Fall 2007

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.

An explication of arithmetic and geometry, including the algebraic operations, number bases, plane and solid figures; and coordinate geometry.

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.

Methods of proof, predicate calculus, set theory, order and equivalence relations, partitions, lattices, functions, cardinality, elementary number theory and combinatories.

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.

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.

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

Incidence and separation properties of planes and spaces. Geometric inequalities, models of Riemannian and hyperbolic geometry. Isometrics, Jordan measure, constructions, and affine geometry.

Cardinals, topologies, separation axioms. Compactness, metrizability, function spaces; completeness; Jordan curve theorem. Recommended preparation: Upper division Math course.

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.

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.

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.

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

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: 501 or 502a or consent of instructor. 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: 501 or 502a or consent of instructor.

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: Consent of Instructor.

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: 407.

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

Brownian motion, stochastic integrals, the Ito formula, stochastic differential equations, analysis of diffusion processes, Girsanov transformation, Feynmann-Kac formula, applications.

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.

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

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.

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.

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.

Hypotheses testing, Neyman-Pearson lemma, generalized likelihood ratio procedures, confidence intervals, consistency, power, jackknife and bootstrap. Monte Carlo Markov chain methods, hidden Markov models. Hypotheses testing, Neyman-Pearson lemma, generalized likeliood

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

Metric spaces, fundamental topological and algebraic concepts, banach and hilbert space theory.

Equivalence of matrices; Jordon canonical form; functions of matrices; diagonalization; singular value decomposition; applications to linear differential equations, stability theory, and Markov processes. Prerequisite: consent of instructor.

Applications of the mathematical, statistical and comutational sciences to data from molecular biology. Statistics for genomic sequence data: DNA sequence assembly, significance of alignment scores, hidden Markov models, genetic mapping, models of sequence evolution, and microarray analysis.

Basic functional analysis in Banach and Hilbert spaces. Weak topologies, linear operators, spectral theory, calculus of vector-valued functions. Banach algebras.

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.

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.