The University of Arizona  1993-95 General Catalog

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Mathematics (MATH)
Mathematics Building, Room 108
(520) 621-6892

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, Lai-Sang 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, Zhen-Su 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 Education--College of Education) and Bachelor of
Science in Education (Secondary Education--College 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 454-455, 454-456, 464-466a, or 475a-475b; 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 upper-division units in
economics. The finance minor should consist of ACCT 200 and 210;
either ECON 201a-201b or 210; FIN 311 and 421; plus six
additional upper-division 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 upper-division 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 Self-Study.
Identical to MATH 116R except taught in a self-study 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 Self-Study. Identical to MATH
117R except taught in a self-study 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. Problem-Solving Laboratory (1) [Rpt./4] I II Development of
creative, mathematical, problem-solving 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. Writing-Emphasis Course. P,
Satisfaction  of the upper-division writing-proficiency
requirement (see "Writing-Emphasis Courses" in the Academic
Guidelines section of this catalog).

330. Topics in Geometry (3) I Topics to be selected from 2- and
3-dimensional 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 1993-94 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 1994-95 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. Advanced-degree 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.

422a-422b.3 Advanced Analysis for Engineers (3-3) 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 522a-522b.

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 1994-95 Topics may include
low-dimensional topology; map coloring in the plane, networks
(graphs) polyhedra, two-dimensional surfaces and their
classification, map coloring on surfaces (Heawood's estimate,
Ringel-Young theory), knots and links or projective geometry. P,
215. May be convened with 530.

431. Calculus of Variations (3) I 1993-94 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, Gauss-Bonnet 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 1994-95 Divisibility properties of
integers, primes, congruences, quadratic residues, number-
theoretic functions. P, 215. May be convened with 546.

447. Combinatorial Mathematics (3) II 1994-95 Enumeration and
construction of arrangements and designs; generating functions;
principle of inclusion-exclusion; 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)

475a-475b. Mathematical Principles of Numerical Analysis (3-3)
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 475a-475b)

479. Game Theory and Mathematical Programming (3) II 1993-94
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 upper-division writing-proficiency 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 1993-94 For a description of
course topics, see 402. Graduate-level 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 1994-95 For a description
of course topics, see 403. Graduate-level 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. Graduate-level 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.

511a-511b. Modern Algebra (3-3) 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. Graduate-level 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.

514a-514b. Algebraic Number Theory (3-3) 1993-94 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. Graduate-level 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. Graduate-level requirements include
more extensive problem sets or advanced projects. P, 415. May be
convened with 416.

517a-517b. Group Theory (3-3) 1994-95 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.

520a-520b. Complex Analysis (3-3) 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. Graduate-level
requirements include more extensive problem sets or advanced
projects. P, 254 or 355. May be convened with 421.

522a-522b.3 Advanced Analysis for Engineers (3-3) For a
description of course topics, see 422a-422b. Graduate-level
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 422a-422b.

523a-523b. Real Analysis (3-3) Lebesque measure and integration,
differentiation, Radon-Nikodym theorem, Lp spaces, applications.
P, 425.

524.3 Elements of Complex Variables (3) I II For a description of
course topics, see 424. Graduate-level 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. Graduate-level 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. Graduate-level requirements include
more extensive problem sets or advanced projects. P, 425. May be
convened with 426.

527a-527b. Principles of Analysis (3-3) Advanced-level 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.

528a-528b. Banach and Hilbert Spaces (3-3) 1994-95 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 1994-95 For a description
of course topics, see 430. Graduate-level 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 1993-94 Poincare duality, fixed
point theorems, characteristics classes, classification of
principal bundles, homology of fiber bundles, higher homotopy
groups, low dimensional manifolds. P, 534a-534b.

534a-534b. Topology-Geometry (3-3) Point set topology, the
fundamental group, calculus on manifolds. Homology, de Rham
cohomology, other topics. Examples will be emphasized. P, 415 and
425.

536a-536b. Algebraic Geometry (3-3) 1994-95 Affine and projective
varieties, morphisms and rational maps. Dimension, degree and
smoothness. Basic coherent sheaf theory and Cech cohomology. Line
bundles, Riemann-Roch theorem. P, 511, 520a, 534a.

537a-537b. Global Differential Geometry (3-3) 1993-94 Surfaces in
R3, structure equations, curvature. Gauss-Bonnet 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, 534a-534b.

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 1993-94 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. Graduate-level 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 1994-95 For a description of course
topics, see 446. Graduate-level requirements include more
extensive problem sets or advanced projects. P, 215. May be
convened with 446.

547. Combinatorial Mathematics (3) II 1994-95 For a description
of course topics, see 447. Graduate-level 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)

553a-553b. Partial Differential Equations (3-3) 1994-95 Theory
and examples of linear equations; characteristics, well-posed
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 1994-95 General theory
of linear systems, Floquet theory. Local theory of nonlinear
systems, stable manifold and Hartman-Grobman 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. Graduate-level
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. Graduate-level
requirements include more extensive problem sets or advanced
projects. P, 322 or 421 or 422a. May be convened with 456.

557a-557b. Dynamical Systems and Chaos (3-3) 1993-94 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, 422a-422b or 454.

559a-559b. Lie Groups and Lie Algebras (3-3) 1994-95
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, 534a-534b, or consent of the instructor.

563a-563b. Probability Theory (3-3) 1994-95 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. Graduate-level requirements include more
extensive problem sets or advanced projects. P, 322 or 323.
(Identical with STAT 564) May be convened with 464.

565a-565b. Stochastic Processes (3-3) 1993-94 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. Graduate-level 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)

575a-575b. Numerical Analysis (3-3) 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 575a-575b)

576a-576b. Numerical Analysis PDE (3-3) 576a: Finite difference,
finite element and spectral discretization methods; semidiscrete,
matrix and Fourier analysis. 576b: Well-posedness, numerical
boundary conditions, nonlinear instability, time-split
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 1993-94 For
a description of course topics, see 479. Graduate-level
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 1993-94 Representations of
special functions, asymptotic methods for integrals and linear
differential equations in the complex domain, applications of
conformal mapping, Wiener-Hopf techniques. P, 422b or 424.

583a-583b. Principles and Methods of Applied Mathematics (3-3)
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. Graduate-level 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. Graduate-level requirements include more
advanced projects. P, 421, CR 475b. May be convened with 485.

586. Case Studies in Applied Mathematics (1-3) [Rpt./6 units] I
II In-depth 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 1994-95
Regular and singular perturbations, boundary layer theory,
multiscale and averaging methods for nonlinear waves and
oscillators. P, 422a-422b 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 1994-95 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 (1-3) [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 1994-95 (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. 422a-422b
will not be considered a two-semester 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.

 


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