Fall 2017

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.

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.

Applications of integration, review of techniques of integration, infinite sequences and series, some beginning linear algebra, ordinary differential equations. Designed for students who earn a score of 4 or 5 on the Advanced Placement Calculus AB Examination, or a score of 3 or 4 on the BC Examination. Admission to course by departmental approval. Duplicates credit in MATH 126.

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. 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 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. Prerequisites: MATH 226 or MATH 227.

Probability, counting, independence, distributions, random variables, simulation, expectation, variance, covariance, transformations, law of large numbers, Central limit theorem, estimation, efficiency, maximum likelihood, Cramer-Rao bound, bootstrap. Prerequisites: MATH 118 or MATH 125 or MATH 126.

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, 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. 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.

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.

The Euclidean plane and its isometries. Models for the hyperbolic metric, isometries, linear fractional maps, geodesics. Gluing constructions. Knot theory and invariants. Prerequisite: MATH 226 or MATH 227.

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

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.

Individual research and readings. Not available for graduate credit.

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.

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.

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

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.

Stochastic processes revisited, Brownian motion, Martingale theory, stochastic differential equations, Feynman-Kac formula, binomial models, basic concepts in arbitrage pricing theory, equivalent Martingale measure. Recommended preparation: Math-225, Math-407. Duplicates credit in the former MATH-503.

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.

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: 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.

Binary classification, empirical risk minimization, support vector machines, voting algorithms and AdaBoost, Vapnik-Chervonenkis combinatorics, concentration-of-measure inequalities, sparse recovery problems, high-dimensional convex geometry.

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

Metric spaces, compactness, completion, fixed-point theorems, abstract spaces, linear operators, functionals, duality, conjugates, Lp spaces, Fourier transform. Applications from Engineering, Natural and Social sciences. Recommended preparation: Linear Algebra and Calculus.

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.

Practical principles for the long-term development of effective teaching within college disciplines. Intended for teaching assistants in Dornsife College. Graded CR/NC.

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

Structure and representation theory of various algebraic structures, such as groups, Lie algebras, Hopf algebras, algebraic groups, or group schemes.

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.