Non-linear Dynamics and Applied Mathematics
We will investigate a multi-degree-of-freedom Duffing system.
In the 'stiffness' part a stochastic parameter will account for
possible measurement error. We shall focus on the first order
perturbation and a Neumann expansion of the matrix operator.
A uniform distribution will
be assumed to compute the statistical moments of responses
up to kurtosis. The response correlation matrix will be analyzed.
Ordinary differential equations, matrix algebra, and probabilistic
calculations will be integrated in a symbolic computer program written in
Mathematica.
Extensive graphics will be employed to display results Possible extension of attaching modules for other probability
distribution, use of sound to auralize covariances and remote
computing over the internet will be undertaken in future projects.
Professor Gautam Dasgupta of Civil Engineering and Engineering
Mechanics; Graduate student leader Elisabeth Malsch
students: Martin Andersen , Kareen Rozen
, Andrew Rudman , Smita Sihag
Other Projects in the Series
<- ! --- From: Henry Pinkham
To: dea13@columbia.edu, ajs87@columbia.edu, jjb53@columbia.edu,
msa25@columbia.edu, arr48@columbia.edu, ss767@columbia.edu,
John Conley , Scott Meltzer ,
David Kagan , Darya Krym ,
max lip , kr178@columbia.edu
Cc: goldfeld@cpw.math.columbia.edu, krichev@cpw.math.columbia.edu,
neumann@cpw.math.columbia.edu, Gautam Dasgupta ,
psorin@cpw.math.columbia.edu, bayer@cpw.math.columbia.edu,
greene@phys.columbia.edu, abhijit@cpw.math.columbia.edu, eam27@columbia.edu,
ytian@cpw.math.columbia.edu, ad@cpw.math.columbia.edu,
chuckh@phys.columbia.edu
Subject: VIGRE research projects --- !- >
- Complex Analysis
Professor Igor Krichever. Graduate student leader: Anton Dzhamay:
ad@math.columbia.edu
This project centers around the Riemann mapping theorem, which says
that any simply connected domain U in the complex plane, which is not
the entire complex plane, is analytically isomorphic to the unit
disk. Recent work of Wiegmann and Zabrodin shows that if U is bounded
by an analytic curve, then it satisfies an interesting differential
equation. This seminar will discuss generalizations of this result.
Preferred background: undergraduate complex analysis.
- Dimensional Topology
Professor Walter Neumann. Graduate student leader: Abhijit
Champanerkar
The project will address computation of so-called A-polynomials of
3-manifolds. The main prerequisites and techniques involve abstract
algebra. The necessary knowledge of three-manifold topology can be picked
up quickly. The A-polynomial is a very important recent invariant for
3-manifolds, but it has been hard to compute, and one of the issues would
be to experiment with numerical and computer algebra approaches to
computing it. It also appears to have deep connections with other
invariants. Experimental testing of such connections, as well as the
theoretical investigation of them, will be part of the project. The
project will thus involve experimentation using such programs as snap and
snappea for 3-manifold computation, and pari/gp, mathematica, macaulay,
and the like for algebraic computation. Some programming will be involved.
Even though the emphasis is computational geometry, the students will
learn about the beautiful interplay between algebraic geometry and
topology from the examples at hand.
- Grobner bases and algebraic geometry (not offered)
Professor Henry Pinkham; Graduate student leader: Ye Tian
The problem we will work on here is elementary, and can be stated
completely: let H be a subgroup of Zn (=the product of n copies of
the integers). Let M be the monoid generated by the intersection of H
with any orthant (= generalization to n dimensions of a quadrant in
the plane). The goal is to find a good algorithm for computing a
basis for M as a monoid, in other words, find a collection of vectors
v_1 ... v_m in M so that any element in M can be written as a_1 v_1 +
a_2 v_2 + ... a_m v_m, where the a_i are *positive* integers. This is
called a Hilbert basis. The current algorithms blow up very quickly.
The techniques used to study this problem involve an interesting
called a Hilbert basis. The current algorithms blow up very quickly.
The techniques used to study this problem involve an interesting
combination of linear algebra, ring theory, geometry and
combinatorics, especially that of polyhedra. A good algorithm would
have many applications in pure and applied mathematics. We will start
out by working through the relevant material in the text: "Ideals,
Varieties and Algorithms" by Cox-Little-O'Shea (2nd edition,
Springer-Verlag). The state of the art is given in either of two
books: "Using algebraic geometry" again by Cox-Little-O'Shea (GTM
185, 1998, Springer) or "Grobner Bases and Convex Polytopes" by B.
Sturmfels (AMS 1995). We will also look at the algorithm proposed by
Loic Pottier in a 3 page paper, "Euclid's algorithm in dimension n"
(ISSAC 96, ACM Press, July 1996). The computer component of the
project will be to use the computer program Macaulay to do
computations in rings.
Prerequisite: undergraduate algebra.
The seminars will get started around June 1 or earlier if the
participants prefer and housing is available. The first 5 or 6 weeks
the seminars will meet
Monday-Tuesday-Wednesday-Thursday. The students will do all of the
lecturing.
For more information contact Henry C. Pinkham .