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**Applied Mathematics
Guide to Undergraduate Programs**

**April 2006**

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Division of Applied Mathematics

Brown University

http://www.dam.brown.edu/

Brown Universitys Division of Applied Mathematics*

Information Resources Describing Professional Opportunities *

Courses in Applied Mathematics *

Numerical Analysis and Scientific Computation *

Statistics: Theory and Applications *

Courses in Other Departments *

Cognitive and Linguistic Sciences *

Program for the Sc.B. degree *

Applied Mathematics - Economics *

Program for the Sc.B. degree *

Applied Mathematics - Computer Science*

Applied Mathematics - Biology*

Guidelines for Applied Mathematics concentrator emphasizing probability, statistics, and operations research *

Guidelines for Applied Mathematics concentrator emphasizing scientific computing *

Courses to be offered in 2005-2006 *

Applied Mathematics Concentration Advisors *

This short guide is intended to give students and faculty an overview of the undergraduate program in Applied Mathematics at Brown University, and to answer some of the commonly raised questions. Applied Mathematics is an interdisciplinary subject involving both mathematics and many areas of application. More so than in many other areas it is important to have an overview of the subject and how it relates to other programs at Brown. Many students come to Brown with little prior understanding about what Applied Mathematics is or the broad range of opportunities that it provides.

The Division first prepared this guide in 1991, and has been updating it since then. Many people have said that they have found it valuable. We will continue to keep it up to date and improve it, and welcome comments or suggestions.

Applied Mathematics is an inherently interdisciplinary subject which covers a wide spectrum of scientific activities. It is the mathematics of problems arising in the physical, life and social sciences as well as in engineering, and provides a broad qualitative and quantitative background for use in these fields. The methods of mathematical modeling and analysis provide a unification and mutual enrichment of ideas from many different areas, and a deeper understanding of the fields to which it is applied. Applied Mathematics draws concepts and methods of mathematics to the fields of application and, in turn, brings ideas, techniques and scientific knowledge back to influence the development of mathematics.

Owing to its nature, Applied Mathematics appeals to people with a variety of different interests, ranging from those with a desire to obtain a good quantitative background for use in some future career, to those who wish to have a better understanding of the basic mathematical aspects of other fields, or to those who are interested in the basic techniques and approaches in themselves. Many of the students in Applied Mathematics courses like mathematics and want to learn more about it, but do not intend to use it professionally. Others want to learn more about a subject that is becoming more and more basic to many fields. Many go on to specialize in economics, finance or the other social sciences. The curriculum of the Division is flexible enough to meet these different desires.

Some examples might help to illustrate the range of Applied Mathematics, and the interaction between applications and mathematics which is involved. The examples are sampled from interests of the faculty.

**Medicine and the Environment
**One surprising example arises in medicine. High frequency ventilation of
the lungs is a procedure now used in surgery when the lungs of a patient must
be immobilized. Oxygen supply to the lungs and the removal of carbon dioxide is
achieved by small "tidal" oscillations where small amounts of air are
blown in and out of the lungs very rapidly. Similar exchange and dispersal
processes take place in tidal estuaries and coastal waters, and this latter problem
has been studied by civil engineers for quite some time. From the point of view
of mathematical modeling, the two types of processes are essentially the same
and applied mathematicians working with engineers and physiologists have
carried out basic developments.

**Computational Vision
**The next example concerns certain types of probabilistic models that have
been used for some time in statistical physics. It was realized recently that
the same mathematical framework could be used to model complex decision
problems in computer (robotic) vision and automatic image processing. The same
underlying mathematical methods have also led to major advances in
computational methods for the optimization of very complex systems (such as
VLSI design) and in speech recognition.

**Optimization Problems in Financial Markets and Manufacturing
**Some of the recent work in the study of general mathematical problems in
optimal decision making under uncertainty has been used to study financial
markets and investment strategies, and these in turn have further focused some
of the mathematical work. The mathematics of risk analysis has led to useful
models for policy decisions for parts of the health care system. Some
manufacturing problems can be modeled as flows of material in networks, subject
to capacity and other constraints (and possibly under some uncertainty or
randomness as well). Optimal network flow theory is now a well-developed
subject. Conversely, specific manufacturing models have often led to new
demands on the mathematics. New types of materials composed of laminates and
composites have unusual and highly desirable properties. An understanding of
their strength, reliability, electrical, optical or
other properties requires a good mathematical description of the component parts
and their interactions, and has in turn led to the study of new types of
differential equations.

**Computer Modeling and Simulation
**The computer has made it possible to attack
problems of a scope not dreamed of several years ago. The design of aircraft,
ships and automobiles all depend on computer modeling. To do the modeling, one needs to simulate the physical laws which govern the
behavior, and these laws are often defined by differential or other
mathematical equations. For large problems, the required modeling is not easy
to do since the computer can provide at best only an approximation, and the
accumulation of small errors often leads to poor results. A major goal of
numerical analysis is the design of methods that maintain accuracy even for
large-scale problems. Without advances in this area, even the most powerful
computers would be useless to solve complex "real life" problems.

**Physical Systems, Mathematical Models and Computer Analysis
**The problem of turbulence in fluids and gases
gives another interesting class of examples. The problem is of major importance
in engineering, and has been characterized as one of the outstanding technical
challenges at this time. The needs of this problem have led to new ideas in
scientific computation, differential equations and statistics. Conversely,
virtually all of the mathematical areas represented in the Division have been
applied to that problem. Supercomputers have been an essential tool in these
investigations, and have been a stimulus for the discovery and rigorous
mathematical investigation of more efficient numerical methods for use in all
branches of science. They allow us to attack more complex and less idealized
problems, and require different mathematical methods for their efficient
utilization. The properties of the physical problem might guide the discovery
of the methods, but mathematical insight is needed in such an investigation,
since it involves a rigorous comparison of the behavior of the ideal model for
the physical problem and the properties of the `approximate' model with which
the computer actually works.

*Brown
Universitys Division of Applied Mathematics*

Brown University is unusual in having a separate department devoted to Applied Mathematics. The Division of Applied Mathematics was created in 1941 in response to the awareness of the contributions that such a program could make to the dramatically increasing national scientific and engineering needs of that period. Since then the scope and interests of the Division have developed as the subject areas have evolved.

The Division of Applied Mathematics provides a special environment for the interaction of people with varied scientific interests. The Division draws together faculty who elsewhere might be dispersed in quite different departments, depending on their primary interests. In fact, Applied Mathematics programs in other Universities are often spread among several departments and lack the coherence and focus of the program at Brown.

The basic mathematical skills of Applied Mathematics come from a variety of sources, which depend on the problems of interest: the theory of ordinary and partial differential equations, matrix theory, statistical sciences, probability and decision theory, risk and insurance analysis, the classical methods for formulating and solving problems in the sciences, operational analysis, optimization theory, the mechanics of solid materials and of fluids flows, numerical analysis, scientific computation and the science of modern computer based modeling.

Courses are regularly offered in all of these areas. Each year, there are special offerings in topics of current interest, such as the mathematics and physics of systems which exhibit "chaotic" behavior, cryptography, and the mathematics of speculation.

The standard Applied Mathematics concentrations lead to either the A.B. or Sc.B. degrees. The program is very flexible. Numerous joint programs with other departments are described below, and individual concentrations suited to particular needs can be arranged. The range of offerings, either within Applied Mathematics alone, or in combination with offerings of other departments, provide almost endless opportunities.

Students take courses in applied mathematics for many reasons, not necessarily with an applied mathematics concentration in mind. The value of learning about applied mathematics goes beyond a career opportunity. It provides an education in the use of quantitative methods in thinking about and solving problems: knowledge that is valuable in all walks of life.

The various concentrations in Applied Mathematics do prepare students for a great variety of career opportunities. In recent years students who have followed one of the undergraduate concentrations in Applied Mathematics have gone into many different areas including: graduate study in applied mathematics, engineering, physical or earth sciences; actuarial work, insurance and investment management; computer consulting and information industries; scientific careers in industry or government service; medical school; teaching; banking and finance; graduate study in economics or business studies; operations research or statistical analysis in industry or government agencies. In particular courses offered in applied mathematics provide the preparation needed for several of the actuarial professional examinations. Business schools often seek graduate applicants with a good background in applied mathematics and economics.

*Information Resources Describing Professional
Opportunities*

The major professional organizations in the mathematical sciences publish information about career and educational opportunities. A short list of these resources and how to access them is given below. These documents are available on the web, and the relevant URLs are included here. (The online version of this document contains active web links.)

*American Mathematical Society (AMS)
*

P.O. Box 6248

Providence, RI 02940-6248

800 321-4267

http://www.ams.org

The AMS fosters mathematics research and its members are predominantly based in colleges and universities. It provides professional services for students and mathematical scientists.

The AMS, in cooperation with SIAM and MAA, and supported by funding from the
Sloan Foundation, maintains a web site of career profiles of professionals
trained in the mathematical sciences. The URL is

http://www.ams.org/careers/

This database is searchable by various characteristics of the individuals included, such as their highest degree in the mathematical sciences. Check out this site.

Each fall, the AMS publishes a guide to support (assistantships, fellowships
and scholarships) entitled *Assistantships and Graduate Fellowships in the
Mathematical Sciences*. This is particularly useful for undergraduates
considering advanced study in mathematical sciences. A copy is available at the
Division for reference.

*American Statistical Association (ASA)*

*1429 Duke Street*

*Alexandria**, **VA**
**22314-3402**
703 684-1221
http://www.amstat.org*

The ASA is the principal professional organization for statisticians in the US.

The ASA publishes three brochures aimed at students who wish to learn about professions in statistics:

*Careers in Statistics**Women and Statistics**Minorities in Statistics.*

Single copies are available for free from the ASA, and an electronic summary
of *Careers in Statistics* can be found at the link for __Education__
on the ASA web site.

*Association for Women in Mathematics (AWM)
*

University of Maryland

College Park, MD 20724-2461

301 405-7892

email: awm@math.umd.edu

http://www.math.neu.edu/awm/mathindex.html

The AWM works to encourage women to study and to have active careers in the mathematical sciences. The AWM publishes two brochures of potential interest to students interested in pursuing mathematical studies:

*Profiles of Women in Mathematics: The Emmy Noether Lectures**Careers that Count: Opportunities in the Mathematical Sciences*

The former publication describes careers and accomplishments of female
research mathematicians and may be of interest to students considering graduate
study. It is accessible in electronic form at the link for __Noether____
Lecture Series__ on the AWM web site. *Careers that Count* is available
for $1.50.

*Casualty Actuarial Society
*

1100 N. Glebe Road

http://www.casact.org

The Casualty Actuarial Society promotes actuarial science applied to
property, casualty, and similar risk exposures. It administers standards for
professional certification of casualty actuaries, including education programs
and exams for Associates and Fellows. Their web site includes a brochure
entitled *An Actuarial Career*.

*Institute for Operations Research and the Management Sciences (INFORMS)
*

http://www.informs.org

INFORMS is the primary professional organization for OR and MS, formed through the merger of the Operations Research Society of America (ORSA) and The Institute of Management Sciences (TIMS). INFORMS publishes two brochures likely to be of interest to students who wish to learn about careers in OR/MS:

*Careers in Operations Research**Educational Programs in Operations Research*

Both of these are available free from INFORMS.

The INFORMS web site includes *The INFORMS Career Booklet: Is a Career in
Operations Research/Management Science Right for You? *at the URL

http://www.informs.org/Edu/Career/booklet.html

This is an interesting and well-organized resource about OR careers.

*Mathematical Association of America (MAA)
*

202 387-5200

http://www.maa.org

The MAA promotes mathematics education and preparation for careers that build on mathematics. It publishes three brochures about professional opportunities in mathematical sciences:

*Careers in the Mathematical Sciences*(free)*Mathematical Scientists at Work*($3.00)*More Careers in the Mathematical Sciences*($.25)

The MAA web site also includes career profiles of individuals with degrees
in mathematical sciences at the URL

http://www.maa.org/careers/

*National Academy of Sciences / National Academy of Engineering /
Institute of Medicine
*

The National Academy Press is the publisher for the National Academies and
the Institute of Medicine. Its web site contains a wealth of information about
science, engineering and health including reports by expert panels about
directions for development of branches of mathematical sciences. One particular
publication that may be of interest to students considering graduate school is *Careers
in Science and Engineering: A Student Planning Guide
to Grad School and Beyond* accessible at the URL

http://www.nap.edu/readingroom/books/careers/

*Society for Industrial and Applied Mathematics (SIAM)
*

215 382-9800

http://www.siam.org

SIAM is a leading professional organization of applied mathematicians in industry and academia. It is devoted to advancing the application of mathematics to science and industry.

The SIAM web site includes a continuing series of articles entitled *Mathematics
That Counts* highlighting advances in applied and computational mathematics
that have led to increased productivity, improvements in product design, and
solutions to problems related to health and the environment. These can be
accessed through the link to __SIAM News__ from the SIAM Home Page or
directly at the URL

http://www.siam.org/siamnews/mtc/mtc.htm

*Society of Actuaries
*

http://www.soa.org

The Society of Actuaries is the principal professional organization of life actuaries who (in contrast with casualty actuaries) are primarily concerned with assessment of risks in the fields of life insurance, health insurance, and pensions and annuities.

The actuarial societies tout the fact that "actuary" was twice
rated the best job in America by the *Jobs Rated Almanac*. The 1988 and
1995 editions put the actuarial profession at the top of the list of 250
professions ranked on criteria such as work environment, job outlook, security,
and stress.

The Society of Actuaries publishes the brochure

*Actuaries Make a Difference*

which is free. Its web site contains information about education programs and actuarial exams for certification as an Associate or Fellow of the Society.

**Choosing Courses**

The courses in Applied Mathematics are designed for students with a wide range of goals and are not limited to the needs of students following an applied mathematics concentration. There are many opportunities for students to explore different subject areas and find which they find most interesting. In this section we list and explain the courses offered by the Division showing how they relate to each other. These courses are offered each year unless stated otherwise. We also list a selection of courses from other departments which are relevant to applied mathematics, either giving further applications or providing additional mathematical background.

When choosing courses consider what your goals are. Do you wish to pursue applied mathematics at a graduate level? Do you wish to gain a good basis in applied mathematics at an undergraduate level, but intend later to pursue some other related area? Are you simply taking courses for general understanding and knowledge?

A general recommendation for students planning to follow graduate study in any subject relating to applied mathematics (engineering, economics, physics, chemistry, computer science, etc.) is to complete the three semester calculus sequence MA 9,10 and 18 or the equivalent; and in addition to complete the linear algebra course MA 52 or MA 54. This recommendation is also sound advice for all students.

Students taking courses in applied mathematics start either with one of the lower level courses (AM 9 or AM 16) or with the Mathematical Methods courses AM 35,36 (or 33,34) which provide many of the basic approaches used in applied mathematics. The Statistical Inference Course AM 165 provides the introduction to the other courses in statistics and operations research, while AM 117 gives a good overview of numerical methods. Beyond these comments, students should select courses so that they pursue specific topic areas in a coherent manner.

Students following the applied mathematics concentration and intending to go on to graduate study in this subject should take some additional mathematics courses, such as those mentioned later in this section.

Faculty concentration advisors will be glad to give advice on course
selections to *all* students.

**Insurance Mathematics: **Although not specifically designed for the
purpose, some of the Division's courses offer preparation for several of the Associateship examinations of the Society of Actuaries. In
particular, AM 165 (Introduction to Mathematical Statistics) and AM 166 (Linear
Models) are ample preparation for the Society's examinations for its Courses
110 (Probability and Statistics) and 120 (Applied Statistical Methods),
although the latter also contains some time series analysis covered, in much
greater detail, in AM 167 (Time Series Analysis). The material in the Society's
Courses 130 (Operations Research) and 135 (Numerical Methods) are more than
completely covered in AM 120 (Operations Analysis: Probabilistic Models), AM
121 (Operations Analysis: Deterministic Models) and AM 117 (Introduction to
Numerical Analysis).

*Courses in Applied Mathematics*

**AM 9.
Introduction to Modeling**

**
**Topics of Applied Mathematics, introduced in the context of practical
applications where defining the problems and understanding what kinds of
solutions they can have is the central issue. Computations are performed in
MATLAB; instruction is provided.

**AM 16. Introduction to Computing Sciences**

For
student in any discipline that may involve numerical computations. Includes instruction for programming in
MATLAB. Applications include solution of
linear equations (with vectors and matrices) and nonlinear equations (by
bisection, iteration, and Newton's method), interpolation, and curve-fitting,
difference equations, iterated maps, numerical differentiation and integration,
and differential equations. Prerequisite: MA 10 or its equivalent.

**AM 41.**** Mathematical Methods in the Brain
Sciences**

Basic mathematical methods commonly used in the cognitive and neural sciences. Topics include: introduction to differential equations, emphasizing qualitative behavior; introduction to probability and statistics, emphasizing hypothesis testing and modern nonparametric methods; introduction to Fourier analysis. Time permitting, also considers some elementary information theory. Examples from biology, psychology, and linguistics. Prerequisite: MA 10 or equivalent.

The courses AM 33/35 and AM 34/36 cover mathematical techniques involving differential equations used in the analysis of physical, biological and economic phenomena. In the sequence AM 33,34 the primary emphasis is placed on the use of established methods rather than on rigorous treatment of the underlying mathematics. AM 35,36 covers similar material (except for introduction to statistics) in more depth. It is intended for students who prefer a more rigorous development of the mathematical foundations of the methods.

Students who are considering one of the concentrations in Applied Mathematics and others who will be taking advanced courses in Applied Mathematics, Mathematics, Physics or Engineering are encouraged to take AM 35, 36.

**AM 33/35.**** Methods of Applied Mathematics
I**

Solution of first order differential equations, including the use of exact
differentials. Solution of second order, linear differential equations.

**AM 34/36.**** Methods of Applied Mathematics
II**

Review of vector algebra and matrix methods, with applications to systems of linear, first order differential equations. Nonlinear problems and stability. Introduction to partial differential equations and Fourier series methods. Boundary value problems and an introduction to Sturm-Liouville systems.

**Note: **AM 34 also contains a short introduction to
probability/statistics.

Prerequisites: MA 9,10. It will be expected that most students taking AM 33,34 and all students taking AM 35,36 will have taken MA 18 (or equivalent) or will be taking it at the same time. Students are strongly encouraged to take MA 52 or MA 54 (Linear algebra). The majority of students in AM 35,36 have taken or are taking a linear algebra course.

Beyond these two basic courses in mathematical methods the Division offers further courses AM 205, 206 that are intended for senior level undergraduates or beginning graduate students. These courses cover additional topics and develops further those introduced in the earlier courses. Major emphases of these courses include complex variable methods and solution methods for partial differential equations. The courses will be of immediate importance to students interested in applications to the physical sciences, engineering and biology, but are also relevant to other application areas.

**AM 205, 206.**** Mathematical Methods of
Applied Science
**Prerequisite: AM 36 or 34, MA 52 or 54 and MA 18 (or equivalent)

Introduces applied mathematics, science and engineering students to a variety of fundamental mathematical methods. Topics include linear algebra, complex variables, Fourier series, Fourier and Laplace transforms and their applications, ordinary differential equations tensors, curvilinear coordinates, partial differential equations and calculus of variations.

There are many applications of mathematics to the description and quantitative study of biological processes. Applied mathematics provides many valuable tools in determining the relative importance of the many factors that may affect biological systems in such diverse areas as population studies, epidemiology, chemical oscillators and the nervous system. Students interested in developing a concentration combining biology and applied mathematics should contact Professor David Rand in Biology, or Professor Elie Bienenstock in Applied Mathematics..

The course AM 107 (Biomed 149) listed below is suitable for students who may not be specialists in biology but have some background in the subject, and are interested in some of the applications of mathematics in the biological sciences. There are also interested related courses in biomechanics offered in Engineering (EN 121, 122).

**AM 107. Quantitative Models of Biological Systems (Biomed
149)**

An introductory course on the use of quantitative modeling techniques in solving problems in biology. Each year one major biological area will be explored in detail from a modeling perspective. The particular topic will vary from year to year. Mathematical techniques will be discussed as they arise in the context of biological problems.

Prerequisites: some introductory level biology; AM 33,34 or 35,36; or written permission.

Offered in alternate years.

Numerical
Analysis and Scientific Computation

The course AM 117 is a valuable, general introduction to numerical methods
that are widely used in many applications. It provides an essential basis for
scientific computation, whatever the area of interest. AM 118 is more
specifically focused on ordinary and partial differential equations. This would
be of value to students with interests in applications to the physical sciences
or engineering.

**AM 117.**** Introduction to
Numerical Analysis**

**
**Devoted
to computational linear algebra.
Tailored to computer science concentrators, but the topics will also
appeal to other science concentrators.
Topics include Newton methods, elements of basic linear algebra, Gauss
elimination and matrix decompositions (Cholesky, LU,
QR, etc.) Methods for
computation of eigenvalues and eigenvectors,
round-off errors and error analysis.
Basic interative methods
such as SOR and conjugate gradient methods.

**AM
118. Introduction to the Numerical Solution of Differential Equations**

**
**Basic
numerical techniques for solving ordinary and partial differential
equations. Topics include Euler, Runge-Kutta, and multistep
method, error analysis, and step-size control for ordinary differential
equations. Methods for partial
differential equations include finite difference and finite element methods for
Poisson equation, the heat equation, and wave problems and various solution
techniques. Prerequisites: AM 33, 34, or 35, 36, AM 117 is recommended,
not required.

Operations Analysis originated with attempts to make optimal decisions about the allocation of scarce resources, the design of efficient distribution networks and the need to make rational, optimal decisions when faced with uncertain information. The field has grown now to include many of the mathematical methods and models which are used for the design, optimization and analysis of management systems in government, business and economics.

Two courses which cover the fundamental ideas and methods of the field are offered: Applied Mathematics 120 is concerned with probabilistic or statistical models, where the system of concern is subject to randomness or uncertainty of some sort. It is an excellent introduction to some of the most widely used models and ideas of probability theory as well as their use in practical problems; Applied Mathematics 121 is concerned with optimization or analysis methods for deterministic problems. The courses deal with both the theory and selected applications.

Students may also be interested in the closely related course EN 132 - Transportation Systems Analysis.

**AM 120. Operational Analysis: Probabilistic Models**

Methods of problem formulation and solution. Introduction to the theory of Markov chains, the probabilistic `analog' of a difference or differential equation. This is the most widely used of the probabilistic processes which evolve over time according to some statistical rule. Birth-death statistical processes and their applications. Queuing, probabilistic service and waiting line theory.

Sequential decision theory via the methods of Dynamic Programming. This is the theory of optimal decisions when a sequence of decisions is to be made over time, each one affecting the situation of those that come later.

Prerequisite: AM 165, or MA 161, or equivalent

**AM 121.**** Operational Analysis:
Deterministic Methods**

An introduction to the basic mathematical ideas and computational methods of optimization. Linear Programming: This is the theory of optimal decision making under linear constraints on resources, and may be the most widely used set of ideas in the field. Applications include decision theory in economics, transportation theory, optimal assignments, production and operations scheduling. The theory of network modeling and flows. The theory of integer programming, which constitute the ideas for decision and optimization when the decision variables are integers (e.g., number of staff to be assigned etc.).

Prerequisites: An introduction to matrix calculations, such as AM 34 or MA 52.

Mechanics

Two undergraduate courses (AM 125 or EN 137) AM 126 are offered in mechanics. The prerequisites for both are an introductory mechanics course such as EN 4, PH 5 PH 7, and mathematical methods AM 33,34. AM 125 covers the mechanics of systems of particles and rigid bodies, including motion in rotating systems. It explains the advanced methods used to study complex systems, and gives an understanding of the unusual characteristics of general rigid body motion. AM 126 is a self-contained, one semester course providing an introduction to the mechanics of fluid motion and the elasticity of solids. The course differs from traditional engineering courses in this area and will emphasize other applications to physics, earth sciences, biomechanics, and other sciences.

Beyond these courses students may also consider courses in Engineering or Geological Sciences which are listed later in this guide, or some of the first-year graduate courses offered in Applied Mathematics or Engineering on mechanics, fluid dynamics and solid mechanics.

Mechanics provides a rich supply of examples of chaotic dynamical systems, which are discussed in the course on Chaotic Dynamics (AM 136). This is an exciting area which is continuing to develop.

The presence of motion in deterministic systems that is effectively unpredictable is now recognized as an essential scientific phenomenon, and modern science is slowly coming to terms with its implications. This so-called "chaos" has challenged mathematicians and there is now a substantial mathematical theory supporting the observations of chaotic behavior in the real world. Despite its recent surge in popularity, an early motivation was the study by Poincare and others at the end of the last century of celestial mechanics. Here supposedly simple systems governed by well-defined equations of motion appeared to have very complex behavior and to be very sensitive to disturbances. As techniques of mathematical modeling have developed for new, more varied applications in economics, biology and chemistry so too has the realization that complex behavior is a common feature of nonlinear systems. The course AM 136 presents in a systematic way the mathematical concepts and definitions used in the study of nonlinear systems.

**AM 136. Topics in Chaotic Dynamics**

Overview and introduction to dynamical systems. Local theory of maps and periodic points: stability and bifurcation. Global theory of maps, attractors, limit sets, Lyapunov exponents, the Sarkovskii-Li-Yorke theorem, and chaos. Dimension theory and fractals: definitions and classical examples (e.g. Cantor set and Koch snowflake). Continuous systems: Lorentz attractor, Hamiltonian systems, fluid dynamics and related PDEs, Hopf bifurcation. Fourier methods and spectral techniques: the power spectrum.

Prerequisites: AM 34 or 36 or Math 111, and Math 52 or 54.

Statistics:
Theory and Applications

Probability and statistics are basic tools in economics, physics, biological modeling, many modern applications of computers (such as to image analysis, speech recognition, and expert systems), epidemiology, and in many industrial applications, such as quality control, factory automation, optimal resource allocation, and risk assessment. The sequence AM 165/166 provides an introduction to the general theory. Other courses, requiring AM 165 (or Math 161) as prerequisites, explore some of the more modern and more powerful statistical tools and some applications. These include AM 167 (Time Series Analysis), AM 168 (Nonparametric Statistics), AM 177 (Information Theory), AM 120 (Operations Research: Probabilistic Models), and various "Senior Seminar Courses" (listed under AM 193 or AM 194) such as Introduction to Pattern Analysis, The Mathematics of Speculation, and Information and Coding Theory.

**AM 165. Statistical Inference I**

AM 165/166 constitute an integrated first course in mathematical statistics. The first third of AM 165 is probability theory, and its last two-thirds are statistics. Specific topics include probability spaces, discrete and continuous random variables, methods for parameter estimation, large and small sample techniques for confidence intervals and hypothesis testing.

Prerequisite: Mathematics 10 or equivalent.

**AM 166.**** Statistical Inference II**

AM 166 is designed as a sequel to AM 165 to form one of the alternative tracks for an integrated years course in mathematical statistics. The main topic is linear models in statistics. Specific topics include likelihood-ratio tests, nonparametric tests introduction to statistical computing, matrix approach to simple-linear and multiple regression, analysis of variance, and design of experiments.

Prerequisite: AM 165, or equivalent, basic linear algebra.

**AM 167.**** Time Series Analysis**

An introduction to stochastic processes - the study of structure and randomness in sequences of observations. Time series analysis is used to model complex interactions among evolving observations in diverse applications, such as Economics (market prices, economic indicators), Biology (nerve cell activities), Engineering (speech and other sound waveforms). Time series models are a mixture of deterministic and random components, which capture structure and fluctuations respectively. The course will cover basic classes of models and some of their applications, parameter estimation, and spectral (Fourier) analysis.

Prerequisite: AM 166.

**AM 168. Non-parametric Statistics**

A systematic treatment of the distribution-free alternatives to classical statistical tests. These nonparametric tests make minimum assumptions about distributions governing the generation of observations, yet are of nearly equal power as the classical alternatives.

Prerequisite: AM 165 or equivalent.

**AM 169. Introduction to Computational Probability and Statistics**

Examination of probability theory and mathematical statistics from the perspective of computing. Topics selected from random number generation, Monte Carlo methods, limit theorems, stochastic dependence, estimation and hypothesis testing.

Prerequisites: linear algebra and Applied Mathematics 165 or equivalent. Some experience in programming is desirable.

Offered in alternate years

**AM 171. Information Theory**

**
**Information theory is the mathematical study of the fundamental limits of
information transmission (or coding) and storage (or compression). This course
offers a broad introduction to information theory and its real-world
applications. A subset of the following is covered: entropy and information;
the asymptotic equipartition property; theoretical
limits of lossless data compression and practical algorithms; communication in
the presence of noise channel coding, channel capacity; source-channel
separation; Gaussian channels; Lossy data
compression.

**Senior
Seminar Courses 2005 - 2006**

Each year the Division offers about two to four senior seminar courses (AM 193 and AM 194), which explore areas of applied mathematics in a manner different from the regular lecture format. Students are encouraged to study more independently and develop specific projects.

Planned Senior Seminar Courses for 2005 - 2006 are given below. The topics of these courses vary regularly.

**AM 193. (S1) Holdem Poker**

A study of the probabilities, the strategies, and the game theory. Algorithms for optimal and near-optimal play will be discussed, and practical issues of search strategies and pruning will be explored. The goal is for each student to write a computer program that plays competitive Holdem Poker. A competition, via simulation, will be held.

Prerequisites: probability (AM0165 or MA0161) and some experience writing computer programs. Limited enrollment.

**AM 194. (S1) Topics in Kinetic Theory**

This course will introduce current mathematical study for Boltzmann equation and Vlasov
equation. We will study large time behavior and hydrodynamic limits for Boltzmann
theory and instability and instabilities in the Vlasov theory. Graduate PDE course
is required.

**AM 194. (S2) Fluid Dynamics and Physical Oceanography**

Introduction to fluid dynamics as applied to the mathematical modeling and
simulation of ocean dynamics and near-shore processes. Oceanography topics include:
overview of atmospheric and thermal forcing of the oceans, ocean circulation, effects
of topography and Earth's rotation, wind-driven currents in the upper ocean, costal
upwelling, the Gulf Stream, tidal flows, wave propagation, tsunamis.

**MA158. CRYPTOGRAPHY **

Topics include symmetric ciphers, public key ciphers, complexity, digital signatures, applications and protocols. Math 153 will not be required for the course. What is needed from abstract algebra and elementary number theory will be covered. Prerequisite: MA 52 or MA 54. Fall Semester 2005-06

Applied Mathematics is an interdisciplinary subject, as noted earlier, and there are many courses taught in other departments at Brown which are relevant to an applied mathematics program, either in providing additional mathematical tools or in developing applications in one of the sciences or engineering. Listed below are some suggested courses that complement those offered by the Division, and which students may consider taking. The list is by no means comprehensive. It is meant to suggest possible directions to explore.

Many of these courses would be accepted as part of an applied mathematics concentration program. Students should ensure first that they have satisfied the prerequisites for a particular course.

All Applied Mathematics majors should complete the calculus sequence MA 9,10,18 or the equivalent, and a course in linear algebra, MA 52 or 54. A student considering a graduate applied mathematics program in the future should go beyond this and take some additional courses from the Mathematics Department.

Suggestions include:

MA 101 |
Analysis |

MA 126 |
Complex Analysis |

MA 127 |
Topics in Functional Analysis |

MA 153 |
Abstract Algebra |

MA 161 |
Probability |

These and other courses in mathematics are accepted for the applied mathematics concentration.

The Physics Department offers several introductory courses such as PH 5,6 or the more advanced PH 7,8. These are suitable for first year students and provide an excellent background in the physical sciences and a good basis for other courses.

Suggested courses include:

PH 5 |
Foundations of Mechanics |

PH 6 |
Foundations of Electromagnetism and Modern Physics |

PH 7 |
Analytical Mechanics |

PH 8 |
Physics of Waves, Relativity and Quantum Mechanics |

PH 47 |
Electricity and Magnetism |

PH 151 |
Advanced Electromagnetic Theory |

PH 50 |
Advanced Classical Mechanics |

PH 79 |
Physics of Matter |

Other courses are relevant as applications areas in applied mathematics but are directed more towards students following a concentration in physics.

The structure of courses in Engineering is directed towards students following an accredited engineering program. However, it is possible for students who are not engineering majors to take upper level engineering courses. For some it is important to have either EN 3,4 or PH 5,6 or the equivalent as preparation.

Suggested courses include:

EN 52 |
Electrical and Optical Systems |

EN 157 |
Linear Systems Analysis |

EN 158 |
Communication Systems |

EN 163 |
Digital Electronics Systems Design |

EN 166 |
Automatic Control Systems |

EN 31 |
Mechanics of Solids and Structures |

EN 175 |
Advanced Mechanics of Solids |

EN 72 |
Thermodynamics |

EN 81 |
Fluid Mechanics |

EN 186 |
Advanced Fluid Mechanics |

EN 121 |
Biomechanics |

EN 122 |
NeuroengineeringŸ Control of Movement |

Cognitive and Linguistic Sciences

CG 41 |
Introduction to Linguistic Theory |

CG 102 |
Neural Modeling Laboratory |

CG 129 |
Understanding the Brain |

The Chemistry Department offers a variety of courses on introductory chemistry, chemical kinetics, inorganic and organic chemistry that are valuable for their general scientific content.

CH 10 |
Introductory Chemistry |

At the 100-level, more advanced courses offered include:

CH 114 |
Physical Chemistry: Quantum Chemistry |

CH 115 |
Physical Chemistry: Thermodynamics and Statistical Mechanics |

GE 135 |
Meteorological Aspects of Climatic Change |

GE 160 |
Environmental and Exploration Geophysics |

GE 161 |
Solid Earth Geophysics |

GE 162 |
Continuum Physics of the Solid Earth |

GE 196 |
Groundwater Penetrating Radar Data Analysis |

There is a standard concentration in Applied Mathematics - Computer Sciences for the Sc.B. degree which lists specific course suggestions.

There are standard concentrations in Applied Mathematics - Economics for both the Sc.B. and A.B. degrees which list specific course suggestions.

We have already mentioned the connections between applied mathematics and the study of biological systems. Students interested in combining studies in biology and applied mathematics should consult a concentration advisor about structuring an independent concentration.

There are two graduate level courses in Biology and Medicine Community Health that are suitable for students who have taken statistics courses in applied mathematics and are interested in public health issues.

BC213 |
Principles of Biostatistics and Data Analysis |

BC216 |
Applied Regression Analysis |

Applied Mathematics offers several standard concentration programs which are listed in detail in the University Catalogue and the Course Announcement. Both the A.B. and Sc.B. concentrations in Applied Mathematics require certain basic courses to be taken, but beyond this there is a great deal of flexibility as to which areas of application are pursued. Students are encouraged to take courses in applied mathematics, mathematics and one or more of the application areas in the natural sciences, social sciences or engineering. Whichever areas are chosen these should be studied in some depth.

Standard program for
the A.B. degree.

Prerequisite: MA 9, 10 or their equivalent.

Program: Ten
additional semester courses approved by the Division of Applied Mathematics,
including MA 18, 52, AM 35, 36 and one of AM 9, 16, or CS 4, 15, or 17. AM 33,
34 will sometimes be accepted as substitutes for AM 35, 36. Of the unspecified
courses four should be chosen from the 100-level courses taught by the Division
of Applied Mathematics. Substitution of alternate courses for the specific
requirements is subject to approval of the division. Concentrators are urged to
consider MA 54 as an alternative to MA 52 and to complete their introductory
programming course before the end of their sophomore year.

Standard program for the Sc.B.
degree.

Eighteen approved semester courses in
mathematics, applied mathematics, engineering, the natural or social sciences,
including MA 9, 10, 18, 52; AM 35, 36, and 193 or 194, and one of AM 9, 16, or
CS 4, 15, or 17. AM 33, 34 will sometimes be accepted as substitutes for AM 35,
36. Of the unspecified courses six should be chosen from the 100-level or
higher level courses. Substitution of alternate courses for the specific
requirements is subject to approval by the division. Concentrators are urged to
consider MA 54 as an alternative to MA 52 and to complete their introductory
programming course before the end of their sophomore year.

Students who have not established
proficiency in a foreign language will be encouraged to take French, German, or
Russian.

Standard program for the Sc.B. degree.

Requirements:

1. AM 35, 36 (or 33, 34), 165, and another approved 100-level course.

2. Four biology courses agreed upon by the student and advisor (see below for some possible areas of emphasis).

3. MA 9, 10 (or 17), 18, and 52 (or an applied algebra course).

4. CH 33. Recommended for some concentrators: organic chemistry and biochemisty.

5. PH 3, 4 (or 5, 6).

6. Two
additional courses in applied math, biology, chemistry, math, or physics. *At
least one of these must be a directed research course *(e.g., AM 193, 194;
BI 195, 196).

Possible areas of emphasis and suggested courses include:

1. Biochemistry: BI 28, 127, and CH 35, 36.

2. Cells, tissues, and organs: BI 80, 117, 110, 118, and/or appropriate bioengineering courses, such as BI 108, 109, 112, 114.

3. Neurosciences: BN courses; AM 41

4. Population biology and ecology: BI 42, 43, 45, 48, 141, 142, 143, 144, 147, 148.

5. Genetics: B1 47, 141.

**Concentration Advisors:** Professor David Rand, 3-1063, Walter Hall 202 or BioMed Center 418.
Professor Elie Bienenstock, 3-1195, 302D Metcalf Chemistry Laboratory.

*Applied
Mathematics - Computer Science*

The standard Sc.B. concentration in Applied Math-Computer Science is designed to provide a foundation of basic concepts and methodology of mathematical analysis and computation and to prepare students for advanced work in computer science, applied mathematics, and scientific computation. Interested students may contact advisors in the Division of Applied Mathematics or the Department of Computer Science. The concentration has the following specific requirements.

**Prerequisites:** Mathematics 10 or 17.

**Required courses:** MA 18 or 35 and 52 or 54; AM 35, 36, and
either AM 117 or 118; CS 15, or 17, 16,
or 18, 22, 31, and 32. (In some cases, substitutions of equivalent courses will
be permitted.) In addition, students must complete three courses in applied
mathematics at the 100-level or higher and three 100-level courses in computer science, as well as a
senior seminar in either department. Concentrators must also complete an
approved English writing course.

Of the 100-level or higher applied mathematics courses, at least two should constitute a standard sequence, or address a common theme. For example, either of the pairs AM 120-121 or AM 165-166 would be suitable. Similarly, at least two of the three additional 100-level computer science courses must constitute an approved sequence, as in the computer science A.B. concentration. In addition, at least one of the three 100-level computer science courses must be a theoretical computer science course. Substitution of courses in mathematics or engineering for courses in applied mathematics or computer science may be permitted, at the discretion of the concentration advisor.

*Applied Mathematics - Economics*

The philosophy of the program is to provide sufficient command of mathematical concepts to allow pursuit of an economics program emphasizing modern research problems. Economic theory has tended in recent decades to use more and more mathematics, and empirical research in economics has witnessed the development of large scale computational routines, utilizing the data processing capabilities of modern computers. The applied mathematics-economics concentration is designed to reflect the mathematical nature of modern economic theory and empirical research. Courses in the concentration include statistics, econometrics, mathematical programming, and applications to research.

Standard program for the A.B. degree

Prerequisites: MA 9, 10, 52.

Course requirements:

1. Applied Mathematics:

a. AM 35; 36 (note that AM 33, 34 may substitute for AM 35, 36 with permission)

b. At least one course from AM 9, 16, or CS 4, 15, or 17

c. AM 165

d. AM 120 or 121 (EN 131)

e. At least one other 100 level course chosen from AM 120, 121, 166 to 171, MA 101, and others, with permission

2. Economics:

a. EC 113 ( or 111 with permission)

b. EC 121

c. EC163

d. At least three other 100 level economics courses. Of those three courses, at least two must be chosen from the "mathematical-economics" group. This group comprises EC 117, 147, 164, 171, 178, 182, 183, 185, 186, and 187.

Standard program for the Sc.B. degree

Prerequisites: MA 9, 10.

Course requirements:

1. Applied Mathematics

a. Am 35; 36 (note that AM 33, 34 may substitute for AM 35, 36 with permission)

b. At least one from AM 9, 16, or CS 4, 15, or 17

c. AM 120 or 121 (EN 131)

d. AM 165

e. MA 52 or 54

f. Two other approved courses. Courses recommended to satisfy these remaining requirements include AM 117, 120, 121, 166 to 171, MA 101, and others, with permission.

2. Economics:

a. EC 113 (or 111 with permission)

b. EC 121

c. EC 163

d. At least five other 100 level economic courses. Of those five courses, at least three must be chosen from the mathematical-economics group. This group comprises EC 117, 147, 164, 171, 178, 182, 185, 186 and 187.

One of the approved courses in the concentration must satisfy the Sc.B. requirement of one semester of independent study, and may be fulfilled by taking AM 193, 194, EC 164, another approved course, or an approved internship.

Students concentrating in Applied Mathematics are able to emphasize particular areas of interest in the mathematical sciences. Standard joint concentrations focus on Biology, Computer Science, Economics, or Psychology. In addition, one of several themes can emerge from within a "straight" Applied Mathematics concentration (Sc.B. or A.B.).

Below are some guidelines for an Applied Mathematics concentrator who might wish to emphasize either the general area of probability/statistics/operations research, or the general area of scientific computing.

*Guidelines for Applied
Mathematics concentrator emphasizing probability, statistics, and operations
research*

The theories of probability, statistics, and operations research are increasingly exploited in modern applications, including problems in medicine (data analysis, medical imaging, epidemiology), economics (pricing and portfolio theories), robotics (machine vision, control theory), biology (genetic codes, theory of evolution, theories about the nervous system), and the actuarial sciences. An Sc.B. can be designed within the Applied Mathematics concentration that develops a basic set of tools in the theories of probability, statistics, and operations research, and explores selected applications.

Required:

- Math 9, 10, 18 (calculus)
- Math 52 (linear algebra)---Math 54 is preferable
- Applied Math 35, 36 (methods of applied mathematics I, II)---or, possibly, Applied Math 33, 34
- Applied Mathematics 9 (introduction to modeling ), or 16 (introduction to computing sciences), or Computer Science 4 (introduction to computing), or 15 (computer programming, problem solving and applications).

Suggested:

- Applied Math 9 or 16 (topics in scientific computing ). We strongly suggest students to take one course no later than in the sophomore year
- Applied Math 117 (introduction to numerical analysis) and 118 (introduction to numerical solution of differential equations)
- A selection of two to three courses from: Applied Math 121 (operations research: deterministic models), 205 (mathematical methods of applied sciences), 206 (mathematical methods of applied science), 165 (introduction to mathematical statistics)
- A selection of two Senior Seminars courses (Applied Math 193 or 194). Recent examples of interest include:

The theory of difference equations

Coding and information theory

Software for mathematical experiments

The mathematics of speculation

Topics in parallel computing

Related courses in other fields---selection of two to four from engineering, computer science, mathematics and physics. Students should check with concentration advisors about suitable courses.

Students in the Applied Mathematics concentration (or in a joint concentration), whose work in the field of the concentration has demonstrated superior quality and culminated in an Honors Thesis of Distinction, will be recommended by the Division to be designated "Honors" upon their graduation by the University.

The GPA in the concentration courses of an honors candidate should be at least 3.6. Exceptions may be made if the GPA is marginal but the record shows an improvement over the years. The honors thesis should be supervised by a faculty member in the Applied Mathematics Division, or from another department as long as the thesis is related to Applied Mathematics, and should be seen and evaluated by another faculty member. Since the work to prepare a thesis involves at least a semester, we strongly recommend students who have the intention to apply for honors to approach a faculty member for supervising a thesis no later than the end of their third year. In April of their fourth year, the faculty member who supervises the thesis and another faculty member who has seen and evaluated the thesis must write a recommendation letter to the Applied Mathematics Division. The Division will evaluate the overall academic record including the quality of the thesis to reach a decision on whether or not to recommend to the University an honors designation.

From time to time the Rohn Truell Premium is awarded to outstanding students graduating in the Applied Mathematics concentration. The prize is named after Professor Truell, a former chairman and professor in Applied Mathematics.

Each year graduating students who have a strong academic record of achievement in the physical, mathematical and life sciences are considered for nomination to the Sigma Xi scientific society.

**Courses to be offered in ****2005-2006**

Course |
Semester |
Course Title |

AM 16 |
II |
Introduction to Computing Sciences |

AM 33 |
I & II |
Methods of Applied Mathematics I |

AM 34 |
I & II |
Methods of Applied Mathematics II |

AM 35 |
I |
Methods of Applied Mathematics I |

AM 36 |
II |
Methods of Applied Mathematics II |

AM 41 |
I |
Mathematical Methods in the Brain Sciences |

AM 65 |
II |
Essential Statistics |

AM 117 |
I |
Introduction to Numerical Analysis |

AM 120 |
II |
Operations Research: Probabilistic Models |

AM 121 |
I |
Operations Research: Deterministic Models |

AM 136 |
II |
Topics in Chaotic Dynamics |

AM 165 |
I |
Statistical Inference I |

AM 166 |
II |
Statistical Inference II |

AM 171 |
I |
Information Theory |

AM 193 |
I |
Holdem Poker |

MA 158 |
I |
Cryptography |

AM 194 |
II |
Topics in Kinetic Theory |

AM 194 |
II |
Fluid Dynamics and Physical Oceanography |

AM 195 |
I |
Independent Study |

AM 196 |
II |
Independent Study |

AM 205 |
I |
Mathematical Methods of Applied Science |

AM 206 |
II |
Mathematical Methods of Applied Science |

**Applied Mathematics Concentration
Advisors****
**

**Please see one of our undergraduate concentration advisors if you have any
questions:**

Professor Martin Maxey (Chair)

Telephone extension 31482, Room
206,

E-mail: maxey@dam.brown.edu

Professor Charles Lawrence

Telephone extension 31479, Room 227,

E-mail: lawrence@dam.brown.edu

Professor Govind Menon

Telephone extension 33793, Room 325,

E-mail: menon@dam.brown.edu

Professor Chau-Hsing Su

Telephone extension 31447, Room 204,

E-mail: Chau-Hsing_Su@brown.edu