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