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Winter Academic Term 2002 Course Guide

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Courses in Mathematics


This page was created at 5:26 PM on Fri, Mar 22, 2002.

Winter Academic Term, 2002 (January 7 - April 26)

Open courses in Mathematics
(*Not real-time Information. Review the "Data current as of: " statement at the bottom of hyperlinked page)

Wolverine Access Subject listing for MATH

Winter Academic Term '02 Time Schedule for Mathematics.


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

Full QR

Credits: (4).

Course Homepage: 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 105 are fully prepared for Math 115. This is a course on analyzing data by means of functions and graphs. The emphasis is on mathematical modeling of real-world applications. The functions used are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Algebra skills are assessed during the term by periodic testing. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab.

TEXT: Functions Modeling Change, Connally, Wiley Publishing.

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MATH 107. Mathematics for the Information Age.

Section 001.

Instructor(s): Karen Rhea (krhea@umich.edu)

Prerequisites & Distribution: Three to four years high school mathematics. (3). (MSA). (QR/1).

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. Grading will be based on homework, at least one written report, a midterm, and a final.

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

Instructor(s):

Prerequisites & Distribution: See Elementary Courses above. Enrollment in Math. 110 is by recommendation of Math. 115 instructor and override only. No credit granted to those who already have 4 credits for pre-calculus mathematics courses. (2). (Excl).

Credits: (2).

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

The course covers data analysis by means of functions and graphs. Math 110 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. The course is a condensed, half-term version of Math 105 (Math 105 covers the same material in a traditional classroom setting) designed for students who appear to be prepared to handle calculus but are not able to successfully complete Math 115. Students who complete 110 are fully prepared for Math 115. Students may enroll in Math 110 only on the recommendation of a mathematics instructor after the third week of classes in the Fall and must visit the Math Lab to complete paperwork and receive course materials.

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MATH 115. Calculus I.

Instructor(s):

Prerequisites & Distribution: Four years of high school mathematics. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/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 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. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

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

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.

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MATH 127. Geometry and the Imagination.

Instructor(s): Bryan Mosher (bmosher@umich.edu)

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

Full QR First-Year Seminar

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math127/

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.

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MATH 147. Introduction to Interest Theory.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

This course is designed for students who seek an introduction to the mathematical concepts and techniques employed by financial institutions such as banks, insurance companies, and pension funds. Actuarial students, and other mathematics concentrators should elect Math 424, which covers the same topics but on a more rigorous basis requiring considerable use of calculus. Topics covered include: various rates of simple and compound interest, present and accumulated values based on these; annuity functions and their application to amortization, sinking funds, and bond values; depreciation methods; introduction to life tables, life annuity, and life insurance values. This course is not part of a sequence. Students should possess financial calculators.

Text: Mathematics of Finance, Zima and Brown, McGraw Hill Publishing.

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MATH 186. Honors Calculus II.

Instructor(s):

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

Full QR

Credits: (4).

Course Homepage: No homepage submitted.

The sequence Math 185-186-285-286 is the honors introduction to the calculus. It is taken by students intending to major in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program.

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

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

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MATH 214. Linear Algebra and Differential Equations.

Instructor(s):

Prerequisites & Distribution: Math. 115 and 116. 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. (4). (MSA). (BS).

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 textbook is Linear Algebra and its Applications, second edition, David Lay, Addison Wesley.

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. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

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

The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a midterm and final exam. Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; line, surface, and volume integrals and applications; vector fields and integration; Green's Theorem and Stokes' Theorem. There is a weekly computer lab using Maple software. Math 285 is a somewhat more theoretical course which covers the same material. For students intending to concentrate in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is Math 217. Students who intend to take only one further mathematics course and need differential equations should take Math 216.

TEXT: STUDENTS HAVE CHOICE OF EITHER: Calculus, 4th edition, James Stewart, Brooks/Cole Publishing, or Multivariable Calculus, 4th 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. Credit can be earned for only one of Math. 216, 256, 286, or 316. (4). (MSA). (BS).

Credits: (4).

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

For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216-417 (or 419) and Math 217-316. The sequence Math 216-417 emphasizes problem-solving and applications and is intended for students of engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217-316. After an introduction to ordinary differential equations, the first half of the course is devoted to topics in linear algebra, including systems of linear algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors. In the second half these tools are applied to the solution of linear systems of ordinary differential equations. Topics include: oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods. There is a weekly computer lab using MATLAB software. This course is not intended for mathematics concentrators, who should elect the sequence 217-316. Math 286 covers much of the same material in the honors sequence. The sequence Math 217-316 covers all of this material and substantially more at greater depth and with greater emphasis on the theory. Math 404 covers further material on differential equations. Math 217 and 417 cover further material on linear algebra. Math 371 and 471 cover additional material on numerical methods.

Text: Differential Equations, Computing and Modeling, 2nd edition, Edwards and Penney, Prentice Hall Publishing.

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MATH 217. Linear Algebra.

Instructor(s):

Prerequisites & Distribution: Math. 215, 255, or 285. 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. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: No homepage submitted.

For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216-417 (or 419) and Math 217-316. The sequence Math 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217-316. These courses are explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved. Therefore the student entering Math 217 should come with a sincere interest in learning about proofs. The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of Rn; linear dependence, bases, and dimension; linear transformations; eigenvalues and eigenvectors; diagonalization; 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.

Text: Linear Algebra and Its Applications, 2nd edition, David Lay, Addison Wesley Publishing.

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MATH 255. Applied Honors Calculus III.

Instructor(s):

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

Credits: (4).

Course Homepage: No homepage submitted.

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

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

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MATH 286. Honors Differential Equations.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

See Math. 186 for a general description of the sequence Math 185-186-285-286.

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

Text: Elementary Differential Equations, 7th edition, Boyce and DiPrima, Wiley.

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MATH 288. Math Modeling Workshop.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 216 or 316, and Math. 217 or 417. (1). (Excl). (BS). Offered mandatory credit/no credit. May be repeated for a total of three credits.

Credits: (1).

Course Homepage: No homepage submitted.

This course is designed to help students understand more clearly how techniques from other undergraduate mathematics courses can be used in concert to solve real-world problems. After the first two lectures the class will discuss methods of attacking problems. For credit a student will have to describe and solve an individual problem and write a report on the solution. Computing methods will be used. During the weekly workshop students will be presented with real-world problems on which techniques of undergraduate mathematics offer insights. They will see examples of how to approach and set up a given modeling problem systematically, how to use mathematical techniques to begin a solution of the problem, what to do about the loose ends that can't be solved, and how to present the solution to others. Students will have a chance to use the skills developed by participating in the UM Undergraduate Math Modeling Meet.

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MATH 289. Problem Seminar.

Instructor(s): Harm Derksen (hdersken@umich.edu)

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

Credits: (1).

Course Homepage: http://www.math.lsa.umich.edu/~hderksen/math289.html

No Description Provided.

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MATH 296. Honors Mathematics II.

Instructor(s): Bryan Mosher (bmosher@umich.edu)

Prerequisites & Distribution: Prior knowledge of first year calculus and permission of the Honors advisor. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (Excl). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math296/

No Description Provided.

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MATH 310. Elementary Topics in Mathematics.

Section 001 Math Games and the Theory of Games.

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

Prerequisites & Distribution: Two years of high school mathematics. (4). (Excl). (BS).

Credits: (4).

Course Homepage: No homepage submitted.

Students electing this course must have sophomore or higher standing, some previous college mathematics, and an interest in games of strategy. No previous knowledge of any particular game is required.

We will study the strategy of several games where mathematics can play a role. Possibilities are "dots and boxes," "tic-tac-toe," "nim," "go-moku," "Dead pigs won't fly," "pegboard solitaire," "Colonel Blotto," etc. Some are one person games (puzzles, really) and some are two person games of strategy. The main part of the course will be an introduction to the (von Neumann-Morgenstern) mathematical theory of games with emphasis on two person games.

This course can be counted toward the mathematics concentration or academic minor in mathematics.

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MATH 312. Applied Modern Algebra.

Instructor(s):

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

Credits: (3).

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

One of the main goals of the course (along with every course in the algebra sequence) is to expose students to rigorous, proof-oriented mathematics. Students are required to have taken Math 217, which should provide a first exposure to this style of mathematics. A distinguishing feature of this course is that the abstract concepts are not studied in isolation. Instead, each topic is studied with the ultimate goal being a real-world application. As currently organized, the course is broken into four parts: 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.

Instructor(s):

Prerequisites & Distribution: Math. 215 and 217. Credit can be earned for only one of Math. 216, 256, 286, or 316. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No homepage submitted.

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

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MATH 333. Directed Tutoring.

Instructor(s): Eugene F Krause

Prerequisites & Distribution: Math. 385 and enrollment in the Elementary Program in the School of Education. (1-3). (Excl). (EXPERIENTIAL). May be repeated for a total of three credits.

Credits: (1-3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 354. Fourier Analysis and its Applications.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

This 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.

Instructor(s):

Prerequisites & Distribution: Engineering 101; one of Math. 216, 256, 286, or 316. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students.

Credits: (3).

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

Course Homepage: No homepage submitted.

This is a survey course of the basic numerical methods which are used to solve practical scientific problems. Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra, and may provide practice in FORTRAN programming and the use of a software library subroutine. Convergence theorems are discussed and applied, but the proofs are not emphasized. Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration and differentiation, solutions to non-linear equations, ordinary differential equations, and polynomial approximations. Other topics may include discrete Fourier transforms, two-point boundary-value problems, and Monte-Carlo methods. Math 471 is a similar course which expects one more year of maturity and is somewhat more theoretical and less practical. The sequence Math 571-572 is a beginning graduate level sequence which covers both numerical algebra and differential equations and is much more theoretical. This course is basic for many later courses in science and engineering. It is good background for Math 571-572.

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MATH 396. Honors Analysis II.

Instructor(s):

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

Credits: (4).

Course Homepage: No homepage submitted.

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

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MATH 399. Independent Reading.

Instructor(s):

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

Credits: (1-6).

Course Homepage: No homepage submitted.

Designed especially for Honors students.

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MATH 412. Introduction to Modern Algebra.

Instructor(s):

Prerequisites & Distribution: Math. 215, 255, or 285; and 217. 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. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 416. Theory of Algorithms.

Section 001.

Instructor(s):

Prerequisites & Distribution: Math. 312 or 412 or CS 203, and CS 281. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 417. Matrix Algebra I.

Instructor(s):

Prerequisites & Distribution: Three courses beyond Math. 110. 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. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/courses/417/

Many problems in science, engineering, and mathematics are best formulated in terms of matrices rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (honors). Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations.

Math 419 is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (despite its lower number) is also a more theoretical course which covers much of the material of 417 at a deeper level. Math 513 is an honors version of this course, which is also taken by some mathematics graduate students. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.

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MATH 419. Linear Spaces and Matrix Theory.

Instructor(s):

Prerequisites & Distribution: Four terms of college mathematics beyond Math. 110. 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. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 422 / BE 440. Risk Management and Insurance.

Instructor(s):

Prerequisites & Distribution: Math. 115, junior standing, and permission of instructor. (3). (Excl). (BS).

Upper-Level Writing

Credits: (3).

Course Homepage: No homepage submitted.

We will explore how much insurance affects the lives of students (automobile insurance, social security, health insurance, theft insurance) as well as the lives of other family members (retirements, life insurance, group insurance). While the mathematical models are important, an ability to articulate why the insurance options exist and how they satisfy the customer's needs are equally important. In addition, there are different options available (e.g., in social insurance programs) that offer the opportunity of discussing alternative approaches.

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MATH 423. Mathematics of Finance.

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

Prerequisites & Distribution: Math. 217 and 425; CS 183. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~conlon/math423/index.html

Prerequisites: A solid background in probability theory at the 400 level, math 425 or equivalent.

Required Text: Options, Futures and Other Derivatives by Hull, fourth edition, Prentice Hall 1999.

Background and Goals: This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. The goal is to understand how the models derive from basic principles of economics, and to provide the necessary mathematical tools for their analysis. A solid background in basic probability theory is necessary.

Contents:

  1. Forwards and Futures, Hedging using Futures, Bills and Bonds, Swaps, Perfect Hedges.
  2. Options-European and American, Trading Strategies, Put-Call Parity, Black-Scholes formula.
  3. Volatility, methods for estimating volatility-exponential, GARCH, maximum likelihood. (d) Dynamic Hedging, stop-loss, Black-Scholes, the Greek letters.
  4. Other Options.

Grading: The grade for the course will be determined from performances on 8 quizzes, a midterm and a final exam. There will be 8 homework assignments. Each quiz will consist of a slightly modified homework problem.

  • 8 quizzes= 8x10=80 points
  • midterm= 60 points
  • final= 80 points
  • Total= 220 points

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MATH 424. Compound Interest and Life Insurance.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

This course explores the concepts underlying the theory of interest and then applies them to concrete problems. The course also includes applications of spreadsheet software. The course is a prerequisite to advanced actuarial courses. It also helps students prepare for the Part 4A examination of the Casualty Actuarial Society and the Course 140 examination of the Society of Actuaries. The course covers compound interest (growth) theory and its application to valuation of monetary deposits, annuities, and bonds. Problems are approached both analytically (using algebra) and geometrically (using pictorial representations). Techniques are applied to real-life situations: bank accounts, bond prices, etc. The text is used as a guide because it is prescribed for the Society of Actuaries exam; the material covered will depend somewhat on the instructor. Math 424 is required for students concentrating in actuarial mathematics; others may take Math 147, which deals with the same techniques but with less emphasis on continuous growth situations. Math 520 applies the concepts of Math 424 together with probability theory to the valuation of life contingencies (death benefits and pensions).

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MATH 425 / STATS 425. Introduction to Probability.

Section 001, 003, 007.

Instructor(s):

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

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. Math 525 is a similar course for students with stronger mathematical background and ability. Stat 426 is a natural sequel for students interested in statistics. Math 523 includes many applications of probability theory.

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MATH 425 / STATS 425. Introduction to Probability.

Section 002.

Instructor(s): Alexander Barvinok (barvinok@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~barvinok/m425.html

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. Math 525 is a similar course for students with stronger mathematical background and ability. Stat 426 is a natural sequel for students interested in statistics. Math 523 includes many applications of probability theory.

Text: D. Stirzaker, Probability and Random Variables. A beginner's guide, Cambridge University Press, reprinted with corrections in 2001.

Grading: The final grade will be computed from the following:

  • First midterm exam: 20 %
  • Second midterm exam: 20 %
  • Final exam: 30 %
  • Homework: 30 %

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MATH 425 / STATS 425. Introduction to Probability.

Section 004, 005, 006.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

See Statistics 425.004.

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MATH 427 / HB 603. Retirement Plans and Other Employee Benefit Plans.

Section 001.

Instructor(s):

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

Upper-Level Writing

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 450. Advanced Mathematics for Engineers I.

Section 001.

Instructor(s): Brent Carswell (carswell@umich.edu)

Prerequisites & Distribution: Math. 215, 255, or 285. (4). (Excl). (BS).

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/~carswell/math450/

No Description Provided.

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MATH 450. Advanced Mathematics for Engineers I.

Section 002.

Instructor(s): Buckley

Prerequisites & Distribution: Math. 215, 255, or 285. (4). (Excl). (BS).

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.

Text: Advanced Engineering Mathematics, 8th edition, Edward Kreyszig Wiley.

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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).

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, 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.

Text: Elementary Analysis, The Theory of Calculus, Kenneth Ross, Springer-Verlag.

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MATH 452. Advanced Calculus II.

Instructor(s):

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

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.

Section 001.

Instructor(s): Smereka

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

Credits: (3).

Course Homepage: 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.

There is no textbook listed for this course.

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MATH 454. Boundary Value Problems for Partial Differential Equations.

Section 002.

Instructor(s): Ralf Wittenberg

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~ralf/math454/index.html

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.

Text: Richard Haberman, "Elementary Applied Partial Differential Equations : with Fourier Series and Boundary Value Problems" (3rd edition), Prentice-Hall (1998).

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MATH 462. Mathematical Models.

Instructor(s): Patrick Nelson (pwn@umich.edu)

Prerequisites & Distribution: Math. 216, 256, 286, or 316; and 217, 417, or 419. Students with credit for Math. 362 must have department permission to elect Math. 462. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~pwn/Math462.html

This course will cover biological models constructed from difference equations and ordinary differential equations. Applications will be drawn from population biology, population genetics, the theory of epidemics, biochemical kinetics, and physiology. Both exact solutions and simple qualitative methods for understanding dynamical systems will be stressed (anticipated text is Mathematical Models in Biology by Leah Edelstein-Keshet).

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MATH 471. Introduction to Numerical Methods.

Section 001.

Instructor(s): Mark Dickinson (dickinsm@umich.edu)

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).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~dickinsm/471w02/

Course description: The aim of this course is to present a variety of basic techniques and algorithms used to solve numerical problems arising mainly from science and engineering. We will learn to implement these methods using MATLAB, and we will also examine issues of accuracy, stability and efficiency for selected methods. The main topics to be covered in this course are: numerical linear algebra, one-dimensional root-finding, polynomial and spline interpolation, quadrature, and numerical solution of ordinary differential equations.

Prerequisites: You should be familiar with basic ideas from calculus, linear algebra and ordinary differential equations; Math 216 and Math 217 provide sufficient background for linear algebra and differential equations. You should also have had some experience of programming in a standard procedural programming language. Please talk to me if you are unsure that you satisfy all of these requirements.

Assessment: The final grade will be based on the coursework along with the midterm and final exams, in the following proportions:

  • Coursework: 40%
  • Midterm exam: 25%
  • Final exam: 35%

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MATH 471. Introduction to Numerical Methods.

Section 002.

Instructor(s): Ralf Wittenberg (ralf@umich.edu)

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).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~ralf/math471/index.html

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.

Text: Richard L. Burden and J. Douglas Faires, Numerical Analysis (7th edition), Brooks/Cole (2001).

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MATH 475. Elementary Number Theory.

Instructor(s):

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

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.

Instructor(s):

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

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 485. Mathematics for Elementary School Teachers and Supervisors.

Instructor(s): Eugene F Krause

Prerequisites & Distribution: One year of high school algebra. No credit granted to those who have completed or are enrolled in Math. 385. (3). (Excl). (BS). May not be included in a concentration plan in mathematics.

Credits: (3; 2 in the half-term).

Course Homepage: No homepage submitted.

The history, development, and logical foundations of the real number system and of numeration systems including scales of notation, cardinal numbers, and the cardinal concept; and the logical structure of arithmetic (field axioms) and relations to the algorithms of elementary school instruction. Simple algebra, functions, and graphs. Geometric relationships. For persons teaching or preparing to teach in the elementary school.

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MATH 486. Concepts Basic to Secondary Mathematics.

Instructor(s): Eugene F Krause

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

Credits: (3).

Course Homepage: No homepage submitted.

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.

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MATH 489. Mathematics for Elementary and Middle School Teachers.

Instructor(s): Eugene F Krause

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

Credits: (3).

Course Homepage: No homepage submitted.

All elementary teaching certificate candidates are required to take two mathematics courses, Math 385 and Math 489, either before or after admission to the School of Education. Math 385 is offered in the Fall, Math 489 in the Winter. The next Spring Term offering of Math 489 will be in 2003. For further information about future course offerings, contact Prof. Krause at 763-1186 or at his office, 3086 East Hall. 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.

Instructor(s):

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

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 498. Topics in Modern Mathematics.

Markov Chains: Theory and Applications.

Instructor(s): Divakar Viswanath

Prerequisites & Distribution: Senior mathematics concentrators and Master Degree students in mathematical disciplines. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No homepage submitted.

Markov chains form an elegant topic with numerous applications. In the first half of the course, we will discuss basic questions: What is a Markov chain? What is its eventual behaviour? How fast does it converge? The second half will make the following relatively advanced topics accessible:

  1. How many times should you shuffle a deck of cards to make the deck completely random? The answer depends upon how you shuffle.
  2. Can you read the encoded message (look for a flier near you) exchanged between inmates of a Texan penitentiary? Something called the Metropolis method was used to decode it.
  3. Ising models and phase transitions, as when ice melts into water.
  4. Random matrix theory, if time permits.

The final project has to make a connection to any area you like. Your options will be plenty: math, physics, compsci, econ, biology, and chemistry are all fair game.

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MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.

Section 001.

Instructor(s): Peter Miller (millerpd@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). (BS). Offered mandatory credit/no credit. May be repeated for a total of 6 credits.

Credits: (1).

Course Homepage: http://www.math.lsa.umich.edu/~millerpd/Courses/501.html

No Description Provided.

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MATH 512. Algebraic Structures.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 513. Introduction to Linear Algebra.

Instructor(s): Mahdi Asgari (asgari@umich.edu)

Prerequisites & Distribution: Math. 412. Two credits granted to those who have completed Math. 214, 217, 417, or 419. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~asgari/ma513_wint02.html

This is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis is on concepts and proofs with some calculations to illustrate the theory. For students with only the minimal prerequisite, this is a demanding course; at least one additional proof-oriented course e.g., Math 451 or 512) is recommended. Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear transformations, bases, and matrices; eigenvalues and eigenvectors; applications to linear and linear differential equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form. Math 419 covers much of the same material using the same text, but there is more stress on computation and applications. Math 217 is similarly proof-oriented but significantly less demanding than Math 513. Math 417 is much less abstract and more concerned with applications. The natural sequel to Math 513 is 593. Math 513 is also prerequisite to several other courses (Math 537, 551, 571, and 575) and may always be substituted for Math 417 or 419.

Text: Linear Algebra, An Introductory Approach, Charles Curtis, Springer-Verlag.

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MATH 521. Life Contingencies II.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 523. Risk Theory.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 525 / STATS 525. Probability Theory.

Instructor(s): Alexander Barvinok (barvinok@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~barvinok/m525.html

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

Text:
G. Grimmett and D. Stirzaker, Probability and Random Processes, Cambridge University Press, third edition, 2001.

Grading:

The final grade will be computed from the following:

  1. First midterm exam: 20 %
  2. Second midterm exam: 20 %
  3. Final exam: 30 %
  4. Homework: 30 %

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MATH 526 / STATS 526. Discrete State Stochastic Processes.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 526 / STATS 526. Discrete State Stochastic Processes.

Section 001.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

See Statistics 526.001.

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MATH 531. Transformation Groups in Geometry.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 550 / CMPLXSYS 510. Introduction to Adaptive Systems.

Section 001 Introduction to Dynamical Systems for Biocomplexity.

Instructor(s): Carl P Simon (cpsimon@umich.edu)

Prerequisites & Distribution: Math. 215, 255, or 285; Math. 217; and Math. 425, and Permission of instructor. Working knowledge of calculus, probability, and matrix algebra. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://precisione.physics.lsa.umich.edu/CSCS/education/CSCS-courses/cscs510-w01.html

See Complex Systems 510.001.

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

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 557. Methods of Applied Mathematics II.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 558. Ordinary Differential Equations.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 559. Selected Topics in Applied Mathematics.

Section 001 Advanced Mathematical Methods for the Biological Sciences Partial Differential Equations in Biology

Instructor(s): Trachette Jackson (tjacks@umich.edu)

Prerequisites & Distribution: Math. 451; and 217 or 419. (3). (Excl). (BS). May be repeated for a total of six credits.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~tjacks/559f.ps

Natural systems behave in a way that reflects an underlying spatial pattern. For example, on the molecular level, rarely do reactions occur in a homogenous environment and the spatial organization does somehow influence the way in which particles interact. In this course, we will discover the way in which spatial variation influences the motion, dispersion, and persistence of species. We shall become aware of the fine balance that exists between interdependent species and demonstrate that spatial diversity can have subtle, but important effects or can lead to the emergence of remarkable spatial patterns from a previously uniform state. The concepts underlying spatially dependent processes and the partial differential equations which model them will be discussed in a general manner with examples taken from the molecular, cellular, and population levels. We will then apply these ideas to more specific cases with the aim of understanding interesting biological phenomena. Topics include: Population dispersal based on diffusion models; Cell movements (e.g., chemotaxis and haptotaxis); Growth of branching organisms; Traveling waves in microorganisms; Transport of biological substances; Models for development and pattern formation; and Age-Structured models of HIV dynamics.

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

Instructor(s):

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

Credits: (3).

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

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 567. Introduction to Coding Theory.

Instructor(s): Harm Derksen (hdersken@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~hderksen/math567.html

No Description Provided.

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MATH 571. Numerical Methods for Scientific Computing I.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 572. Numerical Methods for Scientific Computing II.

Ordinary and Partial Differential Equations

Instructor(s): Robert Krasny (krasny@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~krasny/math572.html

Math 572 is an introduction to numerical methods used in solving di eren- tial equations. The course will focus on finite-difference schemes for ordinary and partial differential equations. Theoretical concepts and practical computing issues will be covered.

Prerequisites.
Advanced calculus, linear algebra, complex variables. Math 571 is not a prerequisite.

Texts.
Numerical Initial Value Problems in Ordinary Differential Equations, by C. W. Gear, Prentice-Hall (available as a coursepack in the Michigan Union Bookstore)

Numerical Solution of Partial Differential Equations, by K. W. Morton and D. F. Mayers, Cambridge University Press

Syllabus.
ODEs: Euler's method, asymptotic expansion of the error, Richardson extrapolation, Runge-Kutta methods, multistep methods, leap-frog method, consistency, stability, con- vergence, root condition, absolute stability, stiff systems, A-stability PDEs: heat equation, wave equation, Laplace equation, nite-di erence schemes, artificial viscosity, Crank-Nicolson, Lax-Wendroff, operator splitting, stability analysis, maximum principle, energy method, discrete Fourier analysis, CFL condition, Lax equivalence theorem, Kriess matrix theorem, discontinuous solutions, Gibbs phenomenon, trigonometric interpolation, pseudospectral method, nonlinear equations

Course Grade.
The course grade will be based on homework (30%), a midterm exam (30%), and a final exam (40%). The homework will include programming exercises for which I recommend using Matlab.

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

MATH 582. Introduction to Set Theory.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 592. Introduction to Algebraic Topology.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 594. Algebra II.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 597. Analysis II.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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