# Program in Applied Mathematics

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 M7. PROBABILITY AND STATISTICS No. of Credits: 3,  no. of ECTS credits: 6 Semester or Time Period of the: Fall Semester Prerequisites: Undergraduate Calculus and Real Analysis Course Level: Introductory course Brief introduction to the course: 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 goals of the course: 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. 9. The learning outcomes of the course: 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. More detailed display of contents (Week by week breakdown): Axioms of probability, conditional probability and independence, Lovász Local Lemma, Borel-Cantelli Lemma, famous problems. Random variables, distribution function, basic discrete probability distributions (Bernoulli, Binomial, Pascal, Poisson) and applications. Absolutely continuous distributions (Uniform, Exponential, Normal) with applications, density functions and moments. Conditional expectation. Laws of large numbers, Central Limit Theorem, large deviations. Statistical space, statistical sample. Basic statistics, empirical distribution function, Glivenko-Cantelli Theorem, histograms. Ordered sample, Kolmogorov-Smirnov Theorems. Sufficiency, Neyman-Fisher factorization. Completeness, exponential family. Theory of estimation: unbiased estimators, efficiency, consistency. Fisher information. Cramer-Rao inequality, Rao-Blackwell-Kolmogorov Theorem. Methods of point estimation: maximum likelihood estimation (asymptotic normality), method of moments, Bayes estimation. Interval estimation, confidence intervals. Theory of hypothesis testing, Neyman-Pearson Lemma for simple alternative and its extension to composite hypotheses. Parametric inference: u, t, F, chi-square, Welch, Bartlett tests. Nonparametric inference: chi-square, Kolmogorov-Smirnov tests. Sequential analysis, Wald-Wolfowitz Theorem, CO-curves. Two-variate normal distribution and common features of methods based on it. Theory of least squares, regression analysis, Gauss-Markov Theorem. Literature: S. Ross, A First Course in Probability. Fifth Edition, Prentice-Hall, 1998. A. Rényi, Probability Theory, Acad. Press, 1978. 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.MATRIX COMPUTATIONS WITH APPLICATIONS Prerequisites: Undergraduate Calculus, Linear Algebra Book(s): G. Golub, C. F. van Loan, Matrix computations, Johns Hopkins University Press, 1996. Commitment: 3 hours/week, 3 credits Contents: 1.  Introduction to Matlab. Matrix manipulations and matlab notations.2.  Matrix multiplication problems3.  Matrix analysis 4.  Linear systems5.  Orthogonalization and least squares 6.  Eigenvalue problems, Lanczos methods 7.  Iterative methods for linear systems8.  Krylov subspaces, matrix functions9.  Applications (using matlab): numerical solution of partial differential equations, nonnegativity preservation, matrix exponential, numerical solution of Maxwell equations, model reduction.  COMPUTATIONS IN ALGEBRA No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: Basic Algebra 1 Course Level:  intermediate Brief introduction to the course: Different computational methods are discussed with an eye for applications. Diversity is a primary aim. The goals of the course: One of the main goals of the course is to introduce the main algorithms and their implementation. The computer algebra package GAP is used and its features are made familiar. The learning outcomes of the course: The students will learn the GAP system and attain fluency in it. They will get acquainted with some important algorithms and their implementations. They will be able to approach various problems with this arsenal. More detailed display of contents: 1. A complexity analysis of the basic algorithms with polynomials2. Discrete Fourier transform and fast Fourier transform of polynomials; fast multiplication of polynomials3. Factorization of univariate and bivariate polynomials over finite fields;4. Berlekamp and Cantor–Zassenhaus algorithms compared5. applications to coding theory (decoding Reed-Solomon codes)6. Lattice basis reduction; the LLL algorithms;7. applications to cryptography (breaking knapsack cryptosystems)8. Factorization of polynomials with integer coefficients9. Computational problems with multivariate polynomials; Groebner bases10. Applications of Groebner bases (solving a system of algebraic equations)11. Applications of Groebner bases (automatic geometric theorem proving; coding theory)12. Various applications based on some computer algebra softwareOptional topics:Categorical approach: products, coproducts, pullback, pushout, functor categories, natural transformations, Yoneda lemma, adjoint functors.Books: 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. CRYPTOLOGY 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 Commitment: 3 hours/week, 3 credits Contents: 1. Computational difficulty, computational indistinguishability2. Pseudorandom function, pseudorandom permutation3. Probabilistic machines, BPP4. Hard problems5. Public key cryptography6. Protocols, ZK protocols, simulation7. Unconditional Security, multiparty protocols8. Broadcast and pairwise channels9. Secret Sharing Schemes, Verifiable SSS10. Multiparty Computation    DIFFERENTIAL GEOMETRY Prerequisites: abstract and linear algebra, general topology, analysis, ordinary differential equations Books: 1. M.P. do Carmo, Differential Geometry of Curves and Surfaces Prentice-Hall, Englewood Cliffs, NJ, 1976.2. W. Klingenberg, A Course in Differential Geometry, Springer, 1978.3. W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Second Edition, Academic Press, 1986. Commitment: 3 hours/week, 3 credits Contents: 1. Curves in R2. 2. Hypersurfaces in R2. Theorema Egregium. Special surfaces.3. Differentiable manifolds, tangent budle, tensor bundles; Lie algebra of vector fields, distributions and Frobenius' theorem; Covariant derivation, the Levi-Civita connection of a Riemannian manifold, parallel transport, holonomy groups; Curvature tensor, symmetries of the curvature tensor, decomposition of the curvature tensor; Geodesic curves, the exponential map, Gauss Lemma, Jacobi fields, the Gauss-Bonnet theorem; Differential forms, de Rham cohomology, integration on manifolds, Stokes' theorem. INTRODUCTION TO DISCRETE MATHEMATICS No. of Credits: 3 and no. of ECTS credits: 6 Prerequisites: Course Level: introductory Brief introduction to the course: 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 ruleWeek 2. Occupancy problems, partitions of integersWeek 3. Solving recurrences, Fibonacci numbersWeek 4. Generating functions, applications to recurrencesWeek 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 functionApplication of sieve formula to Stirling numbers, derangements, and other involved problemsWeek 8. Pigeonhole principle, Ramsey theory, Erdos Szekeres theorem Week 9. Basic definitions of graph theory, treesWeek 10. Special properties of trees, Cayley’s theorem on the number of labeled trees Week 11. Flows in networks, connectivityWeek 12. Graph colorings, Brooks theorem, colorings of planar graphsReferences:1. Fred. S. Roberts, Applied Combinatorics, Prentice Hall, 19842. Fred. S. Roberts, Barry Tesman, Applied Combinatorics, Prentice Hall, 20043. Bela Bollobas, Modern Graph Theory, Springer, 1998GRAPH THEORY AND APPLICATIONS Prerequisities: Book: Reinhard Diestel, Graph Theory, Springer, Berlin, New York, 2005. Commitment: 3 hours/week, 3 credits Contents: 1.    Basics: Graphs, degrees, components, paths and cycles, independent sets, cliques, isomorphism, subgraphs, complement of a graph, trees, Euler tours,  other notions of graphs2.    Hamilton cycles: Sufficient conditions for Hamiltonicity, theorems of Dirac, Ore3.    Coloring: The concept of chromatic number, its relation to the clique number, theorems on chromatic number4.    Coloring edges and matchings: Vizing's theorem, matchings in bipartite graphs, Kõnig--Hall--Frobenius theorem, theorems of Gallai, Tutte's theorem5.    Network flows and connectivity: Networks, Ford-Fulkerson theorem, connectivity numbers, Menger's theorems6.    Planarity: Euler's lemma, Kuratowski's theorem, coloring of planar graphs7.    Turán- and Ramsey-type questions: Turán's theorem and related results, the role of the chromatic number in extremal graph theory, graph Ramsey theorems8.    Probabilistic approach: Some important results showing the power of the probabilistic method in graph theory will be discussed9.    Graph Algorithms: Shortest paths (Dikstra, Bellman-Ford, Floyd), depth first and width first search, mincost spanning trees (Prim, Kruskal), matching in bipartite graphs (Hungarian method)10.   Various applicationsNON-STANDARD ANALYSIS Prerequisities: Complex Function Theory, Functional Analysis. Books: Abraham Robinson, Non-standard Analysis, Princeton Univ. Press, 1995 Commitment: 3 hours/week, 3 credits Contents: 1. Tools from mathematical logic: compactness theorem, higher-order logic2. Enlargement3. Elementary Analysis: differentiation, integration, convergence 4. Topological Spaces: compactness, Tichonov's theorem, Uhrysson's theorem on metrizable spaces5. Theorems of Montel and Kakeya on lacunary polynomials6. Complex Functions: the Picard's theorem, Julia directionDIFFERENCE EQUATIONS  AND APPLICATIONS Prerequisites: Undergraduate Calculus Book: Ronald E. Mickens, Difference Equations. Theory and Applications, Van Nostrand Reinhold, New York, 1990. Commitment: 3 hours/week, 3 credits Contents: 1. The difference calculus2. First-order difference equations3. Linear difference equations4. Linear partial difference equations5. Nonlinear difference equations6. Various applications EVOLUTION EQUATIONS AND APPLICATIONS Prerequisites: Real and Complex Analysis, Functional Analysis 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. Commitment: 3 hours/week, 3 credits Contents: 1. Preliminaries of linear and nonlinear functional analysis2. Existence and regularity of solutions to evolution equations in Hilbert spaces3. Boundedness of solutions on the positive half axis4. Stability of solutions. Strong and weak convergence results. 5. Periodic forcing. The asymptotic dosing problem6. Applications to delay equations, parabolic and hyperbolic boundary value problems. Specific examples.

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