The University of Arizona 199395 General Catalog Catalog Home All UA Catalogs UA Home

Mathematics (MATH) Mathematics Building, Room 108 (520) 6216892 Professors Alan C. Newell, Head, Clark T. Benson, John Brillhart, M.S. Cheema (Emeritus), James R. Clay, Jim M. Cushing, Jack L. Denny (Statistics), William G. Faris, Hermann Flaschka, W.M. Greenlee, Helmut Groemer, Larry C. Grove, Deborah Hughes Hallet, George L. Lamb (Optical Sciences), C. David Levermore, Peter Li, David O. Lomen, John S. Lomont (Emeritus), David Lovelock, Henry B. Mann (Emeritus), Warren May, Jerry Moloney, Donald E. Myers, Yves Pomeau, Alwyn C. Scott, Moshe Shaked, Arthur Steinbrenner (Emeritus), Michael Tabor, Elias Toubassi, William Y. Velez, Stephen S. Willoughby, LaiSang Young, Vladimir Zakharov Associate Professors William E. Conway, Carl L. DeVito, Nicholas M. Ercolani, David Gay, Oma Hamara, Thomas G. Kennedy, Theodore W. Laetsch, Daniel Madden, William G. McCallum, John N. Palmer, Douglas M. Pickrell, Wayne Raskind, ZhenSu She, Frederick W. Stevenson, Richard B. Thompson, Maciej P. Wojtkowski, Bruce Wood, A. Larry Wright (Statistics) Assistant Professors Bruce J. Bayly, Moysey Brio, Kwok Chow, Marta Civil, Samuel Evens, Paul Fan, Leonid Friedlander, Robert S. Maier, Wayne M. Raskind, Marek Rychlik, Douglas Ulmer, Jan Wehr, Xue Xin Lecturers Robert C. Dillon (Emeritus), John L. Leonard, Stephen G. Tellman Mathematics forms a foundation for all technical disciplines and is an excellent preparation for a career or graduate study in many subjects. The department offers courses in pure mathematics, applied mathematics, probability and statistics, computational mathematics, engineering mathematics and mathematics education. Planned minors in numerous professional fields are available; interested persons should consult with a Mathematics Department advisor to help choose the option, minor, and additional course work that best prepares for their chosen career. Mathematics is available as a major for the following degrees: Bachelor of Arts and Bachelor of Science (College of Arts and Sciences), Bachelor of Science in Engineering Mathematics (College of Engineering and Mines), Bachelor of Arts in Education (Elementary EducationCollege of Education) and Bachelor of Science in Education (Secondary EducationCollege of Education), Master of Arts, Master of Science, Master of Education and Doctor of Philosophy. The major for the B.A. and B.S. consists of a core of basic courses and one of five possible options. It must include 33 units in mathematics courses numbered 124 or above. The core courses are C SC 115, MATH 124 or 125a, 125b, 215, 223, 323, and 355. Advanced students need not take lower numbered courses. The comprehensive mathematics option: The core above and 413, 415, 424, and 425. The industrial and applied mathematics option: One of the sequences 454455, 454456, 464466a, or 475a475b; either 424 or 425; one of 410, 413, or 415. The computational science option: Either of the sequences 415, 416 or 475a, 475b; one of 443, 447, or 479; and one more of the above courses or 413. The probability and statistics option: 425, 464, 466a and either 413 or 468. The economics and finance option: 425, 464, either 410 or 413, one of 426, 466a or 479. The minor must be in either economics or finance. The economics minor should consist of ECON 200 or 210; 361 or 411; 300; and 12 additional upperdivision units in economics. The finance minor should consist of ACCT 200 and 210; either ECON 201a201b or 210; FIN 311 and 421; plus six additional upperdivision units in finance. The minor in mathematics with the College of Arts and Sciences: A minimum of 20 units including 124 or 125a, 125b, 215 and at least nine additional upperdivision units. The mathematics education option for the Bachelor of Science in Education (College of Education: The core above and 397, 405, either 315 or 415, either 362 or 464, and either 330 or 430. In order to be accepted into the secondary teacher preparation program of the College of Education with a major in mathematics, a student must have successfully completed the following four mathematics courses: 124 or 125a, 125b, 223 and 215. Furthermore, students who do not have a GPA of 2.5 in those four courses may not enroll in 315, 330 or 397 without special permission. The elementary education major area of specialization: 301 plus 12 units selected in consultation with a mathematics department advisor. The engineering mathematics major: Requirements are given in the College of Engineering section. Prerequisites: Because of the nature of mathematics, the department recommends that students refrain from enrolling in any course that carries prerequisites unless those prerequisites have been completed with a grade of "C" or better. Students without university credit in the prerequisites for 117R, 117S, 118, 119, 121, 123, 124, 125a will be required to have an appropriate score on the math readiness test to be enrolled in these courses. The department strongly recommends that students not enroll in any prerequisite for courses in which they have already received credit. Students must have proof of having taken the math readiness test in order to register for mathematics courses numbered below 125b. Test scores are valid for one year. The department participates in the honors program. 101. Survey of Mathematical Thought (3) A study of the nature of mathematics and its role in civilization, utilizing historical approaches and computational examples. Not applicable to the mathematics major. P, fulfillment of university entrance requirements in math without deficiency. 116R.2 Introduction to College Algebra (3) I II Lecture. Not applicable to the mathematics major or minor. Basic concepts of algebra, linear equations and inequalities, relations and functions, quadratic equations, system of equations. P, two entrance units in algebra or an acceptable score on the math readiness test. 116S.2 Introduction to College Algebra (3) I II SelfStudy. Identical to MATH 116R except taught in a selfstudy tutorial format. Not applicable to the mathematics major or minor. P, two entrance units in algebra or an acceptable score on the math readiness test. 117R.2,4 College Algebra (3) I II Lecture. Not applicable to the mathematics major or minor. Brief review and continuation of MATH 116R/S, functions, mathematical models, systems of equations and inequalities, exponential and logarithmic functions, polynomial and rational functions, sequences and series. Students with credit in 120 will obtain only one unit of graduation credit for 117R. P, 116R or 116S or an acceptable score on the math readiness test. 117S.2,4 College Algebra (3) I II SelfStudy. Identical to MATH 117R except taught in a selfstudy tutorial format. Not applicable to mathematics majors or minors. Students with credit in 120 will obtain only one unit of graduation credit for 117S. P, 116R or 116S or an acceptable score on the math readiness test. 118.1,2 Plane Trigonometry (2) I II Not applicable to the mathematics major or minor. Students with credit in 120 will obtain one unit of graduation credit for 118. P, one entrance unit in geometry, and either 1 1/2 entrance units in algebra, or 116R/S. 119.1 Finite Mathematics (3) I II Elements of set theory and counting techniques, probability theory, linear systems of equations, matrix algebra; linear programming with simplex method, Markov chains. P, 117R/S or an acceptable score on the math readiness test. 120.1,2,4 Calculus Prep (3) I II S Reviews manipulative algebra and trigonometry; covers uses of functional notation, partial fraction decomposition and analytic geometry. For students who have high school credit in college algebra and trigonometry but have not attained a sufficient score on the math readiness test to enter calculus. Students with credit in MATH 117R/S will obtain only one unit of graduation credit. Students with credit in 118 will obtain two units of graduation credit. P, high school credit in college algebra and trigonometry, and an acceptable score on the math readiness test. Graphing calculator will be required in this course. Graphing calculator will be required in this course. 121. Basic Mathematical Procedures (3) I II S Evaluating mathematical expressions, introduction to basic programming, right triangle trigonometry, exponents and logarithms, probability and introduction to statistics. P, 116R/S. 123.2 Elements of Calculus (3) I II Introductory topics in differential and integral calculus. P, 117R/S or an acceptable score on the math readiness test. 124.2,4 Calculus with Applications (5) Introduction to calculus with an emphasis on understanding and problem solving. Concepts are presented graphically and numerically as well as algebraically. Elementary functions, their properties and uses in modelling; the key concepts of derivative and definite integral; techniques of differentiation, using the derivative to understand the behavior of functions; applications to optimization problems in physics, biology and economics. Graphing calculator will be required for this course. Credit allowed for 124 or 125a, but not both. P, 120 or 117R/S and 118, or acceptable score on math readiness test. 125a.2,4 Calculus (3) An accelerated version of 124. Introduction to calculus with an emphasis on understanding and problem solving. Concepts are presented graphically and numerically as well as algebraically. Elementary functions, their properties and uses in modelling; the key concepts of derivative and definite integral; techniques of differentiation, using the derivative to understand the behavior of functions; applications to optimization problems in physics, biology and economics. Graphing calculator will be required in this course. P, an acceptable score on math readinesss test. Credit allowed for 124 or 125a, but not both. 125b.4 Calculus (3) Continuation of 124 or 125a. Techniques of symbolic and numerical integration, applications of the definite integral to geometry, physics, economics, and probability; differential equations from a numerical, graphical, and algebraic point of view; modelling using differential equations, approximations by Taylor series. Graphing calculator will be required in this course. P, 124 or 125a. 129. Calculus with a Computer (2) II Designed to supplement regular calculus courses. The use of computers to solve calculus problems emphasizing numerical and geometrical understanding of calculus. P or CR, 125b. 200. ProblemSolving Laboratory (1) [Rpt./4] I II Development of creative, mathematical, problemsolving skills, with challenging problems taken from calculus, elementary number theory and geometry. P, 125b. 202. Introduction to Symbolic Logic (3) (Identical with PHIL 202) 215. Introduction to Linear Algebra (3) I II Vector spaces, linear transformations and matrices. P, 125b. 223. Vector Calculus (4) I II Vectors, differential and integral calculus of several variables. P. 125b. 243. Discrete Mathematics in Computer Science (3) I II Set theory, logic, algebraic structures; induction and recursion; graphs and networks. P, 125b. 254.4 Introduction to Ordinary Differential Equations (3) I II Solution methods for ordinary differential equations, qualitative techniques; includes matrix methods approach to systems of linear equations and series solutions. P, 223. 301. Understanding Elementary Mathematics (4) I II Development of a basis for understanding the common processes in elementary mathematics related to the concepts of number, measurement, geometry and probability. 3R, 3L. Open to elementary education majors only. P, 117R/S, or 121, or an acceptable score on the math readiness test. 315. Introduction to Number Theory and Modern Algebra (3) II Elementary number theory, complex numbers, field axioms, polynomial rings; techniques for solving polynomial equations with integer and real coefficients. P, 323. 322. Mathematical Analysis for Engineers (3) I II Complex functions and integration, line and surface integrals, Fourier series, partial differential equations. Credit allowed for this course or 422a, but not for both. P, 254 or 355. 323. Intermediate Analysis (3) I II Elementary manipulations with sets and functions, properties of real numbers, topology of the real line, continuity, differentiation, sequences and series of real valued functions of a real variable, with emphasis on proving theorems. P, 215. WritingEmphasis Course. P, Satisfaction of the upperdivision writingproficiency requirement (see "WritingEmphasis Courses" in the Academic Guidelines section of this catalog). 330. Topics in Geometry (3) I Topics to be selected from 2 and 3dimensional combinatorial geometry, postulational Euclidean geometry, Euclidean transformational geometry, symmetry, and 2 dimensional crystallography. P, 215. 344. Foundations of Computing (3) II S (Identical with C SC 344) 355.4 Analysis of Ordinary Differential Equations (3) II Basic solution techniques for linear systems, qualitative behavior of nonlinear systems, numerical methods, computer studies; applications drawn from physical, biological and social sciences. P, 215 and C SC 115 or knowledge of FORTRAN, PASCAL, or another high level computer language. 362. Introduction to Probability Theory (3) I II Sample spaces, random variables and their properties, with considerable emphasis on applications. P, 123 or 125b. 375. Introduction to Numerical Methods (3) I II Rounding error and error propagation, roots of single equations, solving linear systems, curve fitting, numerical integration, numerical solution of ordinary differential equations. P, 215 and knowledge of a scientific programming language. 397. Workshop a. Mathematics Education (1) I II Open only to teaching majors in MATH P, 315 or 330. 402. Mathematical Logic (3) I 199394 Sentential calculus, predicate calculus; consistency, independence, completeness, and the decision problem. Designed to be of interest to majors in mathematics or philosophy. P, 124 or 125a. (Identical with C SC 402) May be convened with 502. 403. Foundations of Mathematics (3) II 199495 Topics in set theory such as functions, relations, direct products, transfinite induction and recursion, cardinal and ordinal arithmetic; related topics such as axiomatic systems, the development of the real number system, recursive functions. P, 215. (Identical with PHIL 403) May be convened with 503. 404. History of Mathematics (3) I The development of mathematics from ancient times through the 17th century, with emphasis on problem solving. The study of selected topics from each field is extended to the 20th century. P, 125b. May be convened with 504. 405. Mathematics in the Secondary School (3) II Not applicable to B.A. or B.S. degrees for math majors. (Identical with TTE 405) 410.4 Matrix Analysis (3) I II General introductory course in the theory of matrices. Advanceddegree credit not available to math majors. P, 254 or 355. 413.4 Linear Algebra (3) II Vector spaces, linear transformations and matrices, eigenvalues, bilinear forms, orthogonal and unitary transformations. P, 215. May be convened with 513. 415. Introduction to Abstract Algebra (3) I Introduction to groups, rings, and fields. P, 323. May be convened with 515. 416. Second Course in Abstract Algebra (3) II A continuation of 415. Topics may include Galois theory, linear and multilinear algebra, finite fields and coding theory. Polya enumeration. P, 415. May be convened with 516. 421. Fourier Series and Orthogonal Functions (3) I Linear spaces, orthogonal functions, Fourier series, Legendre polynomials and Bessel functions. P, 254 or 355. May be convened with 521. 422a422b.3 Advanced Analysis for Engineers (33) Laplace transforms, Fourier series, partial differential equations, vector analysis, integral theorems, matrices, complex variables. Credit allowed for 422a or 322, but not for both. P, 254 or 355. 422a is not prerequisite to 422b. Both 422a and 422b are offered each semester. May be convened with 522a522b. 424.3 Elements of Complex Variables (3) I II Complex numbers and functions, conformal mapping, calculus of residues. P, 223. May be convened with 524. 425. Real Analysis of One Variable (3) I Continuity and differentiation of functions of one variable. Riemann integration, sequences of functions and uniform convergence. P, 223 and 323. May be convened with 525. 426. Real Analysis of Several Variables (3) II Continuity and differentiation in higher dimensions, curves and surfaces; change of coordinates; theorems of Green, Gauss and Stokes; exact differentials. P, 425. May be convened with 526. 430. Second Course in Geometry (3) II 199495 Topics may include lowdimensional topology; map coloring in the plane, networks (graphs) polyhedra, twodimensional surfaces and their classification, map coloring on surfaces (Heawood's estimate, RingelYoung theory), knots and links or projective geometry. P, 215. May be convened with 530. 431. Calculus of Variations (3) I 199394 Euler equations and basic necessary conditions for extrema, sufficiency conditions, introduction to optimal control, direct methods. P, 254 or 355. 434. Introduction to Topology (3) II Properties of metric and topological spaces and their maps; topics selected from geometric and algebraic topology, including the fundamental group. P, 323. 436. Metric Differential Geometry (3) I Differential geometry of surfaces; nonintrinsic geometry: fundamental forms, Gaussian and mean curvatures; intrinsic geometry: Theorema Egregium, geodesics, GaussBonnet theorem. P, 254 or 355. 443. Theory of Graphs and Networks (3) II Undirected and directed graphs, connectivity, circuits, trees, partitions, planarity, coloring problems, matrix methods, applications in diverse disciplines. P, 215 or 223 or 243. (Identical with C SC 443) May be convened with 543. 446. Theory of Numbers (3) I 199495 Divisibility properties of integers, primes, congruences, quadratic residues, number theoretic functions. P, 215. May be convened with 546. 447. Combinatorial Mathematics (3) II 199495 Enumeration and construction of arrangements and designs; generating functions; principle of inclusionexclusion; recurrence relations; a variety of applications. P, 215 or 243. May be convened with 547. 454. Intermediate Ordinary Differential Equations and Stability Theory (3) I General theory of systems of ordinary differential equations, properties of linear systems, stability and boundedness of systems, perturbation of linear systems, Liapunov functions, periodic and almost periodic systems. P, 254 or 355. 455.4 Elementary Partial Differential Equations (3) II Theory of characteristics for first order partial differential equations; second order elliptic, parabolic, and hyperbolic equations. P, 254 or 355. May be convened with 555. 456.4 Applied Partial Differential Equations (3) II Properties of partial differential equations and techniques for their solution: Fourier methods, Green's functions, numerical methods. P, 322 or 421 or 422a. May be convened with 556. 464. Theory of Probability (3) I II Probability spaces, random variables, weak law of large numbers, central limit theorem, various discrete and continuous probability distributions. P, 322 or 323. (Identical with STAT 464) May be convened with 564. 466a. Theory of Statistics (3) I (Identical with STAT 466a) May be convened with 566a. 468. Applied Stochastic Processes (3) II Applications of Gaussian and Markov processes and renewal theory; Wiener and Poisson processes, queues. P, 464. (Identical with STAT 468) May be convened with 568. 473. Automata, Grammars and Language (3) I (Identical with C SC 473) 475a475b. Mathematical Principles of Numerical Analysis (33) 475a: Analysis of errors in numerical computations, solution of linear algebraic systems of equations, matrix inversion, eigenvalues, roots of nonlinear equations, interpolation and approximation. P, 215; 254 or 355; and a knowledge of a scientific computer programming language. 475b: Numerical integration, solution of systems of ordinary differential equations, initial value and boundary value problems. (Identical with C SC 475a475b) 479. Game Theory and Mathematical Programming (3) II 199394 Linear inequalities, games of strategy, minimax theorem, optimal strategies, duality theorems, simplex method. P, 410 or 413 or 415. (Identical with C SC 479) May be convened with 579. 484. Operational Mathematics (3) I Basic concepts of systems analysis, Fourier and Laplace transforms, difference equations, stability criteria. P, 421 and 424 or 422b. May be convened with 584. 485. Mathematical Modelling (3) II Development, analysis, and evaluation of mathematical models for physical, biological, social, and technical problems; both analytical and numerical solution techniques are required. P, 421, CR 475b. May be convened with 585. Writing Emphasis Course. P, satisfaction of the upperdivision writingproficiency requirement (see "Writing Emphasis Courses" in the Academic Guidelines of this catalog). 496. Seminar b. Mathematical Software (3) [Rpt.] I P, 254 or 355, knowledge of "C" programming. May be convened with 596b. 502. Mathematical Logic (3) II 199394 For a description of course topics, see 402. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 124 or 125a or PHIL 325. (Identical with C SC 502) May be convened with 402. 503. Foundations of Mathematics (3) II 199495 For a description of course topics, see 403. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 215. (Identical with PHIL 503) May be convened with 403. 504. History of Mathematics (3) I For a description of course topics, see 404. Graduatelevel requirements include more extensive problem sets or advanced projects. Not applicable to M.A., M.S., or Ph.D. degrees for math majors. P, 125b. May be convened with 404. 511a511b. Modern Algebra (33) Structure of groups, rings, modules, algebras; Galois theory. P, 415 and 416, or 413 and 415. 513. Linear Algebra (3) II For a description of course topics, see 413. Graduatelevel requirements include more extensive problem sets or advanced projects. Not applicable to M.A., M.S., or Ph.D. degrees for math majors. P, 215. May be convened with 413. 514a514b. Algebraic Number Theory (33) 199394 Dedekind domains, complete fields, class groups and class numbers, Dirichlet unit theorem, algebraic function fields. P, 511b. 515. Introduction to Abstract Algebra (3) I For a description of course topics, see 415. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 323. May be convened with 415. 516. Second Course in Abstract Algebra (3) II For a description of course topics, see 416. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 415. May be convened with 416. 517a517b. Group Theory (33) 199495 Selections from such topics as finite groups, noncommutative groups, abelian groups, characters and representations. P, 511b. 518. Topics in Algebra (3) [Rpt./36 units] I II Advanced topics in groups, rings, fields, algebras; content varies. 519. Topics in Number Theory and Combinatorics (3) [Rpt./36 units] I II Advanced topics in algebraic number theory, analytic number theory, class fields, combinatorics; content varies. 520a520b. Complex Analysis (33) 520a: Analyticity, Cauchy's integral formula, residues, infinite products, conformal mapping, Dirichlet problem, Riemann mapping theorem. P, 424. 520b: Rudiments of Riemann surfaces. P, 520a or 582. 521. Fourier Series and Orthogonal Functions (3) I For a description of course topics, see 421. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 254 or 355. May be convened with 421. 522a522b.3 Advanced Analysis for Engineers (33) For a description of course topics, see 422a422b. Graduatelevel requirements include more extensive problem sets or advanced projects. Not applicable to M.A., M.S., or Ph.D degrees for math majors. P, 254 or 355. May be convened with 422a422b. 523a523b. Real Analysis (33) Lebesque measure and integration, differentiation, RadonNikodym theorem, Lp spaces, applications. P, 425. 524.3 Elements of Complex Variables (3) I II For a description of course topics, see 424. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 223. May be convened with 424. 525. Real Analysis of One Variable (3) I For a description of course topics, see 425. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 223 and 323. May be convened with 425. 526. Real Analysis of Several Variables (3) II For a description of course topics, see 426. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 425. May be convened with 426. 527a527b. Principles of Analysis (33) Advancedlevel review of linear algebra and multivariable calculus; survey of real, complex and functional analysis, and differential geometry with emphasis on the needs of applied mathematics. P, 410, 424, and a differential equations course. 528a528b. Banach and Hilbert Spaces (33) 199495 Introduction to the theory of normed spaces, Banach spaces and Hilbert spaces, operators on Banach spaces, spectral theory of operators on Hilbert spaces, applications. P, 523a, 527b, or 583b. 529. Topics in Modern Analysis (3) [Rpt./36 units] I II Advanced topics in measure and integration, complex analysis in one and several complex variables, probability, functional analysis, operator theory; content varies. 530. Second Course in Geometry (3) II 199495 For a description of course topics, see 430. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 215. Not applicable to M.A., M.S., or Ph.D. degrees in Mathematics. May be convened with 430. 531. Algebraic Topology (3) I 199394 Poincare duality, fixed point theorems, characteristics classes, classification of principal bundles, homology of fiber bundles, higher homotopy groups, low dimensional manifolds. P, 534a534b. 534a534b. TopologyGeometry (33) Point set topology, the fundamental group, calculus on manifolds. Homology, de Rham cohomology, other topics. Examples will be emphasized. P, 415 and 425. 536a536b. Algebraic Geometry (33) 199495 Affine and projective varieties, morphisms and rational maps. Dimension, degree and smoothness. Basic coherent sheaf theory and Cech cohomology. Line bundles, RiemannRoch theorem. P, 511, 520a, 534a. 537a537b. Global Differential Geometry (33) 199394 Surfaces in R3, structure equations, curvature. GaussBonnet theorem, parallel transport, geodesics, calculus of variations, Jacobi fields and conjugate points, topology and curvature; Riemannian geometry, connections, curvature tensor, Riemannian submanifolds and submersions, symmetric spaces, vector bundles. Morse theory, symplectic geometry. P, 534a534b. 538. Topics in Geometry and Topology (3) [Rpt./36 units] I II Advanced topics in point set and algebraic topology, algebraic geometry, differential geometry; content varies. 539. Algebraic Coding Theory (3) II 199394 Construction and properties of error correcting codes; encoding and decoding procedures and information rate for various codes. P, 415. (Identical with ECE 539) 543. Theory of Graphs and Networks (3) II For a description of course topics, see 443. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 215 or 223 or 243. (Identical with C SC 543) May be convened with 443. 546. Theory of Numbers (3) I 199495 For a description of course topics, see 446. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 215. May be convened with 446. 547. Combinatorial Mathematics (3) II 199495 For a description of course topics, see 447. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 215 or 243. May be convened with 447. 550. Mathematical Population Dynamics (4) II (Identical with ECOL 550) 553a553b. Partial Differential Equations (33) 199495 Theory and examples of linear equations; characteristics, wellposed problems, regularity, variational properties, asymptotics. Topics in nonlinear equations, such as shock waves, diffusion waves, and estimates in Sobolev spaces. P, 523b or 527b or 583b. 554. Ordinary Differential Equations (3) I 199495 General theory of linear systems, Floquet theory. Local theory of nonlinear systems, stable manifold and HartmanGrobman theorems. Poincare Bendixxson theory, limit cycles, Poincare maps. Bifurcation theory, including the Hopf theorem. P, 413, 426 or permission of instructor. 555.4 Elementary Partial Differential Equations (3) II For a description of course topics, see 455. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 254 or 355. May be convened with 455. 556.4 Applied Partial Differential Equations (3) II For a description of course topics, see 456. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 322 or 421 or 422a. May be convened with 456. 557a557b. Dynamical Systems and Chaos (33) 199394 Qualitative theory of dynamical systems, phase space analysis, bifurcation, period doubling, universal scaling, onset of chaos. Applications drawn from atmospheric physics, biology, ecology, fluid mechanics and optics. P, 422a422b or 454. 559a559b. Lie Groups and Lie Algebras (33) 199495 Correspondence between Lie groups and Lie algebras, structure and representation theory, applications to topology and geometry of homogeneous spaces, applications to harmonic analysis. P, 511a, 523a, 534a534b, or consent of the instructor. 563a563b. Probability Theory (33) 199495 563a: Introduction to measure theory, strong law of large numbers, characteristic functions, the central limit theorem, conditional expectations, and discrete parameter martingales. P, 464. 563b: A selection of topics in stochastic processes from Markov chains, Brownian motion, the functional central limit theorem, diffusions and stochastic differential equations, martingales. P, 563a, 468 recommended. 564. Theory of Probability (3) I II For a description of course topics, see 464. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 322 or 323. (Identical with STAT 564) May be convened with 464. 565a565b. Stochastic Processes (33) 199394 Stationary processes, jump processes, diffusions, applications to problems in science and engineering. P, 468. 566a. Theory of Statistics (3) I (Identical with STAT 566a) May be convened with 466a. 568. Applied Stochastic Processes (3) II For a description of course topics, see 468. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 464. (Identical with STAT 568) May be convened with 468. 573. Theory of Computation (3) II (Identical with C SC 573) 575a575b. Numerical Analysis (33) Error analysis, solution of linear systems and nonlinear equations, eigenvalues interpolation and approximation, numerical integration, initial and boundary value problems for ordinary differential equations, optimization. P, 475b and 455 or 456. (Identical with C SC 575a575b) 576a576b. Numerical Analysis PDE (33) 576a: Finite difference, finite element and spectral discretization methods; semidiscrete, matrix and Fourier analysis. 576b: Wellposedness, numerical boundary conditions, nonlinear instability, timesplit algorithms, special methods for stiff and singular problems. P, 413, 456, 575b. 577. Topics in Applied Mathematics (3) [Rpt./36 units] I II Advanced topics in asymptotics, numerical analysis, approximation theory, mathematical theory of mechanics, dynamical systems, differential equations and inequalities, mathematical theory of statistics; content varies. 578. Computational Methods of Algebra (3) II Applications of machine computation to various aspects of algebra, such as matrix algorithms, character tables and conjugacy classes for finite groups, coset enumeration, integral matrices, crystallographic groups. P, 415 and a knowledge of scientific computer programming language. (Identical with C SC 578) 579. Game Theory and Mathematical Programming (3) II 199394 For a description of course topics, see 479. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 410 or 413 or 415. (Identical with C SC 579) May be convened with 479. 582. Applied Complex Analysis (3) II 199394 Representations of special functions, asymptotic methods for integrals and linear differential equations in the complex domain, applications of conformal mapping, WienerHopf techniques. P, 422b or 424. 583a583b. Principles and Methods of Applied Mathematics (33) Boundary value problems; Green's functions, distributions, Fourier transforms, the classical partial differential equations (Laplace, heat, wave) of mathematical physics. Linear operators, spectral theory, integral equations, Fredholm theory. P, 424 or 422b or CR, 520a. 584. Operational Mathematics (3) I For a description of course topics, see 484. Graduatelevel requirements include more extensive problem sets or advanced projects. P, 421 and 424, or 422b. May be convened with 484. 585. Mathematical Modelling (3) II For a description of course topics, see 485. Graduatelevel requirements include more advanced projects. P, 421, CR 475b. May be convened with 485. 586. Case Studies in Applied Mathematics (13) [Rpt./6 units] I II Indepth treatment of several contemporary problems or problem areas from a variety of fields, but all involving mathematical modeling and analysis; content varies. 587. Perturbation Methods in Applied Mathematics (3) I 199495 Regular and singular perturbations, boundary layer theory, multiscale and averaging methods for nonlinear waves and oscillators. P, 422a422b or 454. 588. Topics in Mathematical Physics (3) [Rpt./36 units] I II Advanced topics in field theories, mathematical theory of quantum mechanics, mathematical theory of statistical mechanics; content varies. 589. Nonlinear Wave Motion (3) II 199495 Nonlinear partial differential equations describing wave phenomena in water, gases, plasmas, lasers; shocks, modulated wave trains, parametric resonance, solutions and exactly solvable equations. P, 422b or 456 or 455. 595. Colloquium a. Math Instruction (1) [Rpt./12 units] I II b. Research in Mathematics (1) [Rpt./4] I II c. Research in Applied Mathematics (1) [Rpt./4] I II 596. Seminar a. Topics in Mathematics (13) [Rpt./12] S b. Mathematical Software (3) [Rpt.] I P, 254 or 355, knowledge of "C" programming. May be convened with 496b. 636. Information Theory (3) II 199495 (Identical with ECE 636) 667. Theory of Estimation (3) I (Identical with STAT 667) 668. Theory of Testing Hypothesis (3) II (Identical with STAT 668) 697. Workshop a. Problems in Computational Science (3) I II [Rpt./1] (Identical with PHYS 697a) 1 Students without university credit in the prerequisites for these courses will be required to have an appropriate score on the math readiness test to be enrolled in these courses. 2 Credit will not be given for this course if the student has credit in a higher level math course; these students will be dropped by the Registrar's Office. Students with unusual circumstances can petition the department for exemption from this rule. This policy does not infringe on the student's rights granted by the university policy on repeating a course. 3 Credit will be allowed for only one of 424 or 422b. 422a422b will not be considered a twosemester course at the 400 level in the Master of Arts degree program. 4 Credit will be allowed for only one from each of the following groups: 117R/S or 120; 124 or 125a; 125b; 254 or 355; 455 or 456; 410 or 413. 
Catalog Home All UA Catalogs UA Home Email catalog comments and suggestions to catalog@arizona.edu. Page last updated: April 30, 2002
