CEU, Department of Mathematics and its Applications

This is a two-year program in Applied Mathematics, including coursework, research and thesis components.
The M.S. regular courses are delivered during the two main terms, Fall and Winter Term, that start end of September and early January, respectively (and each of them has a duration of 12 weeks). We also schedule special courses, invited lecture series or seminars beyond these two terms, especially during the Spring Term.

GRADUATION REQUIREMENTS

M.S. students are required to earn a total of 45 credits in mandatory and elective courses (see Section Schedule of Coursework below). In addition, 5 research credits must be earned by participation in seminars.

All M.S. students are required to attend the Departmental Seminar, starting the Winter Term of their first year. In addition, there is a required Spring MS Seminar for first year students as well as a Winter MS Seminar for second year students. MS seminars aim to introduce students to various topics in mathematics and its applications and facilitate their research work toward the MS Thesis (see Section Thesis below).

First year M.S. students are required to write a preliminary thesis proposal of 3-5 pages and present it in a session of the Spring MS Seminar. This proposal should eventually be improved and/or extended and then re-discussed during their second year.

As a final condition for graduation, M.S. students are required to write a thesis (worth 10 credits) which is normally defended by the end of the program. The thesis must be submitted 3 weeks in advance. The defense comprises the presentation of the thesis as well as a comprehensive exam on the area of the thesis, covering the material of 3 courses, including at least 2 electives (i.e., either 3 electives or 2 electives + 1 mandatory).

The minimum passing grade for every exam, including thesis, is C+ (worth 2.33). Course grades count towards the final Grade Point Average (GPA) as per credit and constitute 75% of the final GPA. More precisely, the

Final GPA = 0.75 × ån_ig_i / ån_i + 0.25 × thesis grade,

where n_irepresents the number of credits attributed to course c_i and g_i is the grade obtained in the corresponding exam. The minimum final GPA required for the M.S. degree is 2.67. Note that our university uses the US credit system.

Therefore, the requirements to qualify for the M.S. degree are:
•    passing all mandatory courses;
•    earning at least 45 credits for the overall coursework;
•    earning 5 credits for research;
•    successful thesis;
•    final GPA = 2.67 or higher.

For additional details, please visit http://www.ceu.hu/studentlife/current/student-policies, Section Student Rights, Rules, and Academic Regulations. This includes in particular standards and assessment techniques. Moreover, specific assessment methods, including specific homework assignments, computer simulations, presentations on various topics, etc. may be required by each instructor, depending on the nature of the course.

Remark. It should be pointed out that our M.S. program is totally in line with the European standards, taking into account that 1 CEU credit = 2 ECTS credits. Recall that the so-called European Credit Transfer and Accumulation System (ECTS) is a standard for comparing the study attainment and performance of students of higher education across the European Union and other collaborating European countries.

COURSE LIST

Below is a list of courses we are offering. We assume flexibility in choosing different courses from the list of elective courses (see below), depending on the interests of the students. However, students will be assisted in identifying sets of courses which correspond to specific areas of specialization, such as Applied Differential Equations, Applied Statistics, Financial Mathematics, Mathematical Biology, or various combinations of them.
We should also take into account the needs of the job market.
Our course offer exceeds the demand and not all the courses are taken by a given class. On the other hand, further elective courses may be added to the list, if necessary.
Occasionally, we offer non-credit introductory bridge courses to help first year M.S. students meet the level required by the program.

Mandatory Courses

M1. Basic Algebra 1 (Pal Hegedus)
M2. Basic Algebra 2 (Pal Hegedus)
M3. Real Analysis (Laszlo Csirmaz)
M4. Complex Function Theory (Robert Szoke)
M5. Functional Analysis and Differential Equations (Gabor Elek or Gheorghe Morosanu)
M6. Introduction to Computer Science (Istvan Miklos)
M7. Probability and Statistics (Marianna Bolla)

Elective Courses

•    Matrix Computations with Applications (Robert Horvath)
•    Computations in Algebra (Pal Hegedus)
•    Cryptology (Laszlo Csirmaz)
•    Differential Geometry (Balazs Csikos)
•    Introduction to Discrete Mathematics (Ervin Gyori)
•    Graph Theory and Applications (Balazs Patkos)
•    Non-standard Analysis (Laszlo Csirmaz)
•    Difference Equations and Applications (Gheorghe Morosanu)
•    Evolution Equations and Applications (Gheorghe Morosanu)
•    Applied Partial Differential Equations (Gheorghe Morosanu)
•    Difference Methods for Partial Differential Equations (Gheorghe Morosanu)
•    Control of Dynamic Systems (Gheorghe Morosanu)
•    Combinatorial Optimization (Ervin Gyori)
•    Optimization in Economics (Alexandru Kristaly)
•    Quantitative Financial Risk Analysis (Balazs Janecsko and Imre Kondor)
•    Nonlinear Optimization (Sandor Bozoki)
•    Topics in Financial Mathematics (Alexandru Kristaly)
•    Approximation Theory (Jozsef Szabados)
•    Applied Numerical Analysis (Robert Horvath)
•    Mathematical Models in Biology and Ecology (Paul Georgescu)
• The Mathematical Theory of Infectious Disease Propagation (Paul Georgescu)
•    Evolutionary Games and Population Dynamics (Paul Georgescu)
•    Bioinformatics (Istvan Miklos)
•    Computational Neuroscience (Tamas Kiss, Krisztina Szalisznyo and Laszlo Zalanyi)
•    Probabilistic Models of the Brain and the Mind (Máté Lengyel, József Fiser)
•    Computational Number Theory (Laszlo Csirmaz)
•    Protocols (Laszlo Csirmaz)
•    Mathematical Methods in Natural Language Processing (Andras Sereny)
•    Stochastic Processes and Applications (Balazs Szekely)
•    Statistics of Stochastic Processes (Balazs Szekely)
•    Multivariate Statistical Inference (Marianna Bolla or Tamas Rudas)
•    Survey Methodology (Tamas Rudas)
•    Topics in Actuarial Statistics (TBA)

For short descriptions, see Section Syllabi below.

SCHEDULE OF COURSEWORK

Students are required to earn a total of 45 course credits, out of which (at least) 27 credits should be taken in the first year, including all the mandatory courses M1-M7 (see the list above). Note that each M.S. regular course has 3 credits (1 credit = 12 x 50 minutes = 600 teaching minutes). The remaining necessary course credits are taken in the second year.
To acquire enough general information, among the 45 course credits, students are encouraged to earn some credits in courses from other CEU departments. They are also allowed to take some Ph.D. courses, subject to the course instructor's agreement. Every Ph.D. course is counted as an M.S. course, with the same number of credits.

THESIS

Students are required to write a thesis on a topic related to the specific area of specialization, under close supervision. The thesis should not necessarily include original results. However, since our M.S. students are supposed to become solvers of real world problems, their dissertations should address significant applications related to biological, ecological, economic, industrial, political, social phenomena. They should be able to handle mathematical and computational methods in solving specific problems. Students may also address more theoretical topics in their theses. In such a case, a good survey of a certain modern area in mathematics is expected as well as possible original contributions.

Normally, an M.S. student completes her/his coursework in two years, prepares the thesis during the second year, and defends it by the end of the second year. However, according to the general CEU rules, the thesis may be submitted within a maximum of two years of finishing the coursework of the program, with the director of the program’s prior agreement if this has not been done in due course. Additional courses may be offered to some of the students, beyond the first two years of studies, in order to help them write a good thesis and get better prepared in their area of interest.

Basically, our permanent faculty members serve as advisers. In the second year, when the interests of the students get more specific, they are encouraged to choose advisers who are closer to their specific area of interest among the available faculty, including adjunct professors, if they accept and have enough time for consultation.

Besides close and specialized supervision, students can benefit from workshops, seminars, summer schools.

Normally, the thesis is defended by the end of the student’s second year. The thesis must be submitted 3 weeks in advance. The defense comprises the presentation of the thesis as well as a comprehensive exam on the area of the thesis, covering the material of 3 courses, including at least 2 electives (i.e., either 3 electives or 2 electives + 1 mandatory).

SYLLABI

Basic concepts and theorems are presented. Emphasis is put on familiarizing with the aims and methods of abstract algebra. Interconnectedness is underlined throughout. Applications are presented.

One of the main goals of the course is to introduce students to the most important concepts and fundamental results in abstract algebra. A second goal is to let them move confidently between abstract and concrete phenomena.

The students will learn some basic notions and results of abstract algebra. More importantly, they will gain expertise in using it in various areas of mathematics.

Resultants, polynomials in non-commuting variables, twisted polynomials, subdirect products, subdirectly irreducible algebras, subdirect representation. Categorical approach: products, coproducts, pullback, pushout, functor categories, natural transformations, Yoneda lemma, adjoint functors.

1. N Jacobson, Basic Algebra I-II, WH Freeman and Co., San Francisco, 1974/1980.

2. I M Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994

Further concepts and theorems are presented. Emphasis is put on difference of questions at different areas of abstract algebra and interconnectedness is underlined throughout. Applications are presented.

One of the main goals of the course is to introduce the main distinct areas of abstract algebra and the fundamental results therein. A second goal is to let them move confidently between abstract and concrete phenomena.

The students will learn in some depth the theories in the three main areas of abstract algebra: groups, rings and fields. More importantly, they will gain some expertise in using them in various areas of mathematics.

1. Groups: composition series, Jordan-Hölder Theorem,
2. conjugation, centralizer, normalizer, class equation, p-groups,
3. nilpotent groups, Frattini subgroup, Frattini argument,
4. direct product, Krull-Schmidt Theorem, semidirect product, groups of small order.
5. Commutative rings: unique factorization, principal ideal domains, Euclidean domains,
6. finitely generated modules over principal ideal domains, Fundamental Theorem of finite abelian groups, Jordan normal form of matrices,
7. Noetherian rings, Hilbert Basis Theorem, operations with ideals.
8. Fields: algebraic and transcendental extensions, transcendence degree,
9. splitting field, algebraic closure, the Fundamental Theorem of Algebra, normal extensions, finite fields, separable extensions,
10. Galois group, Fundamental Theorem of Galois Theory, cyclotomic fields,
11. radical expressions, insolvability of the quintic equation, traces and norms: Hilbert’s Theorem,
12. Artin-Schreier theorems, ordered and formally real fields.

Optional topics:

Categorical approach: products, coproducts, pullback, pushout, functor categories, natural transformations, Yoneda lemma, adjoint functors.

Books:
1. N Jacobson, Basic Algebra I-II, WH Freeman and Co., San Francisco, 1974/1980.
2. I M Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994

Introduction to Lebesgue integration theory; measure, σ-algebra, σ-finite measures. Different notion of convergences; product spaces, signed measure, Radon-Nikodym derivative, Fubini and Riesz theorems; Weierstrass approximation theorem. Solid foundation in the Lebesgue integration theory, basic techniques in analysis. It also enhances student’s ability to make their own notes.

At the end of the course students are expected to understand the difference between ”naive” and rigorous modern analysis. Should have a glimpse into the topics of functional analysis as well. They must know and recall the main results, proofs, definition. As a conclusion of the course, students take an oral exam where all acquired knowledge is checked and graded.

2. lim inf and lim sup of sets; their measure. The Borel-Cantelli lemma. Complete measure

3. Caratheodory outer measure on a metric space. Borel sets. Lebesgue measure. Connection between Lebesgue measurable sets and Borel sets

4. Measurable functions. Measurable functions are closed under addition and multiplication. Continuous functions are measurable. Example where the composition of measurable functions is not measurable

6. Egoroff's theorem: if f_iconverges pointwise a.e to f then it converges uniformly with an exceptional set of measure <ε. Convergence in measure; pointwise convergence for a subsequence.

7. Lusin's theorem: a Lebesgue measurable function is continuous with an exceptional set of measure <ε. Converging to a measurable function by simple functions.

8. Definition of the integral; conditions on a measurable function to be integrable

9. Fatou's lemma, Monotone Convergence Theorem; Lebesgue's Dominated Convergence Theorem. Counterexample: a sequence of functions tends to f, but the integrals do not converge to the integral of f.

11. Riesz-Fischer theorem: L^p is complete, conjugate spaces, basic properties

14. Product measure, Fubini's theorem. Counterexample where the order of integration cannot be exchanged

Fundamental concepts and themes of classic function theory in one complex variable are presented: complex derivative of complex valued functions, contour integration, Cauchy's integral theorem, Taylor and Laurent series, residues, applications, conformal maps.

The goal of the course is to acquaint the students with the fundamental concepts and results of classic complex function theory.

The students will learn some basic notion and theorems (with some applications) of classic complex function theory.

The basic definitions and results of functional analysis will be presented closely following the classic book of Reed and Simon.

The main goal of the course is to provide a foundation for further studies in operator theory, mathematical physics as well as differential equations.

Book: Reed-Simon: Functional Analysis (Methods of Modern Mathematical Physics Vol. I), Academic Press, 1980.

Greedy and dynamic programming algorithms. Famous tricks in computer science. the most important data structures in computer science. The Chomsky hierarchy of grammars, parsing of grammars, relationship to automaton theory. Computers, Turing machines, complexity classes P and NP, NP-complete. Stochastic Turing machines, important stochastic complexity classes. Counting classes, stochastic approximation with Markov chains.

To learn dynamic programming algorithms
To get an overview of standard tricks in algorithm design
To learn the most important data structures like chained lists, hashing, etc.
To learn the theoretical background of computer science (Turing machines, complexity classes)
To get an introduction in stochastic computing

The students will be able to read and understand moderately involved scientific papers related to the topic.

Lecture 1.
Theory: The O, W and Q notations. Greedy and dynamic programming algorithms. Kruskal’s algorithm for minimum spanning trees, the folklore algorithm for the longest common subsequence of two strings.
Practice: The money change problem and other famous dynamic programming algorithms

Lecture 2.
Theory: Dijstra’s algorithm and other algorithms for the shortest path problem.
Preactice: Further dynamic programming algorithms.

Lecture 3.
Theory: Divide-and-conqueror and checkpoint algorithms. The Hirshberg’s algorithm for aligning sequences in linear space
Practice: Checkpoint algorithms. Reduced memory algorithms.

Lecture 4.
Theory: Quick sorting. Sorting algorithms.
Practice: Recursive functions. Counting with inclusion-exclusion.

Lecture 5.
Theory: The Knuth-Morrison-Pratt algorithm. Suffix trees.
Practice: String processing algorithms. Exact matching and matching with errors.

Lecture 6.
Theory: Famous data structures. Chained lists, reference lists, hashing.
Practice: Searching in data structures.

Lecture 7.
Theory: The Chomsky-hierarchy of grammars. Parsing algorithms. Connections to the automaton theory.
Practice: Regular expressions, regular grammars. Parsing of some special grammars between regular and context-free and between context-free and context-dependent classes.

Lecture 8.
Theory: Introduction to algebraic dynamic programming and the object-oriented programming.
Practice: Algebraic dynamic programming algorithms.

Lecture 9.
Theory: Computers, Turing-machines, complexity and intractability, complexity of algorithms, the complexity classes P and NP. 3-satisfiability, and NP-complete problems.
Practice: Algorithm complexities. Famous NP-complete problems.

Lecture 10.
Theory: Stochastic Turing machines. The complexity class BPP. Counting problems, #P, #P-complete, FPRAS.
Practice: Stochastic algorithms.

Lecture 11.
Theory: Discrete time Markov chains. Reversible Markov chains, Frobenious theorem. Relationship between the second largest eigenvalue modulus and convergence of Markov chains. Upper and lower bounds on the second largest eigenvalue.
Practice: Upper and lower bounds on the second largest eigenvalue.

Lecture 12.
Theory: The Sinclair-Jerrum theorem: relationship between approximate counting and sampling.

Practice: Some classical almost uniform sampling (unrooted binary trees, spanning trees).

Students will learn basic probability models with applications. Laws of large numbers, central limit and large deviation theorems will be introduced together with the notion of conditional expectation that plays a crucial role in statistics.

While probability theory describes random phenomena, mathematical statistics teaches us how to behave in the face of uncertainties, according to the famous mathematician Abraham Wald. Roughly speaking, we will learn strategies of treating randomness in everyday life.

The first part of the course gives an introduction to probability models and basic notion of conditional distributions, while the second part to the theory of estimation and hypothesis testing.

The main concept is that our inference is based on a randomly selected sample from a large population, and hence, our observations are treated as random variables. Through the course we intensively use laws of large numbers and the Central Limit Theorem. On this basis, applications are also discussed, mainly on a theoretical basis, but we make the students capable of solving numerical exercises.

Students will be able to identify probability models, further to find the best possible estimator for a given parameter by investigating the bias, efficiency, sufficiency, and consistency of an estimator on the basis of theorems and theoretical facts. Students will gain familiarity with basic methods of estimation and will be able to construct statistical tests for simple and composite hypotheses. They will become familiar with applications to real-world data and will be able to choose the most convenient method for given real-life problems.

W. Feller, An Introduction to Probability. Theory and Its Applications, Wiley, 1966.

C.R. Rao, Linear statistical inference and its applications. Wiley, New York, 1973.

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

M.G. Kendall, A. Stuart, The Theory of Advanced Statistics I-III. Griffin, London, 1966.

Handouts: tables of notable distributions and percentile values of basic test distributions.

· Book(s): G. Golub, C. F. van Loan, Matrix computations, Johns Hopkins University Press, 1996.

1. Introduction to Matlab. Matrix manipulations and matlab notations.
2. Matrix multiplication problems
3. Matrix analysis
4. Linear systems
5. Orthogonalization and least squares
6. Eigenvalue problems, Lanczos methods
7. Iterative methods for linear systems
8. Krylov subspaces, matrix functions
9. Applications (using matlab): numerical solution of partial differential equations, nonnegativity preservation, matrix exponential, numerical solution of Maxwell equations, model reduction.

· Brief introduction to the course:

The course will cover many basic algorithms that are still used by computer algebra systems today. We shall cover mostly problems connected to abstract and linear algebra, but some outside areas will be touched especially if it mathematically related. The computer algebra system GAP will be used many times.

· The goals of the course:

One of the main goals of the course is to introduce students to the most important concepts and fundamental results in computational algebra. A second goal is to make the student more comfortable with abstract algebra itself.

The students will learn some basic notions and results of computational algebra. More importantly, they will gain expertise in using it in various contexts in mathematics. Some expertise in using GAP will be earned.

10-12. Practical sessions in the PhD study room will be evenly distributed among the weeks.

1. J von zur Gathen and J Gerhard, Modern Computer Algebra, Cambridge University Press, 1999.
2. Th Becker and V Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra, Graduate Texts in Mathematics, Springer, 1998.

· Prerequisities: Basic Algebra 1, Introduction to Computer Science, Probability and Statistics

· Books:
1. Ivan Damgard (Ed), Lectures on Data Security, Springer 1999
2. Oded Goldreich, Modern Cryptography, Probabilistic Proofs and Pseudorandomness, Springer 1999

1. Computational difficulty, computational indistinguishability
2. Pseudorandom function, pseudorandom permutation
3. Probabilistic machines, BPP
4. Hard problems
5. Public key cryptography
6. Protocols, ZK protocols, simulation
7. Unconditional Security, multiparty protocols
8. Broadcast and pairwise channels
9. Secret Sharing Schemes, Verifiable SSS
10. Multiparty Computation

This course is split into three parts. In the first two parts we give an introduction to the classical roots of modern differential geometry, the theory of curves and hypersurfaces in n-dimensional Euclidean spaces. In the thrird part foundations of manifold theory are laid.

Differential geometry is a powerful combination of geometry and analysis. It has various applications within many branches of mathematics (theory of ordinary and partial differential equations, calculus of variations, algebraic geometry, ...), as well as in mathematical physics, optics, mechanics, engineering, etc.

Modern differential geometry has developed from the analytical theory of curves and surfaces in the Euclidean space. Pioneers of classical differential geomety were Huygens, Euler, Claireaut, Monge, Dupin, and Gauss. Manifods were introduced first by Poincaré as k-dimensional “surfaces” in an n-dimensional linear space. The notion of abstract manifolds appeared later in connection with the development of the notion of a topological space. This course gives an introduction to differential geometry, following the historical development of the subject.

The course will provide the most important basics of differential geometry, which enables the students to find applications of differential geometry in other areas of mathematics. They will also get a good basis to continue their study with more advanced branches of differential geometry, like Riemannian or Lorentzian geometry, symplectic or contact geometry, complex manifolds, Lie groups and symmetric spaces etc.

Week 1: Parameterized curves. (Length, reparameterizations, natural reparameterization. Tangent line and osculating affine subspaces.)

Week 2: Frenet theory of curves. (Frenet frame, curvatures, Frenet equations. Fundamental Theorem of curve theory.)

Week 3: Some applications. (Osculating circle, evolute, involute. Envelope of a family of planar curves, and other optional applications.)

Week 4: Hypersurfaces. (Tangent hyperplane, Gauss map. Normal curvature, Meusnier’s theorem. Fundamental forms, principal curvatures and principal directions, Euler’s formula, Weingarten map, Gaussian and Minkowski curvature.)

Week 5: Applications. (Surfaces of revolution, ruled and developable surfaces, and other optional applications.)

Week 6: Fundamental equations of hypersurface theory. (Gauss and Codazzi-Mainardi equations. Intrinsic geometry of a hypersurface, Theorema Egregium.)

Week 7: The Gauss-Bonnet formula. (Integration on hypersurfaces, geodesic curvature of curves on a hypersurface, local and global versions of the Gauss-Bonnet formula.)

Week 8: Differentiable manifolds. (Definitions. Examples, submanifolds of a manifold. Smooth maps. Tangent vectors of a manifold. The derivative of a smooth map.)

Week 9: Lie algebra of vector fields. (Definition and properties of the Lie bracket, the flow generated by a vector field. Geometrical meaning of the Lie bracket.)

Week 10: Connections. (Definition. Christoffel symbols with respect to a chart. Torsion. Parallel transport. Compatibility with a Riemannian metric. Levi-Civita connection.)

Week 11: Curvature tensor. (Definiton. Linearity over smooth functions. Symmetry properties. Derived curvature quantities: sectional curvature, Ricci curvature, scalar curvature.

Week 12: Geodesics. (Definition. Exponential map. Normal coordinates. Gauss lemma. Formula for the first variation of the length. Short geodesic segments minimize the length.)

Fundamental concepts and results of combinatorics and graph theory. Main topics: counting, recurrences, generating functions, sieve formula, pigeonhole principle, Ramsey theory, graphs, flows, trees, colorings.

· The goals of the course:

The main goal is to study the basic methods of discrete mathematics via a lot of problems, to learn combinatorial approach of problems. Problem solving is more important than in other courses!

· The learning outcomes of the course:

Knowledge of combinatorial techniques that can be applied not just in discrete mathematics but in many other areas of mathematics. Skills in solving combinatorial type problems.

· More detailed display of contents:

Week 1. Basic counting problems, permutations, combinations, sum rule, product rule
Week 2. Occupancy problems, partitions of integers
Week 3. Solving recurrences, Fibonacci numbers
Week 4. Generating functions, applications to recurrences
Week 5. Exponential generating functions, Stirling numbers, derangements
Week 6. Advanced applications of generating functions (Catalan numbers, odd partitions)
Week 7. Principle of inclusion and exclusion (sieve formula), Euler function
Application of sieve formula to Stirling numbers, derangements, and other involved problems
Week 8. Pigeonhole principle, Ramsey theory, Erdos Szekeres theorem
Week 9. Basic definitions of graph theory, trees
Week 10. Special properties of trees, Cayley’s theorem on the number of labeled trees
Week 11. Flows in networks, connectivity
Week 12. Graph colorings, Brooks theorem, colorings of planar graphs

References:
1. Fred. S. Roberts, Applied Combinatorics, Prentice Hall, 1984
2. Fred. S. Roberts, Barry Tesman, Applied Combinatorics, Prentice Hall, 2004
3. Bela Bollobas, Modern Graph Theory, Springer, 1998

GRAPH THEORY AND APPLICATIONS

In recent years in the study of networks (web, VLSI, etc.) graph theory became central in applications of mathematical methods to everyday problems. The course is to review the most important questions related to graphs emphasizing on subjects with practical applications as well as applications in other areas of mathematics. Furthermore, we are going to deal with the algorithmic aspects, though we are not to cover all details of implementation, etc.

The course is designed for students oriented to applied mathematics as well as to pure mathematics.

The main goal of the course is to introduce students to some important graph theoretical methods and to show their applicability to various problems. We intend to discuss graph algorithms as well as theoretical results.

The students will learn the basic concepts and methods, which are very useful for applied mathematicians. Even more, they will learn how to use these tools in solving specific problems.

Week 6: Matchings in bipartite graphs, matchings in arbitrary graphs, Tutte’s theorem, matching algorithms

Week 12: Graph algorithms, minimum cost spanning trees, DFS and BFS spanning trees and their applications

1. Tools from mathematical logic: compactness theorem, higher-order logic
2. Enlargement
3. Elementary Analysis: differentiation, integration, convergence
4. Topological Spaces: compactness, Tichonov's theorem, Uhrysson's theorem on metrizable spaces
5. Theorems of Montel and Kakeya on lacunary polynomials
6. Complex Functions: the Picard's theorem, Julia direction

· Book: Ronald E. Mickens, Difference Equations. Theory and Applications, Van Nostrand Reinhold, New York, 1990.

1. The difference calculus
2. First-order difference equations
3. Linear difference equations
4. Linear partial difference equations
5. Nonlinear difference equations
6. Various applications

EVOLUTION EQUATIONS AND APPLICATIONS

· Books:
1. H. Brezis, Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North Holland, Amsterdam, 1973.
2. V.-M. Hokkanen and G. Morosanu, Functional Methods in Differential Equations, Chapman & Hall/CRC, 2002.
3. G. Morosanu, Nonlinear Evolution Equations and Applications, Reidel, 1988.

· Contents:

1. Preliminaries of linear and nonlinear functional analysis
2. Existence and regularity of solutions to evolution equations in Hilbert spaces
3. Boundedness of solutions on the positive half axis
4. Stability of solutions. Strong and weak convergence results.
5. Periodic forcing. The asymptotic dosing problem
6. Applications to delay equations, parabolic and hyperbolic boundary value problems. Specific examples.

APPLIED PARTIAL DIFFERENTIAL EQUATIONS

· Prerequisites: Undergraduate Calculus, Linear Algebra, Real and Complex Analysis

· Books:
1. L.C. Evans, Partial Differential Equations, Graduate Studies in Math. 19, AMS, Providence, Rhode Island, 1998.
2. R. Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, Fourth Edition, Pearson Education, Inc. Pearson Prentice Hall, 2004.
3. R.M.M. Mattheij, S.W. Rienstra and J.H.M. ten Thije Boonkkamp, Partial Differential Equations. Modeling, Analysis, Computation, SIAM, Philadelphia, 2005.

· Contents:

1. Various models involving linear and nonlinear partial differential equations
2. Elliptic equations. Maximum principles
3. Variational solutions for elliptic boundary value problems
4. Parabolic equations
5. Hyperbolic equations and systems. Vibrating strings and membranes
6. Theory for nonlinear partial differential equations. Variational and nonvariational techniques
7. Conservation laws
8. Laplace transform solution of partial differential equations

DIFFERENCE METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

· Books:
1. J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Second Edition,

SIAM

, 2004.
2. J.W. Thomas, Numerical Partial Differential Equations. Finite Difference methods, Texts in Appl. Math. 22, Springer, 1995.
3. A. Tveito and R. Winther, Introduction to Partial Differential Equations. A Computational Approach, Texts in Appl. Math. 29, Springer, 1998.

· Contents:

1. Introduction to finite differences
2. Convergence, consistency, stability
3. Difference schemes for parabolic equations
4. Difference schemes for hyperbolic equations
5. Difference schemes for systems of partial differential equations
6. Dispersion and dissipation
7. Various applications and examples

· Brief introduction to the course:

Basic principles and methods of control theory are discussed. The main concepts (observability, controllability, stabilizability, optimality conditions, etc.) are addressed, with special emphasis on linear differential systems and quadratic functionals. Many applications are discussed in detail.
The course is designed for students oriented to Applied Mathematics.

· The goals of the course:

The main goal of the course is to introduce students to the theory of optimal control for differential systems. We also intend to discuss specific problems which arise from down-to-earth applications in order to illustrate this remarkable theory.

· The learning outcomes of the course:

The students will learn some basic concepts and results in control theory, which are very useful for applied mathematicians, economists, engineers, physicists. Even more, they will learn how to use these tools in solving specific real world problems.

Week 1: Linear Differential Systems (existence of solutions, variation of constants formula, continuous dependence of solutions on data, exercises)

Week 2: Nonlinear Differential Systems (local and global existence of solutions for the Cauchy problem, continuous dependence on data, differential inclusions, exercises)

Week 3: Basic Stability Theory (concepts of stability, stability of the equilibrium, stability by linearization, Lyapunov functions, applications)

Week 4: Observability of linear autonomous systems (definition, observability matrix, necessary an sufficient conditions for observability, examples)

Week 5: Observability of linear time varying systems (definition, observability matrix, numerical algorithms for observability, examples)

Week 6: Input identification for linear systems (definition, the rank condition in the case of autonomous systems, examples)

Week 7: Controllability of linear systems (definition, controllability of autonomous systems, controllability matrix, Kalman’s rank condition, the case of time varying systems, applications)

Week 8: Controllability of perturbed systems (perturbations of the control matrix, nonlinear autonomous systems, time varying systems, examples)

Week 9: Stabilizability (definition, state feedback, output feedback, applications)

Week 10: Introduction to optimal control theory (Meyer’s problem, Pontryagin’s Minimum Principle, examples)

Week 11: Linear quadratic regulator theory (introduction, the Riccati equation, perturbed regulators, applications)

Week 12: Time optimal control (general problem, linear systems, bang-bang control, applications)

Books:
1. N.U. Ahmed, Dynamic Systems and Control with Applications, World Scientific, 2006.
2. E.B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley, 1967.

Basic concepts and theorems are presented. Some significant applications are analyzed to illustrate the power and the use of combinatorial optimization. Special attention is paid to algorithmic questions.

One of the main goals of the course is to introduce students to the most important results of combinatorial optimization. A further goal is to discuss the applications of these results to particular problems, including problems involving applications in other areas of mathematics and practice. Finally, computer science related problems are to be considered too.

The students will learn some basic notions and results of combinatorial optimization. They will learn how to use these tools in solving every day life problems as well as in software developing.

More detailed display of contents.

Week 1: Typical optimization problems, complexity of problems, graphs and digraphs
Week 2: Connectivity in graphs and digraphs, spanning trees, cycles and cuts, Eulerian and Hamiltonian graphs
Week 3: Planarity and duality, linear programming, simplex method and new methods
Week 4: Shortest paths, Dijkstra method, negative cycles
Week 5: Flows in networks
Week 6: Matchings in bipartite graphs, matching algorithms
Week 7: Matchings in general graphs, Edmonds’ algorithm
Week 8: Matroids, basic notions, system of axioms, special matroids
Week 9: Greedy algorithm, applications, matroid duality, versions of greedy algorithm
Week 10: Rank function, union of matroids, duality of matroids
Week 11: Intersection of matroids, algorithmic questions
Week 12: Graph theoretical applications: dedge disjoint and coverong spanning trees, directed cuts

Book: E.L. Lawler, Combinatorial Optimization: Networks and Matroids, Courier Dover Publications, 2001 or earlier edition: Rinehart and Winston, 1976

In the last decades mathematical methods have become indispensable in the study of many economical problems, in particular, in the optimization of certain real-life phenomena. For instance, J. F. Nash received the Nobel Prize in Economics (1994) for his outstanding contributions in the field of Economics via mathematical tools. Our aim here is to emphasize the importance of Mathematics in the study of a broad range of economical problems. Many applications/examples will be discussed in detail.

The main goal of the present course is to introduce Students into the most important concepts and fundamental results of Economics by using various tools from Mathematics as calculus of variations, critical points, matrix-algebra, or even Riemannian-Finsler geometry. Starting with basic economical problems, our final purpose is to describe some recent research directions concerning certain optimization problems in Economics.

The Students will learn how to use well-known mathematical tools to treat both theoretical and practical economical problems.

Lecture 1. Introduction and motivation: some basic problems from Economics via optimization.

Lecture 2. Economic applications of one-variable calculus (demand and marginal revenue, elasticity of price, cost functions, profit-maximizing output).

Lecture 3. Economic applications of multivariate calculus (consumer choice theory, production theory, the equation of exchange in Macroeconomics, Pareto-efficiency, application of the least square method).

Lectures 4. Linear programming (application of the geometric, simplex and dual simplex method).

Lecture 5. Linear economical problems (diet problem, Ricardian model of international trade).

Lecture 6. Comparative statics I (equilibrium comparative statics in one and two dimensions; comparative statics with optimization, perfectly competitive firms, Cournot duopoly model).

Lecture 7. Comparative statics II: n variables with and without optimization (equilibrium comparative statics in n dimensions, Gross-substitute system, perfectly competitive firms).

Lecture 8. Comparative statics III: Optimization under constraints (Lagrange-multipliers, specific utility functions, expenditure minimization problems).

Lecture 10. Nash equilibrium points (existence, location, dynamics, and stability).

Lecture 11. Optimal placement of a deposit between markets: a Riemann-Finsler geometrical approach.

Lecture 12. Economical problems via best approximations.

· Prerequisites: Probability and Statistics, Real Analysis, Complex Function Theory, Functional analysis and Differential Equations

· Books:
1. R.N. Mantegna and H.E. Stanley: An Introduction to Econophysics, Correlations and Complexity in Finance, Cambridge University Press, 2000
2. M. K. Ong: Internal Credit Risk Models, Capital Allocation and Performance Measurement, Risk Books, 2000
3. Credit Suisse First Boston: CreditRisk+, A Credit Risk Management Framework, 1997

· Contents:

1. Market risk measurement
2. Time independent fat tailed distributions of market price (FX rates, interest rates, stock and commodity prices) fluctuations
3. Volatility clusters in stock exchanges, GARCH models
4. Filtered historical simulation
5. Best practice for calculating Value at Risk for market risk related problems
6. Credit portfolio risk models
7. Mathematical background of the Basel II regulatory model
8. Granularity adjustment for undiversified idiosyncratic risk
9. CreditRiskPlus as a realistic and implementable portfolio model
10. Comparison of CreditRiskPlus and CreditMetrics models
11. Probability of Default (PD) Estimation
12. Low default problem

· Brief introduction to the course:

The course provides an introduction to the nonlinear optimization problems. Main topics are the first- and second-order, necessary and sufficient optimality conditions; convex optimization; quasiconvex and pseudoconvex functions; Lagrange duality, weak and strong duality theorems, saddle point theorem; Newton’s method in optimization, theorems of convergence.

The aim of the course is to encourage students to the use of nonlinear optimization techniques in many areas of their interest and to gain theoretical and practical knowledge. Students are proposed to know the elementary theorems and proofs of nonlinear optimization and also to use the corresponding tools and commands in Matlab and/or Maple.

· The learning outcomes of the course:

At the end of the course students can identify, model and classify nonlinear optimization problems and can solve some of them by using Lagrange multipliers or Newton’s method. Students will have a toolbox of basic nonlinear optimization routines as well as the ability of implementing elementary algorithms.

·        More detailed display of contents:

1. Modeling of nonlinear optimization problems – examples, well known mathematical problems written as nonlinear optimization problems, alternative ways for modeling the same problem

2. First- and second-order, necessary and sufficient optimality conditions – and solution of numerical exercises

3. Convex optimization – theorems of convex optimization, applications in inequalities

4. An introduction to the generalized convexity: quasiconvex and pseudoconvex functions – with examples and counterexamples

5. Lagrange duality – relation to the primal problem, solution of numerical exercises

6. Duality theorems

7. Saddle point theorem

8. Newton’s method in optimization, theorems of convergence

9. The implementation of Newton’s method in one and two dimensions – in Matlab and/or Maple

10. Newton’s method and fractals

Lecture notes:
•    Tamás Rapcsák, Smooth Nonlinear Optimization in Rn, Kluwer Academic Publishers, 1997.

•    Pascal Sebah, Xavier Gourdon: Newton’s method and high order iterations
o    http://numbers.computation.free.fr/Constants/Algorithms/newton.html
o    http://numbers.computation.free.fr/Constants/Algorithms/newton.ps

TOPICS IN FINANCIAL MATHEMATICS

· Books:
1. J. V. Allen, Lectures Notes on Actuarial Mathematics, manuscript, 2005.
2. J.-P. Aubin, Analyse Non Lineaire et ses Motivations Economiques, Masson, 1984.

· Contents:

1. Financial markets, financial derivatives, payoff functions
2. Asset price model
3. Black-Scholes analysis; American and European options; Black-Scholes formula
4. Variations on Black-Scholes models; Future options
5. Numerical methods: Monte Carlo method, binomial method, finite difference method, fast algorithms for solving linear systems
6. Exotic options
7. Path-dependent options: General method, average strike options, look-back options
8. Bonds and interest rate derivatives: Bond models, interest models

APPROXIMATION THEORY

· Textbooks:
1. G.G. Lorentz: Approximation of Functions, Holt, Rinehart and Winston, 1966
2. R. DeVore and G. G. Lorentz, Constructive Approximation, Springer, 1993

· Contents:

Best polynomial and rational approximation, moduli of smoothness, Jackson-Timan type quantitative direct estimates for polynomial approximation, converse theorems, positive linear operators, Korovkin theorems, interpolation (Lagrange, Newton, Hermite), Fourier series.

· Books:
1. A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Series: Springer Series in Computational Mathematics, Vol. 23, 1997.
2. L. J. Segerlind, Applied Finite Element Analysis, J. Wiley & Sons, New York 1987.
3. M.N.O. Sadiku, Numerical Techniques in Electromagnetics, 2nd edition, CRC Press 2001.
4. K. Kunz, R. Lubbers, The Finite Difference Time Domain Method for Electromagnetics, CRC Press 1993.

· Contents:

1. Approximations of differential equations (finite difference, finite element, Galerkin and collocation methods)
2. Applications (Heat transfer, Maxwell equations, air-pollution transport model, torsion of noncircular sections, irrotational flows, Black-Scholes' equation)
3. Operator splitting techniques, matrix exponentials and their applications to air-pollution transport models and to Maxwell equations
4. Qualitative properties of mathematical and numerical models
5. Computer examples and implementations

MATHEMATICAL MODELS IN BIOLOGY AND ECOLOGY

· Books:
1. L. Edelstein-Keshet, Mathematical Models in Biology, SIAM Classics in Applied Mathematics 46, 2004
2. F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2001

· Contents:

1. Discrete and continuous single species models. Exponential and logistic growth. The delayed logistic equation
2. Multi-species communities: competition, comensualism, coexistence.
3. Predator-prey models. The Lotka-Volterra model and more complicated models (Gause, Kolmogorov). Prey-dependent and ratio-dependent predation.
4. Chemical reaction kynetics: Michaelis-Menten theory
5. Simple oscillatory reactions. Nerve impulses and Hodgkin-Huxley theory. FitzHugh-Nagumo model.
6. Reaction-diffusion equations. Convection, advection. Chemotaxis.
7. Ecological epidemiology: integrated pest management strategies.
8. Numerical simulations and visualisations by means of XPP, Phaser, Maple (or equivalent)

· Books:
1. F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2001
2. V. Capasso, Mathematical structures of Epidemic Systems, Lecture Notes in Biomathematics, Springer Verlag, Berlin 1993.

1. Basic concepts of mathematical epidemiology. Deterministic models. Compartmental models.
2. Single population models with constant population size. Models with no immunity.
3. Models with nonconstant population size and immunity effects. Basic reproduction number of a disease. Stability and persistence.
4. Infective periods of fixed length. Models with delay. Arbitrarily distributed infective periods.
5. Seasonality and periodicity. Orbital stability of periodic solutions.
6. Models with pulse vaccination.
7. Multigroup models (models with patchy structure).
8. Numerical simulations and visualisations by means of XPP, Phaser, Maple (or equivalent).

EVOLUTIONARY GAME THEORY AND POPULATION DYNAMICS

· Prerequisites: Basic Calculus, Ordinary Differential Equations, Mathematical Models in Population Dynamics

· Book: J. Hofbauer and K. Zigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.

· Contents:

1. Evolutionary stability. Normal form games. Evolutionarily stable strategies. Population games
2. Replicator dynamics. The equivalence of the replicator equation to the Lotka-Volterra equation. The rock-scissors-paper game. Partnetship games and gradients
3. Other game dynamics. Imitation dynamics. Monotone selection dynamics. Best-response dynamics. Adjustment dynamics. A universally cyclic game.
4. Adaptive dynamics. The repeated Prisoner's Dilemma. Adaptive dynamics and gradients.
5. Asymmetric games and replicator dynamics for them.
6. Population dynamics and game dynamics
7. Game dynamics for Mendelian populations
8. Numerical simulations and visualisations

Stochastic models: HMMs, SCFGs and time-continuous Markov models and their algorithmic aspects.

To learn the stochastic transformational grammars, especially HMMs and SCFGs
To learn time-continuous Markov models describing sequence evolution
To learn the algorithmic background of these models
To learn the statistical background and tools, like Maximum Likelihood and Expectation Maximization

The students will be able to read and understand scientific papers related to the topic.

Lecture 1.
Theory: Score based dynamic programming algorithms. Linear, concave and affine gap penalties.

Lecture 2.
Theory: Conditional probability, Bayes theorem. Unbiased, consistent estimations. Statistical testing. Local alignment, extreme value distributions for local alignments, p and E value estimations.

Lecture 3.
Theory: Hidden Markov Models. Parsing algorithms: Forward, Backward and Viterbi. Posterior probabilities. Expectation Maximization. The Baum-Welch algorithm.

Lecture 4.
Theory: Profile HMMs. Aligning sequences via profile-HMMs. Pair-HMMs.
Practice: HMM topology design.

Lecture 5.
Theory: Substitution models. Felsenstein’s algorithm for fast likelihood calculation of a tree.

Lecture 6.
Theory: Predicting protein secondary structures with profile HMMs and evolutionary models. Gene prediction with HMMs.

Lecture 7.
Theory: Modeling insertions and deletions with time-continuous Markov models: The Thorne-Kishino-Felsenstein models.

Lecture 8.
Theory: Describing the TKF models as pair-HMMs. Extension to many sequences: multiple-HMMs. The transducer theory for evolving sequences on an evolutionary tree.

Lecture 9.
Theory: Stochastic transformational grammars. Stochastic regular grammars are HMMs. Stochastic Context-Free Grammars. Parsing algorithms for SCFGs: Inside, Outside and CYK.

Lecture 10.
Theory: Posterior decoding of SCFGs. Expectation Maximization. Combining SCFGs with evolutionary models: the Knudsen-Hein algorithm.

Lecture 11.
Theory: Covarion Models as ‘profile-SCFGs’. The RFam database. Predicting tRNAs in the human genome.

Lecture 12.
Theory: The Zuker-Tinoco model for RNA secondary structures. Calculating the partition function of the Boltzmann distribution and other moments of the Boltzmann distribution.

· Prerequisites: Undergraduate Calculus, Elementary Linear Algebra, Basic knowledge of Differential Equations,
Further useful skills: programming in Matlab/Scilab/Octave, NEURON, XPPaut

· Books:
1.Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, Dayan P, L.F. Abbott, MIT press, 2001.
2.Spikes: Exploring the Neural Code, Rieke F, Warland D, van Steveninck RR, Bialek W, MIT press,1997.
3. Spiking Neuron Models, Wulfram Gerstner and Werner M. Kistler Cambridge University Press 2002.
4. Neural Organization: Structure, Function and Dynamics. Arbib MA, Érdi P, Szentágothai J, MIT Press,1997.

· Contents:

1. General introduction to Computational Neuroscience
2. General introduction to the anatomy, evolution and cellular basis of the nervous system
3. Basics of nerve cell electrochemistry and electrophysiology. Conductance based models of neurons
4. Parallel conductance model. Mechanism of action potential generation. The Hodgkin-Huxley model. Ionic currents, ionchannels, gate kinetics
5. Simplified neuron models. Simplifications of the Hodgkin-Huxley model: the FitzHugh-Nagumo-Rinzel model, phase-space analysis. Explanation of bursting by bifurcation analysis. Abstract models: phase model, rate model, McCulloch-Pitts neuron, integrate and Fire neuron model
6. Beyond the Hodgkin_Huxley model. Diverse voltage- and ligand gated kinetics in single-compartment models. Role of cellular morphology, dendritic effects. The cable-equation and multi compartmental models. What is detailed modeling good for? Taxonomy of neuron models. Synapses and synaptic plasticity. Detailed, simplified and phenomenological models od the synaptic function
7. Cellular bases of learning: synaptic plasticity. The Hebbian rule of learning. variations for the Hebbian rule. Long term synaptic potentiation and depression. Synaptic plasticity on different time scales. Metaplasticity. Basics of modeling neural networks. The two (three) levels of neural dynamics. Learning rules: reinforcement, supervised and unsupervised learning. Basic neural architectures: feedforward and feedback structures, lateral connections, attractor networks
8. Windows to the World: traditional and modern measuring and data processing techniques: EEG, PET, fMRI, electrodes, intra- and extracellular measurements, patch-clamp. Fourier- and wavelet transformations, EEG/MEG imaging, spike-sorting
9. Neural oscillations: generation of oscillations an the cellular and network level. Oscillation based neural computations: timing and dynamic linking. Oscillations in memory models
10. The hyppocampus: modeling memory and spatial navigation. Place cells and place fields. Phase and rate coding. Dynamic modes of the hyppocampus
11. Modeling neurological and psychiatric disorders. Epilepsy, Parkinson's disease, Alzheimer's disease, schisophrenia.

· Prerequisites: Undergraduate Calculus, Elementary Linear Algebra, Probability and Statistics

· Books:
1. Dayan & Abbott. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT press, 2001.
2. MacKay. Information Theory, Inference & Learning Algorithms, Cambridge University Press, 2002.
3. Sutton & Barto. Reinforcement Learning: An Introduction, MIT Press, 1998.
4. Doya et al. Bayesian Brain: Probabilistic Approaches to Neural Coding, MIT Press, 2007.
5. Rao et al. Probabilistic Models of the Brain: Perception and Neural Function, MIT Press, 2002.

Machine learning, unsupervised learning, Bayesian networks, reinforcement learning, sampling algorithms, variational methods, computer vision, Cognitive science,
inductive reasoning, statistical learning, semantic memory, vision as analysis by synthesis,
sensorimotor control, classical and instrumental conditioning, behavioural economics,
Neuroscience, neural representations of uncertainty, probabilistic neural networks,
probabilistic population codes, natural scene statistics and efficient coding,
neuroeconomics, neuromodulation

COMPUTATIONAL NUMBER THEORY

· Book: Richard Crandall, and Carl Pomerance, Prime Numbers - A Computational Perspective, Springer,2000

· Contents:

Primes, primes of special form, the prime number theorem
Sieving
Arithmetic on large numbers
Primality test: Fermat and Frobenius test
Proving primality: the polynomial algorithm
Factoring primes: Pollard rho method, Baby-step Giant-step method
Solving the discrete logarithm problem
Subexponential pactoring algorithm, quadratic sieve
Elliptic curves, using elliptic curves in factoring

PROTOCOLS

· Book: Colin Boyd and Anish Maturia, Protocols for Authentication and Key Establishment, Springer, 2003.

· Contents:

What protocols are; properties, attacks agains protocols
Turing Machines, Oracle, computational and decisional Diffie-Hellman
problems
Protocols involving two or more parties, security notions, simulability
ZK protocols, concurrent ZK, resettable ZK protocols, AM protocols
Public key and symmetric key protocols
Protocols for key establishment
Identity based protocols, signatures
Famous attacks against famous protocols

· Prerequisites: A proper understanding of elementary probability theory is necessary. Familiarity with statistics and formal languages might be helpful, but not required.

· Books:
1. D. Jurafsky and J.H. Martin: Speech and Language Processing, Second Edition, Prentice Hall Inc., to appear in 2008, available online.
2. C.D. Manning and H. Schűtze: Foundations of Statistical Natural Language Processing, MIT Press, 1999.
3. A. Kornai: Mathematical Linguistics, Springer, 2007.

· Contents:

1. Finite-state automata and transducers, rules in phonology and morphology
2. Counting words in corpora, Zipf's law
3. Hidden Markov Models, training and decoding algorithms
4. Speech recognition architecture, low-level processing, feature extraction
5. Discriminative training of Hidden Markov Models
6. Language modeling: n-gram and factored language models
7. Maximum entropy modeling
8. Document classification

The most common classes of stochastic processes are presented that are important in applications an stochastic modeling. Several real word applications are shown. Emphasis is put on learning the methods and the tricks of stochastic modeling.

The main goal of the course is to learn the basic tricks of stochastic modeling via studying many applications. It is also important to understand the theoretical background of the methods.

The students will learn the most common methods in stochastic processes and their applications.

1. S. M. Ross, Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 1970.

· Books:
1. T. W. Anderson, The Statistical Analysis of Time Series, Wiley, 1971.
2. S. M. Ross, Applied Probability Models with Optimization Applications, Holden-Day, 1970.
3. O. Cappé, E. Moulines, T. Rydén, Inference in Hidden Markov Models, Springer, 2005.
4. H. Jaeger, Discrete-time, discrete-valued observable operator models: a tutorial, on-line notes, 2000.

· Contents:

1. Stationary processes, ARMA processes
2. Time series, trend and seasonality analysis
3. Spectrum analysis, parameter estimation of stationary processes
4. Markov decision processes, semi-Markov decision processes
5.Inventory theory, continuous time optimization models
6. Hidden Markov Models and their applications
7. Observable Operator Models

· Brief introduction to the course:

The course is a continuation of the Probability and Statistics course, and generalizes the concepts studied there to multivariate observations and multidimensional parameter spaces. Students will be introduced to basic models of multivariate analysis with applications. We also aim at developing skills to work with real-world data.

· The goals of the course:

The first part of the course gives an introduction to the multivariate normal distribution
and deals with spectral techniques to reveal the covariance structure of the data. In the second part dimension reduction methods will be introduced (factor analysis and canonical correlation analysis) together with linear models, regression analysis and analysis of variance. In the third part students will learn classification and clustering methods to establish connections between the observations. Finally, algorithmic models are introduced for large data sets. Applications are also discussed, mainly on a theoretical basis, but we make the students capable of using statistical program packages.

· The learning outcomes of the course:

Students will be able to identify multivariate statistical models, analyze the results and make further inferences on them. Students will gain familiarity with basic methods of dimension reduction and classification (applied to scale, ordinal or nominal data). They will become familiar with applications to real-world data sets, and will be able to choose the most convenient method for given real-life problems.

1. K.V. Mardia, J.T. Kent, and M. Bibby, Multivariate analysis. Academic Press, New York, 1979.

2. C.R. Rao, Linear statistical inference and its applications. Wiley, New York, 1973.

3. R.A. Johnson, G.K. Bhattacharyya, Statistics. Principles and Methods. Wiley, New York, 1992.

4. M. G. Kendall, A. Stuart, The theory of advanced statistics I-III. Griffin, London, 1966.

SURVEY METHODOLOGY

· Books:
1. E.K. Foreman, Survey Sampling Principles, Marcel Dekker, 1991.
2. D. Freedman, R. Pisani, R. Purves, A. Adhikari, Statistics 2nd ed, Norton 1991.
3. M.H. Hansen, W.G. Hurwitz, W.G. Madow, Sample Survey Methods and Theory, Vol 1, Wiley, 1993.

·        Course Description:

Every empirical investigation in the social sciences requires valid and reliable data, and the application of carefully selected statistical methods. The typical form of data collection is conducting a survey, and well designed surveys can provide the researcher with good data, even based on surprisingly small sample sizes. The course will discuss the most important concepts and techniques in survey design. A clear understanding of these methods is necessary for any scientist who is engaged in data collection, but it is also useful for the researcher who analyses or interprets data.
Topics:
•    Surveys and censuses
•    Probability versus non-probability samples
•    Role of the sample size, accuracy of estimates
•    Sampling and nonsampling errors
•    Sample-based and model-based approaches to surveys
•    Questionnaire design
•    Sample survey design
•    Main sampling techniques:
-simple random sampling
-stratified sampling
-cluster sampling
•    Handling of missing data

MS Program