501 Numerical Analysis and Computation (3, FaSpSm) Linear equations and matrices, Gauss elimination, error estimates, iteration techniques; contrac-tive mappings, Newton's method; matrix eigenvalue problems; least-squares approximation, Newton-Cotes and Gaussian quadratures; finite difference methods. Prerequisite: linear algebra and calculus.

502ab Numerical Analysis (a: 3, Fa; b: 3, Sp) Computational linear algebra; solution of general nonlinear systems of equations; approximation theory using functional analysis; numerical solution of ordinary and partial differential equations. Prerequisite: MATH 425a and MATH 471.

504ab Numerical Solution of Ordinary and Partial Differential Equations (a: 3, Sp; b: 3, Fa) a: Initial value problems; multistep methods, stability, convergence and error estimation, automatic stepsize control, higher order methods, systems of equations, stiff problems; boundary value problems; eigenproblems. Prerequisite: MATH 501 or MATH 502a or departmental approval. b: Computationally efficient schemes for solving PDE numerically; stability and convergence of difference schemes, method of lines; fast direct and iterative methods for elliptic equations. Prerequisite: MATH 501 or MATH 502a or departmental approval.

505ab Applied Probability (a: 3, Fa; b: 3, Sp) a: Populations, permutations, combinations, random variables, distribution and density functions conditional probability and expectation, binomial, Poisson, and normal distributions; laws of large numbers, central limit theorem. Prerequisite: departmental approval. b: Markov processes in discrete or continuous time; renewal processes; martingales; Brownian motion and diffusion theory; random walks, inventory models, population growth, queueing models, shot noise. Prerequisite: departmental approval.

506 Stochastic Processes (3) Basic concepts of stochastic processes with examples illustrating applications; Markov chains and processes; birth and death processes; detailed treatment of 1-dimensional Brownian motion. Prerequisite: MATH 407.

507ab Theory of Probability (a: 3, Fa; b: 3, Sp) a: Probability spaces; distributions and characteristic functions; laws of large numbers, central limit problems; stable and infinitely divisible laws; conditional distributions. Prerequisite: MATH 525a or MATH 570. b: Dependence, martingales, ergodic theorems, second-order random functions, harmonic analysis, Markov processes.

508 Filtering Theory (3) Theory of random differential equations and stochastic stability; optimum linear and nonlinear filtering, with discussion of asymptotic behavior of filter. Prerequisite: MATH 507a.

509 Stochastic Differential (3) Brownian motion, stochastic integrals, the Ito formula, stochastic differential equations, analysis of diffusion processes, Girsanov transformation, Feynmann-Kac formula, applications. Prerequisite: MATH 505ab or MATH 507ab.

510ab Algebra (a: 3, Fa; b: 3, Sp) a: Group Theory: Isomorphism theorems, group actions, Sylow's theorems, simple and solvable groups; Field Theory: Galois correspondence, radical extensions, algebraic and transcendental extensions, finite fields. b: Commutative Algebra: Integrality, Hilbert Basis theorem, Hilbert Nullstellensatz; Modules: modules over PIDs, chain conditions, tensor products; Noncommutative Rings: Jacobson radical, Artin-Wedderburn theorem, Maschke's theorem. Prerequisite: MATH 410, MATH 471.

511L Data Analysis (4) (Enroll in PM 511L)

520 Complex Analysis (3, Sp) Theory of analytic functions Ñ power series and integral representations, calculus of residues, harmonic functions, normal families, approximation theorems, conformal mapping, analytical continuation. Prerequisite: MATH 425ab.

525ab Real Analysis (a: 3, Fa; b: 3, Sp) a: Measure and integration over abstract measure spaces, Radon-Nikodym theorem, Fubini's theorem, convergence theorems, differentiation. Prerequisite: MATH 425ab. b: Metric spaces, contraction principle, category, Banach spaces, Riesz representation theorem, properties of Lp Hilbert spaces, orthogonal expansions, Fourier series and transforms, convolutions. Prerequisite: MATH 525a.

530 Analytic Number Theory (3, Fa) Additive and multiplicative number theory, exponential and character sums, cyclotomy, the Riemann zeta function, primes in an arithmetic progression, the prime number theorem, sieve methods. Prerequisite: MATH 520.

532 Combinatorial Analysis (3, Fa) Inversion formulas, generating functions and recursions, partitions, Stirling numbers, distinct representatives, Ramsey's theorem, graph theory, block designs, difference sets, finite geometrics, Latin squares, Hadamard matrices.

533 Combinatorial Analysis and Algebra (3, Sp) Advanced group theory; algebraic automata theory; graph theory; topics in combinatorial analysis.

535ab Differential Geometry (a: 3, Fa; b: 3, Sp) Elementary theory of manifolds, Lie groups, homogeneous spaces, fiber bundles and connections. Riemannian manifolds, curvature and conjugate points, second fundamental form, other topics. Prerequisite: MATH 440.

540 Topology (3, Sp) Initial and final topologies, function spaces, algebras in C(Y), homotopy, fundamental group, fiber spaces and bundles, smashes, loop spaces, groups of homotopy classes, cw-complexes. Prerequisite: MATH 440.

541ab Introduction to Mathematical Statistics (a: 3, Sp; b: 3, Fa) a: Exponential families, sufficiency. Estimation: methods of estimation, maximum likelihood, least squares, comparison of estimators, unbiased estimation, optimality, theory, information inequality, asymptotic efficiency, confidence intervals. Prerequisite: MATH 505a or MATH 407 or MATH 408. b: Testing: Neyman-Pearson lemma, consistency, power, linear models, regression, analysis of variance, discrete data, nonparametric methods. Prerequisite: MATH 541a.

542L Analysis of Variance and Design (3, Sp) Least squares estimation in the linear model, analysis of variance and covariance, F-test, multiple comparisons, multiple regression, selection of variables; introduction to experimental design. Includes laboratory. Prerequisite: MATH 225, MATH 226, and MATH 208x.

543L Nonparametric Statistics (3) Distribution-free methods for comparisons of two or more samples, tests of randomness, independence, goodness of fit; classification, regression. Comparison with parametric techniques. Includes laboratory. Prerequisite: MATH 226, MATH 208x.

544L Multivariate Analysis (3) (Enroll in PM 544L)

545L Introduction to Time Series (3, Fa) Transfer function models; stationary, nonstationary processes; moving average, autoregressive models; spectral analysis; estimation of mean, autocorrelation, spectrum; seasonal time series. Includes laboratory. Prerequisite: MATH 225, MATH 226, and MATH 208x.

546 Statistical Computing (3) (Enroll in PM 546)

547 Methods of Statistical Inference (3, Fa) Statistical decision theory: game theory, loss and risk functions; Bayes, minimax, admissible rules; sufficiency, invariance, tests of hypotheses, optimality properties. Inference for stochastic processes. Prerequisite: MATH 407 or MATH 408.

548 Sequential Analysis (3) Sequential decision procedures: sequential probability-ratio tests, operating characteristic, expected sample size, two-stage procedures, optimal stopping, martingales, Markov processes; applications to gambling, industrial inspection. Prerequisite: MATH 407 or MATH 408.

549 Bayesian Statistics (3) Bayes risk, Bayes decision functions, prior, posterior distributions, factorization, conjugate families, likelihood function, improper prior distributions, convergence of posterior distributions; Bayes methods of estimation, testing. Prerequisite: MATH 225, MATH 226 and MATH 208x.

550 Sample Surveys (3, Sp) Theory of sampling and design of sample surveys; bias and precision; finite populations; stratification; cluster sampling; multistage, systematic sampling; non-sampling errors. Prerequisite: MATH 208x.

551L Analysis of Discrete Observations (3, Sp) Standard discrete distributions, probability generating functions, branching processes, birth, death processes; goodness of fit, contingency tables, chi-square, likelihood ratio tests; regression, probit, logit models. Laboratory. Prerequisite: MATH 225, MATH 226 and MATH 208x.

555ab Partial Differential Equations (a: 3, Fa; b: 3, Sp) Second-order partial differential equations of elliptic, parabolic, and hyperbolic type; in particular, potential and wave equations. Prerequisite: MATH 425ab.

565ab Ordinary Differential Equations (a: 3, Fa; b: 3, Sp) Existence, uniqueness and continuation of solutions, differential inequalities, linear systems, Sturm-Liouville theory, boundary value problems, Poincare-Bendixson theory, periodic solutions, perturbations, stability, fixed point techniques. Prerequisite: MATH 425ab.

570ab Methods of Applied Mathematics (a: 3, FaSp; b: 3, SpSm) a: Metric spaces, fundamental topological and algebraic concepts, Banach and Hilbert space theory. Prerequisite: MATH 425a or departmental approval. b: Hilbert spaces, normal, self-adjoint and compact operators, geometric and spectral analysis of linear operators, elementary partial differential equations. Prerequisite: MATH 570a.

572 Applied Algebraic Structures (3, Fa) Elementary predicate logic, model theory, axiomatic set theory; relations, functions, equivalences; algebraic and relational structures; graph theory; applications of lattices, Boolean algebras; groups, rings, field. Prerequisite: departmental approval.

574 Applied Matrix Analysis (3, Fa) Equivalence of matrices; Jordon canonical form; functions of matrices; diagonalization; singular value decomposition; applications to linear differential equations, stability theory, and Markov processes. Prerequisite: departmental approval.

576 Applied Complex Analysis and Integral Transforms (3, Fa) Review of basic complex analysis; integral transforms of Laplace, Fourier, Mellin, and Hankel; applications to solutions of ordinary and partial differential equations; Wiener-Hopf technique. Prerequisite: MATH 475 or MATH 520.

578 DNA and Protein Sequence Analysis (3, Sp) Genetic and physical mapping of genomes, restriction mapping of DNA, reconstruction of evolutionary trees from sequence data, algorithms and statistics for sequence comparisons, secondary structure. Prerequisite: MATH 425a and MATH 407, or departmental approval.

580 Introduction to Functional Analysis (3, Fa) Basic functional analysis in Banach and Hilbert spaces. Weak topologies, linear operators, spectral theory, calculus of vector-valued functions. Banach algebras. Prerequisite: MATH 525ab.

585ab Mathematical Theory of Optimal Control (a: 3, Fa; b: 3, Sp) a: Deterministic control: calculus of variations; optimal control; Pontryagin principle; multiplier rules and abstract nonlinear programming; existence and continuity of controls; problem of Mayer; dynamic programming. Prerequisite: MATH 570 and MATH 525a. b: Stochastic control: Markov diffusion processes; backward-forward equations; linear systems; Kalman-Bucy filtering; optimal stochastic control; stochastic dynamic programming; linear regulator problem; separation principle. Prerequisite: MATH 585a and either MATH 505ab or MATH 507ab.

587ab Mathematical Models of Neurons and Neural Networks (a: 3, Fa; b: 3, Sp) a: Dynamics of discrete and analog neural networks; qualitative and numerical analysis; computer simulation; learning algorithms and convergence; Kolmagorov theory of feed-forward networks. Prerequisite: MATH 465 and either MATH 501 or MATH 502a. b: Nernst-Planck and Goldman-Hodgkin-Katz equations; Hodgkin-Huxley theory; cable theory; compartment models of dendritic structures; McCulloch-Pitts networks; perceptron theory. Prerequisite: MATH 587a.

590 Directed Research (1-12, FaSpSm) Research leading to the master's degree. Maximum units which may be applied to the degree to be determined by the department. Graded CR/NC.

594abz Master's Thesis (2-2-0, FaSpSm) Credit on acceptance of thesis. Graded IP/CR/NC.

597 Mathematical Packages and Scientific Computing (3) Use of mathematical software packages in scientific computing (linear algebra, differential equations, etc.): reliability; portability; influence of different computer architectures; techniques for large, sparse systems. Prerequisite: MATH 501 or MATH 502a or departmental approval.

599 Special Topics (2-4, max 8, FaSpSm) Course content will be selected each semester to reflect current trends and developments in the field of mathematics.

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