar coordinates, spherical coordinates. Integration along a path, and along a surface. Divergence theorem, Stokes theorem, Green theorems, (the proof of these theorems is only sketched).
Function sequences and series: Pointwise convergence of a sequence or series of functions. Absolute convergence of a series of functions. Uniform and locally uniform convergence. The space of continuous functions with the sup-norm. Weierstrass criterion. Interchanging differentiation and the limit, integration and the limit. Term-by-term differentiability and integrability of a series of functions. Recall: properties of power series.
Fourier series: Fourier coefficients, Fourier series. The Fourier series of a twice continuously differentiable periodic function converges uniformly.

Literature:
– F.R. Giordano, J. Hass, G.B. Thomas, M.D. Weir: Thomas-féle kalkulus 3, Typotex kiadó, 2008, ISBN: 978-963-279-438-9
– G.B. Thomas: Thomas' calculus, Addison Wesley (2004)
– Laczkovich Miklós, T. Sós Vera: Valós analízis 2, Typotex kiadó, 2012, ISBN: 978-963-2797-33-5.

Code

Title

III

BMETEAGBsMBALG-00

Introduction to Algebra

Course coordinator: Dr. Pál Hegedüs

Preliminary requirement: Vector and Matrix Algebra for Mathematicians

Description: The mathematics of integers: divisibility, division with remainder, greatest common divisor, Euclidean algorithm, irreducible and prime numbers, the fundamental theorem of number theory. Linear Diophantine equations, modular arithmetic, complete and reduced residue systems, solving linear congruences. Fields of prime order. Irreducibility of polynomials and unique factorization. Schönemann-Eisenstein criterion. Multivariate polynomials, complete and elementary symmetric polynomials, relations between roots and coefficients.
Cayley-Hamilton theorem. Bilinear forms, symmetric and symplectic bilinear functions. Standard form, signature, principal axis theorem. Quadratic forms. Classification of local extrema, geometric applications and illustration. Unitary and normal matrices, complex spectral theorem. Polar decomposition, applications of SVD, pseudoinverse and its properties. Normal forms of matrices, existence, uniqueness and computation, generalized eigenvectors, Jordan chain and Jordan basis. Norms of real and complex vectors, matrix norms, basic properties and computation, functions of matrices (convergence only mentioned and illustrated), exponential functions of matrices. Vector spaces over arbitrary fields. Existence of basis, dimension, infinite dimensional examples (function spaces, etc.), isomorphism of vector spaces. Notion, properties, isomorphism of Euclidean space. Dual space. Applications of vector spaces over a finite field in coding theory, cryptography, combinatorics.
S Roman: Advanced Linear Algebra. Springer 2008.
R. Irving: Integers, Polynomials, and Rings - A Course in Algebra. Springer 2004.

Literature:
– S. Roman: Advanced Linear Algebra. Springer 2008.
– R. Irving: Integers, Polynomials, and Rings - A Course in Algebra. Springer 2004.

Code

Title

III

BMEVISZA025

Combinatorics and Graph Theory 1

Course coordinator: Dr. Tamás Fleiner

Descripton: Enumerative combinatorics (permutations and combinations, binomial theorem, theorems on the binomial coefficients). Significant methods for enumeration, pigeonhole principle and the sieve. Basic Graph Theoretical notions (vertex, edge, degree, isomorphism, path, cycle, connectivity). Trees, Cayley's formula, Prüfer-sequences. Kruskal's greedy algorithm. Characterization of bipartite graphs. Matchings, theorems of Kőnig, Hall and Frobenius, Tutte theorem, Gallai's theorems. Network flows, the Ford-Fulkerson algorithm, Edmonds-Karp algorithm. Menger's theorems, higher vertex and edge connectivity of graphs, Dirac's theorem. Euler's result on Eulerian tours and trails. Hamiltonian cycles and paths, necessary condition for the existence. Sufficient conditions (theorems of Dirac, Ore, Pósa and Chvátal). Planarity, relation to embeddability on the sphere and the torus, stereographic projection, Euler polyhedron theorem, Kuratowski's theorem, Fáry theorem. BFS and DFS algorithms for shortest paths (Dijkstra, Ford, Floyd), PERT.

Literature:
– R. Diestel: Graph Theory, online available.
– J.A. Bondy,U.S.R. Murty: Graph Theory with Applications.

Code

Title

III

BMETEAGBsMGE2E-00

Geometry 2e

Course coordinator: Dr. Szilárd Szabó

Preliminary requirement: Geometry 1e

Description: Isometries: planar and spatial classification, matrix representation, homogeneous coordinates, classification of similarities; Regular polygons and polyhedra: Euler theorem, Platonic and Archemedean solids, Cauchy rigidnes theorem; Conic sections: Dandelin spheares, excentricity, classification by quadratic forms; Introduction to projective geometry: axioms, Desargues theorem, Pappus-Pascal-theorem, perspectivity and projectivity; Classical Eucledean theorems from higher geometry: Ceva and Menelaus theorems.

Literature:
– Á.G. Horváth: Wonderful Geometry, Typotex 2015.
– H.S.M. Coxeter: Non-Euclidean Geometry, The Univ. of Toronto Press, 1947.
– J.T. Smith: Methods of Geometry , Wiley and Sons, Inc. 2000

Code

Title

III

BMETE91AM43

Informatics 2

Course coordinator: Dr. Gábor Nagy

Preliminary requirement: Informatics 1

Descripton: The course aims to learn the programming through understanding the Python language. Introduction to programming and Python language, data types, expressions, input, output. Control structures: if, while. Flowchart, structogram, Jackson figures. Complex control structures. Fundamental algorithms (sum, selection, search extrema, decision..., many practical examples). Lists. For cycle. Newer algorithms (sorting, splitting into two lists...). Exception handling. Abstraction of a part of the program, name it, using as a building block = function. Function call process, parameters, local variables, passing by value. Abstraction: complex data types from simple ones, for example fraction (numerator + denominator), complex numbers (real & imaginary part). OOP concepts: object, method. File management. Command-line arguments. Recursion (painting of an area, building a labyrinth). Algorithms efficiency, quick sorting, binary search versus linear search, O(n). Data structures: binary tree (algorithms), effectiveness: search trees (Morse tree). Mathematical libraries. Modules.

Code

Title

III

BMEGT35A410

Accounting

Course coordinator: Dr. Elvira Böcskei

Description: 1. Accounting and information systems. Accounting Concept. Users of Accounting Information. Classification of Accounting. 2. Accounting Principles. The Financial Statements. Limitation of Accounting Data. Requirements for Accounting Information. 3. The Accounting Equation. The Financial Statements. The Balance Sheet. The Income Statement. 4. The breakdown of the Balance Sheet. Practice and problem solving. The Financial Statements. The Statement of Retained Earnings. The Statement of Cash Flows. 5. The Accounting Cycle. Accounting Principles. Analysis of Business Transactions. The Recording Process. The Account. Classification of Accounts. 6. The Journal and the Ledger. Stages in the Accounting Cycle. Opening Stage. Development Stage. Adjustment Stage (Deferrals, Accruals, Depreciation). Closing Stage. 7. Practice and problem solving. Preparing financial statements. Working papers and their uses. Illustrative working papers. Statements prepared from working papers. Adjusting entries. Closing entries. Accrual adjustments and entries. 8. The Income Statement. The analysis of the Profit and Loss Statement. The breakdown of the Profit and Loss Statement. 9. The Profit and Loss Statement: practice and problem solving. Inventories. Merchandising Business. Types of Inventories. Inventory Costing Methods. Specific Unit Cost. Weighted-average Cost. Accounting for Inventories. Permanent and Periodic Inventory System. 10. Managerial accounting and Financial accounting relationship Managerial accounting and corporate strategy relationship. Strategic and operative controlling. Cost accounting and profit-and-loss relation. Examination and analysis of the income and financial state of the company. 11. Managerial decision-making in cost accounting. Case study: examination of factors influencing contribution margin. 12. Corporate TAX. VAT

Code

Title

III

BMETEAOBsMAN1E-00

Analysis 1e

Course coordinator: Dr. József Pitrik

Preliminary requirement: Calculus

Description: 1. The topology of real numbers. Open, closed and bounded sets. Interior and the closure of a set. Dense set. Rational numbers are dense in reals. Cantor's intersection theorem for bounded closed sets. Compact sets. Borel-Lebesgue theorem in the set of real numbers. Bolzano-Weierstrass theorem about convergent subsequence and compact sets in the reals. Cauchy sequences and Cauchy's criterion for convergence.
2. Functions on the real numbers. Power series and the Cauchy-Hadamard theorem. Definition of elementary function with power series. Euler's formula. Limits and continuity of functions. The topological characterization of continuity. Weierstrass extreme value theorem. Intermediate value theorem for continuous functions. Uniform continuity and the Heine-Cantor theorem about uniform continuity.
3. Metric and normed spaces. 3.1. Scalar product and the induced Euclidean norm on finite dimensional vector spaces. Cauchy-Schwarz inequality. Norm on vector spaces. The basic properties of the norm. Example: p-norm. 3.2. Metric spaces. Metric induced by norm. Open, closed and bounded sets in metric spaces. Interior and the closure of a set. Dense set. Cauchy sequences and complete metric spaces. Compact sets. Cantor's intersection theorem. Borel-Lebesgue theorem for finite dimensional Euclidean spaces. Bolzano-Weierstrass theorem. Limits and continuity of functions. The topological characterization of continuity. Weierstrass extreme value theorem. Contraction mapping. Banach fixpoint theorem. 3.3. Scalar product and the induced norm. Hilbert space. Orthonormal and complete set of vectors. Fourier decomposition of a vector with respect to a complete orthonormal set of vectors. Bessel inequality and Parseval identity. 3.4. Convex sets, separation of convex sets and an exterior point and separation of two disjoint convex bodies in finite dimensional Euclidean spaces.

Literature:
– T. Tao: Analysis II, Springer Singapore, 2016.
– E. Zakon: Mathematical Analysis I, The Trillia Group, 2004.
– A.N. Kolmogorov, S.V. Fomin: Introductory Real Analysis, Dover Publications, 1975.

Code

Title

III

BMETE91AM38

Algebra 1

Course coordinator: Dr. Sándor Kiss

Preliminary requirement: Introduction to Algebra

Descripton: Groups, semigroups. Basic properties of groups, group homomorphism, subgroups, cosets. Langrange's Theorem. Examples: diherdral groups, quaternion group, symmetric groups, alternating groups. Decomposition of permutations into disjoint cycles, transpositions. Permutation groups, group actions, transitivity, Cayley's Theorem. Cyclic groups, order of a group element. Cauchy's Theorem. Direct product of groups. Normal subgroups, factor group, Homomorphism Theorem, Noether's Isomorphism Theorems. Important subgroups: derived subgroup, centre, class equation. Subgroup chains, Sylow's Theorems, description of the structure of groups of small size. Nilpotent groups. Fundamental Theorem of Finite Abelian Groups. Free groups. Free algebras over rings, ideals, maximal and prime ideals. Description of the polynomial ring R[x]. Principal ideal domains. Noether rings, unique factorization domains (UFD). Factor rings, field extensions, construction of finite fields. Modules over rings, submodules, module homomorphisms. Semisimple modules and rings. The structure of matrix algebras over division rings. Vector space and module constructions: factor module, direct product, direct sum, tensor product. Linear fuction and the dual space.

Literature:
– P.J. Cameron: Introduction to Algebra, Oxford Science Publications, 1998.
– Atiyah, Macdonald: Introduction to commutative algebra, online textbook.

Code

Title

III

BMETE95AM29

Probability Theory 1

Course coordinator: Dr. Péter Bálint

Preliminary requirements: Multivariable Calculus

Descripton: Introduction: empirical background, sample space, events, probability as a set function. Enumeration problems, inclusion-exclusion formula, urn models, problems of geometric origin. Conditional probability: basic concepts, multiplication rule, law of total probability, Bayes formula, applications. Independence. Discrete random variables: probability mass function, Bernoulli, geometric, binomial, hypergeometric and negative binomial distributions. Poisson approximation of the binomial distribution, Poisson distribution, Poisson process, applications. General theory of random variables: (cumulative) distribution function and its properties, singular continuous distributions, absolutely continuous distributions and probability density functions. Important continuous distributions: uniform, exponential, normal (Gauss), Cauchy. Distribution of a function of a random variable, transformation of probability densities. Quantities associated to distributions: expected value, moments, median, variance and their properties. Computation for the important distributions. Steiner formula. Applications. Joint distributions: joint distibution, mass and density functions, marginal and conditional distributions. Important joint distributions: polynomial, polyhypergoemetric, uniform and mutlidimensional normal distribution. Conditional distribution and density functions. Conditional expectation and prediction, conditional variance. Vector of expected values, Covariance matrix, Cauchy-Schwartz inequality, correlation. Indicator random variables. Weak Law of Large Numbers: Bernoull Law of Large Numbers, Markov and Chebyshev inequality. Weak Law of Large numbers in full generality. Application: Weierstrass approximation theorem. Normal approximation of binomial distribution: Stirling formula, de Moivre-Laplace theorem. Applications. Normal fluctuations. Central Limit Theorem.

Literature:
– Ross, Sheldon: A First Course in Probability, 8th Edition, Pearson Education International, 2010.

Code

Title

III

BMETE91AM56

Programming Exercises for Probability Theory

Course coordinator: Dr. Gábor Nagy

Preliminary requirements: Informatics 2 AND Probability Theory 1 [parallel]

Descripton: The aim of the course is to maintain the students' programming skills through programming problems associated with the topics of Probability Theory course helping the understanding of the basic concepts of probability simulations of random events at the same time.

Code

Title

III

BMETE93AM15

Differential Equations 1

Course coordinator: Dr. Márton Kiss

Preliminary requirements: Multivariable Calculus

Descripton: Ordinary differential equations. Explicitly solvable equations, exact and linear equations. Well-posedness of the initial value problem, existence, uniqueness, continuous dependence on initial values. Approximate solution methods. Linear systems of equations, variational system. Elements of stability theory, stability, asymptotic stability, Lyapunov functions, stability by the linear approximation. Phase portraits of planar autonomous equations. Laplace transform, application to solve differential equations. Discrete-time dynamical systems.

Literature:
– W.E. Boyce, R.C. DiPrima: Elementary Differential Equations and Boundary Value Problems, Wiley 2008.
– J.C. Robinson: An Introduction to Ordinary Differential Equations, Cambridge University Press 2003.

Code

Title

III

BMETE91AM44

Informatics 3

Course coordinator: Dr. Gábor Nagy

Preliminary requirements: Informatics 2

Descripton: The aim of the course is to understand the basic elements of C++ language fundamental in effective scientific calculations. Compiling C++ programs, programming environments for C++. Input/Output. Built-in data types: int, double, char, bool, complex. Control commands: if, switch, for, while, do. Exception handling (recall Python). Functions. Extending operators (fractions struct), references (a += b, cout << fraction, cin >> fractions). Object-oriented programming in C++: object, class, encapsulation, member functions, constructors, destructors (in complex class with re + im or r + fi data members). Using arrays in C++. Pointers, relationship with arrays. File management. Basic algorithms: search, sort, etc. Command-line arguments. Dynamic memory management, new[], delete[]. Inheritance. Templates. Libraries. Header files.

Literature:
– E. Scheinerman: C++ for Mathematicians. An Introduction for Students and Professionals, CRC Press.

Code

Title

III

BMETE95AM31

Mathematical Statistics 1

Course coordinator: Dr. Marianna Bolla

Preliminary requirement: Probability Theory 1 (signature)

Descripton: Statistical sample, descriptive statistics, empirical distributions.Most frequently used probabilictic models, likelihood function, sufficiency, maximum likelihood principle. Theory of point estimation: unbiased and asymptotically unbiased estimators, efficiency, consistency. Methods of point estimation: maximum likelihood, method of moments, Bayes principle. Interval estimation, confidence intervals. Theory of hypothesis testing, likelihood ratios. Parametric inference: u, t, F tests, comparing two treatments. Two-way classified data, contingency tables, chi-square test. Nonparametric inference: Wilcoxon and sign tests, Spearman correlation. Regression analysis. Linear regression, method of least squares, Pearson correlation. Multivariate regression, multiple correlation. Linear models, analysis of variance for one- and two-way classified data. Practical considerations: selecting the sample size, test for normality, resampling methods.

Literature:
– R.A. Johnson, G.K. Bhattacharyya: Statistics: Principles and methods, Wiley, New York, 1992.

Code

Title

III

BMETEAOBsMAN2E-00

Analysis 2e

Course coordinator: Dr. József Pitrik

Preliminary requirement: Analysis 1e

Description: 1. Continuous liner maps. Operator norm and the properties of the space of continuous linear maps. Linear maps with finite dimensional domains are continuous. Norms are equivalent on finite dimensional vector spaces. Borel-Lebesgue theorem for finite dimensional vector spaces.
2. Sequences and series of functions. Pointwise limit of a sequence or series of functions. Pointwise and uniform convergence of a sequence or series of functions. Absolute convergence of a series of functions. Weierstrass criterion. Power series and Cauchy-Hadamard theorem. Interchanging differentiation and the limit, integration and the limit. Term-by-term differentiability and integrability of a series of functions. Function of diagonalizable matrices, function of normal matrices.
3. Fourier series. Fourier coefficients, Fourier series. The Fourier series of a twice continuously differentiable periodic function converges uniformly to the function.
4. Theory of complex functions. Holomorphic functions, harmonic functions, Cauchy-Riemann equations. Piecewise continuously differentiable curves, complex path integrals. Primitive function. Newton-Leibniz theorem. Goursat's lemma. The index of a point with respect to a curve, properties of the index function. Homotopy equivalence of curves. Simply connected sets. Cauchy's integral formulas. Every holomorphic function is analytic. Taylor series expansion, radius of convergence. Riemann's theorem on removable singularities. Liouville theorem. The fundamental theorem of algebra. Zeroes of holomorphic functions, multiplicity of zeroes. Maximum principle. Laurent series. Poles of holomorphic function: removable, with finite order and essential. Residue theorem.

Literature:
– R.B. Ash, W.P. Novinger: Complex Variables: Second Edition, Dover Publications, 2007.
– A.N. Kolmogorov, S.V. Fomin: Elements of the Theory of Functions and Functional Analysis, Vol. I: Metric and Normed Spaces, Dover Publications, 1996.
– MIT, OpenCourseWare, Spring 2018, Undergraduate: Complex Variables With Applications.

Code

Title

III

BMETE94AM26

Differential Geometry 1

Course coordinator: Dr. Szilárd Szabó

Preliminary requirements: Geometry 2e AND Multivariable Calculus

Descripton: Curves, reparameterization, length. Tangent line, osculating planes, curves of general type. Frenet frame, Frenet's formulas, curvatures. The fundamental theorem of curve theory. Plane curves: osculating circle, evolute, involutes, parallel curves. Rotation number, Hopf's theorem. Convex curves, the four vertex theorem. Curves in space: osculating, normal and rectifying planes, geometrical interpretation of curvatures. Hypersurfaces, parameterization, tangent plane, normal curvature, Meusnier's theorem. Fundamental forms, Weingarten map. Principal Axis Theorem, principal curvatures, Gaussian and mean curvature. Umbilical points, surfaces of rotation, ruled surfaces. Gauss frame, Christoffel symbols, Gauss and Codazzi–Mainardi equations. The fundamental theorem of hypersurface theory, Theorema Egregium. Tensor fields, Riemannian curvature tensor, Bianchi identity.

Literature:
– M. Do Carmo: Differential Geometry of Curves and Surfaces.
– B. Csikós: Differential Geometry

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Title

III

BMETE93AM19

Operations Research

Course coordinator: Dr. Pál Burai

Preliminary requirements: Introduction to Algebra

Descripton: Introduction to operations research; convex sets, polyhedron, polytope Krein-Milmann theorem. Separation, Farkas' lemma. Linear programming problem, basis, basic solution, optimal solution. Simplex algorithm. Two-phase simplex algorithm, degeneration, index selection rules. Modified simplex algorithm. Sensitivity testing. Weak and strong duality theorem. Network flow problems, algorithms. Network simplex algorithm. Transportation problem, assignment problem, the Hungarian method. Integer programming: Branch and bound method, dynamic programming, cutting plane procedures. Game theory: matrix games.

Literature:
– K.G. Murty: Linear and combinatorial programming, John Wiley and Sons., New York, 1976.
– V. Chvatal: Linear programming, W.H. Freeman & Co Ltd, 1983.

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Title

III

BMEVISZAB03

Theory of Algorithms

Course coordinator: Dr. Katalin Friedl

Preliminary requirements: Combinatorics and Graph Theory 1 AND Informatics 2 AND Programming Exercises for Theory of Algorithms [parallel]

Descripton: Pattern matching: naive algorithm, the fingerprinting method of Rabin and Karp, solution by finite automata. Deterministic and non-deterministic finite automata and their equivalence. Regular expressions, regular languages, and their connections to finite automata. Finite automaton as lexical analyser. Context free grammars. Parse tree, left and right derivation. Ambiguous words, grammars, languages. The importance of unambiguous grammars for algorithms. Pushdown automaton. Connection between pushdown automata and context free grammars, how to get a PDA from a CF grammar. The main task of a parser. The general automaton: Turing machine. Church-Turing thesis. The classes P, NP, coNP, their relations. Karp reduction and the notion of NP completeness. Theorem of Cook and Levin. 3SAT, 3COLOR are NP complete languages. Further NP complete languages: MAXSTABLE, HAM-CYCLE, HAM-PATH, TSP, 3DH, SUBSETSUM, PARTITION, KNAPSACK, SUBGRAPHISO. The problem of GRAPHISO. Linear and integer programming. LP is in P (without proof), IP is in NP. LP and IP as algorithmic tools, translation of combinatorial problems to integer programming. Another tool: branch and bound. Dynamical programming (example: knapsack, longest common substring). The objective in approximation algorithms. Bin packing has fast and good approximations (FF, FFD, theorem of Ibarra and Kim). Fro the TSP even the approximation s hard in general but there is efficient 2-approximation in the euclidean case. Comparison based sorting: bubble sort, insertion sort, merge sort, quick sort. Lower bound for the number of comparisons. Other sorting methods: counting sort, bin sort, radix sort. Linear and binary search. The binary search is optimal in the number of comparisons. Notion of search tree, their properties and analysis. Red-black tree as a balanced search tree. The 2-3 tree, and its generalization, the B tree. Comparisons of the different data structures.

Literature:
– T. Corman, C. Leiserson, R. Rivest, C. Stein: Introduction to Algorithms, MIT Press.
– M. Sipser: Introduction to the Theory of Computing, Thomson.

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III

BMETE91AM57

Programming Exercises for Theory of Algorithms

Course coordinator: Dr. Gábor Nagy

Preliminary requirements: Informatics 2 AND Theory of Algorithms [parallel]

Descripton: The aim of the course is to maintain the students' programming skills through programming problems associated with the topics of Algorithm Theory course helping the understanding of the basic concepts of algorithms.

Literature:
– M.L. Hetland: Python Algorithms, Mastering Basic Algorithms in the Python Language, Apress, 2010.

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III

BMETE95AM12

Creating Mathematical Models

Course coordinator: Dr. Roland Molontay

Preliminary requirements: Multivariable Calculus AND Algebra 1

Descripton: The aim of the seminar to present case studies on results, methods and problems from applied mathematics for promoting. The spreading of knowledge and culture of applied mathematics. The development of the connections and cooperation of students and professors of the Mathematical Institute, on the one hand, and of personal, researchers of other departments of the university or of other firms, interested in the applications of mathematics. The speakers talk about problems arising in their work. They are either applied mathematicians or non-mathematicians, during whose work the mathematical problems arise. An additional aim of this course to make it possible for interested students to get involved in the works presented for also promoting their long-range carrier by building contacts that can lead for finding appropriate jobs after finishing the university.

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III

BMEGT30A410

Micro- and Macroeconomics

Course coordinator: Dr. Zsolt Gilányi

Description: Selected topics and analytical techniques in micro- and macroeconomics tailored for engineering students. Introduction to microeconomics. Some basic economic concepts and analytical tools. Scarcity: source of eternal struggle or the foundation of all economic systems? How does “choice” determine everyday life, and what role does it play in the operation of businesses? Opportunity cost, sunk cost, normal profit. How does the product market work? Consumer choice: what are the options on the demand side, what are the goals of the consumer and how they are achieved? The forms and aims of businesses. Basics of accounting and finance. Cost and profit analysis. Competition and market systems. Introduction to macroeconomics. How does government policy interact with the decisions, profitability and life cycle of businesses? The main issues of macroeconomic study: gross domestic product, changes in the price level, unemployment ratio. Governmental policies: tools and effects. Fiscal policy: direct intervention to the life of the households and firms. Monetary policy: changes in the regulations, workings and major indicators of the financial market, and their effect on the households and firms. Economic growth and productivity. Issues of international trade: exchange rate and exchange rate policy.

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Title

III

BMEGT35A411

Finance

Course coordinator: Dr. László Vértesy

Description: 1. The system of finances, history and formation of modern money, money substitutes. 2. Money circulation between economic sectors, money demand -supplement, money market balance. 3. Economic Cycles. 4. Fiscal policy - structure of public finances, balance and balance, public debt. 5. Fiscal policy - budget and tax system. 6. Monetary policy - money creation, inflation, reserves. 7. Monetary policy – instruments: account management, open market operations, own securities, exchange rate and interest rate influence, securities discount, mandatory reserve ratio. 8. Financial markets, financial intermediation system (banks, insurers, stock exchange). 9. Active, Passive and Indifferent Banking, Insurance Products. 10. Securities, investments and stock market operations. 11. Balance of payments. 12. International Financial Systems (IMF, EU).

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III

BMETE90AM47

BSc Thesis Project

Course coordinator: Dr. Miklós Horváth

Preliminary requirement: at least 144 credits

Descripton: This course is for graduate students to prepare their graduate thesis in which they prove that they can use the acquired knowledge independently and creatively.

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III

BMETE91AM39

Algebra 2

Course coordinator: Dr. Erzsébet Lukács

Preliminary requirement: Algebra 1

Descripton: Field extensions, construction and uniqueness of simple algebraic extensions, finite and algebraic extensions. Normal extensions, splitting field, separable extension, finite fields, Wedderburn's theorem, Galois group, irreducibility of the cyclotomic polynomials, Galois groups of radical extensions, Galois correspondence, Fundamental theorem of Galois theory. Applications of Galois theory: Fundamental theorem of algebra, ruler and compass constructions, solvability of equations by radicals, Abel–Ruffini theorem. Existence and uniqueness of algebraic closure, transcendental extensions, transcendence of e, Gelfand-Schneider theorem. - Review of the basic concepts of number theory, Euler ϕ function. Linear congruences and systems of congruences, binomial congruences of higher degree, discrete logarithm, congruences of prime power moduli. Quadratic congruences, Legendre and Jacobi symbol, quadratic reciprocity. Prime numbers: Euclid's theorem, gaps between primes, Chebyshev's theorem, harmonic series of primes, Dirichlet's theorem for (nk + 1). Arithmetic functions: d(n), σ(n), ϕ(n). Multiplicativity, convolution, Möbius function, the Möbius inversion formula. Prime number theorem, magnitude of the nth prime, prime tests, Rabin–Miller test, RSA function. Diophantine equations: linear diophantine equations, Pythagorean triples, Fermat's two squares theorem, Gaussian integers.

Literature:
– I. Stewart: Galois Theory, CRC Press, 2003.
– Niven, Zuckerman, Montgomery: An Introduction to the Theory of Numbers, John Wiley & Sons, 1960.
– M.B. Nathanson: Elementary Methods in Number Theory, Springer, 2000.

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III

BMETE92AM57

Functional Analysis 1

Course coordinator: Dr. Máté Matolcsi

Preliminary requirement: Analysis 2e

Descripton: Metric spaces, completeness, compactness (Baire theorem, Arzela–Ascoli theorem). Linear spaces, Hahn–Banach theorem on linear spaces. Normed spaces, Banach spaces. Bounded linear operators and functionals. Hahn–Banach theorem in normed spaces. Fundamental theorems of functional analysis: uniform boundedness theorem, open mapping theorem, closed graph theorem, and their applications. Dual spaces, specific examples, reflexivity. Weak and weak* topology. The compactness of the unit ball in different topologies (without proof for weak and weak* topologies). Spectrum of a bounded linear operator. Spectral theory of compact operators. Hilbert spaces, bounded linear operators on Hilbert spaces. Spectral theory of self-adjoint compact operators.

Literature:
– Reed, Simon: Functional Analysis

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III

BMETE95AM34

Stochastic Processes

Course coordinator: Dr. Károly Simon

Preliminary requirement: Probability Theory 1

Descripton: Basic notions: finite dimensional marginals, Kolmogorov’s fundamental theorem, strongly and weakly stationary processes, processes with stationary and/or independent increments. Discrete Markov chains: linear algebra of stochastic matrices, classification of states. Finite Markov chains: stationary measures and ergodic behaviour. Reversibility, random walk on graphs. Urn models. Countable Markov chains: transience, null-recurrence, positive-recurrence. Random walks on Zd: Polya’s theorem. Random walks on countable graphs, branching processes, discrete time birth-and-death processes, queuing problems. Random walks on Z1: the reflection principle and limit distribution of the maximum, difference equations. Continuous time, discrete space Markov processes: the Poisson process, jump rates, exponential clocks. Stochastic semigroup: Kolmogorov-Chapman equations, infinitesimal generator. Complements of measure theory: filtrations, adapted processes, natural filtration. The general notion of conditional expectation (Kolmogorov’s theorem), fundamental properties. Discrete time martingales: sub/super/martingales, stopping times, stopped martingales. Optional stopping theorem, Wald identity, martingale convergence theorem, submartingale inequality, maximal inequality. Azuma-Hoffding inequality, applications. The Brownian motion: defining properties, covariances. Sketch of Paul Levy’s construction, basic analytic properties. Applications.

Literature:
– Essentials of Stochastic Processes (2nd edition), Springer, 2012.
– R. Durrett: Probability Theory with Examples, 4th edition, Cambridge U. Press, 2010.

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III

BMEVISZA026

Combinatorics and Graph Theory 2

Course coordinator: Dr. Tamás Fleiner

Preliminary requirement: Combinatorics and Graph Theory 1

Descripton: Geometric and abstract duality, weak isomorphism (2-isomorphism) and the Whitney theorems. Vertex and edge coloring, Mycielsky's construction, Brooks' theorem. 5-colour theorem, Vizing's theorem, connection of edge-colouring to matchings, Petersen's theorem. List colouring of graphs, Galvin's theorem. Perfect graphs, interval graphs and the perfect graph theorem. Ramsey's theorem, Erdős-Szekeres theorem, Erdős' lower bound and the probabilistic method. Turán's theorem, Erdős-Stone theorem, Erdős-Simonovits theorem. Hypergraphs, Erdős-Ko-Rado theorem, Sperner's theorem and the LYM inequality. De Bruijn-Erdős theorem, finite planes, construction from finite field, and from difference sets. Generating functions, Fibonacci numbers, Catalan numbers. Posets, Dilworth's theorem.

Literature:
– R. Diestel: Graph Theory, online available.
– J.A. Bondy, U.S.R. Murty: Graph Theory with Applications.

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III

BMETE90AM48

Individual Research Project 1

Course coordinator: Dr. Lajos Rónyai

Descripton: Under the guidance of a chosen tutor, the student works on understanding a paper or a book chapter about contemporary mathematics. The goal is to get familiar with basic methods and abilities of research like exact understanding of mathematics in English, use of libraries and of the net etc. At the end of the semester the student makes a written English summary in a few pages and gives a short presentation in a seminar talk.

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III

BMETE13AM16

Physics 1 for Mathematicians

Course coordinator: Dr. László Udvardi

Preliminary requirement: Multivariable Calculus

Descripton: Review of the physics we learned in secondary school: Newton's laws, Conservation laws. Inertial frame of reference, general transformation between two Inertial frame of reference. Galilei transformation, Lorentz transformation. Introduction to special relativity: Lorentz contraction, time dilation, proper time, invariant quantities. Four vectors. Accelerated Reference Frames, Fictitious force: Coriolis force, Foucault pendulum, centrifugal force. Demonstration experiments. Primer to geometrical optics, Fermat's principle, Euler-Lagrange equation. Hamilton's principle, Lagrange function, equation of motion. Relation between the symmetry of the Lagrangian and the conservation laws, Noether's theorem. Application of the law of conservation, motion in central field. Kepler problem.

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III

BMETE95AM33

Tools of Modern Probability Theory

Course coordinator: Dr. Imre Tóth

Preliminary requirement: Probability Theory 1

Description: The goal of the course is to teach the most important tools that modern probability theory uses from combinatorics, linear algebra, real analysis, measure theory, complex analysis, functional analysis and geometry.
We demonstrate the use of these tools through examples, but the emphasis is on developing the tools. A part of the knowledge acquired will be utilised in the masters program.
Combinatorics: method of generator functions. Stirling formula. Euler gamma function.
Topology: convergence on metric spaces and topological spaces. Compactness. Product space, product topology. Tychonoff's theorem.
Linear algebra: inner product spaces. Cauchy-Schwartz inequality. Calculating powers of matrices, analytic matrix-calculus. (Application: Markov transition probabilities.)
Transformations of functions: Laplace transform. Fourier expansion, Fourier transformation. Discrete Fourier tranformation. (Application: characteristic function.) Legendre transform.
Measure theory: exchanging integral and derivative. Uniform convergence and continuity. (Application: differentiability of the characteristic function.) Jensen inequality. Absoulte continuity, Radon-Nikodym theorem. (Application: conditional expectation.) Push-forward of measures, integration by substitution. (Application: distribution of random variables, expectation of random variables.) Product space, product measure. Fubini's theorem. (Application: independence.) Decomposition of measures, conditional measure, factor measure.
Complex analysis: Residue theorem, Laurent expansion. (Application: calculating convolutions and characteristic functions.) Analytic extensin, Vitali's theorem.
Functional analysis: spectrum of bounded operators, resolvent, spectral radius. Hahn-Banach theorem. C^k spaces, Arsela-Ascoli theorem. Continuous linear functionals, Riesz-Markov theorem. Dual spaces, weak star topology, tightness. Fourier transform once again, Riesz-Fischer theorem.

Literature:
Járai Antal: Mérték és integrál (in Hungarian)
Rudin: Functional Analysis

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III

BMETEAOBsMMATR-00

Matrix Analysis

Course coordinator: Dr. Milán Mosonyi

Description: 1. Finite-dimensional Hilbert spaces, Dirac formalism, bra and ket vectors. Trace and the Hilbert-Schmidt inner product. 2. Special operators, spectral decomposition, functional calculus. Generators of B(H), self-adjoint subalgebras. Quantum states and measurements, Born rule. 3. Absolute value, partial isometries, polar decomposition, singular values. 4. Positive semi-definite order, minimum and maximum of self-adjoint operators. Trace minimum and maximum, optimal success and error probabilities of quantum state discrimination, max-relative entropy radius and center. Binary case, operational interpretation of the trace norm distance. 5. Perspective function, classical f-divergences, convexity and monotonicity, variational distance, classical relative entropy and Rényi divergences. 6. Monotonicity, convexity, and subadditivity of trace functions; Courant-Weyl-Fischer minimax theorem, Jensen inequality with operator weights. Neumann entropy, quantum Rényi entropies. 7. Operator convex and operator monotone functions, basic examples, special integral representations. 8. Tensor product of Hilbert spaces and operators. Asymptotic binary i.i.d. state discrimination problem, Audenaert’s inequality, attainability parts of the Stein, Chernoff and Hoeffding error exponents. Petz-type Rényi divergences, Umegaki relative entropy. 9. Positive semi-definite block operators, Schur complement. Absolutely continuous part. Inequalities for positive and 2-positive super-operators. 10. Operator perspective function, Kubo-Ando means, Petz-type and maximal quantum f-divergences, quantum Rényi divergences and relative entropies. 11. Operator Jensen inequality, joint convexity/concavity of Kubo-Ando means and f-divergences. 12. Discrete Weyl operators, partial trace via twirling. Completely positive maps in Kraus form, Stinespring dilation. Monotonicity of the Petz-type and the maximal f-divergences under CPTP maps. 13. First and second derivatives of operator functions, characterization of operator monotonicity and convexity via derivatives.

Literature:
– R. Bhatia: Matrix Analysis
– F. Hiai: Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization
– F. Hiai, D. Petz: Introduction to matrix analysis and applications

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III

BMETEAOBsMBKIE-00

Introduction to Quantum Information Theory

Course coordinator: Dr. Milán Mosonyi

Description: 1. Finite-dimensional classical models, states and measurements, representation by diagonal matrices. 2. Finite-dimensional Hilbert spaces, Dirac formalism. Orthonormal bases, trace, Hilbert-Schmidt inner product. Special operators, spectral decomposition, functional calculus. 3. Finite-dimensional operator algebraic models, states, measurements, Born rule. Quantum bit, Bloch ball. 4. Extremal states and measurements, quantum indeterminism. 5. Noiseless information transmission, perfect state discrimination. Quantum key distribution. 6. Composite systems, tensor product of Hilbert spaces and observable algebras. 7. Marginal states, partial trace. Product, separable and entangled states. Schmidt decomposition. Purification of states. Maximally entangled states, Bell bases. 8. Mathematical description of time evolution. Completely positive maps and their representations, Choi criterion, Kraus decomposition, Stinespring dilation. Naimark dilation of POVMs. Description of closed and open quantum systems. 9. Cloning and broadcasting of quantum states, no cloning theorem. Superdense coding, quantum teleportation. 10. Classical, quantum, and no-signaling correlations, non-local games, CHSH game, pseudo-telepathy games.

Literature:
– A.S. Holevo: Probabilistic and statistical aspects of quantum theory, North-Holland 1982
– A.S. Holevo: Quantum Systems, Channels, Information, De Gruyter 2012
– M.A. Nielsen, I. Chuang: Quantum Computation and Quantum Information, Cambridge University Press, 2000

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III

BMETE91AM40

Set Theory

Course coordinator: Dr. Gábor Sági

Description: Equivalency of sets. A set and its powerset are not equivalent. Naive definition of cardinality and its inconsistency. The ZFC axiom system. Extending the language (introducing new relation and function symbols). The notions of ordered pairs, directs products, relations, functions. Ordered sets, and their initial segments. Ordinals and their basic properties. Ordinals form a proper class. Successor and limit ordinals. Transfinite induction and recursion. The axiom of choice and some of its equivalents. Cardinality operations. The fundamental theorem of cardinal arithmetic. The cofinality operation. Some famous statements which are independent from ZFC. ZFC is essentially incomplete. On models of set theory.

Literature:
– Hajnal András, Hamburger Péter: Halmazelmélet, Tankönyvkiadó, 1983.

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III

BMETE91AM52

Mathematical Logic

Course coordinator: Dr. András Simon

Description: The language of first order logic, an outlook to higher order languages. Formalization. Structure, valuation. The sets of true valuations. Logical consequence and comparing with the operation implication. Deduction theorem, and characterizations of logical consequence. Normal forms: conjuctive, prenex, Skolem. Compactness theorem and its applications. – Proof theory. Deductive and refutation calculi. Analitic tableaux and its semantical background. Completeness theorem and its importance. Examples for semantical and proof theoretical approaches of some logical properties. The model method. Theorems of Löwenheim-Skolem types. Model constructions. Standard and non-standard models, on the concepts on non-standard real numbers, integers, infinitesimals. Categoricity, and completeness. – Discrete and density orderings. On the limits of first order logic, incompleteness and undecidableness, the famous results of Gödel and Church. On the connection of propositonal logic and Boolean algebras.

Literature:
– H.B. Enderton: A Mathematical Introduction to Logic, Academic Press, 2001.
– M. Ben-Ari: Mathematical Logic for Computer Science, Springer, 2012
– M. Ferenczi, M. Szőts: Mathematical Logic for Applications, Typotex, 2016
– M. Ferenczi, A. Pataricza, L. Rónyai: Formal Methods in Computing, Kluwer-Akadémia Kiadó, 2005

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III

BMETE95AM36

Introduction to Data Science 1

Course coordinator: Dr. Roland Molontay

Preliminary requirement: Informatics 2 AND Statistics 1

Description: The aim of the course is to introduce the basic concepts of data science in a practical approach, building on previously acquired mathematical knowledge. From the beginning, students will gain precise theoretical and practical hands-on knowledge by experiencing the knowledge through real-life application examples, in a spiral progression from the beginning to the end. The theoretical knowledge is backed up by machine learning algorithms, while the practical exercises build on the knowledge of the Python language.
History, examples, case studies, disciplines that can be classified as data science. History, history, data science, history, history, history of data science. Least squares method. Linear Regression. Gradient method, maximum-likelihood estimation. Polynomial regression, logistic regression, Perceptron, Newton's method, Naive-Bayes. Generalized linear models (Exponential family), learning/validation/testing set, cross-validation, Bias-Variance tradeoff, regularization, Precision-Recall, F1-score, ROC curve. SVM, linear SVM, kernel trick. Neural networks. Decision trees. Random forests. Boosting. Unsupervised learning. Clustering. K-means clustering. EM algorithm. PCA, ICA. Larger case studies, insights.
Practical: steps of data manipulation, predictive analysis, visualization with real data (e.g. kaggle) mainly using Python packages (pandas, scikit-learn, matplotlib, ggplot) and R.

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III

BMETE95AM43

Programming Exercises for Data Science

Course coordinator: Dr. Roland Molontay

Preliminary requirement: Introduction to Data Science 1 [parallel]

Description: A tárgy célja a Bevezetés az adattudományba 1 tárgyban kevésbé tárgyalt adattudományi fogalmak, algoritmusok a korábban megszerzett matematikai ismeretekre épülő, gyakorlati megközelítésű megismertetése. Az adatmanipulálás, prediktív analízis, megjelenítés lépései valódi adatokkal, elsősorban Python-csomagok (pandas, scikit- learn, matplotlib, ggplot) és R használatával, ismerkedés más adattudományi szoftverek használatával. Bayes-hálók, Együttes módszerek osztályozásra (véletlen erdő, bagging, boosting), Klaszterezés (k-közép, hierarchikus, DBSCAN, EM algoritmus), Ajánlórendszerek, Dimenziócsökkentés (PCA) Asszociációs szabályok, Anomáliák (outlierek) detektálása. Nagyobb esettanulmányok, kitekintés. Az adatmanipulálás, prediktív analízis, megjelenítés lépései valódi adatokkal, elsősorban Python-csomagok (pandas, scikit- learn, matplotlib, ggplot) és R használatával, ismerkedés más adattudományi szoftverek használatával.

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III

BMETE92AM54

Applied Numerical Methods with Matlab

Course coordinator: Dr. Róbert Horváth

Preliminary requirements: Introduction to Algebra AND Differential Equations 1

Descripton: Usage of MATLAB (all discussed numerical methods will be introduced and tested in MATLAB ). The discussed topics are: error calculation, direct and iterative solution of linear systems of equations: Gauss elimination, Gauss transform factorizations of matrices, conditionality of linear systems of equations, Jacobi, Seidel and SOR iteration; convergence of the iteration, error estimation, optimization type methods for solving linear systems of equations, estimation of the eigenvalue, power method for the eigenvalue, eigenvector problem of matrices, inverse power method, transforming matrices to special forms, Jacobi method for determining eigenvalues and eigenvectors, QR method for determining eigenvalues, simple interpolation with polynomials, Hermite interpolation, interpolation with third degree spline, approximation according to least squares with polynomials and trigonometric polynomials, trigonometric interpolation, basics of fast Fourier transform, numerical integration, Newton-Cotes formula and its usage, Gaussian quadrature, solution of non linear systems of equations, roots of polynomials, numerical solution to the initial value problems of ordinary differential equations, basic terms of one step methods, Runge-Kutta methods, stability, convergence and error estimation of one step methods, multi step methods.

Literature:
– S.C. Chapra: Applied Numerical Methods with MATLAB - for engineers and scientists, McGraw Hill, 2008.
– W. Cheney, D. Kincaid, Numerical Mathematics and Computing, Brooks/Cole, Cangage learning, 2013.

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III

BMETE91AM59

Number Theory

Course coordinator: Dr. Sándor Kiss

Preliminary requirements: Calculus

Descripton: Basic Number Theory: Divisibility, greatest common divisor, Euclid's algorithm, congruences, Chinese remainder theorem, Hensel lifting, primitive roots, discrete logarithm, quadratic residues, Legendre and Jacobi symbol. Law of quadratic reciprocity.
Analytic Number Theory: Prime numbers and its properties, primes of special forms. Primes in arithmetic progressions, gaps between primes, Bertrand's postulate, the Prime Number Theorem. The Riemann zeta function, Riemann Hypothesis, Dirichlet characters. The generating function and its applications, partitions. Sieve methods, application of Brun's sieve to estimate the number of twin primes, Goldbach's conjecture. Additive and multiplicative arithmetic functions. Additive Number Theory: Sumsets, direct and inverse problems. Sum-product estimates.
Combinatorial Number Theory: Schnirelman density, Schur's theorem, van der Waerden's theorem, Szemerédi's theorem about arithmetic progressions. Zero-sum combinatorics: the polynomial method, Combinatorial Nullstellensatz, applications.
Diophantine equations: sum of two, three, four squares, representations as the sums of k-th powers, Waring problem. Fermat's last theorem. Mordell equation. The abc conjecture.
Miscellaneous modern topics (sketch only): Number Theory in Cryptography: The RSA and the ElGamal scheme. Primality tests. Diophantine Approximation Theory: Continued fractions. Pell equation. Wiener attack against RSA. p-adic numbers.

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III

BMETE95AM30

Probability Theory 2

Course coordinator: Dr. Balázs Bárány

Preliminary requirement: Probability Theory 1 AND Analysis 2e

Descripton:
Sums of independent random variables and convolution of distributions (with introduction to Riemann-Stieltjes integral). Applications: Gaussian, Cauchy, exponential, Gamma, etc.
The generating function. Applications: branching processes, hitting times and occupation times of random walks, weak convergence of discrete distributions and Poisson approximation, etc.
The Weak Law of Large Numbers: Chebyshev's and Markov's inequalities and the WLLN. Applications.
Tail and large deviation estimates for sums of independent random variables: Bernstein, Hoeffding, Chernoff and Cramér bounds applications.
Convergence in probability and almost sure convergence. The Borel-Cantelli Lemma. Application: the Strong Law of Large Numbers assuming fourth moment.
Kolmogorov's inequality and the Two Series Theorem, Kolmogorov's Strong Law of Large Numbers (in full detail). Kolmogorov's 0-1 Law.
The Characteristic Function 1: definition; basic properties; moments of rv and derivatives of its chf; smoothness of pdf and decay of chf; inversion.
Weak Convergence of Probability Distribution Functions: definition and characterizations; tightness and subsequential weak convergence.
The Characteristic Function 2: pointwise convergence of chf-s and weak convergence of pdf-s. Application: The Central Limit Theorem in its full strength.

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III

BMETE92AM45

Partial Differential Equations

Course coordinator: Dr. Mihály Kovács

Preliminary requirements: Differential Equations 1 AND Analysis 2e

Descripton: Classification of partial differential equations (PDEs). First order linear PDEs. Convection transport processes. First order quasilinear PDEs. Parabolic Cauchy problems. Heat conduction problem, qualitative properties. Hyperbolic Cauchy problems. Wave equation in one space dimension: vibrating string, travelling and standing waves. Wave equation in two and three space dimensions using surface integral. Elliptic boundary value problems. Elliptic models: stationary heat distribution, elastic torsion. Uniqueness of the solution. The problem of the notion of solution. Theoretical background: Hilbert spaces, Fourier series, symmetric operators. Fourier series expansion for elliptic boundary value problems using eigenfuctions. Theoretical background: distributions, Sobolev spaces. Weak solution of elliptic problems. Weak eigenvalue problem. Parabolic and és hyperbolic initial-boundary value problems. Elliptic fundamental solution, mathematical description of the potential for a point source, Green’s function.

Literature:
– L.C. Evans: Partial Differential Equations, AMS, 2010.

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III

BMETE94AM22

Convex Geometry

Course coordinator: Dr. Zsolt Lángi

Preliminary requirements: Geometry 2e AND Introduction to Algebra

Descripton: Introduction: affine and convex sets, affine dependence, independence, affine and convex combinations, affine hull, isolation theorem, characterization of closed, convex sets as the intersection of closed half spaces. Convex hull, theorems of Radon, Helly and Carathéodory, their applications. Linear functionals and their connection with hyperplanes, Minkowski sum, separation of convex sets with hyperplanes, supporting hyperplanes, faces of a convex body, extremal and exposed points, theorems of Krein-Milman and Straszewicz. Indicator function, algebras of closed/compact convex sets, valuations, Euler characteristic and the proof of its existence. Convex polytopes and polyhedral sets, their connection, face structure of polytopes, combinatorial equivalence. The f-vector of polytopes, Euler characteristic of polytopes, theorem of Euler. Polar of a set, fundamental properties of polarity, properties of the polar of a polytope, dual polytope. Moment curve, cyclic polytopes and their face structure, Gale’s evenness condition. Hausdorff distance of convex bodies. Affine transformations, Banach-Mazur distance. Ellipsoid as an affine ball. Unique existence of largest volume inscribed, and smallest volume circumscribed ellipsoid of a convex body. The Löwner-John ellipsoid, John’s theorem for general, and centrally symmetric convex bodies.

Literature:
– B. Grünbaum: Convex Polytopes, Graduate Texts in Mathematics 221, Springer, New York, 2003.