MATH-MATHEMATICS -- 2003-05 Catalog
Mathematics Department

Satisfactory completion of the Entry Level Mathematics (ELM) requirement is a prerequisite for enrollment in all mathematics courses except MATH 100 and MATH 104.

MATH 100 Beginning Algebra Review (3) (CR/NC)

Review of basic algebra skills at the beginning algebra level intended primarily to prepare students for MATH 104. Course open only to students who have taken the ELM examination and are not qualified for MATH 104. Not for baccalaureate credit. Credit/No Credit grading only. 3 lectures. Prerequisite: Two years of high school algebra.

MATH 104 Intermediate Algebra (3) (CR/NC)

Review of basic algebra skills at the intermediate algebra level intended primarily to prepare students for MATH 116. Not for baccalaureate credit. Credit/No Credit grading only. 3 lectures. Prerequisite: Two years high school algebra and appropriate score on the ELM examination, or credit in MATH 100.

MATH 110 Beginning Algebra Laboratory (1) (CR/NC)

Facilitated study and discussion of the theory, problems, and applications of beginning algebra. Not for baccalaureate credit. Credit/No Credit grading only. 1 laboratory. Corequisite: Concurrent enrollment in the associated section of MATH 100.

MATH 112 The Nature of Modern Mathematics (4) GE B1

Topics from contemporary mathematics, their development, applications, and role in society. Some typical topics, to be chosen by the instructor: graph theory, critical path analysis, statistical inference, coding, game theory, and symmetry. 4 lectures. Prerequisite: Appropriate score on ELM examination or an ELM exemption or credit in MATH 104.

MATH 114 Intermediate Algebra Laboratory (1) (CR/NC)

Facilitated study and discussion of the theory, problems, and applications of intermediate algebra. Not for baccalaureate credit. Credit/No Credit grading only. 1 laboratory. Corequisite: Concurrent enrollment in the associated section of MATH 104.

1 MATH 116, 117 Pre-Calculus Algebra I, II (3) (3)
For MATH 116 and 117: GE B1

Pre-calculus college algebra without trigonometry. Special products and factoring, exponents and radicals. Fractional and polynomial equations. Matrices, determinants, and systems of equations. Polynomial, rational, exponential, and logarithmic functions. Graphing, inequalities, absolute value, and complex numbers. MATH 116 and MATH 117 are equivalent to MATH 118, but are taught at a slower pace. Upon completion of MATH 116 and MATH 117, a student will receive 4 units of GE credit for Area B1. Not open to students with credit in MATH 118 or MATH 120. 3 lectures. MATH 116 prerequisite: Appropriate score on ELM examination, or an ELM exemption, or credit in MATH 104. MATH 117 prerequisite: MATH 116 with a grade of C- or better or consent of instructor.

MATH 118 Pre-Calculus Algebra (4) GE B1

Pre-calculus algebra without trigonometry. Special products and factoring, exponents and radicals. Fractional and polynomial equations. Matrices, determinants, and systems of equations. Polynomial, rational, exponential, and logarithmic functions. Graphing, inequalities, absolute value, and complex numbers. MATH 118 is equivalent to MATH 116 and MATH 117. Not open to students with credit in MATH 117 or MATH 120. 4 lectures. Prerequisite: Appropriate score on ELM examination, or an appropriate ELM exemption.

MATH 119 Pre-Calculus Trigonometry (4) GE B1

Rectangular and polar coordinates. Trigonometric functions, fundamental identities. Inverse trigonometric functions and relations. Vectors, complex numbers, conic sections, and analytic geometry. Not open to students with credit in MATH 120. 4 lectures. Prerequisite: Completion of ELM requirement and passing score on appropriate Mathematics Placement Examination, or MATH 117, or MATH 118 or equivalent.

MATH 120 Pre-Calculus Algebra and Trigonometry (5) GE B1

An integrated review course in pre-calculus algebra and trigonometry covering function concepts and symbols, rectangular coordinates, linear, quadratic, polynomial, and rational functions, inequalities, trigonometric functions, inverse trigonometric functions, exponential and logarithmic functions, systems of equations, complex numbers, and analytic geometry. MATH 120 is equivalent to MATH 118 and MATH 119. Not open to students with credit in MATH 117, MATH 118, or MATH 119. 5 lectures. Prerequisite: Completion of ELM requirement and passing score on appropriate Mathematics Placement Examination and high school trigonometry or equivalent.

MATH 126 Pre-Calculus Algebra I Laboratory (1) (CR/NC)

Facilitated study and discussion of the theory, problems, and applications of pre-calculus algebra. Credit/No Credit grading only. 1 laboratory. Co-requisite: Concurrent enrollment in the associated section of MATH 116.

MATH 127 Pre-Calculus Algebra II Laboratory (1) (CR/NC)

Facilitated study and discussion of the theory, problems, and applications of pre-calculus algebra. Credit/No Credit grading only. 1 laboratory. Co-requisite: Concurrent enrollment in the associated section of MATH 117.

MATH 128 Pre-Calculus Algebra Laboratory (1) (CR/NC)

Facilitated study and discussion of the theory, problems, and applications of pre-calculus algebra. Credit/No Credit grading only. 1 laboratory. Co-requisite: Concurrent enrollment in the associated section of MATH 118.

MATH 129 Pre-Calculus Trigonometry Laboratory (1) (CR/NC)

Facilitated study and discussion of the theory, problems, and applications of pre-calculus trigonometry. Credit/No Credit grading only. 1 laboratory. Corequisite: Concurrent enrollment in the associated section of MATH 119.

1 MATH 131, 132, 133 Technical Calculus (4) (4) (4)

Functions, their graphs and limits; techniques and applications of differential and integral calculus; introduction to applied differential

equations. Designed principally for technology students and others interested in an applied three-quarter calculus sequence. Not open to students with credit in MATH 142, MATH 143, MATH 318 (respectively) or equivalents. 4 lectures. Prerequisite: ELM requirement, and passing score on Mathematics Placement Examination, or MATH 118 and MATH 119, or equivalent.

1 MATH 141, 142, 143 Calculus I, II, III (4) (4) (4) GE B1
(Also listed as HNRS 141, 142, 143)

Limits, continuity, differentiation, integration. Techniques of integration, applications to physics, transcendental functions. Infinite sequences and series, vector algebra, curves. 4 lectures. MATH 141 prerequisite: Completion of ELM requirement and passing score on appropriate Mathematics Placement Examination, or MATH 118 and MATH 119 or equivalent. MATH 142 prerequisite: MATH 141 with a grade of C- or better or consent of instructor. MATH 143 prerequisite: MATH 142.

MATH 151, 152, 153 Calculus Laboratories I, II, III (1) (1) (1) (CR/NC)

Facilitated study and discussion of the theory, problems, and applications of calculus. Credit/No Credit grading only. 1 laboratory. Corequisite: Concurrent enrollment in the associated section of MATH 141, MATH 142, or MATH 143.

MATH 161, 162 Calculus for the Life Sciences I, II (4) (4) GE B1

Review of exponential, logarithmic, and trigonometric functions. Differential and integral calculus with applications to the biological sciences. Introduction to differential equations and mathematical modeling. Examples, exercises and applications to emphasize problems in life sciences. Not open to students with credit in MATH 141, 142 respectively. 4 lectures. Prerequisite: Completion of ELM requirement and passing score on appropriate Mathematics Placement Examination, or MATH 118 or equivalent.

MATH 182 Calculus for Architecture and Construction Management (4) GE B1

Integral calculus with applications to architecture and construction management. The algebra of vectors. Polar, cylindrical, and spherical coordinate systems. Not open to students with credit in MATH 142. 4 lectures. Prerequisite: MATH 141 or equivalent.

MATH 202 Orientation to the Mathematics Major (1) (CR/NC)

Career opportunities in the field of mathematics, preparing a field of study, and a survey of departmental facilities and procedures related to research, study and graduation. Credit/No Credit grading only. 1 lecture. Corequisite: Sophomore standing or consent of instructor.

MATH 206 Linear Algebra I (4)

Matrices, inverses, linear systems, determinants, eigenvalues, eigenvectors, vector spaces, linear transformations, applications. 4 lectures. Prerequisite: MATH 143 or consent of instructor.

MATH 211, 212 Computational Mathematics I, II (4) (4)

Fundamentals of procedural programming in C/C++ and selected applications to problems in integral and differential calculus, matrix analysis, geometry, and physics. 4 lectures. Prerequisite: MATH 141 or consent of instructor.

MATH 221 Calculus for Business and Economics (4) GE B1

Polynomial calculus for optimization and marginal analysis, and elementary integration. Not open to students with credit in MATH 142. 4 lectures. Prerequisite: Completion of ELM requirement and passing score on appropriate Mathematics Placement Examination, or MATH 118 or equivalent.

MATH 231 Calculus for Business and Economics Laboratory (1) (CR/NC)

Facilitated study and discussion of the theory, problems, and applications of business calculus. Credit/No Credit grading only. 1 laboratory. Coreq-uisite: Concurrent enrollment in the associated section of MATH 221.

MATH 241 Calculus IV (4)

Partial derivatives, multiple integrals, introduction to vector analysis. 4 lectures. Prerequisite: MATH 143.

MATH 242 Differential Equations (4)

Ordinary differential equations: introduction with selected applications; classification of equations and their analytic solutions; study of interrelationships between differential systems, graphs, and physical problems. Systems of ordinary differential equations. Not open to CENG students. 4 lectures. Prerequisite: MATH 241.

MATH 244 Linear Analysis I (4)

Separable and linear ordinary differential equations with selected applications; numerical and analytical solutions. Linear algebra: vectors in n-space, matrices, linear transformations, eigenvalues, eigenvectors, diagonalization; applications to the study of systems of linear differential equations. 4 lectures. Prerequisite: MATH 143 or consent of instructor.

MATH 248 Methods of Proof in Mathematics (4)

Methods of proof (direct, contradiction, conditional, contraposition); valid and invalid arguments. Examples from set theory. Quantified statements and their negations. Functions, indexed sets, set functions. Proofs in number theory, algebra, geometry and analysis. Proof by induction. Equivalence and well-defined operations and functions. The axiomatic method. 4 lectures. Prerequisite: MATH 143 or consent of instructor.

MATH 300 Technology in Mathematics Education (4)

Examination of existing hardware and software designed for educational uses. Discussion of mathematical topics appropriate for computer enhancement. Special methods and techniques for educational uses of computers. Emphasis on activity learning and applications. Computer as a classroom management device. 4 lectures. Prerequisite: MATH 141 or MATH 329, and a course in CSC or MATH 211, or consent of instructor.

MATH 304 Vector Analysis (4) GE B6

Algebra of free vectors with applications. Differential and integral calculus of vectors. Development of theory and application of vector operations. Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. 4 lectures. Prerequisite: MATH 241 or consent of instructor.

MATH 306 Linear Algebra II (4)

Inner product spaces, orthogonality, Fourier series and orthogonal bases, linear transformations and similarity, eigenvalues and diagonalization. 4 lectures. Prerequisite: MATH 206 and MATH 242, or MATH 241 and MATH 244, and a C- or better in MATH 248, or consent of instructor.

1 MATH 327, 328, 329 Mathematics for Elementary Teaching I, II, III (4) (4) (4)

Introduction to set theory, number theory, real numbers, probability, statistics, and geometry. Computer applications. 2 lectures, 2 activities. MATH 327 prerequisite: Completion of ELM requirement, and passing score on appropriate Mathematics Placement Examination, or MATH 118 or equivalent. MATH 328 prerequisite: MATH 327 with a grade of C- or better or consent of instructor. MATH 329 prerequisite: MATH 328.

MATH 331 Topics in Mathematics for Teachers (1-6) (CR/NC)

Topics in mathematics for practicing credentialed teachers. Content will vary according to teaching level. Class Schedule will list topic selected. Total credit limited to 12 units. Credit/No Credit grading only. 1-6 activities. Prerequisite: Multiple Subject or Single Subject teaching credential or consent of instructor.

MATH 333 Numerical Analysis I (4)

Topics in interpolation and approximation methods, initial value problems, and boundary value problems of ordinary differential equations. 4 lectures. Prerequisite: MATH 242, or MATH 241 and MATH 244, or consent of instructor.

MATH 335 Graph Theory (4)

Introduction to graph theory and its applications: isomorphism, paths and searching, connectedness, trees, tournaments, planarity, graph colorings, matching theory, network flow, adjacency and incidence matrices. Further topics to be selected from the theory of finite state machines, Ramsey theory, extremal theory, and graphical enumeration. 4 lectures. Prerequisite: Junior standing or consent of instructor.

MATH 336 Combinatorial Mathematics (4)

Methods of enumerative combinatorics: sum, product, and division rules, bijective and recursive techniques, inclusion and exclusion, generating functions, and the finite difference calculus. Advanced topics to be selected from the theory of partitions, Polya theory, designs, and codes. 4 lectures. Prerequisite: Junior standing or consent of instructor.

MATH 341 Theory of Numbers (4)

Properties of numbers. Euclid’s Algorithm, greatest common divisors, diophantine equations, prime numbers, congruences, number theoretic

functions, the quadratic reciprocity laws, primitive roots and indices. 4 lectures. Prerequisite: MATH 248 with a grade of C- or better or consent of instructor.

MATH 344 Linear Analysis II (4) GE B6

Linear methods applied to the solution of differential equations. Laplace transforms. Series solutions to ordinary differential equations. Orthogonality in n-space, Gram-Schmidt orthogonalization and least squares methods. Orthogonal bases in function spaces, Sturm-Liouville theory. Fourier series and transforms. Special functions of applied mathematics. 4 lectures. Prerequisite: MATH 206 and MATH 242, or MATH 241 and MATH 244, or consent of instructor.

MATH 350 Mathematica (4)

Problem-solving using Mathematica in a UNIX environment. 4 lectures. Prerequisite: MATH 241.

MATH 370 Putnam Exam Seminar (2)

Directed group study of mathematical problem solving techniques. Open to undergraduate students only. Class members are expected to participate in the annual William Lowell Putnam Mathematical Competition. Course may be repeated up to eight units. 2 seminars. Prerequisite: Consent of instructor.

MATH 371 Math Modeling Seminar (2)

Directed group study of mathematical modeling techniques. Open to undergraduate students only. Class members are

expected to participate in the annual Mathematical Competition in Modeling. Total credit limited to 8 units. 2 seminars. Prerequisite: Consent of instructor.

MATH 372 Mathematical Community Service Projects (2) (CR/NC)

Directed group mathematical research in support of volunteer community service projects. Total credit limited to 8 units. 2 seminars. Prerequisite: consent of instructor and consent of department chair.

MATH 400 Special Problems for Advanced Undergraduates (1-4)

Individual investigation, research, studies, or surveys of selected problems. Total credit limited to 8 units. Prerequisite: Junior standing and consent of department chair.

MATH 404 Introduction to Differential Geometry and Topology (4)

Theory of curves and surfaces in space. Topics such as curvature, geodesics, Gauss map, Gauss-Bonnet Theorem, combinatorial topology, point set topology. 4 lectures. Prerequisite: MATH 206 or MATH 244, and MATH 304, or consent of instructor.

MATH 406 Linear Algebra III (4)

Complex vector spaces, unitary and self-adjoint matrices, Spectral Theorem, Jordan canonical form. Selected topics in linear programming, convexity, numerical methods, and functional analysis. 4 lectures. Prerequisite: MATH 306 or consent of instructor.

1 MATH 408, 409 Complex Analysis I, II (4) (4) MATH 408: GE B6

Elementary analytic functions and mappings. Cauchy’s Integral Theorem; Poisson’s Integral Formula. Taylor and Laurent series, theory of residues, and the evaluation of integrals. Harmonic functions, conformal mappings. 4 lectures. Prerequisite: MATH 242, or MATH 241 and MATH 244, or consent of instructor.

MATH 412 Introduction to Analysis I (4)

Introduction to concepts and methods basic to real analysis. Topics such as the real number system, sequences, continuity, uniform continuity and differentiation. 4 lectures. Prerequisite: MATH 248 with a grade of C- or better or consent of instructor. Strongly recommended: successful completion of MATH 306, MATH 341, MATH 442, or MATH 481.

1 MATH 413, 414 Introduction to Analysis II, III (4) (4)

A continuation of Introduction to Analysis I covering such topics as integration, infinite series, uniform convergence and functions of several variables. Highly recommended for students planning to enter graduate programs or secondary teaching and those interested in applied mathematics. 4 lectures. Prerequisite: MATH 206 or MATH 244, and MATH 412, or consent of instructor.

MATH 417 Introduction to Dynamical Systems (4)

Theory of dynamical systems in one and two dimensions. Topics such as bifurcation theory, chaos, attractors, limit cycles, nonlinear dynamics. 4 lectures. Prerequisite: MATH 242, or MATH 241 and MATH 244, or consent of instructor.

MATH 418 Partial Differential Equations (4)

Mathematical formulation of physical laws. Separation of variables. Orthogonal functions and generalized Fourier series. Bessel functions, Legendre polynomials. Sturm-Liouville problem. Boundary value problems; nonhomogeneous techniques. Applications to heat flow, potential theory, vibrating strings and membranes. 4 lectures. Prerequisite: MATH 344 or consent of instructor.

MATH 419 Introduction to the History of Mathematics (4)

Evolution of mathematics from earliest to modern times. Major trends in mathematical thought, the interplay of mathematical and technological innovations, and the contributions of great mathematicians. Appropriate for prospective and in-service teachers. 4 lectures. Prerequisite: MATH 248 with a grade of C- or better and at least one upper division course in mathematics, or consent of instructor.

MATH 424 Organizing and Teaching Mathematics (4)

Organization, selection, presentation, application and interpretation of subject matter in mathematics. Introduction to current issues in mathematics education. For students who will be teaching in secondary schools. 4 lectures. Prerequisite: Senior standing or consent of instructor.

1 MATH 431, 432 Mathematical Optimization I, II (4) (4)

Classical optimization. Maximum/minimum of functions, linear and nonlinear optimization problems, duality, constrained optimization. Model building and applications to various fields. 4 lectures. Prerequisite: MATH 206 or MATH 244, and MATH 241, or consent of instructor.

MATH 433 Numerical Analysis II (4)

Numerical techniques for solving partial differential equations of the parabolic, hyperbolic and elliptic type. 4 lectures. Prerequisite: MATH 333 or equivalent.

MATH 437 Game Theory (4)

Development of the mathematical concepts, techniques, and models used to investigate optimal strategies in competitive situations; games in extensive, normal, and characteristic form, Nash equilibrium points and

Nash Bargaining Model. 4 lectures. Prerequisite: MATH 206 or MATH 244 or consent of instructor. Recommended: MATH 335 and MATH 431.

MATH 442 Euclidean Geometry (4)

Foundations of Euclidean geometry, finite geometries, congruence, similarities, polygonal regions, circles and spheres. Constructions, mensuration, the parallel postulate. Appropriate for prospective and in-service mathematics teachers. 4 lectures. Prerequisite: MATH 248 with a grade of C- or better or consent of instructor.

MATH 443 Modern Geometries (4)

Non-Euclidean and projective geometries. Properties of parallels, biangles, Saccheri and Lambert quadrilaterals, angle-sum and area. Limiting curves, hyperbolic trigonometry, duality, perspectivity, quadrangles, fundamental theorems of projective geometry, conics. 4 lectures. Prerequisite: MATH 442.

MATH 459 Undergraduate Seminar (4)

Written and oral analysis and presentations by students on topics from mathematical modeling. 4 seminars. Prerequisite: Junior standing, MATH 206 and MATH 242, or MATH 241 and MATH 244, or consent of instructor.

1 MATH 461, 462 Senior Project (2) (2)

Selection and completion of a project under faculty supervision. Projects typical of problems which graduates must solve in their fields of employment. Project results are presented in a formal report. Minimum 120 hours total time. Prerequisite: MATH 459.

MATH 470 Selected Advanced Topics (1-4)

Directed group study of selected topics for advanced students. Open to undergraduate and graduate students. Class Schedule will list topic selected. Total credit limited to 8 units. 1 to 4 lectures. Prerequisite: Junior standing and consent of instructor.

1 MATH 481, 482 Abstract Algebra I, II (4) (4)

Fundamental algebraic structures and types of algebras, including operations within them and relations among them. Groups, rings and fields. 4 lectures. Prerequisite: MATH 248 with a grade of C- or better or consent of instructor. Strongly recommended: successful completion of MATH 306, MATH 341, MATH 412, or MATH 442.

MATH 485 Cooperative Education Experience (6) (CR/NC)

Part-time work experience in business, industry, government, and other areas of student career interest. Positions are paid and usually require relocation and registration in course for two consecutive quarters. Formal report and evaluation by work supervisor required. Total credit limited to 16 units. Credit/No Credit grading only. Prerequisite: Sophomore standing and consent of instructor.

MATH 495 Cooperative Education Experience (12) (CR/NC)

Full-time work experience in business, industry, government, and other areas of student career interest. Positions are paid and usually require relocation and registration in course for two consecutive quarters. Formal report and evaluation by work supervisor required. Total credit limited to 16 units. Credit/No Credit grading only. Prerequisite: Sophomore standing and consent of instructor.

MATH 500 Individual Study (1-4)

Individual research or advanced study planned and completed under the direction of a departmental faculty member. Open only to graduate students demonstrating ability to do independent work. Prerequisite: Graduate standing and consent of department chair.

1 MATH 501, 502 Methods of Applied Mathematics I, II (4) (4)

Introduction to advanced methods of mathematics useful in the analysis of engineering problems. Theory of vector fields, Fourier analysis, Sturm-Liouville theory, functions of a complex variable. Selected topics in asymptotic analysis, special functions, perturbation theory. Not open to students in major or master’s degree program in mathematics. MATH 501: Distance Learning Lab fee may be required–see Class Schedule. 4 lectures. Prerequisite: MATH 344 or equivalent, and graduate standing or consent of instructor.

MATH 510 Survey of Modern Mathematics (4)

Selected topics from the field of modern mathematics. Projective and synthetic geometry, topology, logic, matrices, vectors. Theory of games, probability, linear and modern algebra and convex sets. Boolean algebras, graph theory, lattice theory, geometry of complex numbers. 4 seminars. Prerequisite: Graduate standing or consent of instructor.

1 MATH 520, 521 Applied Analysis I, II (4) (4)

Advanced mathematical methods of analysis in science and engineering, integrated with modeling of physical phenomena. Topics include applications of complex analysis, Fourier analysis, ordinary and partial differential equations. Additional topics to be drawn from perturbation methods, asymptotic analysis, dynamical systems, numerical methods, optimization, and the calculus of variations. 4 lectures. Prerequisite: MATH 408, MATH 412, MATH 418 and graduate standing, or consent of the instructor.

MATH 522 Applied Analysis III (4)

Selected topics in applied analysis. 4 lectures. Prerequisite: MATH 521 and graduate standing, or consent of the instructor.

1 MATH 530, 531 Discrete Mathematics with Applications I, II (4) (4)

Advanced mathematical methods of discrete mathematics with applications. Topics will include probability theory with generating functions, difference equations and number theory. Additional topics to be drawn from the theory of algorithms, coding theory, set theory, and the relation of discrete mathematics to complex analysis. 4 lectures. Prerequisite: MATH 481, MATH 306 and graduate standing, or consent of instructor.

MATH 532 Discrete Mathematics with Applications III (4)

Selected advanced topics in discrete mathematics. These topics may include foundations, numerical and computational methods of discrete mathematics, finite geometries or current problems in discrete mathematics. 4 lectures. Prerequisite: MATH 531 and graduate standing, or consent of the instructor.

MATH 540 Introduction to Topology (4)

Basic ideas of general topology, metric spaces, homeomorphisms and the separation axioms. 4 seminars. Prerequisite: Satisfactory completion of the Graduate Written Examination in Analysis or consent of the Graduate Committee.

MATH 550 Real Analysis (4)

Introduction to Lebesgue measure and integration, convergence theorems, L1 spaces, Radon-Nikodym Theorem and Fubini’s Theorem. 4 seminars. Prerequisite: Satisfactory completion of the Graduate Written Examination in Analysis or consent of the Graduate Committee.

MATH 560 Field Theory (4)

Polynomial rings, field extensions, normal and separable extensions, automorphisms of fields, fundamental theorem of Galois theory, solvable groups, solution by radicals, insolvability of the quintic. 4 lectures. Prerequisite: Satisfactory completion of the Graduate Written Examination in Algebra or consent of the Graduate Committee.

MATH 580 Seminar (1–4)

Built around topics in advanced mathematics chosen according to the common interests and needs of the students enrolled. Each seminar will have a subtitle according to the nature of the content. Total credit limited to 12 units. 1–4 seminars. Prerequisite: Graduate standing and consent of instructor.

MATH 596 Thesis (3)

Serious research endeavor devoted to the development, pedagogy or learning of mathematics. Course to be taken twice for a total of 6 units. Prerequisite: Graduate standing and consent of instructor.

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1 Each course in a combined listing of sequentially numbered courses is a prerequisite to its successor in the same listing.