Basic existence and stability theory. Difference equations. Boundary value problems.
MATH A. Introduction to Numerical Optimization: Linear Programming 4. Linear optimization and applications. Linear programming, the simplex method, duality. Selected topics from integer programming, network flows, transportation problems, inventory problems, and other applications. Three lectures, one recitation. Introduction to Numerical Optimization: Nonlinear Programming 4. Convergence of sequences in Rn, multivariate Taylor series. Bisection and related methods for nonlinear equations in one variable. Equality-constrained optimization, Kuhn-Tucker theorem.
Inequality-constrained optimization. Introduction to convexity: convex sets, convex functions; geometry of hyperplanes; support functions for convex sets; hyperplanes and support vector machines. Linear and quadratic programming: optimality conditions; duality; primal and dual forms of linear support vector machines; active-set methods; interior methods. Convex constrained optimization: optimality conditions; convex programming; Lagrangian relaxation; the method of multipliers; the alternating direction method of multipliers; minimizing combinations of norms.
Conjoined with MATH Floating point arithmetic, direct and iterative solution of linear equations, iterative solution of nonlinear equations, optimization, approximation theory, interpolation, quadrature, numerical methods for initial and boundary value problems in ordinary differential equations. Mathematical background for working with partial differential equations. Survey of finite difference, finite element, and other numerical methods for the solution of elliptic, parabolic, and hyperbolic partial differential equations.
Formerly MATH Graduate students do an extra paper, project, or presentation, per instructor. Mathematical models of physical systems arising in science and engineering, good models and well-posedness, numerical and other approximation techniques, solution algorithms for linear and nonlinear approximation problems, scientific visualizations, scientific software design and engineering, project-oriented.
Graduate students will do an extra paper, project, or presentation per instructor. Probability spaces, random variables, independence, conditional probability, distribution, expectation, variance, joint distributions, central limit theorem. Prior or concurrent enrollment in MATH is highly recommended. Random vectors, multivariate densities, covariance matrix, multivariate normal distribution. Random walk, Poisson process. Other topics if time permits. Markov chains in discrete and continuous time, random walk, recurrent events. If time permits, topics chosen from stationary normal processes, branching processes, queuing theory.
Multivariate distribution, functions of random variables, distributions related to normal. Parameter estimation, method of moments, maximum likelihood. Estimator accuracy and confidence intervals. Hypothesis testing, type I and type II errors, power, one-sample t-test. Hypothesis testing. Linear models, regression, and analysis of variance. Goodness of fit tests. Nonparametric statistics. Prior enrollment in MATH is highly recommended. Topics covered may include the following: classical rank test, rank correlations, permutation tests, distribution free testing, efficiency, confidence intervals, nonparametric regression and density estimation, resampling techniques bootstrap, jackknife, etc.
Statistical learning refers to a set of tools for modeling and understanding complex data sets. It uses developments in optimization, computer science, and in particular machine learning. This encompasses many methods such as dimensionality reduction, sparse representations, variable selection, classification, boosting, bagging, support vector machines, and machine learning.
Analysis of trends and seasonal effects, autoregressive and moving averages models, forecasting, informal introduction to spectral analysis. Design of sampling surveys: simple, stratified, systematic, cluster, network surveys. Sources of bias in surveys. Estimators and confidence intervals based on unequal probability sampling.
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Design and analysis of experiments: block, factorial, crossover, matched-pairs designs. Analysis of variance, re-randomization, and multiple comparisons. Introduction to probability. Discrete and continuous random variables—binomial, Poisson and Gaussian distributions. Central limit theorem. Data analysis and inferential statistics: graphical techniques, confidence intervals, hypothesis tests, curve fitting. Introduction to the theory and applications of combinatorics.
Enumeration of combinatorial structures permutations, integer partitions, set partitions. Statistical analysis of data by means of package programs. Regression, analysis of variance, discriminant analysis, principal components, Monte Carlo simulation, and graphical methods. Emphasis will be on understanding the connections between statistical theory, numerical results, and analysis of real data. This course will cover discrete and random variables, data analysis and inferential statistics, likelihood estimators and scoring matrices with applications to biological problems.
Introduction to Binomial, Poisson, and Gaussian distributions, central limit theorem, applications to sequence and functional analysis of genomes and genetic epidemiology. An introduction to the basic concepts and techniques of modern cryptography. Classical cryptanalysis. Probabilistic models of plaintext.
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Monalphabetic and polyalphabetic substitution. The one-time system. Caesar-Vigenere-Playfair-Hill substitutions. The Enigma. Modern-day developments. The Data Encryption Standard. Public key systems. Security aspects of computer networks. Data protection. Electronic mail. Recommended preparation: programming experience. Renumbered from MATH The object of this course is to study modern public key cryptographic systems and cryptanalysis e.
We also explore other applications of these computational techniques e. A rigorous introduction to algebraic combinatorics. Basic enumeration and generating functions. Enumeration involving group actions: Polya theory. Posets and Sperner property. Partitions and tableaux. An introduction to various quantitative methods and statistical techniques for analyzing data—in particular big data. Quick review of probability continuing to topics of how to process, analyze, and visualize data using statistical language R. Further topics include basic inference, sampling, hypothesis testing, bootstrap methods, and regression and diagnostics.
Offers conceptual explanation of techniques, along with opportunities to examine, implement, and practice them in real and simulated data. An introduction to point set topology: topological spaces, subspace topologies, product topologies, quotient topologies, continuous maps and homeomorphisms, metric spaces, connectedness, compactness, basic separation, and countability axioms. Students who have not completed prerequisites may enroll with consent of instructor.
Examples of all of the above. Instructor may choose further topics such as deck transformations and the Galois correspondence, basic homology, compact surfaces. Students who have not completed the listed prerequisite may enroll with consent of instructor. Topics to be chosen by the instructor from the fields of differential algebraic, geometric, and general topology.
Prerequisites: MATH or consent of instructor. Probabilistic Foundations of Insurance. Short-term risk models. Survival distributions and life tables. Introduction to life insurance. Life Insurance and Annuities. Analysis of premiums and premium reserves. Introduction to multiple life functions and decrement models as time permits. Introduction to the mathematics of financial models. Basic probabilistic models and associated mathematical machinery will be discussed, with emphasis on discrete time models. Concepts covered will include conditional expectation, martingales, optimal stopping, arbitrage pricing, hedging, European and American options.
Students will be responsible for and teach a class section of a lower-division mathematics course. They will also attend a weekly meeting on teaching methods. Does not count toward a minor or major. Prerequisites: consent of instructor. A variety of topics and current research results in mathematics will be presented by guest lecturers and students under faculty direction. Prerequisites: upper-division status. Subject to the availability of positions, students will work in a local company under the supervision of a faculty member and site supervisor. Units may not be applied toward major graduation requirements.
Prerequisites: completion of ninety units, two upper-division mathematics courses, an overall 2. Department stamp required. Independent reading in advanced mathematics by individual students. Three periods. Prerequisites: permission of department. Honors thesis research for seniors participating in the Honors Program. Research is conducted under the supervision of a mathematics faculty member. Prerequisites: admission to the Honors Program in mathematics, department stamp. Group actions, factor groups, polynomial rings, linear algebra, rational and Jordan canonical forms, unitary and Hermitian matrices, Sylow theorems, finitely generated abelian groups, unique factorization, Galois theory, solvability by radicals, Hilbert Basis Theorem, Hilbert Nullstellensatz, Jacobson radical, semisimple Artinian rings.
Recommended for all students specializing in algebra. Basic topics include categorical algebra, commutative algebra, group representations, homological algebra, nonassociative algebra, ring theory. May be taken for credit six times with consent of adviser as topics vary. Introduction to algebra from a computational perspective. Groups, rings, linear algebra, rational and Jordan forms, unitary and Hermitian matrices, matrix decompositions, perturbation of eigenvalues, group representations, symmetric functions, fast Fourier transform, commutative algebra, Grobner basis, finite fields.
Prerequisites: graduate standing or consent of instructor. Second course in algebra from a computational perspective. Third course in algebra from a computational perspective. Introduction to algebraic geometry. Topics chosen from: varieties and their properties, sheaves and schemes and their properties. May be taken for credit up to three times. Second course in algebraic geometry. Continued exploration of varieties, sheaves and schemes, divisors and linear systems, differentials, cohomology.
Third course in algebraic geometry. Continued exploration of varieties, sheaves and schemes, divisors and linear systems, differentials, cohomology, curves, and surfaces. First course in graduate-level number theory. Local fields: valuations and metrics on fields; discrete valuation rings and Dedekind domains; completions; ramification theory; main statements of local class field theory.
Second course in graduate-level number theory.
Third course in graduate-level number theory. Zeta and L-functions; Dedekind zeta functions; Artin L-functions; the class-number formula and generalizations; density theorems. Topics in algebraic and analytic number theory, such as: L-functions, sieve methods, modular forms, class field theory, p-adic L-functions and Iwasawa theory, elliptic curves and higher dimensional abelian varieties, Galois representations and the Langlands program, p-adic cohomology theories, Berkovich spaces, etc.
May be taken for credit nine times. Prerequisites: graduate standing. Introduction to varied topics in algebraic geometry. Topics will be drawn from current research and may include Hodge theory, higher dimensional geometry, moduli of vector bundles, abelian varieties, deformation theory, intersection theory. Nongraduate students may enroll with consent of instructor. Continued development of a topic in algebraic geometry.
May be taken for credit three times with consent of adviser as topics vary. Introduction to varied topics in algebra. In recent years, topics have included number theory, commutative algebra, noncommutative rings, homological algebra, and Lie groups. Various topics in algebraic geometry. Various topics in number theory. Complex variables with applications. Linear algebra and functional analysis.
Fourier analysis of functions and distributions in several variables. In recent years, topics have included applied complex analysis, special functions, and asymptotic methods. May be repeated for credit with consent of adviser as topics vary. Various topics in the mathematics of biological systems. Cauchy theorem and its applications, calculus of residues, expansions of analytic functions, analytic continuation, conformal mapping and Riemann mapping theorem, harmonic functions. Dirichlet principle, Riemann surfaces. Introduction to varied topics in several complex variables.
In recent years, topics have included formal and convergent power series, Weierstrass preparation theorem, Cartan-Ruckert theorem, analytic sets, mapping theorems, domains of holomorphy, proper holomorphic mappings, complex manifolds and modifications. Continued development of a topic in several complex variables. Topics include formal and convergent power series, Weierstrass preparation theorem, Cartan-Ruckert theorem, analytic sets, mapping theorems, domains of holomorphy, proper holomorphic mappings, complex manifolds and modifications.
Existence and uniqueness theorems. Cauchy-Kowalewski theorem, first order systems. Hamilton-Jacobi theory, initial value problems for hyperbolic and parabolic systems, boundary value problems for elliptic systems. Introduction to varied topics in differential equations. In recent years, topics have included Riemannian geometry, Ricci flow, and geometric evolution.
Continued development of a topic in differential equations. Topics include Riemannian geometry, Ricci flow, and geometric evolution. Lebesgue integral and Lebesgue measure, Fubini theorems, functions of bounded variations, Stieltjes integral, derivatives and indefinite integrals, the spaces L and C, equi-continuous families, continuous linear functionals general measures and integrations.
Metric spaces and contraction mapping theorem; closed graph theorem; uniform boundedness principle; Hahn-Banach theorem; representation of continuous linear functionals; conjugate space, weak topologies; extreme points; Krein-Milman theorem; fixed-point theorems; Riesz convexity theorem; Banach algebras. Prerequisites: Math A-B-C or consent of instructor. In recent years, topics have included Fourier analysis in Euclidean spaces, groups, and symmetric spaces.
Various topics in functional analysis. Convex sets and functions, convex and affine hulls, relative interior, closure, and continuity, recession and existence of optimal solutions, saddle point and min-max theory, subgradients and subdifferentials. Recommended preparation: course work in linear algebra and real analysis. Optimality conditions, strong duality and the primal function, conjugate functions, Fenchel duality theorems, dual derivatives and subgradients, subgradient methods, cutting plane methods.
Convex optimization problems, linear matrix inequalities, second-order cone programming, semidefinite programming, sum of squares of polynomials, positive polynomials, distance geometry. Introduction to varied topics in real analysis. In recent years, topics have included Fourier analysis, distribution theory, martingale theory, operator theory. May be taken for credit six times with consent of adviser. Continued development of a topic in real analysis. Topics include Fourier analysis, distribution theory, martingale theory, operator theory. Various topics in real analysis. Differential manifolds, Sard theorem, tensor bundles, Lie derivatives, DeRham theorem, connections, geodesics, Riemannian metrics, curvature tensor and sectional curvature, completeness, characteristic classes.
Differential manifolds immersed in Euclidean space. Lie groups, Lie algebras, exponential map, subgroup subalgebra correspondence, adjoint group, universal enveloping algebra. Structure theory of semisimple Lie groups, global decompositions, Weyl group. Geometry and analysis on symmetric spaces. Prerequisites: MATH and or consent of instructor. Various topics in Lie groups and Lie algebras, including structure theory, representation theory, and applications. Introduction to varied topics in differential geometry.
In recent years, topics have included Morse theory and general relativity.
Continued development of a topic in differential geometry. Topics include Morse theory and general relativity. May be taken for credit three times with consent of adviser. Various topics in differential geometry. Riemannian geometry, harmonic forms. Lie groups and algebras, connections in bundles, homotopy sequence of a bundle, Chern classes.
Applications selected from Hamiltonian and continuum mechanics, electromagnetism, thermodynamics, special and general relativity, Yang-Mills fields. Prerequisites: graduate standing in mathematics, physics, or engineering, or consent of instructor. Propositional calculus and first-order logic. Feasible computability and complexity.
Peano arithmetic and the incompleteness theorems, nonstandard models. Introduction to the probabilistic method.
Combinatorial applications of the linearity of expectation, second moment method, Markov, Chebyschev, and Azuma inequalities, and the local limit lemma. Introduction to the theory of random graphs. Introduction to probabilistic algorithms. Game theoretic techniques. Applications of the probabilistic method to algorithm analysis.
Markov Chains and Random walks. Applications to approximation algorithms, distributed algorithms, online and parallel algorithms. Advanced topics in the probabilistic combinatorics and probabilistic algorithms. Random graphs. Spectral Methods. Network algorithms and optimization. Statistical learning. Introduction to varied topics in combinatorial mathematics. In recent years topics have included problems of enumeration, existence, construction, and optimization with regard to finite sets. Recommended preparation: some familiarity with computer programming desirable but not required. Continued development of a topic in combinatorial mathematics.
Topics include problems of enumeration, existence, construction, and optimization with regard to finite sets. Topics from partially ordered sets, Mobius functions, simplicial complexes and shell ability. Enumeration, formal power series and formal languages, generating functions, partitions. Lagrange inversion, exponential structures, combinatorial species. Finite operator methods, q-analogues, Polya theory, Ramsey theory.
Representation theory of the symmetric group, symmetric functions and operations with Schur functions. Introduction to varied topics in mathematical logic. Topics chosen from recursion theory, model theory, and set theory. Discover the best of shopping and entertainment with Amazon Prime. Prime members enjoy FREE Delivery on millions of eligible domestic and international items, in addition to exclusive access to movies, TV shows, and more. Back to top. Get to Know Us. English Choose a language for shopping. Audible Download Audio Books. Alexa Actionable Analytics for the Web.
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