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Undergraduate Courses
105 | 115 | 116 | 127 | 128 | 147 | 156 | 174 | 175 | 176 | 185 | 186 | 214 | 215 | 216 | 217 | 255 | 256 | 285 | 286 | 288 | 289 | 295 | 296 | 310 | 312 | 316 | 327 | 351 | 354 | 362 | 371 | 385 | 389 | 395 | 396 | 404 | 412 | 416 | 417 | 419 | 420 | 422 | 423 | 424 | 425 | 427 | 431 | 432 | 433 | 450 | 451 | 452 | 454 | 462 | 463 | 464 | 471 | 472 | 475 | 476 | 481 | 486 | 489 | 490 | 497 | 498 |
512 | 513
| 520 | 521 | 522
| 523 | 524 | 525
| 526 | 528 | 531
| 532 | 535 | 537
| 547 | 548 | 550 | 551 | 555
| 556 | 557 | 558
| 559 | 561 | 562
| 563
| 565 | 566 | 567
| 571 | 572 | 575
| 582 | 590 | 591
| 592 | 593 | 594
| 596 | 597
*For a list of current Math textbooks, please click here.
Math 105 - Data, Functions and Graphs
- Prerequisites: 3-4 years HS math
- Frequency: Fall (I), Winter (II), Summer (IIIb)
- Student Body: First-year students
- Credit: 4 Credits. No credit granted for those who have completed any Math course numbered 110 or higher.
- Recent Texts: Functions Modeling Change (Connally, Hughes-Hallett and Gleason)
- Background and Goals: 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 successfully
complete 105 are fully prepared for Math 115.
- Content: This course presents the concepts of precalculus from four points of view: geometric (graphs), numeric (tables), symbolic (formulas), and written (verbal descriptions). The emphasis is on the mathematical modeling of real-life problems using linear, polynomial, exponential, logarithmic, and trigonometric functions. Students develop their reading, writing, and questioning skills in an interactive classroom setting.
- Alternatives: Math 110 (Pre-Calculus (Self-Paced)) is a condensed half-term
version of the same material offered as a self-study course through
the Math Lab.
- Subsequent Courses: The course prepares students for Math 115 (Calculus I).
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Math 115 - Calculus I
- Prerequisites: 3-4 years HS math including trigonometry
- Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb)
- Student Body: First-year students
- Credit: 4 Credits
- Recent Texts: Calculus: Single Variable (Hughes-Hallett, et al.)
- Background and Goals: 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 two uniform midterms and a final exam.
- Content: The course presents the concepts of calculus from four points of view: geometric (graphs), numeric (tables), symbolic (formulas), and verbal descriptions. Students will develop their reading, writing and questioning skills, as well as their ability to work cooperatively. Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and an introduction to integration. The classroom atmosphere is interactive and cooperative. Both individual and team homework is assigned.
- Alternatives: Math 185 (Honors Calculus I ) is a somewhat more theoretical course which covers some of the same material. Math 175 (Intro to Cryptology) is a non-calculus alternative for students with a good command of first-semester calculus. Math 295 (Honors Mathematics I) is a much more intensive and rigorous course. A student whose preparation is insufficient for Math 115 should take Math 105 (Data, Functions and Graphs).
- Subsequent Courses: Math 116 (Calculus II) 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 (Honors Calculus II).
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Math 116 - Calculus II
- Prerequisites: Math 115 or AP credit
- Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb)
- Student Body: First-year students
- Credit: 4 Credits
- Recent Texts: Calculus: Single Variable (Hughes-Hallett, et al.)
- Background and Goals: 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 two uniform midterms and a final exam.
- Content: The course presents the concepts of calculus from four points of view: geometric (graphs), numeric (tables), symbolic (formulas), and verbal descriptions. Students will develop their reading, writing and questioning skills, as well as their ability to work cooperatively. Topics include techniques of integration, applications of integration, Taylor series, an introduction to differential equations, and infinite series. The classroom atmosphere is interactive and cooperative. Both individual and team homework is assigned.
- Alternatives: Math 156 (Applied Honors Calculus II) also covers much of the
same material; emphasizing applications to science
and engineering. Math 186 (Honors Calculus II) is a somewhat
more theoretical course which covers much of the same material.
- Subsequent Courses: Math 215 (Calculus III) 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 255 (Applied Honors Calculus III) or Math 285 (Honors Calculus III).
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Math 127 - Geometry and the Imagination
- Prerequisites: 3 years HS math
- Frequency: Winter (II)
- Student Body: First-year students and sophomores
- Credit: 4 Credits. No credit after completing any 200+ level math course
except 385, 489, or 497
- Recent Texts: Beyond the Third Dimension: Geometry, Computer
Graphics and Higher Dimensions (Banchoff)
- Background and Goals: This course introduces students to the ideas and some of the basic results in Euclidean and non-Euclidean geometry. Beginning with geometry in ancient Greece, the course includes the construction of new geometric objects from old ones by projecting and by taking slices. The course is intended for students who want an introduction to mathematical ideas and culture. Emphasis is on conceptual thinking — students will do hands-on experimentation with geometric shapes, patterns and ideas.
- Content: The 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: coordinization — the mathematician’s tool for studying higher dimensions; construction of higher-dimension analogues of some familiar objects like spheres and cubes; discussion of the proper higher-dimensional analogues of some geometric notions (length, angle, orthogonality, etc.).
- Alternatives: none
- Subsequent Courses: This course does not provide preparation for
any further study of mathematics.
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Math 128 - Explorations in Number Theory
- Prerequisites: 3 years HS math
- Frequency: Fall (I)
- Student Body: First-year students and sophomores
- Credit: 4 Credits. No credit after completing any 200+ level math course
except 385, 489, or 497.
- Recent Texts: coursepack
- Area: Number Theory
- Background and Goals: 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 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.
- Content: After studying the 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.
- Alternatives: none
- Subsequent Courses: This course does not provide preparation for
any further study of mathematics.
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Math 147 - Introduction to Interest Theory
- Prerequisites: Math 112 or 115
- Frequency: Fall (I), Winter (II)
- Student Body: First-year students and sophomores
- Credit: 3 Credits. no credit after completing any 200+ level math course
except 385, 489, or 497
- Recent Texts: Mathematics of Finance (Zima and Brown)
- Area: Actuarial & Financial
- Background and Goals: 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. The course
is not part of a sequence. Students should possess financial calculators.
- Content: 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.
- Alternatives: Math 424 (Compound Interest and Life Ins) covers the
same material in greater depth and with a higher level of mathematical
content.
- Subsequent Courses: none
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Math 156 - Applied Honors Calculus II
- Prerequisites: Score of 4 or 5 on Advanced Placement AB or BC Calculus exam, or permission of instructor
- Frequency: Fall (I)
- Student Body: First-year students
- Credit: 4 Credits.
- Recent Texts: Single Variable Calculus (Stewart)
- Background and Goals: Math 156 is part of the applied honors calculus sequence for engineering and science concentrators. The course is an alternative to Math 116 for students with strong mathematics ability. Applications and concepts receive equal treatment. Theorems are stated precisely and are derived, but technical details are omitted. Examples are given to illustrate the theory. Critical thinking and class participation are encouraged. The goal is to provide students with a solid background needed for subsequent courses in mathematics, engineering, and science.
- Content: Riemann sums, definite integral, fundamental theorem of calculus, applications of integral calculus (e.g. arclength, surface area, work, center of mass, probability density functions), improper integrals, infinite sequences and series, geometric series, alternating series, power series, Taylor series, differential equations, complex numbers. Students are introduced to MAPLE.
- Alternatives: Math 116 (Calculus II) or Math 186 (Honors Calculus II).
- Subsequent Courses: Math 255 (Applied Honors Calculus III) is the
natural sequel.
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Math 174 - Proofs in Geometry
- Prerequisites: Permission of honors advisor
- Frequency: Fall (I)
- Student Body: First-year students
- Credit: 4 Credits. No credit granted to those who have completed a 200-level or higher Math course.
- Recent Texts: none
- Background and Goals: The course will be very interactive, eliciting suggestions towards proof from the students so that all the problems are eventually solved by a joint effort between the students and the instructor. The format has worked well in the past for honors courses. To enhance the visualization, we plan to develop software for two-dimensional geometric constructions. This software will be able to produce multi-color pictures if geometric configurations. In the long run, such software will save us time in creating problem sets, handouts and perhaps slides. Additional topics may be added depending on the interest and abilities of the students.
- Content: A good text for the course is already available: the classic "Geometry Revisited" by Coxter and Greitzer, which contains a wonderful exposition of the material and has suitable exercises. As a precursor to the mathematics, the course will use familiar games such as the old game Mastermind where player A has a code which player B has to use. Students will pair off and play the game, with the important additional feature that the guesser must write down what(s) he knows and can deduce after each guess, and therefore motivate his/her next guess. This should help set the mood and instill the idea of analyzing the facts at hand and making logical deductions. After this the course will develop some basic theorems of Euclidean geometry. An example of such a theorem is that the angle bisectors (or medians, or altitudes, or perpendicular bisectors) of a triangle are concurrent. These results are fairly straightforward but exemplify the spirit of the course by providing a good introduction to rigorous proofs, Then we move to some more difficult but beautiful theorems from geometry such as Ceva's theorem, the Euler line, the nine-point circle theorem, Ptolemy's theorem and Morley's theorem.
- Alternatives: none
- Subsequent Courses: none
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Math 175 - Introduction to Cryptology
- Prerequisites: Permission of honors advisor
- Frequency: Fall (I)
- Student Body: First-year students
- Credit: 4 Credits
- Recent Texts: coursepack; Invitation to Cryptology (Barr)
- Background and Goals: This course is an alternative to Math 185 as an entry to the honors sequence. Students are expected to have previous experience with the basic concepts and techniques of first-semester 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 semester 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.
- Content:This course gives a historical introduction to Cryptology and introduces a number of mathematical ideas and results involved in the development and analysis of secret codes. The course begins with the study of permutation-based codes: substitutional ciphers, transpositional codes, and more complex polyalphabetic substitutions. The mathematical subjects treated in this section include enumeration, modular arithmetic and some elementary statistics. The subject then moves to bit stream encryption methods. These include block cipher schemes such as the Data Encryption Standard. The mathematical concepts introduced here are recurrence relations and some more advanced statistical results. The final part of the course is devoted to public key encryption, including Diffie-Hellman key exchange, RSA and Knapsack codes. The mathematical tools come from elementary number theory.
- Alternatives: Math 115 (Calculus I), Math 185 (Honors Calculus I), or Math 295 (Honors Mathematics I).
- Subsequent Courses:Math 176 (Dynamical Systems and Calculus), Math 186 (Honors Calculus II), or Math 116 (Calculus II).
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Math 176 - Dynamical Systems and Calculus
- Prerequisites: Math 175 or permission of instructor
- Frequency: Sporadically
- Student Body: First-year students
- Credit: 4 Credits
- Recent Texts: coursepack; Nonlinear Dynamics and Chaos (Strogatz)
- Background and Goals: 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 semester 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.
- Content: The general theme of the course will be discrete-time and continuous-time dynamical systems. Examples of dynamical systems arising in the sciences are used as motivation. Topics include: iterates of functions, simple ordinary differential equations, fixed points, attracting and repelling fixed points and periodic orbits, ordered and chaotic motion, self-similarity, and fractals. Tools such as limits and continuity, Taylor expansions of functions, exponentials, logarithms, eigenvalues, and eigenvectors are reviewed or introduced as needed. There is a weekly computer work-station lab.
- Alternatives: Math 116 (Calculus I) or Math 186 (Honors Calculus II)
- Subsequent Courses: Math 285 (Honors Calculus III)
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Math 185 - Honors Calculus I
- Prerequisites: Permission of honors advisor; not intended for students receiving a score of 5 on the AB, or 4 or 5 on the BC Advanced Placement exam.
- Frequency: Fall (I)
- Student Body: First-year students
- Credit: 4 Credits
- Recent Texts: Calculus 5th edition (Stewart)
- Background and Goals:The sequence Math 185-186-285-286 is an introduction to calculus at the honors level. It is not appropriate for students who have received scores of 5 on the AB, or 4 or 5 on the BC Advanced Placement exam (those students should elect Math 156 or Math 295). 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.
- Content: 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.
- Alternatives: Math 115 (Calculus I) is a somewhat less theoretical
course which covers much of the same material. Math 295 (Honors
Mathematics I) gives a much more theoretical treatment of much
of the same material.
- Subsequent Courses: Math 186 (Honors Calculus II) is
the natural sequel.
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Math 186 - Honors Calculus II
- Prerequisites: Permission of honors advisor
- Frequency: Winter (II)
- Student Body: First-year students
- Credit: 4 Credits
- Recent Texts: Calculus 5th edition (Stewart)
- Background and Goals: The sequence Math 185-186-285-286 is an introduction to calculus at the honors level. 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.
- Content: Topics covered include transcendental functions; techniques of integration; applications of calculus such as elementary differential equations, simple harmonic motion, and center of mass; conic sections; polar coordinates; infinite sequences and series including power series and Taylor series. Other topics, often an introduction to matrices and vector spaces, will be included at the discretion of the instructor.
- Alternatives: Math 116 (Calculus II) is a somewhat less theoretical
course which covers much of the same material. Math 156 (Applied
Honors Calculus II) is more application based, but covers much
of the same material.
- Subsequent Courses: Math 285 (Honors Calculus III) is the natural sequel.
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Math 214 - Linear Algebra and Differential Equations
- Prerequisites: Math 115 and 116
- Frequency: Fall (I), Winter (II)
- Student Body: Engineering students, particularly in Industrial
and Operations Engineering
- Credit: 4 Credits. Credit is granted for only one course among Math 214,
217, 417, and 419. No credit granted to those who have completed
or are enrolled in Math 513. Two credits granted after completing
Math 216.
- Recent Texts: Linear Algebra with Applications (Bretscher)
- Background and Goals:An introduction to matrices and linear algebra. This course covers the basics needed to understand a wide variety of applications that use the ideas of linear algebra, from linear programming to mathematical economics. The emphasis is on concepts and problem solving. The course is designed as an alternative to Math 216 for students who need more linear algebra and less differential equations background than provided in 216.
- Content:An introduction to the main concepts of linear algebra… matrix
operations, echelon form, solution of systems of linear equations,
Euclidean vector spaces, linear combinations, independence and
spans of sets of vectors in Euclidean space, eigenvectors and
eigenvalues, similarity theory. There are applications to discrete
Markov processes, linear programming, and solutions of linear
differential equations with constant coefficients.
- Alternatives: Math 419 (Linear Spaces and Matrix Theory) has a somewhat
more theoretical emphasis. Math 217 is a more theoretical course
which covers much of the material of Math 214 at a deeper level.
Math 513 (Intro. to Linear Algebra) is an honors version of this
course. Mathematics concentrators are required to take Math 217 or Math
513.
- Subsequent Courses: Math 420 (Matrix Algebra II), Math 561 (Linear Programming I), Math 462 (Mathematical Models), Math 571 (Numer. Meth. for Sci. Comput. I).
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Math 215 - Calculus III
- Prerequisites: Math 116, 156, or 186
- Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb)
- Student Body: Sophomores
- Credit: 4 Credits. Credit is granted for only one course among Math 215,
255, and 285.
- Recent Texts: Multivariable Calculus 5th edition (Stewart)
- Background and Goals: 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.
- Content: 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.
- Alternatives: Math 285 (Honors Calculus III) is a somewhat more theoretical course which covers the same material. Math 255 (Applied Honors
Calculus III) is also an alternative.
- Subsequent Courses: 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 (Linear Algebra). Students who intend to take only one further mathematics course and need differential equations should take Math 216 (Intro. to Differential Equations).
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Math 216 - Introduction to Differential Equations
- Prerequisites: Math 116, 156, or 186
- Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb)
- Student Body: Sophomore engineering students
- Credit: 4 Credits. 2 credits granted to those who have completed or are
enrolled in Math 214; credit is granted for only one course among
Math 216, 256, 286, and 316.
- Recent Texts: Differential Equations: Computing and Modeling 3rd edition (Edwards and Penney)
- Background and Goals: For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations: 216-417 (or 419) and 217-316. The sequence 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Mathematics concentrators and other students who have some interest in the theory of mathematics should elect the sequence 217-316.
- Content: Math 216 is a basic course on differential equations, intended for engineers and other scientists who need to apply the techniques in their work. The lectures are accompanied by a computer lab and recitation section where students have the opportunity to discuss problems and work through computer experiments to further develop their understanding of the concepts of the class. Topics covered include some material on complex numbers and matrix algebra, first and second order linear and non-linear systems with applications, introductory numerical methods, and elementary Laplace transform techniques.
- Alternatives: Math 286 (Honors Differential Equations) 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 256 (Applied Honors Calculus IV) is also an alternative.
- Subsequent
Courses: Math 404 (Intermediate Diff. Eq.) covers further material on differential equations. Math 217 (Linear Algebra) and Math 417 (Matrix Algebra I) cover further material on linear algebra. Math 371 (Engin. 303 - Numerical Methods) and Math 471 (Intro. to Numerical Methods) cover additional material on numerical methods.
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Math 217 - Linear Algebra
- Prerequisites: Math 215, 255, or 285
- Frequency: Fall (I), Winter (II)
- Student Body: Sophomore prospective mathematics concentrators
- Credit: 4 Credits. Credit is granted for only one course among Math 214, 217, 417, and 419. No credit granted to those who have completed or are enrolled in Math 513.
- Recent Texts: Linear Algebra and Its Applications (Lay)
- Background and Goals: For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations: 216-417 (or 419) and 217-316. The sequence 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Mathematics concentrators and other students who have some interest in the theory of mathematics should elect the sequence 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.
- Content: 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.
- Alternatives: Math 214, 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.
- Subsequent Courses: The intended course to follow Math 217 is Math
316 (Differential Equations). Math 217 is also prerequisite for
Math 312 (Applied Modern Algebra), Math 412 (Introduction to Modern
Algebra) and all more advanced courses in mathematics.
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Math 255 - Applied Honors Calculus III
- Prerequisites: Math 156, or permission of instructor
- Frequency: Winter (II)
- Student
Body: First-year students
- Credit: 4 Credits. Credit is granted for only one course among Math 215,
255, and 285.
- Recent Texts: Multivariable Calculus 5th edition (Stewart)
- Background and Goals: Math 255 is part of the applied honors calculus sequence for engineering and science concentrators. Applications and concepts receive equal treatment. Theorems are stated precisely and are derived, but technical details are omitted. Examples are given to illustrate the theory. Critical thinking and class participation are encouraged. The goal is to provide students with a solid background needed for subsequent courses in mathematics, engineering, or science.
- Content: Analytic geometry of lines and planes using vector notation, parametric representation of curves and surfaces, multivariable calculus, line surface and volume integrals, vector fields, Green’s theorem, Stokes’ theorem, divergence theorem, applications (e.g. electromagnetic fields, fluid dynamics). MAPLE will be used throughout.
- Alternatives: Math 215 (Calculus III) or Math 285 (Honors Calculus III).
- Subsequent
Courses: Math 256 (Applied Honors Calculus IV)
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Math 256 - Applied Honors Calculus IV
- Prerequisites: Math 255
- Frequency: Fall (I)
- Student
Body: Sophomores
- Credit: 4 Credits. Credit is granted for only one course among Math 216,
256, 286, and 316.
- Recent Texts: Elementary Differential Equations and Boundary Value Problems (Boyce and DiPrima)
- Background and Goals: Math 256 is part of the applied honors calculus sequence for engineering science concentrators. Applications and concepts receive equal treatment. Theorems are stated precisely and are derived, but technical details are omitted. Examples are given to illustrate the theory. Critical thinking and class participation are encouraged. The goal is to provide students with a solid background needed for subsequent courses in mathematics, engineering, or science.
- Content: Linear algebra, matrices, systems of differential equations, initial value problems, qualitative theory of dynamical systems (e.g. equilibria, phase space, stability, bifurcations), nonlinear equations, numerical methods. Optional: boundary value problems. Students will learn to use MATLAB for computer simulations.
- Alternatives: Math 216 (Intro. to Differential Equations) or Math 286
(Honors Differential Equations)
- Subsequent
Courses: Many upper-level courses
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Math 285 - Honors Calculus III
- Prerequisites: Math 156, 176 or 186 or permission
- Frequency: Fall (I)
- Student
Body: Sophomores and first-year students with suitable
background
- Credit: 4 Credits. Credit is granted for only one course among Math 215,
255, and 285.
- Recent Texts: Multivariable Calculus 5th edition (Stewart)
- Background
and Goals: The sequence Math 185-186-285-286 is an introduction to calculus at the honors level. 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.
- Content: 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.
- Alternatives: Math 215 (Calculus III) is a less theoretical course
which covers the same material. Math 255 (Applied Honors Calc.
III) is an applications-oriented honors course which covers much
of the same material.
- Subsequent
Courses: Math 216 (Intro. to Differential Equations),
Math 286 (Honors Differential Equations) or Math 217 (Linear Algebra).
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Math 286 - Honors Differential Equations
- Prerequisites: Math 285 or permission
- Frequency: Winter (II)
- Student
Body: Sophomores
- Credit: 3 Credits. Credit is granted for only one course among Math 216,
256, 286, and 316.
- Recent Texts: Elementary Differential Equations and Boundary Value Problems (Boyce and DiPrima)
- Background
and Goals: The sequence Math 185-186-285-286 is an introduction to calculus at the honors level. 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.
- Content: Topics include first-order differential equations, higher-order linear differential equations with constant coefficients, an introduction to linear algebra, linear systems, the Laplace Transform, series solutions and other numerical methods (Euler, Runge-Kutta). If time permits, Picard’s Theorem will be proved.
- Alternatives: Math 216 (Intro. to Differential Equations) and Math
316 (Differential Equations) cover much of the same material.
Math 256 (Applied Honors Calculus IV) is also an alternative.
- Subsequent
Courses: Math 471 (Intro. to Numerical Methods) and/or
Math 572 (Numer. Meth. for Sci. Comput. II) are natural sequels
in the area of differential equations, but Math 286 is also preparation
for more theoretical courses such as Math 451 (Advanced Calculus
I).
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Math 288 - Math Modeling Workshop
- Prerequisites: Math 216, 256, 286, or 316, and Math 217, 417, or 419
- Frequency: Sporadically
- Student
Body: Junior and Senior mathematics concentrators
- Credit: 1 Credit.
- Recent Texts: none
- Background
and Goals: This course is designed to help students understand more clearly how techniques from other undergraduate mathematics courses can be used in concert to solve real-world problems. After the first two lectures the class will discuss methods of attacking problems. For credit a student will have to describe and solve an individual problem and write a report on the solution. Computing methods will be used.
- Content: During the weekly workshop students will be presented with real-world problems on which techniques of undergraduate mathematics offer insights. They will see examples of (1) how to approach and set up a given modeling problem systematically, (2) how to use mathematical techniques to begin a solution of the problem, (3) what to do about the loose ends that can’t be solved, and (4) how to present the solution to others. Students will have a chance to use the skills developed by participating in the U-M Undergraduate Math Modeling Meet.
- Alternatives: Math 462 (Mathematical Models) is a formal course in
mathematical modeling.
- Subsequent
Courses: This course may be repeated for credit.
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Math 289 - Problem Solving
- Prerequisites: permission
- Frequency: Fall (I), Winter (II)
- Student
Body: Junior and Senior mathematics concentrators
- Credit: 1 Credit.
- Recent Texts: none
- Background
and Goals: 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 competition.
- Content: 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.
- Alternatives: none
- Subsequent
Courses: This course may be repeated for credit.
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Math 295 - Honors Mathematics I
- Prerequisites: permission of honors advisor
- Frequency: Fall (I)
- Student
Body: First-year students
- Credit: 4 Credits.
- Recent Texts: Calculus (M. Spivak)
- Background
and Goals: Math 295-296-395-396 is the most theoretical
and demanding honors calculus sequence. The emphasis is on concepts,
problem solving, as well as the underlying theory and proofs of
important results. It provides an excellent background for advanced
courses in mathematics. The expected background is high school
trigonometry and algebra (previous calculus is not required, but
helpful). This sequence is not restricted to students enrolled
in the LS&A Honors program. Math 295 and 296 may be substituted
for any Math 451 requirement. Math 296 and 395 may be substituted
for any Math 513 requirement.
- Content: Real functions, limits, elementary topology of the real line, continuous functions, derivatives, indefinite and definite integrals.
- Alternatives: Math 156 (Applied Honors Calc II), Math 175 (Intro. to Cryptology) and Math 185 (Honors Calculus I) are
alternative honors courses.
- Subsequent
Courses: Math 296 (Honors Mathematics II)
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Math 296 - Honors Mathematics II
- Prerequisites: Math 295
- Frequency: Winter (II)
- Student
Body: First-year students
- Credit: 4 Credits.
- Recent Texts: Calculus (Spivak)
- Background
and Goals: Math 295-296-395-396 is the most theoretical
and demanding honors calculus sequence. The emphasis is on concepts,
problem solving, as well as the underlying theory and proofs of
important results. It provides an excellent background for advanced
courses in mathematics. The expected background is high school
trigonometry and algebra (previous calculus is not required, but
helpful). This sequence is not restricted to students enrolled
in the LS&A Honors program.
- Content: Infinite series, power series, vector spaces, structure of linear maps, duality, eigenvalues, normed vector spaces, higher-dimensional derivatives (Chain rule, inverse/implicit function theorems).
- Alternatives: none
- Subsequent
Courses: Math 395 (Honors Analysis I)
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Math 310 - Elementary Topics: Math Games and Theory of Games
- Prerequisites: Sophomore standing and one previous university math course
- Frequency: Winter (II)
- Student Body:
- Credit: 3 Credits.
- Recent Texts: none
- Background and Goals: The Elementary Topics course may focus on any one of several topics. The material is presented at a level appropriate for sophomores and juniors without extensive coursework in math. The current offering of the course focuses on game theory.
- Content: Students study the strategy of several games where mathematical ideas and concepts can play a role. Most of the course will be occupied with the structure of a variety of two person games of strategy: tic tac toe, tic tac toe misere, the French military game, hex, nim, the penny dime game, and many others. If there is sufficient interest students can study: dots and boxes, go moku, and some aspects of checkers and chess. There will also be a brief introduction to the classical Von Neuman/Morgenstern theory of mixed strategy games.
- Alternatives: none
- Subsequent Courses: none
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Math 312 - Applied Modern Algebra
- Prerequisites: Math 215, 255, or 285 and Math 217
- Frequency: Sporadically
- Student Body: Sophomore and Junior mathematics and computer science concentrators
- Credit: 3 Credits. 1 credit after Math 412
- Recent Texts: A Concrete Introduction to Higher Algebra (Childs)
- Background and Goals: One of the main goals of the course (along
with every course in the algebra sequence) is to expose students
to rigorous, proof-oriented mathematics. Students are required
to have taken Math 217, which should provide a first exposure
to this style of mathematics. A distinguishing feature of this
course is that the abstract concepts are not studied in isolation.
Instead, each topic is studied with the ultimate goal being a
real-world application.
- Content:
Sets and functions, relations and graphs, rings, Boolean algebras, semigroups, groups, and lattices. Applications from areas such as switching, automata, and coding theory, and may include finite and minimal state machines, algebraic decompositions of logic circuits, semigroup machines, binary codes, and series and parallel decomposition of machines.
- Alternatives:
Math 412 (Introduction to Modern Algebra) is a more abstract
and proof-oriented course with less emphasis on applications and
is better preparation for most pure mathematics courses. Math
567 is a more advanced course on coding theory.
- Subsequent
Courses: Math 312 is one of the alternative prerequisites
for Math 416 (Theory of Algorithms), and several advanced EECS
courses make substantial use of the material of Math 312. Another
good follow-up course is Math 475 (Elementary Number Theory).
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Math 316 - Differential Equations
- Prerequisites:
Math 215, 255, or 285; and Math 217
- Frequency:
Fall (I), Winter (II)
- Student
Body: A mix of undergraduate mathematics, science, and economics concentrators
- Credit:
3 Credits. Credit is granted for only one course among Math 216,
256, 286, and 316.
- Recent Texts: Elementary Differential Equations and Boundary Value Problems (Boyce and DiPrima)
- Background
and Goals: 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.
- Content:
First-order equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvector-eigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability. Applications to physical problems are considered throughout.
- Alternatives:
Math 216 covers somewhat less material without presupposing
linear algebra and with less emphasis on theory. Math 286 (Honors
Differential Equations) is the honors version of Math 316.
- Subsequent
Courses: Math 471 (Intro. to Numerical Methods) and/or
Math 572 (Numer. Meth. For Sci. Comput. III) are natural sequels
in the area of differential equations, but Math 316 is also preparation
for more theoretical courses such as Math 451 (Advanced Calculus
I).
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Math 327 - Evolution of Mathematical Concepts
- Prerequisites:
Math 116 or Math 186
- Frequency:
Sporadically
- Student
Body: Juniors and seniors interested in mathematics and
the history of science
- Credit:
3 Credits.
- Recent Texts: none
- Background
and Goals: This course examines the evolution of major
mathematical concepts from mathematical and historical points
of view. The course's goal is to throw light on contemporary mathematics
by retracing the history of some of the major mathematical discoveries.
- Content:
This course follows the evolution of three mathematical
ideas in geometry, analysis and algebra. Typical choices of subject
are: Euclid's parallel postulate and the development of non-Euclidean
geometries, the notions of limit and infinitesimals, and the development
of the theory of equations culminating with Galois theory.
- Alternatives:
none
- Subsequent
Courses: none
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Math 351 - Principles of Analysis
- Prerequisites: Math 215 and 217 or permission of instructor
- Frequency: Sporadically
- Student Body: Sophomores and juniors
- Credit: 3 Credits. No credit after 451
- Recent Texts: Understanding Analysis (Abbott)
- Area: Analysis
- Background and Goals: The course content is similar to that of Math 451, but Math 351 assumes less background. This course covers topics that might be of greater use to students considering a Mathematical Sciences concentration or a minor in Mathematics.
- Content: Analysis of the real line, rational and irrational numbers, infinity - large and small, limits, convergence, infinite sequences and series, continuous functions, power series, and differentiation.
- Alternatives: Math 451 (Advanced Calculus I) covers similar topics while assuming more background than 351.
- Subsequent Courses: none
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Math 354 - Fourier Analysis and its Applications
- Prerequisites:
Math 216, 256, 286, or 316
- Frequency:
Sporadically
- Student
Body: Junior and Senior math and non-math concentrators
- Credit:
3 Credits. No credit after 454
- Recent Texts: Fourier and Laplace Transforms (Beerends, et al.)
- Area: Analysis
- Background
and Goals: This course is an introduction to Fourier analysis with emphasis on applications. The course also can be viewed as a way of deepening one's understanding of the 100-and 200-level material by applying it in interesting ways.
- Content:
This is an introduction to Fourier analysis at an elementary level, emphasizing applications. The main topics are Fourier series, discrete Fourier transforms, and continuous Fourier transforms. A substantial portion of the time is spent on both scientific/technological applications (e.g. signal processing, Fourier optics), and applications in other branches of mathematics (e.g. partial differential equations, probability theory, number theory). Students will do some computer work, using MATLAB, an interactive programming tool that is easy to use, but no previous experience with computers is necessary.
- Alternatives:
Math 454 (Bound Val. Probs. for Part. Diff. Eq.) covers
some of the same material with more emphasis on partial differential
equations.
- Subsequent
Courses: This course is good preparation for Math 451
(Advanced Calculus I), which covers the theory of calculus in
a mathematically rigorous way.
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Math 362 - Applications of Calculus and Linear Algebra
- Prerequisites:
Math 215 and Math 216, 256, 286, or 316; or 217; or Math 396
- Frequency:
Sporadically
- Student
Body: Junior and Senior math and non-math concentrators
- Credit:
3 Credits. Credit after 462 only with advisor approval
- Recent Texts: Mathematics in Medicine and the Life Sciences (Hoppensteadt
and Peskin)
- Area: Applied/NA
- Background
and Goals: This is an introduction to applied mathematics,
making use of calculus and linear algebra. While the specific
content is very much instructor dependent, the course will always
focus on applications of mathematics in the natural or social
sciences. Any knowledge required to understand the applications
is taught in the course. The goal is to deepen the students' understanding
of calculus and linear algebra and motivate them to pursue mathematics
further.
- Content:
Instructor dependent. Examples of suitable topics are
mathematical biology or mathematical fluid dynamics. No background
in the specific application is needed, but often some high-school
level knowledge of physics will be helpful or even needed (when
in doubt, check with the instructor!) Students will often use
computers in this course, but absolutely no previous experience
with computers is assumed.
- Alternatives:
Math 462 (Mathematical Models) or Math 463 (Math Modeling in Biology) are similar courses at
a somewhat more advanced level.
- Subsequent
Courses: This course is not a specific prerequisite for
any subsequent course.
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Math 371 (Engin 303) - Numerical Methods
- Prerequisites:
Engin 101; and Math 216, 255, 286, or 316
- Frequency:
Fall (I), Winter (II)
- Student
Body: Sophomore, Junior, and Senior engineering students
- Credit:
3 Credits. No credit after Math 471.
- Recent Texts: A Friendly Introduction to Numerical Analysis (Bradie)
- Area: Applied/NA
- Background
and Goals: 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.
- Content:
Floating point arithmetic, Gaussian elimination, polynomial
interpolation, spline approximations, numerical integration and
differentiation, solutions to non-linear equations, ordinary
differential equations, polynomial approximations. Other topics
may include discrete Fourier transforms, two-point boundary-value
problems, and Monte-Carlo methods.
- Alternatives:
Alternatives: Math 471 (Numerical Analysis) provides
a more in-depth study of the same topics, with a greater emphasis
on analyzing the accuracy and stability of the numerical methods.
Math 571 (Numerical Linear Algebra) is a detailed study of the
solution of systems of linear equations and eigenvalue problems,
with some emphasis on large-scale problems. Math 572 (Numerical
Methods for Differential Equations) covers numerical methods for
both ordinary and partial differential equations. (Math 571 and
572 can be taken in either order).
- Subsequent
Courses: This course is basic for many later courses
in science and engineering. It is good background for 571-572.
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Math 385 - Math for Elementary School Teachers
- Prerequisites:
One year each of HS algebra and geometry
- Frequency:
Fall (I), Spring (IIIa) of even-numbered years
- Student
Body: Undergraduate concentrators in the Teaching Certificate
Program
- Credit:
3 Credits. No credit after 485
- Recent Texts: Elementary Mathematics for Teachers Complete Package (Parker and Baldridge)
- Area: Teaching
- Background
and Goals: This course, together with its sequel Math 489, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable.
- Content:
Topics covered include problem solving, sets and functions, numeration systems, whole numbers (including some number theory) and integers. Each number system is examined in terms of its algorithms, its applications, and its mathematical structure.
- Alternatives:
none
- Subsequent
Courses: Math 489 (Math for Elem. and Middle Sch. Teach.)
is the required sequel.
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Math 389 - Explorations in Mathematics
- Prerequisites: None formally; see instructor beforehand.
- Frequency: Winter (II)
- Student Body: Undergraduates
- Credit: 3 Credits.
- Recent Texts: none
- Background and Goals: The course is designed to show you how new mathematics is actually created: how to take a problem, make models and experiment with them, and search for underlying structure. The format involves little formal lecturing, and much more laboratory work and student presentation of partial results and approaches.
- Content: Problems for projects are drawn from a wide variety of mathematical areas, pure and applied. Problems are chosen to be accessible to undergraduates. Please see the course website for more information.
- Alternatives:
none
- Subsequent
Courses: After this course students should be ready for a variety of courses and research experiences.
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Math 395 - Honors Analysis I
- Prerequisites:
Math 296
- Frequency:
Fall (I)
- Student
Body: First-year students and Sophomores
- Credit:
4 Credits.
- Recent Texts: Analysis on Manifolds (Munkres)
- Background
and Goals: This course is a continuation of the sequence
Math 295-296 and has the same theoretical emphasis. Students are
expected to understand and construct proofs.
- Content:
Structure of bilinear forms, tensor products, metric spaces, function spaces, topology of vector spaces, higher-dimensional integrals, change of variables formula, partitions of unity.
- Alternatives:
none
- Subsequent
Courses: Math 396 (Honors Analysis II), Math 512 (Algebraic Structures), Math 525 (Probability Theory)
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Math 396 - Honors Analysis II
- Prerequisites:
Math 395
- Frequency:
Winter (II)
- Student
Body: Sophomores
- Credit:
4 Credits.
- Recent Texts: Analysis on Manifolds (Munkres)
- Background
and Goals: This course is a continuation of Math 395
and has the same theoretical emphasis. Students are expected to
understand and construct proofs.
- Content:
Geometry on manifolds, differential forms, vector fields, Stokes' theorem, deRham cohomology.
- Alternatives:
none
- Subsequent
Courses: Students who have successfully completed the
sequence Math 295-396 are generally prepared to take a range of
advanced undergraduate and graduate courses such as Math 512 (Algebraic Structures), Math 513 (Intro. to Linear Algebra), Math 525 (Probability Theory), Math 590 (Intro. to Topology), and many others.
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Math 404 - Intermediate Differential Equations
- Prerequisites:
Math 216, 256, 286, or 316
- Frequency:
Fall (I) sporadically
- Student
Body: undergraduate and graduate students from engineering
and LS&A
- Credit:
3 Credits.
- Recent Texts: Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering (Strogatz)
- Area: Applied/NA
- Background
and Goals: This course is an introduction to the modern qualitative theory of ordinary differential equations with emphasis on geometric techniques and visualization. Much of the motivation for this approach comes from applications. Examples of applications of differential equations to science and engineering are a significant part of the course. There are relatively few proofs.
- Content:
Geometric representation of solutions, autonomous systems, flows and evolution, linear systems and phase portraits, nonlinear systems, local and global behavior, linearization, stability, conservation laws, periodic orbits. Applications: free and forced oscillations, resonance, relaxation oscillations, competing species, Zeeman's models of heartbeat and nerve impulse, chaotic orbits, strange attractors.
- Alternatives:
Math 558 (Applied Nonlinear Dynamics) covers some of
the same material at a faster pace and includes additional topics.
- Subsequent
Courses: Math 454 (Boundary Value Problems for Partial
Differential Equations) is a natural sequel.
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Math 412 - Introduction to Modern Algebra
- Prerequisites:
Math 215, 255, or 285, and Math 217
- Frequency:
Fall (I), Winter (II)
- Student
Body: Mainly undergraduate mathematics concentrators with
some graduate students from other departments
- Credit:
3 Credits. 1 credit after Math 312
- Recent Texts: Abstract Algebra: an Introduction (T. Hungerford)
- Area: Algebra
- Background
and Goals: 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 and 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.
- Content:
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: 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.
- Alternatives:
Math 312 (Applied Modern Algebra) is a somewhat less
abstract course which replaces some of the material on rings and
fields of Math 412 with additional applications to areas such
as switching and coding theory.
- Subsequent
Courses: A student who successfully completes this course
will be prepared to take a number of other courses in abstract
mathematics : Math 416 (Theory of Algorithms), Math 451 (Advanced
Calculus I), Math 475 (Elementary Number Theory), Math 575 (Intro.
to Theory of Numbers), Math 513 (Introduction to Linear Algebra),
Math 481 (Intro. to Mathematical Logic), and Math 582 (Intro. to
Set Theory). All of these courses will extend and deepen the student's
grasp of modern abstract mathematics.
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Math 416 - Theory of Algorithms
- Prerequisites:
Math 312 or 412 or EECS 203, and EECS 281 or permission of instructor
- Frequency:
Sporadically
- Student
Body: Largely computer science concentrators with a few graduate students from other fields
- Credit:
3 Credits.
- Recent Texts: Algorithm Design (Kleinberg and Tardos)
- Background
and Goals: Many common problems from mathematics and computer science may be solved by applying one or more algorithms- well-defined procedures that accept input data specifying a particular instance of the problem and produce a solution. Students entering Math 416 typically have encountered some of these problems and their algorithmic solutions in a programming course. The goal here is to develop the mathematical tools necessary to analyze such algorithms with respect to their efficiency (running time) and correctness. Different instructors will put varying degrees of emphasis on mathematical proofs and computer implementation of these ideas.
- Content:
Typical problems considered are: sorting, searching, matrix multiplication, graph problems (flows, travelling salesman), and primality and pseudo-primality testing (in connection with coding questions). Algorithm types such as divide-and-conquer, backtracking, greedy, and dynamic programming are analyzed using mathematical tools such as generating functions, recurrence relations, induction and recursion, graphs and trees, and permutations. The course often includes a section on abstract complexity theory including NP completeness.
- Alternatives:
This course has substantial overlap with EECS 586 (Design and Analysis of Algorithms)- more or less depending on the instructors. In general, Math 416 will put more emphasis on the analysis aspect in contrast to design of algorithms.
- Subsequent
Courses: Math 516 (Topics in Theory of Algorithms) (given
infrequently) and EECS 574 (Computational Complexity) and 575 (Advanced Cryptography) include some topics which follow those of this course.
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Math 417 - Matrix Algebra I
- Prerequisites:
Three mathematics courses beyond Math 110
- Frequency:
Fall (I), Winter (II), Spring (IIIa), Summer (IIIb)
- Student
Body: largely engineering and science students, both undergraduate
and graduate
- Credit:
3 Credits. Credit is granted for only one course among Math 214,
217, 417, and 419. No credit granted to those who have completed
or are enrolled in Math 513.
- Recent Texts: Linear Algebra with Applications (Bretscher)
- Area: Algebra
- Background
and Goals: Many problems in science, engineering, and mathematics are best formulated in terms of matrices - rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (honors).
- Content:
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.
- Alternatives:
Math 419 (Lin. Spaces and Matrix Thy.) is an enriched
version of Math 417 with a somewhat more theoretical emphasis.
Math 217 (Linear Algebra) (despite its lower number) is also a
more theoretical course which covers much of the material of 417
at a deeper level. Math 513 (Introduction to Linear Algebra) is
an honors version of this course, which is also taken by some
mathematics graduate students. Mathematics concentrators are required
to take Math 217 or Math 513.
- Subsequent
Courses: Math 420 (Matrix Algebra II) is the natural
sequel but this course serves as prerequisite to several courses:
Math 452 (Advanced Calculus II), Math 462 (Mathematical Models),
Math 561 (Linear Programming I), and Math 571 (Numer. Meth. for
Sci. Comput. I).
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Math 419 - Linear Spaces and Matrix Theory
- Prerequisites:
4 mathematics courses beyond Math 110
- Frequency:
Fall (I), Winter (II)
- Student
Body: largely engineering graduate students and undergraduates; some mathematics undergraduates
- Credit:
3 Credits. Credit is granted for only one course among Math 214,
217, 417, and 419. No credit granted to those who have completed
or are enrolled in Math 513.
- Recent Texts: Linear Algebra with Applications (Bretscher)
- Area: Algebra
- Background
and Goals: Math 419 covers much of the same ground as
Math 417 (Matrix Algebra I) but presents the material in a somewhat
more abstract way in terms of vector spaces and linear transformations
instead of matrices. There is a mix of proofs, calculations, and
applications with the emphasis depending somewhat on the instructor.
A previous proof-oriented course is helpful but by no means necessary.
- Content:
Basic notions of vector spaces and linear transformations:
spanning, linear independence, bases, dimension, matrix representation
of linear transformations; determinants; eigenvalues, eigenvectors,
Jordan canonical form, inner-product spaces; unitary, self-adjoint,
and orthogonal operators and matrices, applications to differential
and difference equations.
- Alternatives:
Math 417 (Matrix Algebra I) is less rigorous and theoretical
and more oriented to applications. Math 217 (Linear Algebra) is
similar to Math 419 but slightly more proof-oriented. Math 513
(Introduction to Linear Algebra) is much more abstract and sophisticated. Mathematics concentrators are required to
take Math 217 or Math 513.
- Subsequent
Courses: Math 420 (Matrix Algebra II) is the natural
sequel but this course serves as prerequisite to several courses:
Math 452 (Advanced Calculus II), Math 462 (Mathematical Models),
Math 561 (Linear Programming I) , and Math 571 (Numer. Meth. for
Sci. Comput. I).
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Math 420 - Matrix Algebra II
- Prerequisites:
Math 214, 217, 417, or 419
- Frequency:
Varies
- Student
Body: largely engineering and economics graduate students and undergraduates; some mathematics undergraduates
- Credit:
3 Credits.
- Recent Texts: Linear Algebra and its Applications 3rd edition (G. Strang)
- Area: Algebra
- Background
and Goals: This course is designed to improve the student's understanding of the fundamental results of linear algebra and to provide an introduction to some if its most important concepts.
- Content:
Gauss elimination and LDU-factorization. Vector spaces and dimension. Orthogonality, including least-squares approximations, Gram-Schmidt orthogonalization and QR-factorization. Review of determinants. Eigenvalue decompositions and Jordan canonical forms-applications to difference and differential equations. Positive-definite and semi-definite matrices. Singular value decompositions, pseudoinverses, and optimal (minimal norm) solutions.
- Alternatives:
none
- Subsequent
Courses: Math 571 (Numerical Methods for Scientific Computing
I).
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Math 422 (BE 440) - Risk Management and Insurance
- Prerequisites:
Math 115, junior standing and permission of instructor
- Frequency:
Winter (II)
- Student
Body: Junior and senior mathematics concentrators; some business
undergraduates
- Credit:
3 Credits.
- Recent Texts: Introduction to Risk Management and Insurance (Dorfman)
- Area: Actuarial & Financial
- Background
and Goals: This course is designed to allow students to explore the insurance mechanism as a means of replacing uncertainty with certainty. A main goal of the course is to explain, using mathematical models from the theory of interest, risk theory, credibility theory and ruin theory, how mathematics underlies many important individual and societal problems.
- Content:
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 consumer'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. This course may be used to satisfy the LS&A upper-level writing requirement.
- Alternatives:
none
- Subsequent
Courses: none
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Math 423 - Mathematics of Finance
- Prerequisites:
Math 217, Math 425 and EECS 183 or equivalents
- Frequency:
Fall (I), Winter (II)
- Student
Body: Junior and senior mathematics concentrators; some business undergraduates
- Credit:
3 Credits.
- Recent Texts: Mathematics for Finance: An Introduction to Financial Engineering (Capinksi and Zastawniak)
- Area: Actuarial & Financial
- 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.
- Content:
Topics include risk and return theory, portfolio theory, capital asset pricing model, random walk model, stochastic processes, Black-Scholes Analysis, numerical methods and interest rate models.
- Alternatives:
none
- Subsequent
Courses: Math 523 (Risk Theory) and Math 623 (Computational
Finance).
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Math 424 - Compound Interest and Life Insurance
- Prerequisites:
Math 215, 255, or 285
- Frequency:
Fall (I), Winter (II)
- Student
Body: undergraduate and graduate students in the Actuarial
Mathematics Program, or students interested in exploring the concepts
underlying the theory of interest.
- Credit:
3 Credits.
- Recent Texts: Mathematical Interest Theory (Daniel and Vaaler)
- Area: Actuarial & Financial
- Background
and Goals: 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 some of the professional actuarial exams.
- Content:
The course covers compound interest (growth) theory and its application to valuation of monetary deposits, annuities, and bonds. Problems are approached both analytically (using algebra) and geometrically (using pictorial representations). Techniques are applied to real-life situations: bank accounts, bond prices, etc. The text is used as a guide because it is prescribed for the professional examinations; the material covered will depend somewhat on the instructor.
- Alternatives:
Math 424 is required for students concentrating in actuarial
mathematics; others may take Math 147 (Introduction to Interest
Theory), which deals with the same techniques but with less emphasis
on continuous growth situations.
- Subsequent
Courses: Math 520 (Life Contingencies I) applies the
concepts of Math 424 together with probability theory to the valuation
of life contingencies (death benefits and pensions).
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Math 425 (Stat 425) - Introduction to Probability
- Prerequisites:
Math 215, 255, or 285
- Frequency:
Fall (I), Winter (II), Spring (IIIa), Summer (IIIb)
- Student
Body: about 80% undergraduate mathematics, engineering, and computer science concentrators with a few graduate students
- Credit:
3 Credits.
- Recent Texts: A First Course in Probability 7th edition (S. Ross)
- Area: Analysis
- Background
and Goals: 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.
- Content:
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.
- Alternatives:
Math 525 (Probability Theory) is a similar course for
students with stronger mathematical background and ability.
- Subsequent
Courses: Stat 426 (Intro. to Theoretical Stat.) is a natural
sequel for students. Math 423 (Mathematics of Finance) and Math
523 (Risk Theory) include many applications of probability theory.
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Math 427 - Retirement Plans and Other Employee Benefits
- Prerequisites:
Junior standing or permission of instructor
- Frequency:
Winter (II)
- Student
Body: Mainly Actuarial Mathematics students, but also some non-mathematics students
- Credit:
3 Credits.
- Recent Texts: Pension Planning: Pensions, Profit-Sharing, and Other Deferred Compensation Plans (Allen et al.)
- Area: Actuarial & Financial
- Background
and Goals: 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. This course is certified for
satisfaction of the upper-level writing requirement.
- Content:
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.
- Alternatives:
none
- Subsequent
Courses: Math 521 (Life Contingencies II) and/or Math
522 (Act. Theory of Pensions and Soc. Sec.) (which can be taken
independently of each other) provide more in-depth examination
of the actuarial techniques used in employee benefit plans.
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Math 431 - Topics in Geometry for Teachers
- Prerequisites:
Math 215
- Frequency:
Fall (I)
- Student
Body: mainly teaching certificate candidates; a few other
mathematics and engineering students
- Credit:
3 Credits.
- Recent Texts: Euclidean and Non-Euclidean Geometry (Greenberg)
- Area: Teaching
- Background
and Goals: This course is a study of the axiomatic foundations
of Euclidean and non-Euclidean geometry. Concepts and proofs are
emphasized; students must be able to follow as well as construct
clear logical arguments. For most students this is an introduction
to proofs. A subsidiary goal is the development of enrichment
and problem materials suitable for secondary geometry classes.
- Content:
Topics selected depend heavily on the instructor but
may include classification of isometries of the Euclidean plane;
similarities; rosette, frieze, and wallpaper symmetry groups;
tesselations; triangle groups; finite, hyperbolic, and taxicab
non-Euclidean geometries.
- Alternatives:
Alternative geometry courses at this level are Math 432
(Projective Geometry) and Math 433 (Intro to Differential Geometry).
- Subsequent
Courses: Although it is not strictly a prerequisite,
Math 431 is good preparation for Math 531 (Transformation Groups
in Geom.).
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Math 432 - Projective Geometry
- Prerequisites:
Math 215, 255, or 285
- Frequency:
sporadically
- Student
Body:
- Credit:
3 Credits.
- Recent Texts: Projective Geometry 2nd edition (H.M.S. Coxeter)
- Area: Geometry/Topology
- Background
and Goals: The primary goal of the course is to introduce
students to non-standard concepts of geometry. A secondary goal
is to provide students with experience with proofs and to make
them more comfortable with abstract notions. Concepts and proofs
are emphasized; there is almost no calculational component. Of
course additional background is helpful, but the course is entirely
self-contained.
- Content:
Content will vary with the instructor; one recent version
included the following topics. Combinatorial Geometries: incidence
structures, geometric lattices, affine and linear subspaces, graphic
matroids; Projective Spaces: axioms, duality, harmonic relation,
Pappas' and Desargues' Theorems, projectivities, conics; Projective
Planes: finite fields, constructing and coordinatizing projective
planes. Another version provided examples of the application of
linear algebra to geometric problems.
- Alternatives:
The experience with proofs in Math 432 can be useful
in other proof-oriented courses such as Math 412 (Introduction
to Modern Algebra) and Math 451 (Advanced Calculus I).
- Subsequent
Courses: none
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Math 433 - Introduction to Differential Geometry
- Prerequisites:
Math 215, 255, or 285; and Math 217
- Frequency:
Fall (I)
- Student
Body: Half undergraduate mathematics concentrators, half graduate students from EECS and physics
- Credit:
3 Credits.
- Recent Texts: Elements of Differential Geometry (Millman and Parker)
- Area: Geometry/Topology
- Background
and Goals: This course is about the analysis of curves
and surfaces in 2- and 3-space using the tools of calculus and
linear algebra. There will be many examples discussed, including
some which arise in engineering and physics applications. Emphasis
will be placed on developing intuitions and learning to use calculations
to verify and prove theorems. Students need a good background
in multivariable calculus (215) and linear algebra (preferably
217). Some exposure to differential equations (216 or 316) is
helpful but not absolutely necessary.
- Content:
Curves and surfaces in three-space using calculus. Curvature
and torsion of curves. Curvature, covariant differentiation, parallelism,
isometry, geodesics, and area on surfaces. Gauss-Bonnet Theorem.
Minimal surfaces.
- Alternatives:
Math 537 is a substantially more advanced course which
requires a strong background in topology (590), linear algebra
(513) and advanced multivariable calculus (452). It treats some
of the same material from a more abstract and topological perspective
and introduces more general notions of curvature and covariant
derivative for spaces of any dimension.
- Subsequent
Courses: Math 635 (Differential Geometry) and Math 636
(Topics in Differential Geometry) further study Riemannian manifolds
and their topological and analytic properties. Physics courses
in general relativity and gauge theory will use some of the material
of this course.
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Math 450 - Advanced Mathematics for Engineers I
- Prerequisites:
Math 215, 255, or 285, and Math 216, 256, 286, or 316
- Frequency:
Fall (I), Winter (II), Summer (IIIb)
- Student
Body: Undergraduate mathematics and engineering students; engineering graduate students
- Credit:
4 Credits. No credit after Math 454.
- Recent Texts: Advanced Engineering Mathematics (Greenberg)
- Area: Analysis
- Background
and Goals: This course is an introduction to some of the main mathematical techniques in engineering and physics. It is intended to provide some background for courses in those disciplines with a mathematical requirement that goes beyond calculus. Model problems in mathematical physics are studied in detail. Applications are emphasized throughout.
- Content:
Topics covered include: Fourier series and integrals; the classical partial differential equations (the heat, wave and Laplace's equations) solved by separation of variables; an introduction to complex variables and conformal mapping with applications to potential theory. A review of series and series solutions of ODEs will be included as needed. A variety of basic diffusion, oscillation and fluid flow problems will be discussed.
- Alternatives:
This course overlaps with 454 and, to a much lesser extent,
with 555. The coverage of PDEs in 450 is not as in-depth as 454;
for example, in 450 coverage of special functions is reduced to
the simplest Bessel functions. Those students needing a more thorough
discussion of PDEs and boundary-value problems should take 454.
On the other hand, 450 should provide a broader introduction to
applied methods.
- Subsequent
Courses: Math 555 (Complex Variables) and Math 556 (Methods
of Applied Math I) are graduate-level courses that further develops
both the theory and applications covered in 450.
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Math 451 - Advanced Calculus I
- Prerequisites: Math 285, or Math 215 and one subsequent course
- Frequency:
Fall (I), Winter (II), Spring (IIIa)
- Student
Body: Half undergraduate mathematics concentrators, half science and engineering graduate students
- Credit:
3 Credits. No credit after 351.
- Recent Texts: Elementary Analysis: The Theory of Calculus (K. Ross)
- Area: Analysis
- Background
and Goals: 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.
- Content:
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.
- Alternatives:
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. Math 351 covers similar topics with much less rigor.
- Subsequent
Courses: The natural sequel to Math 451 is 452, which
extends the ideas considered here to functions of several
variables. In a sense, Math 451 treats the theory behind Math
115-116, while Math 452 does the same for Math 215 and a part
of Math 216. Math 551 (Advanced Multivariable Calculus) is a more
advanced version of Math 452. Math 451 is also a prerequisite
for several other courses: Math 575 (Intro. to Theory of Numbers),
Math 590 (Intro. to Topology), Math 596 (Analysis I) (Complex)),
and Math 597 (Analysis II (Real)).
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Math 452 - Advanced Calculus II
- Prerequisites:
Math 217, 417, or 419 (may be concurrent) and Math 451
- Frequency:
Winter (II)
- Student
Body: A majority of mathematics undergraduates with some non-mathematics graduate students
- Credit:
3 Credits.
- Recent Texts: Analysis on Manifolds (Munkres), Advanced Calculus of Several Variables (Edwards)
- Area: Analysis
- Background
and Goals: 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.
- Content:
Topics include (1) partial derivatives and differentiability,
(2) gradients, directional derivatives, and the chain rule, (3)
implicit function theorem, (4) surfaces, tangent plane, (5) max-min
theory, (6) multiple integration, change of variable, etc. (7)
Green's and Stokes' theorems, differential forms, exterior derivatives.
- Alternatives:
none
- Subsequent
Courses: Math 452 is prerequisite to Math 572 and is
good general background for any of the more advanced courses in
analysis (Math 596, 597) or differential geometry or topology
(Math 537, 635).
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Math 454 - Boundary Value Problems for Partial Differential Equations
- Prerequisites:
Math 216, 256, 286, or 316
- Frequency:
Fall (I), Winter (II), Spring (IIIa)
- Student
Body: Some mathematics undergraduates, but more non-mathematics graduate students
- Credit:
3 Credits. 1 credit after Math 354. No credit after Math 450.
- Recent Texts: Applied Partial Differential Equations (Haberman)
- Area: Applied/NA
- Background
and Goals: This course is devoted to the use of Fourier
series and other orthogonal expansions in the solution of initial-value
and boundary-value problems for second-order linear partial differential
equations. Emphasis is on concepts and calculation. The official
prerequisite is ample preparation.
- Content:
Classical representation and convergence theorems for
Fourier series; method of separation of variables for the solution
of the one-dimensional heat and wave equation; the heat and wave
equations in higher dimensions; eigenfunction expansions; spherical
and cylindrical Bessel functions; Legendre polynomials; methods
for evaluating asymptotic integrals (Laplace's method, steepest
descent); Laplace's equation and harmonic functions, including
the maximum principle. As time permits, additional topics will
be selected from: Fourier and Laplace transforms; applications
to linear input-output systems, analysis of data smoothing and
filtering, signal processing, time-series analysis, and spectral
analysis; dispersive wave equations; the method of stationary
phase; the method of characteristics.
- Alternatives:
Both Math 455 (Bound Val. Probs. and Complex Anal.) and
Math 554 (Advanced Math for Engineers) cover many of the same
topics but are very seldom offered.
- Subsequent
Courses: Math 454 is prerequisite to Math 571 (Numer.
Meth. for Sci. Comput. I) and Math 572 (Numer. Meth. for Sci.
Comput. II). Although it is not a formal prerequisite, it is good
background for Math 556 (Methods of Applied Math I).
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Math 462 - Mathematical Models
- Prerequisites:
Math 216, 256, 286, or 316, and Math 217, 417, or 419
- Frequency:
Winter (II) sporadically
- Student
Body: Mainly Junior and Senior mathematics concentrators; students from engineering, biology, physics, and medicine
- Credit:
3 Credits. 1-3 credits after 362 depending on overlap
- Recent Texts: Mathematical Models (R. Haberman)
- Area: Applied/NA
- Background
and Goals: The focus of this course is the application
of a variety of mathematical techniques to solve real-world problems.
Students will learn how to model a problem in mathematical terms
and use mathematics to gain insight and eventually solve the problem.
Concepts and calculations, using applied analysis and numerical
simulations, are emphasized.
- Content:
Construction and analysis of mathematical models in physics,
engineering, economics, biology, medicine, and social sciences.
Content varies considerably with instructor. Recent versions:
Use and theory of dynamical systems (chaotic dynamics, ecological
and biological models, classical mechanics), and mathematical
models in physiology and population biology.
- Alternatives:
Students who are particularly interested in biology should
considered Math 463 (Math Modeling in Biology).
- Subsequent
Courses: any higher-level course in differential equations
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Math 463 - Math Modeling in Biology
- Prerequisites:
Math 217, 417 or 419; and 216, 286 or 316
- Frequency:
Fall (I)
- Student
Body: Juniors, seniors, and Master's level; half Engineering and half LS&A
- Credit:
3 Credits.
- Recent Texts: Mathematical Models in Biology (Keshet-Edelstein); Mathematical Biology (Murray)
- Area: Applied/NA
- Background
and Goals:It is widely anticipated that Biology and Medicine will be the premier sciences of the 21st century. The complexities of the biological sciences make interdisciplinary involvement essential and the increasing use of mathematics in biology is inevitable as biology becomes more quantitative. Mathematical biology is a fast growing and exciting modern application of mathematics that has gained worldwide recognition. In this course, mathematical models that suggest possible mechanisms that may underlie specific biological processes are developed and analyzed. Another major emphasis of the course is illustrating how these models can be used to predict what may follow under currently untested conditions. The course moves from classical to contemporary models at the population, organ, cellular, and molecular levels. The goals of this course are: (1) Critical understanding of the use of differential equation methods in mathematical biology and (2) Exposure to specialized mathematical and computations techniques which are required to study ordinary differential equations that arise in mathematical biology. By the end of this course students will be able to derive, interpret, solve, understand, discuss, and critique discrete and differential equation models of biological systems.
- Content:
This course provides an introduction to the use of continuous and discrete differential equations in the biological sciences. Biological topics may include single species and interacting population dynamics, modeling infectious and dynamic diseases, regulation of cell function, molecular interactions and receptor-ligand binding, biological oscillators, and an introduction to biological pattern formation. There will also be discussions of current topics of interest such as Tumor Growth and Angiogenesis, HIV and AIDS, and Control of the Mitotic Clock. Mathematical tools such as phase portraits, bifurcation diagrams, perturbation theory, and parameter estimation techniques that are necessary to analyze and interpret biological models will also be covered. Approximately one class period each week will be held in the mathematics computer laboratory where numerical techniques for finding and visualizing solutions of differential and discrete systems will be discussed.
- Alternatives:
none
- Subsequent
Courses: Math 563
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Math 464 - Inverse Problems
- Prerequisites:
Math 217, 417, or 419, and Math 216, 256, 286, or 316
- Frequency:
Sporadic
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