# Objectives of the Cartan-Lie Group

For more on
Elie Cartan

For more on
Sophus Lie

This is a small working group which started meeting in the Fall of 1995.
The goal is to discuss research problems
in computer science involving
In the Fall semester 95, we
tried to see how the theory of Lie groups could be applied
to certain problems in computer vision and motion
planning. This led us to
study differential forms. This study was continued in the
Spring, with fibre bundles and connections.
Other topics were considered later on.

This semester (Spring 2000), we are planning to meet
every Friday, 4:03-4:33pm, in Moore 554

I will need volonteer speakers.
As in the past, we will also hear people
talk about their current pet problems.

## Next Meeting:

Friday April 21, 3:03-4:33pm

Moore 554

## Topic:

Structure of Linear Shift-invariant Operators on Sequences
## Speaker:

Paul Hughett

University of Pennsylvania, Neuropsychiatry Section
Abstract:
The theory of digital signal processing is based on three fundamental
results about the structure of a linear shift-invariant (LSI) operator
on sequences. (1) The action of any LSI operator can be represented
as convolution by its impulse response. (2) Every complex exponential
sequence is an eigenfunction for every LSI operator. (3) The
frequency response of an LSI operator is the Z-transform of its
impulse response. Recent work has shown, however, that the
traditional proofs of these facts are not valid if non-causal
operators or two-sided sequences are admitted and that it is possible
to construct LSI operators not representable by their impulse
responses and whose frequency responses are unrelated to their impulse
responses. (On the other hand, the exponential eigenfunction property
can be shown rigorously, even though the traditional proof is
invalid.) In this talk, I will exhibit counterexamples to the
traditional structure theorems, explain conditions under which the
traditional theorems (or close analogs) remain valid, and discuss the
structure of general LSI operators on two-sided sequences.

## Old Problems

1. Find an efficient method for defining triangular surface splines
meeting with C^k-continuity. Find some kind of de Boor algorithm
(contributed by Jean Gallier).
2. Find an efficient method for determining the intersection of a quadric
(say a sphere) with a parametric surface of bidegree <2, 2>
(contributed by Deepak Tolani).

## To Join the Cartan-Lie Group:

If you want to be added to (or dropped from!) the mailing list, please contact
Jean Gallier: jean@saul.cis.upenn.edu

Jean Gallier's home page

A Cartan groupie (courtesy of
Andy Hicks)