I call the rule AB.CD = AC.BD Mid-Quarter-Swap. The axioms AA = A, AB = BA, and AB.CD = AC.BD are complete for identities involving the ordinary two-input Average. Distributivity, AB.C = AC.BC, follows easily. Dropping commutativity covers weighted averages with an unspecified weight W. Specific W will satisfy an additional axiom or two. For W = 2/3, Avg(A,B) = (A+2B)/3. additional axiom: AB.BC = AC.CB For W = 1/phi = .618..., the additional axiom is A.BC = C.BA, one of the Dozen Cousins of Associativity. If we introduce Cancellation, [ AX = AY -> X = Y ], into the regular Average axioms, then Inverse, [ AX=B has a solution X ] follows for finite systems, and we can proceed to define + . This leads to odd-order abelian groups. Three-input average axioms are AAA=A, ABC=BAC=BCA, ABC.DEF.G = ABD.CEF.G Contrast with Median (aka Majority): AAB=A, ABC=BAC=BCA, AB.CDE = ABC.ABD.E Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of R. William Gosper Sent: Sat 4/30/2005 1:35 AM To: math-fun@mailman.xmission.com Subject: [math-fun] elliptic mean
Wrt Means, Gosper's objection was a bit more subtle: should we require that
avg( avg(a,b), avg(c,d) ) = same thing with b and c swapped? Bingo. "Association" was a poor choice of words. Dyadic symmetry?
This *is* true for ordinary averages, and geometric, harmonic, etc. means. And the resulting symmetric expressions make it obvious how to define avg(a,b,c). A nonobvious def in terms of dyadic avg is the root of
avg(avg(a,b),avg(c,x)) = x . This does *not* work with elliptic mean. One plausible def is to iterate [a,b,c] <- [em(b,c),em(c,a),em(a,b)]. But I think em should be classified among the unscrupulous means. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun