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Open courses in Mathematics
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Wolverine Access Subject listing for MATH
Winter Term '01 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 upperlevel 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 halfterm version of the same material offered as a selfstudy 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 halfterm 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 115116215 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 156255256, 175176285286, 185186285286, and 295296395396 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 185285 covers much of the material of Math 115215 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 175176 assumes a knowledge of calculus roughly equivalent to Math 115 and covers a substantial amount of socalled combinatorial mathematics (see course description) as well as calculusrelated 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 295396 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 217316 (Linear AlgebraDifferential 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 – UNIFORM EVENING EXAMS ON WEDS., FEB. 9 AND MAR. 22, 68 P.M.
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).
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 realworld 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 halfterm version of the same material offered as a selfstudy course through the Math Lab.
Text: Functions Modeling Change
Connally Wiley.
TI83 Graphing Calculator
Texas Instruments
MATH 110. PreCalculus (SelfStudy).
Section 001.
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 precalculus 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, halfterm 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.
MATH 115. Calculus I.
Section – UNIFORM EVENING EXAMS ON WEDS., FEB. 2 AND MAR. 15, 6 – 8 P.M.
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).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/115/
The sequence Math 115116215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to major 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 reallife problems in various fields, and definite integrals. The classroom atmosphere is interactive and cooperative and homework is done in groups. 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 TI83 is recommended).
Text: Calculus, 2nd Edition
HughesHallet/Gleason Wiley.
TI83 Graphing Calculator
Texas Instruments.
Student Solutions Manual
HughesHallet/Gleason Wiley.
MATH 116. Calculus II.
Section – UNIFORM EVENING EXAMS ON WEDS., FEB. 9 AND MAR. 22, 6 – 8 P.M.
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).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/116/index.shtml
See Math 115 for a general description of the sequence Math 115116215.
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.
Text: Calculus, 2nd Edition
HughesHallet/Gleason Wiley.
TI83 Graphing Calculator
Texas Instruments.
Student Solutions Manual
HughesHallet/Gleason Wiley.
MATH 127. Geometry and the Imagination.
Section 001.
Prerequisites & Distribution: Three years of high school mathematics including a geometry course. Only firstyear students, including those with sophomore standing, may preregister for FirstYear 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).
FirstYear Seminar,
Credits: (4).
Course Homepage: No Homepage Submitted.
This course introduces students to the ideas and some of the basic results in Euclidean and nonEuclidean 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 nonEuclidean 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 higherdimensional analogues of some familiar objects like spheres and cubes; discussion of the proper higherdimensional 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 handson experimentation with geometric shapes, patterns, and ideas. Grades based on homework and a final project. No exams. Text: Beyond the Third Dimension (Thomas Banchoff, 1990).
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.
Text: Mathematics of Finance
Zima and Brown McGraw Hill
MATH 176. Dynamical Systems and Calculus.
Section 001.
Prerequisites & Distribution: Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: No Homepage Submitted.
The sequence Math 175176 is a twoterm introduction to Combinatorics, Dynamical Systems, and Calculus. The topics are integrated over the two terms, although the first term will stress combinatorics and the second term will stress the development of calculus in the context of dynamical systems. Students are expected to have some previous experience with the basic concepts and techniques of calculus. The course stresses discovery as a vehicle for learning. Students will be required to experiment throughout the course on a range of problems and will participate each term in a group project. UNIX workstations will be a valuable experimental tool in this course, and students will run preset lab routines on them using Matlab and MAPLE. The general theme of the course will be discretetime and continuoustime dynamical systems. Examples of dynamical systems arising in the sciences are used as motivation. Topics include: iterates of functions, simple ordinary differential equations, fixed points, attracting and repelling fixed points and periodic orbits, ordered and chaotic motion, selfsimilarity, and fractals. Tools such as limits and continuity, Taylor expansions of functions, exponentials, logarithms, eigenvalues, and eigenvectors are reviewed or introduced as needed. There is a weekly computer workstation lab.
MATH 186. Honors Calculus II.
Section.
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).
Credits: (4).
Course Homepage: No Homepage Submitted.
No Description Provided
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MATH 214. Linear Algebra and Differential Equations.
Section.
Prerequisites & Distribution: Math 115 and 116. Credit can be earned for only one of Math. 214, 217, 417, or 419. Two credits granted to those who have completed or are enrolled in Math. 216. No credit granted to those who have completed or are enrolled in Math 513. (4). (MSA). (BS).
Credits: (4).
Course Homepage: No Homepage Submitted.
No Description Provided
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MATH 215. Calculus III.
Section – UNIFORM EVENING EXAMS ON MON, FEB 7, & THURS, MAR 23, 68 P.M.
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).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/215/
The sequence Math 115116215 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: Choice of either:
Calculus, 4th edition
James Stewart Brooks/Cole
Multivariable Calculus, 4th edition
James Stewart Brooks/Cole. course pack
Uribe, Gavosto HaydenMcNeil.
MATH 216. Introduction to Differential Equations.
Section – UNIFORM EVENING EXAMS ON THURS., FEB. 10 AND MAR. 23, 6:00 – 8:00 P.M.
Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math 214. (4). (MSA). (BS).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/216/
For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216417 (or 419) and Math 217316. The sequence Math 216417 emphasizes problemsolving 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 217316. 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 217316. Math 286 covers much of the same material in the Honors sequence. The sequence Math 217316 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.
MATH 217. Linear Algebra.
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 513. (4). (MSA). (BS). (QR/1).
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 216417 (or 419) and Math 217316. The sequence Math 216417 emphasizes problemsolving 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 217316. 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 R^{n}; 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.
Text: Linear Algebra and Its Applications, 2nd edition
David Lay Addison Wesley
MATH 255. Applied Honors Calculus III.
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
MATH 286. Honors Differential Equations.
Section 001.
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 185186285286.
Topics include firstorder differential equations, higherorder linear differential equations with constant coefficients, an introduction to linear algebra, linear systems, the Laplace Transform, series solutions and other numerical methods (Euler, RungeKutta). 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
MATH 289. Problem Seminar.
Section 001 – problem solving in elementary mathematics
Prerequisites & Distribution: (1). (Excl). (BS). May be repeated for credit with permission.
Credits: (1).
Course Homepage: https://coursetools.ummu.umich.edu/2001/winter/math/289/001.nsf
This course deals with aspects of problem solving in elementary mathematics. The textbook for this course is "Mathematics and Plausible Reasoning" by George Polya (volumes 1&2). We will
go through various techniques of problem solving and apply them in class. There will be graded
HW assignments and a final exam. The final grade will depend on homeworks, the final exam and class participation.
MATH 296. Honors Mathematics II.
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. 116, 119, 156, 176, 186, and 296. (4). (Excl). (BS). (QR/1).
Credits: (4).
Course Homepage: No Homepage Submitted.
The sequence Math 295296395396 is a more intensive Honors sequence than 185186285286. The material includes all of that of the lower sequence and substantially more. The approach is theoretical, abstract, and rigorous. Students are expected to learn to understand and construct proofs as well as do calculations and solve problems. The expected background is a thorough understanding of high school algebra and trigonometry. No previous calculus is required, although many students in this course have had some calculus. Students completing this sequence will be ready to take advanced undergraduate and beginning graduate courses. This sequence is not restricted to students enrolled in the LS&A Honors Program. The precise content depends on material covered in 295 but will generally include topics such as infinite series, power series, Taylor expansion, metric spaces. Other topics may include applications of analysis, Weierstrass Approximation theorem, elements of topology, introduction to linear algebra, complex numbers.
Text: Calculus, 3rd edition
James Stewart Brooks/Cole
MATH 312. Applied Modern Algebra.
Section 001.
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/~brichert/class/312.html
This course is an introduction to modern algebra and its applications. You will learn about some of the central concepts of algebra in a rigorous and prooforiented manner. 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 of a realworld application. The four main topics to be covered in this course are: (1) the
integers "modulo n" and linear algebra over the integers modulo p, with applications to error correcting codes; (2) some number theory, with applications to publickey cryptography; (3) polynomial algebra, with an
emphasis on factoring algorithms over various fields, and (4) permutation groups, with applications to enumeration of discrete structures "up to automorphisms", which is also known as Pòlya Theory.
Text: L.N. Childs. A Concrete Introduction to Higher Algebra. Second Edition. Springer, 1995.
Coursework: There will be weekly homework assignments. We will have two inclass midterms, and a final exam.
Grades: Grades will be based on the exams and homework. Performance in class may be taken into account. Each midterm counts 20%, the final 35% and the homework 25%.
MATH 316. Differential Equations.
Section 001.
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. Firstorder equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvectoreigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higherorder 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.
Text: Elementary Differential Equations, 7th edition
Boyce and DiPrima Wiley
MATH 333. Directed Tutoring.
Prerequisites & Distribution: Math. 385 and enrollment in the Elementary Program in the School of Education. (13). (Excl). (EXPERIENTIAL). May be repeated for a total of three credits.
Credits: (13).
Course Homepage: No Homepage Submitted.
An experiential mathematics course for exceptional upperlevel students in the elementary teacher certification program. Students tutor needy beginners enrolled in the introductory courses (Math 385 and Math 489) required of all elementary teachers.
MATH 354. Fourier Analysis and its Applications.
Section 001.
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: https://coursetools.ummu.umich.edu/2001/winter/math/354/001.nsf
Fourier Analysis is a powerful tool for solving problems in signal processing, optics, heat conduction, sound propagation, and CAT scanning. 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. We will spend a substantial portion of the time on some of the scientific and technological applications described above as well as the applications to other branches of mathematics, such as partial differential equations and signal processing. Students will do some computer work using Matlab and Maple, interactive programming tools that are easy to use, but no pervious experience with computers is necessary.
Text: Fourier Series and Integral Transforms
Pinkus & Zafrany Cambridge Univ. course pack
MATH 371/Engin. 371. Numerical Methods for Engineers and Scientists.
Section 001.
Prerequisites & Distribution: Engineering 101; one of Math. 216, 256, 286, or 316. (3). (Excl). (BS). CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: http://www.math.lsa.umich.edu/~zduan/class/
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 nonlinear equations, ordinary differential equations, polynomial approximations. Other topics may include discrete Fourier transforms, twopoint boundaryvalue problems, and MonteCarlo 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 571572 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 571572.
Text: Elementary Numerical Analysis, Kendall E. Atkinson
MATH 396. Honors Analysis II.
Section 001.
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 295396 are generally prepared to take a range of advanced undergraduate and graduate courses such as Math 512, 513, 525, 590, and many others.
MATH 399. Independent Reading.
Prerequisites & Distribution: (16). (Excl). (INDEPENDENT). May be repeated for credit.
Credits: (16).
Course Homepage: No Homepage Submitted.
Designed especially for Honors students.
MATH 412. Introduction to Modern Algebra.
Section 001.
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).
Credits: (3).
Course Homepage: No Homepage Submitted.
This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions of the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc. and their proofs. Math 217 or equivalent required as background. The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, complex numbers). These are then applied to the study of particular types of mathematical structures such as groups, rings, and fields. These structures are presented as abstractions from many examples such as the common number systems together with the operations of addition or multiplication, permutations of finite and infinite sets with function composition, sets of motions of geometric figures, and polynomials. Notions such as generator, subgroup, direct product, isomorphism, and homomorphism are defined and studied.
Math 312 is a somewhat less abstract course which substitutes material on finite automata and other topics for some of the material on rings and fields of Math 412. Math 512 is an Honors version of Math 412 which treats more material in a deeper way. A student who successfully completes this course will be prepared to take a number of other courses in abstract mathematics: Math 416, 451, 475, 575, 481, 513, and 582. All of these courses will extend and deepen the student's grasp of modern abstract mathematics.
Text: Abstract Algebra, An Introduction, 2nd edition
Thomas Hungerford Saunders College
MATH 417. Matrix Algebra I.
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 problemsolving, 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.
Text: Linear Algebra with Applications, 3rd edition
Otto Bretscher Prentice Hall
MATH 417. Matrix Algebra I.
Section 001, 003.
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/~moy/math417/
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 problemsolving, 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.
Textbook:
Otto Bretscher, Linear algebra with applications, Prentice Hall, 1997. You should be able to find this book at Ulrich's, at the Michigan Union, and possibly elsewhere.
MATH 419/EECS 400/CS 400. Linear Spaces and Matrix Theory.
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). CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: No Homepage Submitted.
Math 419 covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous prooforiented course is helpful but by no means necessary. Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations; determinants; eigenvalues, eigenvectors, Jordan canonical form, innerproduct spaces; unitary, selfadjoint, and orthogonal operators and matrices, applications to differential and difference equations.
Math 417 is less rigorous and theoretical and more oriented to applications. Math 217 is similar to Math 419 but slightly more prooforiented. Math 513 is much more abstract and sophisticated. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.
Text: Linear Algebra with Applications, 3rd edition
Otto Bretscher Prentice Hall
MATH 420. Matrix Algebra II.
Section 001.
Prerequisites & Distribution: Math. 217, 417, or 419. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~gorokhov/math420.htm
 Systems of Linear Equation and Gaussian Elimination:
Gaussian Elimination (a.k.a. row reduction), matrices,
LU factorization.
 Vector Spaces and Subspaces:
Sets of solutions of linear systems, Definition and examples of vector
spaces and subspaces, Basis and dimension, Subspaces associated with
a matrix, Linear transformations.
 Inner Products and Orthogonality:
Projections, Least squares solutions, GramSchmidt orthogonalization,
Orthogonal bases, Orthogonal matrices.
 Review of determinants
 Eigenvalues and Eigenvectors:
Diagonalization of a matrix, Application to differential and difference equations,
Complex matrices, Jordan canonical form.
 Positive definite matrices:
Positive definite matrices, minimax principle,
Singular value decomposition
Text: Linear Algebra and Its Applications, by G. Strang, 3rd edition.
Grades: will be based on
Quizzes: 15%,
First Exam: 20%,
Second Exam: 25%,
Final: 40%
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).
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.
MATH 423. Mathematics of Finance.
Section 001.
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
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: (a) Forwards and Futures, Hedging using Futures, Bills and Bonds, Swaps, Perfect Hedges. (b) OptionsEuropean and American, Trading Strategies, PutCall Parity, BlackScholes formula. (c) Volatility, methods for estimating volatilityexponential, GARCH, maximum likelihood. (d) Dynamic Hedging, stoploss, BlackScholes, the Greek letters. (e)
Required Text: Options, Futures and Other Derivatives by Hull, fourth edition, Prentice Hall 1999.
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
MATH 424. Compound Interest and Life Insurance.
Section 001.
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 reallife 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).
MATH 425/Stat. 425. Introduction to Probability.
Section 001.
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (MSA). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
See Statistics 425.001.
MATH 425/Stat. 425. Introduction to Probability.
Section 002, 003, 006.
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (MSA). (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.
Text: A First Course in Probability
Sheldon Ross
5th Edition
PrenticeHall
ISBN: 0137463146
MATH 425/Stat. 425. Introduction to Probability.
Section 004, 005.
Instructor(s): Yanhong Wu
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (MSA). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
See Statistics 425.004.
MATH 425/Stat. 425. Introduction to Probability.
Section 006.
Instructor(s): P Jeganathan
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (MSA). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
See Statistics 425.004.
MATH 427/Human Behavior 603 (Social Work). Retirement Plans and Other Employee Benefit Plans.
Section 001.
Prerequisites & Distribution: Junior standing. (3). (Excl).
Credits: (3).
Course Homepage: No Homepage Submitted.
An overview of the range of employee benefit plans, the considerations (actuarial and others) which influence plan design and implementation practices, and the role of actuaries and other benefit plan professionals and their relation to decision makers in management and unions. Particular attention will be given to government programs which provide the framework, and establish requirements, for privately operated benefit plans. Relevant mathematical techniques will be reviewed, but are not the exclusive focus of the course. Math 521 and/or 522 (which can be taken independently of each other) provide more indepth examination of the actuarial techniques used in employee benefit plans. No textbook
MATH 450. Advanced Mathematics for Engineers I.
Section 001.
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 3space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems, derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. Math 450 is an alternative to Math 451 as a prerequisite for several more advanced courses. Math 454 and 555 are the natural sequels for students with primary interest in engineering applications.
Text: Advanced Engineering Mathematics, 8th edition
Edward Kreyszig Wiley
MATH 450. Advanced Mathematics for Engineers I.
Section 002.
Prerequisites & Distribution: Math. 215, 255, or 285. (4). (Excl). (BS).
Credits: (4).
Course Homepage: http://www.math.utsa.edu/~gokhman/courses/2001_1/math450.html
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 3space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems, derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. Math 450 is an alternative to Math 451 as a prerequisite for several more advanced courses. Math 454 and 555 are the natural sequels for students with primary interest in engineering applications.
Book: E. Kreyszig, Advanced Engineering Mathematics, 8th ed.
Wiley, ISBN: 0471154962 Material: Chapters 1013, 16 and, time permitting, parts of 14, 15.
Exams:
There will be three exams (2 midterms and a final).
The exams are closed book, but standard formulas will be provided and you can bring notes on one 3" x 5" card.
No makeups.
The final exam will cover all of the material.
The total exam score will be the maximum of the combined percentage on the midterms and the percentage on the
final.
MATH 451. Advanced Calculus I.
Section 001.
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: http://www.math.lsa.umich.edu/~carswell/math451/
This course has two complementary goals: (1) a rigorous development of the fundamental ideas of calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc. and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits and continuity; differentiation; integration, the Fundamental Theorem of Calculus; infinite series; sequences and series of functions.
There is really no other course which covers the material of Math 451. Although Math 450 is an alternative prerequisite for some later courses, the emphasis of the two courses is quite distinct. The natural sequel to Math 451 is 452, which extends the ideas considered here to functions of several variables. In a sense, Math 451 treats the theory behind Math 115116, 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 SpringerVerlag
MATH 451. Advanced Calculus I.
Section 002.
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; 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 115116, 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 SpringerVerlag
MATH 452. Advanced Calculus II.
Section 001.
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) maxmin theory; (6) multiple integration, change of variable, etc., (7) Green's and Stokes' theorems, differential forms, exterior derivatives. Math 551 is a higherlevel 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).
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).
Credits: (3).
Course Homepage: https://coursetools.ummu.umich.edu/2001/winter/math/354/001.nsf
This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of boundaryvalue problems for secondorder 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 onedimensional heat and wave equation; the heat and wave equations in higher dimensions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; applications to linear inputoutput systems, analysis of data smoothing and filtering, signal processing, timeseries 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.
no textbook
MATH 462. Mathematical Models.
Section 001 – Topic?
Prerequisites & Distribution: Math. 216, 256, 286, or 316; and 217, 417, or 419. Students with credit for 362 must have department permission to elect 462. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
Check Times, Location, and Availability
MATH 463. Mathematical Modeling in Biology.
Section 001.
Prerequisites & Distribution: Math. 217, 417, or 419; 286, 256, or 316. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
This course will concentrate on the applications of ordinary differential
equations to physiological systems. Partial differential equations will
not be covered in detail. Thus, a course in ODEs such as 216 or 316 will
be sufficient preparation for this course.
Who could take the course? Basically anybody who is interested
in applying mathematical methods to the biological sciences. For instance, students from Biology, Chemistry, Physics, Complex Systems, Biophysics, Biomedical Engineering, Mathematics, Chemical Engineering, Physiology, Microbiology, Epidemiology.
What kind of background will you need? Basically a course in differential
equations, such as 216 or 316. If you have never seen a differential equation
before, you may have trouble with the course. You will also need to be familiar
and comfortable with computers, as a lot of the work in the course will
have to be done on a computer. You will not need to be an expert in biology, as we will learn most of what we need to know as we go
How will the course be graded? I will assign projects to do throughout the term, and there will also be written assignments. The projects will
require a lot of computing, and students will be able to choose between
a selection of projects. There will also be a final examination, probably
a takehome exam, but I have not yet decided this for sure.
Topics
 Review of phaseplane methods for ordinary differential equations
 PredatorPrey models
 Models of disease transmission. SIR models, and epidemics.
 Enzyme kinetics
 Pseudosteadystate hypothesis.
 Cooperativity.
 MonodWymanChangeux models.
 Cellular homeostasis
 The membrane potential. The Nernst equation. Electrodiffusion. The
GoldmanHodgkinKatz equation. The constant field approximation.
 Osmosis
 Control of cell volume
 Sodium transport and cell volume control
 Review of limit cycles and oscillations. The Hopf bifurcation.
 Excitability
 The HodgkinHuxley model and action potentials
 Twovariable models. FitzHughNagumo model
 Phaseplane analysis of the models
 The circulatory system
 Blood flow and compliance
 Guyton's model
 Cardiac regulation
 Fetal circulation
 The Windkessel model
 Blood
 Myoglobin and hemoglobin
 Cooperativity and oxygen transport
 Carbon dioxide transport. Bohr and Haldane effects.
 Muscle
 The Hill model
 Crossbridge theory and the Huxley model
 Discrete binding site models
The last few topics are flexible. Depending on student interest, we could
cover any topic from the textbook, including the visual system, hearing, pulsatile hormone secretion, pattern formation, or wave propagation.
Text
Mathematical
Physiology by Keener and Sneyd, Springer, 1998. This has a lot of
additional material, but does cover the majority of the syllabus. A lot
of the lecture material (but not all) will be taken from this book, as will
lots of the problems.
Other Recommended Books
Mathematical Models in Biology, by EdelsteinKeshet, McGrawHill,1988.
Very expensive book, but covers a lot of basic material, concentrating on
population biology and ecology sorts of things. Also covers a lot of basic
math stuff, although not as well as many other books.
Mathematical Biology, by J. Murray, Springer, 1989. Covers a huge
amount of stuff. An excellent book, but mostly too advanced for this course.
Great for reference though, for all students interested in the field. One
of the great books in the field.
Mathematics in Medicine and the Life Sciences, by Hoppensteadt
and Peskin, Springer. A bit more elementary, but lots of good stuff.
Nonlinear Dynamics and Chaos, by S. Strogatz. This is probably the best book for revising phase plane theory, elementary bifurcations and differential equations. Has lots of applications, including some from biology.
Differential Equations and Their Applications, by Braun. This
is another excellent book for learning differential equations, with applications
to all kinds of things, including disease models, the theory of conflict, population models etc. Review of phaseplane methods for ordinary
differential equations.
MATH 471. Introduction to Numerical Methods.
Section 001.
Prerequisites & Distribution: Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one highlevel computer language. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~zduan/class/471/
This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Topics may include computer arithmetic, Newton's method for nonlinear 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, 2point boundary value problems, Dirichlet problem for the Laplace equation. Math 371 is a less sophisticated version intended principally for sophomore and junior engineering students; the sequence Math 571572 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: An Introduction to Numerical Analysis, Kendall E. Atkinson
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 highlevel computer language. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Topics may include computer arithmetic, Newton's method for nonlinear 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, 2point boundary value problems, Dirichlet problem for the Laplace equation. Math 371 is a less sophisticated version intended principally for sophomore and junior engineering students; the sequence Math 571572 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: An Introduction to Numerical Analysis
Kendall Atkinson Wiley
MATH 475. Elementary Number Theory.
Section 001.
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 numbertheoretic 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 selfcontained, 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.
MATH 476. Computational Laboratory in Number Theory.
Section 001.
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 numbertheoretic purposes, e.g., for factoring. No exams.
MATH 486. Concepts Basic to Secondary Mathematics.
Section 001.
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
Check Times, Location, and Availability
MATH 489. Mathematics for Elementary and Middle School Teachers.
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.
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 precalculus 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 problemsolving strategies are used throughout the course.
MATH 490. Introduction to Topology.
Section 001.
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 pointset and algebraic topology. Although much of the presentation is theoretical and prooforiented, the material is wellsuited 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. Pointset topology, examples of topological spaces, orientable and nonorientable surfaces, fundamental groups, homotopy, 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.
MATH 498. Topics in Modern Mathematics.
Section 001 – Boundary Value Problems II
Prerequisites & Distribution: Senior mathematics concentrators and Master Degree students in mathematical disciplines. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~timcall/
This course is a continuation of Math 454, although the
offcial prerequisite is adequate preparation. Topics to be covered (time permitting) include continuous spectra (unbounded
domains and the Dirac delta function), coherent and incoherent scattering and radiation, perturbation theory, adiabatic
invariants, spatially inhomogeneous media (e.g., dielectrics), anisotropic media, inhomogeneous equations and Green's functions, boundary layers, the nonlinear wave equation (characteristics and shock waves) and nonlinear PDE's.
As with Math 454, this is a very applied course, with
examples from electromagnetism, mechanics, aerodynamics, financial engineering, etc. There will be several homework sets, a midterm and a final.
MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.
Section 001.
Prerequisites & Distribution: At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). (Excl). Offered mandatory credit/no credit. May be repeated for a total of 6 credits.
Credits: (1).
Course Homepage: No Homepage Submitted.
No Description Provided
Check Times, Location, and Availability
MATH 512. Algebraic Structures.
Section 001.
Prerequisites & Distribution: Math. 451 or 513. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
This is one of the more abstract and difficult courses in the undergraduate program. It is frequently elected by students who have completed the 295396 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, and fields) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Exercises tend to be quite challenging. Students should have some previous exposure to rigorous prooforiented mathematics and be prepared to work hard. Students from Math 285 are strongly advised to take some 400500 level course first, for example, Math 513. Some background in linear algebra is strongly recommended.
The course covers basic definitions and properties of groups, rings, and fields, including homomorphisms, isomorphisms, and simplicity. Further topics are selected from (1) Group Theory: Sylow theorems, Structure Theorem for finitelygenerated Abelian groups, permutation representations, the symmetric and alternating groups; (2) Ring Theory: Euclidean, principal ideal, and unique factorization domains, polynomial rings in one and several variables, algebraic varieties, ideals; and (3) Field Theory: statement of the Fundamental Theorem of Galois Theory, Nullstellensatz, subfields of the complex numbers and the integers mod p.
Math 412 is a substantially lowerlevel course over about half of the material of Math 512. The sequence Math 593594 covers about twice as much Group and Field Theory as well as several other topics and presupposes that students have had a previous introduction to these concepts at least at the level of Math 412. Together with Math 513, this course is excellent preparation for the sequence Math 593594.
MATH 513. Introduction to Linear Algebra.
Section 001.
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: No Homepage Submitted.
No Description Provided
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MATH 521. Life Contingencies II.
Section 001.
Prerequisites & Distribution: Math. 520. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
This course extends the single decrement and single life ideas of Math 520 to multidecrement and multiplelife applications directly related to life insurance. The sequence Math 520521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Concepts and calculation are emphasized over proof. Topics include multiple life models – joint life, last survivor, contingent insurance; multiple decrement models – disability, withdrawal, retirement, etc.; and reserving models for life insurance. Math 522 is a parallel course covering mathematical models for prefunded retirement benefit programs.
MATH 523. Risk Theory.
Section 001.
Prerequisites & Distribution: Math. 425. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~conlon/math523/index.html
Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in finance, risk management, or insurance and provides background for the professional examinations in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society. Students should have a basic knowledge of common probability distributions (Poisson, exponential, gamma, binomial, etc. and have at least junior standing. Two major problems will be considered: (1) modeling of payouts of a financial intermediary when the amount and timing vary stochastically over time; and (2) modeling of the ongoing solvency of a financial intermediary subject to stochastically varying capital flow. These topics will be treated historically beginning with classical approaches and proceeding to more dynamic models. Connections with ordinary and partial differential equations will be emphasized. Classical approaches to risk including the insurance principle and the riskreward tradeoff. Review of probability. Bachelier and Lundberg models of investment and loss aggregation. Fallacy of time diversification and its generalizations. Geometric Brownian motion and the compound Poisson process. Modeling of individual losses which arise in a loss aggregation process. Distributions for modeling size loss, statistical techniques for fitting data, and credibility. Economic rationale for insurance, problems of adverse selection and moral hazard, and utility theory. The three most significant results of modern finance: the Markowitz portfolio selection model, the capital asset pricing model of Sharpe, Lintner and Moissin, and (time permitting) the BlackScholes option pricing model.
Text: Loss Models – From Data to Decisions
Klugman, Panjer, et al Wiley
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).
Credits: (3).
Course Homepage: No Homepage Submitted.
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 510511 are natural sequels.
An Introduction to Probability Theory and Its Applications, 3rd
edition
William Feller Wiley
recommended – Introduction to Probability Theory
Hoel, Port, Stone HoughtonMifflin
MATH 526/Stat. 526. Discrete State Stochastic Processes.
Section 001.
Prerequisites & Distribution: Math. 525 or EECS 501. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
The course this academic term will be about probabilistic models of proteins and nucleic acids, and their
uses in molecular biology. The topics will include a review of basic concepts of probability and very
rudimentary molecular biology; probability and the design of similarity scoring functions; optimal local
and global alignments of sequences: dynamic programming, SmithWaterman algorithm, other
algorithms available on the Web (BLAST and FastA, etc.), probabilistic (heuristic) versus rigorous
algorithms; significance of scores and simulation; dependence of scoring functions and optimal
alignments on parameters, comparison of standard tables; hidden Markov models and neural
network models; multiple sequence alignment methods and algorithms, families of proteins;
phylogenetic tree determinations; structure of proteins and recognizable patterns in amino acid
sequences (motif recognition). Guest lecturers will address the class on applications in the
pharmaceutical industry, as well as some earlier examples of these techniques applied to problems
in linguistics and speech recognition.
Students will be expected to complete three to four problem sets, most of which will hopefully be
group projects, some of which will involve using Webbased tools. If the class demographics work out
favorably, we will be mixing students with biological background and mathematical background in
each group. Every effort will be made to accommodate students from diverse backgrounds.
Text:
 R. Durbin, S. Eddy, A. Krogh and G. Mitchison, Biological Sequence Analysis, Cambridge, C.U.P., 1998. [required]
 M. Waterman, Introduction to Computational Biology, London, Chapman and Hall, 1995. [optional]
MATH 528. Topics in Casualty Insurance.
Section 001 – Topic?
Prerequisites & Distribution: Math 217, 417, or 419. (3). (Excl).
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
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MATH 531. Transformation Groups in Geometry.
Section 001.
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math531/
This course lies in the intersection of group theory and geometry and will develop the foundation for many beautiful and fascinating topics. Among these topics are symmetries of polygons and polyhedra, wallpaper
groups (popularized by Escher, for example), crystallographic groups, and sphere packing.
Along the way we will learn about Euclidean and hyperbolic geometry and the group theory and linear algebra that encode the symmetries of these spaces.
There is no required text, but the course will follow parts of Armstrong, Groups and symmetry (Springer). The following books may pique your interest: Coxeter, Introduction to geometry; Coxeter, Regular polytopes;
Cromwell, Polyhedra; Beardon, The geometry of discrete groups.
The prerequisite for the course is Math 215 or 285. We will develop the necessary algebra at a pace determined by the backgrounds of the students. Problem sets will be assigned.
MATH 550/Complex Systems 510. Introduction to Adaptive Systems.
Section 001 – Introduction to Dynamical Systems for Biocomplexity
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/CSCScourses/cscs510w01.html
See Complex Systems 510.001.
MATH 555. Introduction to Functions of a Complex Variable with Applications.
Section 001.
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.
Section 001.
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.
Section 001.
Prerequisites & Distribution: Math. 450 or 451. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
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MATH 561/SMS 518 (Business Administration)/IOE 510. Linear Programming I.
Section 001.
Prerequisites & Distribution: Math. 217, 417, or 419. (3). (Excl). (BS). CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: http://wwwpersonal.engin.umich.edu/~murty/510/index.html
Background Required: Elementary matrix algebra (concept of linear independence, bases, matrix inversion, pivotal methods for solving linear equations), geometry of R^{n}
including convex sets and affine spaces.
Reference Books:
 K. G. Murty, Operations Research: Deterministic Optimization Models, Prentice Hall, 1995.
 K. G. Murty, Linear Programming, Wiley, 1983.
 M.S. Bazaraa, J. J. Jarvis, and H. D. Shirali, Linear Programming and Network Flows, Wiley, 1990.
 R. Saigal, Linear Programming: A Modern Integrated Analysis, Kluwer, 1995.
 D. Bertsimas and J. N. Tsitsiklis,Introduction to Linear Optimization, Athena, 1997.
 R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Scientific Press, 1993.
Course Content:
 Linear Programming models and their various applications. Separable piecewise linear convex function minimization problems, uses in curve fitting and linear parameter estimation. Approaches for solving multiobjective linear programming models, the Goal programming technique.
 What useful planning information can be derived from an LP model (marginal values and their planning uses).
 Pivot operations on systems of linear equations, basic vectors, basic solutions, and bases. Brief review of the geometry of convex polyhedra.
 Duality and optimality conditions for LP.
 Revised primal and dual simplex methods for LP.
 Infeasibility analysis, marginal analysis, cost coefficient and right hand side constant ranging, and other sensitivity analyses.
 Algorithm for transportation models.
 Bounded variable primal simplex method.
 Brief review of Interior point methods for LP.
Work:
Weekly Homework Assignments; Midterm;
Final Exam;
Two Computational Projects to be solved using AMPL.
Appoximate weights for determining final grade are: Homeworks(15%), Midterm(20%), Final Exam(50%), Computer Projects(15%).
MATH 566. Combinatorial Theory.
Section 001.
Prerequisites & Distribution: Math. 216, 256, 286, or 316. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
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MATH 567. Introduction to Coding Theory.
Section 001.
Prerequisites & Distribution: One of Math 217, 419, 513. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
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).
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
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MATH 572. Numerical Methods for Scientific Computing II.
Section 001.
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 582. Introduction to Set Theory.
Section 001.
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.
Section 001.
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.
Section 001.
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.
Section 001.
Prerequisites & Distribution: Math. 451 and 513. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
Check Times, Location, and Availability
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