2008-2009 Bulletin – PDF - SEAS Bulletin - Columbia University

APAM E9900x and y, and S9900
Doctoral dissertation
0 pts. Members of the faculty.
A candidate for the doctorate may be required to
register for this course every term after the
course work has been completed, and until the
dissertation has been accepted.
COURSES IN APPLIED
MATHEMATICS
APMA E2101y Introduction to applied
mathematics
Lect: 3. 3 pts. Professor Keyes.
Prerequisite: Calculus III. A unified, single-semester
introduction to differential equations and linear
algebra with emphases on (1) elementary analytical
and numerical technique and (2) discovering
the analogs on the continuous and discrete sides
of the mathematics of linear operators: superposition,
diagonalization, fundamental solutions.
Concepts are illustrated with applications using
the language of engineering, the natural sciences,
and the social sciences. Students execute scripts
in Mathematica and MATLAB (or the like) to illustrate
and visualize course concepts (programming
not required).
APMA E3101x Linear algebra
Lect: 3. 3 pts. Professor Spiegelman.
Matrix algebra, elementary matrices, inverses,
rank, determinants. Computational aspects of
solving systems of linear equations: existenceuniqueness
of solutions, Gaussian elimination,
scaling, ill-conditioned systems, iterative techniques.
Vector spaces, bases, dimension.
Eigenvalue problems, diagonalization, inner
products, unitary matrices.
APMA E3102y Partial differential equations
Lect: 3. 3 pts. Professor Bal.
Prerequisite: MATH E1210 or the equivalent.
Introduction to partial differential equations; integral
theorems of vector calculus. Partial differential
equations of engineering in rectangular, cylindrical,
and spherical coordinates. Separation of the
variables. Characteristic-value problems. Bessel
functions, Legendre polynomials, other orthogonal
functions; their use in boundary value problems.
Illustrative examples from the fields of electromagnetic
theory, vibrations, heat flow, and fluid
mechanics.
APMA E3900x and y Undergraduate research
in applied mathematics
0 to 4 pts. Members of the faculty.
This course may be repeated for credit, but no
more than 6 points may be counted toward the
satisfaction of the B.S. degree requirements.
Candidates for the B.S. degree may conduct an
investigation in applied mathematics or carry out
a special project under the supervision of the
staff. Credit for the course is contingent upon the
submission of an acceptable thesis or final report.
APMA E4001y Principles of applied mathematics
Lect: 3. 3 pts. Professor Schmitt.
Prerequisites: Introductory linear algebra required.
Ordinary differential equations recommended.
Review of finite-dimensional vector spaces and
elementary matrix theory. Linear transformations,
change of basis, eigenspaces. Matrix representation
of linear operators and diagonalization.
Applications to difference equations, Markov
processes, ordinary differential equations, and stability
of nonlinear dynamical systems. Inner product
spaces, projection operators, orthogonal bases,
Gram-Schmidt orthogonalization. Least squares
method, pseudo-inverses, singular value decomposition.
Adjoint operators, Hermitian and unitary
operators, Fredholm Alternative Theorem. Fourier
series and eigenfunction expansions. Introduction
to the theory of distributions and the Fourier
Integral Transform. Green’s functions. Application
to partial differential equations.
APMA E4101x Introduction to dynamical
systems
Lect: 3. 3 pts. Professor Weinstein.
Prerequisites: APMA E2101 (or MATH V1210) and
APMA E3101 or their equivalents, or the instructor’s
permission. An introductiion to the analytic
and geometric theory of dynamical systems; basic
existence, uniqueness, and parameter dependence
of solutions to ordinary differential equations; constant
coefficient and parametrically forced systems;
fundamental solutions; resonance; limit points,
limit cycles, and classification of flows in the plane
(Poincare-Bendixson Theory); conservative and
dissipative systems; linear and nonlinear stability
analysis of equilibria and periodic solutions; stable
and unstable manifolds; bifurctions, e.g. Andronov-
Hopf; sensitive dependence and chaotic dynamics;
selected applications.
APMA E4150x Applied functional analysis
Lect: 3. 3 pts. Professor Courdurier.
Prerequisites: Advanced calculus and a course in
basic analysis, or the instructor’s permission.
Introduction to modern tools in functional analysis
that are used in the analysis of deterministic and
stochastic partial differential equations and in the
analysis of numerical methods: metric and
normed spaces, Banach space of continuous
functions, measurable spaces, the contraction
mapping theorem, Banach and Hilbert spaces,
bounded linear operators on Hilbert spaces and
their spectral decomposition, and (time permitting)
distributions and Fourier transforms.
APMA E4200x Partial differential equations
Lect: 3. 3 pts. Instructor to be announced.
Prerequisite: a course in ordinary differential equations.
Techniques of solution of partial differential
equations. Separation of the variables. Orthogonality
and characteristic functions, nonhomogeneous
boundary value problems. Solutions in orthogonal
curvilinear coordinate systems. Applications of
Fourier integrals, Fourier and Laplace transforms.
Problems from the fields of vibrations, heat conduction,
electricity, fluid dynamics, and wave propagation
are considered.
APMA E4204x Functions of a complex variable
Lect: 3. 3 pts. Professor Polvani.
Prerequisite: MATH V1202 or the equivalent.
Complex numbers, functions of a complex variable,
differentiation and integration in the complex
plane. Analytic functions, Cauchy integral theorem
and formula, Taylor and Laurent series, poles and
residues, branch points, evaluation of contour
integrals. Conformal mapping. Schwarz-Christoffel
transformation. Applications to physical problems.
APMA E4300y Introduction to numerical methods
Lect: 3. 3 pts. Instructor to be announced.
Prerequisites: MATH V1201, MATH E1210, and
APMA E3101, or their equivalents. Some programming
experience and Matlab will be extremely
useful. Introduction to fundamental algorithms
and analysis of numerical methods commonly
used by scientists, mathematicians, and engineers.
Ths course is designed to give a fundamental
understanding of the building blocks of
scientific computing that will be used in more
advanced courses in scientific computing and
numerical methods for PDEs. Topics include
numerical solutions of algebraic systems, linear
least-squares, eigenvalue problems, solution of
nonlinear systems, optimization, interpolation,
numerical integration and differentiation, initial
value problems, and boundary value problems for
systems of ODEs. All programming exercises will
be in Matlab.
APMA E4301x Numerical methods for partial
differential equations
Lect: 3. 3 pts. Professor Keyes.
Prerequisites: APMA E4300 and E3102 or E4200,
or their equivalents. Numerical solution of partial
differential equtions (PDE) arising in various physical
fields of application. Finite difference, finite
element, and spectral methods. Elementary finite
volume methods for conservation laws. Time
stepping, method of lines, and simultaneous
space-time discretization. Direct and iterative
methods for boundary-value problems. Applied
numerical analysis of PDE, including sources of
numerical error and notions of convergence and
stability, to an extent necessary for successful
numerical modeling of physical phenomena.
Applications will include the Poisson equation, heat
equation, wave equation, and nonlinear equations
of fluid, solid, and gas dynamics. Homework
assignments will involve substantial programming.
AMCS E4302x Parallel scientific computing
Lect: 3. 3 pts. Offered in alternate years.
Not given in 2008–2009.
Prerequisites: APMA E3101, E3102, and E4300,
or their equivalents. Corequisites: APMA E4301
and programming ability in C/C++ or FORTRAN/F90.
An introduction to the concepts, the hardware and
software environments, and selected algorithms
and applications of parallel scientific computing,
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SEAS 2008–2009