Turns out Google gives 693 hits for "product integral", and it appears that the concept was first invented by Volterra in 1887. --Dan
=Daz Turns out Google gives 693 hits for "product integral", and it appears that the concept was first invented by Volterra in 1887.
Cool. There's some interesting papers out there (eg http://www.math.uu.nl/people/gill/Preprints/prod_int_0.pdf ) Browsing them reminds me: I forgot to mention that you can extend prodigals to matrices (with sufficient care--guess you have to use the analog of Lebesgue-Stieltjes integration or something). I wonder if there are continuous versions of Gosper's path-invariant (matrix) (summation) identities? This suggests considering prodigals upon encountering discrete products of almost ANY kind. The lack of accepted terminology and notation for these simple ideas surely limits insight and discourse.
Marc LeBrun <mlb@fxpt.com> wrote:
I wonder if there are continuous versions of Gosper's path-invariant (matrix) (summation) identities?
Back in the early eighties I showed how to view Gosper's path-invariant formalism in terms of product integration. -Bill Dubuque
At 04:18 PM 11/7/2002 -0800, Marc LeBrun wrote:
I forgot to mention that you can extend prodigals to matrices (with sufficient care--guess you have to use the analog of Lebesgue-Stieltjes integration or something).
This seems to be related to a way to handle continuous composition of rotations, for example. If you have two rotation matrices A and B, they don't in general commute, but lim n->oo ((A^1/n)(B^1/n))^n = lim n->oo ((B^1/n)(A^1/n))^n. [ One way to see this is that lim n->oo A^1/n = lim n->oo B^1/n = Identity ] How is this related to the product integral? If A and B are thought of as matrix-valued functions of a parameter t where, e.g., A(1/n) ^ n = A(1), B(1/n) ^ n = B(1), which makes easy to see geometric sense for constant rate rotations, then the product integral P (A(t)B(t))^dt seems to be the continuously combined rotation (at least according to one way of defining it), and seems to be equal to P (B(t)A(t))^dt. Taking the definite product integral between 0 and 1 would give the same value as the limits above. (I guess there's a multiplicative constant of integration in the indefinite product integral). Perhaps this extends to more general matrices, though it is easier to visualize for rotations (first, at least). Also, does this suggest that there is any more general ability to rearrange what would be ordinarily be non-commutative operators "under the P"? Shel
At 01:52 PM 11/8/2002 -0800, I wrote:
then the product integral P (A(t)B(t))^dt seems to be the continuously combined rotation (at least according to one way of defining it), and seems to be equal to P (B(t)A(t))^dt.
I also see that this doesn't quite work out according to the way of defining things that is most similar to the ordinary integral (because (AB)^dx is not the same as (A^dx)(B^dx), although it would be awfully nice to have some operator that works like this, for "simultaneous" applications of transformations.
The Putnam is really getting tough. When I took it, we had several hours. Steve Gray
I'd like to say a couple of things about "prodigals". A. Countable products in non-abelian groups are also quite interesting and useful. Absolute convergence makes sense: given a left-invariant metric d on the group, you can ask that the sum of d(identity, g_i) converges. With this condition, if the group is complete w.r.t. the metric, then the product converges in any order, but the answer depends on the ordering. The right way to define these products is to allow an index set with **any** countable linearly order, not just the natural numbers. E.g. there are a number of important constructions in topology that depend on the product using the natural order on the negative integers. I've had occasion to use this construction for sets of group elements indexed by an order-dense set, e.g. the rational numbers. You get an uncountable, but not arbitrary, set of values for the product of an absolutely convergent countable set of group elements, depending on the order. (There are probably additional condiitons needed to guarantee convergence in general, but if the metric is say a Riemannian metric on a Lie group I think absolute convergence is enough.) B. I don't want to discourage this nice discussion of prodigals, but just inject a little mainstream mathematical terminology. Another name for the integral-like product in GL(n,C) is linear differential equation (or if you prefer, a system thereof), which may or may not have constant coefficients. The infinitesimal elements of a Lie group constitute its Lie algebra. In the constant coefficient case, the prodigal is also called the exponential map, and it has been extensively studied in many variations. In the variable-coefficient case, these are the same as connections of G-bundles over an interval. The Campbell-Hausdorff formula is a power series expressing corrections to commutativity, as below. The curvature of a connection of a G-bundle over a square is one well-developed differential geometry context to understand non-commutativity in generally. Control theory is another area that addresses many types of questions about this stuff. I think this discussion is partly symptomatic of the malaise in mathematics that things like differential equations are usually taught in a formal way that not many students internalize. On Sat, Nov 09, 2002 at 04:31:33AM -0800, Shel Kaphan wrote:
At 01:52 PM 11/8/2002 -0800, I wrote:
then the product integral P (A(t)B(t))^dt seems to be the continuously combined rotation (at least according to one way of defining it), and seems to be equal to P (B(t)A(t))^dt.
I also see that this doesn't quite work out according to the way of defining things that is most similar to the ordinary integral (because (AB)^dx is not the same as (A^dx)(B^dx), although it would be awfully nice to have some operator that works like this, for "simultaneous" applications of transformations.
participants (6)
-
asimovd＠aol.com -
Bill Dubuque -
Bill Thurston -
Marc LeBrun -
Shel Kaphan -
Steve Gray