**
Elementary Courses.** In order to accommodate diverse backgrounds
and interests, several course options are available to beginning
mathematics students. All courses require three years of high
school mathematics; four years are strongly recommended and more
information is given for some individual courses below. Students
with College Board Advanced Placement credit and anyone planning
to enroll in an upper-level class should consider one of the Honors
sequences and discuss the options with a mathematics advisor.

Students who need additional preparation for calculus are tentatively identified by a combination of the math placement test (given during orientation), college admissions test scores (SAT or ACT), and high school grade point average. Academic advisors will discuss this placement information with each student and refer students to a special mathematics advisor when necessary.

Two courses preparatory to the calculus, Math 105 and Math
110, are offered. Math 105 is a course on data analysis, functions
and graphs with an emphasis on problem solving. Math 110 is a
condensed half-term version of the same material offered as a
self-study course through the Math Lab and directed towards students
who are unable to complete a first calculus course successfully.
Election of Math 110 is by recommendation of a Math 115 instructor
**only**. A maximum total of 4 credits may be earned
in courses numbered 110 and below. Math 103 is offered exclusively
in the Summer half-term for students in the Summer Bridge Program.

Math 127 and 128 are courses containing selected topics from
geometry and number theory, respectively. They are intended for
students who want exposure to mathematical culture and thinking through a single course. They are neither prerequisite nor preparation
for any further course. * No credit will be received for the
election of Math 127 or 128 if a student already has received
credit for a 200- (or higher) level mathematics course. *

Each of Math 112, 115, 185, and 295 is a first course in calculus
and generally credit can be received
for only one course from this list. Math 112 is designed for students
of business and the social sciences who require only one term
of calculus. It neither presupposes nor covers any trigonometry.
The sequence 115-116-215 is appropriate for most students who
want a complete introduction to calculus. One of Math 215, 285, or 395 is prerequisite to most more advanced courses in Mathematics. * Math 112 does not provide preparation for any subsequent course. *

Students planning a career in medicine should note that some medical schools require a course in calculus. Generally either Math 112 or 115 will satisfy this requirement, although most science concentrations require at least a year of calculus. Math 112 is accepted by the School of Business Administration, but Math 115 is prerequisite to concentration in Economics and further math courses are strongly recommended.

The sequences 156-255-256, 175-176-285-286, 185-186-285-286, and 295-296-395-396 are Honors sequences. All students must have the permission of an Honors advisor to enroll in any of these courses, but they need not be enrolled in the LS&A Honors Program. All students with strong preparation and interest in mathematics are encouraged to consider these courses; they are both more interesting and more challenging than the standard sequences.

Math 185-285 covers much of the material of Math 115-215 with more attention to the theory in addition to applications. Most students who take Math 185 have taken a high school calculus course, but it is not required. Math 175-176 assumes a knowledge of calculus roughly equivalent to Math 115 and covers a substantial amount of so-called combinatorial mathematics (see course description) as well as calculus-related topics not usually part of the calculus sequence. Math 175 and 176 are taught by the discovery method: students are presented with a great variety of problems and encouraged to experiment in groups using computers. The sequence Math 295-396 provides a rigorous introduction to theoretical mathematics. Proofs are stressed over applications and these courses require a high level of interest and commitment. Most students electing Math 295 have completed a thorough high school calculus course. The student who completes Math 396 is prepared to explore the world of mathematics at the advanced undergraduate and graduate level.

Students with strong scores on either the AB or BC version of the College Board Advanced Placement exam may be granted credit and advanced placement in one of the sequences described above; a table explaining the possibilities is available from advisors and the Department. In addition, there are two courses expressly designed and recommended for students with one or two semesters of AP credit, Math 119 and Math 156. Both will review the basic concepts of calculus, cover integration and an introduction to differential equations, and introduce the student to the computer algebra system MAPLE. Math 119 will stress experimentation and computation, while Math 156 is an Honors course intended primarily for science and engineering concentrators and will emphasize both applications and theory. Interested students should consult a mathematics advisor for more details.

In rare circumstances and * with permission of a Mathematics
advisor * reduced credit may be granted for Math 185 or 295
after one of Math 112 or 115. A list of these and other cases
of reduced credit for courses with overlapping material is available
from the Department. To avoid unexpected reduction in credit, students should always consult a advisor before switching from
one sequence to another. In all cases a maximum total of 16 credits
may be earned for calculus courses Math 112 through Math 396, and no credit can be earned for a prerequisite to a course taken
after the course itself.

Students completing Math 116 who are principally interested
in the application of mathematics to other fields may continue
either to Math 215 (Analytic Geometry and Calculus III) or to
Math 216 (Introduction to Differential Equations) – these two
courses may be taken in either order. Students intending to take
more advanced courses in mathematics, however,** must **follow the sequence 116-215-217-316. Math 217 (or the Honors version, Math 513) is required for a concentration in Mathematics; it both
serves as a transition to the more theoretical material of advanced
courses and provides the background required for optimal treatment
of differential equations in Math 316. Math 216 is not intended
for mathematics concentrators.

NOTE: WL:2 for all courses.

A * maximum total of 4 credits * may be earned in Mathematics
courses numbered 110 and below. A * maximum total of 16 credits *
may be earned for calculus courses Math 112 through Math 396, and no credit can be earned for a prerequisite to a course taken
after the course itself.

**105. Data, Functions, and Graphs. *** Students with credit for Math. 103 can elect
Math. 105 for only 2 credits. (4). (MSA). (QR/1).
*Math 105 serves both as a preparatory class to the calculus
sequences and as a terminal course for students who need only this level of mathematics. Students who complete 105 are fully
prepared for Math 115. This is a course on analyzing data by means
of functions and graphs. The emphasis is on mathematical modeling
of real-world applications. The functions used are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Algebra
skills are assessed during the term by periodic testing. Math
110 is a condensed half-term version of the same material offered
as a self-study course through the Math Lab. WL:2

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**110. Pre-Calculus (Self-Study).
*** See Elementary Courses above. Enrollment
in Math 110 is by recommendation of Math 115 instructor and override
only. No credit granted to those who already have 4 credits for
pre-calculus mathematics courses. (2). (Excl).
*The course covers data analysis by means of functions and graphs. Math 110 serves both as a preparatory class to the calculus
sequences and as a terminal course for students who need only this level of mathematics. The course is a condensed, half-term
version of Math 105 (Math 105 covers the same material in a traditional
classroom setting.) designed for students who appear to be prepared
to handle calculus but are not able to successfully complete Math
115. Students who complete 110 are fully prepared for Math 115.
Students may enroll in Math 110 only on the recommendation of
a mathematics instructor after the third week of classes in the
Winter and must visit the Math Lab to complete paperwork and receive
course materials. WL:2

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**112. Brief Calculus. *** See
Elementary Courses above. Credit is granted for
only one course from among Math. 112, 113, 115, 185, and 295.
(4). (MSA). (BS).
*This is a one-term survey course that provides the basics
of elementary calculus. Emphasis is placed on intuitive understanding
of concepts and not on rigor. Topics include differentiation with
application to curve sketching and maximum-minimum problems, antiderivatives
and definite integrals. Trigonometry is not used. This course
does not mesh with any of the courses in the other calculus sequences.
WL:2

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**115. Calculus I. *** Four
years of high school mathematics. See Elementary Courses
above. Credit usually is granted for only one course from among
Math. 112, 115, 185, and 295. (4). (MSA). (BS). (QR/1).
*The sequence Math 115-116-215 is the standard complete introduction
to the concepts and methods of calculus. It is taken by the majority
of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields.
The emphasis is on concepts and solving problems rather than theory
and proof. All sections are given a uniform midterm and final
exam. The course presents the concepts of calculus from three
points of view: geometric (graphs); numerical (tables); and algebraic
(formulas). Students will develop their reading, writing, and questioning skills.

Topics include functions and graphs, derivatives and their
applications to real-life problems in various fields, and definite
integrals. Math 185 is a somewhat more theoretical course which
covers some of the same material. Math 175 includes some of the
material of Math 115 together with some combinatorial mathematics.
A student whose preparation is insufficient for Math 115 should
take Math 105 (Data, Functions, and Graphs). Math 116 is the natural
sequel. A student who has done very well in this course could
enter the Honors sequence at this point by taking Math 186. The
cost for this course is over $100 since the student will need
a text (to be used for 115 and 116) and a graphing calculator
(the Texas Instruments TI-82 is recommended). WL:2

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**116. Calculus II. *** Math.
115. Credit is granted for only one course from among Math. 116, 119, 156, 186, and 296. (4). (MSA). (BS). (QR/1).
*See Math 115 for a general description of the sequence Math
115-116-215.

Topics include the indefinite integral, techniques of integration, introduction to differential equations, infinite series. Math
186 is a somewhat more theoretical course which covers much of the same material. Math 215 is the natural sequel. A student who
has done very well in this course could enter the Honors sequence
at this point by taking Math 285. WL:2

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**127. Geometry and the Imagination.
*** Three years of high school mathematics including
a geometry course. No credit granted to those who have completed
a 200- (or higher) level mathematics course. (4). (MSA). (BS).
(QR/1).
*This course introduces students to the ideas and some of the basic results in Euclidean and non-Euclidean geometry. Beginning
with geometry in ancient Greece, the course includes the construction
of new geometric objects from old ones by projecting and by taking
slices. The next topic is non-Euclidean geometry. This section
begins with the independence of Euclid's Fifth Postulate and with the construction of spherical and hyperbolic geometries in which the Fifth Postulate fails; how spherical and hyperbolic geometry
differs from Euclidean geometry. The last topic is geometry of
higher dimensions: coordinatization – the mathematician's tool
for studying higher dimensions; construction of higher-dimensional
analogues of some familiar objects like spheres and cubes; discussion
of the proper higher-dimensional analogues of some geometric notions
(length, angle, orthogonality,

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**147. Introduction to Interest
Theory. *** Math. 112 or 115. No credit granted to those
who have completed a 200- (or higher) level mathematics course.
(3). (MSA). (BS).
*This course is designed for students who seek an introduction
to the mathematical concepts and techniques employed by financial
institutions such as banks, insurance companies, and pension funds.
Actuarial students, and other mathematics concentrators, should
elect Math 424 which covers the same topics but on a more rigorous
basis requiring considerable use of the calculus. Topics covered
include: various rates of simple and compound interest, present
and accumulated values based on these; annuity functions and their
application to amortization, sinking funds and bond values; depreciation
methods; introduction to life tables, life annuity, and life insurance
values. This course is not part of a sequence. Students should
possess financial calculators. WL:2

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**186. Honors Analytic Geometry
and Calculus II. *** Permission of the Honors advisor.
Credit is granted for only one course from among Math. 114, 116, 119, 156, 186, and 296. (4). (MSA). (BS). (QR/1).
*The sequence Math 185-186-285-286 is the Honors introduction
to the calculus. It is taken by students intending to major in
mathematics, science, or engineering as well as students heading
for many other fields who want a somewhat more theoretical approach.
Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also
included. This sequence is

Topics covered include transcendental functions; techniques
of integration; applications of calculus such as elementary differential
equations, simple harmonic motion, and center of mass; conic sections;
polar coordinates; infinite sequences and series including power
series and Taylor series. Other topics, often an introduction
to matrices and vector spaces, will be included at the discretion
of the instructor. Math 116 is a somewhat less theoretical course
which covers much of the same material. Math 285 is the natural
sequel. WL:2

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**215. Calculus III. *** Math.
116, 156, or 186. (4). (MSA). (BS). (QR/1).
*The sequence Math 115-116-215 is the standard complete introduction
to the concepts and methods of calculus. It is taken by the majority
of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields.
The emphasis is on concepts and solving problems rather than theory
and proof. All sections are given a midterm and final exam. Topics
include vector algebra and vector functions; analytic geometry
of planes, surfaces, and solids; functions of several variables
and partial differentiation; line, surface, and volume integrals
and applications; vector fields and integration; Green's Theorem
and Stokes' Theorem. There is a weekly computer lab using

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**216. Introduction to Differential
Equations. *** Math. 215. (4). (MSA). (BS).
*For a student who has completed the calculus sequence, there
are two sequences which deal with linear algebra and differential
equations, Math 216-417 (or 419) and Math 217-316. The sequence
Math 216-417 emphasizes problem-solving and applications and is
intended for students of engineering and the sciences. Math concentrators
and other students who have some interest in the theory of mathematics
should elect the sequence Math 217-316. After an introduction
to ordinary differential equations, the first half of the course
is devoted to topics in linear algebra, including systems of linear
algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors.
In the second half these tools are applied to the solution of
linear systems of ordinary differential equations. Topics include:
oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods.
There is a weekly computer lab using

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**217. Linear Algebra. *** Math.
215, 255, or 285. No credit granted to those who have completed
or are enrolled in Math. 417, 419, or 513. (4). (MSA). (BS). (QR/1).
*For a student who has completed the calculus sequence, there
are two sequences which deal with linear algebra and differential
equations, Math 216-417 (or 419) and Math 217-316. The sequence
Math 216-417 emphasizes problem-solving and applications and is
intended for students of Engineering and the sciences. Math concentrators
and other students who have some interest in the theory of mathematics
should elect the sequence Math 217-316. These courses are explicitly
designed to introduce the student to both the concepts and applications
of their subjects and to the methods by which the results are
proved. Therefore the student entering Math 217 should come with
a sincere interest in learning about proofs. The topics covered
include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of R

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**255. Applied Honors Calculus
III. *** Math. 156. (4). (MSA). (BS).
*Multivariable calculus, line, surface, and volume integrals;
vector fields, Green's theorem, Stokes theorem; divergence theorem, applications.

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**286. Honors Differential
Equations. *** Math. 285. (3). (MSA). (BS).
*See Math. 186 for a general description of the sequence Math
185-186-285-286.

Topics include first-order differential equations, higher-order
linear differential equations with constant coefficients, an introduction
to linear algebra, linear systems, the Laplace Transform, series
solutions and other numerical methods (Euler, Runge-Kutta). If
time permits, Picard's Theorem will be proved. Math 216 and 316
cover much of the same material. Math 471 and/or 572 are natural
sequels in the area of differential equations, but Math 286 is
also preparation for more theoretical courses such as Math 451.
WL:2

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**288. Math Modeling Workshop.
*** Math. 216 or 316, and Math. 217 or 417. (1). (Excl).
(BS). Offered mandatory credit/no credit. May be repeated for
a total of three credits.
*This course is designed to help students understand more
clearly how techniques from other undergraduate mathematics courses
can be used in concert to solve real-world problems. After the
first two lectures the class will discuss methods of attacking
problems. For credit a student will have to describe and solve
an individual problem and write a report on the solution. Computing
methods will be used. During the weekly workshop students will
be presented with real-world problems on which techniques of undergraduate
mathematics offer insights. They will see examples of (1) how
to approach and set up a given modeling problem systematically, (2) how to use mathematical techniques to begin a solution of the problem, (3) what to do about the loose ends that can't be
solved, and (4) how to present the solution to others. Students
will have a chance to use the skills developed by participating
in the UM Undergraduate Math Modeling Meet. WL:2

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**289. Problem Seminar. *** (1).
(Excl). (BS). May be repeated for credit with permission.
*One of the best ways to develop mathematical abilities is
by solving problems using a variety of methods. Familiarity with
numerous methods is a great asset to the developing student of
mathematics. Methods learned in attacking a specific problem frequently
find application in many other areas of mathematics. In many instances
an interest in and appreciation of mathematics is better developed
by solving problems than by hearing formal lectures on specific
topics. The student has an opportunity to participate more actively
in his/her education and development. This course is intended
for superior students who have exhibited both ability and interest
in doing mathematics, but it is not restricted to Honors students.
This course is excellent preparation for the Putnam exam. Students
and one or more faculty and graduate student assistants will meet
in small groups to explore problems in many different areas of
mathematics. Problems will be selected according to the interests
and background of the students. WL:2

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**296(196). Honors Mathematics
II. *** Prior knowledge of first year calculus and permission
of the Honors advisor. Credit is granted for only one course from
among Math. 116, 119, 156, 186, and 296. (4). (Excl). (BS). (QR/1).
*The sequence Math 295-296-395-396 is a more intensive Honors
sequence than 185-186-285-286. The material includes all of that
of the lower sequence and substantially more. The approach is theoretical, abstract, and rigorous. Students are expected to
learn to understand and construct proofs as well as do calculations
and solve problems. The expected background is a thorough understanding
of high school algebra and trigonometry. No previous calculus
is required, although many students in this course have had some
calculus. Students completing this sequence will be ready to take
advanced undergraduate and beginning graduate courses. This sequence
is

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**312. Applied Modern Algebra.
*** Math. 217. Only 1 credit granted to those who have
completed Math. 412. (3). (Excl). (BS).
*One of the main goals of the course (along with every course
in the algebra sequence) is to expose students to rigorous, proof-oriented
mathematics. Students are required to have taken Math 217, which
should provide a first exposure to this style of mathematics.
A distinguishing feature of this course is that the abstract concepts
are not studied in isolation. Instead, each topic is studied with the ultimate goal being a real-world application. As currently
organized, the course is broken into four parts. (1) the integers
"mod n" and linear algebra over the integers mod p, with applications to error correcting codes; (2) some number theory, with applications to public-key cryptography; (3) polynomial algebra, with an emphasis on factoring algorithms over various fields, and (4) permutation groups, with applications to enumeration of
discrete structures "up to automorphisms" (a.k.a. Pólya
Theory). Math 412 is a more abstract and proof-oriented course
with less emphasis on applications. EECS 303 (Algebraic Foundations
of Computer Engineering) covers many of the same topics with a
more applied approach. Another good follow-up course is Math 475
(Number Theory). Math 312 is one of the alternative prerequisites
for Math 416, and several advanced EECS courses make substantial
use of the material of Math 312. Math 412 is better preparation
for most subsequent mathematics courses. WL:2

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**316. Differential Equations.
*** Math. 215 and 217. Credit can be received for only
one of Math. 216 or Math. 316, and credit can be received for
only one of Math. 316 or Math. 404. (3). (Excl). (BS).
*This is an introduction to differential equations for students
who have studied linear algebra (Math 217). It treats techniques
of solution (exact and approximate), existence and uniqueness theorems, some qualitative theory, and many applications. Proofs
are given in class; homework problems include both computational
and more conceptually oriented problems. First-order equations:
solutions, existence and uniqueness, and numerical techniques;
linear systems: eigenvector-eigenvalue solutions of constant coefficient
systems, fundamental matrix solutions, nonhomogeneous systems;
higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium
points, stability. Applications to physical problems are considered throughout. Math 216 covers somewhat less material without the
use of linear algebra and with less emphasis on theory. Math 286
is the Honors version of Math 316. Math 471 and/or 572 are natural
sequels in the area of differential equations, but Math 316 is
also preparation for more theoretical courses such as Math 451.
WL:2

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**333. Directed Tutoring.
*** Math. 385 and enrollment in the Elementary Program
in the School of Education. (1-3). (Excl). (EXPERIENTIAL). May
be repeated for a total of three credits.
*An experiential mathematics course for exceptional upper-level
students in the elementary teacher certification program. Students
tutor needy beginners enrolled in the introductory courses (Math
385 and Math 489) required of all elementary teaachers. Permission
of instructor. WL:2

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**354. Fourier Analysis and its Applications. *** Math. 216, 316, or 286. No credit
granted to those who have completed or are enrolled in Math. 454.
(3). (Excl). (BS).
*This is an introduction to Fourier analysis at an elementary
level emphasizing applications. The main topics are Fourier series, discrete Fourier transforms, and continuous Fourier transforms.
A substantial portion of the time is spent on both scientific/technological
applications

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**371/Engin. 371. Numerical
Methods for Engineers and Scientists. *** Engineering
101, and Math. 216. (3). (Excl). (BS).
*This is a survey course of the basic numerical methods which
are used to solve practical scientific problems. Important concepts
such as accuracy, stability, and efficiency are discussed. The
course provides an introduction to

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**399. Independent Reading.
*** (1-6). (Excl). (INDEPENDENT). May be repeated for
credit.
*Designed especially for Honors students. WL:2

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**403. Mathematical Modeling
Using Computer Algebra Systems. *** Math. 116 and junior
standing. (3). (MSA). (BS). (QR/1).
*Many fields of study including the natural sciences, engineering, economics, and statistics use mathematics regularly and extensively
both as a tool and as a means for modeling phenomena. Since the
realistic models usually lead to problems not solvable by simple
analytic techniques – either because they involve too many parameters
or are highly nonlinear – new methods are needed to give the students
insight into the problem. One rather new powerful technique for
doing this is the so-called Computer Algebra (CA) system. These
systems manipulate symbols as easily as hand-held calculators
manipulate numbers. So, for example,

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**412. Introduction to Modern
Algebra. *** Math. 215 or 285; and 217. No credit granted
to those who have completed or are enrolled in 512. Students with
credit for 312 should take 512 rather than 412. One credit granted
to those who have completed 312. (3). (Excl). (BS).
*This course is designed to serve as an introduction to the
methods and concepts of abstract mathematics. A typical student
entering this course has substantial experience in using complex
mathematical (calculus) calculations to solve physical or geometrical
problems, but is unused to analyzing carefully the content of
definitions of the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite
distinct from those of calculus, an important goal of the course
is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems
(propositions, lemmas,

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**417. Matrix Algebra I.
*** Three courses beyond Math. 110. No credit granted
to those who have completed or are enrolled in 217, 419, or 513.
(3). (Excl). (BS).
*Many problems in science, engineering, and mathematics are
best formulated in terms of matrices – rectangular arrays of numbers.
This course is an introduction to the properties of and operations
on matrices with a wide variety of applications. The main emphasis
is on concepts and problem-solving, but students are responsible
for some of the underlying theory. Diversity rather than depth
of applications is stressed. This course is not intended for mathematics
concentrators, who should elect Math 217 or 513 (Honors). Topics
include matrix operations, echelon form, general solutions of
systems of linear equations, vector spaces and subspaces, linear
independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least
squares method (regression), discrete Markov processes, linear
programming, and differential equations. Math 419 is an enriched
version of Math 417 with a somewhat more theoretical emphasis.
Math 217 (despite its lower number) is also a more theoretical
course which covers much of the material of 417 at a deeper level.
Math 513 is an Honors version of this course, which is also taken
by some mathematics graduate students. Math 420 is the natural
sequel but this course serves as prerequisite to several courses:
Math 452, 462, 561, and 571. WL:2

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**419/EECS 400/CS 400. Linear
Spaces and Matrix Theory. *** Four terms of college mathematics
beyond Math 110. No credit granted to those who have completed
or are enrolled in 217 or 513. One credit granted to those who
have completed Math. 417. (3). (Excl). (BS).
*Math 419 covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector
spaces and linear transformations instead of matrices. There is
a mix of proofs, calculations, and applications with the emphasis
depending somewhat on the instructor. A previous proof-oriented
course is helpful but by no means necessary. Basic notions of
vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations;
determinants; eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators
and matrices, applications to differential and difference equations.
Math 417 is less rigorous and theoretical and more oriented to
applications. Math 217 is similar to Math 419 but slightly more
proof-oriented. Math 513 is much more abstract and sophisticated.
Math 420 is the natural sequel, but this course serves as prerequisite
to several courses: Math 452, 462, 561, and 571. WL:2

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**420. Matrix Algebra II.
*** Math. 217, 417 or 419. (3). (Excl). (BS).
*This course is intended to sharpen the student's skills in the manipulations and applications of linear algebra. Some proofs
are given, and the object is to learn to use the theory to solve
problems. One previous proof-oriented course is recommended, although
not required. Similarity theory, Euclidean and unitary geometry, applications to linear and differential equations, interpolation theory, least squares and principal components, B-splines. WL:2

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**424. Compound Interest
and Life Insurance. *** Math. 215. (3). (Excl). (BS).
*This course explores the concepts underlying the theory of
interest and then applies them to concrete problems. The course
also includes applications of spreadsheet software. The course
is a prerequisite to advanced actuarial courses. It also helps
students prepare for the Part 4A examination of the Casualty Actuarial
Society and the Course 140 examination of the Society of Actuaries.
The course covers compound interest (growth) theory and its application
to valuation of monetary deposits, annuities, and bonds. Problems
are approached both analytically (using algebra) and geometrically
(using pictorial representations). Techniques are applied to real-life
situations: bank accounts, bond prices,

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**425/Stat. 425. Introduction
to Probability. *** Math. 215, 255, or 285. (3). (MSA).
(BS).
Sections 001 and 002. * This course introduces students to
useful and interesting ideas of the mathematical theory of probability
and to a number of applications of probability to a variety of
fields including genetics, economics, geology, business, and engineering.
The theory developed together with other mathematical tools such
as combinatorics and calculus are applied to everyday problems.
Concepts, calculations, and derivations are emphasized. The course
will make essential use of the material of Math 116 and 215. Math
concentrators should be sure to elect sections of the course that
are taught by Mathematics (not Statistics) faculty. Topics include the basic results and methods of both discrete and continuous
probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. Different instructors will vary the emphasis.
Math 525 is a similar course for students with stronger mathematical
background and ability. Stat 426 is a natural sequel for students
interested in statistics. Math 523 includes many applications
of probability theory. WL:2

*Sections 003 and 004. * See Statistics
425.

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**450. Advanced Mathematics
for Engineers I. *** Math. 216, 286, or 316. (4). (Excl).
(BS).
*Although this course is designed principally to develop mathematics
for application to problems of science and engineering, it also
serves as an important bridge for students between the calculus
courses and the more demanding advanced courses. Students are
expected to learn to read and write mathematics at a more sophisticated
level and to combine several techniques to solve problems. Some
proofs are given, and students are responsible for a thorough
understanding of definitions and theorems. Students should have
a good command of the material from Math 215, and 216 or 316, which is used throughout the course. A background in linear algebra,

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**451. Advanced Calculus
I. *** Math. 215 and one course beyond Math. 215; or
Math. 285. Intended for concentrators; other students should elect
Math. 450. (3). (Excl). (BS).
*This course has two complementary goals: (1) a rigorous development
of the fundamental ideas of calculus, and (2) a further development
of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor"
and "proof"; almost all of the material of the course
consists in understanding and constructing definitions, theorems
(propositions, lemmas,

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**452. Advanced Calculus
II. *** Math. 217, 417, or 419; and Math. 451. (3). (Excl).
(BS).
*This course does a rigorous development of multivariable
calculus and elementary function theory with some view towards
generalizations. Concepts and proofs are stressed. This is a relatively
difficult course, but the stated prerequisites provide adequate
preparation. Topics include (1) partial derivatives and differentiability;
(2) gradients, directional derivatives, and the chain rule; (3)
implicit function theorem; (4) surfaces, tangent plane; (5) max-min theory; (6) multiple integration, change of variable,

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**454. Boundary Value Problems
for Partial Differential Equations. *** Math. 216, 286
or 316. Students with credit for Math. 354, 455 or 554 can elect
Math. 454 for 1 credit. (3). (Excl). (BS).
*This course is devoted to the use of Fourier series and other
orthogonal expansions in the solution of boundary-value problems
for second-order linear partial differential equations. Emphasis
is on concepts and calculation. The official prerequisite is ample
preparation. Classical representation and convergence theorems
for Fourier series; method of separation of variables for the
solution of the one-dimensional heat and wave equation; the heat
and wave equations in higher dimensions; spherical and cylindrical
Bessel functions; Legendre polynomials; methods for evaluating
asymptotic integrals (Laplace's method, steepest descent); Fourier
and Laplace transforms; applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series
analysis, and spectral analysis. Both Math 455 and 554 cover many
of the same topics but are very seldom offered. Math 454 is prerequisite
to Math 571 and 572, although it is not a formal prerequisite, it is good background for Math 556. WL:2

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**462. Mathematical Models.
*** Math. 216, 286 or 316; and 217, 417, or 419. Students
with credit for 362 must have department permission to elect 462.
(3). (Excl). (BS).
*This course will cover biological models constructed from
difference equations and ordinary differential equations. Applications
will be drawn from population biology, population genetics, the theory of epidemics, biochemical kinetics, and physiology. Both
exact solutions and simple qualitative methods for understanding
dynamical systems will be stressed (anticipated text is

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**464/BiomedE 464. Inverse
Problems. *** One of Math. 217, 417, or 419; and one
of Math. 216, 256, 286, or 316. (3). (Excl). (BS).
Section 001 – Mathematics and Medical Imaging. * Solution of
an inverse problem is a central component of fields such as medical
tomography, geophysics, non-destructive testing, and control theory.
The solution of any practical inverse problem is an interdisciplinary
task. Each such problem requires a blending of mathematical constructs
and physical realities. Thus, each problem has its own unique
components; on the other hand, there is a common mathematical
framework for these problems and their solutions. The course content
is often motivated by a particular inversion problem from a field
such as medical tomography (transmission, emission), geophysics
(remote sensing, inverse scattering, tomography), or non-destructive
testing. Mathematical topics include ill-posedness (existence, uniqueness, stability), regularization

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**471. Introduction to Numerical
Methods. *** Math. 216, 286, or 316; and 217, 417, or
419; and a working knowledge of one high-level computer language.
(3). (Excl). (BS).
*This is a survey of the basic numerical methods which are
used to solve scientific problems. The emphasis is evenly divided
between the analysis of the methods and their practical applications.
Some convergence theorems and error bounds are proved. The course
also provides an introduction to

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**475. Elementary Number Theory.
*** At least three terms of college mathematics are recommended.
(3). (Excl). (BS).
*This is an elementary introduction to number theory, especially
congruence arithmetic. Number theory is one of the few areas of
mathematics in which problems easily describable to a layman (is
every even number the sum of two primes?) have remained unsolved
for centuries. Recently some of these fascinating but seemingly
useless questions have come to be of central importance in the
design of codes and cyphers. The methods of number theory are
often elementary in requiring little formal background. In addition
to strictly number-theoretic questions, concrete examples of structures
such as rings and fields from abstract algebra are discussed.
Concepts and proofs are emphasized, but there is some discussion
of algorithms which permit efficient calculation. Students are
expected to do simple proofs and may be asked to perform computer
experiments. Although there are no special prerequisites and the
course is essentially self-contained, most students have some
experience in abstract mathematics and problem solving and are
interested in learning proofs. A Computational Laboratory (Math
476, 1 credit) will usually be offered as an optional supplement
to this course. Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder
Theorem, Hensel's Lemma, Diophantine equations, arithmetic in
polynomial rings, primitive roots, quadratic reciprocity and quadratic
fields. Math 575 moves much faster, covers more material, and requires more difficult exercises. There is some overlap with
Math 412 which stresses the algebraic content. Math 475 may be
followed by Math 575 and is good preparation for Math 412. All
of the advanced number theory courses, Math 675, 676, 677, 678, and 679, presuppose the material of Math 575, although a good
student may get by with Math 475. Each of these is devoted to
a special subarea of number theory. WL:2

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**476. Computational Laboratory
in Number Theory. *** Prior or concurrent enrollment
in Math. 475 or 575. (1). (Excl). (BS).
*Students will be provided software with which to conduct
numerical explorations. Students will submit reports of their
findings weekly. No programming necessary, but students interested
in programming will have the opportunity to embark on their own
projects. Participation in the laboratory should boost the student's
performance in Math 475 or Math 575. Students in the lab will
see mathematics as an exploratory science (as mathematicians do).
Students will gain a knowledge of algorithms which have been developed
(some quite recently) for number-theoretic purposes,

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**486. Concepts Basic to
Secondary Mathematics. *** Math. 215. (3). (Excl). (BS).
*This course is designed for students who intend to teach
junior high or high school mathematics. It is advised that the
course be taken relatively early in the program to help the student
decide whether or not this is an appropriate goal. Concepts and proofs are emphasized over calculation. The course is conducted
in a discussion format. Class participation is expected and constitutes
a significant part of the course grade. Topics covered have included
problem solving; sets, relations and functions; the real number
system and its subsystems; number theory; probability and statistics;
difference sequences and equations; interest and annuities; algebra;
and logic. This material is covered in the course pack and scattered
points in the text book. There is no real alternative, but the
requirement of Math 486 may be waived for strong students who
intend to do graduate work in mathematics. Prior completion of
Math 486 may be of use for some students planning to take Math
312, 412, or 425. WL:2

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**489. Mathematics for Elementary
and Middle School Teachers. *** Math. 385 or 485. May
not be used in any graduate program in mathematics. (3). (Excl).
*This course, together with its predecessor Math 385, provides
a coherent overview of the mathematics underlying the elementary
and middle school curriculum. It is required of all students intending
to earn an elementary teaching certificate and is taken almost
exclusively by such students. Concepts are heavily emphasized
with some attention given to calculation and proof. The course
is conducted using a discussion format. Class participation is
expected and constitutes a significant part of the course grade.
Enrollment is limited to 30 students per section. Although only
two years of high school mathematics are required, a more complete
background including pre-calculus or calculus is desirable. Topics
covered include fractions and rational numbers, decimals and real
numbers, probability and statistics, geometric figures, and measurement.
Algebraic techniques and problem-solving strategies are used throughout the course. WL:2

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**490. Introduction to Topology.
*** Math. 412 or 451 or equivalent experience with abstract
mathematics. (3). (Excl). (BS).
*This course in an introduction to both point-set and algebraic
topology. Although much of the presentation is theoretical and proof-oriented, the material is well-suited for developing intuition
and giving convincing proofs which are pictorial or geometric
rather than completely rigorous. There are many interesting examples
of topologies and manifolds, some from common experience (combing
a hairy ball, the utilities problem). In addition to the stated
prerequisites, courses containing some group theory (Math 412
or 512) and advanced calculus (Math 451) are desirable although
not absolutely necessary. The topics covered are fairly constant
but the presentation and emphasis will vary significantly with the instructor. Point-set topology, examples of topological spaces, orientable and non-orientable surfaces, fundamental groups, homotopy, covering spaces. Metric and Euclidean spaces are emphasized. Math
590 is a deeper and more difficult presentation of much of the
same material which is taken mainly by mathematics graduate students.
Math 433 is a related course at about the same level. Math 490
is not prerequisite for any later course but provides good background
for Math 590 or any of the other courses in geometry or topology.
WL:2

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**498. Topics in Modern Mathematics.
*** Senior mathematics concentrators and Master Degree
students in mathematical disciplines. (3). (Excl). (BS).
*The course concentrates on topics in modern mathematics not
represented in the standard course list. The choice of topics
varies from term to term depending on faculty and student interests.
WL:2

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**512. Algebraic Structures.
*** Math. 451 or 513. No credit granted to those who
have completed or are enrolled in 412. Math. 512 requires more
mathematical maturity than Math. 412. (3). (Excl). (BS).
*This is one of the more abstract and difficult courses in the undergraduate program. It is frequently elected by students
who have completed the 295-396 sequence. Its goal is to introduce
students to the basic structures of modern abstract algebra (groups, rings, and fields) in a rigorous way. Emphasis is on concepts
and proofs; calculations are used to illustrate the general theory.
Exercises tend to be quite challenging. Students should have some
previous exposure to rigorous proof-oriented mathematics and be
prepared to work hard. Students from Math 285 are strongly advised
to take some 400-500 level course first, for example, Math 513.
Some background in linear algebra is strongly recommended The
course covers basic definitions and properties of groups, rings, and fields, including homomorphisms, isomorphisms, and simplicity.
Further topics are selected from (1) Group Theory: Sylow theorems, Structure Theorem for finitely-generated Abelian groups, permutation
representations, the symmetric and alternating groups; (2) Ring
Theory: Euclidean, principal ideal, and unique factorization domains, polynomial rings in one and several variables, algebraic varieties, ideals; and (3) Field Theory: statement of the Fundamental Theorem
of Galois Theory, Nullstellensatz, subfields of the complex numbers
and the integers mod

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**513. Introduction to Linear
Algebra. *** Math. 412. Two credits granted to those
who have completed Math. 417; one credit granted to those who
have completed Math 217 or 419. (3). (Excl). (BS).
*This is an introduction to the theory of abstract vector
spaces and linear transformations. The emphasis is on concepts
and proofs with some calculations to illustrate the theory. For
students with only the minimal prerequisite, this is a demanding
course; at least one additional proof-oriented course

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**521. Life Contingencies
II. *** Math. 520. (3). (Excl). (BS).
*This course extends the single decrement and single life
ideas of Math 520 to multi-decrement and multiple-life applications
directly related to life insurance. The sequence Math 520-521
covers the Part 4A examination of the Casualty Actuarial Society
and covers the syllabus of the Course 150 examination of the Society
of Actuaries. Concepts and calculation are emphasized over proof.
Topics include multiple life models – joint life, last survivor, contingent insurance; multiple decrement models – disability, withdrawal, retirement,

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**523. Risk Theory. *** Math.
425. (3). (Excl). (BS).
*Risk management is of major concern to all financial institutions
and is an active area of modern finance. This course is relevant
for students with interests in finance, risk management, or insurance
and provides background for the professional examinations in Risk
Theory offered by the Society of Actuaries and the Casualty Actuary
Society. Students should have a basic knowledge of common probability
distributions (Poisson, exponential, gamma, binomial,

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**524. Topics in Actuarial
Science II. *** Math. 424, 425, and 520; and Stat. 426.
(3). (Excl). (BS). May be repeated for a total of 9 credits.
*Topics covered are: the nature and properties of survival
models, including both parametric and tabular models; methods
of estimating tabular models from both complete and incomplete
data samples, including the actuarial, moment, and maximum likelihood
estimation techniques; methods of estimating parametric models
from both complete and incomplete data samples, including parametric
models with concomitant variables; evaluation of estimators from
sample data; valuation schedule exposure formulas; and practical
issues in survival model estimation. WL:2

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**525/Stat. 525. Probability
Theory. *** Math. 450 or 451. Students with credit for
Math. 425/Stat. 425 can elect Math. 525/Stat. 525 for only 1 credit.
(3). (Excl). (BS).
*This course is a thorough and fairly rigorous study of the
mathematical theory of probability. There is substantial overlap
with 425, but here more sophisticated mathematical tools are used
and there is greater emphasis on proofs of major results. Math
451 is preferable to Math 450 as preparation, but either is acceptable.
Topics include the basic results and methods of both discrete
and continuous probability theory. Different instructors will
vary the emphasis between these two theories. EECS 501 also covers
some of the same material at a lower level of mathematical rigor.
Math 425 is a course for students with substantially weaker background
and ability. Math 526, Stat 426, and the sequence Stat 510-511
are natural sequels. WL:2

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**526/Stat. 526. Discrete
State Stochastic Processes. *** Math. 525 or EECS 501.
(3). (Excl). (BS).
*See Statistics 526. (Woodroofe)

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**555. Introduction to Functions
of a Complex Variable with Applications. *** Math. 450
or 451. Students with credit for Math. 455 or 554 can elect Math.
555 for one hour credit. (3). (Excl). (BS).
*This course is an introduction to the theory of complex valued
functions of a complex variable with substantial attention to
applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized
over proofs, but arguments are rigorous. The prerequisite of a
course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper
real integrals, fluid dynamics. Math 596 covers all of the theoretical
material of Math 555 and usually more at a higher level and with
emphasis on proofs rather than applications. Math 555 is prerequisite
to many advanced courses in science and engineering fields. WL:2

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**557. Methods of Applied
Mathematics II. *** Math. 556. (3). (Excl). (BS).
*Although this is a continuation of Math 556, this course
is not really required as a prerequisite. A strong background
in complex analysis and linear algebra is essential. There is
somewhat less emphasis on proofs than in Math 556. Topics include
transform methods for partial differential equations, asymptotic
expansions, regular and singular perturbation problems, non-linear
stability theory, bifurcations, non-linear evolution equations, and associated phenomena. WL:2

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**561/SMS 518 (Business Administration)/IOE
510. Linear Programming I. *** Math. 217, 417, or 419.
(3). (Excl). (BS).
*Formulation of problems from the private and public sectors
using the mathematical model of linear programming. Development
of the simplex algorithm; duality theory and economic interpretations.
Postoptimality (sensitivity) analysis; applications and interpretations.
Introduction to transportation and assignment problems; special
purpose algorithms and advanced computational techniques. Students
have opportunities to formulate and solve models developed from
more complex case studies and use various computer programs. WL:2 (Murty)

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**566. Combinatorial Theory.
*** Math. 216, 356, 286, or 316. (3). (Excl). (BS).
*This course is a rigorous introduction to classical combinatorial theory. Concepts and proofs are the foundation, but there are
copious applications to modern industrial problem-solving. Permutations, combinations, generating functions, and recurrence relations.
The existence and enumeration of finite discrete configurations.
Systems of representatives, Ramsey's Theorem and extremal problems.
Construction of combinatorial designs. There is no real alternative, although there is some overlap with Math 565. Sequels are Math
664-665 and Math 669. WL:2

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**567. Introduction to Coding
Theory. *** Math 217 or 419. (3). (Excl).
*This course is an introduction to coding theory, focusing
on the mathematical background for linear error-correcting codes.
It will begin with a discussion of Shannon's theorem and channel
capacity. The definition of linear codes will be given along with
a review of necessary tools from linear algebra and an introduction
to abstract algebra and finite fields. Basic examples of codes
will be studied including the Hamming, BCH, cyclic, Melas, Reed-Muller, and Ree-Solomon codes. An introduction to the problem of decoding
will be included, starting with syndrome decoding and covering
weight enumerator polynomials and the Mac-Williams Sloane identity.
Further topics to be included range from consideration of asymptotic
parameters and bounds to a discussion of algebraic geometric codes
in their simplest form. WL:2

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**571. Numerical Methods
for Scientific Computing I. *** Math. 217, 419, or 513;
and 454. (3). (Excl). (BS).
*This course is a rigorous introduction to numerical linear
algebra with applications to 2-point boundary value problems and the Laplace equation in two dimensions. Both theoretical and computational
aspects of the subject are discussed. Some of the homework problems
require computer programming. Students should have a strong background
in linear algebra and calculus, and some programming experience.
The topics covered usually include direct and iterative methods
for solving systems of linear equations: Gaussian elimination, Cholesky decomposition, Jacobi iteration, Gauss-Seidel iteration, the SOR method, an introduction to the multigrid method, conjugate
gradient method; finite element and difference discretizations
of boundary value problems for the Poisson equation in one and two dimensions; numerical methods for computing eigenvalues and eigenvectors. Math 471 is a survey course in numerical methods
at a more elementary level. Math 572 covers initial value problems
for ordinary and partial differential equations. Math 571 and 572 may be taken in either order. Math 671 (Analysis of Numerical
Methods I) is an advanced course in numerical analysis with varying
topics chosen by the instructor. WL:2

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**572. Numerical Methods
for Scientific Computing II. *** Math. 217, 419, or 513;
and 454. (3). (Excl). (BS).
*This is one of the basic courses for students beginning study
towards the Ph.D. degree in mathematics. Graduate students from
engineering and science departments and strong undergraduates
are also welcome. The course is an introduction to numerical methods
for solving ordinary differential equations and hyperbolic and parabolic partial differential equations. Fundamental concepts
and methods of analysis are emphasized. Students should have a
strong background in linear algebra and analysis, and some experience
with computer programming. Content varies somewhat with the instructor.
Numerical methods for ordinary differential equations; Lax's equivalence theorem; finite difference and spectral methods for linear time
dependent PDEs: diffusion equations, scalar first order hyperbolic
equations, symmetric hyperbolic systems. There is no real alternative;
Math 471 covers a small part of the same material at a lower level.
Math 571 and 572 may be taken in either order. Math 671 (Analysis
of Numerical Methods I) is an advanced course in numerical analysis
with varying topics chosen by the instructor. WL:2

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**582. Introduction to Set
Theory. *** Math. 412 or 451 or equivalent experience
with abstract mathematics. (3). (Excl). (BS).
*One of the great discoveries of modern mathematics was that
essentially every mathematical concept may be defined in terms
of sets and membership. Thus set theory plays a special role as
a foundation for the whole of mathematics. One of the goals of this course is to develop some understanding of how set theory
plays this role. The analysis of common mathematical concepts

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**592. Introduction to Algebraic
Topology. *** Math. 591. (3). (Excl). (BS).
*This is one of the basic courses for students beginning study
towards the Ph.D. degree in mathematics. The approach is theoretical
and rigorous, and emphasizes abstract concepts and proofs. Fundamental
group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, Eilenberg-Maclane axioms, Brouwer's and Lefschetz' fixed-point theorems, and other topics. WL:2

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**594. Algebra II. *** Math.
593. (3). (Excl). (BS).
*This is one of the basic courses for students beginning study
towards the Ph.D. degree in mathematics. The approach is theoretical
and rigorous and emphasizes abstract concepts and proofs. Topics
include group theory, permutation representations, simplicity
of alternating groups for

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**597. Analysis II. *** Math.
451 and 513. (3). (Excl). (BS).
*This is one of the basic courses for students beginning study
towards the Ph.D. degree in mathematics. The approach is theoretical
and rigorous and emphasizes abstract concepts and proofs. Topics
include Lebesgue measure on the real line; measurable functions
and integration on

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