Spring 2024

Review of Algebra. Real numbers, linear equations and inequalities, functions, graphs, systems of equations, exponents, polynomials, factoring, rational expressions, radicals, quadratic equations and solutions. Graded CR/NC.

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.

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

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

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

Trigonometric functions; applications of integration; techniques of integration; indeterminate forms; infinite series; Taylor series; polar coordinates. Engineering and physics applications. Prerequisite: MATH 125. (Duplicates credit in MATH 126 and MATH 127)

Descriptive statistics, probability, discrete and continuous random variables, expectation and variance, sampling, Central Limit Theorem, estimation, hypothesis testing, correlation and regression. Emphasis on health science. Prerequisite: MATH 118xg or MATH 125.

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

Vectors, vector valued functions; differential and integral calculus of functions of several variables; Green's theorem, Divergence theorem, Stoke's theorem.

A continuation of 129; vectors, vector valued functions; differential and integral calculus of functions of several variables; Greens theorem. Engineering and physics applications. Prerequisite: MATH 126, or MATH 127, or MATH 129.(Duplicates credit in MATH 226 and MATH 227)

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

Preparation for reading and writing proofs with applications from major branches of mathematics, focusing on axiomatic structures and transformations that preserve structure.

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.

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

Principles for testing hypotheses and estimation, confidence intervals, methods of moments, maximum likelihood, information inequality, likelihood ratio tests, goodness of fit and nonparametric methods.

Sets; relations; groups; homomorphisms; symmetric groups; Abelian groups; Sylow's theorems; introduction to rings and fields.

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

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.

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.

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.

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.

Rounding errors in digital computation; solution of linear algebraic systems; Newton's method for nonlinear systems; matrix eigenvalues; polynomial approximation; numerical integration; numerical solution of ordinary differential equations.

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.

Methods for static, dynamic, unconstrained, constrained optimization. Gradient, conjugate gradient, penalty methods. Lagrange multipliers, least squares, linear, nonlinear dynamic programming. Application to control and estimation.

Polynomial rings, vector spaces, linear transformations, canonical forms, inner product spaces.

Complex numbers and functions. Power series, analytic functions, harmonic functions. Complex integration, Cauchy's theorem, Cauchy's formula. Conformal mapping. Applications in physics and 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.

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.

Statistical methods for analysis of categorical data including dichotomous, ordinal, multinomial and count data, using Stata package. Includes laboratory.

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.

Combinatorics motivated by Representation Theory and Algebraic Geometry. Topics may include: representations of the symmetric group, symmetric functions, Young tableaux and plane partitions, permutation enumeration.

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.

Transfer function models; stationary, nonstationary processes; moving average, autoregressive models; spectral analysis; estimation of mean, autocorrelation, spectrum; seasonal time series.

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.

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.

Part-time or full-time, practical work experience in the students field of study. The internship must be located at an off-campus facility. Students are individually supervised by faculty. May not be taken until the student has completed at least one semester of enrollment in the graduate program with a cumulative 3.0 GPA.

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

Topics to be chosen from the following: geometry of complex manifolds, relations between topology and curvature, homogeneous spaces, symmetric spaces, geometry of submanifolds.

Topics to be chosen from the following: Elliptic, Parabolic, Hyperbolic, and Dispersive PDEs, Conservation Laws, Mathematical Fluid Dynamics and Variational Methods. Recommended preparation: Math-555a.

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.