1. Precalculus. Lecture, three hours; discussion, one hour. Preparation: three years of high school mathematics. Requisite: successful completion of Mathematics Diagnostic Test. Function concept. Linear and polynomial functions and their graphs, applications to optimization. Inverse, exponential, and logarithmic functions. Trigonometric functions. P/NP or letter grading.
2. Finite Mathematics. Lecture, three hours; discussion, one hour. Preparation: three years of high school mathematics. Finite mathematics consisting of matrices, Gauss/Jordan method, combinatorics, probability, Bayes theorem, and Markov chains. P/NP or letter grading.
3A. Calculus for Life Sciences Students. Lecture, three hours; discussion, one hour. Preparation: three and one-half years of high school mathematics (including trigonometry). Requisite: successful completion of Mathematics Diagnostic Test or course 1 (C - or better). Not open for credit to students with credit in another calculus sequence. Techniques and applications of differential calculus. Introduction to the integral. P/NP or letter grading.
3B. Calculus for Life Sciences Students. Lecture, three hours; discussion, one hour. Requisite: course 3A (C - or better). Techniques and applications of integral calculus, logarithmic and exponential functions, introduction to differential equations. P/NP or letter grading.
3C. Calculus for Life Sciences Students. Lecture, three hours; discussion, one hour. Requisite: course 3B (C - or better). Functions of several variables, vectors, partial differentiation, and vector-valued functions. P/NP or letter grading.
31A. Calculus and Analytic Geometry. Lecture, three hours; discussion, one hour. Preparation: at least three and one-half years of high school mathematics (including some coordinate geometry and trigonometry). Requisite: successful completion of Mathematics Diagnostic Test or course 1 (C - or better). Differential calculus and applications; introduction to integration.
31B. Calculus and Analytic Geometry. Lecture, three hours; discussion, one hour. Requisite: course 31A (C - or better). Transcendental functions; methods and applications of integration.
31BH. Calculus and Analytic Geometry (Honors). Lecture, three hours; discussion, one hour. Honors course parallel to course 31B.
31E. Calculus for Economics Students. Lecture, three hours; discussion, one hour. Requisite: course 31A (C - or better). Not open for credit to students with credit for course 3B, 3C, or 31B. Calculus with applications to economics. Partial differentiation, differentials, implicit functions, exponential and logarithmic functions, extrema, optimization, constrained extrema, first-order linear differential equations with constant coefficients. P/NP or letter grading.
32A. Calculus of Several Variables. Lecture, three hours; discussion, one hour. Requisite: course 31B (C - or better). Introduction to differential calculus of several variables.
32AH-32BH. Calculus of Several Variables (Honors). Lecture, three hours; discussion, one hour. Requisite: course 31B (B or better). Honors sequence parallel to courses 32A, 32B.
32AL. Calculus Computer Laboratory (1 unit). Corequisite: course 32A. Prior knowledge of computers not required. Application of mathematical software to calculus of curves and surfaces. P/NP or letter grading.
32B. Calculus of Several Variables. Lecture, three hours; discussion, one hour. Requisite: course 32A (C - or better). Introduction to integral calculus of several variables, vector field theory, line and surface integrals. P/NP or letter grading.
32BL. Calculus Computer Laboratory (1 unit). Requisite: course 32AL. Corequisite: course 32B. Application of mathematical software to calculus of curves and surfaces. P/NP or letter grading.
33A. Matrices and Differential Equations. Lecture, three hours; discussion, one hour. Requisite: course 32A (C - or better). Introduction to matrix theory, differential equations, and systems of differential equations.
33AH-33BH. Matrices, Differential Equations, and Infinite Series (Honors). Lecture, three hours; discussion, one hour. Honors sequence parallel to courses 33A, 33B. P/NP or letter grading.
33B. Infinite Series. Lecture, three hours; discussion, one hour. Requisite: course 33A (C - or better). Infinite sequences and series; applications.
38A-38B. Fundamentals of Mathematics for Elementary Teachers. Not open to freshmen or for credit to students with credit for any course from Mathematics 110A through 199. May not be applied toward Letters and Science general education requirements. Courses 38A, 38B, and 104 form one-year sequence for prospective elementary teachers in Diversified Liberal Arts Program. P/NP or letter grading. 38A. Lecture, three hours; discussion, one hour. Counting numbers and other subsystems of real numbers; sets; operations, relations, algorithms; applications and problem solving. Emphasis on understanding arithmetic procedures. 38B. Lecture, three hours; discussion, one hour; laboratory, one hour. Requisite: course 38A. Continuation of course 38A. Elementary number theory; probability and statistics; the microcomputer and simple instructional programs; measurement and approximation; coordinate geometry. Other topics appropriate for elementary classroom.
61. Introduction to Discrete Structures. Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B, Program in Computing 10A or 3. Not open for credit to students with credit for course 113. Discrete structures commonly used in computer science and mathematics, including sets and relations, permutations and combinations, graphs and trees, induction, Boolean algebras.
Mathematics 113, 115A, 117, 131A, 132, 142, 151A, 164, 167, and Statistics 154A-154B are offered each term. The remaining upper division courses are usually offered once or twice each year. The tentative class schedule for the forthcoming academic year is posted in the Student Services Office in February.
104. Fundamental Concepts of Geometry. Lecture, three hours; discussion, one hour. Requisites: courses 38A, 38B. Designed for prospective elementary teachers. Informal geometry and topology, motion geometry, measurement of geometric figures, LOGO computer language, models and constructions appropriate for elementary classrooms.
106. History of Mathematics. Requisite: course 3A or 31A. Roots of modern mathematics in ancient Babylonia and Greece, development of algebra through Middle Ages to Fermat and Abel, invention of analytic geometry and calculus, selected topics in modern mathematics. P/NP or letter grading.
110A-110B. Algebra. Lecture, three hours; discussion, one hour. Requisite: course 115A. 110A. Not open for credit to students with credit for course 117. Ring of integers, integral domains, fields, polynomial domains, unique factorization. 110B. Groups, structure of finite groups.
110AH-110BH. Algebra (Honors). (Formerly numbered 110AH-110BH-110CH.) Lecture, three hours; discussion, one hour. Honors sequence parallel to courses 110A-110B.
110C. Algebra. Lecture, three hours; discussion, one hour. Requisites: courses 110A-110B. Field extensions, Galois theory, applications to geometric constructions, and solvability by radicals.
111. Theory of Numbers. (Formerly numbered 111A-111B-111C.) Lecture, three hours; discussion, one hour. Requisites: courses 110A or 117, 115A. Divisibility, congruences, Diophantine analysis, selected topics in theory of primes, algebraic number theory, Diophantine equations.
M112. Introduction to Set Theory. (Formerly numbered M112A.) (Same as Philosophy M134.) Lecture, three hours; discussion, one hour. Requisite: course 31B or Philosophy 32. Axiomatic set theory as framework for mathematical concepts; relations and functions, numbers, cardinality, axiom of choice, transfinite numbers. P/NP or letter grading.
113. Combinatorics. Lecture, three hours; discussion, one hour. Prerequisites: courses 32B, 33B. Permutations and combinations, counting principles, recurrence relations and generating functions, combinatorial designs, graphs and trees, with applications including games of complete information. Combinatorial existence theorems, Ramsey theorem.
114A-114B. Logic and Computability. (Formerly numbered 114A-114B-114C.) Lecture, three hours; discussion, one hour. Requisite: course 115A. Propositional and predicate logic; syntax and semantics; formal deductions; completeness and compactness; Herbrand expansions. Effectively computable, Turing computable, and recursive functions; thesis of Church. Universal functions; unsolvability results. Recursive and recursively enumerable sets; recursive enumerability of valid sentences. Formal number theory; definability of recursive functions; incompleteness and undecidability; theorems of Gödel, Tarski, Church. P/NP or letter grading.
115A-115B. Linear Algebra. Lecture, three hours; discussion, one hour. P/NP or letter grading. 115A. Prerequisite: course 33A. Abstract vector spaces, linear transformations, and matrices; determinants; inner product spaces; eigenvector theory. 115B. Prerequisite: course 115A. Linear transformations, conjugate spaces, duality; theory of a single linear transformation, Jordan normal form; bilinear forms, quadratic forms; Euclidean and unitary spaces, symmetric skew and orthogonal linear transformations, polar decomposition.
115AH. Linear Algebra (Honors). Lecture, three hours; discussion, one hour. Honors course parallel to course 115A.
117. Algebra for Applications. Lecture, three hours; discussion, one hour. Prerequisite: course 115A. Not open for credit to students with credit for course 110A. Integers, congruences; fields, applications of finite fields; polynomials; permutations, introduction to groups.
120A-120B. Differential Geometry. Lecture, three hours; discussion, one hour. Prerequisites: courses 32B, 33B, 115A, 131A. Curves in 3-space, Frenet formulas, surfaces in 3-space, normal curvature. Gaussian curvature. Congruence of curves and surfaces. Intrinsic geometry of surfaces, isometries, geodesics, Gauss/Bonnet theorem.
121. Introduction to Topology. Requisite: course 131A. Metric and topological spaces, completeness, compactness, connectedness, functions, continuity, homeomorphisms, topological properties.
123. Foundations of Geometry. Lecture, three hours; discussion, one hour. Requisite: course 115A. Axioms and models, Euclidean geometry, Hilbert axioms, neutral (absolute) geometry, hyperbolic geometry, Poincaré model, independence of parallel postulate.
131A-131B. Analysis. Lecture, three hours; discussion, one hour. 131A. Requisite: course 33B. Rigorous introduction to foundations of real analysis; real numbers, point set topology in Euclidean space, functions, continuity. 131B. Requisites: courses 33B, 115A, 131A. Derivatives, Riemann integral, sequences and series of functions, power series, Fourier series.
131AH-131BH. Analysis (Honors). Lecture, three hours; discussion, one hour. Honors sequence parallel to courses 131A-131B.
131C. Topics in Analysis. Lecture, three hours; discussion, one hour. Requisites: courses 131A-131B. Advanced topics in analysis, such as Lebesgue integral, integration on manifolds, harmonic analysis. Content varies from year to year. May be repeated for credit by petition.
132. Complex Analysis for Applications. Lecture, three hours; discussion, one hour. Prerequisites: courses 32B, 33B. Introduction to basic formulas and calculation procedures of complex analysis of one variable relevant to applications. Topics include Cauchy/Riemann equations, Cauchy integral formula, power series expansion, contour integrals, residue calculus.
135A-135B. Ordinary Differential Equations. Lecture, three hours; discussion, one hour. Prerequisites: courses 33A, 33B, 115A. Systems of differential equations; linear systems with constant coefficients, analytic coefficients, periodic coefficients, and linear systems with regular singular points; existence and uniqueness results; linear boundary and eigenvalue problems; two-dimensional autonomous systems, phase-plane analysis; stability and asymptotic behavior of solutions.
136. Partial Differential Equations. Lecture, three hours; discussion, one hour. Requisites: courses 33A, 33B. Linear partial differential equations, boundary and initial value problems; wave equation, heat equation, and Laplace equation; separation of variables, eigenfunction expansions; selected topics, as method of characteristics for nonlinear equations.
142. Mathematical Modeling. Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B. Introduction to fundamental principles and spirit of applied mathematics. Emphasis on manner in which mathematical models are constructed for physical problems. Illustrations from many fields of endeavor, such as physical sciences, biology, economics, and traffic dynamics.
143. Analytic Mechanics. Lecture, three hours; discussion, one hour. Prerequisites: courses 32B, 33B. Foundations of Newtonian mechanics, kinematics and dynamics of a rigid body, variational principles and Lagrange equations; calculus of variations, variable mass; related topics in applied mathematics.
146. Methods of Applied Mathematics. Lecture, three hours; discussion, one hour. Prerequisite: course 33B. Integral equations, Green's function, and calculus of variations. Selected applications from control theory, optics, dynamical systems, and other engineering problems.
149. Mathematics of Computer Graphics. Lecture, three hours; discussion, one hour. Prerequisites: course 115A, and Program in Computing 10A or equivalent knowledge of programming in either PASCAL or C language. Study of homogeneous coordinates, projective transformations, interpolating and approximating curves, representation of surfaces, and other mathematical topics useful for computer graphics.
151A-151B. Applied Numerical Methods. (Formerly numbered 141A-141B.) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, Program in Computing 3 or 10A. Introduction to numerical methods with emphasis on algorithms, analysis of algorithms, and computer implementation issues. 151A. Solution of nonlinear equations. Numerical differentiation, integration, and interpolation. Direct methods for solving linear systems. 151B. Numerical solution of differential equations. Approximation theory, iterative solutions of linear equations, solution of nonlinear systems, two-point boundary value problems, optimization.
153. Numerical Methods for Partial Differential Equations. (Formerly numbered 148A.) Lecture, three hours; discussion, one hour. Requisites: courses 151A-151B. Introduction to first- and second-order linear partial differential equations. Finite difference and finite element solution of elliptic, hyperbolic, and parabolic equations. Method of lines and Rayleigh/Ritz procedures. Concepts of stability and accuracy.
157. Software Techniques for Scientific Computation. Lecture, three hours; discussion, one hour. Requisites: course 151A, Program in Computing 10C. Software structures, concepts, and conventions that support object-oriented programming. Identification of class structure, problem partitioning, and abstraction. Design and implementation of computer applications requiring scientific computation, visualization, and GUI components. Interlanguage interfacing. P/NP or letter grading.
164. Linear Programming. (Formerly numbered 144.) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for Electrical Engineering 136. Principles of linear programming, duality theorem, simplex methods; applications to industrial and business problems. Additional topics such as sensitivity analysis, integer programming, distribution and transportation algorithms, and applications to game theory.
167. Game Theory. (Formerly numbered 147.) Lecture, three hours; discussion, one hour. Requisite: course 115A. Games in extensive form, strategic equilibrium, matrix games and minimax theorem, cooperative and noncooperative solutions of bimatrix games and Lemke/Howson algorithm. Possible additional topics include combinatorial games, stochastic games, coalitional games and the core, marriage problem, and cost allocation. P/NP or letter grading.
M170A. Probability Theory. (Formerly numbered M150A.) (Same as Statistics M152A.) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B. Not open to students with credit for Statistics M152A, 154A, or Electrical Engineering 131A. Probability distributions, random variables and vectors, expectation. P/NP or letter grading.
170B. Probability Theory. (Formerly numbered 150B.) Lecture, three hours; discussion, one hour. Requisite: course M170A or Statistics M152A. Convergence in distribution, normal approximation, laws of large numbers, Poisson processes, random walks.
171. Stochastic Processes. (Formerly numbered 151.) Lecture, three hours; discussion, one hour. Requisite: course M170A or Statistics M152A. Discrete Markov chains, continuous-time Markov chains, renewal theory.
172A-172B. Actuarial Mathematics. Lecture, three hours; discussion, one hour. 172A. Prerequisite: course 70. Survival distributions and life tables, life insurance, life annuities, net premiums, net premium reserves. 172B. Prerequisites: course 172A, Statistics 154A-154B. Multiple life functions, multiple decrement models, valuation theory for pension plans, insurance models, nonforfeiture benefits and dividends.
190. Honors Mathematics Seminar. Seminar, three hours. Participating seminar on advanced topics in mathematics. Content varies from year to year. May be repeated for credit by petition.
199. Special Studies in Mathematics (1 to 4 units). At discretion of chair and subject to availability of staff, individuals or groups may study topics suitable for undergraduate course credit but not specifically offered as separate courses. May be repeated for credit, but no more than one 199 course may be applied toward upper division courses required for a major offered by Mathematics Department.
201A-201B-201C. Topics in Algebra and Analysis. Prerequisite: bachelor's degree in mathematics or equivalent. Designed for students in mathematics/education program. Important ideas of algebra, geometry, and calculus leading effectively from elementary to modern mathematics. Approaches to number system, point sets, geometric interpretations of algebra and analysis, integration, differentiation, series and analytic functions. May not be applied toward M.A. degree requirements.
202A-202B. Mathematical Models and Applications. Prerequisite: bachelor's degree in mathematics or equivalent. Designed for students in mathematics/education program. Development of mathematical theories describing various empirical situations. Basic characterizing postulates; development of a logical structure of theorems. Modern topics such as operations research, linear programming, game theory, learning models, models in social and life sciences. May not be applied toward M.A. degree requirements.
205A-205B-205C. Number Theory. Prerequisites: courses 210A and 246A, or consent of instructor. Topics from analytic algebraic and geometric number theory, including distribution of primes and factorization in algebraic number fields. Selected topics from additive number theory, Diophantine approximation, partitions, class-field theory, lattice point problems, valuation theory, etc.
206A-206B. Combinatorial Theory. Prerequisite: consent of instructor. Generating functions. Probabilistic methods. Polya theorem. Enumerative graph theory. Partition theory. Number theoretical applications. Structure of graphs, matching theory, duality theorems. Packings, pavings, coverings, statistical designs, difference sets, triple systems, finite planes. Configurations, polyhedra. Ramsey theory, finite and transfinite, and applications.
210A-210B-210C. Algebra. Requisites: courses 110A-110B, 110C. Students with credit for courses 110B and/or 110C cannot receive M.A. degree credit for courses 210B and/or 210C. Group theory, including theorems of Sylow and Jordan/Holder/Schreier; rings and ideals, factorization theory in integral domains, modules over principal ideal rings, Galois theory of fields, multilinear algebra, structure of algebras.
211. Structure of Rings. Prerequisite: course 210A or consent of instructor. Radical, irreducible modules and primitive rings, rings and algebras with minimum condition.
212. Homological Algebra. Prerequisite: course 210A or consent of instructor. Modules over a ring, homomorphisms and tensor products of modules, functors and derived functors, homological dimension of rings and modules.
213A-213B. Theory of Groups. Prerequisite: course 210A or consent of instructor. Topics include representation theory, transfer theory, infinite Abelian groups, free products and presentations of groups, solvable and nilpotent groups, classical groups, algebraic groups.
214A-214B. Introduction to Algebraic Geometry. Prerequisite: course 210A or consent of instructor. Basic definitions and first properties of algebraic varieties in affine and projective space: irreducibility, dimension, singular and smooth points. More advanced topics, such as sheaves and their cohomology, or introduction to theory of Riemann surfaces, as time permits.
215A-215B. Commutative Algebra. Prerequisite: course 210A or consent of instructor. Topics from commutative ring theory, including techniques of localization, prime ideal structure in commutative Noetherian rings, principal ideal theorem, Dedekind rings, modules, projective modules, Serre conjecture, regular local rings.
216. Further Topics in Algebraic Geometry. Prerequisites: courses 214A-214B or consent of instructor. Closer examination of areas of current research in algebraic geometry. Variable content may include algebraic surfaces, Abelian varieties, invariant theory, Hodge theory, or geometry over finite fields. May be repeated for credit by petition.
220A-220B-220C. Mathematical Logic and Set Theory. Lecture, three hours. Requisite: course M112. Model theory: compactness theorem; Lowenheim/Skolem theorems; definability; ultraproducts; preservation theorems; interpolation theorems. Recursion function theory: thesis of Church; recursively enumerable sets; hierarchies; degrees. Formal proofs: completeness and incompleteness theorems; decidable and undecidable theories; quantifier elimination. Set theory: Zermelo/Fraenkel and von Neumann/Gödel axioms; cardinal and ordinal numbers; continuum hypothesis; constructible sets; independence results and forcing. S/U or letter grading.
222A-222B. Lattice Theory and Algebraic Systems. Lecture, three hours. Prerequisite: course 210A or consent of instructor. Partially ordered sets, lattices, distributivity, modularity; completeness, interaction with combinatorics, topology, and logic; algebraic systems, congruence lattices, subdirect decomposition, congruence laws, equational bases, applications to lattices.
223A. Model Theory. Prerequisites: courses 220A-220B-220C. Topics include ultraproducts, preservation theorems, interpolation theorems, saturated models, omitting types, categoricity, two cardinal theorems, enriched languages, soft model theory, and applied model theory.
223B. Set Theory. Prerequisites: courses 220A-220B-220C. Topics include constructibility theory, Cohen extensions, large cardinals, and combinatorial set theory.
223C. Recursion Theory. Prerequisites: courses 220A-220B-220C. Topics include degrees of unsolvability, recursively enumerable sets, undecidable theories, inductive definitions, admissible sets and ordinals, and recursion in higher types.
223D. Descriptive Set Theory. Prerequisites: courses 220A-220B-220C. Classical descriptive set theory: Borel and projective sets. Effective descriptive set theory. Consequences of strong set-theoretic hypotheses.
225A. Differentiable Manifolds. Lecture, three hours. Prerequisites: courses 121 and 131A-131B, or consent of instructor. Smooth manifolds and maps, basic examples and properties, orientability, tangent and cotangent spaces, embeddings and immersions, Sard theorem and transversality, vector fields and integral curves, Lie brackets and Frobenius theorem, Lie derivative, tensors, differential forms and exterior derivative, Stokes theorem on manifolds.
225B. Introduction to Algebraic Topology. Lecture, three hours. Prerequisite: course 225A or consent of instructor. Elementary concepts of homotopy theory; covering spaces and fundamental group. Singular homology theory, axioms of homology, Mayer/Vietoris sequence, calculation of homology of standard spaces, applications, Betti numbers and Euler characteristic, cell complexes and cellular homology.
225C. Further Topics in Geometry and Topology. Lecture, three hours. Prerequisites: courses 225A and 225B, or consent of instructor. Topics may include cohomology (singular, cellular, de Rham), duality theorems, de Rham theorem, degree theory, cup products, higher homotopy groups, transversality theory, Morse theory, Riemannian metric.
226A-226B-226C. Differential Geometry. Lecture, three hours. Prerequisite: course 225A or consent of instructor. Manifold theory; connections, curvature, torsion, and parallelism. Riemannian manifolds; completeness, submanifolds, constant curvature. Geode-sics; conjugate points, variational methods, Myers theorem, nonpositive curvature. Further topics such as pinched manifolds, integral geometry, Kahler manifolds, symmetric spaces.
227A-227B. Algebraic Topology. Lecture, three hours. Prerequisite: course 225B or consent of instructor. CW complexes, fiber bundles, homotopy theory, cohomology theory, spectral sequences.
229A-229B-229C. Lie Groups and Lie Algebras. Prerequisite: knowledge of basic theory of topological groups and differentiable manifolds. Lie groups, Lie algebras, subgroups, subalgebras. Exponential map. Universal enveloping algebra. Campbell/Hausdorff formula. Nilpotent and solvable Lie algebras. Cohomology of Lie algebras. Theorems of Weyl, Levi-Mal'cev. Semi-simple Lie algebras. Classification of simple Lie algebras. Representations. Compact groups. Weyl character formula.
233. Partial Differential Equations on Manifolds. Lecture, three hours. Prerequisites: courses 226A and 251A, or consent of instructor. Topics may include Laplacian operator on a Riemannian manifold, eigenvalues, Atiyah/Singer index theorem, isoperimetric inequalities, elliptic estimates, harmonic functions, function theory on manifolds, Green's function, heat equation, minimal hypersurfaces, prescribed curvature equations, harmonic maps, Yang/Mills equation, Monge/Ampere equations.
234. Topics in Differential Geometry. Lecture, three hours. Prerequisites: courses 226A-226B or consent of instructor. Complex and Kahler geometry, Hodge theory, homogeneous manifolds and symmetric spaces, finiteness and convergence theorems for Riemannian manifolds, almost flat manifolds, closed geodesics, manifolds of positive scalar curvature, manifolds of constant curvature. Topics vary from year to year. May be repeated for credit by petition.
235. Topics in Manifold Theory. Lecture, three hours. Prerequisites: courses 225A and 225B, or consent of instructor. Emphasis on low-dimensional manifolds. Structure and classification of manifolds, automorphisms of manifolds, submanifolds (e.g., knots and links). Topics vary from year to year. May be repeated for credit by petition.
236. Topics in Geometric Topology. Lecture, three hours. Prerequisites: courses 225A and 225B, or consent of instructor. Decomposition spaces, surgery theory, group actions, dimension theory, infinite dimensional topology. Topics vary from year to year. May be repeated for credit by petition.
237. Topics in Algebraic Topology. Lecture, three hours. Prerequisites: courses 227A-227B or consent of instructor. Fixed-point theory, fiber spaces and classifying spaces, characteristic classes, generalized homology and cohomology theories. Topics vary from year to year. May be repeated for credit by petition.
238A-238B. Dynamical Systems. Lecture, three hours. Recommended preparation: first-year analysis courses. Topics include qualitative theory of differential equations, bifurcation theory, and Hamiltonian systems; differential dynamics, including hyperbolic theory and quasiperiodic dynamics; ergodic theory; low-dimensional dynamics. S/U or letter grading.
240. Methods of Set Theory. Lecture, three hours. Prerequisites: courses 110A-110B, 121 or equivalent, 131A-131B. Naive, axiomatic set theory, axiom of choice and its equivalents, well-orderings, transfinite induction, ordinal and cardinal arithmetic. Applications to algebra: Hamel bases, Stone representation theorem. Applications to analysis and topology: Cantor/Bendixson theorem, counterexamples in measure theory, Borel and analytic sets, Choquet theorem.
245A-245B-245C. Real Analysis. Lecture, three hours. Requisites: courses 121, 131A-131B. Students with credit for former course 134 cannot receive M.A. degree credit for course 245A. Basic measure theory. Measure theory on locally compact spaces. Fubini theorem. Elementary aspects of Banach and Hilbert spaces and linear operators. Function spaces. Radon/Nikodym theorem. Fourier transform and Plancherel on Rn and Tn.
246A-246B-246C. Complex Analysis. Requisites: courses 131A-131B. Students with credit for course 132 cannot receive M.A. degree credit for course 246A. Cauchy/Riemann equations. Cauchy theorem. Cauchy integral formula and residue calculus. Power series. Normal families. Harmonic functions. Linear fractional transformations. Conformal mappings. Analytic continuation. Examples of Riemann surfaces. Infinite products. Partial fractions. Classical transcendental functions. Elliptic functions.
247A-247B. Classical Fourier Analysis. Lecture, three hours. Prerequisites: courses 245A-245B, 246A. Distribution on Rn and Tn. Principal values; other examples. Distributions with submanifolds as supports. Kernel theorem. Convolution; examples of singular integrals. Tempered distributions and Fourier transform theory on Rn. Distributions with compact or one-sided supports and their complex Fourier transforms.
250A. Ordinary Differential Equations. Prerequisite: course 246A or consent of instructor. Basic theory of ordinary differential equations. Existence and uniqueness of solutions. Continuity with respect to initial conditions and parameters. Linear systems and nth order equations. Analytic systems with isolated singularities. Self-adjoint boundary value problems on finite intervals.
250B. Nonlinear Ordinary Differential Equations. Prerequisite: course 250A. Asymptotic behavior of nonlinear systems. Stability. Existence of periodic solutions. Perturbation theory of two-dimensional real autonomous systems. Poincaré/Bendixson theory.
250C. Advanced Topics in Ordinary Differential Equations. Prerequisites: courses 250A, 250B. Selected topics, such as spectral theory or ordinary differential operators, nonlinear boundary value problems, celestial mechanics, approximation of solutions, and Volterra equations.
251A. Introductory Partial Differential Equations. Prerequisite: consent of instructor. Classical theory of heat, wave, and potential equations; fundamental solutions, characteristics and Huygens principle, properties of harmonic functions. Classification of second-order differential operators. Maximum principles, energy methods, uniqueness theorems. Additional topics as time permits.
251B-251C. Topics in Partial Differential Equations. Prerequisite: consent of instructor. In-depth introduction to topics of current interest in partial differential equations or their applications.
252A-252B. Topics in Complex Analysis. (Formerly numbered 252A-252B-252C.) Lecture, three hours. Requisites: courses 245A-245B-245C, 246A-246B-246C. Potential theory, subharmonic functions, harmonic measure; Hardy spaces; entire functions; univalent functions; Riemann surfaces; extremal length, variational methods, quasi-conformal mappings. Topics vary from year to year. S/U or letter grading.
253A-253B. Several Complex Variables. Prerequisites: courses 245A-245B-245C and 246A-246B-246C, or consent of instructor. Introduction to analytic functions of several complex variables. The[partialdiff]- problem, Cousin problems, domains of holomorphy, complex manifolds.
254A-254B. Topics in Real Analysis. Prerequisites: courses 245A-245B-245C, 246A-246B-246C. Selected topics in analysis and its applications to geometry and differential equations. Topics may vary from year to year. May be repeated for credit by petition.
255A. Functional Analysis. Prerequisites: courses 245A-245B or 265A-265B, and 246A, or consent of instructor. Banach spaces, basic principles. Weak topologies. Compact operators. Fredholm operators. Special spaces including Hilbert spaces and C(X).
255B-255C. Topics in Functional Analysis. Prerequisite: course 255A. Topics include Banach algebras, operators on Banach spaces and Hilbert space, semigroups of operators, linear topological vector spaces, and other related areas.
256A-256B. Topological Groups and Their Representations. (Formerly numbered 256A-256B-256C.) Lecture, three hours. Requisite: course 255A. Topological groups and their basic properties. Haar measure. Compact groups and their representations. Duality and Fourier analysis on locally compact abelian groups. Induced representations, Frobenius reciprocity. Representations of special groups (Lorentz, Galilean, etc.). Projective representations. Representations of totally disconnected groups. S/U or letter grading.
259A-259B. Operator Algebras in Hilbert Space. Prerequisites: courses 255A, 255B-255C. Selected topics from theories of C* and von Neumann algebras. Applications.
260. Introduction to Applied Mathematics. Prerequisite: course 142 or consent of instructor. Construction, analysis, and interpretation of mathematical models of problems which arise outside of mathematics.
M261. Game Theory. (Formerly numbered 261.) (Same as Economics M214B and Political Science M208A.) Lecture, three hours. Prerequisite: graduate standing in mathematics or consent of instructor. Bargaining theory, the core, the value, other solution concepts. Applications to oliogopoly, general exchange and production economies, and allocation of joint costs. S/U or letter grading.
264. Applied Complex Analysis. Prerequisite: course 246A or consent of instructor. Topics include contour integration conformal mapping, differential equations in complex plane, special functions, asymptotic series, Fourier and Laplace transforms, singular integral equations.
265A-265B. Real Analysis for Applications. Prerequisites: courses 131A-131B or consent of instructor. Not open for credit to students with credit for courses 245A-245B-245C. Lebesgue measure and integration on real line, absolutely continuous functions, functions of bounded variation, L2- and Lp- spaces. Fourier series. General measure and integrations, Fubini and Radon/Nikodym theorems, representation of functionals, Fourier integrals.
266A. Applied Ordinary Differential Equations. Lecture, three hours. Requisites: courses 131A-131B, 132, and 135A-135B or 146. Spectral theory of regular boundary value problems and examples of singular Sturm/Liouville problems, related integral equations, phase/plane analysis of nonlinear equations. S/U or letter grading.
266B-266C. Applied Partial Differential Equations. Prerequisite: course 266A or consent of instructor. Classification of equations, classical potential theory, Dirichlet and Neumann problems. Green's functions, spectral theory of Laplace equation in bounded domains, first-order equations, wave equations, Cauchy problem, energy conservation, heat equation, fundamental solution, equations of fluid mechanics and magnetohydrodynamics.
266D-266E. Applied Differential Equations. Prerequisites: courses 266A, 266B-266C. Advanced topics in linear and nonlinear partial differential equations, with emphasis on energy estimates, numerical methods, and applications to fluid mechanics. Additional topics include dispersive waves, systems with multiple time scales, and applications to fluid mechanics.
268A. Applied Functional Analysis. Lecture, three hours. Prerequisites: courses 115A-115B, 131A-131B, and 132, or consent of instructor. Topics may include Hilbert spaces, distributions, Fourier transforms, L2- space, the Laplacian, linear operators, spectrum and resolvent, self-adjoint and unitary operators, problems of evolution in Banach spaces, well-posed initial value problems, semigroups, applications to applied problems.
268B-268C. Topics in Applied Functional Analysis. Prerequisite: course 255A. Topics include spectral theory with applications to ordinary differential operators, eigenvalue problems for differential equations, generalized functions, and partial differential equations.
269A-269B-269C. Advanced Numerical Analysis. Lecture, three hours. Requisites: courses 115A, 135A, 151A-151B. Numerical solution for systems of ordinary differential equations; initial and boundary value problems. Numerical solution for elliptic, parabolic, and hyperbolic partial differential equations. Topics in computational linear algebra. S/U or letter grading.
270A-270F. Mathematical Aspects of Scientific Computing. Lecture, three hours. Requisites: courses 115A, 151A-151B, Program in Computing 10A. S/U or letter grading:
270A. Techniques of Scientific Computing. Mathematical modeling for computer applications, scientific programming languages, software development, graphics, implementation of numerical algorithms on different architectures, case studies.
270B-270C. Computational Linear Algebra. Direct, fast, and iterative algorithms, overdetermined systems; singular value decomposition, regularization, sparse systems, algebraic eigenvalue problem.
270D-270E. Computational Fluid Dynamics. Basic equations, finite difference, finite element, pseudo-spectral, and vortex methods; stability, accuracy, shock capturing, and boundary approximations.
270F. Parallel Numerical Algorithms. Prerequisites: courses 270B-270C. Recommended: courses 270A, 270D-270E. Design, analysis, and implementation of numerical algorithms on modern vector and parallel computers. Discussion of classical numerical algorithms and novel parallel algorithms. Emphasis on applications to PDEs.
271A. Tensor Analysis. Prerequisite: course 131A or consent of instructor. Algebra and calculus of tensors on n-dimensional manifolds. Curvilinear coordinates and coordinate-free methods. Covariant differentiation. Green/Stokes theorem for differential forms. Applications to topics such as continuum and particle mechanics.
271B. Analytical Mechanics. Prerequisites: course 271A, prior knowledge of mechanics. Newtonian and Lagrangian equations. Hamilton principle. Principle of least action. Holonomic and nonholonomic systems. Hamilton canonical equations, contact transformations, applications.
271C. Introduction to Relativity. Prerequisites: course 271A, prior knowledge of mechanics. Restricted theory of relativity. Extensions to general theory. Relativistic theory of gravitation.
271D. Wave Mechanics. General concepts of mechanical systems (states, space-time, "logics," etc.). Classical and quantum examples. Correspondence principle. Spinors.
272A. Foundations of Continuum Mechanics. Lec-ture, three hours. Prerequisite: consent of instructor. Kinematic preliminaries, conservation laws for mass, momentum and energy, entropy production, constitutive laws. Linear elasticity, inviscid fluid, viscous fluid. Basic theorems of fluid mechanics. Simple solutions. Low Reynolds number flow, Stokes drag. High Reynolds number flow, boundary layers. Two-dimensional potential flow, simple aerofoil. Compressible flow, shocks.
272B. Mathematical Aspects of Fluid Mechanics. Lecture, three hours. Prerequisite: course 272A or consent of instructor. Review of basic theory of mov-ing continua, fluid equations, integral theorems. Simple solutions, flow created by slowly moving bodies, flows where viscosity is negligible, vortices, boundary layers and their separation, water waves, ship waves, compressional waves, shock waves, turbulence theory (overview).
272C. Magnetohydrodynamics. Lecture, three hours. Prerequisites: course 272A, consent of instructor. Basic electromagnetism. Steady flows, Hartmann layers. Alfvén theorem and waves. Compressible media. Magnetostatic equilibria and stability.
272D. Rotating Fluids and Geophysical Fluid Dynamics. Lecture, three hours. Prerequisite: consent of instructor. Effects of Coriolis forces on fluid behavior. Inviscid flows, Taylor/Proudman theorem, Taylor columns, motions of bodies, inertial waves in spheres and spherical shells, Rossby waves. Ekman layers, spin-up. Shallow-water theory, wind-driven ocean circulation. Effects of stratification, Benard convection. Baroclinic instability, Eady model. S/U or letter grading.
273. Optimization, Calculus of Variations, and Control Theory. Prerequisite: consent of instructor. Application of abstract mathematical theory to optimization problems of calculus of variations and control theory. Abstract nonlinear programming and applications to control systems described by ordinary differential equations, partial differential equations, and functional differential equations. Dynamic programming.
274A. Asymptotic Methods. Lecture, three hours. Requisite: course 132. Fundamental mathematics of asymptotic analysis, asymptotic expansions of Fourier integrals, method of stationary phase. Watson lemma, method of steepest descent, uniform asymptotic expansions, elementary perturbation problems. S/U or letter grading.
274B-274C. Perturbation Methods. Lecture, three hours. Prerequisite: course 266A or equivalent. Boundary layer theory, matched asymptotic expansions, WKB theory. Problems with several time scales: Poincaré method, averaging techniques, multiple-scale analysis. Application to eigenvalue problems, nonlinear oscillations, wave propagation, and bifurcation problems. Examples from various fields of science and engineering.
275A-275B. Probability Theory. Prerequisite: course 245A or 265A. Connection between probability theory and real analysis. Weak and strong laws of large numbers, central limit theorem, conditioning, ergodic theory, martingale theory.
275C. Stochastic Processes. Lecture, three hours. Prerequisite: course 275B or consent of instructor. Brownian motion, continuous-time martingales, Markov processes, potential theory. S/U or letter grading.
275D. Stochastic Calculus. Lecture, three hours. Prerequisite: course 275C or consent of instructor. Stochastic integration, stochastic differential equations, Ito formula and its applications. S/U or letter grading.
275E. Stochastic Particle Systems. Lecture, three hours. Prerequisite: course 275C or consent of instructor. Interacting particle systems, including contact process, stochastic Ising model, and exclusion processes; percolation theory. S/U or letter grading.
276A-276B. Statistical Theory. Lecture, three hours. Prerequisite: Statistics 152C or consent of instructor. 276A. Sufficiency, exponential families, least squares, maximum likelihood estimation, Fisher information, Cramér/Rao inequality, confidence intervals. 276B. Asymptotic properties of tests and estimates, consistency and efficiency, likelihood ratio tests, chi-squared tests.
276C. Statistical Decision Theory. Prerequisite: course 276A. Invariant estimates and tests; best unbiased and locally best tests; multiple decision problems; application to general linear model; other topics.
277. Data Analysis. Lecture, three hours. Prerequisites: course 276A and Statistics M153A, or consent of instructor. Outline of principles of applied statistics, followed by survey of specific data analyses from physical, life, and social sciences. Methods include regression, analysis of variance and covariance, survival analysis, categorical data analysis, and simple time-series analysis. Illustration of transformations, plotting, model selection and evaluation, and estimation and decision procedures.
278A. Multivariate Analysis. Lecture, three hours. Prerequisite: course 276B or consent of instructor. Distributions in several dimensions, partial and multiple correlation. Normal distribution theory, Wishart distribution, Hotelling T2. Principal components, canonical correlation, discriminant analysis. Introduction to linear structural relations and factor analysis.
278B. Nonparametric and Robust Statistics. Lecture, three hours. Prerequisite: course 276B or consent of instructor. Development of nonparametric and robust procedures for hypothesis testing, estimation in one- and two-sample problems, linear and nonlinear regression, multiple classification, density estimation.
278C. Decision Theory. Lecture, three hours. Prerequisites: courses 131A and 276B, or consent of instructor. Bayes, admissible, and minimax decision rules. Invariant tests and estimates, best unbiased tests, locally best tests. Application to general linear model.
278D. Sequential Analysis. Lecture, three hours. Prerequisites: courses 131A and 276B, or consent of instructor. Bayes sequential decision problems, stopping rule problems, optimality of sequential probability ratio test, Wald identity, asymptotic theory, and other topics.
M279A-M279B. Linear Statistical Models. (Same as Biostatistics M250A-M250B.) Lecture, three hours; discussion, one hour. Preparation: one upper division three-term theoretical statistics course. Topics include linear algebra applied to linear statistical models, distribution of quadratic forms, Gauss/Markov theorem, fixed and random component models, balanced and unbalanced designs. S/U or letter grading.
M280. Statistical Computing. (Same as Biomathematics M280 and Biostatistics M280.) Lecture, three hours. Prerequisites: course 115A, Statistics 152C, or equivalent. Introduction to theory and design of statistical programs: computing methods for linear and nonlinear regression, dealing with constraints, robust estimation, and general maximum likelihood methods.
285A-285N. Seminars. (Formerly numbered 285A-285L.) Lecture, three hours. Prerequisite: consent of instructor. No more than two 285 courses may be applied toward M.A. degree requirements except by prior consent of graduate vice chair. Topics in various branches of mathematics and their applications by means of lectures and informal conferences with staff members. S/U or letter grading:
285A. History and Development of Mathematics.
285B. Number Theory.
285C. Algebra.
285D. Logic.
285E. Geometry.
285F. Topology.
285G. Analysis.
285H. Differential Equations.
285I. Functional Analysis.
285J. Applied Mathematics.
285K. Probability.
285L. Statistics.
285N. Dynamical Systems
290. Seminar: Current Literature. Intended for Ph.D. candidates. Readings and presentations of papers in mathematical literature under supervision of a staff member.
296A-296N. Participating Seminars (1 to 4 units each). (Formerly numbered 296A-296M.) Seminars and discussion by staff and students. S/U grading:
296A. History and Development of Mathematics.
296B. Number Theory.
296C. Algebra.
296D. Logic.
296E. Geometry.
296F. Topology.
296G. Analysis.
296H. Differential Equations.
296I. Functional Analysis.
296J. Applied Mathematics.
296K. Probability.
296L. Statistics.
296M. Mathematics.
296N. Dynamical Systems.
370A-370B. Teaching of Mathematics. (Formerly numbered 370.) Lecture, three hours; discussion, one hour. Prerequisites: course 33B, upper division standing. Course 370A is prerequisite to 370B. Topics in geometry, algebra, number theory, discrete mathematics, and functions presented from a problem-solving and student participation point of view, with emphasis on historical context and appropriate role of proof. S/U or letter grading.
375. Teaching Apprentice Practicum (1 to 4 units). Preparation: apprentice personnel employment as a teaching assistant, associate, or fellow. Teaching apprenticeship under active guidance and supervision of a regular faculty member responsible for curriculum and instruction at the University. May be repeated for credit. S/U grading.
495. Teaching College Mathematics (2 units). Discussion, one hour; two-day intensive training at beginning of Fall Quarter. Required of all new teaching assistants and new doctoral students. Special course for teaching assistants designed to deal with problems and techniques of teaching college mathematics. S/U grading.
501. Cooperative Program (2 to 8 units). Prerequisite: consent of UCLA department chair and graduate dean, and host campus instructor, department chair, and graduate dean. Used to record enrollment of UCLA students in courses taken under cooperative arrangements with USC. S/U grading.
596. Directed Individual Study or Research (2 to 8 units). Supervised individual reading and study on project approved by a faculty member, which may be preparation for M.A. examination. May be repeated for credit, but only two 596 courses (eight units) may be applied toward M.A. degree unless departmental consent is obtained.
599. Research in Mathematics (2 to 12 units). Prerequisite: advancement to doctoral candidacy. Study and research for Ph.D. dissertation. May be repeated for credit.