From: Chairman Maoi
Newsgroups: sci.math
Subject: "AGGREGATION" OF SEVERAL VARIABLES INTO FEWER VARIABLES
Date: Mon, 13 Jul 1998 13:56:05 -0700
I'm posting this message for a friend of mine. Please forward responses
to him at:
marschak@socrates.berkeley.edu
This query has to do with a smooth (C-infinity) function of n real
variables, m of which can be "aggregated" into r < m variables. (This
arises
in mathematical economics, specifically in models of efficient
communication
in organizations).
Given: an open set Q in the reals and a smooth real-valued function
F on the n-fold Cartesian product of Q. When do there exist smooth
real-valued functions G,A_1,...,A_r such that
F(x_1,...,x_n) =
G(A_1(x_1,...,x_m),...,A_r(x_1,...,x_m),x_{m+1},...x_n),
where r r.
If that's not the case, then I'm stuck.
EXAMPLE: n=4
F =(x_1 x_2 x_3 x_4) + (x_1+x_2+x_3+x_4) + (1/x_1 + 1/x_2 +
1/x_3 + 1/x_4)^2.
CONJECTURE: there do not exist smooth real-valued functions G,A_1,A_2
such that
F = G(A_1(x_1,x_2,x_3), A_2(x_1,x_2,x_3) , x_4).
Here r=2, m= 3, n-m+1 = 2. The bordered Hessian has two columns and so
the
necessary condition "rank at most 2" is incapable of being violated. The
sufficient condition "rank exactly 2 when we delete the "F" column" is
incapable of being satisfied.
How, then, can one approach the conjecture?
[Note that if we remove the exponent 2 from the third term, then we can
do
it: We let A_1 = x_1 x_2 x_3, A_2 = x_1 + 1/x_1 + x_2
+1/x_2 + x_3 + 1/x_3
and G(A_1,A_2,x_4) = x_4 A_1 + x_4 + 1/x_4 + A_2. ]
Any suggestions, or citations, would be gratefully received.
Tom
(Haas School of Business, univ. of Calif., Berkeley)
marschak@socrates.berkeley.edu
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: "AGGREGATION" OF SEVERAL VARIABLES INTO FEWER VARIABLES
Date: 20 Jul 1998 21:41:30 GMT
In article <35AA7465.BA9F5F15@popd.netcom.com>,
>Given: an open set Q in the reals and a smooth real-valued function
>F on the n-fold Cartesian product of Q. When do there exist smooth
>real-valued functions G,A_1,...,A_r such that
>
> F(x_1,...,x_n) =
> G(A_1(x_1,...,x_m),...,A_r(x_1,...,x_m),x_{m+1},...x_n),
>
>where r R and want to know when F = G o (A o pi_1, pi_2) for
some G and A, where pi_1 and pi_2 are the projections onto the factors.
Here U is an open subset of R^m and V a subset of R^(n-m) (respectively
Q^m and Q^(n-m) in your notation).
Suppose there were such functions A_i and G. Then for any fixed p0 in U,
consider the points p in U having F(p,q) = F(p0,q) for all q. On the one
hand, this set includes all the points p with A(p) = A(p0); since A is
assumed to be a smooth mapping R^m -> R^r with r < m, this set is a
(m-r)-dimensional manifold, almost everywhere in a precise sense (Sard's
theorem). On the other hand, the set of equations F(p,q) = F(p0,q) (for q in
V) describe a smooth variety whose dimension is m - dim span { dF/dp } where
the derivatives are evaluated at p=p0 and q=arbitrary in V. So the
condition that A and G exist seems to be that the vectors dF/dp be
almost everywhere of rank only r; equivalently, that there exist for almost
every p0 at least m-r linearly independent vectors v_i(p) perpendicular
to dF/dq for all q.
So in response to the question
>EXAMPLE: n=4
>
> F =(x_1 x_2 x_3 x_4) + (x_1+x_2+x_3+x_4) + (1/x_1 + 1/x_2 +
>
> 1/x_3 + 1/x_4)^2.
>
>CONJECTURE: there do not exist smooth real-valued functions G,A_1,A_2
>such that
>
> F = G(A_1(x_1,x_2,x_3), A_2(x_1,x_2,x_3) , x_4).
the answer is that such A_i cannot exist: if they did, we would need for
almost every p=(x_1,x_2,x_3) a nonzero vector v(p) = (v_1,v_2,v_3)
perpendicular to (dF/dx_1, dF/dx_2, dF/dx_3) for all x_4. This in turn
would certainly require that v(p) be perpendicular to this derivative
for x_4 = 1, 2, and 3, a condition which is equivalent to the vanishing
of a 3x3 determinant, and which Maple tells me can only happen if
some x_i = x_j, and which therefore certainly does not occur almost
everywhere in U.
Your original post considered the rank of the matrix of second derivatives
d^2F/ dp dq. This roughly speaking amounts to the same condition I have
described above: if the vectors dF/dp stay in a subspace, then of course
the derivatives d(dF/dp)/dq stay there as well, and the Hessian matrix
has low enough rank. Conversely, if the Hessian has low rank, then the
vectors dF/dp show no first-order change as the coordinates of q
change; this does not quite imply that all the values of dF/dp lie in
a proper subspace, although I didn't see a counterexample offhand if
the Hessian is assumed to have low rank for _all_ p.
dave