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Topics in Applied Mathematics

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Applied Math 252b :Topics in Applied Mathematics1 Some basic informationTime: Mon., Wed. and Fri. 10:30-11:20 am.I may need to cancel a class in February. We will try to find atime to make it up.Place: DL 102. Dunham Lab is located at 10 Hillhouse Ave.Instructor: Mokshay Madiman.Room 206, Statistics Dept., 24 Hillhouse Avenue.Phone: 2-7602.Email: [email protected] text: Introduction to Applied Mathematics by Gilb e rtStrang. This is an unusual textbook, but it do e s a good job ofcohesively presenting a diverse amount of material.Prerequisite: After linear algebra (MATH 222a or b or 225bor equivalent) and differential equations (AMTH 251a or ENAS194a or b or MATH 246a or b). Knowing a little about probabil-ity will help, but we will cover what we need for the class. If youare not sure you are ready to take this class, come talk to me.Course Webpage: I will be using this to post homeworks andannouncements. Look for Amth252b on the Classes.v2 server athttp://classesv2.yale.eduOffice house: 5:30-7:00 pm on Tuesdays, and 1:30-3:00 pm onFridays. I will b e mostly in my office during those times. Youcan also email me to set up a meeting if you cannot make thesetimes.Grading: Homework (about weekly), a mid-term exam, and ei-ther a final exam or a “project” (to be decided later based onclass interest).12 About the CourseWelcome to Applied Math 252b! This is a basic course for those who wish to learn aboutthe applications of mathematics in the real world. The main prerequisite is linear algebraand calculus, and some familiarity with differential equations. While the course is primarilyfor undergraduates, graduate students are also welcome. My hope is that the course will beinteresting and useful to students in a variety of fields and with a variety of backgrounds:from physics to finance, economics to engineering, and math to biology.The official course description from the Yale Bulletin reads: “Topics in applied mathemat-ics including partial differential equations, optimization, variational calculus, and control.”Since this is singularly unilluminating, here is a more detailed description.The first major theme of the course is optimization, i.e., situations in which our goal isto maximize or minimize some quantity subject to some laws that govern that quantity. Forexample, we may wish to maximize profit by finding the most efficient way for a firm totransport goods; or we may wish to find the configuration of gas molecules in a room thathas minimum energy (the “equilibrium” configuration); or we may wish to find the modelthat best fits certain observed data from any field of application (the problem of regression instatistics). All of these problems are instances of “minimum principles”, and in most cases,they reduce to solving linear equations of various kinds. We will spend a good amount oftime in the course studying the common ideas underlying optimization problems from manyfields, and in particular, understanding why they give rise to linear equations.The second major theme of the course is linear equations of various kinds:• linear algebraic equations (hopefully you are already familiar with how to solve these)• linear differential e quations (hopefully you have seen these before)• linear partial differential e quations or PDE’sWhile the motivation for s tudying many of these comes from optimization problems, thetools used to analyze and solve them are completely different. The key idea is that ofdiagonalization or eigenvalues, and we will use this idea to understand the following tools:• The QΛQTdecomposition of a symmetric matrix that leads to the solution of lineardifferential equations.• The Fourier transform or decomposition of a function into oscillations that leads to thesolution of many linear PDE’s.As we study these analytical tools, we will also look at practical ways in which solutions canbe found using a computer. In particular, there will be a few computational problems in yourhomeworks in addition to the “theory” problems.A third important theme is the effect of nonlinearity, although we will only scratch thesurface of this difficult subject. Both for general nonlinear optimization and for nonlineardifferential equations, practical numerical methods are often as important as the theory, andthis will be reflected in the way we study them.In addition to these major themes, we may also explore several fun “extra” topics. Pos-sibilities include:2• Nonlinear dynamics and chaos.• An analysis of different voting methods in elections, which turns out to be closely relatedto many other phenomena.• Dynamics of random systems, i.e., how do we study systems whose “differential equa-tions” have some randomness in them?Due to time limitations, we will have to leave out at least some of these topics, but I willmake every effort to cover those topics that are of special interest to students in the


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