From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math.num-analysis
Subject: Re: Hilbert's 13th conjecture
Date: 9 Dec 1998 23:49:52 GMT
Yuan Si wrote:
>Does anyone have any idea about Hilbert's 13th conjecture?
>
>I am interested in the approximation of a two-variable function f(x,y) by
>using single variable functions Xi(x) and Yi(y), i=1,2,....
>
>It will be highly appreciated if anyone can recommend some references.
Well, you need _some_ functions of two variables; you can't express
f(x,y) = x + y by putting x and y into a stack and then
continuing to apply functions of _one_ variable to the top element of
the stack.
But that's really all you need. Allow me to quote verbatim from
"Mathematical Developments Arising from the Hilbert Problems",
Proc Symp Pure Math AMS, vol 28 (1976) p 420 (G.G. Lorentz, section author):
Observing that there appears to be no 2-variable function really different
from addition, he writes
"The astonishing result of Kolmogorov (1957) confirms this. Kolmogorov proved:
Theorem A: There exist fixed continuous increasing functions phi_{p,q}(x)
on [0,1] so that each continuous function f on I^n can be written in
the form
f(x1, ..., xn) = Sum_{q=1}^{2n+1} g_q( Sum_{p=1}^{n} phi_{p,q}(x_p) )
where g_q are properly chosen continuous functions of one variable."
He goes on to comment about improvements in the nature of the g_q and
phi_{p,q}. Thus Hilbert's 13th conjecture is false in the continous category.
On the other hand, it's true in the smooth category. Quoting again:
"Theorem B. [For each r >=1, ] there are r-times continuously differentiable
functions of n >=2 variables not representable by superpositions of
r-times continuously differentiable functions of less than n variables..."
It's not at all clear that you really have a project which is related to
Hilbert's problem. Perhaps you can clarify just what you'd like to do?
dave
==============================================================================
From: "Yuan Si"
To: "Dave Rusin"
Subject: Re: Hilbert's 13th conjecture
Date: Thu, 10 Dec 1998 10:31:57 +0800
>It's not at all clear that you really have a project which is related to
>Hilbert's problem. Perhaps you can clarify just what you'd like to do?
>
Thank you very much for your kind response. Actually, I am not a
mathematician. Instead, I am more engineering oriented. My research area
is computational mechanics. One of my recent interests is the extended
Kantorovich method originally proposed by A. D. Kerr. The method is roughly
like the following.
Take the Poisson equation as an example. we assume the solution to be of
the form
u(x,y) = X_1(x)Y_1(y) + X_2(x)Y_2(y) + ... + X_n(x)Y_n(y)
or in vector form
u(x,y) = transpose{X} {Y}
First, we assume the variation of {X} and solve for {Y} by the classical
Kantorovich method. And then we fix {Y} and relax {X} to be unknown and
solve for {X} by another round of Kantorovich method. This procedure is
repeated until the solution converges to an satisfactory level. Kerr only
used one term while we included more terms. Practical numerical experiments
have shown that in most cases 3-4 terms are quite sufficient to give nearly
exact solutions, which leads us expecting more from this method.
Essentially, this method solves a PDE by using a finite number of ODEs and,
it seems to me, is essentially the same idea of approximating a 2-variable
function using single-variable functions. If a 2-var. function can be
represented exactly by finite number of 1-var. functions, then that implies
the solution of a PDE can be exactly given by solution of a finite number of
ODEs. Is that a dream or a future reality?
Any comments will be highly appreciated.
S. Yuan
P.S.
For your information, hear are two of our recent related papers (in
English):
Si Yuan, Yan Jin; "Computation of elastic buckling loads of rectangular thin
plates using the extended Kantorovich method", Computers & Structures, 1998,
Vol. 66, No. 6, pp. 861-867.
Si Yuan, Yan Jin, Fred W. Williams; "Bending Analysis of Mindlin Plates by
Extended Kantorovich Method", Journal of Engineering Mechanics, ASCE,
Dec.,1998, Vol. 124, No. 12, pp. 1339-1345.