Applied Mathematics, ELTE University Budapest
List and description of the courses
First year
Nr.
Name of the course
1
Analysis I.
2
Algebra I.
3
Discrete Mathematics I.
4
Number Theory I.
5
Computer Science
6
Maple Course
7
Introduction to Computers
8
9
10
11
12
13
14
Analysis II.
Algebra II.
Numerical Methods of Linear Algebra
Discrete Mathematics II.
Number Theory II.
Geometry
LaTeX Course
Second year
Nr.
Name of the course
1
Analysis III.
2
Algebra III.
3
Numerical Analysis I.
4
Operations Research
7
Mathematical Applications I.
6
Autocad Course
7
8
9
10
11
12
13
Analysis IV.
Theory of Algorithms & Data Structures I.
C Programming
Differential Geometry
Differential Equations
Introduction to Financial Management
Linux
Third year
Nr.
1
2
3
4
5
6
7
8
Name of the course
Numerical Analysis II.
Probability
Theory of Algorithms & Data Structures II.
Complex Analysis I.
Dynamical Systems
Functional Analysis
Digital Convexity
Hajós Geometry Seminar
third year, continuation
9
Stochastic Processes
10 Computational Complexity
11 Statistics
12 Fourier Analysis
13 Mathematical Logic
14 Applications of Probability and Statistics
15 Partial Differential Equations I.
16 Modelling River Pollution
Fourth year
Nr.
Name of the course
1
Discrete Dynamical Systems
2
Continuous Dynamical Systems
3
Formulating Mathematical Models I.
4
Partial Differential Equations II.
5
Numerical Solution of Elliptic Equations
6
Computational Number Theory
7
Computer Algebra I.
8
Transformations Mathematics I.
9
10
11
12
13
14
15
16
17
Partial Differential Equations III.
Special Functions
Cryptography
Computer Algebra
Finite Element Method
Formulating Mathematical Models
Analysis of Mathematical Models
Applications of Functional Analysis
Transformations in Mathematics II.
Short description of the courses
First year
1. Analysis I-II. Sequences, convergence, Bolzano-Weierstrass, series, multiplication, convergence
criteria, limit of mappings, continuity, open, closed and compact sets, differentiation, Darboux
property, Taylor-formula, Riemann, Riemann-Stieltjes integration, functions of bounded variation,
curves, function series, differentiation in higher dimension, inverse and implicit functions.
2. Algebra I-II. Complex numbers, (symmetric) polynomials, linear algebra, vector spaces,
eigenvectors, Jordan, Cayley-Hamilton. Bilinearity, quadratic forms, Gram-Schmidt, Sylvester,
Euclidean spaces and linear transformations, groups, free group, Dyck, Jordan-Hölder, solvable groups,
direct product, finite Abelian groups, rings, number theory in rings, Euclidean rings.
3. Discrete Maths I.-II. Elementary algorithms, Dijsktra, graphs, Hall condition, Ramsey,
König lemma, convex polygons, extremal problems, Turán, graphs in the plane, sieve method, Stirling
numbers, normal and perfect graphs, Menger.
4. Number Theory I-II. Primes, congruences, congruence systems, Diophantine equations,
Pythagorean triplets, Gauss integers, reduction of Fermat’s Conjecture.Thue, Waring problem.
Diophantine approximation theory, Liouville, exponential sums.
5. Computer Science Unix, Maple.
7. Maple Course
8. Introduction to Computers Windows, Office, Internet
10. Numerical Methods of Linear Algebra Implementation of numbers in computers, matrix
decompositions (LU, Cholesky, QR, singular, Schur), Gauss’ Algorithm, norms, errors of solution of
linear equation, generalized solution, iterative methods, relaxations.
13. Geometry Affin and projective geometry
Second year
1. Analysis III-IV. Jordan measure, integration, transformation of integrals, Fubini, Gauss-Green,
primitive, Green-formulae, Poisson kernel, harmonic functions. Abstract measure theory, extension,
Lebesgue-Stieltjes measures, Jordan, Hahn, Radon-Nikodym, differentiation of measures, interval
functions, density. Lp spaces, convolution.
2. Algebra III. Commutative fields, extensions, Galois Theory, applications. Hamming and BCH
codes. Lattices, Jordan-Dedekind, Kurosh-Ore. Noether rings. Ideal theory, radicals, Lasker-Noether,
Dedekind rings.
3. Numerical Analysis Direct and iterative methods for calculating eigenvectors and eigenvalues.
Interpolations, Chebyshev, Hermite. Uniform approximation. Splines, orthogonal polynomials
4. Operations Research Linear algebra, polyhedra, cones, Caratheodory, Farkas lemma,
duality, polar cones, unimodular matrices, integer programming, simplex algorithm,
currents, MFMC, Edmonds-Karp, Hungarian method, shortest walks, applications
in geometry: Helly, Kirchberger, applications in the theory of games
6. Autocad Course Brief introduction to Autocad
7. Applications I. Introduction to Corporational Economy. Mathematics of Elections.
8. Theory of Algorithms & Data Structures I-II. Exercise, P and NP class, program verification,
ADT & ADS, sorting (bubble, pan, radix, Batcher, tournament, heap, quicksort, shell, run), priority
sequence, lists, stacks, sequences. Seeking. Coding. B-trees. Simultaneous min and max search.Median
search, pattern fitting, Rabin-Karp, Knuth-Morris-Pratt.
9. C Programming ANSI C with standard libraries
10. Differential Geometry Curves, Lagrange, curvature, torsion, supporting trihedron, Serret-Fernet.
Surfaces, first and second-order Gauss’ main curvature values, equivalence, classification, normal
intersection, Meusnier, Euler, main directions, median curve.
11. Differential Equations Gronwall lemma, existence and unicity theorems, differentiable
dependence, linear differential equations, solution, dynamical systems, stability, Ljapunoff,
Barbasin-Krasovski, asymptotic behaviour, Poincaré-Bendixson, Floquet theory, HartmanGrobman,boundary problems, Sturm separation theorems, calculus of variation.
12. Introduction to Financial Management Treynor-Balck, CAP-M, bond pricing, options.
13. Linux Introduction to Linux from the viewpoint of a user and single-machine-administrator.
Third year
1. Numerical Analysis II. Numerical integration, Simpson, Newton-Cotes, Gauss. Numerical solution
of differential equations, higher order methods, Runge-Kutta, Taylor, Euler, Mylne-Simpson. Local
and global error, 0-stability, A-stability.
2. Probability General random variables, and their functions (cdf, pdf, expected value, deviation,
correlation, covariance etc.) independence, Kolmogoroff’s 0-1 law,
stochastic, Lp convergence, metrics, Levy’s inequality weak convergence, Cramer-Sluckij, Prohoroff,
Doob lemma, central limit theorems, Laws of Large Numbers (Khinchine, Bernstein, Kolmogoroff),
Characteristic functions, Central Limit, inversion, Lindeberg-Feller, Ljapunoff, conditional
expectation, convergence theorems, Jensen, martingales, Doob, Kolmogoroff, regular martingale,
Kronecker lemma.
4. Functional Analysis Hilbert spaces, bounded operators, compact operators, Banach spaces, open
mapping, closed mapping, Hahn-Banach, Banach-Steinhaus.
5. Dynamical Systems Equivalences, bifurcations, classification of linear systems. Smale. Special
attractors.
6. Complex Analysis Holomorphic functions, Cauchy formulae, Laurent series, isolated singularities,
residues, Rouche, Schwartz, etc., Weierstrass, conformal mappings, Hurwitz, Riemann mapping
theorem, analytic continuation, Schwartz-Christoffel, Mittag-Leffler, Picard, Borel.
Entire functions of finite order, harmonic functions.
7. Digital Convexity Affin and lattice geometry. Polyhedra, polytops, cones, separability.
Carathedory, Radon, Helly, Gale in affin spaces. Pick, Chevtál-Frisk, Minkowski, Blichfelt, Voronoi,
Caratheodory, Helly, Hermite constant, Rogers on lattices.
8. Hajós Geometry seminar Various lectures from Geometry.
9. Stochastic Processes Finite-state Markoff processes. Classification of states, recurrence, positive
states, the limit of transition probability, stationary disribution. Poisson processes. Recovery processes,
recovery function, recovery equation. Elementary recovery theorem, asymptotics of number of
recoveries. Wiener processes, its constructionInvariance (Donsker). Characteristics of trajectories (nondifferentiability, quadratic variance, Hölder continuity). Kolmogoroff’s continuity and its application.
10. Computational Complexity Finite-state automata, regular languages, non-deterministic
automata,Turing machines, recursive and recursively enumerable languages,
recursive functions, universality,Rice. Time and memory complexity, polynomial algorithms, Cook,
NP-complete problems (SAT, 3-SAT, 3-colouring, set partition, travelling salesman)
11. Statistics Estimations of the distribution function and the density, sufficiency, Fisher
information, completeness, exponential families, point estimation, unbiasedness, admissibility,
minimaxity, efficiency, blackwellization, information inequality. Asymptotic properties of estimators,
consistency. Empirical estimators, method of moments, maximum likelihood estimation. Bayes
estimation, sampling from finite population. Hypothesis testing, Neyman-Pearson lemma. One-sided
and two-sided alternatives in exponential families, classical parametric tests and their optimality,
likelihood ratio test, confidence sets and intervals, parameter estimation for the multivariate normal
distribution, normal linear model.
12. Fourier Analysis topological groups, Haar-measure, character and dual of a group, Fourier
transformations (TFT, TFC, TFS, DFT), elementary characteristics, unit approximation, wavelet
operator, inversion, L2, Plancherel, differentiation, convergence, FT in AC, summations ((C,1), De La
Vallé Poussin, Riesz, Rogosinski, Weierstrass, Abel, θ, Fejér, Poisson), consistency, Jackson
polynomials. Approximations, Stone-Weierstrass, Jackson.
13. Mathematical Logic Russel paradoxon, cardinality, Cantor-Bernstein-Schröder, Cantor power set,
continuum hypothesis, Brouwer-Sperner, computabililty, recursive functions, Church, primitive
functions, logical operations, zero and first-order languages, formulae, Gödel incompleteness.
15. Partial differential equations Examples from Physics, classification of linear, second order PDE
with constant coefficients, test functions, distributions, smooth functions, differentiation of
distributions, direct product, convolution, fundamental solutions, classical and generalized Cauchy
problem in the parabolic case, Green-formulae. Soboljev spaces, embedding to L2. Classical
and generalized eigenvector problem, Fourier method.
Fourth year
1. Discrete Dynamical Systems Topological transitivity, minimality. Symbolic dynamics, Bernoulli
shift. Revolting the unit circle. General revolting number. Invariant measure, Riesz representation,
Kryloff-Bogoluboff. Cantor and ω-limit sets, uniquely ergodynamicality and minimality, Haarmeasure, irrationally revolting maps are uniquely ergodynamical. Unimodality, ω-limit periodicity,
ordering and weak equivalence, topological conjugatedness. Entropy, monotonicity, entropy of power
map, equivalent definitions, zig-zag number, Misiuvewicz-Szlenk, Milnor-Thurston. Markoff graphs,
Sharkovskii.
2. Continuous Dynamical Systems Manifolds. General Stabil Manifold Theorem. General Centre
Manifold Theorem. General Hartman-Grobman Theorem. Hamiltonian Systems.
3. Formulating Mathematical Models I-II. Sample models of population and evolution dynamics.
Transport processes. Models of marketing.
4. Partial Differential Equations II-III. Temperated distributions. General and partial Fourier
transformation. Inversion, transformation of distributions with compact support. L 1&L2. Paley-Wiener.
Application. Locally rectangular-like surfaces, trace operator, surfacial integral, embedding theorem,
compactivity. More general elliptic equations, ellipticity, uniform ellipticity. Eigenvalue and boundary
value problem. Variational analysis. Ritz, Galerkin methods. More general hyperbolic equation,
unicity, Fourier method, Galerkin method. More general parabolic equations, unicity, Fourier method,
Galerkin method. Non-linear equations, the abstract problem, Galerkin method. Bounded, finite
dimensional, coercive, monotonic, pseudo-monotonic operators from reflexive, separable Banach
spaces to its dual. Wk,p as special cases.
5. Numerical Solution of Elliptic Equations Comparison of methods. Differential schemes,
stability, convergence, consistency, Lax. Comparison of solution methods for linear systems,
preconditioning. Multigrid methods, interpolations, relaxations, smoothing possibilities. 2 nd –order
boundary condition approximation. Convergence. Friedrichs, 1D example, FFT, eigenvalues. General
domain, Hilbert-Schmidt core.
6. Computational Number Theory Time of elementary operations. Erathostenes’ sieve, calculating
gcd. Congruencies, calculating multiplicative inverse, solving linear system of congruences. RSA,
modular power. Factorization, Mersenne primes. Finite fields, nth unit roots in F q. Quadratic residues,
Jacobi symbol, calculating Legendre symbol. Evolving radicals mod p. Error correcting codes.
Cryptography, discrete logarithm, Diffie-Hellman. Deterministic and stochastic prime test, pseudo
primes, Carmichel numbers, Solovay-Strassen. Fermat factorization, factor base, quadratic sieve.
Elliptic curves, over finite fields, Hasse, Diffie-Hellman.
7. Computer Algebra I-II. Orderings of polynomials, Dickson, Hilbert, Gröbner base, Buchberger.
P-adic numbers, Ostrakowski, Hensel. Primes, factorizations, long multiplication. Resultant,
discriminant, supporting matrix, measure, reducibility, square-free factorization, Berlekamp,
Kronecker. Symbolic integration, differential fields, logarithmic extension, Hermite, Horowitz,
Rothstein-Tragen.
8. Transformations in Mathematics I-II. Bessel’s Inequality, Riemann-Lebesgue Lemma, Fejér.
Calculating area, curvature, arclength, moments, centre of mass with Fourier coefficients.
Parametrization, symmetries. Two-terms. Fourier series of piecewise linear paths, special cases. DFT.
Corresponding algorithms. Contouring. Zak transformation, L2, surjectivity, convolution. Bases,
biorthogonality. Frames, reconstruction, Weyl-Heisenberg. Gábor and Wavelet transformations. Voice
transformation, Haar-measure, Heisenberg group. Peter-Weyl. Disc algebra. Rademacher,
Kolmogoroff. Riesz base.
9. Special Functions Γ, ζ, Bessel, spherical functions
10. Cryptography Classic codings (transposition, substitution, block and polyalphabetic versions),
breaking. DES, AES.Entropy. Public key coding, RSA, Rabin.
11. Finite Element Method Comparison. 1D example. Realization, choice of the base, Nitsche, Aubin,
Oganesjan-Ruhovec). Multivariate case, hp methods, implementation.