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July 19

Extrema Finding and Integral Transforms

Is there a transform which "simplifies" finding of minima or maxima? Or to put the question more generally, what are the "frequency domain" analogues for extrema finding for various transforms, such as the Fourier, Laplace, and Mellin transforms? (Or any others, if a straightforward analogue exists.) I'm looking at a class of non-trivial minimization problems, and am wondering if a transform might help with conceptualizing them, much like the Fourier transform helps with conceptualizing wave behavior with its time domain/frequency domain duality. -- 76.201.158.47 (talk) 00:37, 19 July 2009 (UTC)[reply]

Almost certainly not, without extra information. A local minimum or maximum is a local property of the function (that is to say, dependent on behavior within a small neighborhood of the point), and just about any integral transform (that is to say, the family of transforms involving integration against some kernel) "smears" that information out across the entire area. If you're looking for a "transform" that only takes into account local information, you're really talking about something like taking a derivative. Ray^Talk 02:23, 19 July 2009 (UTC)[reply]

Note that the transformations you are quoting are linear ones, that is, quite too rigid to really change the view of a nonlinear problem. In any case there is no transformation that works like a panacea for all minimization problems. Nevertheless, special classes of problems may possibly be treated by a special transformation. The first important example of what you have in mind is maybe the use of the Legendre-Fenchel transformation in convex minimization; for instance it is the way one passes from the Lagrangian to the Hamiltonian formalism in mechanics.--pma (talk) 06:56, 19 July 2009 (UTC)[reply]

Π-like notation for tupling

So I find myself wanting to write some ordered n-tuples of the form $(a_{1},\ldots ,a_{n})\in S_{1}\times \cdots \times S_{n}$ , except that in place of $a_{i}$ I have a longer expression, long enough that I don't want to write it twice, and the context in which this appears is such that I can't easily write a separate "where" clause as I usually would ( $(a_{1},\ldots ,a_{n}){\mbox{ where }}a_{i}=\cdots$ ). If I had a similar problem on the right hand side I could solve it easily by writing $\prod _{i}\,\!S_{i}$ , but I don't think I've ever seen an analogous notation for the elements of a Cartesian product, even though one ought to exist. $({\underset {i}{,}}\,a_{i})$ just looks silly. Has any prominent source ever defined a notation for this, or, failing that, can anyone suggest something that looks good? I have the full resources of LaTeX at my disposal. -- BenRG (talk) 11:35, 19 July 2009 (UTC)[reply]

How about

(a_{i})_{1\leq i\leq n}

?. -- Meni Rosenfeld (talk) 12:35, 19 July 2009 (UTC)[reply]

I once used (X_i : i ∈ B) in a published paper.

(X_{i}:i\in B)\,

The idea was that B was some subset of the index set {1, ..., n). E.g. if B = {2, 4, 9} then

(X_{i}:i\in B)=(X_{2},X_{4},X_{9}).\,

That may not be exactly what you need, since in the "tuples" I was using the order didn't actually matter and since B was simply a set, I couldn't have written

(X_{4},X_{2},X_{9})\,

in this notation. But I didn't need that. Maybe you can play with variations on this theme and find one that suits your purpose. I didn't explain the notation; it seemed self-explanatory in the context in which I used it. Michael Hardy (talk) 14:38, 19 July 2009 (UTC)[reply]

Thanks. I think I'll go with

(a_{i})_{i\in B}\in \prod _{i\in B}S_{i}

where

B=\{1,\ldots ,n\}

, which is compact enough for my purposes. But it still bugs me that the notation on the left doesn't parallel the notation on the right as it should. -- BenRG (talk) 20:04, 19 July 2009 (UTC)[reply]

Sometimes it's convenient to adopt the function notation for n-tuples, that, is just

\textstyle a\in \prod _{i\in B}S_{i}

, with

\textstyle a(i)

instead of

\textstyle a_{i}

. Precisely,

\textstyle a:B\to \coprod _{i\in B}S_{i}

as a section of the natural

\textstyle \coprod _{i\in B}S_{i}\to B

. --pma (talk) 21:14, 19 July 2009 (UTC)[reply]

The function notation is not just notation, it's what elements of the Cartesian product are according to its definition. — Emil J. 12:34, 20 July 2009 (UTC)[reply]

I don't understand the question. Could you be looking for the projection map $\mathrm {proj} _{j}\!$ ? 67.117.147.249 (talk) 23:28, 19 July 2009 (UTC)[reply]

I doubt that's what was meant. Michael Hardy (talk) 03:46, 20 July 2009 (UTC)[reply]