Winter '00 Course Guide

Courses in Mathematics (Division 428)

Winter Term, 2000 (January 5 April 26, 2000)

Take me to the Winter Term '00 Time Schedule for Mathematics.


Elementary Mathematics 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. 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 115, 185, and 295 is a first course in calculus and generally credit can be received for only one course from this list. 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.

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 Math 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 an advisor before switching from one sequence to another. In all cases a maximum total of 16 credits may be earned for calculus courses Math 115 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 who have greater interest in theory or who intend to take more advanced courses in mathematics should continue with Math 215 followed by the sequence Math 217-316 (Linear Algebra-Differential Equations). 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.

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.


Math. 105. Data, Functions, and Graphs.

Section 170 and 171 only by Permission of CSP.

Instructor(s):

Prerequisites & Distribution: Students with credit for Math. 103 can elect Math. 105 for only 2 credits. No credit granted to those who have completed any Mathematics course numbered 110 or higher. (4). (MSA). (QR/1).

Full QR

Credits: (4).

Course Homepage: No Homepage Submitted.

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.

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Math. 110. Pre-Calculus (Self-Study).

Section 001 Students In Math 110 Receive Individualized Self-Paced Instruction In the Mathematics Laboratory.

Instructor(s):

Prerequisites & Distribution: 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).

Credits: (2).

Course Homepage: http://www.math.lsa.umich.edu/~meggin/math110.html

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.

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Math. 115. Calculus I.

Section 170 and 173 Only by Permission of CSP

Instructor(s):

Prerequisites & Distribution: 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. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/~meggin/math115/

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-83 is recommended).

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Math. 115. Calculus I.

Section 100 Students must Go to the Math Lab During the First Full Week of Classes.

Instructor(s):

Prerequisites & Distribution: 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. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).

No Description Provided

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Math. 116. Calculus II.

Instructor(s):

Prerequisites & Distribution: Math. 115. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/courses/116/index.shtml

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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Math. 127. Geometry and the Imagination.

Section 001.

Instructor(s):

Prerequisites & Distribution: Three years of high school mathematics including a geometry course. Only first-year students, including those with sophomore standing, may pre-register for First-Year Seminars. All others need permission of instructor. No credit granted to those who have completed a 200- (or higher) level mathematics course. (4). (MSA). (BS). (QR/1).

Full QR First-Year Seminar,

Credits: (4).

Course Homepage: No Homepage Submitted.

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, etc. This course is intended for students who want an introduction to mathematical ideas and culture. Emphasis on conceptual thinking students will do hands-on experimentation with geometric shapes, patterns, and ideas. Grades based on homework and a final project. No exams. Text: Beyond the Third Dimension (Thomas Banchoff, 1990).

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Math. 147. Introduction to Interest Theory.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 112 or 115. No credit granted to those who have completed a 200- (or higher) level mathematics course. (3). (MSA). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

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 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.

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Math. 176. Dynamical Systems and Calculus.

Section 001.

Instructor(s):

Prerequisites & Distribution: Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: No Homepage Submitted.

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. UNIX workstations will be a valuable experimental tool in this course, and students will run preset lab routines on them using Matlab and MAPLE. The general theme of the course will be discrete-time and continuous-time dynamical systems. Examples of dynamical systems arising in the sciences are used as motivation. Topics include: iterates of functions, simple ordinary differential equations, fixed points, attracting and repelling fixed points and periodic orbits, ordered and chaotic motion, self-similarity, and fractals. Tools such as limits and continuity, Taylor expansions of functions, exponentials, logarithms, eigenvalues, and eigenvectors are reviewed or introduced as needed. There is a weekly computer work-station lab.

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Math. 186. Honors Calculus II.

Instructor(s):

Prerequisites & Distribution: Permission of the Honors advisor. Credit is granted for only one course from among Math. 114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: No Homepage Submitted.

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 not restricted to students enrolled in the LS&A Honors Program.

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.

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Math. 214. Linear Algebra and Differential Equations.

Section 001.

Instructor(s): Morton Brown (mbrown@umich.edu)

Prerequisites & Distribution: Math 115 and 116. No credit granted to those who have completed or are enrolled in Math 216. (4). (MSA).

Credits: (4).

Course Homepage: No Homepage Submitted.

Winter Term 2000 the Department of Mathematics will offer a new 4-credit course, Math 214, Linear Algebra and Differential Equations. The prerequisite is Math 116 or equivalent. The course is intended for second-year students who might otherwise take Math 216 (Introduction to Differential Equations) but who have a greater need or desire to study Linear Algebra. This may include some Engineering students, particularly from Industrial and Operations engineering (IOE), as well as students of Economics and other quantitative social sciences. Students intending to concentrate in Mathematics must continue to elect Math 217.

While Math 216 includes 3-4 weeks of Linear Algebra as a tool in the study of Differential Equations, Math 214 will include roughly 3 weeks of Differential Equations as an application of Linear Algebra. The textbook is Linear Algebra and its Applications by David Lay.

The following is a tentative outline of the course:

Regular problem sets and exams.

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Math. 215. Calculus III.

Instructor(s):

Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/courses/215/

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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Math. 216. Introduction to Differential Equations.

Instructor(s):

Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math 214. (4). (MSA). (BS).

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/courses/216/

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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Math. 217. Linear Algebra.

Instructor(s):

Prerequisites & Distribution: 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).

Full QR

Credits: (4).

Course Homepage: No Homepage Submitted.

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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Math. 255. Applied Honors Calculus III.

Instructor(s):

Prerequisites & Distribution: Math. 156. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS).

Credits: (4).

Course Homepage: No Homepage Submitted.

Multivariable calculus, line, surface, and volume integrals; vector fields, Green's theorem, Stokes theorem; divergence theorem, applications. Maple will be used throughout.

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Math. 286. Honors Differential Equations.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 285. Credit can be earned for only one of Math. 216, 256, 286, or 316. (3). (MSA). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

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.

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Math. 289. Problem Seminar.

Section 001 (Drop/Add deadline=January 25).

Instructor(s):

Prerequisites & Distribution: (1). (Excl). (BS). May be repeated for credit with permission.

Mini/Short course

Credits: (1).

Course Homepage: No Homepage Submitted.

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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Math. 296. Honors Mathematics II.

Section 001.

Instructor(s):

Prerequisites & Distribution: 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, 176, 186, and 296. (4). (Excl). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: No Homepage Submitted.

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 not restricted to students enrolled in the LS&A Honors Program. The precise content depends on material covered in 295 but will generally include topics such as infinite series, power series, Taylor expansion, metric spaces. Other topics may include applications of analysis, Weierstrass Approximation theorem, elements of topology, introduction to linear algebra, complex numbers.

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Math. 312. Applied Modern Algebra.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 217. Only one credit granted to those who have completed Math. 412. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/courses/312/

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.

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Math. 316. Differential Equations.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 215 and 217. Credit can be earned for only one of Math. 216, 256, 286, or 316. No credit granted to those who have completed Math. 404. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

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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Math. 333. Directed Tutoring.

Instructor(s):

Prerequisites & Distribution: 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.

Credits: (1-3).

Course Homepage: No Homepage Submitted.

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 teachers.

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Math. 354. Fourier Analysis and its Applications.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math. 454. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

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 e.g., signal processing, Fourier optics), and applications in other branches of mathematics e.g., partial differential equations, probability theory, number theory). Students will do some computer work, using MATLAB, an interactive programming tool that is easy to use, but no previous experience with computers is necessary.

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Math. 371/Engin. 371. Numerical Methods for Engineers and Scientists.

Section 001.

Instructor(s):

Prerequisites & Distribution: Engineering 101, and Math. 216. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students.

Credits: (3).

Lab Fee: CAEN lab access fee required for non-Engineering students.

Course Homepage: No Homepage Submitted.

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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Math. 396. Honors Analysis II.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 395. (4). (Excl). (BS).

Credits: (4).

Course Homepage: No Homepage Submitted.

This course is a continuation of Math 395 and has the same theoretical emphasis. Students are expected to understand and construct proofs. Differential and integral calculus of functions on Euclidean spaces. Students who have successfully completed the sequence Math 295-396 are generally prepared to take a range of advanced undergraduate and graduate courses such as Math 512, 513, 525, 590, and many others.

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Math. 399. Independent Reading.

Instructor(s):

Prerequisites & Distribution: (1-6). (Excl). (INDEPENDENT). May be repeated for credit.

Credits: (1-6).

Course Homepage: No Homepage Submitted.

Designed especially for Honors students.

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Math. 404. Intermediate Differential Equations and Dynamics.

Section 001.

Instructor(s): Dmitry Zenkov (zenkov@umich.edu)

Prerequisites & Distribution: Math. 216, 256 or 286, or Math. 316. No credit granted to those who have completed Math. 256, 286, or 316. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Math 404 is a second course in ordinary differential equations at an intermediate level designed for advanced undergraduate and beginning graduate math/physics/engineering students.

Since their invention by I. Newton, differential equations have been consistently used in mechanics and physics. In our days the range of applications of differential equations extends from physics ad engineering to chemistry and biology.

This course is an introduction to the modern qualitative theory of ordinary differential equations with emphasis on geometric technique and visualization. Much of the motivation for this approach comes from applications. Examples of applications of differential equations to science and engineering are a significant part of this course.

Outline:

Course requirements: Homework, midterm, and final exams.

Text: Dynamical Systems by D. K. Arrowsmith and C.M. Place, Chapman and Hall, 1992

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Math. 412. Introduction to Modern Algebra.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 215, 255, 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).

No Description Provided

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Math. 417. Matrix Algebra I.

Instructor(s):

Prerequisites & Distribution: Three courses beyond Math. 110. No credit granted to those who have completed or are enrolled in 217, 419, or 513. (3). (Excl). (BS).

No Description Provided

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Math. 419/EECS 400/CS 400. Linear Spaces and Matrix Theory.

Instructor(s):

Prerequisites & Distribution: 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). CAEN lab access fee required for non-Engineering students.

No Description Provided

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Math. 419/EECS 400/CS 400. Linear Spaces and Matrix Theory.

Section 004.

Instructor(s): Dmitry Zenkov (zenkov@umich.edu)

Prerequisites & Distribution: 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). CAEN lab access fee required for non-Engineering students.

No Description Provided

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Math. 424. Compound Interest and Life Insurance.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).

No Description Provided

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Math. 425/Stat. 425. Introduction to Probability.

Instructor(s):

Prerequisites & Distribution: Math. 215, 255, or 285. (3). (MSA). (BS).

No Description Provided

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Math. 425/Stat. 425. Introduction to Probability.

Instructor(s):

Prerequisites & Distribution: Math. 215, 255, or 285. (3). (MSA). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

See Statistics 425..

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Math. 427/Human Behavior 603 (Social Work). Retirement Plans and Other Employee Benefit Plans.

Instructor(s):

Prerequisites & Distribution: Junior standing. (3). (Excl).

No Description Provided

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Math. 450. Advanced Mathematics for Engineers I.

Instructor(s):

Prerequisites & Distribution: Math. 216, 256, 286, or 316. (4). (Excl). (BS).

No Description Provided

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Math. 451. Advanced Calculus I.

Instructor(s):

Prerequisites & Distribution: Math. 215 and one course beyond Math. 215; or Math. 255 or 285. Intended for concentrators; other students should elect Math. 450. (3). (Excl). (BS).

No Description Provided

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Math. 452. Advanced Calculus II.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 217, 417, or 419; and Math. 451. (3). (Excl). (BS).

No Description Provided

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Math. 454. Boundary Value Problems for Partial Differential Equations.

Section 001.

Instructor(s): Timothy Callahan (timcall@umich.edu)

Prerequisites & Distribution: Math. 216, 256, 286, or 316. Students with credit for Math. 354 can elect Math. 454 for one credit. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2000/winter/lsa/math/454/001.nsf

This course covers methods of solving partial differential equations (e.g., the heat, wave, Helmholtz and Laplace equations), with specified boundary conditions in various geometries. We will cover separation of variables, Fourier series, Bessel functions, spherical harmonics, orthogonal polynomials, Sturm Liouville theory, eigenfunctions of the Laplacian in several different coordinate systems, Fourier and Bessel transforms, conformal mapping, etc. These methods have applications in fields as diverse as mechanics, quantum mechanics, thermodynamics, aerodynamics, finance, electromagnetism, and many others, and we will take our examples from such disciplines.

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Math. 463. Mathematical Modeling in Biology.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 217, 417, or 419; 286, 256, or 316. (3). (Excl). (BS).

No Description Provided

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Math. 471. Introduction to Numerical Methods.

Instructor(s):

Prerequisites & Distribution: Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (3). (Excl). (BS).

No Description Provided

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Math. 475. Elementary Number Theory.

Section 001.

Instructor(s):

Prerequisites & Distribution: At least three terms of college mathematics are recommended. (3). (Excl). (BS).

No Description Provided

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Math. 476. Computational Laboratory in Number Theory.

Section 001.

Instructor(s):

Prerequisites & Distribution: Prior or concurrent enrollment in Math. 475 or 575. (1). (Excl). (BS).

No Description Provided

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Math. 486. Concepts Basic to Secondary Mathematics.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).

No Description Provided

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Math. 489. Mathematics for Elementary and Middle School Teachers.

Instructor(s):

Prerequisites & Distribution: Math. 385 or 485. May not be used in any graduate program in mathematics. (3). (Excl).

No Description Provided

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Math. 490. Introduction to Topology.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).

No Description Provided

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Math. 512. Algebraic Structures.

Section 001.

Instructor(s): Pasha Belorousski (belorous@umich.edu)

Prerequisites & Distribution: 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).

Credits: (3).

Course Homepage: No Homepage Submitted.

Text: M. Artin, Algebra.

This course will provide a rigorous introduction to the basic structures of algebra: groups, rings and fields. These structures are central in any field of modern mathematics and are ubiquitous in its application to subjects such as crystallography, quantum mechanics, and chemistry, to name a few. The emphasis in the course will be on concepts and proofs. Concrete topics such as symmetry and ruler/compass constructions in the plane will be used to illustrate the more abstract material.

Here is a keyword outline of the route: groups, homomorphisms, quotient groups, rigid motions and symmetry of regular solids, rings, factorization, principal ideal domains, unique factorization domains, fields, algebraic extensions, finite fields, geometric constructions in the plane using ruler and compass. Time permitting, we will make further excursions into the theory of groups: class equation, Sylow theorems, simple groups.

Expected workload: Weekly homework assignments, a midterm and a final exam.

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Math. 513. Introduction to Linear Algebra.

Section 001.

Instructor(s): Natasa Macura (nmacura@umich.edu)

Prerequisites & Distribution: Math. 412 or permission of Honors advisor. 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).

Credits: (3).

Course Homepage: No Homepage Submitted.

Text: C. Curtis, Linear Algebra, An Introductory Approach, Springer

Math 513 is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis of the course will be on concepts and proofs with some calculations to illustrate the theory.

Math 513 is the linear algebra course for students in the Honors Mathematics Program. It is also appropriate for students who have completed one or more proof oriented courses such as math 412, 451, or 512 and who seek a sophisticated course in linear algebra. One of the most important topics in Math 513 is the Jordan Canonical Form, the study of which is an excellent preparation for Math 593.

The course will begin with the discussion of matrices and determinants. We shall than discuss vector spaces over arbitrary fields (including finite fields), linear transformations, diagonalization, applications to linear and linear differential equations, Jordan Canonical Form and the Spectral Theorem. If time permits, we shall study bilinear and quadratic forms.

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Math. 521. Life Contingencies II.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 520. (3). (Excl). (BS).

No Description Provided

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Math. 523. Risk Theory.

Section 001.

Instructor(s): Joseph Conlon (conlon@umich.edu)

Prerequisites & Distribution: Math. 425. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~conlon/math523/index.html

Background and Goals: 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. It provides background for the professional exams in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society.

Contents: Standard distributions used for claim frequency models and for loss variables, theory of aggregate claims, compound Poisson claims model, discrete time and continuous time models for the aggregate claims variable, the Chapman-Kolmogorov equation for expectations of aggregate claims variables, the Brownian motion process, estimating the probability of ruin, reinsurance schemes and their implications for profit and risk.

Credibility theory, classical theory for independent events, least squares theory for correlated events, examples of random variables where the least squares theory is exact.

Grading: The grade for the course will be determined from performances on homeworks, a midterm and a final exam.

Required Text: Loss Models-from Data to Decisions, by Klugman, Panjer and Willmot, Wiley 1998.

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Math. 524. Topics in Actuarial Science II.

Section 001 Survival Modes and construction of Mortality Tables

Instructor(s): Curtis Huntington (chunt@umich.edu)

Prerequisites & Distribution: Math. 424, 425, and 520; and Stat. 426. (3). (Excl). (BS). May be repeated for a total of 9 credits.

Credits: (3).

Course Homepage: No Homepage Submitted.

The nature and properties of survival models, including both parametric and tabular models; methods of estimating tabular modes 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; estimation of estimators from sample data; valuation schedule exposure formulae; and practical issues on survival mode estimation.

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Math. 525/Stat. 525. Probability Theory.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 450 or 451. Students with credit for Math. 425/Stat. 425 can elect Math. 525/Stat. 525 for only one credit. (3). (Excl). (BS).

No Description Provided

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Math. 526/Stat. 526. Discrete State Stochastic Processes.

Instructor(s):

Prerequisites & Distribution: Math. 525 or EECS 501. (3). (Excl). (BS).

No Description Provided

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Math. 526/Stat. 526. Discrete State Stochastic Processes.

Section 001 Biological Applications: probabilistic models of proteins and nucleic acids

Instructor(s): Daniel Burns (dburns@umich.edu)

Prerequisites & Distribution: Math. 525 or EECS 501. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

The course this semester will be about probabilistic models of proteins and nucleic acids, and their uses in molecular biology. The topics will include a review of basic concepts of probability and very rudimentary molecular biology; probability and the design of similarity scoring functions; optimal local and global alignments of sequences: dynamic programming, Smith-Waterman algorithm, other algorithms available on the Web (BLAST and FastA, etc.), probabilistic (heuristic) versus rigorous algorithms; significance of scores and simulation; dependence of scoring functions and optimal alignments on parameters, comparison of standard tables; hidden Markov models and neural network models; multiple sequence alignment methods and algorithms, families of proteins; phylogenetic tree determinations; structure of proteins and recognizable patterns in amino acid sequences (motif recognition). Guest lecturers will address the class on applications in the pharmaceutical industry, as well as some earlier examples of these techniques applied to problems in linguistics and speech recognition.

Students will be expected to complete three to four problem sets, most of which will hopefully be group projects, some of which will involve using Web-based tools. If the class demographics work out favorably, we will be mixing students with biological background and mathematical background in each group. Every effort will be made to accommodate students from diverse backgrounds.

Text:

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Math. 528. Topics in Casualty Insurance.

Section 001.

Instructor(s): Curtis Huntington (chunt@umich.edu)

Prerequisites & Distribution: Math 217, 417, or 419. (3). (Excl).

Credits: (3).

Course Homepage: No Homepage Submitted.

The insurance policy is the contract describing the services and protection which the insurance company provides to the insured. This course will develop an understanding of the nature of the coverages provided and the bases of exposure used in the respective product lines. It will explore the basic purpose and principles of the underwriting function, how products are designed and modified and the different marketing systems. It will also look at how claims are settled since this determines losses which are key components for insurance ratemaking and reserving. Finally, the course will explore basic ratemaking principles and concepts of loss reserving.

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Math. 531. Transformation Groups in Geometry.

Section 001.

Instructor(s): Kenneth Bromberg (kbromber@umich.edu)

Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Isometries and congruences in the Euclidean plane as generated by reflections, translations and half-turns. Tilings, affine and hyperbolic geometries, Poincaré model of the hyperbolic plane. Selected applications to ornamental design crystallography, and regular polytopes.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2

Math. 555. Introduction to Functions of a Complex Variable with Applications.

Section 001.

Instructor(s): John Erik Fornaess (fornaess@umich.edu)

Prerequisites & Distribution: Math. 450 or 451. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Text: Complex Variables and Applications, 5th ed. (Churchill and Brown);

Student Body: largely engineering and physics graduate students with some math and engineering undergrads

Background and Goals: This course is an introduction to the theory of complex valued functions of a complex variable. Concepts and calculations are emphasized over proofs.

Content: Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals. This corresponds to Chapters 1-9 of Churchill.

Alternatives: Math 596 (Analysis I (Complex)) covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than calculations.

Subsequent Courses: Math 555 is prerequisite to many advanced courses in science and engineering fields.

There will be homework, midterm and a final.

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Math. 557. Methods of Applied Mathematics II.

Section 001.

Instructor(s): Paul Federbush (pfed@umich.edu)

Prerequisites & Distribution: Math 217, 419, or 513; 451 and 555. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

In applied mathematics, we try to understand a physical process by formulating and analyzing a mathematical model. Many models consist of a differential equation with initial and boundary conditions. Most of the time especially if the equation is nonlinear, an explicit formula for the solution is not available. Even if we are clever or lucky enough to find such a formula, it may be difficult to extract useful information from it. In practice, we must settle for a sufficiently accurate approximate solution obtained by numerical simulation or asymptotic analysis (or a combination of the two). This course is an introduction to asymptotic analysis. It is aimed at graduate and advanced undergraduate students in engineering, mathematics and science. The main prerequisite is complex analysis (e.g., Math 55 or Math 596). Math 556 is not a prerequisite. Murray's text will occupy 2/3 of the course. In the remaining time, I will present topics in PDE and fluid dynamics.

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Math. 561/SMS 518 (Business Administration)/IOE 510. Linear Programming I.

Section 001.

Instructor(s): Katta Murty (murty@umich.edu)

Prerequisites & Distribution: Math. 217, 417, or 419. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students.

No Description Provided

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Math. 566. Combinatorial Theory.

Section 001.

Instructor(s): Thomas Storer

Prerequisites & Distribution: Math. 216, 256, 286, or 316. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

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.

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Math. 567. Introduction to Coding Theory.

Section 001.

Instructor(s): Trevor Wooley (wooley@umich.edu)

Prerequisites & Distribution: One of Math 217, 419, 513. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Discover the secrets underlying the "Digital Age''! Error Correcting Codes are fundamental to modern data transmission (telecommunications, the Internet) and data storage (compact disks, DVD's, etc.) ... this is a course which introduces the mathematical theory of coding, and prepares you for the next millennium.

This course focuses on the mathematical background for linear error correcting codes. We begin with a discussion of Shannon's theorem and channel capacity. The definition of linear codes will be given along with a review of the necessary tools from linear algebra (including a review of/introduction to abstract algebra and finite fields). Basic examples of codes will be discussed, including the Hamming, BCH, cyclic, Reed-Muller and Reed-Solomon codes.

We also discuss the problem of decoding, starting with syndrome decoding and covering weight enumerator polynomials and the MacWilliams-Sloane identity. Following these basic topics, we will discuss topics of interest to the audience and instructor, amongst which may be a consideration of asymptotic parameters and bounds, algebraic-geometric codes, and a brief introduction to cryptography.

There will be at most 9 problem sets throughout the academic term, an in-class midterm, and a take-home final.

Primary Text: J. H. van Lint, Introduction to Coding Theory, 3rd Edition, Springer-Verlag GTM 86, 1999. Useful sources: "Coding Theory: the essentials", by Hoffman, Leonard, Lindner, Phelps, Rodger and Wall, Marcel Dekker, 1991. "Foundations of Coding Theory", by Jiri Adamek, Wiley, 1991.

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Math. 571. Numerical Methods for Scientific Computing I.

Section 001.

Instructor(s): Alexander Kurganov (kurganov@umich.edu)

Prerequisites & Distribution: Math. 217, 417, 419, or 513; and one of Math. 450, 451, or 454. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

This course is an introduction to numerical methods for solving linear systems of equations (Ax=b) and for computing eigenvalues of a matrix. Topics: singular value decomposition, QR factorization, Gram-Schmidt orthogonalization, least squares problems, condition number, Gaussian elimination, iterative methods (Arnoldi, GMRES, conjugate gradient), preconditioning, methods for computing eigenvalues (e.g., power method, inverse iteration, QR algorithm, shifts).

Text: Numerical Linear Algebra, L.N. Trefethen & D. Bau, SIAM

Prerequisites: a course in linear algebra on the level of Math 417, 419, or 513, computer programming (any language)

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Math. 572. Numerical Methods for Scientific Computing II.

Section 001.

Instructor(s): Peter Smereka (psmereka@umich.edu)

Prerequisites & Distribution: Math. 217, 417, 419, or 513; and one of Math. 450, 451, or 454. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

This is an introduction to numerical methods for initial value problems. The course will cover numerical methods for ordinary differential equations and both linear parabolic and hyperbolic partial differential equations. We also plan to discuss the numerical solution of nonlinear hyperbolic equations. This course should be useful to students in mathematics, physics, and engineering. Homework assignments will be a crucial part of the class; students must know how to program and use elementary computer graphics.

Background Required: Strong background in advanced calculus and linear algebra is needed. It would be preferable if the student has taken Math 454 or equivalent. It is mandatory that the students have a knowledge of computing programming in Fortran, c, or even Matlab.

There will be no textbook but the following would be good references:

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Math. 582. Introduction to Set Theory.

Section 001.

Instructor(s): Peter Hinman (pgh@umich.edu)

Prerequisites & Distribution: Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Set Theory is at the same time (1) a branch of mathematics, (2) a tool used in practically every other branch of mathematics, and (3) the best medium for understanding the foundations of mathematics. This course is mainly a study of (1), but much of the motivation comes from (2) and (3) and these aspects will be covered from time to time. Everyone who has taken a course in any sort of abstract mathematics, be it algebra, analysis, or topology, has used notions such as "set", "function" "equivalence relation", "linear ordering", etc.; a low-level goal of the course is to improve familiarity and comfort with these common mathematical notions. Deeper topics are well-orderings, ordinal numbers, cardinal numbers, and their properties. Set theory as a separate discipline really began with Cantor's discovery (in the late 19th century) that infinite sets can have different sizes, and the consequences and refinements of this fact will be a centerpiece of the course. We will also discuss historically troublesome assertions such as the Axiom of Choice and the Continuum Hypothesis.

All of these will be considered from both the non-axiomatic and axiomatic perspectives. The axiomatic approach is both more necessary in set theory than is other branches of mathematics and more fruitful. It is necessary partly because of the discovery that intuitions about sets can easily go astray and lead to paradox and contradiction. It is fruitful because a relatively simple set of axioms suffices to generate all of the theorems of set theory. Since essentially all mathematical notions can be expressed in terms of sets, the axiomatization of set theory is in effect an axiomatization of all of mathematics. Hence the context of axiomatic set theory is well-suited for dealing with the philosophical issue of what it means for a mathematical assertion to be true or provable. These considerations lead to a necessarily brief discussion of consistency and independence results.

The announced prerequisites of Math 412 or 451 have more to do with general level of mathematical sophistication than specific content. The course is well-suited to math majors, Honors or not, beginning graduate students, and mathematically minded students of philosophy or computer science. If you have any doubts about the level of the course, please talk with me. A course in mathematical logic is not presupposed. We will follow the book of Y.N. Moschovakis, Notes on Set Theory (Springer-Verlag, ISBN 0-387-94180-0 and 3-540-94180-0). There will be several problem assignments and perhaps a take-home final exam.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2

Math. 592. Introduction to Algebraic Topology.

Section 001.

Instructor(s): Igor Kriz (ikriz@umich.edu)

Prerequisites & Distribution: Math. 591. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

This is the beginning graduate course in algebraic topology. It is an introduction to basic distinguishing characteristics between topological spaces, called topological invariants. How do we prove, for example, that the 2-dimensional sphere is topologically different from the surface of a donut? The topological invariants to be discussed in this course are the fundamental group and homology. The prerequisite for this class is Math 591. There is no single comprehensive textbook, but suggested texts are W.S. Massey: A basic course in algebraic topology, and J.R. Munkres: Elements of algebraic topology.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2

Math. 594. Algebra II.

Section 001.

Instructor(s): Robert Lazarfeld (rlaz@umich.edu)

Prerequisites & Distribution: Math. 593. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Math 594 is the second half of the introductory ("alpha") graduate course in algebra. We will cover essentially the material on group theory and Galois theory that is covered in the Qualifying Review Exams. The formal text will be: I.M. Isaacs, Algebra (A Graduate Course), Brooks/Cole, although I will not necessarily follow the text very closely.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2

Math. 596. Analysis I.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 451. (3). (Excl). (BS). Students with credit for Math. 555 may elect Math 596 for two credits only.

No Description Provided

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Math. 597. Analysis II.

Section 001.

Instructor(s): Lizhen Ji (lji@umich.edu)

Prerequisites & Distribution: Math. 451 and 513. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Lebesgue measure on the real line. Measurable functions and integration on R. Differentiation theory, fundamental theorem of calculus. Function spaces, Lp(R), C(K), Hölder and Minkowski inequalities, duality. General measure spaces, product measures, Fubini's theorem. Radon-Nikodym theorem, conditional expectation, signed measures.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2

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