Spring 2015

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.

An introduction to the basic tools of statistics. Descriptive statistics; probability; expected value; normal approximation sampling; chance models; tests of significance. Recommended preparation: MATH 040x or math placement exam. Not available for major credit to Math students.

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. Prerequisites: MATH 108 or MATH 117 or placement exam in MATH.

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. Prerequisites: MATH 126 or MATH 127.

A continuation of 126; vectors, vector valued functions; differential and integral calculus of functions of several variables; Green's theorem. Prerequisites: MATH 126 or MATH 127.

A continuation of MATH 127; vectors and vector spaces functions of several variables, partial differential equations, optimization theory, multiple integration; Green's Stokes', divergence theorems. Duplicates credit in MATH 226.

First-order differential equations; second-order linear differential equations; determinants and matrices; systems of linear differential equations; Laplace transforms. Prerequisites: MATH 226 or MATH 227.

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. Prerequisites: MATH 125. Recommended preparation: MATH 126 or MATH 127 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. Prerequisites: MATH 226 or MATH 227, and MATH 225 or MATH 245.

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 or MATH 127.

Probability spaces, discrete and continuous distributions, moments, characteristic functions, sequences of random variables, laws of large numbers, central limit theorem, special probability laws. Prerequisite: MATH 226 or MATH 227.

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 MATH 470. Prerequisite: MATH 225. Recommended preparation: one 400-level Mathematics course, excluding MATH-450.

The real number system, metric spaces, limits, continuity, derivatives and integrals, infinite series. Prerequisites: MATH 226 or MATH 227. Recommended preparation: One 400-level Mathematics course, excluding MATH 450.

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

Introduction to the theory of numbers, including prime factorization, congruences, primitive roots, N-th power residues, number theoretic functions, and certain diophantine equations. Prerequisites: MATH 126 or MATH 127.

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: MATH 225 or MATH 226 or MATH 227.

Vectors, elements of vector analysis, applications to curves and surfaces, standard material of differential geometry. Prerequisite: MATH 226 or MATH 227.

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: (MATH-225 and MATH-226 or MATH-227) or MATH-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.

Nonlinear models, bifurcations, discrete and continuous dynamics, stability, dissipative and reversible systems, Poincare maps, Lyapunov exponents, fractal dimension. Applications stressed: Economics, Engineering, Biology, Physics. Prerequisites: MATH 225 or MATH 245.

Formulation and study of models arising in population dynamics, growth of plankton, pollution of rivers, highway traffic, morphogenesis and tidal dynamics: stability, oscillations, bifurcations, chaos. The lab will consist of computer simulation of models using commercially available software.

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. Prerequisites: MATH 226 or MATH 227.

Individual research and readings. Not available for graduate credit.

Lectures on advanced material not covered in regularly scheduled courses. No more than two registrations allowed. Prerequisite: Departmental Approval.

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.

Experimental laboratory trading for financial markets using double auctions: handling statistical packages for data analysis. Practical training in virtual market environments, using financial trading system software.

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.

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, 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.