Vol. 43, No. 3, 1972

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Invariant functions of an iterative process for maximization of a polynomial

Peter F. Stebe

Vol. 43 (1972), No. 3, 765–783
Abstract

Let P be a polynomial with real non-negative coefficients and variables xij,i = 1,,k,j = 1,,ni. Let d = k1ni. Let Rd be the d-dimensional real vector space. Let M be the subset of Rd defined by

                       ∑i
M˜ = {x|x ∈ Rd,xi,j ≧ 0,n   xi,j = 1}
j=1

where the symbols xi,j denote the components of x. If x is a vector in the interior of M, define τ(x) as the vector in M with components xi,jgiven by

         x  ∂∂Px-
x′i,j = ∑i--i,j--ij----.
h=1x  ∂∂xPi,h n
ih

The expression on the right is evaluated at x. The transformation τ is defined on the boundary of M by the same formula if the denominators do not vanish.

Let F be the set of fixed points of τ in M. It is shown that if τ is a homeomorphism of M onto itself, there is a set of dk functions f1, , fdk defined on MF such that fi(x) = fi(τ(x)) for x MF. The functions fi are continuous and independent on an open dense subset of MF. Explicit expressions for certain invariant functions are also obtained.

Mathematical Subject Classification 2000
Primary: 58E15
Secondary: 54H15
Milestones
Received: 18 February 1971
Revised: 20 July 1972
Published: 1 December 1972
Authors
Peter F. Stebe