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Winter Academic Term 2004 Course Guide

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Courses in Mathematics


This page was created at 7:54 PM on Wed, Jan 21, 2004.

Winter Academic Term, 2004 (January 6 - April 30)


Mathematics is the language and tool of the sciences, a cultural phenomenon with a rich historical traditions, and a model of abstract reasoning. Historically, mathematical methods and thinking have proved extraordinarily successful in physics, and engineering. Nowadays, it is used successfully in many new areas, from computer science to biology and finance. A Mathematics concentration provides a broad education in various areas of mathematics in a program flexible enough to accommodate many ranges of interest.

The study of mathematics is an excellent preparation for many careers; the patterns of careful logical reasoning and analytical problem solving essential to mathematics are also applicable in contexts where quantity and measurement play only minor roles. Thus students of mathematics may go on to excel in medicine, law, politics, or business as well as any of a vast range of scientific careers. Special programs are offered for those interested in teaching mathematics at the elementary or high school level or in actuarial mathematics, the mathematics of insurance. The other programs split between those which emphasize mathematics as an independent discipline and those which favor the application of mathematical tools to problems in other fields. There is considerable overlap here, and any of these programs may serve as preparation for either further study in a variety of academic disciplines, including mathematics itself, or intellectually challenging careers in a wide variety of corporate and governmental settings.

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 admission 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 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 taught through the Math Lab and is only open to students in MATH 115 who find that they need additional preparation to successfully complete the course. A maximum total of 4 credits may be earned in courses numbered 103, 105, and 110. 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 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. Students need not be enrolled in the LS&A Honors Program to enroll in any of these courses but must have the permission of an Honors advisor. Students with strong preparation and interest in mathematics are encouraged to consider these courses.

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 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 problem 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. MATH 295-396 is excellent preparation for 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 is one course expressly designed and recommended for students with one or two semesters of AP credit, MATH 156. 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, student 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 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 Equation -- 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 to optimal treatment of differential equations in MATH 316. MATH 216 is not intended for mathematics concentrators.

Special Departmental Policies. All prerequisite courses must be satisfied with a grade of C- or above. Students with lower grades in prerequisite courses must receive special permission of the instructor to enroll in subsequent courses.


MATH 105. Data, Functions, and Graphs.

Instructor(s):

Prerequisites & Distribution: (4). (MSA). (QR/1). May not be repeated for credit. 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. A maximum of four credits may be earned in MATH 101, 103, 105, and 110.

Full QR

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/courses/105/

MATH 105 serves both as a preparatory course to the calculus sequences and as a terminal course for students who need only this level of mathematics. Students who complete MATH 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.

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

MATH 107. Mathematics for the Information Age.

Section 001.

Instructor(s): Karen Rhea

Prerequisites & Distribution: Three to four years high school mathematics. (3). (MSA). (QR/1). May not be repeated for credit.

Full QR

Credits: (3).

Course Homepage: No homepage submitted.

From computers and the Internet to playing a CD or running an election, great progress in modern technology and science has come from understanding how information is exchanged, processed and perceived.

Typical topics: cryptography, error-correcting codes, data compression, fairness in politics, voting systems, population growth, and biological modeling.

No prerequisites. The course is particularly suited for students wanting to satisfy distribution requirements and not necessarily heading for calculus.

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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. (2). (Excl). May not be repeated for credit. No credit granted to those who already have 4 credits for pre-calculus mathematics courses. A maximum of four credits may be earned in MATH 101, 103, 105, and 110.

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 MATH 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.

ENROLLMENT IN MATH 110 IS BY PERMISSION OF MATH 115 INSTRUCTOR ONLY. COURSE MEETS SECOND HALF OF THE TERM. STUDENTS WORK INDEPENDENTLY WITH GUIDANCE FROM MATH LAB STAFF.

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MATH 115. Calculus I.

Instructor(s):

Prerequisites & Distribution: Four years of high school mathematics. See Elementary Courses above. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit usually is granted for only one course from among 115, 185, and 295. No credit granted to those who have completed MATH 175.

Full QR

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/courses/115/

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 MATH 115 and 116) and a graphing calculator (the Texas Instruments TI-83 is recommended).

TEXT: Calculus, 3rd edition, Hughes-Hallet, Wiley Publishing.
TI-83 Graphing Calculator, Texas Instruments.

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MATH 116. Calculus II.

Instructor(s):

Prerequisites & Distribution: MATH 115. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit is granted for only one course from among MATH 116, 156, 176, and 186.

Full QR

Credits: (4).

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

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, and 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.

Text: Calculus, 3rd Edition, Hughes-Hallet/Gleason, Wiley Publishing.
TI-83 Graphing Calculator, Texas Instruments.

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

MATH 127. Geometry and the Imagination.

Section 001.

Instructor(s): Emina Alibegovic

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. (4). (MSA). (BS). (QR/1). May not be repeated for credit. No credit granted to those who have completed a 200- (or higher) level mathematics course (except for MATH 385 and 485).

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.

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

MATH 147. Introduction to Interest Theory.

Section 001.

Instructor(s): Yulai Yang

Prerequisites & Distribution: MATH 115. (3). (MSA). (BS). May not be repeated for credit. No credit granted to those who have completed a 200- (or higher) level mathematics course.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/147/001.nsf

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 186. Honors Calculus II.

Instructor(s):

Prerequisites & Distribution: Permission of the Honors advisor. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit is granted for only one course from among MATH 116, 156, 176, 186, and 296.

Full QR

Credits: (4).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 5: Students in LSA College Honors may request overrides from the Honors Office; other students may request them from the Math Dept Office, 2084 East Hall.

MATH 214. Linear Algebra and Differential Equations.

Instructor(s):

Prerequisites & Distribution: MATH 115 and 116. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in MATH 513.

Full QR

Credits: (4).

Course Homepage: No homepage submitted.

This 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 following is a tentative outline of the course:

  • Systems of linear equations, matrices, row operations, reduced row echelon form, free variables, basic variables, basic solution, parametric description of the solution space. Rank of a matrix.
  • Vectors, vector equations, vector algebra, linear combinations of vectors, the linear span of vectors.
  • The matrix equation Ax = b. Algebraic rules for multiplication of matrices and vectors.
  • Homogeneous systems, principle of superposition.
  • Linear independence.
  • Applications, Linear models.
  • Matrix algebra, dot product, matrix multiplication.
  • Inverse of a matrix.
  • Invertible matrix theorem.
  • Partitioned matrices.
  • 2-dimensional discrete dynamical systems.
  • Markov process, steady state.
  • Transition matrix, eigenvector, steady state lines (affine hulls).
  • Geometry of two and three dimensions: affine hulls, linear hulls, convex hulls, half planes, distance from point to a plane, optimization.
  • Introduction to linear programming.
  • The geometry of transition matrices in 2 dimensions (rotations, shears, ellipses, eigenvectors).
  • Transition matrices for 3-D (rotations, orthogonal matrices, symmetric matrices)
  • Determinants.
  • 2- and 3-dimensional determinant as area and volume.
  • Eigenvectors and Eigenvalues.
  • Eigenvectors.
  • Complex numbers including Euler's formula.
  • Complex eigenvalues and their geometric meaning.
  • Review of ordinary differential equations.
  • Systems of ordinary differential equations in 2 dimensions.

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. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 215, 255, or 285.

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.

TEXT: STUDENTS HAVE CHOICE OF EITHER:
Calculus, 5th edition, James Stewart, Brooks/Cole Publishing,
or
Multivariable Calculus, 5th edition, James Stewart, Brooks/Cole Publishing.

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MATH 216. Introduction to Differential Equations.

Instructor(s):

Prerequisites & Distribution: MATH 116, 119, 156, 176, 186, or 296. Not intended for Mathematics concentrators. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 216, 256, 286, or 316.

Full QR

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 MATH 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. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in MATH 513.

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; and 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. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 215, 255, or 285.

Full QR

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.

Text: Multivariable Calculus, 4th edition, James Stewart, Brooks/Cole.

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MATH 286. Honors Differential Equations.

Instructor(s): Berit Stensones

Prerequisites & Distribution: MATH 285. (3). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 216, 256, 286, or 316.

Full QR

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

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

MATH 289. Problem Seminar.

Section 001.

Instructor(s): Hendrikus Gerardus Derksen

Prerequisites & Distribution: (1). (Excl). (BS). May be repeated for credit. Repetition requires permission of the department.

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): Brian D Conrad

Prerequisites & Distribution: Prior knowledge of first year calculus and permission of the Honors advisor. (4). (Excl). (BS). (QR/1). May not be repeated for credit. Credit is granted for only one course from among MATH 156, 176, 186, and 296.

Full QR

Credits: (4).

Course Homepage: No homepage submitted.

The sequence MATH 295-296-395-396 is a more intensive honors sequence than MATH 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 MATH 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.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 5: Students in LSA College Honors may request overrides from the Honors Office; other students may request them from the Math Dept Office, 2084 East Hall.

MATH 310. Elementary Topics in Mathematics.

Section 001 — Math Games & Theory of Games.

Instructor(s): Morton Brown

Prerequisites & Distribution: Two years of high school mathematics. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

The current offering of the course focuses on game theory. Students study the strategy of several games where mathematical ideas and concepts can play a role. Most of the course will be occupied with the strructure of a variety of two person games of strategy: tic-tac-toe, tic-tac-toe misere, the French military game, hex, nim, the penny dime game, and many others. If there is sufficient interest students can study: dots and boxes, go moku, and some aspects of checkers and chess. There will also be a brief introduction to the classical Von Neuman/Morgenstern theory of mixed strategy games.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 1, 5: Permission of Instructor

MATH 312. Applied Modern Algebra.

Section 001.

Instructor(s): Patricia Lynn Hersh

Prerequisites & Distribution: MATH 217. (3). (Excl). (BS). May not be repeated for credit. Only one credit granted to those who have completed MATH 412.

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: the integers "mod n" and linear algebra over the integers mod p, with applications to error correcting codes; some number theory, with applications to public-key cryptography; polynomial algebra, with an emphasis on factoring algorithms over various fields, and 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): Arthur G Wasserman

Prerequisites & Distribution: MATH 215 and 217. (3). (Excl). (BS). May not be repeated for credit. Credit can be earned for only one of MATH 216, 256, 286, or 316.

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 MATH 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 327. Evolution of Mathematical Concepts.

Section 001.

Instructor(s): Alejandro Uribe-Ahumada

Prerequisites & Distribution: MATH 116 or 186. (3). (Excl). (BS). May not be repeated for credit.

Upper-Level Writing

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/327/001.nsf

No Description Provided. Contact the Department.

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MATH 333. Directed Tutoring.

Instructor(s):

Prerequisites & Distribution: Enrollment in the secondary teaching certificate program with concentration in mathematics. Permission of instructor required. (1-3). (Excl). (EXPERIENTIAL). May be repeated for credit for a maximum of 3 credits. Offered mandatory credit/no credit.

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.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 5, Permission of Instructor

MATH 351. Principles of Analysis.

Section 001.

Instructor(s): Morton Brown

Prerequisites & Distribution: MATH 215 and 217. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 451.

Credits: (3).

Course Homepage: No homepage submitted.

The content of this course is similar to that of MATH 451 but MATH 351 assumes less background. This course covers topics that might be of greater use to students considering a Mathematical Sciences concentration or a minor in Math. Course content includes: analysis of the real line, rational and irrational numbers, infinity — large and small, limits, convergence, infinite sequences and series, continuous functions, power series and differentiation.

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MATH 354. Fourier Analysis and its Applications.

Section 001.

Instructor(s): Mahdi Asgari

Prerequisites & Distribution: MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 450 or 454.

Credits: (3).

Course Homepage: No homepage submitted.

This course 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 / ENGR 371. Numerical Methods for Engineers and Scientists.

Section 001.

Instructor(s): David Gammack

Prerequisites & Distribution: ENGR 101; one of MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in Math 471. 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. Convergence theorems are discussed and applied, but the proofs are not emphasized.

Objectives of the course

  • Develop numerical methods for approximately solving problems from continuous mathematics on the computer
  • Implement these methods in a computer language (MATLAB)
  • Apply these methods to application problems

Computer language: In this course, we will make extensive use of Matlab, a technical computing environment for numerical computation and visualization produced by The MathWorks, Inc. A Matlab manual is available in the MSCC Lab. Also available is a MATLAB tutorial written by Peter Blossey.

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MATH 396. Honors Analysis II.

Section 001.

Instructor(s): Mario Bonk

Prerequisites & Distribution: MATH 395. (4). (Excl). (BS). May not be repeated for credit.

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.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: Students in LSA College Honors may request overrides from the Honors Office; other students may request them from the Math Dept Office, 2084 East Hall.

MATH 399. Independent Reading.

Instructor(s):

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

Credits: (1-6).

Course Homepage: No homepage submitted.

Designed especially for Honors students.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 5, Permission of instructor/department

MATH 412. Introduction to Modern Algebra.

Instructor(s):

Prerequisites & Distribution: MATH 215, 255, or 285; and 217. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 512. Students with credit for MATH 312 should take MATH 512 rather than 412. One credit granted to those who have completed MATH 312.

Credits: (3).

Course Homepage: No homepage submitted.

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 or 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, and 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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MATH 417. Matrix Algebra I.

Instructor(s):

Prerequisites & Distribution: Three courses beyond MATH 110. (3). (Excl). (BS). May not be repeated for credit. Credit can be earned for only one of MATH 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled MATH 513.

Credits: (3).

Course Homepage: No homepage submitted.

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 MATH 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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MATH 419. Linear Spaces and Matrix Theory.

Instructor(s):

Prerequisites & Distribution: Four terms of college mathematics beyond MATH 110. (3). (Excl). (BS). May not be repeated for credit. Credit can be earned for only one of MATH 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in MATH 513.

Credits: (3).

Course Homepage: No homepage submitted.

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, and 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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MATH 422 / BE 440. Risk Management and Insurance.

Section 001.

Instructor(s): Curtis E Huntington

Prerequisites & Distribution: MATH 115, junior standing, and permission of instructor. (3). (Excl). (BS). May not be repeated for credit.

Upper-Level Writing

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 1 and Permission of Instructor. If you are interested in this class, please put your name on the waitlist and email Prof. Huntington (chunt@umich.edu).

MATH 423. Mathematics of Finance.

Section 001.

Instructor(s): Moore

Prerequisites & Distribution: MATH 217 and 425; EECS 183. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/423/001.nsf

No Description Provided. Contact the Department.

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MATH 423. Mathematics of Finance.

Section 002.

Instructor(s): Kausch

Prerequisites & Distribution: MATH 217 and 425; EECS 183. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/423/002.nsf

No Description Provided. Contact the Department.

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MATH 424. Compound Interest and Life Insurance.

Section 001.

Instructor(s): David Towler Kausch

Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/424/001.nsf

No Description Provided. Contact the Department.

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MATH 425 / STATS 425. Introduction to Probability.

Instructor(s): Mathematics faculty

Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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. 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. STATS 426 is a natural sequel for students interested in statistics. MATH 523 includes many applications of probability theory.

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MATH 425 / STATS 425. Introduction to Probability.

Instructor(s): Statistics faculty

Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

See STATS 425.

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MATH 425 / STATS 425. Introduction to Probability.

Section 003, 007.

Instructor(s): Sergey Fomin (fomin@umich.edu)

Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~fomin/425w04.html

This course introduces students to the theory of probability and to a number of applications. 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.

There will be approximately 10 problem sets. Grade will be based on two 1-hour midterm exams, 20% each; 20% homework; 40% final exam. pText (required): Sheldon Ross, A First Course in Probability, 6th edition, Prentice-Hall, 2002.

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MATH 450. Advanced Mathematics for Engineers I.

Instructor(s):

Prerequisites & Distribution: MATH 215, 255, or 285. (4). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 354 or 454.

Credits: (4).

Course Homepage: No homepage submitted.

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; and 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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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). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 351.

Credits: (3).

Course Homepage: No homepage submitted.

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, and the Fundamental Theorem of Calculus; infinite series; and 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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MATH 452. Advanced Calculus II.

Section 001 — Multivariable Calculus and Elementary Function Theory.

Instructor(s): Lukas I Geyer

Prerequisites & Distribution: MATH 217, 417, or 419; and MATH 451. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course does a rigorous development of multivariable calculus and elementary function theory with some view towards generalizations. Concepts and proofs are stressed. This is a relatively difficult course, but the stated prerequisites provide adequate preparation. Topics include:

  1. partial derivatives and differentiability;
  2. gradients, directional derivatives, and the chain rule;
  3. implicit function theorem;
  4. surfaces, tangent plane;
  5. max-min theory;
  6. multiple integration, change of variable, etc.; and
  7. Green's and Stokes' theorems, differential forms, exterior derivatives.

MATH 551 is a higher-level course covering much of the same material with greater emphasis on differential geometry. Math 450 covers the same material and a bit more with more emphasis on applications, and no emphasis on proofs. MATH 452 is prerequisite to MATH 572 and is good general background for any of the more advanced courses in analysis (MATH 596, 597) or differential geometry or topology (MATH 537, 635).

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MATH 454. Boundary Value Problems for Partial Differential Equations.

Instructor(s):

Prerequisites & Distribution: MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. Students with credit for MATH 354 can elect MATH 454 for one credit. No credit granted to those who have completed or are enrolled in MATH 450.

Credits: (3).

Course Homepage: No homepage submitted.

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; and 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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MATH 462. Mathematical Models.

Section 001.

Instructor(s): David Bortz

Prerequisites & Distribution: MATH 216, 256, 286, or 316; and MATH 217, 417, or 419. (3). (Excl). (BS). May not be repeated for credit. Students with credit for MATH 362 must have department permission to elect MATH 462.

Credits: (3).

Course Homepage: No homepage submitted.

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.

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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). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 371 or 472.

Credits: (3).

Course Homepage: No homepage submitted.

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 proven. 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, and the 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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MATH 475. Elementary Number Theory.

Section 001.

Instructor(s): Muthukrishnan Krishnamurthy

Prerequisites & Distribution: At least three terms of college mathematics are recommended. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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

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MATH 476. Computational Laboratory in Number Theory.

Section 001.

Instructor(s): Muthukrishnan Krishnamurthy

Prerequisites & Distribution: Prior or concurrent enrollment in MATH 475 or 575. (1). (Excl). (BS). May not be repeated for credit.

Credits: (1).

Course Homepage: No homepage submitted.

Students will be provided software with which to conduct numerical explorations. Students will submit reports of their findings weekly. No programming necessary, but students interested in programming will have the opportunity to embark on their own projects. Participation in the laboratory should boost the student's performance in MATH 475 or MATH 575. Students in the lab will see mathematics as an exploratory science (as mathematicians do). Students will gain a knowledge of algorithms which have been developed (some quite recently) for number-theoretic purposes, e.g., for factoring. No exams.

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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). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/486/001.nsf

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

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

MATH 489. Mathematics for Elementary and Middle School Teachers.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

This course, together with its predecessor MATH 385, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable. Topics covered include fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement. Algebraic techniques and problem-solving strategies are used throughout the course.

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MATH 490. Introduction to Topology.

Section 001 — An Introduction to Point-Set and Algebraic Topology.

Instructor(s): Elizabeth A Burslem

Prerequisites & Distribution: MATH 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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

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MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.

Section 001.

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

Prerequisites & Distribution: At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). (Excl). May be repeated for credit for a maximum of 6 credits. Offered mandatory credit/no credit.

Credits: (1).

Course Homepage: No homepage submitted.

The Applied and Interdisciplinary Mathematics (AIM) student seminar is an introductory and survey course in the methods and applications of modern mathematics in the natural, social, and engineering sciences. Students will attend the weekly AIM Research Seminar where topics of current interest are presented by active researchers (both from U-M and from elsewhere). The other central aspect of the course will be a seminar to prepare students with appropriate introductory background material. The seminar will also focus on effective communication methods for interdisciplinary research. MATH 501 is primarily intended for graduate students in the Applied & Interdisciplinary Mathematics M.S. and Ph.D. programs. It is also intended for mathematically curious graduate students from other areas. Qualified undergraduates are welcome to elect the course with the instructor's permission.

Student attendance and participation at all seminar sessions is required. Students will develop and make a short presentation on some aspect of applied and interdisciplinary mathematics.

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MATH 512. Algebraic Structures.

Section 001 — Basic Structures of Modern Abstract Algebra.

Instructor(s): Robert L Griess Jr

Prerequisites & Distribution: MATH 451 or 513. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Student Body: mainly undergrad math concentrators with a few grad students from other fields

Background and Goals: This is one of the more abstract and difficult courses in the undergraduate program. It is frequently elected by students who have completed the 295--396 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, and fields) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Exercises tend to be quite challenging. Students should have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard. Students from Math 285 are strongly advised to take some 400-500 level course first, for example, Math 513. Some background in linear algebra is strongly recommended

Content: The course covers basic definitions and properties of groups, rings, and fields, including homomorphisms, isomorphisms, and simplicity. Further topics are selected from (1) Group Theory: Sylow theorems, Structure Theorem for finitely-generated Abelian groups, permutation representations, the symmetric and alternating groups (2) Ring Theory: Euclidean, principal ideal, and unique factorization domains, polynomial rings in one and several variables, algebraic varieties, ideals, and (3) Field Theory: statement of the Fundamental Theorem of Galois Theory, Nullstellensatz, subfields of the complex numbers and the integers mod p.

Alternatives: Math 412 (Introduction to Modern Algebra) is a substantially lower level course covering about half of the material of Math 512. The sequence Math 593--594 covers about twice as much Group and Field Theory as well as several other topics and presupposes that students have had a previous introduction to these concepts at least at the level of Math 412.

Subsequent Courses: Together with Math 513, this course is excellent preparation for the sequence Math 593 — 594.

Text Book: Abstract Algebra, Second Edition by David Dummit and Richard Foote.

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MATH 513. Introduction to Linear Algebra.

Section 001 — Theory of Abstract Vector Spaces and Linear Transformations.

Instructor(s): William E Fulton

Prerequisites & Distribution: MATH 412. (3). (Excl). (BS). May not be repeated for credit. Two credits granted to those who have completed MATH 214, 217, 417, or 419.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Math 412 or Math 451 or permission of the instructor

Background and Goals: 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.

Content: 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. This corresponds to most of the first text with the omission of some starred sections and all but Chapters 8 and 10 of the second text.

Alternatives: Math 419 (Lin. Spaces and Matrix Thy) covers much of the same material using the same text, but there is more stress on computation and applications. Math 217 (Linear Algebra) is similarly proof-oriented but significantly less demanding than Math 513. Math 417 (Matrix Algebra I) is much less abstract and more concerned with applications.

Subsequent Courses: The natural sequel to Math 513 is Math 593 (Algebra I). 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.

Text: Curtis: Linear Algebra, An Introductory Approach (4th edition, Springer-Verlag)

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MATH 521. Life Contingencies II.

Section 001.

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

Prerequisites & Distribution: MATH 520. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course is a continuation of MATH520 (a year-long sequence). It covers the topics of reserving models for life insurance; multiple-life models including joint life and last survivor contingent insurances; multiple-decrement models including disability, retirement and withdrawal; insurance models including expenses; and business and regulatory considerations.

Text: Actuarial Mathematics (2nd Edition) by Bowers, Gerber, Hickman, Jones and Nesbitt (Society of Actuaries).

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MATH 523. Risk Theory.

Section 001 — Risk Management.

Instructor(s): Conlon

Prerequisites & Distribution: MATH 425. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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

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MATH 525 / STATS 525. Probability Theory.

Section 001.

Instructor(s): Gautam Bharali

Prerequisites & Distribution: MATH 451 (strongly recommended) or 450. MATH 425 would be helpful. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Background: This course is a fairly rigorous study of the mathematical basis of probability theory. There is some overlap of topics with Math 425, but in Math 525, there is a greater emphasis on the proofs of major results in probability theory. This course and its sequel - Math 526 - are core courses for the Applied and Interdisciplinary Mathematics (AIM) program.

Content: The notion of a probability space and a random variable, discrete and continuous random variables, independence and expectation, conditional probability and conditional expectations, generating functions and moment generating functions, the Law of Large Numbers, and the Central Limit Theorem comprise the essential core of this course. Further topics, to be decided later (and, if feasible, selected according to audience interest), will be covered in the last month of the semester.

Alternatives: EECS 501 covers some of the above material at a lower level of mathematical rigor. Math 425 (Introduction to Probability) is recommended for students with substantially less mathematical preparation.

Text: Introduction to Probability Models by Sheldon Ross

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MATH 526 / STATS 526. Discrete State Stochastic Processes.

Section 001.

Instructor(s): Charles R Doering

Prerequisites & Distribution: MATH 525 or EECS 501. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

See STATS 526.001.

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MATH 528. Topics in Casualty Insurance.

Section 001 — Risk Management.

Instructor(s): Virginia R Young

Prerequisites & Distribution: MATH 217, 417, or 419. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/528/001.nsf

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 insurance, risk management, or finance. We will cover the following topics: advanced topics in credibility theory, risk measures and premium principles, optimal (re)insurance, reinsurance products, and reinsurance pricing.

I assume that you have taken MATH 523, Risk Theory. In fact, one can think of this course as a continuation of MATH 523 with emphasis on applying the material learned in Risk Theory to more practical settings.

The official text for the course is a set of notes available at UM.CourseTools. In addition, an excellent book concerning modern reinsurance products is Integrating Corporate Risk Management by Prakash Shimpi, published by Texere. I suggest that you buy this book, but I do not require that you do so.

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MATH 531. Transformation Groups in Geometry.

Section 001.

Instructor(s): Emina Alibegovic

Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: MATH 412 or 512 would be helpful, but neither is necessary. Text required: None. Text recommended: Armstrong, Groups and Symmetry; Lyndon: Groups and Geometry. textbook comment: Your class notes and my handouts will be sufficient. The books I listed contain some of the material we will cover, but not all of it. Course description: The purpose of this course is to explore the close ties between geometry and algebra. We will study Euclidean and hyperbolic spaces and groups of their isometries. Our discussions will include, but will not be limited to, free groups, triangle groups, and Coxeter groups. We will talk about group actions on spaces, and in particular group actions on trees.

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

Section 001.

Instructor(s): Divakar Viswanath

Prerequisites & Distribution: MATH 450 or 451. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Recent Texts: Complex Variables and Applications, 6th ed. (Churchill and Brown); Student Body: largely engineering and physics graduate students with some math and engineering undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: 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. This course is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program. Content: Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. 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 applications. Subsequent Courses: Math 555 is prerequisite to many advanced courses in science and engineering fields.

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MATH 557. Methods of Applied Mathematics II.

Section 001.

Instructor(s): Peter D Miller (millerpd@umich.edu)

Prerequisites & Distribution: MATH 217, 419, or 513; 451 and 555. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: (1) one of the following: Math 217, 419, or 513 (i.e. a course in linear algebra); (2) one of the following: Math 216, 256, 286, 316, or 404 (i.e. a course in differential equations); (3) Math 451 (or an equivalent course in advanced calculus); (4) Math 555 (or an equivalent course in complex variables). Text: There is no required text. Lecture notes will be made available to students from the instructor's website. Recommended texts will be announced in class. Audience: Graduate students and advanced undergraduates in applied mathematics, engineering, or the natural sciences. Background and Goals: In applied mathematics, we often try to understand a physical process by formulating and analyzing mathematical models which in many cases consist of differential equations with initial and/or 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 an explicit formula, it may be difficult to extract useful information from it and in practice, we must settle for a sufficiently accurate approximate solution obtained by numerical or asymptotic analysis (or a combination of the two). This course is an introduction to the latter of these two approximation methods. The material covered in the textbook includes the nature of asymptotic approximations, asymptotic expansions of integrals and applications to transform theory (Fourier and Laplace), regular and singular perturbation theory for differential equations including transition point analysis, the use of matched expansions, and multiple scale methods. The time remaining after studying these topics will be devoted to the derivation of several famous canonical model equations of applied mathematics (e.g. the Korteweg-de Vries equation and the nonlinear Schroedinger equation) using multiscale asymptotics. Students will come to understand how these equations arise again and again from fields of study as diverse as water wave theory, molecular dynamics, and nonlinear optics. Grading: Students will be evaluated on the basis of homework assignments and also participation and lecture attendance.

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MATH 558. Ordinary Differential Equations.

Section 001.

Instructor(s): Andrew J Christlieb

Prerequisites & Distribution: MATH 450 or 451. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Basic Linear Algebra, Ordinary Differential Equations (math 216), Multivariable Calculus (215) and Either Advanced Calculus (math 451) or an advanced mathematical methods course (e.g. Math 454); preferably both. Course Objective: This course is an introduction to the modern qualitative theory of ordinary differential equations with emphasis on geometric techniques 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 the course. There are relatively few proofs. Course Description: Nonlinear differential equations and iterative maps arise in the mathematical description of numerous systems throughout science and engineering. Such systems may display complicated and rich dynamical behavior. In this course we will focus on the theory of dynamical systems and how it is used in the study of complex systems. The goal of this course is to provide a broad overview of the subject as well as an in-depth analysis of specific examples. The course is intended for students in mathematics, engineering, and the natural sciences. Topics covered will include aspects of autonomous and driven two variable systems including the geometry of phase plane trajectories, periodic solutions, forced oscillations, stability, bifurcations and chaos. Applications to problems from physics, engineering and the natural sciences will arise in the course by way of examples in lecture ad through the homework problems. We will cover material from Chapters 1-5 and 8-13 of the text. Textbook Nonlinear Ordinary Differetial Equations, Oxford Press. by: D.W. Jordan and P. Smith References Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer. John Guckenheimer and Philip Holmes Nonlinear Differential Equations and Dynamical Systems, Springer. Ferdinand Verhulst Applications of Centre Manifold Theory, Springer. J. Carr Nonlinear Systems, Chambridge. P.G. Drazin

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MATH 561 / IOE 510 / OMS 518. Linear Programming I.

Section 001.

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

Prerequisites & Distribution: MATH 217, 417, or 419. (3). (Excl). (BS). May not be repeated for credit. CAEN lab access fee required for non-Engineering students.

Credits: (3).

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

Course Homepage: http://www.engin.umich.edu/class/ioe510/

Background required: Elementary matrix algebra (concept of linear independence, bases, matrix inversion, pivotal methods for solving linear equations), geometry of Rn including convex sets and affine spaces.

Course Objectives: To provide first-year graduate students with basic understanding of linear programming, its importance, and applications. To discuss algorithms for linear programming, available software and how to use it intelligently.

Recommended Books: K. G. Murty, Linear Programming, Wiley, 1983.
Also, R. Saigal, Linear Programming: A Modern Integrated Analysis, Kluwer, 1995, can be used as a reference book.

Course Content:

  1. LP models, various applications. Separable piece-wise linear convex function minimization problems, uses in curve fitting and linear parameter estimation. Approaches for solving multiobjective linear programming models, Goal programming.
  2. What useful planning information can be derived from an LP model (marginal values and their planning uses).
  3. Review of Pivot operations, basic vectors, basic solutions, and bases. Brief review of polyhedral geometry.
  4. Duality and optimality conditions for LP.
  5. Revised primal and dual simplex methods for LP.
  6. Infeasibility analysis, marginal analysis, cost coefficient and RHS constant ranging, other sensitivity analyses.
  7. Algorithm for transportation models.
  8. Other topics time permitting.

There will be weekly homework assignments.

Grading (approximate): Midterm Exam 20% Final Exam 50% Computational Project 15% Homework 15%

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MATH 566. Combinatorial Theory.

Section 001.

Instructor(s): John R Stembridge

Prerequisites & Distribution: MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisite: MATH 512 or an equivalent level of mathematical maturity.

This course will be an introduction to algebraic combinatorics. Previous exposure to combinatorics will not be necessary, but experience with proof-oriented mathematics at the introductory graduate or advanced undergraduate level, and linear algebra, will be needed.

Most of the topics we cover will be centered around enumeration and generating functions. But this is not to say that the course is only about enumeration — questions about counting are a good starting point for gaining a deeper understanding of combinatorial structure.

Some of the topics to be covered include sieve methods, the matrix-tree theorem, Lagrange inversion, the permanent-determinant method, the transfer matrix method, and ordinary and exponential generating functions. Recommended text: R. Stanley, Enumerative Combinatorics, Vol. I Cambridge Univ. Press, 1997.

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MATH 567. Introduction to Coding Theory.

Section 001.

Instructor(s): Hendrikus Gerardus Derksen

Prerequisites & Distribution: One of MATH 217, 419, 513. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Student Body: Undergraduate math majors and EECS graduate students

Background and Goals: This course is designed to introduce math majors to an important area of applications in the communications industry. From a background in linear algebra it will cover the foundations of the theory of error-correcting codes and prepare a student to take further EECS courses or gain employment in this area. For EECS students it will provide a mathematical setting for their study of communications technology.

Content: Introduction to coding theory focusing on the mathematical background for error-correcting codes. Shannon's Theorem and channel capacity. Review of tools from linear algebra and an introduction to abstract algebra and finite fields. Basic examples of codes such and Hamming, BCH, cyclic, Melas, Reed-Muller, and Reed-Solomon. Introduction to decoding starting with syndrome decoding and covering weight enumerator polynomials and the Mac-Williams Sloane identity. Further topics range from asymptotic parameters and bounds to a discussion of algebraic geometric codes in their simplest form.

Alternatives: none

Subsequent Courses: Math 565 (Combinatorics and Graph Theory) and Math 556 (Methods of Applied Math I) are natural sequels or predecessors. This course also complements Math 312 (Applied Modern Algebra) in presenting direct applications of finite fields and linear algebra.

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MATH 571. Numerical Methods for Scientific Computing I.

Section 001.

Instructor(s): James F Epperson

Prerequisites & Distribution: MATH 217, 417, 419, or 513; and one of MATH 450, 451, or 454. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course is an introduction to numerical linear algebra, a core subject in scientific computing. Three general problems are considered: (1) solving a system of linear equations, (2) computing the eigenvalues and eigenvectors of a matrix, and (3) least squares problems. These problems often arise in applications in science and engineering, and many algorithms have been developed for their solution. However, standard approaches may fail if the size of the problem becomes large or if the problem is ill-conditioned, e.g. the operation count may be prohibitive or computer roundoff error may ruin the answer. We'll investigate these issues and study some of the accurate, efficient, and stable algorithms that have been devised to overcome these difficulties.

The course grade will be based on homework assignments, a midterm exam, and a final exam. Some homework exercises will require computing, for which Matlab is recommended.

Topics:

  1. warmup: vector and matrix norms, orthogonal matrices, projection matrices, singular value decomposition (SVD);
  2. least squares problems: QR factorization, Gram-Schmidt orthogonalization, Householder triangularization, normal equations;
  3. backward error analysis: stability, condition number, IEEE floating point arithmetic;
  4. direct methods for Ax=b: Gaussian elimination, LU factorization, pivoting, Cholesky factorization;
  5. eigenvalues and eigenvectors: Schur factorization, reduction to Hessenberg and tridiagonal form, power method, inverse iteration, shifts, Rayleigh quotient iteration, QR algorithm;
  6. iterative methods for Ax=b: Krylov subspace, Arnoldi iteration, GMRES, conjugate gradient method, preconditioning;
  7. applications: image compression using the SVD, least squares data fitting, finite-difference schemes for a two-point boundary value problem, Dirichlet problem for the Laplace equation

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MATH 572. Numerical Methods for Scientific Computing II.

Section 001.

Instructor(s): Divakar Viswanath

Prerequisites & Distribution: MATH 217, 417, 419, or 513; and one of MATH 450, 451, or 454. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Math 572 is an introduction to numerical methods for solving differential equations. These methods are widely used in science and engineering. The four main segments of the course will cover the following topics:

1. Ordinary differential equations (Runge-Kutta methods, multiple time-step methods, stiffness)

2. Finite difference methods (von Neumann stability analysis, ADI, CFL, Lax equivalence theorem)

3. Spectral methods (Fourier methods, method of lines)

4. Finite element methods (2-point boundary value problems and 2-dimensional elliptic problems) If time permits, we will go into the multigrid method for solving linear systems.

OPTIONAL TEXT: Numerical Solution Of Partial Differential Equations, K.W. Morton and D.F. Mayers, Cambridge University Press

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MATH 592. Introduction to Algebraic Topology.

Section 001.

Instructor(s): Igor Kriz

Prerequisites & Distribution: MATH 591. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

The purpose of this course is to introduce basic concepts of algebraic topology, in particular fundamental group, covering spaces and homology. These methods provide the first tools for proving that two topological spaces are not topologically equivalent (example: the bowling ball is topologically different from the teacup). Other simple applications of the methods will also be given, for example fixed point theorems for continuous maps. Prerequisites: basic knowledge of point set topology, such as from 590 or 591. Books: There is no ideal text covering all this material on exactly the level needed (basic but rigorous). Recommended texts include Munkres: Elements of Algebraic topology (for homology) and J.P.May: A concise course in algebraic topology (for fundamental group and covering spaces). Both texts include topics which will not be covered in 592, and are also suitable textbooks for the next course in algebraic topology, 695.

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MATH 594. Algebra II.

Section 001.

Instructor(s): Igor Dolgachev (idolga@umich.edu)

Prerequisites & Distribution: MATH 593. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisite: MATH 593. I. Group theory: Group actions on sets. Linear groups. Sylow theorems. Solvable and nilpotent groups. Free groups and presentations. Linear representations of groups. Character tables. II Field extensions: Algebraic and transcendental extensions. Algebraic functions. Luroth theorem. III. Galois theory. Galois correspondence. Kummer's and Schreier's extensioins. Solutions of equations in radicals. Computation of Galois groups of equations. Text-book: M. Artin, Algebra. Prentice Hall. 1991.

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MATH 597. Analysis II.

Section 001.

Instructor(s): David E Barrett (demeyer@umich.edu)

Prerequisites & Distribution: MATH 451 and 513. (3). (Excl). (BS). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Topics will include: Lebesgue measure on the real line and in Rn; general measures; Hausdorff dimension; measurable functions; integration; monotone convergence theorem; Fatou's lemma; dominated convergence theorem; Fubini's theorem; function spaces; Holder and Minkowski inequalities; functions of bounded variation; differentiation theory; Fourier analysis. Additional topics such as Sobolev spaces to be covered as time permits.

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