Re: Re: A Sum-like notation for iteration

• To: mathgroup at smc.vnet.net
• Subject: [mg102438] Re: [mg102415] Re: A Sum-like notation for iteration
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Sun, 9 Aug 2009 06:06:44 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <h5bk7g\$hn3\$1@smc.vnet.net> <h5ebii\$1rc\$1@smc.vnet.net> <200908080840.EAA01381@smc.vnet.net>

```I don't recall whether anybody has pointed out the following in connect
with this thread.  Another possibility for such things is to use the
EllipsisSequence package that is part of David Park's Presentations
application:

http://home.comcast.net/~djmpark/DrawGraphicsPage.html

For example:

Needs["Presentations`EllipsisSequences`"]

mysum = iSum[1/i^2, {i, 1, 2, , k, Null}, tag1]
1 + 1/4 + ... + 1/k^2 + ...
(*  output displays in 2D form *)

mysum // iSumNormalize[tag1, {i, 1, Infinity}] // Simplify
Pi^2/6
(* output displays in 2D form, with the symbol for Pi there *)

That's an example with an infinite sum, but finite sums are handled in a
similar way.

For writing, one can thereby keep the sum displayed, but evaluate the
sum when required.  Products with ellipses are handled similarly.  And
one can use sequences with ellipses, too.

magma wrote:
> Being neither Greek nor Trojan, but rather a Roman, I shall try to
> bring a Pax Romana between the 2 arguing parties :-)
>
> 1- The Theorema project is a great Mathematica package about which I would
> like to see much more discussion. Indeed the logicographic symbols
> technology is an excellent example of the kind of sophisticated and
> innovative notation that is possible to achieve in Mathematica.
> As Carlos correctly points out, they invented a notation where no
> standard notation existed before. So why not for Table too?
>
> 2-Carlos, can we see what you have done - "up to 3 iterators" - so
> far ? Please post the code. I always enjoy reading some nice notation-
> tweaking before dinner.
> The Notation package has some limitations as David Park first pointed
> out to me 1.5 years ago. It's OK for little things, but the real
> connoisseurs do not use it, preferring to sculpt their notations from
> virgin Boxes. Of course "boxing" is not for the puny, so beginners
> should stick to the Notation Package until their Mathematica muscles are
> strong enough to enter the ring.
>
> 3- But then...why stop at Table? The Underoverscript notation could be
> used also for other "iterative" constructions. For example the Plot-
> type of things.
>
> 4-  "Timeo Danaos et dona ferentis" (Latin: I fear the Greeks even
> bearing gifts) says Lacoon in Virgil's Aeneid, but pfalloon (who
> rhymes with Lacoon :-)) does not need to be so cautious in this case.
> Introducing an extra 2D notation for Table & Co. through the gates of
> Mathematica, will not bring down its powerful walls.
> Users who do not like it , will simply avoid it in input, plodding
> along with the usual 1D notation, while the more visually driven ones
> will boldly press another button in a palette to create the 2D
> template.
> The (unevaluated) output could be 1D or 2D as is now with Sum (though,
> it seems to me that unevaluated Sums are always displayed 2D,
> regardless of how you input them.
> see for ex.
> Sum[f[i], {i, 1, Infinity}]
> but there might be other situations where the output is 1D)
>
> Quote: how do we format something like Table[f[x], {x, 1, 100, 2}]?
> Or,
> worse, Table[f[x], {x, {1,2,3,4,5}}]? Whatever solution is chosen, the
> point is that it is no longer "intuitive", at least not in the sense
> that one could readily guess what it should look like.
>
> 5- Sum and Table use the standard Mathematica iteration specification.
> If we look carefully, we will see that Sum offers a 2D notation only
> for some simple iterator forms, the most general being {i,imin,imax}.
> For more complicated type of iteration - for example with a step size
> - there is no 2D notation supported by Sum. So the same limitation
> could apply to Table
>
> Quote:The whole point of using 2D inputs (for powers,
> fractions, named special functions, sums, products, and so on) is that
> they should be general constructs recognized universally.
>
> 6- The brain of T-Rex had special receptors for movement detection so
> they could capture hiding preys, the brain of the Mathematician has
> special receptors for "form" detection, so they can capture hiding
> symmetries.
> There is hardly any "form" in 1D. So Mathematicians created 2D "forms"
> to please their minds. That is the whole point of 2D notations.
> So there is no point in denying them this whole point :-)
>
> Summing up point 1....6  (no pun intended....really!): a more visually
> appealing notation for "iterative" constructs like  Table, Plot, ect
> is certainly welcomed. It does not need to interfere with the 20 year
> old habits of the "flatliners" and it can and should be developed
> right away without waiting for the uncertain approval/implementation
> of the Royal House.
>
>

--
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

```

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