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 LSA 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 majors 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:3 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. The course prepares
students for Math 115
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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).
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. Students who complete 110 are fully
prepared for Math 115. The course is a condensed, half-term version
of Math 105 designed for students who appear to be prepared to
handle calculus but are not able to successfully complete Math
115. Students may enroll in Math 110 only on the recommendation
of a mathematics instructor after the third week of classes in the Fall and must visit the Math Lab to complete paperwork and receive course materials. The course covers data analysis by means
of functions and graphs. Math 105 covers the same material in
a traditional classroom setting. The course prepares students
for Math 115.
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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.
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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).
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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.
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119. Calculus II Using MAPLE. Math. 115. Credit is granted for only one course from among Math. 114, 116, 119, 186, and 296. (4). (MSA). (BS). (QR/1).
MAPLE is a symbolic algebra computer software program which aids the student in visualization, computation, and organization. Students who complete Math 119 and continue to Calculus III should elect Math 219 which will be a special MAPLE-oriented version of Math 215.
The sequence Math 119-219 is intended for students who have
earned a score of 3 or better on either the AB or BC version of the Advanced Placement Exam in Mathematics. No familiarity with
computers is necessary. The material covered will be approximately that of Math 116. In addition, students are taught to use the
computer algebra system MAPLE (on the Macintosh) – a symbolic
algebra program which aids the student in visualization, computation, and organization – as a tool to do routine calculations, to visualize
and to explore. MAPLE is thoroughly integrated into the course, and the use of MAPLE is permitted (encouraged) on homework and tests. Students are presented with challenging, unstructured problems
done in groups. Learning to work well with others is an important
(and satisfying) part of the course. The emphasis is on concepts
and problem-solving rather than theory and proof. Topics include
applications of the definite integral, separable differential
equations, inverse functions, infinite sequences and series, conics
and parametric curves. Math 156 (Fall) is a quite similar course
in the Honors sequence with greater emphasis on applications to the physical sciences and engineering. Math 255 is the natural
sequel to Math 156. Students who complete Math 119 and continue
to Calculus III should elect Math 219 which is a special MAPLE-oriented
version of Math 215. A student who has done very well in this
course could enter the Honors sequence at this point by taking
285.
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128. Explorations in Number Theory. High school mathematics through at least Analytic Geometry. No credit granted to those who have completed a 200- (or higher) level mathematics course. (4). (MSA). (BS). (QR/1).
This course is intended for non-science concentrators and students
in the pre-concentration years with no intended concentration, who want to engage in mathematical reasoning without having to
take calculus first. Students will be introduced to elementary
ideas of number theory, an area of mathematics that deals with
properties of the integers. Students will make use of software
provided for IBM PCs to conduct numerical experiments and to make
empirical discoveries. Students will formulate precise conjectures, and in many cases prove them. Thus the students will, as a group, generate a logical development of the subject. After studying
factorizations and greatest common divisors, emphasis will shift
to the patterns that emerge when the integers are classified according
to the remainder produced upon division by some fixed number ('congruences').
Once some basic tools have been established, applications will
be made in several directions. For example, students may derive
a precise parameterization of Pythagorean triples a2
+ b2 = c2. Students who like math but don't
especially like calculus will want to enroll in this first-year
MSA seminar. Students will do hands-on experimentation with numerical
patterns and will tackle numerical riddles and brainteasers as they focus on empirical discovery and proof of theorems. Students
will write their own text in number theory and enjoy their growing
ability to think like mathematicians.
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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. The course is not part of a sequence. Students should
possess financial calculators.
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156. Applied Honors Calculus II. Score of 4 or 5 on the AB or BC Advanced Placement calculus exam. Credit is granted for only one course among Math 114, 116, 119, 156 and 296. (4). (MSA). (BS). (QR/1).
The sequence 156-255-256 is an Honors calculus sequence for
engineering and science concentrators who scored 4 or 5 on the
AB or BC Advanced Placement calculus exam. Topics include Riemann
sums, the definite integral, fundamental theorem of calculus, applications of integral calculus (e.g. arclength, surface
area, work, hydrostatic pressure, center of mass), improper integrals, infinite sequences and series, differential equations, complex
numbers. MAPLE will be used throughout.
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175. Combinatorics and Calculus. Permission of Honors advisor. (4). (MSA). (BS). (QR/1).
This course is an alternative to Math 185 as an entry to the
Honors sequence. The sequence Math 175-176 is a two-term introduction
to Combinatorics, Dynamical Systems, and Calculus. The topics
are integrated over the two terms although the first term will
stress combinatorics and the second term will stress the development
of calculus in the context of dynamical systems. Students are
expected to have some previous experience with the basic concepts
and techniques of calculus. The course stresses discovery as a
vehicle for learning. Students will be required to experiment throughout the course on a range of problems and will participate
each term in a group project. Grades will be based on homework
and projects with a strong emphasis on homework. Personal computers
will be a valuable experimental tool in this course, and students
will be asked to learn to program in either BASIC, PASCAL, or
FORTRAN. There are two major topic areas: enumeration theory and graph theory. The section on enumeration theory will emphasize
classical methods for counting including (1) binomial theorem
and its generalizations; (2) solving recursions; (3) generating
functions; and (4) the inclusion-exclusion principle. In the process, we will discuss infinite series. The section on graph theory will
include basic definitions and some of the more interesting and useful theorems of graph theory. The emphasis will be on topological
results and applications to computer science and will include
(1) connectivity; (2) trees, Prufer codes, and data structures;
(3) planar graphs, Euler's formula and Kuratowski's Theorem; and (4) coloring graphs, chromatic polynomials, and orientation. This
material has many applications in the field of Computer Science.
Math 176 is the standard sequel.
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185. Honors Analytic Geometry and Calculus I. Permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. (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 concentrate
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 not restricted to
students enrolled in the LS&A Honors Program. Topics covered
include functions and graphs, limits, derivatives, differentiation
of algebraic and trigonometric functions and applications, definite
and indefinite integrals and applications. Other topics will be
included at the discretion of the instructor. Math 115 is a somewhat
less theoretical course which covers much of the same material.
Math 186 is the natural sequel.
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203. Introduction to MAPLE and MATHEMATICA. Prior or concurrent enrollment in one term of calculus. No programming experience is assumed. No credit granted to those who have completed Math. 119. (1). (Excl). (BS). Offered mandatory credit/no credit. May be repeated for a total of two credits.
This course is designed to provide the student with an introduction
to two powerful Computer Algebra Systems (MAPLE and MATHEMATICA)
for doing Algebra, Calculus and Statistical and Graphical Analysis.
Recent years have seen the development of several powerful software
packages, known as Computer Algebra Systems, for doing mathematics
on the computer. These programs have the capacity to solve problems
numerically, graphically, and symbolically in calculus, linear
algebra, differential equations, statistics, and many areas of
science and engineering. This one-credit mini-course is a brief
introduction to the two most popular of these systems, Maple
and Mathematica. It will be of interest to all students
whose career interests require mathematical skills. No programming
experience is assumed. Students should have taken or be concurrently
enrolled in a first course in calculus. The elementary features
of Maple and Mathematica will be introduced
and applied to various types of problems in algebra and calculus.
Math 403 is a more thorough introduction to either Maple
and Mathematica. This course introduces the student to
a tool which can be useful in almost any course which uses mathematics.
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215. Calculus III. Math. 116 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 MAPLE
software. Math 285 is a somewhat more theoretical course which
covers the same material. For students intending to concentrate
in mathematics or who have some interest in the theory of mathematics
as well as its applications, the appropriate sequel is Math 217.
Students who intend to take only one further mathematics course
and need differential equations should take Math 216.
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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 MATLAB software. This
course is not intended for mathematics concentrators, who should
elect the sequence 217-316. Math 286 covers much of the
same material in the Honors sequence. The sequence Math 217-316
covers all of this material and substantially more at greater
depth and with greater emphasis on the theory. Math 404 covers
further material on differential equations. Math 217 and 417 cover
further material on linear algebra. Math 371 and 471 cover additional
material on numerical methods.
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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 Rn; linear
dependence, bases, and dimension; linear transformations; eigenvalues
and eigenvectors; diagonalization; inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. Math 417 and 419 cover similar material
with more emphasis on computation and applications and less emphasis
on proofs. Math 513 covers more in a much more sophisticated way.
The intended course to follow Math 217 is 316. Math 217 is also
prerequisite for Math 412 and all more advanced courses in mathematics.
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256. Applied Honors Calculus IV. Math. 255. (4). (MSA). (BS).
Linear algebra, matrices, systems of differential equations, initial and boundary value problems, qualitative theory of dynamical
systems, nonlinear equations, numerical methods, MAPLE.
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285. Honors Analytic Geometry and Calculus III. Math. 186 or permission of the Honors advisor. (4). (MSA). (BS).
See Math. 186 for a general description of the sequence Math 185-186-285-286.
Topics include vector algebra and vector functions; analytic
geometry of planes, surfaces, and solids; functions of several
variables and partial differentiation, maximum-minimum problems;
line, surface, and volume integrals and applications; vector fields
and integration; curl, divergence, and gradient; Green's Theorem
and Stokes' Theorem. Additional topics may be added at the discretion
of the instructor. Math 215 is a less theoretical course which
covers the same material.
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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 elected for a total of 3 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.
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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.
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295(195). Honors Mathematics I. Prior knowledge of first year calculus and permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. (4). (MSA). (BS). (QR/1).
Math 295-296-395-396 is the main Honors calculus sequence.
It is aimed at talented students who intend to major in mathematics, science, or engineering. The emphasis is on concepts, problem
solving, as well as the underlying theory and proofs of important
results. Students interested in taking advanced mathematical courses
later should definitely start with this sequence. The expected
background is high school trigonometry and algebra (previous calculus
not required). This sequence is not restricted to students enrolled
in the LS&A Honors program. Real functions, limits, continuous
functions, limits of sequences, complex numbers, derivatives, indefinite integrals and applications, some linear algebra. Math
175 and Math 185 are lower-level Honors courses. Math 296 is the
intended sequel.
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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.
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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 elementary teachers.
Students would tutor elementary (Math. 102) or intermediate (Math.
104) algebra in the Math. Lab. They would also participate in
a weekly seminar to discuss mathematical and methodological questions.
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350/Aero. 350. Aerospace Engineering Analysis. Math. 216 or 316. (3). (Excl). (BS).
This is a three-hour lecture course in engineering mathematics
which continues the development and application of ideas introduced
in Math 215 and 216. The course is required in the Aerospace Engineering
curriculum, and covers subjects needed for subsequent departmental
courses. The major topics discussed include vector analysis, Fourier
series, and an introduction to partial differential equations, with emphasis on separation of variables. Some review and extension
of ideas relating to convergence, partial differentiation, and integration are also given. The methods developed are used in the formulation and solution of elementary initial- and boundary-value
problems involving, e.g., forced oscillations, wave motion, diffusion, elasticity, and perfect-fluid theory. There are two
or three one-hour exams and a two-hour final, plus about ten homework
assignments, or approximately one per week, consisting largely
of problems from the text. The text is Mathematical Methods
in the Physical Sciences by M.L. Boas.
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371/Engin. 371. Numerical Methods for Engineers and Scientists. Engineering 103 or 104, 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 MATLAB, an interactive program
for numerical linear algebra, and may provide practice in FORTRAN
programming and the use of a software library subroutine. Convergence theorems are discussed and applied, but the proofs are not emphasized.
Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration and differentiation, solutions to non-linear equations, ordinary differential equations, polynomial approximations. Other topics may include discrete Fourier
transforms, two-point boundary-value problems, and Monte-Carlo
methods. Math 471 is a similar course which expects one more year
of maturity and is somewhat more theoretical and less practical.
The sequence Math 571-572 is a beginning graduate level sequence
which covers both numerical algebra and differential equations
and is much more theoretical. This course is basic for many later
courses in science and engineering. It is good background for
571-572.
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385. Mathematics for Elementary School Teachers. One year each of high school algebra and geometry. No credit granted to those who have completed or are enrolled in 485. (3). (Excl).
This course, together with its sequel Math 489, 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 problem solving, sets and functions, numeration
systems, whole numbers (including some number theory) and integers.
Each number system is examined in terms of its algorithms, its
applications, and its mathematical structure. There is no alternative
course. Math 489 is the required sequel.
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395(295). Honors Analysis I. Math. 296 or permission of the Honors advisor. (4). (Excl). (BS).
This course is a continuation of the sequence Math 295-296
and has the same theoretical emphasis. Students are expected to
understand and construct proofs. This course studies functions
of several real variables. Topics are chosen from elementary linear
algebra: vector spaces, subspaces, bases, dimension, solutions
of linear systems by Gaussian elimination; elementary topology:
open, closed, compact, and connected sets, continuous and uniformly
continuous functions; differential and integral calculus of vector-valued
functions of a scalar; differential and integral calculus of scalar-valued
functions on Euclidean spaces; linear transformations: null space, range, matrices, calculations, linear systems, norms; differential
calculus of vector-valued mappings on Euclidean spaces: derivative, chain rule, implicit and inverse function theorems.
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Times, Location, and Availability
399. Independent Reading. (1-6). (Excl). (INDEPENDENT). May be repeated for credit.
Designed especially for Honors students.
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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, MATHEMATICA (the CA system
used in this course) can compute the indefinite integral of tan
x, expand sin x in power series, find the general solution of
y" + y = cos t, and so on. In essence, MATHEMATICA is an
"expert" mathematical assistant. Using MATHEMATICA easily
and productively is the primary goal of Math 403. There are no
final exams but rather students work in teams to produce a term
project using MATHEMATICA. There are two hours of lecture and one hour of actual computer work per week. Weekly demonstrations
of computer competency in using MATHEMATICA amounts to 50% of the term grade. The term project comprises the remaining 50%.
No previous computer programming is required or needed. (Goldberg)
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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, etc.) and their proofs. Math 217
or equivalent required as background. The initial topics include
ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, complex numbers). These are then applied to the
study of particular types of mathematical structures such as groups, rings, and fields. These structures are presented as abstractions
from many examples such as the common number systems together
with the operations of addition or multiplication, permutations
of finite and infinite sets with function composition, sets of
motions of geometric figures, and polynomials. Notions such as
generator, subgroup, direct product, isomorphism, and homomorphism
are defined and studied. Math 312 is a somewhat less abstract
course which substitutes material on finite automata and other
topics for some of the material on rings and fields of Math 412.
Math 512 is an Honors version of Math 412 which treats more material
in a deeper way. A student who successfully completes this course
will be prepared to take a number of other courses in abstract
mathematics: Math 416, 451, 475, 575, 481, 513, and 582. All of these courses will extend and deepen the student's grasp of modern
abstract mathematics.
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413. Calculus for Social Scientists. Not open to freshmen, sophomores or mathematics concentrators. (3). (Excl). (BS).
A one-term course designed for students who require an introduction
to the ideas and methods of the calculus. The course begins with
a review of algebra and then surveys analytic geometry, derivatives, maximum and minimum problems, integrals, integration, and partial
derivatives. Applications to business and economics are given
whenever possible, and the level is always intuitive rather than
highly technical. This course should not be taken by those who
have had a previous calculus course or plan to take more than
one or two further courses in mathematics. The course is specially
designed for graduate students in the social sciences.
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416. Theory of Algorithms. Math. 312 or 412 or CS 303, and CS 380. (3). (Excl). (BS).
Many common problems from mathematics and computer science
may be solved by applying one or more algorithms
– well-defined procedures that accept input data specifying a
particular instance of the problem and produce a solution. Students
entering Math 416 typically have encountered some of these problems
and their algorithmic solutions in a programming course. The goal
here is to develop the mathematical tools necessary to analyze
such algorithms with respect to their efficiency (running time)
and correctness. Different instructors will put varying degrees
of emphasis on mathematical proofs and computer implementation
of these ideas. Typical problems considered are: sorting, searching, matrix multiplication, graph problems (flows, travelling salesman), and primality and pseudo-primality testing (in connection with
coding questions). Algorithm types such as divide-and-conquer, backtracking, greedy, and dynamic programming are analyzed using
mathematical tools such as generating functions, recurrence relations, induction and recursion, graphs and trees, and permutations. The
course often includes a section on abstract complexity theory
including NP completeness. This course has substantial overlap
with EECS 586 – more or less depending on the instructors. In
general, Math 416 will put more emphasis on the analysis
aspect in contrast to design of algorithms. Math 516
(given infrequently) and EECS 574 and 575 (Theoretical Computer
Science I and II) include some topics which follow those of this
course.
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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.
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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.
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422. Topics in Actuarial Mathematics I. Math. 216 or 316. (3). (Excl). (BS).
We will explore how much insurance affects the lives of students
(automobile insurance, social security, health insurance, theft
insurance) as well as the lives of other family members (retirements, life insurance, group insurance). While the mathematical models
are important, an ability to articulate why the insurance options
exist and how they satisfy the customer's needs are equally important.
In addition, there are different options available (e.g. in social
insurance programs) that offer the opportunity of discussing alternative
approaches.
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423. Mathematics of Finance. Math. 217 and 425; CS 183. (3). (Excl). (BS).
This course is an introduction to the mathematical models used
in finance and economics with particular emphasis on models for
pricing derivative instruments such as options and futures. The
goal is to understand how the models derive from basic principles
of economics, and to provide the necessary mathematical tools
for their analysis. A solid background in basic probability theory
is necessary. Topics include risk and return theory, portfolio theory, capital asset pricing model, random walk model, stochastic
processes, Black-Scholes Analysis, numerical methods, and interest
rate models.
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425/Stat. 425. Introduction
to Probability. Math. 215, 255, or 285. (3). (MSA).
(BS).
Sections 001 and 002. See Statistics
425.
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Sections 003 and 004. 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.
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Times, Location, and Availability
431. Topics in Geometry for Teachers. Math. 215. (3). (Excl). (BS).
This course is a study of the axiomatic foundations of Euclidean
and non-Euclidean geometry. Concepts and proofs are emphasized;
students must be able to follow as well as construct clear logical
arguments. For most students this is an introduction to proofs.
A subsidiary goal is the development of enrichment and problem
materials suitable for secondary geometry classes. Topics selected
depend heavily on the instructor but may include classification
of isometries of the Euclidean plane; similarities; rosette, frieze, and wallpaper symmetry groups; tessellations; triangle groups;
finite, hyperbolic, and taxicab non-Euclidean geometries. Alternative
geometry courses at this level are 432 and 433. Although it is
not strictly a prerequisite, Math 431 is good preparation for
531.
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433. Introduction to Differential Geometry. Math. 215, or 255 or 285, and Math. 217(3). (Excl). (BS).
This course is about the analysis of curves and surfaces in
2- and 3-space using the tools of calculus and linear algebra.
There will be many examples discussed, including some which arise
in engineering and physics applications. Emphasis will be placed
on developing intuitions and learning to use calculations to verify
and prove theorems. Students need a good background in multivariable
calculus (215) and linear algebra (preferably 217). Some exposure
to differential equations (216 or 316) is helpful but not absolutely
necessary. Topics covered include (1) curves: curvature, torsion, rigid motions, existence and uniqueness theorems; (2) global properties
of curves: rotation index, global index theorem, convex curves, 4-vertex theorem; (3) local theory of surfaces: local parameters, metric coefficients, curves on surfaces, geodesic and normal curvature, second fundamental form, Christoffel symbols, Gaussian and mean
curvature, minimal surfaces, classification of minimal surfaces
of revolution. 537 is a substantially more advanced course which
requires a strong background in topology (590), linear algebra
(513) and advanced multivariable calculus (551). It treats some
of the same material from a more abstract and topological perspective
and introduces more general notions of curvature and covariant
derivative for spaces of any dimension. Math 635 and Math 636
(Topics in Differential Geometry) further study Riemannian manifolds
and their topological and analytic properties. Physics courses
in general relativity and gauge theory will use some of the material
of this course.
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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, e.g., Math
217, is highly desirable, as is familiarity with Maple software.
Topics include a review of curves and surfaces in implicit, parametric, and explicit forms; differentiability and affine approximations;
implicit and inverse function theorems; chain rule for 3-space;
multiple integrals; scalar and vector fields; line and surface
integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems, derivation
of continuity and heat equation. Some instructors include more
material on higher dimensional spaces and an introduction to Fourier
series. Math 450 is an alternative to Math 451 as a prerequisite
for several more advanced courses. Math 454 and 555 are the natural
sequels for students with primary interest in engineering applications.
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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, etc.), and proofs. This is considered
one of the more difficult among the undergraduate mathematics
courses, and students should be prepared to make a strong commitment
to the course. In particular, it is strongly recommended that
some course which requires proofs (such as Math 412) be taken
before Math 451. Topics include: logic and techniques of proof;
sets, functions, and relations; cardinality; the real number system
and its topology; infinite sequences, limits and continuity; differentiation;
integration, the Fundamental Theorem of Calculus; infinite series;
sequences and series of functions. There is really no other course
which covers the material of Math 451. Although Math 450 is an
alternative prerequisite for some later courses, the emphasis
of the two courses is quite distinct. The natural sequel to Math
451 is 452, which extends the ideas considered here to functions
of several variables. In a sense, Math 451 treats the theory behind
Math 115-116, while Math 452 does the same for Math 215 and a
part of Math 216. Math 551 is a more advanced version of Math
452. Math 451 is also a prerequisite for several other courses:
Math 575, 590, 596, and 597.
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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.
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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 Mathematical
Models in Biology by Leah Edelstein-Keshet).
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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 MATLAB, an interactive program
for numerical linear algebra, as well as practice in computer
programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Topics may include
computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear
equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential
equations, Monte Carlo methods, 2-point boundary value problems, Dirichlet problem for the Laplace equation. Math 371 is a less
sophisticated version intended principally for sophomore and junior
engineering students; the sequence Math 571-572 is mainly taken
by graduate students, but should be considered by strong undergraduates.
Math 471 is good preparation for Math 571 and 572, although it
is not prerequisite to these courses.
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481. Introduction to Mathematical Logic. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
All of modern mathematics involves logical relationships among
mathematical concepts. In this course we focus on these relationships themselves rather than the ideas they relate. Inevitably this
leads to a study of the (formal) languages suitable for expressing
mathematical ideas. The explicit goal of the course is the study
of propositional and first-order logic; the implicit goal is an
improved understanding of the logical structure of mathematics.
Students should have some previous experience with abstract mathematics
and proofs, both because the course is largely concerned with theorems and proofs and because the formal logical concepts will
be much more meaningful to a student who has already encountered these concepts informally. No previous course in logic is prerequisite.
In the first third of the course the notion of a formal language
is introduced and propositional connectives (and, or, not, implies), tautologies and tautological consequence are studied.
The heart of the course is the study of first-order predicate
languages and their models. The new elements here are quantifiers
('there exists' and 'for all'). The study of the notions of truth, logical consequence, and provability lead to the completeness
and compactness theorems. The final topics include some applications
of these theorems, usually including non-standard analysis. Math
681, the graduate introductory logic course, also has no specific
logic prerequisite but does presuppose a much higher general level
of mathematical sophistication. Philosophy 414 may cover much
of the same material with a less mathematical orientation. Math
481 is not explicitly prerequisite for any later course, but the
ideas developed have application to every branch of mathematics.
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497. Topics in Elementary Mathematics. Math. 489. (3). (Excl). (BS). May be repeated for a total of six credits.
This is an elective course for elementary teaching certificate
candidates that extends and deepens the coverage of mathematics
begun in the required two-course sequence Math 385-489. Topics
are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problem-solving are emphasized.
The class meets three times per week in recitation sections. Grades
are based on class participation, two one-hour exams, and a final
exam. Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and in-service elementary, middle, or junior-high school teachers. Content will vary from
term to term.
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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 (e.g.,
Math 451 or 512) is recommended. Topics are selected from: vector
spaces over arbitrary fields (including finite fields); linear
transformations, bases, and matrices; eigenvalues and eigenvectors;
applications to linear and linear differential equations; bilinear
and quadratic forms; spectral theorem; Jordan Canonical Form.
Math 419 covers much of the same material using the same text, but there is more stress on computation and applications. Math
217 is similarly proof-oriented but significantly less demanding than Math 513. Math 417 is much less abstract and more concerned
with applications. The natural sequel to Math 513 is 593. Math
513 is also prerequisite to several other courses (Math 537, 551, 571, and 575) and may always be substituted for Math 417 or 419.
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520. Life Contingencies I. Math. 424 and Math. 425. (3). (Excl). (BS).
The goal of this course is to teach the basic actuarial theory
of mathematical models for financial uncertainties, mainly the
time of death. In addition to actuarial students, this course
is appropriate for anyone interested in mathematical modeling
outside of the physical sciences. Concepts and calculation are
emphasized over proof. The main topics are the development of
(1) probability distributions for the future lifetime random variable, (2) probabilistic methods for financial payments depending on
death or survival, and (3) mathematical models of actuarial reserving.
523 is a complementary course covering the application of stochastic
process models. Math 520 is prerequisite to all succeeding actuarial
courses. Math 521 extends the single decrement and single life
ideas of 520 to multi-decrement and multiple-life applications
directly related to life insurance and pensions. The sequence
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. Math 522 applies the models of 520
to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc. Recommended
text: Actuarial Mathematics (Second Editions) by Bowles et al. (Huntington)
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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, etc.)
and have at least Junior standing. Two major problems will
be considered: (1) modeling of payouts of a financial intermediary
when the amount and timing vary stochastically over time, and (2) modeling of the ongoing solvency of a financial intermediary
subject to stochastically varying capital flow. These topics will
be treated historically beginning with classical approaches and proceeding to more dynamic models. Connections with ordinary and partial differential equations will be emphasized. Classical approaches
to risk including the insurance principle and the risk-reward
tradeoff. Review of probability. Bachelier and Lundberg models
of investment and loss aggregation. Fallacy of time diversification
and its generalizations. Geometric Brownian motion and the compound
Poisson process. Modeling of individual losses which arise in
a loss aggregation process. Distributions for modeling size loss, statistical techniques for fitting data, and credibility. Economic
rationale for insurance, problems of adverse selection and moral
hazard, and utility theory. The three most significant results
of modern finance: the Markowitz portfolio selection model, the
capital asset pricing model of Sharpe, Lintner and Moissin, and (time permitting) the Black-Scholes option pricing model.
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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.
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Times, Location, and Availability
526/Stat. 526. Discrete State Stochastic Processes. Math. 525 or EECS 501. (3). (Excl). (BS).
This is a course on the theory and applications of stochastic
processes, mostly on discrete state spaces. It is a second course
in probability which should be of interest to students of mathematics
and statistics as well as students from other disciplines in which
stochastic processes have found significant applications. The
material is divided between discrete and continuous time processes.
In both, a general theory is developed, and detailed study is
made of some special classes of processes and their applications.
Some specific topics include generating functions; recurrent events
and the renewal theorem; random walks; Markov chains; branching
processes; limit theorems; Markov chains in continuous time with
emphasis on birth and death processes and queuing theory; an introduction
to Brownian motion; stationary processes and martingales. This
course is similar to EECS 502 and IOE 515, although the latter
course tends to be somewhat more oriented to applications. The
next courses in probability are Math 625 and 626, which presuppose
substantial additional background (Math 597).
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537. Introduction to Differentiable
Manifolds. Math. 513 and 590. (3). (Excl). (BS).
This course in intended for students with a strong background
in topology, linear algebra, and multivariable advanced calculus
equivalent to the courses 590, 513, and 551. Its goal is to introduce the basic concepts and results of Differential Topology and Differential
Geometry. Topics may include: Inverse and Implicit function theorem
in Rn, differentiable manifolds, tangent and cotangent
bundles, exterior differential forms, vector fields, partitions
of unity, integration on manifolds, Stokes' Theorem, the divergence theorem. Topics in Riemannian Geometry include Riemannian metrics, covariant differentiation and connections, torsion tensor, Levi-Civita
connection, Riemann curvature tensor, Gaussian, sectional, Ricci, scalar and mean curvatures, 2-dimensional case, hypersurface case, Gauss and Codazzi equations, length and energy of curves, geodesics, completeness (Hopf-Rinow Theorem), exponential map, Cartan-Hadamard
Theorem. Math 433 is an undergraduate version which covers much
less material in a less sophisticated way.
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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.
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556. Methods of Applied Mathematics I. Math. 555 or 554. (3). (Excl). (BS).
This is an introduction to methods of applied analysis with
emphasis on Fourier analysis for differential equations. Initial
and boundary value problems are covered. Students are expected
to master both the proofs and applications of major results. The
prerequisites include linear algebra, advanced calculus and complex
variables. Topics may vary with the instructor but often include:
Fourier series; separation of variables for partial differential
equations; heat conduction, wave motion, electrostatic fields;
Sturm-Liouville problems; Fourier transform; Green's functions;
distributions; Hilbert space, complete orthonormal sets; integral
operators; spectral theory for compact self-adjoint operators.
Math 454 is an undergraduate course on the same topics.
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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. (Murty)
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Times, Location, and Availability
562/IOE 511/Aero. 577/EECS 505/CS 505. Continuous Optimization Methods. Math. 217, 417 or 419. (3). (Excl). (BS).
Survey of continuous optimization problems. Unconstrained optimization
problems: unidirectional search techniques, gradient, conjugate
direction, quasi-Newtonian methods; introduction to constrained
optimization using techniques of unconstrained optimization through
penalty transformation, augmented Lagrangians, and others; discussion
of computer programs for various algorithms. (Saigal)
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565. Combinatorics and Graph Theory. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
This course has two somewhat distinct halves devoted to Graph
Theory and Enumerative Combinatorics. Proofs, concepts, and calculations
play about an equal role. Students should have taken at least
one proof-oriented course. Graph Theory topics include Trees; k -connectivity; Eulerian and Hamiltonian graphs; tournaments;
graph coloring; planar graphs, Euler's formula, and the 5-Color
Theorem; Kuratowski's Theorem; and the Matrix-Tree Theorem. Enumeration
topics include fundamental principles, bijections, generating
functions, binomial theorem, Catalan numbers, tableaux, partitions
and q -series, linear recurrences and rational generating
functions, and Pólya theory. There is a small overlap with
Math 566, but these are the only courses in combinatorics. 416
is somewhat related but much more concerned with algorithms.
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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.
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575. Introduction to Theory of Numbers I. Math. 451 and 513. Students with credit for Math. 475 can elect Math. 575 for 1 credit. (3). (Excl). (BS).
Many of the results of algebra and analysis were invented to
solve problems in number theory. This field has long been admired
for its beauty and elegance and recently has turned out to be
extremely applicable to coding problems. This course is a survey
of the basic techniques and results of elementary number theory.
Students should have significant experience in writing proofs
at the level of Math 451 and should have a basic understanding
of groups, rings, and fields, at least at the level of Math 412
and preferably Math 512. Proofs are emphasized, but they are often
pleasantly short. A Computational Laboratory (Math 476, 1 credit)
will usually be offered as a supplement to this course. Standard
topics which are usually covered include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder
Theorem, Diophantine equations, primitive roots, quadratic reciprocity
and quadratic fields, application of these ideas to the solution
of classical problems such as Fermat's last 'theorem'. Other topics
will depend on the instructor and may include continued fractions, p-adic numbers, elliptic curves, Diophantine approximation, fast
multiplication and factorization, Public Key Crytography, and transcendence. Math 475 is a non-Honors version of Math 575 which
puts much more emphasis on computation and less on proof. Only the standard topics above are covered, the pace is slower, and the exercises are easier. All of the advanced number theory courses
Math 675, 676, 677, 678, and 679 presuppose the material of Math
575. Each of these is devoted to a special subarea of number theory.
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590. Introduction to Topology. Math. 451. (3). (Excl). (BS).
This is an introduction to topology with an emphasis on the
set-theoretic aspects of the subject. It is quite theoretical
and requires extensive construction of proofs. Topological and metric spaces, continuous functions, homeomorphism, compactness
and connectedness, surfaces and manifolds, fundamental theorem
of algebra, and other topics. Math 490 is a more gentle introduction that is more concrete, somewhat less rigorous, and covers parts
of both Math 590 and 591. Combinatorial and algebraic aspects
of the subject are emphasized over the geometrical. Math 591 is
a more rigorous course covering much of this material and more.
Both Math 591 and 537 use much of the material from Math 590.
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591. General and Differential Topology. Math. 451. (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. Topological
and metric spaces, continuity, subspaces, products and quotient
topology, compactness and connectedness, extension theorems, topological
groups, topological and differentiable manifolds, tangent spaces, vector fields, submanifolds, inverse function theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, classification of surfaces. Math 592 is the natural
sequel.
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593. Algebra I. Math. 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. Students
should have had a previous course equivalent to 512. Topics include
rings and modules, Euclidean rings, principal ideal domains, classification
of modules over a principal ideal domain, Jordan and rational
canonical forms of matrices, structure of bilinear forms, tensor
products of modules, exterior algebras.
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596. Analysis I. Math. 451. (3). (Excl). (BS). Students with credit for Math. 555 may elect Math 596 for two credits only.
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. Review
of analysis in R2 including metric spaces, differentiable maps, Jacobians; analytic functions, Cauchy-Riemann
equations, conformal mappings, linear fractional transformations;
Cauchy's theorem, Cauchy integral formula; power series and Laurent
expansions, residue theorem and applications, maximum modulus theorem, argument principle; harmonic functions; global properties
of analytic functions; analytic continuation; normal families, Riemann mapping theorem. Math 595 covers some of the same material
with greater emphasis on applications and less attention to proofs.
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