Fall '99 Course Guide

Courses in Mathematics (Division 428)

Fall Term, 1999 (September 8 December 22, 1999)

Take me to the Fall Term '99 Time Schedule for Mathematics.


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

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

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

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

Each of Math 115, 185, and 295 is a first course in calculus and generally credit can be received for only one course from this list. The sequence 115-116-215 is appropriate for most students who want a complete introduction to calculus. One of Math 215, 285, or 395 is prerequisite to most more advanced courses in Mathematics.

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

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

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

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

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

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


Math. 105. Data, Functions, and Graphs.

Section Joint Evening Examinations for All Sections of Math 105, 6-8 P.M., Tues., Oct. 6 and Wed., Nov. 4. also A Joint Final.

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

Full QR

Credits: (4).

Course Homepage: No Homepage Submitted.

Math 105 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. Students who complete 105 are fully prepared for Math 115. This is a course on analyzing data by means of functions and graphs. The emphasis is on mathematical modeling of real-world applications. The functions used are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Algebra skills are assessed during the term by periodic testing. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab.

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

Section Students in Math 110 Receive Individualized Self-Paced Instruction in the Mathematics Laboratory, in East Hall. Students Must Go to The Math Lab During The First Full Week of Classes.

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

Credits: (2).

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

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

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Math. 115. Calculus I.

Section There Will be Joint Evening Examinations for All sections of Math 115, 6-8 P.M., Thurs, Oct 1 and Nov. 12. Also A Joint Final.

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

Full QR

Credits: (4).

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

The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing, and questioning skills.

Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. Math 185 is a somewhat more theoretical course which covers some of the same material. Math 175 includes some of the material of Math 115 together with some combinatorial mathematics. A student whose preparation is insufficient for Math 115 should take Math 105 (Data, Functions, and Graphs). Math 116 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking Math 186. The cost for this course is over $100 since the student will need a text (to be used for 115 and 116) and a graphing calculator (the Texas Instruments TI-83 is recommended).

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Math. 115. Calculus I.

Section 100 Students in Math 115 Section 100 Receive Individualized Self-Paced Instruction in the Mathematics Laboratory in Room B860 E H. Students Must Go to The Math Lab During The First Full Week of Classes.

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

No Description Provided.

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Math. 116. Calculus II.

Section There Will be Joint Evening Examinations for All Sections of Math 116, 6-8 P.M., Tues., Oct. 6 and Wed., Nov. 4. Also A Joint Final.

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: No Homepage Submitted.

See Math 115 for a general description of the sequence Math 115-116-215.

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

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Math. 128. Explorations in Number Theory.

Section 001.

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

Full QR First-Year Seminar,

Credits: (4).

Course Homepage: No Homepage Submitted.

This course is intended for non-science concentrators and students in the pre-concentration years with no intended concentration, who want to engage in mathematical reasoning without having to take calculus first. Students will be introduced to elementary ideas of number theory, an area of mathematics that deals with properties of the integers. Students will make use of software provided for IBM PCs to conduct numerical experiments and to make empirical discoveries. Students will formulate precise conjectures, and in many cases prove them. Thus the students will, as a group, generate a logical development of the subject. After studying factorizations and greatest common divisors, emphasis will shift to the patterns that emerge when the integers are classified according to the remainder produced upon division by some fixed number ('congruences'). Once some basic tools have been established, applications will be made in several directions. For example, students may derive a precise parameterization of Pythagorean triples a2 + b2 = c2. Students who like math but don't especially like calculus will want to enroll in this first-year MSA seminar. Students will do hands-on experimentation with numerical patterns and will tackle numerical riddles and brainteasers as they focus on empirical discovery and proof of theorems. Students will write their own text in number theory and enjoy their growing ability to think like mathematicians.

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Math. 147. Introduction to Interest Theory.

Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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

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Math. 156. Applied Honors Calculus II.

Section There Will be Joint Evening Examinations for All Sections of Math 156, 6-8 P.M. Dates to be Announced. also A Joint Final.

Prerequisites & Distribution: Score of 4 or 5 on the AB or BC Advanced Placement calculus exam. Credit is granted for only one course among Math 114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/~krasny/math156.html

The sequence 156-255-256 is an Honors calculus sequence for engineering and science concentrators who scored 4 or 5 on the AB or BC Advanced Placement calculus exam. Topics include Riemann sums, the definite integral, fundamental theorem of calculus, applications of integral calculus (e.g. arclength, surface area, work, hydrostatic pressure, center of mass), improper integrals, infinite sequences and series, differential equations, complex numbers. MAPLE will be used throughout.

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Math. 175. Combinatorics and Calculus.

Section 001.

Prerequisites & Distribution: Permission of Honors advisor. No credit granted to those who have completed a 200-level or higher Mathematics course. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: No Homepage Submitted.

This course is an alternative to Math 185 as an entry to the Honors sequence. The sequence Math 175-176 is a two-term introduction to Combinatorics, Dynamical Systems, and Calculus. The topics are integrated over the two terms although the first term will stress combinatorics and the second term will stress the development of calculus in the context of dynamical systems. Students are expected to have some previous experience with the basic concepts and techniques of calculus. The course stresses discovery as a vehicle for learning. Students will be required to experiment throughout the course on a range of problems and will participate each term in a group project. Grades will be based on homework and projects with a strong emphasis on homework. Personal computers will be a valuable experimental tool in this course, and students will be asked to learn to program in either BASIC, PASCAL, or FORTRAN. There are two major topic areas: enumeration theory and graph theory. The section on enumeration theory will emphasize classical methods for counting including: (1) binomial theorem and its generalizations; (2) solving recursions; (3) generating functions; and (4) the inclusion-exclusion principle. In the process, we will discuss infinite series. The section on graph theory will include basic definitions and some of the more interesting and useful theorems of graph theory. The emphasis will be on topological results and applications to computer science and will include: (1) connectivity; (2) trees, Prufer codes, and data structures; (3) planar graphs, Euler's formula and Kuratowski's Theorem; and (4) coloring graphs, chromatic polynomials, and orientation. This material has many applications in the field of Computer Science. Math 176 is the standard sequel.

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Math. 185. Honors Calculus I.

Prerequisites & Distribution: Permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 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: No Homepage Submitted.

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

Topics covered include functions and graphs, limits, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics will be included at the discretion of the instructor. Math 115 is a somewhat less theoretical course which covers much of the same material. Math 186 is the natural sequel.

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Math. 215. Calculus III.

Section There Will be Joint Evening Examinations for All Sections of Math 215, 6-8 P.M., Wed. Oct. 7 and Nov, 11. Also A Joint Final. Students Electing Math 215, Lecture 010 Must Enroll in One of the Laboratory Sections 011-014.

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

Full QR

Credits: (4).

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

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

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Math. 216. Introduction to Differential Equations.

Section There Will be Joint Evening Examinations for All Sections of Math 216, 6-8 P.M., Mon. Oct 5 and Nov. 9. Also A Joint Final.

Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. 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.

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Math. 217. Linear Algebra.

Prerequisites & Distribution: Math. 215, 255, or 285. No credit granted to those who have completed or are enrolled in Math. 417, 419, or 513. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: No Homepage Submitted.

For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216-417 (or 419) and Math 217-316. The sequence Math 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217-316. These courses are explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved. Therefore the student entering Math 217 should come with a sincere interest in learning about proofs. The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of Rn; linear dependence, bases, and dimension; linear transformations; eigenvalues and eigenvectors; diagonalization; inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. Math 417 and 419 cover similar material with more emphasis on computation and applications and less emphasis on proofs. Math 513 covers more in a much more sophisticated way. The intended course to follow Math 217 is 316. Math 217 is also prerequisite for Math 412 and all more advanced courses in mathematics.

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Math. 256. Applied Honors Calculus IV.

Prerequisites & Distribution: Math. 255. Credit can be earned for only one of Math. 216, 256, 286, or 316. (4). (MSA). (BS).

No Description Provided.

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Math. 285. Honors Calculus III.

Section 002.

Prerequisites & Distribution: Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS).

Credits: (4).

Course Homepage: No Homepage Submitted.

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

Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation, maximum-minimum problems; line, surface, and volume integrals and applications; vector fields and integration; curl, divergence, and gradient; Green's Theorem and Stokes' Theorem. Additional topics may be added at the discretion of the instructor. Math 215 is a less theoretical course which covers the same material.

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Math. 285. Honors Calculus III.

Section 003.

Instructor(s): Peter Selinger (selinger@umich.edu)

Prerequisites & Distribution: Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS).

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/~selinger/285/info.html

Course Description: Multivariable calculus is a fascinating subject which combines ideas from geometry and analysis. Functions of several variables provide a much richer and more interesting structure than can be seen in the one variable case, and they also give rise to appealing two- and three-dimensional visualizations. Our objects of study are curves in space, scalar fields, and vector fields. We will study derivatives in higher dimensions, including directional and partial derivatives, as well as gradients, divergence, and curl. Higher-dimensional integrals include line, surface, and volume integrals. All these concepts are intricately related to each other, and the culmination of the course will be when we put them together in the beautiful theorems of Green and Stokes.

Course Work: We will have two in-class (1 hour) midterms. The final exam will be on Tuesday, Dec. 21, from 10:30-12:30. A weekly homework assignment will be given on Tuesdays, to be handed in the following Monday. Generally, we will have lectures on Mondays, Wednesdays, and Fridays, and a problem session on Tuesdays.

Grading: Grades will be based on the exams and homework. Performance in class may be taken into account. Each midterm counts 25%, the final 35%, and the homework 15%.

Textbook: James Stewart. Multivariable Calculus, 4th edition.

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Math. 289. Problem Seminar.

Section 001.

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

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. 295. Honors Mathematics I.

Section 001.

Prerequisites & Distribution: Prior knowledge of first year calculus and permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. (4). (MSA). (BS). (QR/1).

Full QR

Credits: (4).

Course Homepage: No Homepage Submitted.

Math 295-296-395-396 is the main Honors calculus sequence. It is aimed at talented students who intend to major in mathematics, science, or engineering. The emphasis is on concepts, problem solving, as well as the underlying theory and proofs of important results. Students interested in taking advanced mathematical courses later should consider seriously starting with this sequence. The expected background is high school trigonometry and algebra (previous calculus not required). This sequence is not restricted to students enrolled in the LS&A Honors Program. Real functions, limits, continuous functions, limits of sequences, complex numbers, derivatives, indefinite integrals and applications, some linear algebra. Math 175 and Math 185 are less intensive Honors courses. Math 296 is the intended sequel.

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Math. 316. Differential Equations.

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

No Description Provided.

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Math. 333. Directed Tutoring.

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.

No Description Provided.

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

Section 001.

Instructor(s): Zhong-Hui Duan (zduan@umich.edu)

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

No Description Provided.

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Math. 385. Mathematics for Elementary School Teachers.

Section 001.

Prerequisites & Distribution: One year each of high school algebra and geometry. No credit granted to those who have completed or are enrolled in 485. (3). (Excl).

No Description Provided.

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Math. 395. Honors Analysis I.

Section 001.

Prerequisites & Distribution: Math. 296 or permission of the Honors advisor. (4). (Excl). (BS).

No Description Provided.

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Math. 399. Independent Reading.

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

No Description Provided.

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Math. 412. Introduction to Modern Algebra.

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

Prerequisites & Distribution: Math. 215, 255, or 285; and 217. No credit granted to those who have completed or are enrolled in 512. Students with credit for 312 should take 512 rather than 412. One credit granted to those who have completed 312. (3). (Excl). (BS).

No Description Provided.

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Math. 413. Calculus for Social Scientists.

Section 001.

Prerequisites & Distribution: Not open to freshmen, sophomores or mathematics concentrators. (3). (Excl). (BS).

No Description Provided.

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Math. 416. Theory of Algorithms.

Section 001.

Prerequisites & Distribution: Math. 312 or 412 or CS 303, and CS 380. (3). (Excl). (BS).

No Description Provided.

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Math. 417. Matrix Algebra I.

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

No Description Provided.

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

Prerequisites & Distribution: Four terms of college mathematics beyond Math 110. No credit granted to those who have completed or are enrolled in 217 or 513. One credit granted to those who have completed Math. 417. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students.

No Description Provided.

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Math. 422/Business Economics and Public Policy 440. Risk Management and Insurance.

Section 001.

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

No Description Provided.

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Math. 423. Mathematics of Finance.

Section 001.

Instructor(s): Ronnie Sircar (sircar@umich.edu)

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

Credits: (3).

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

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 reflect observed market features, and to provide the necessary mathematical tools for their analysis and implementation. The course will introduce the stochastic processes used for modeling particular financial instruments. However, the students are expected to have a solid background in basic probability theory.

Specific Topics:

  1. Review of basic probability.
  2. The one-period binomial model of stock prices used to price futures.
  3. Arbitrage, equivalent portfolios and risk-neutral valuation.
  4. Multiperiod binomial model.
  5. Options and options markets; pricing options with the binomial model.
  6. Early exercise feature (American options).
  7. Trading strategies; hedging risk.
  8. Introduction to stochastic processes in discrete time. Random walks.
  9. Markov property, martingales, binomial trees.
  10. Continuous-time stochastic processes. Brownian motion.
  11. Black-Scholes analysis, partial differential equation and formula.
  12. Numerical methods and calibration of models.
  13. Interest-rate derivatives and the yield curve.
  14. Limitations of existing models. Extensions of Black-Scholes.
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Math. 424. Compound Interest and Life Insurance.

Section 001.

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

No Description Provided.

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

Section 001, 003.

Instructor(s): Jerome Wolbert (wolbert@umich.edu)

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

No Description Provided.

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

Section 002.

Instructor(s): Andrius Jankunas

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

Credits: (3).

Course Homepage: No Homepage Submitted.

See Statistics 425.002.

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

Section 004.

Instructor(s): Mielniczuk

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

Credits: (3).

Course Homepage: No Homepage Submitted.

See Statistics 425.002.

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

Section 005.

Instructor(s): P Jeganathan (jegan@umich.edu)

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

Credits: (3).

Course Homepage: No Homepage Submitted.

See Statistics 425.005.

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

Section 006.

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

Upper-Level Writing

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. Math concentrators should be sure to elect sections of the course that are taught by Mathematics (not Statistics) faculty. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. Different instructors will vary the emphasis. Math 525 is a similar course for students with stronger mathematical background and ability. Stat 426 is a natural sequel for students interested in statistics. Math 523 includes many applications of probability theory.

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

Math. 431. Topics in Geometry for Teachers.

Section 001.

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

No Description Provided.

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Math. 433. Introduction to Differential Geometry.

Section 001.

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

No Description Provided.

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Math. 450. Advanced Mathematics for Engineers I.

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

No Description Provided.

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Math. 451. Advanced Calculus I.

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

No Description Provided.

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

Section 001.

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

No Description Provided.

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Math. 471. Introduction to Numerical Methods.

Section 001.

Instructor(s): Zhong-Hui Duan (zduan@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).

No Description Provided.

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Math. 471. Introduction to Numerical Methods.

Section 002.

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

No Description Provided.

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Math. 481. Introduction to Mathematical Logic.

Section 001.

Instructor(s): Peter Selinger (selinger@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~selinger/481/

Course Description: Logic is the study of the formal principles of reasoning. In this course, we will study symbolic logic. We will introduce the formal languages of propositional and first-order logic, and we will learn how to formalize the notions of truth and proof. Logic is unlike any other branch of mathematics, in that we do not just prove things, but we reason about properties of proofs. For this reason, logic has sometimes been called meta-mathematics. On the other hand, the study of modern formal logic uses some of the same methods and techniques that are used in other branches of mathematics, and thus logic can also be regarded as a mathematical discipline.

Topics: In the first part of the course, we will introduce the notion of a formal language. We will study the propositional connectives, tautologies, and tautological consequences. The heart of the course is the study of first order predicate logic and its models. We will study formal proofs, establish soundness and completeness theorems, and explore some of their applications. We will see how to formalize elementary number theory. By the end of the course we should be able to state and understand Gödel's First Incompleteness Theorem.

Prerequisites: The official prerequisite, "Math 412 or 451 or equivalent experience with abstract mathematics," means that students should be comfortable with writing mathematical proofs. No specific knowledge of formal logic will be presupposed.

Course Work: There will be two in-class (1 hour) midterms and one final exam. There will also be regular homework assignments, which will be collected in class.

Grading: Grades will be based on exam performance. Each midterm counts 30% and the final 50%. Out of these 110%, the lowest 10% will be dropped. In borderline cases, homework will be the tie-breaker.

Textbook: Herbert B. Enderton. A Mathematical Introduction to Logic. Academic Press.

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

Math. 485. Mathematics for Elementary School Teachers and Supervisors.

Section 001.

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

No Description Provided.

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Math. 497. Topics in Elementary Mathematics.

Section 001.

Prerequisites & Distribution: Math. 489. (3). (Excl). (BS). May be repeated for a total of six credits.

No Description Provided.

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Math. 513. Introduction to Linear Algebra.

Section 001.

Prerequisites & Distribution: Math. 412 or permission of Honors advisor. Two credits granted to those who have completed Math. 417; one credit granted to those who have completed Math 217 or 419. (3). (Excl). (BS).

No Description Provided.

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Math. 520. Life Contingencies I.

Section 001.

Prerequisites & Distribution: Math. 424 and Math. 425. (3). (Excl). (BS).

No Description Provided.

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Math. 523. Risk Theory.

Section 001.

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

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

Credits: (3).

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

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

Required Texts: Actuarial Mathematics by Bowers, Gerber, Hickman, Jones and Nesbitt, Society of Actuaries, 1986. Introduction to Credibility Theory by Herzog, Actex, 1994.

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.

Content: (a) Utility theory, stop-loss insurance, theory of aggregate claims, compound Poisson claims model, estimating the probability of ruin, reinsurance schemes and their implications for profit and risk. (b) 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 8 quizzes, a midterm and a final exam. There will be 8 homework assignments. Each quiz will consist of a slightly modified homework problem.

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

Math. 524. Topics in Actuarial Science II.

Section 001.

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

No Description Provided.

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Math. 525/Stat. 525. Probability Theory.

Section 001.

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

No Description Provided.

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Math. 535. Introduction to Algebraic Curves.

Instructor(s): Pasha Belorousski

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~belorous/teaching/535.fall.99/

Algebraic Geometry is a thriving and beautiful field which is secretly taking over the rest of mathematics. It connects and unifies many seemingly distant fields such as Number Theory, Symplectic Geometry, Analysis, and Coding Theory. In recent times, Algebraic Geometry has been playing a prominent role in mathematical physics and string theory. It has, however, an undeserved reputation of being difficult for the beginner to enter. The goal of this one semester course is to give a low prerequisite introduction to the basic ideas and concepts of Algebraic Geometry through one of its gems the theory of algebraic curves. It can serve either as a first course in the field, providing preparation and motivation for further study, or as the only course, providing "liberal education" in Algebraic Geometry.

The approach will be via the theory of plane curves. This approach is very concrete and makes the connection between geometry and algebra transparent. A plane curve is defined as the zero-set of an equation, and much of the course will be devoted to exploring the link between the algebra of the equations and the geometry of the curves. For that reason, even though the course will start with a brief review, a basic familiarity with rings and ideals will be assumed. Additional commutative algebra will be developed along the way.

Here is a keyword outline of the route: affine and projective varieties, intersection numbers, Bezout's theorem, rational maps, resolution of curve singularities, divisors on curves, Riemann-Roch theorem.

Prerequisites: basic familiarity with rings, ideals, and polynomials.

Text: W. Fulton, Algebraic Curves. (Even though the book is currently out of print, copies will be provided as the course proceeds.)

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

Math. 537. Introduction to Differentiable Manifolds.

Section 001.

Prerequisites & Distribution: Math. 513 and 590. (3). (Excl). (BS).

No Description Provided.

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

Section 001.

Instructor(s): Charles Doering (doering@umich.edu)

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

No Description Provided.

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Math. 556. Methods of Applied Mathematics I.

Section 001.

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

No Description Provided.

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

Section 001.

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

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

Credits: (3).

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

Course Homepage: http://www-personal.engin.umich.edu/~murty/510/

Prerequistes: A course in linear or matrix algebra.

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.

Reference Books:

  1. K. G. Murty, Operations Research: Deterministic Optimization Models , Prentice Hall, 1995.
  2. K. G. Murty, Linear Programming , Wiley, 1983.
  3. M.S. Bazaraa, J. J. Jarvis, and H. D. Shirali, Linear Programming and Network Flows , Wiley, 1990.
  4. R. Saigal, Linear Programming: A Modern Integrated Analysis , Kluwer, 1995.
  5. D. Bertsimas and J. N. Tsitsiklis, Introduction to Linear Optimization , Athena, 1997.
  6. R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming , Scientific Press, 1993.

Course Content:

  1. Linear Programming models and their various applications. Separable piece-wise linear convex function minimization problems, uses in curve fitting and linear parameter estimation. Approaches for solving multi-objective linear programming models, the Goal programming technique.
  2. What useful planning information can be derived from an LP model (marginal values and their planning uses).
  3. Pivot operations on systems of linear equations, basic vectors, basic solutions, and bases. Brief review of the geometry of convex polyhedra.
  4. Duality and optimality conditions for LP.
  5. Revised primal and dual simplex methods for LP.
  6. Infeasibility analysis, marginal analysis, cost coefficient and right hand side constant ranging, and other sensitivity analyses.
  7. Algorithm for transportation models.
  8. Bounded variable primal simplex method.
  9. Brief review of Interior point methods for LP.

Work:

Appoximate weights for determining final grade are: Homeworks(15%), Midterm(20%), Final Exam(50%), Computer Projects(15%).

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

Math. 562/IOE 511/Aero. 577. Continuous Optimization Methods.

Section 001.

Instructor(s): Marina Epelman (mepelman@umich.edu)

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

No Description Provided.

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Math. 565. Combinatorics and Graph Theory.

Section 001.

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

No Description Provided.

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Math. 571. Numerical Methods for Scientific Computing I.

Section 001.

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

No Description Provided.

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Math. 575. Introduction to Theory of Numbers I.

Section 001.

Prerequisites & Distribution: Math. 451 and 513. Students with credit for Math. 475 can elect Math. 575 for 1 credit. (3). (Excl). (BS).

No Description Provided.

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Math. 590. Introduction to Topology.

Section 001.

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

No Description Provided.

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Math. 591. General and Differential Topology.

Section 001.

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

No Description Provided.

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Math. 593. Algebra I.

Section 001.

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

No Description Provided.

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Math. 596. Analysis I.

Section 001.

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

No Description Provided.

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