Tien Yien Li

Period Three Implies Chaos

The way phenomena or processes evolve or change in time is often described by differential equations or difference equations. One of the simplest mathematical situations occurs when the phenomenon can be described by a single number as, for example, when the number of children susceptible to some disease at the beginning of a school year can be estimated purely as a function of the number for the previous year. That is, when the number x n+1, at the beginning of the n + 1st year (or time period) can be written

A Constructive Proof of the Brouwer Fixed-Point Theorem and Computational Results

{x_{n + 1}} = F({x_n}),

Numerical Solution of Polynomial Systems by Homotopy Continuation Methods

(1.1) where F maps an interval J into itself. Of course such a model for the year by year progress of the disease would be very simplistic and would contain only a shadow of the more complicated phenomena. For other phenomena this model might be more accurate. This equation has been used successfully to model the distribution of points of impact on a spinning bit for oil well drilling, as mentioned if [8, 11] knowing this distribution is helpful in predicting uneven wear of the bit. For another example, if a population of insects has discrete generations, the size of the n + 1st generation will be a function of the nth. A reasonable model would then be a generalized logistic equation

The cheater's homotopy: an efficient procedure for solving systems of polynomial equations

{x_{n + 1}} = r{x_n}[1 - {x_n}/K].

A Rank-Revealing Method with Updating, Downdating, and Applications

(1.2)

Asymptotic periodicity of the iterates of Markov operators

Let P ( x ) = 0 be a system of n polynomial equations in n unknowns. Denoting P = ( p 1 ,…, p n ), we want to find all isolated solutions of for x = ( x 1 ,…, x n ). This problem is very common in many fields of science and engineering, such as formula construction, geometric intersection problems, inverse kinematics, power flow problems with PQ-specified bases, computation of equilibrium states, etc. Elimination theory-based methods, most notably the Buchberger algorithm (Buchberger 1985) for constructing Grobner bases, are the classical approach to solving (1.1), but their reliance on symbolic manipulation makes those methods seem somewhat unsuitable for all but small problems.