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 2021

Code

Title

Parameters*

ECTS credits
per semester

III

Obligatory courses (153 ECTS credits)

BMETE91AM35

Basics of Mathematics

BMETE92AM36

Calculus 1

BMETE91AM36

Introduction to Algebra 1

BMETE94AM24

Geometry 1

BMETE91AM42

Informatics 1

BMETE92AM37

Calculus 2

BMETE91AM37

Introduction to Algebra 2

BMEVISZA025

Combinatorics and Graph Theory 1

BMETE94AM25

Geometry 2

BMETE91AM43

Informatics 2

BMEGT35A410

Accounting

BMETE92AM55

Analysis 1

BMETE91AM38

Algebra 1

BMETE95AM29

Probability Theory 1

BMETE91AM56

Programming Exercises for Probability Theory

BMETE93AM15

Differential Equations 1

BMETE91AM44

Informatics 3

BMETE95AM31

Mathematical Statistics 1

BMETE92AM56

Analysis 2

BMETE94AM26

Differential Geometry 1

BMETE93AM19

Operations Research

BMEVISZAB03

Theory of Algorithms

BMETE91AM57

Programming Exercises for Theory of Algorithms

BMETE91AM39

Algebra 2

BMETE92AM57

Functional analysis 1

BMETE95AM12

Creating Mathematical Models

BMEGT30A410

Micro- and Macroeconomics

BMETE94AM20

Differential Geometry 2

BMEGT35A411

Finance

BMETE90AM47

BSc Thesis Project

Specialization courses (18 ECTS credits minimum)

BMETE95AM41

Stochastic Processes

BMEVISZA026

Combinatorics and Graph Theory 2

BMETE90AM48

Individual Research Project 1

BMETE13AM16

Physics 1 for Mathematicians

BMETE92AM59

Matrix analysis

BMETE92AM60

Introduction to quantum information theory

BMETE91AM60

Set Theory and Mathematical Logic

BMETE92AM54

Applied Numerical Methods with Matlab

BMETE91AM59

Number Theory

BMETE94AM31

Differential Geometry 3 (in odd years)

BMETE94AM30

Tools of Computer Aided Geometry (in even years)

BMETE92AM58

Programming Exercises for Differential Equations

BMETE95AM30

Probability 2

BMETE92AM45

Partial Differential Equations

BMETE94AM22

Convex Geometry

BMETE90AM49

Individual Research Project 2

Elective courses (9 ECTS credits must be completed)

All courses

*Parameters:
Lc = lecture, Pr = practice, Lb = laboratory (hours per week);
Rq = requirement or exam type (v = examination, f = midterm exam);
Cr = ECTS credits.
Preliminary requirements: see below.

Previous Curriculums:

2019

2017

DESCRIPTION OF SUBJECTS

Code

Title

III

BMETE91AM35

Basics of Mathematics

Course coordinator: Dr. András Simon

Descripton: Notations, formal languages, formalism in mathematics. Mathematics and the deductive systems. Propositional logic. The language of propositional logic. Logical operations, tautologies, logical equivalences. A calculus in propositional logic. Completeness and its importance. First order logic. Language of first order logic: terms, formulas, quantifiers, equality. Structure, model, algebra. Valuation in a model. The concept of logical consequence. Axioms and theorems. Standard and non-standard models. Calculus, deductive and refutation systems. Completeness. Direct and indirect proofs. On the concepts induction and recursion. The real numbers as ordered field with suprema. The construction of the real numbers. Non-standard real numbers, infinitesimals. Set theory. Ordered pairs, relations, functions. Equivalence- and ordering relations. Equivalence of sets. Countable and non-countable cardinalities. Cantor’s diagonalization procedure. Continuum hypothesis. Classes, Russel paradoxon. Well-ordering. The axiom of choice and its importance.

Literature:
– R.G. Exner: An Accompaniment to Higher Mathematics, Springer, 1996.

Code

Title

III

BMETE92AM36

Calculus 1

Course coordinator: Dr. Miklós Horváth

Descripton: Real numbers, sets and mappings. Important inequalities. Real sequences and limits. Convergent and divergent sequences. Monotone and bounded sequences. Subsequences, accumulation points. Theorems of Bolzano and Weierstrass, limsup, liminf. Cauchy theorem. Important limits. Numerical series; convergence and properties. Series of positive numbers. Comparison test, ratio test, nth root test. Absolute and non-absolut convergent series. Alternating series, Leibniz series. Estimations for series. Product of series. Theorem of Mertens and Abel. Real functions. Limits and continuity. Continous functions on bounded closed intervals. Theorems of Bolzano and Weiersrass. Uniformly continous functions, Heine's theorem. Differentiation. Properties of derivatives. Inverse functions. Higher derivatives. Mean value theorems. Elementary functions. Polinomials, exponential, logarithm, trigonometric functions. Function tests, sketching the graphs of functions. Taylor polinomial. Indefinite integral (antiderivatives). Techniqus of integraton. Integration by parts, substitutions, trigonometric integrals, partial fractions. Riemann integral. Propertiesof the integral, upper, lower sums and oscillation sums. Connection with the derivative, Newton-Leibniz rule. Applications of the integral. Mean value theorem. Improper integral.

Literature:
– P.D. Lax, M.S. Terrell: Calculus with applications.
– S. Lang: A first course in calculus.

Code

Title

III

BMETE91AM36

Introduction to Algebra 1

Course coordinator: Dr. Gábor Sági

Descripton: Elementary number theory: integers, divisibility, division with remainders, greatest comon divisor, Euclidean algorithm, irreducible numbers and prime numbers. Fundamental Theorem of Arithmetic. Linear Diophantine equations, modular arithmetic, complete and reduced remainder systems, solution of linear congruences. Complex numbers, algebraic and trigonometric forms, Binomial Theorem. Relation between the complex numbers and the geometry of the plane. Roots of unity, primitive roots of unity. Polynomials with one variable, operations, Horner-scheme, rational root test, Fundamental Theorem of Algebra. Irreducibility of polinomials, Schönemann-Eisenstein criterion. Multivariate polynomials, complete and elementary symmetric polynomials, Viete formulas, roots of cubic polynomials. Systems of linear equations in two and three variables, Gaussian and Gauss-Jordan elimination. R^n and its subspaces. Linear combinations, linear independence, spanned subspace, basis, dimension. Coordinate systems, row space, column space, nullspace of a matrix. Subspace of solutions, solutions in the row space. Matrix operations, inverse matrix, base change matrix. Operations with special matrices, PLU decomposition. Solution of systems of equations with the help of PLU decomposition. Determinant as the volume of the parallelepiped. Basic properties, determinant of a matrix. The notion of permutations, transpositions, cycles, expansion of the determinant. Laplace Expansion Theorem, Mutiplication Theorem of Matrices, formula for the inverse of a matrix, Cramer's Rule. Basic properties of matrix rank. Linear maps and their matrices: the matirx of a projection to a subspace. Similar matrices. Optimal solution of inconsistent systems of linear equations, normal equation, solution in the row space and its minimality. Moore-Penrose generalized inverse.

Literature:
– W. Sierpinski: Elementary theory of numbers, North Holland, 1987.
– P. Halmos: Finite dimensional vector spaces, Springer, 1967.
– V.V. Prasolov: Problems and Theorems in Linear Algebra, AMS, 1994.

Code

Title

III

BMETE94AM24

Geometry 1

Course coordinator: Dr. Jenő Szirmai

Descripton: Euclid's Axioms and Postulates, Hilbert's axioms, points, straight lines, planes, distances, angles etc. Euclidean plane: Geometric transformations, synthetically. Vector geometry, linearly dependent, linearly independent vectors, scalar and cross product, Cartesian coordinate system, Lagrange-Jacobi identities. Coordinate geometry, analytic description of planes and straight lines, distances, angles, etc. Euclidean space: Geometric transformations (congruences), analytically. Homogeneous coordinates, uniform treatment of geometric transformations. Affinities, similarities. Spherical geometry: geodesic curves, angles, angle-sum formula for spherical triangles, spherical trigonometry. Definition of polyhedra, Euler theorem. Special polyhedra: convex, regular polyhedra, Archimedean solids, Catalan solids etc. Cauchy's rigidity theorem, and other interesting polyhedra.

Literature:
– G.A. Jennings: Modern geometry with applications, Springer-Verlag.
– H.S.M. Coxeter: Introduction to Geometry, New York, Wiley.

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

BMETE92AM37

Calculus 2

Course coordinator: Dr. József Pitrik

Preliminary requirement: Calculus 1

Descripton: Finite dimensional normed vector spaces. Sequences in normed vector spaces, convergence. Theorems of Bolzano and Weierstrass. Multivariable calculus. Continuity. Partial derivatives, directional derivatives. Differentiability and the chain rule. The differential of a function and its geometrical meaning, linear approximation.Tangent plane and the gradient. Higher derivatives. Schwarz's theorem. Extremas of multivariable functions. Absolute minima and maxima. Maxima and minima with subsidiary conditions, Lagrange's method of undetermined multipliers. Inverse and imlicit functions. Multiple integrals, fundamental rules. Jordan-measurable sets and their measure. Double integrals, polar transform. Integrals over regions in three and more dimensions. Transformations of multiple integrals. Vector fields and their analysis. Differential calculus of vector fields. Curves and surfaces in three dimension. Line integrals of vector fields. The fundamental theorem of line integrals, independence of path. Potential function. Green's theorem. The Curl and Divergence of a vector field. Parametric surfaces and their areas. Oriented surfaces. Surface integrals of vector fields. Stokes' theorem. The divergence theorem. Sequences and series of functions. Pointwise and uniform convergence. Weierstrass M-test. Consequences of uniform convergence. Power series. Taylor series, binomial series. Fourier series. Inner products on periodic functions. The Fourier and Plancherel theorem. Periodic convolution. Applications.

Literature:
– S. Lang: Undergraduate Analysis.
– E.M. Stein, R. Shakarchi: Fourier Analysis, An Introduction.

Code

Title

III

BMETE91AM37

Introduction to Algebra 2

Course coordinator: Dr. Erzsébet Lukács

Preliminary requirement: Introduction to Algebra 1

Descripton: Scalar product and its properties in R^n. Orthogonal and orthonormal bases, Gram-Schmidt ortogonatization process, orthogonal matrices, orthogonal transformations. Householder reflections, Givens rotations. The existence of QR decomposition and its calculation. Optimal solution of systems of linear equations with the help of QR decomposition. Scalar product in C^n. Unitary, normal and selfadjoint matrices and transformations. Eigenvalues, eigenvectors and eigenspaces of matrices and linear transformations. Characteristic equation, solution of the eigenvalue problem. Applications. Algebraic and geometric multiplicity, eigenvalues of special matrices, eigenvalues of similar matrices. Cayley-Hamilton Theorem. Diagonilizability of matrices and its equivalent formulations, (real and complex cases), diagonalizibility of special matrices, relation to the eigenvalues. Unitary and orthogonal diagonalizibility. Schur decomposition, spectral decomposition. Bilinear functions, standard form, signature, Main Axis Theorem. Quandratic forms, definity. Classification of local extrema of a function, geometric applications, graphical presentation. Multilinear functions and maps, total derivative as multilinear map, multivariate Taylor formula, determinant as multilinear function. Singular Value Decomposition of matrices, polar decomposition, applications of SVD, generalized inverse from the SVD. Normal forms of matrices, existence, unicity, determination of the normal form. Generalized eigenvectors, Jordan chain, Jordan basis. Norms of real and complex vectors, matrix norms, basic properties, calculation of norms. Matrix functions (convergence just mentioned, and illustrated), matrix exponential functions. Vector spaces over arbitrary fields. Existence of basis, dimension, infinite dimensional vector spaces (e.g. function spaces), isomorphic vector spaces. Notion of Euclidean space, properties, isomorphism between Euclidean spaces. Dual space. Application of vector spaces over finite fields in coding theory, cryptography and combinatorics.

Literature:
– C.D. Meyer: Matrix Analysis and Applied Linear Algebra, online textbook.
– P. Halmos: Finite dimensional vector spaces, Springer, 1967.
– V.V. Prasolov: Problems and Theorems in Linear Algebra, AMS, 1994.

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

BMETE94AM25

Geometry 2

Course coordinator: Dr. Jenő Szirmai

Preliminary requirement: Geometry 1

Descripton: Axiomatic methods, introduction to the absolute geometry, hyperbolic, spherical and projective planes. n-dimensional Euclidean geometry, convex polytopes, regular polytopes. n-dimensional classification of surfaces of second-order.

Literature:
– M. Berger: Geometry I-II, Springer 1994.

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

BMETE92AM55

Analysis 1

Course coordinator: Dr. Attila Andai

Preliminary requirement: Calculus 1

Descripton: Theory of complex functions. Holomorphic functions, harmonic functions, Cauchy–Riemann equations. Piecewise continuously differentiable curves, complex path integrals. Index of a curve with respect to a point, properties of the index function. Primitive function on a domain. Goursat lemma. Cauchy’s integral theorem, Cauchy integral formulas. Taylor series expansion, radius of convergence, Liouville theorem. The fundamental theorem of algebra. Maximum principle, principle of argument, Rouche theorem. Evaluation of real integrals by complex function theory.
Metric spaces, normed spaces. Definition of metric spaces (in particular: definition of normed spaces and the induced metric), examples. Ball, neighbourhood, basic topological notions. Compact sets, Cantor intersection theorem. Sequences in metric spaces, characterization of closed sets with sequences. Cauchy sequences, complete metric spaces (in particular: Banach spaces). Bolzano–Weierstrass theorem (characterization of compact sets with sequences), completion of metric spaces (without proof). Connected and path-connected sets in metric and normed spaces. Functions between metric spaces: limits, continuity. Properties of continuous functions on compact sets: continuous image of a compact set is compact, Weierstrass min-max theorem, continuity of the inverse, Heine theorem on uniform continuity. Lipschitz continuity, contraction. Banach fixpoint theorem.
Normed spaces, Banach spaces, bounded linear operators. Equivalence of norms in finite dimension (specifically: p-norms in Rn). Series in normed spaces, absolute convergence. Continuity and boundedness of linear operators between normed spaces, operator norm. Carl–Neumann series. Diagonalizable operators and their functions. Differentiability of functions between normed spaces, examples. Approximation with Bernstein polynomials. Stone theorem, Stone-Weierstrass theorem.
Hilbert spaces. Scalar products, and the induced norm. Orthogonal decomposition of Hilbert spaces by a closed subspace. Orthogonal projection. Riesz representation of bounded functionals. Bessel inequality, Parseval identity.

Literature:
– T. Tao: Analysis II.

Code

Title

III

BMETE91AM38

Algebra 1

Course coordinator: Dr. Sándor Kiss

Preliminary requirement: Introduction to Algebra 2

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: Calculus 2

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: Calculus 2

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 AND Programming Exercises for Probability Theory

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

BMETE92AM56

Analysis 2

Course coordinator: Dr. Attila Andai

Preliminary requirement: Analysis 1

Description: Measures and measurability. Sigma algebras, definition of measure, measurable space, measure space. Sigma algebra generated by a set system. Outer measures, Caratheodory extension theorem. Lebesgue measure, Lebesgue-Stieltjes measures. Finite, sigma-finite, and probability measures. Complete measures. Measurable functions, and their sum, product, maximum, minimum, reciprocal, and limit of sequences. Borel sets of metric spaces, Borel measurability, Lebesgue measurability. Existence of non-Lebesgue-measurable sets.
Integration in measure spaces. Definition and basic properties of the integral. Interchanging the limit and integration: Beppo–Levi theorem, Fatou-lemma, Lebesgue dominated convergence theorem. Integration with respect to Lebesgue–Stieltjes measures, integration by parts, integration by substitution, Newton–Leibniz formula. Product of measure spaces, Fubini theorem. Parametric integrals. Lp spcaes, basic properties, Holder and Minkowski inequalities. Convolution.
Fourier series. Convergence of the Fourier coefficients of integrable functions. Partial sums of Fourier series, Dirichlet kernel. Dirichlet theorem on the convergence of Fourier series of integrable functions. Cesaro-summable series. Fejer theorem for the Fourier series of continuous functions, Fejer kernel.
Fourier transform. Schwartz space, Fourier transform on the Schwartz space, on L1 and L2.

Literature:
– W. Rudin: Real and complex analysis.

Code

Title

III

BMETE94AM26

Differential Geometry 1

Course coordinator: Dr. Szilárd Szabó

Preliminary requirements: Geometry 2 AND Calculus 2

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

Code

Title

III

BMETE93AM19

Operations Research

Course coordinator: Dr. Pál Burai

Preliminary requirements: Calculus 2

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.

Code

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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Title

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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Title

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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Title

III

BMETE92AM57

Functional Analysis 1

Course coordinator: Dr. Máté Matolcsi

Preliminary requirement: Analysis 2

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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Title

III

BMETE95AM12

Creating Mathematical Models

Course coordinator: Dr. Roland Molontay

Preliminary requirements: Calculus 2 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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Title

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

BMETE94AM20

Differential Geometry 2

Course coordinator: Dr. Szilárd Szabó

Preliminary requirement: Differential Geometry 1

Descripton: Differentiable manifolds, tangent space, tangent bundle. Integral curve of a vector field. Vector bundles and related algebraic constructions (direct sum, tensor product, dual, homomorphisms). Differential forms, pull-back, exterior product, exterior derivation. Integration on compact oriented manifolds, Stokes' theorem. Lie-derivative, Lie-Cartan formula. Riemannian metric, examples. Geodetics, exponential map. Lie groups and algebras. Hopf-Rinow theorem and its consequences. Connections on a vector bundle, parallel transport, integrability. Levi-Civita connection, the Riemann curvature tensor. Properties of the curvature tensor, Ricci curvature. First and second variation of arc length, Jacobi vector fields.

Literature:
– P. Petersen: Riemannian geometry, Graduate Texts in Mathematics, 171. Springer.
– S. Gallot, D. Hulin, J. Lafontaine: Riemannian geometry, Universitext, Berlin, Springer.
– M. Berger, B. Gostiaux: Differential geometry: manifolds, curves, and surfaces, Graduate Texts in Mathematics.

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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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Title

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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Title

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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Title

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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Title

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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Title

III

BMETE13AM16

Physics 1 for Mathematicians

Course coordinator: Dr. László Udvardi

Preliminary requirement: Calculus 2

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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Title

III

BMETE92AM59

Matrix Analysis

Course coordinator: Dr. Milán Mosonyi

Descripton: 1. Finite-dimensional complex Hilbert spaces. Special operators, spectral decomposition, functional calculus. Polar decomposition, singular values. 2. Tensor product, symmetric and anti-symmetric tensor product. 3. Monotonicity and convexity of trace functions, quantum entropies. 4. Operator monotone and operator convex functions, basic examples, special integral representations. Operator Jensen inequality. 5. First and second order derivatives of operator functions, connections to operator monotonicity and convexity. 6. Positive super-operators. 7. Majorization and weak majorization, equivalent characterizations. 8. Mixedness of quantum states, entropy of Gibbs states, entropic characterization of quantum entanglement, conditional quantum entropy. 9. Monotone norms, unitarily invariant norms, Hölder inequality. Famous trace inequalities. 10. Quantum Rényi divergences.

Literature:
– Rajendra Bhatia: Matrix Analysis, Springer, 1997.
– Fumio Hiai: Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization.
– Fumio Hiai, Dénes Petz: Introduction to matrix analysis and applications.

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Title

III

BMETE92AM60

Introduction to Quantum Information Theory

Course coordinator: Dr. Milán Mosonyi

Descripton:
1. Finite-dimensional classical models, pure and mixed states, extremal measurements. 2. Finite-dimensional complex Hilbert spaces, special operators, spectral decomposition, functional calculus. Dirac formalism. Trace functional. 3. Finite-dimensional quantum models, pure and mixed states, measurements, Born rule, quantum uncertainty. 4. Hilbert-Schmidt inner product, discrete Weyl operators, Pauli operators. State space of the quantum bit, spin measurement. 5. Perfect and unambiguous state discrimination. Basics of quantum criptography. 6. Composite systems. Tensor product of Hilbert spaces and finite-dimensional operator algebras. 7. Marginal states, partial trace. Product, separable and entangled states. Schmidt decomposition. Purification of states. Maximally entangled states, Bell bases. 8. Quantum state evolution, completely positive maps and their representations. Quantum instruments, post-measurement state, Naimark dilation. Comparison of closed and open quantum systems. 9. Cloning and broadcasting of states. Dense coding and teleportation. 10. Classical and quantum correlations, non-local games, pseudo-telepathy. 11. Linear entanglement witnesses. 12. Error-free quantum communication over a noisy quantum channel.

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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Title

III

BMETE91AM60

Set Theory and Mathematical Logic

Course coordinator: Dr. Gábor Sági

Descripton: Logic: The language of zero-order and first-order logic. An overview of higher-order languages. Semantics of zero-order and first-order languages. Normal forms. Galois connection between model classes and sets of formulas. Logical consequence and its relation to implication. Deduction theorem, characterization of consequence by the notion of inconsistency. On proof theory: derivation and refutation systems. Hilbert type calculus for propositional logic, zero-order completeness theorem. Some (refutation-)complete calculi of first-order logic (resolution, analytic trees, etc.). Compactness theorem. Löwenheim-Skolem theorems. Isomorphism and elementary equivalence. Decidable and undecidable theories. Outline of the arithmetization of derivability. The theorems of Church and Gödel.
Set theory: The equivalence of sets. The axiom of choice and some basic consequences; countable unions of countable sets are countable. Bernstein's antisymmetry theorem. A set and its power set are never equivalent. The naive definition of cardinality leads to a contradiction. The ZFC axioms. Introducing new operations and relations. Ordered pairs, relations, functions. Partial orders, total orders, well-orders. Ordinals and their basic properties. Ordinals form a proper class. Each ordinal has a successor ordinal, each set of ordinals has a supremum. The existence of limit ordinals. Transfinite induction and transfinite recursion. The axiom of choice and two equivalent characterizations: Zorn's lemma and Zermelo's well-ordering theorem (every set has a well-ordering). The definition of the cardinality operation and some of it's basic properties. Every infinite cardinal is a limit ordinal. Examples. Cardinal arithmetic: addition, multiplication and exponentiation of cardinals and some of their basic properties. The general distributive law. The cofinality operation. Theorems about infinite cardinalities. The fundamental theorem of cardinal arithmetic. The aleph-operateion. Examples: the cardinalities of some well-known mathematical objects. The continuum-problem. Some well-known statements which are independent from ZFC. The cumulative hierarchy of sets; the rank operation and its basic properties. The axiom of regularity is equivalent to the statement that every set has a rank.
Outlook:
– Ultraproducts, axiomatizability, nonstandard real numbers.
– Connections between categoricity and completeness. Countably categorical theories and their characterization by the automorphism group of theirs models. Random graphs, 0-1-laws.
– Algebraic methods in logic.
– Transfinite methods outside of set theory: the transfinite construction of algebraically closed fields; the plane has a subset which intersects each line in exactly 2 points, etc.
– The existence of a Hamel-basis and some interesting consequences.
– Fundementals of partition calculus; some introductory results in infinite Ramsey-theory.

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Title

III

BMETE92AM54

Applied Numerical Methods with Matlab

Course coordinator: Dr. Róbert Horváth

Preliminary requirements: Introduction to Algebra 2 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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Title

III

BMETE91AM59

Number Theory

Course coordinator: Dr. Sándor Kiss

Preliminary requirements: Calculus 1

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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Title

III

BMETE94AM30

Computational Geometry

Course coordinator: Dr. Szilvia Béla

Preliminary requirement: Theory of Algorithms (signature) AND Differential Geometry 1 (signature)

Descripton: Mathematical description of 3-dimensional data structures: discrete data structures of polyhedrons, boundary representation of polyhedral surfaces, edge and face oriented structures, two-manifold surfaces. Solid modeling: description of volumetric data structures with CSG-trees (using primitives or octree structures), computation of Boolean set operations and representation of dynamic models. Analytic data systems of surfaces, algebraic surfaces, analytic CSG trees.
Spline techniques: description of smooth curves using cubic polynomials, properties of Hermite and Bezier curve segments, the effect of parameter transformations, fitting of curve segments, description and fitting of surface patches, introduction to B-splines. Subdivision and level-set methods. Description of offset and blending surfaces.
Visualization: description of affine mappings with homogeneous coordinates. Parallel projection and computation of the images. Oblique and orthogonal axonometries. Matrix of central projection using homogeneous coordinates, stereo-techniques. Window-based data managing techniques. Computer graphics packages.
Algorithms of visibility: model based hidden-line techniques, painting, scan-line algorithms, image-space z-buffer algorithm, ray-tracing, visualizing objects represented by CSG-trees. Physical model of illumination (rendering), shading, theoretical background of coloring, textures, motion and camera moving.
Data cloud (handling of non-structured data sets): triangulation techniques, construction of triangulated meshes. Interpolation techniques. Computation of convex hull, inclusion and collision problems. Distance measuring and neighboring problems. Processing of measured data, data searching according to attributes, sorting and visualization.

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Title

III

BMETE92AM58

Programming Exercises for Differential Equations

Course coordinator: Dr. Szilvia Béla

Preliminary requirement: Informatics 2 AND Differential Equations 1

Descripton: The aim of the course is to maintain the students' programming skills through programming problems associated with the topics of the Differential Equations course helping the understanding of the theory of ordinary differential equations. The course provides an outlook on related applications and simulation techniques.

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Title

III

BMETE95AM30

Probability 2

Course coordinator: Dr. Balázs Bárány

Preliminary requirement: Probability Theory 1 AND Analysis 2

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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Title

III

BMETE92AM45

Partial Differential Equations

Course coordinator: Dr. Mihly Kovács

Preliminary requirements: Differential Equations 1 AND Analysis 2

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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Title

III

BMETE94AM22

Convex Geometry

Course coordinator: Dr. Zsolt Lángi

Preliminary requirements: Geometry 2 AND Introduction to Algebra 2

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.

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Title

III

BMETE90AM49

Individual Research Project 2

Course coordinator: Dr. Lajos Rónyai