# Manual Probability on Discrete Structures volume 110

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Introduction to regression analysis, one of the most widely-used statistical techniques. Simple and multiple linear regression, nonlinear regression, analysis of residuals and model selection. One-way and two-way factorial experiments, random and fixed effects models. Prerequistes: MH and MH Introductory course on the main ideas and results of topology. The emphasis is on beautiful, intuitive material such as surfaces, graphs and knots, rather than technicalities of point-set topology. Continuous functions. Euler characteristics. Maps and graphs on surfaces. Vector fields on surfaces.

Fixed-point theorems and their applications to economics. Basics of homotopy theory. MH - Data Analysis with Computer. Data collection and analysis processes; graphical and numerical methods for describing data.

Summarizing bivariate data; probability and population distributions; estimation and hypothesis testing using a single sample; comparing two population or treatments. Analysis of categorical data and goodness-of-fit tests. Discrete-time Markov chains, examples of discrete-time Markov chains, classification of states, irreducibility, periodicity, first passage times, recurrence and transience, convergence theorems and stationary distributions.

Random walks and Poisson processes. Introduction to the theory and applications of numerical approximation techniques. Commonly used numerical algorithms, and how to implement them. Computational errors. Numerical methods for solving systems of linear equations. Iterative methods for systems of linear equations. Polynomial interpolation. Numerical integration. Numerical solutions of nonlinear equations. First course on optimization and operations research. Introduction to optimization models: objective and constraints, convex sets and functions, polyhedron and extreme points.

Introduction to linear programming LP : solving 2-variable LP via graphical methods; simplex method; dual LP and sensitivity analysis. Karush-Kuhn-Tucker optimality conditions, optimal solution via optimality conditions, Duality theory. Network optimization: shortest path, maximum flow, minimum cost flow, assignment problem, transportation problem, network simplex method. Prerequisite: MH or MH Riemann integral, integrability, fundamental theorem of calculus, improper integrals. Convergent series, absolute convergence, tests of convergence.

Sequence and series of functions, uniform convergence.

## Course Descriptions

First-order equations, quasi-linear equations, general first-order equation for a function of two variables, Cauchy problem. Wave equation, wave equation in two independent variables, Cauchy problem for hyperbolic equations in two independent variables.

Heat equation, the weak maximum principle for parabolic equations, Cauchy problem for heat equation, regularity of solutions to heat equation. Laplace equation, Green's formulas, harmonic functions, maximum principle for Laplace equation, Dirichlet problem, Green's function and Poisson's formula. MH is useful but not required. Unique factorization domains, Euclidean domains, principal ideal domains. Modules, submodules, homomorphisms, quotient modules, modules over principal ideal domains. Field extensions, automorphisms of fields, spilitting fields, normal and separable extensions.

Galois extensions, Galois groups, Galois correspondence, finite fields. MH - Combinatorics. Recursions and generating functions. Partitions and tableaux. Designs, Latin squares, combinatorial designs and projective geometries. Extremal combinatorics, asymptotic analysis. MH - Set Theory and Logic. Partially-ordered sets, well-orderings and order-types, induction and recursion on ordinals, ordinal arithmetic, cardinals, cardinal arithmetic. Axiom of choice and its equivalences, axiom of determinacy. Propositional calculus, truth tables, validity and contradictions.

MH - Coding Theory. Introduction to the theory of error-correcting codes, and their applications in data storage and telecommunication. Error detection, correction and decoding, Hamming distance. Basic facts about finite fields. Linear codes, Hamming weight, generator and parity-check matrices, encoding, and decoding. Construction of codes, Reed-Muller codes.

Computer implementation of efficient coding and decoding. MH - Cryptography.

Classical ciphers, cryptanalysis, linear complexity. The RSA cryptosystem, primality testing and factorization of integers. Discrete logarithms.

## Course Descriptions – Mathematics Department

Signatures; the Digital Signature Standard. Introduction to specialized advanced topics related to information theory, coding theory and cryptography. The choice of the topic depends on the instructor. Prerequisite: division approval. MH - Computational Economics. Introduction to game theory and the theory of computation. Computation of equilibria Nash equilibrium, market equilibrium, Walrasian equilibrium, etc.

• Regulating Securitized Products: A Post Crisis Guide.
• College of Arts and Sciences.
• Steklov Mathematical Institute.
• Discrete Mathematics and Applications?
• List of Publications.

Algorithmic Mechanism Design, Auction Theory Vickrey auction, combinatorial auctions, digital goods auctions, sponsored search auctions. Profit maximization, cost sharing mechanisms. MH - Time Series Analysis. Introduction to time series models and their applications in economics, engineering and finance. Trend fitting, autoregressive and moving average models, spectral analysis. Seasonality, forecasting and estimation.

Use of computer package to analyze real data sets.

## School of Physical and Mathematical Sciences

MH - Multivariate Analysis. Distribution theory: multivariate normal distribution, Hotelling's T2 and Wishart distributions.

Inference on the mean and covariance, principal components and canonical correlation. Factor analysis, discrimination and classification. Topics covered include: Optimal decision rules and K-nearest neighbors methods. Linear models for regression. Generalized linear models for classification.

Cross-validation and bootstrap methods. Artificial neural networks.

Classification and regression trees and ensemble methods. Clustering methods.