Credit Exclusions: Credit is granted for only one course from among MATH 112, 115, and 185. No credit granted to those who have completed MATH 175.

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 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.

Content: The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing and questioning skills. Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. The classroom atmosphere is interactive and cooperative and homework is done in groups.

Alternatives: MATH 185 (Honors Anal. Geom. and Calc. I ) is a somewhat more theoretical course which covers some of the same material. Math 175 (Combinatorics and Calculus) 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 Anal. Geom. and Calc. II).

Advisory Prerequisite: Four years of high school mathematics.

Credit Exclusions: Credit is granted for only one course from among MATH 112, 115, and 185. No credit granted to those who have completed MATH 175.

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 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.

Content: The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing and questioning skills. Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. The classroom atmosphere is interactive and cooperative and homework is done in groups.

Alternatives: MATH 185 (Honors Anal. Geom. and Calc. I ) is a somewhat more theoretical course which covers some of the same material. Math 175 (Combinatorics and Calculus) 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 Anal. Geom. and Calc. II).

Advisory Prerequisite: Four years of high school mathematics.

Credit Exclusions: Credit is granted for only one course among MATH 116, 119, 156, 176, and 186

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

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

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

Advisory Prerequisite: MATH 115.

Credit Exclusions: Credit can be earned for only one of MATH 215, 255, or 285.

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 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.

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 major 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).

Advisory Prerequisite: MATH 116

Credit Exclusions: Credit can be earned for only one of MATH 215, 255, or 285.

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 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.

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 major 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).

Advisory Prerequisite: MATH 116

Credit Exclusions: Credit can be earned for only one of MATH 216, 256, 286, or 316.

Background and Goals: For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, MATH 216-417 (or MATH 419) and MATH 217-316. The sequence MATH 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Math majors and other students who have some interest in the theory of mathematics should elect the sequence MATH 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 (Linear Algebra)-MATH 316 (Differential Equations) 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 ((ENGR 303) Numerical Methods) and MATH 471 (Intro. To Numerical Methods) cover additional material on numerical methods.

Advisory Prerequisite: MATH 116, 119, 156, 176, 186, or 296.

An experiential mathematics course for students enrolled in the Secondary Teaching Certificate Program with a concentration in mathematics. Students would tutor pre-calculus (MATH 105) or calculus (MATH 115) in the Math. Lab. They would also participate in a weekly seminar to discuss mathematical and methodological questions.

Advisory Prerequisite: Enrollment in the secondary teaching certificate program with concentration in Mathematics and permission of instructor.

Designed especially for Honors students.

Advisory Prerequisite: Permission of instructor.

Credit Exclusions: 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.

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

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

Advisory Prerequisite: MATH,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 in MATH 513.

This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. The goal is to understand how the models reflect observed market features, and to provide the necessary mathematical tools for their analysis and implementation. The course will introduce the stochastic processes used for modeling particular financial instruments. However, the students are expected to have a solid background in basic probability theory.

Specific Topics

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

Advisory Prerequisite: MATH 217 and 425; EECS 183 or equivalent.

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

Advisory Prerequisite: MATH 215, 255, or 285 or permission of instructor.

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: STATS 426 (Intro. To Math. Stat.) is a natural sequel for students. MATH 423 (Mathematics of Finance) and MATH 523 (Risk Theory) include many applications of probability theory.

Advisory Prerequisite: MATH 215

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: STATS 426 (Intro. To Math. Stat.) is a natural sequel for students. MATH 423 (Mathematics of Finance) and MATH 523 (Risk Theory) include many applications of probability theory.

Advisory Prerequisite: MATH 215

Credit Exclusions: No credit granted to those who have completed or are enrolled in MATH 351.

This course has two complementary goals: (1) a rigorous development of the fundamental ideas of calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc.) and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as MATH 412) be taken before MATH 451. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits, and continuity; differentiation; integration, and the Fundamental Theorem of Calculus; infinite series; and sequences and series of functions.

There is really no other course which covers the material of MATH 451. Although MATH 450 is an alternative prerequisite for some later courses, the emphasis of the two courses is quite distinct. The natural sequel to MATH 451 is 452, which extends the ideas considered here to functions of several variables. In a sense, MATH 451 treats the theory behind MATH 115-116, while MATH 452 does the same for MATH 215 and a part of MATH 216. MATH 551 is a more advanced version of Math 452. MATH 451 is also a prerequisite for several other courses: MATH 575, 590, 596, and 597.

Advisory Prerequisite: Previous exposure to abstract mathematics, e.g. MATH 217 and 412

Credit Exclusions: Students with credit for MATH 354 can elect MATH 454 for one credit. No credit granted to those who have completed or are enrolled in MATH 450.

This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample preparation. Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; and applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis. Both MATH 455 and 554 cover many of the same topics but are very seldom offered. MATH 454 is prerequisite to MATH 571 and 572, although it is not a formal prerequisite, it is good background for MATH 556.

Advisory Prerequisite: 216,316/286

This course is intended for graduate students in fields other than mathematics who require mathematical skills not otherwise available though existing courses.

Advisory Prerequisite: Graduate standing in a field other than Mathematics and permission of instructor.

Background and Goals: A fundamental problem is the allocation of constrained resources such as funds among investment possibilities or personnel among production facilities. Each such problem has as it's goal the maximization of some positive objective such as investment return or the minimization of some negative objective such as cost or risk. Such problems are called Optimization Problems. Linear Programming deals with optimization problems in which both the objective and constraint functions are linear (the word "programming" is historical and means "planning" rather that necessarily computer programming). In practice, such problems involve thousands of decision variables and constraints, so a primary focus is the development and implementation of efficient algorithms. However, the subject also has deep connections with higher-dimensional convex geometry. A recent survey showed that most Fortune 500 companies regularly use linear programming in their decision making. This course will present both the classical and modern approaches to the subject and discuss numerous applications of current interest.

Content: Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis; algorithmic complexity; the ellipsoid method; scaling algorithms; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and use various computer programs.

Alternatives: Cross-listed as IOE 510.

Subsequent Courses: IOE 610 (Linear Programming II) and IOE 611 (Nonlinear Programming)

Advisory Prerequisite: MATH 217, 417, or 419

Designed for individual students who have an interest in a specific topic (usually that has stemmed from a previous course). An individual instructor must agree to direct such a reading, and the requirements are specified when approval is granted.

Advisory Prerequisite: Graduate standing and permission of instructor.

Election for dissertation work by doctoral student not yet admitted as a Candidate.

Advisory Prerequisite: Election for dissertation work by doctoral student not yet admitted as a Candidate. Graduate standing.

Graduate School authorization for admission as a doctoral Candidate. N.B. The defense of the dissertation (the final oral examination) must be held under a full term Candidacy enrollment period.

Enforced Prerequisites: Graduate School authorization for admission as a doctoral Candidate