BSc in Mathematics | Budapest University of Technology and Economics

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

Code

Title

Parameters*

ECTS credits
per semester

III

Obligatory courses (159 ECTS credits)

BMETE91AM35

Basics of Mathematics

BMETE92AM36

Calculus 1

BMETE91AM36

Introduction to Algebra 1

BMETE94AM17

Introduction to Geometry

BMETE91AM42

Informatics 1

BMETE92AM37

Calculus 2

BMETE91AM37

Introduction to Algebra 2

BMEVISZA025

Combinatorics and Graph Theory 1

BMETE94AM18

Geometry

BMETE91AM43

Informatics 2

BMETE13AM16

Physics 1 for Mathematicians

BMEGT35A410

Accounting

BMETE92AM38

Analysis 1

BMETE91AM38

Algebra 1

BMETE95AM29

Probability Theory 1

BMETE91AM46

Programming Exercises for Probability Theory

BMETE93AM15

Differential Equations 1

BMETE91AM44

Informaitcs 3

BMETE95AM31

Mathematical Statistics 1

BMETE92AM39

Analysis 2

BMETE94AM19

Differential Geometry 1

BMETE93AM19

Operations Research

BMEVISZAB01

Theory of Algorithms

BMETE91AM47

Programming Exercises forTheory of Algorithms

BMETE91AM39

Algebra 2

BMETE93AM16

Optimization Models

BMETE95AM34

Stochastic Processes

BMETE95AM12

Creating Mathematical Models

BMEGT30A410

Micro- and Macroeconomics

BMETE92AMxx

Applied Numerical Methods with Matlab

BMETE94AM20

Differential Geometry 2

BMEGT35A411

Finance

BMETE90AM47

BSc Thesis Project

Specialization courses (12 ECTS credits must be completed)

BMETE95AM33

Tools of Modern Probability Theory

BMETE92AM42

Measure Theory

BMETE90AM45

Individual Research Project 2

BMETE92AM45

Partial Differential Equations

BMETE94AM22

Convex Geometry

BMEVISZA026

Combinatorics and Graph Theory 2

BMETE90AM48

Individual Research Project 1

Elective courses (9 ECTS credits must be completed)

Physical Training (must be completed in 2 semesters)

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

DESCRIPTION OF SUBJECTS

Code

Title

III

BMETE91AM35

Basics of Mathematics

Course coordinator: Dr. Miklós Ferenczi

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. Erzsébet Horváth

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

BMETE94AM17

Introduction to Geometry

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. Ferenc Wettl

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

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. Alex Küronya

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

BMETE94AM18

Geometry

Course coordinator: Dr. Ákos G. Horváth

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. Ferenc Wettl

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. Ágnes Laáb

Code

Title

III

BMETE92AM38

Analysis 1

Course coordinator: Dr. Attila Andai

Descripton: Metrics and metric spaces. Topology of metric spaces. Basic properties of metric and normed spaces. Metric subspaces and isometrics. Sequences in metric spaces. Convergence of sequences in metric spaces. Separable metric spaces. Convergent sequences in normed spaces. Product of metric and normed spaces. Compact sets, relative compact sets and their its basic properties in metric spaces. Characterization of compact metric spaces. Cantor's intersection theorem. Bolzano-Weierstrass theorem. Product of compact metric spaces. Equivalence of norms in finite-dimensional vector spaces. Limit of functions in metric spaces. Definition of continuity in terms of epsilon-delta and limits, and their equivalence. Topological characterization of continuity. Homeomorphism. Uniform continuity. Basic properties of continuous functions on compact spaces. Weierstrass's maximum-minimum principle. Characterization of compact sets in finite-dimensional normed spaces. Fundamental theorem of algebra. Approximation by Bernstein polynomials. Complete metric spaces. Contractions and Banach fixed point theorem in metric spaces. Totally bounded metric spaces and the Hausdorff characterisation theorem. Completeness of finite-dimensional normed spaces. Connected and path-connected metric spaces. Nowhere dense sets and Baire's category theorem. Banach spaces. Characterization of Banach spaces with absolutely convergent series. Linear and multi-linear maps between normed spaces and their continuity and norm. The normed space of linear and multi-linear maps between normed spaces. Positive, negative, definite and indefinite multi-linear maps. Bounded linear operators and functionals. Hahn-Banach theorem and some consequences. Banach-Steinhaus theorem. Open mapping theorem. Closed graph theorem. Bounded inverse theorem. Derivation of functions between normed spaces.

Literature:
– T. Tao: Analysis II.

Code

Title

III

BMETE91AM38

Algebra 1

Course coordinator: Dr. Alex Küronya

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

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

BMETE91AM46

Programming Exercises for Probability Theory

Course coordinator: Dr. Ferenc Wettl

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. Katalin Nagy

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

Informaitcs 3

Course coordinator: Dr. Alex Küronya

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

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

BMETE92AM39

Analysis 2

Course coordinator: Dr. Attila Andai

Descripton: Ring, σ-ring, and σ-algebra of sets. Set functions. Concept of Lebesgue measure. Outer measure. Measurable sets. Measure generated by an outer measure. Example for not Lebesgue-measurable set. Measure space, measurable functions. Null sets. The concept of convergence in measure and almost everywhere (ae) and relations between them. Integral of measurable functions. Beppo-Levi theorem, Fatou's lemma, Lebesgue's dominated convergence theorem. Lp-spaces, and Hölder and Minkowski inequality. Absolute continuity of the integral. Riemann sphere. Limits and properties of complex valued sequences. Limit and continuity of complex functions. Power series of elementary functions. Euler's formula. Complex logarithm function. Differentiability of complex functions. Cauchy-Riemann equations. Regularity of complex functions and elementary properties of regular functions. Harmonic functions, harmonic conjugate. Complex integral, integration by substitution. Newton-Leibniz formula. Goursat lemma. Cauchy's integral theorem and integral formula on convex domain. Index of a curve. Simply-connected subsets. Cauchy integral theorem and integral formula. Primitive functions. Morera's theorem. Power series of regular functions. Liouville theorem and fundamental theorem of algebra. Multiplicity of roots. Laurent series. Isolated, removable and essential singularities of complex functions. Laurent series. Concept of residue and the residue theorem. Residue theorem with logarithmic functions. Argument principle. Rouche's theorem. Open mapping theorem. Maximum and minimum principles.

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

Code

Title

III

BMETE94AM19

Differential Geometry 1

Course coordinator: Dr. Brigitta Szilágyi

Descripton: A görbe fogalma, paraméterezése, átparaméterezés, ívhossz. Ívhossz izometriával szembeni invarianciája, az érintővektor fogalma, a görbület fogalma, általános görbületfogalom, Fox-Milnor-tétel. A normálvektor fogalma, az előjeles görbület fogalma, totális görbület és konvexitás. Globális tételek: négy csúcspont tétele, izoperimetrikus egyenlőtlenség. Frenet-formulák, torzió, a görbeelmélet alaptétele. – A felület fogalma. A Gauss-görbület, főgörbületek. Intrinsic geometria, felületek izometriái. Theorema Egregium. Christoffel-szimbólumok, PMC-egyenletek. A felületelmélet alaptétele. Kovariáns deriválás, Lie-zárójel, Riemann-féle görbületi tenzor. A geodetikus görbület. A Gauss–Bonnet-tétel és alkalmazásai.

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

Code

Title

III

BMETE93AM19

Operations Research

Course coordinator: Dr. Marianna Eisenberg-Nagy

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

BMEVISZAB01

Theory of Algorithms

Course coordinator: Dr. Katalin Friedl

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.

Code

Title

III

BMETE91AM47

Programming Exercises forTheory of Algorithms

Course coordinator: Dr. Ferenc Wettl

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.

Code

Title

III

BMETE91AM39

Algebra 2

Course coordinator: Dr. Erzsébet Lukács

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.

Code

Title

III

BMETE93AM16

Optimization Models

Course coordinator: Dr. Boglárka Gazdag-Tóth

Descripton: Introduction to mathematical modeling, to mathematical programming problems, and their classification. Model reformulations: rewrite complex transportation problem to simple transportation problem, rewrite maximum flow problem to minimum cost maximal flow problem. Modeling problems in economy. Integer modeling tricks, set covering, set partitioning problems. Modeling Facility Location problems. Numerical errors. Dynamic programming. Scheduling problems, heuristics, approximations, online versions. Decision Theory. Inventory tasks.

Literature:
– R. Fourer, D.M. Gay, B.W. Kernighan: AMPL: A Modeling Language for Mathematical Programming, Second edition.
– L. Liberti: Problems and Exercises in Operations Research, École Polytechnique, 2007.
– W.L. Winston: Operations Research: Applications and Algorithms. 4th edition, Thomson Learning.

Code

Title

III

BMETE95AM34

Stochastic Processes

Course coordinator: Dr. Károly Simon

Descripton: Alapfogalmak: sztochasztikus folyamat, peremeloszlások, Kolmogorov-alaptétel, stacionárius, stacionárius növekményű, független növekményű folyamatok, Brown-mozgás, Poisson-folyamat. Véges Markov-láncok: átmenet valószínűségek, sztochasztikus mátrixok lineáris algebrája, félcsoport tulajdonság, hatás előre függvényeken, hatás hátra mértékeken, állapotok osztályozása, irreducibilitás, periódus, P spektruma, konvergencia egyensúlyhoz, spektráslis rés becslése (Doeblin) . Megszámlálható Markov-láncok: pozitív és null-rekurrencia, tranziencia, bolyongások Zd-n: Pólya-tétel, születési-halálozási folyamatok, sorbanállási problémák, elágazó folyamatok. 1-dimenziós bolyongás: tükrözési elv és következményei, tranziencia nem-szimmetrikus esetben, gambler’s ruin, differenciaegyenletek. Felújítási folyamatok: felújítási egyenlet, Laplace-transzformáció alkalmazásai, felújítási paradoxon. Folytonos idejű Markov-láncok: fenomenologikus leírás, ugrási ráták, független exponenciális órák, átmenet-valószínűségek félcsoportja, Komogorov–Chapman-egyenlet, a félcsoport mátrix-analízise, infinitezimális generátor, folytonos idejű Markov-láncok megszámlálható állapottéren. Mértékelméleti kiegészítések: filtrációk, sztochasztikus folyamat természetes filtrációja, feltételes várhatóérték. Martingálok: filtráció, adaptált folyamat, szub-/szuper-/martingál, megállási idők, opcionális megállási tétel (Doob), diszkrét sztochasztikus integrálás, martingál konvergencia tétel (Doob), maximális egyenlőtlenség (Doob), Höffding–Azuma-egyenlőtlenség, iterált logaritmus tétel. Brown-mozgás, Wiener folyamat: fenomenologikus leírás, alaptulajdonságok, Wiener-féle konstrukció vázlata, Paul–Lévy- és Ciesielski–de Feriet-féle konstrukció, skála, önhasonlóság, iterált logaritmus tétel, időinverzió, nem-differenciálhatóság, kapcsolat a hőegyenlettel.

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

Code

Title

III

BMETE95AM12

Creating Mathematical Models

Course coordinator: Dr. Domokos Szász

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.

Code

Title

III

BMETE95AM12

Creating Mathematical Models

Course coordinator: Dr. Katalin Petró

Code

Title

III

BMETE92AMxx

Applied Numerical Methods with Matlab

Course coordinator: Dr. Róbert Horváth

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.

Code

Title

III

BMETE94AM20

Differential Geometry 2

Course coordinator: Dr. Szilárd Szabó

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.

Code

Title

III

BMEGT35A411

Finance

Course coordinator: Dr. Imre Tarafás

Code

Title

III

BMETE90AM47

BSc Thesis Project

Course coordinator: Dr. Miklós Horváth

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.

Code

Title

III

BMETE95AM33

Tools of Modern Probability Theory

Course coordinator: Imre Péter Tóth

Descripton: A tárgy célja a modern valószínűségszámításban használt legfontosabb kombinatorikai, lineáris algebrai, valós függvénytani, mértékelméleti, komplex függvénytani, funkcionálanalízisbeli és geometriai eszközök megtanítása. – Példákon keresztül bemutatjuk ezek valószínűségszámításbeli alkalmazását, de a hangsúly az eszköztár kifejlesztésén van. A megszerzett tudás egy részét az MSc képzésben fogjuk hasznosítani. – Kombinatorika: Generátorfüggvény-módszer. Stirling-formula, Euler Gamma-függvény. Topológia: Konvergencia metrikus téren és topológikus téren. Kompaktság. Szorzattér, szorzat-topológia, Tyihonov-tétel. Lineáris algebra: Belső szorzatterek, Cauchy–Schwarz-egyenlőtlenség. Mátrixok hatványozása, analitikus mátrix-kalkulus. (Alkalmazás: Markov-átmenetvalószínűségek.) Függvénytranszformációk: Laplace-transzformáció. Fourier-sorfejtés, Fourier-transzformáció, diszkrét Fourier-transzformáció. (Alkalmazás: karakterisztikus függvény.) Legendre-transzformáció. Mértékelmélet: Integrálás és deriválás felcserélhetősége. Egyenletes konvergencia és folytonosság. (Alkalmazás: karakterisztikus függvény differenciálhatósága.) Jensen-egyenlőtlenség. Abszolut folytonosság, Radon–Nikodym-tétel. (Alkalmazás: feltételes várható érték.) Mértékek előretoltja, helyettesítéses integrál. (Alkalmazás: Valószínűségi változók eloszlása, eloszlások várható értéke.) Szorzattér, szorzatmérték. Fubini-tétel. (Alkalmazás: függetlenség.) Mértékek dekompozíciója, feltételes mérték, faktormérték. Komplex függvénytan: Reziduum-tétel, Laurent-sorfejtés. (Alkalmazás: konvolóciók és karakterisztikus függvények számolása.) Analitikus kiterjesztés, Vitali tétel. Funkcionálanalízis: Korlátos operátorok spektruma, rezolvens, spektrálsugár. Hahn–Banach-tétel. Ck terek, Arsela–Ascoli-tétel. Folytonos lineáris funkcionálok, Riesz–Markov-tétel. Duális terek, gyenge csillag topológia, feszesség. Fourier-transzformáció még egyszer, Riesz–Fischer-tétel.

Code

Title

III

BMETE92AM42

Measure Theory

Course coordinator: Dr. Miklós Horváth

Descripton: Recapitulation: sigma-algebra, outer measure, measure. Signed measure, Hahn decomposition. Radon measures, approximation theorem. Lebesgue-Stieltjes measure. Measurable functions. Convergence in measure. Theorems of Egoroff and Lusin. Integration in measure spaces. Absolute continuity of the integral. Integration of sequences of functions: theorems of Beppo-Levi, Fatou and Lebesgue. Products of measure spaces, Fubini theorem. Lp spaces. Absolutely continuous and singular measures, Radon-Nikodym derivative, Lebesgue decomposition. Absolutely continuous functions, Newton-Leibniz formula. Total variation. Functions of bounded variation, decomposition into absolutely continuous and singular parts.

Literature:
– T. Tao: An introduction to measure theory, http://terrytao.files.wordpress.com/2011/01/measure-book1.pdf
– P.R. Halmos: Measure Theory, Springer 1978.

Code

Title

III

BMETE90AM45

Individual Research Project 2

Course coordinator: Dr. Miklós Horváth

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.

Code

Title

III

BMETE92AM45

Partial Differential Equations

Course coordinator: Dr. János Karátson

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.

Code

Title

III

BMETE94AM22

Convex Geometry

Course coordinator: Dr. Zsolt Lángi

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.

Code

Title

III

BMEVISZA026

Combinatorics and Graph Theory 2

Course coordinator: Dr. Tamás Fleiner

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.

Code

Title

III

BMETE90AM48

Individual Research Project 1

Course coordinator: Dr. Miklós Horváth