Spring 2013

Intensive review of arithmetic and algebra. Graded CR/NC. (Not available for degree credit.)

Equations and inequalities; functions; graphs; polynomial and rational functions; exponential, logarithmic, and trigonometric function; analytic geometry.

Finite mathematics with application to the social sciences; elementary set theory and logic; counting techniques; probability; statistics; matrices and systems of linear equations. Selected topics.

Functions, graphs, polynomial and rational functions, exponential and logarithmic functions, matrices, systems of linear equations.

Derivatives; extrema. Definite integral; fundamental theorem of calculus. Extrema and definite integrals for functions of several variables. Not available for credit toward a degree in mathematics.

Limits; continuity, derivatives and applications; antiderivatives; the fundamental theorem of calculus; exponential and logarithmic functions.

A continuation of 125: trigonometric functions; applications of integration; techniques of integration; indeterminate forms; infinite series; Taylor series; polar coordinates.

Descriptive statistics, probability concepts, discrete and continuous random variables, mathematical expectation and variance, probability sampling, Central Limit Theorem, estimation and hypothesis testing, correlation and regression. Not available for major credit to mathematics majors. Prerequisite: 118x or 125.

Basic probability, discrete and continuous distributions, expectation and variance, independence. Sampling, estimation, confidence intervals, hypothesis testing.

Matrices, systems of linear equations, vector spaces, linear transformations, eigenvalues, systems of linear differential equations.

A continuation of 126; vectors, vector valued functions; differential and integral calculus of functions of several variables; Green's theorem.

First-order differential equations; second-order linear differential equations; determinants and matrices; systems of linear differential equations; Laplace transforms.

Differential calculus of multivariable functions, optimization, elementary linear algebra and matrix theory, principal component analysis, elementary differential equations, systems, qualitative theory, numerical methods, scientific computation. Recommended preparation: Math-126 or equivalent or AP credit for Calculus BC.

Confidence intervals, hypothesis testing, p-values, likelihood ratio, nonparametrics, descriptive statistics, regression, multiple linear regression, experimental design, analysis of variance, categorical data, chi-squared tests, Bayesian statistics.

Induction, Euclidean algorithm, factorization, congruence classes, Rings, RSA algorithm, Chinese remainder theorem, codes, polynomials, fundamental theorem of algebra, polynomial multiplication, Fourier transform, and other topics.

Systematic approach to solving non-standard and competition level math problems on inequalities, infinite sums and products, combinatorics, number theory, and games. Recommended preparation: MATH 126

Probability spaces, discrete and continuous distributions, moments, characteristic functions, sequences of random variables, laws of large numbers, central limit theorem, special probability laws. Prerequisite: 226 or 226H.

Principles for testing hypotheses and estimation, small sample distributions, correlation and regression, nonparametric methods, elements of statistical decision theory. Prerequisite: 407.

Sets; relations; groups; homomorphisms; symmetric groups; Abelian groups; Sylow's theorems; introduction to rings and fields. Not open to students with credit in 470. Prerequisite: 225.

The real number system, metric spaces, limits, continuity, derivatives and integrals, infinte series.

Implicit function theorems, Jacobians, transformations, multiple integrals, line integrals.

Mathematical induction, counting principles, arrangements, selections, binomial coefficients, generating functions, recurrence relations, inclusion-exclusion, symmetric groups, graphs, Euler and Hamiltonian circuits, trees, graph algorithms; applications. Prerequisite: 225 or 226 or 226H or consent of instructor.

Vectors, elements of vector analysis, applications to curves and surfaces, standard material of differential geometry. Prerequisite: 226 or 226H.

Vector field theory; theorems of Gauss, Green, and Stokes; Fourier series and integrals; complex variables; linear partial differential equations; series solutions of ordinary differential equations. Prerequisite: 245.

Evolution of mathematical ideas and techniques as seen through a study of the contributions of eminent mathematicians to the formulation and solution of celebrated problems. Recommended preparation: upper division MATH course.

Linear systems, phase plane analysis, existence and uniqueness, stability of linear and almost linear systems, Lyapunov's method, nonlinear oscillations, flows, invariant surfaces, and bifurcation.

Polynomial rings, vector spaces, linear transformations, canonical forms, inner product spaces. Recommended preparation: MATH 410.

Limits and infinite series; line integrals; conformal mapping; single-valued functions of a complex variable; applications. Primarily for advanced students in engineering.

Individual research and readings. Not available for graduate credit.

Lectures directed to mathematics graduate students by faculty of the department and by outside speakers. Problem solving workshops. Graded CR/NC.

Linear equations and matrices, Gauss elimination, error estimates, iteration techniques; contractive mappings, Newton's method; matrix eigenvalue problems; least-squares approximation, Newton-Cotes and Gaussian quadratures; finite difference methods. Prerequisite: linear algebra and calculus.

Computational linear algebra; solution of general nonlinear systems of equations; approximation theory using functional analysis; numerical solution of ordinary and partial differential equations.

Markov processes in discrete or continuous time; renewal processes; martingales; brownian motion and diffusion theory; random walks, inventory models, population growth, queuing models, shot noise.

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

Theory of random differential equations and stochastic stability; optimum linear and nonlinear filtering, with discussion of asymptotic behavior of filter.

B: Commutative Algebra: Integrality, Hilbert Basis theorem, Hilbert Nullstellensatz; Modules: modules over PID's, chain conditions, tensor products; Noncommutative Rings: Jacobson radical, Artin-Wedderburn theorem, Maschke's theorem.

Major parametric and non-parametric statistical tools used in biomedical research, computer packages including SAS. Includes laboratory. Lecture, 2 hours; laboratory, 2 hours.

Exploratory data analysis, detection of outliers, fitting data with linear and nonlinear regression models, computer packages including S-PLUS and SPSS.

Theory of analytic functions--power series and integral representations, calculus of residues, harmonic functions, normal families, approximation theorems, conformal mapping, analytical continuation.

a: Measure and integration over abstract measure spaces, Radon-Nikodym theorem, Fubini's theorem, convergence theorems, differentiation. Prerequisite: 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: 525a.

Advanced topics in stochastic analysis, asset pricing in continuous time, stochastic control, Hamilton-Jacobi-Bellman equations, incomplete markets, American options, exotic options, term structure of interest rates. Duplicates credit in the former MATH-506.

Elementary theory of manifolds, Lie groups, homogeneous spaces, fiber bundles and connections. Riemannian manifolds, curvature and conjugate points, second fundamental form, other topics. Prerequisite: 440.

Parametric families of distributions, sufficiency. Estimation: methods of moments, maximum likelihood, unbiased estimation. Comparison of estimators, optimality, information inequality, asymptotic efficiency. EM algorithm, jacknife and bootstrap.

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, 226 or 226H, and 208x or equivalent.

Second-order partial differential equations of elliptic, parabolic, and hyperbolic type; in particular, potential and wave equations.

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.

Practical experience in computational molecular biology applications. Mathematical and statistical software packages relevant for genomic analysis. Retrieval and analysis of genomic data from databases. Recommended preparation: higher level programming language.

Applications of the mathematical, statistical and computational sciences to data from molecular biology. Algorithms for genomic sequence data: sequence and map assembly and alignment, RNA secondary structure, protein structure, gene-finding, and tree construction. Recommended preparation: familiarity with the concepts of basic molecular biology as covered in BISC 320.

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.

Credit on acceptance of thesis.

Credit on acceptance of thesis.

Credit on acceptance of thesis. Graded CR/NC.

Course content will be selected each semester to reflect current trends and developments in the field of mathematics.

Necessary and sufficient conditions for existence of extrema with equality constraints; gradient methods; Ritz methods; eigenvalue problems; optimum control problems; inequality constraints; mathematical programming.

Research leading to the doctorate. Maximum units which may be applied to the degree to be determined by the department. Graded CR/NC.

Credit on acceptance of dissertation.

Credit on acceptance of dissertation.

Credit on acceptance of dissertation.

Credit on acceptance of dissertation.

Credit on acceptance of dissertation.