Below are the reports for the original draft. All the
minor errors and typos pointed out below, and that were
justified, have been
corrected in the final version, that got published in the Elec.
J. of Comb.
---begin reports-------
From simpson@spider.math.ilstu.edu Mon Nov 28 12:33:32 1994
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From: simpson@spider.math.ilstu.edu
Date: Mon, 28 Nov 94 11:33:46 CST
Message-Id: <9411281733.AA03954@spider.math.ilstu.edu>
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To: zeilberg@euclid.math.temple.edu
X-Sender: simpson@spider.math.ilstu.edu
Status: RO
>Dear Doron,
Here's my checker's report. There were no problems with my sub^5lemma, but
I noticed a typo in one of its offspring. The cheque from Temple arrived
last week. Thanks, and thanks very much for your hospitality.
Best wishes, Jamie.
> I have now completed checking (sub)^5lemma 1.2.1.2.1.1.
>To the best of my judgment, it is correct, provided
>all its offspring (sub)^6-lemmas are correct.
>
>Needless to say, I don't know (and don't care!) whether
>the proofs of these offspring (sub)^[]-lemmas are actually
>correct.
>
>and (optional)
>
>I have found the following typos or other errors:
In the statement of sub^6lemma 1.2.1.2.1.1.2 (although this is none of my
business) "h" should be "i".
> Best Wishes,
>
> JAMIE SIMPSON
>
>P.S. I like the epithet you wrote for me
>
>OR
>
> I rather write my own, which is enclosed below (Up to four lines,
>in eight Roman, please.):[ ]
>
>
>
>---------end checkers form----------------
>
>
>
>
>
>
>
>
Jamie Simpson | Illinois State University
E-Mail: simpson@math.ilstu.edu | Mathematics Department
Phone: 309-438-7007 | 4520 Math
| Normal, IL 61790-4520
From foda@maths.mu.OZ.AU Sat Feb 11 02:54:28 1995
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Message-Id: <199502110754.27783@mundoe.maths.mu.OZ.AU>
To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
In-Reply-To: Your message of Tue, 24 Jan 95 09:48:41 -0500.
<199501241448.JAA22043@euclid.math.temple.edu>
Date: Sat, 11 Feb 95 18:54:53 +1100
From: Omar Foda
Status: RO
Dear Doron,
Firstly, my apologies for this delay in answering your
e-mail. These just happen to be a rather hectic times
for me.
> I believe that to show the equivalence of 1.4 and 1.4's one
> should take the signed permutation g to be
> g=(\pi^{-1}, \eps(\pi)}
^
unballanced brackets
You clearly mean:
g=(\pi^{-1}, \eps(\pi))
> where \eps(\pi) is the sign-assignemnt that has
> \eps(\pi) (i)= \eps (\pi(i)
^ ^
something's and an unballanced
missing, I bracket!
think.
I think that you mean:
\eps(\pi)_(i)= \eps( \pi(i) )
> Then, after renaming the variables x_1 to x_{\pi(1)}
> ... x_k to x_{\pi(k)} , we should get Eq. (1.4')
> Please let me know if you agree now.
At this point, I do not agree. The above looks illegal:
you changed the definition of a signed permutation.
I think that the statement (1.4') as it stands is perf-
ectly correct. It is simply a re-labeling of the vari-
ables of (1.4).
My problem is with the notation of (1.4.1).
Why do you use \pi(u) as a subscript for the variable
whose sign-assignment will get undone?
Why not just use u?
Think of it from the point of view of the original variables:
1. Undo the relabelling of the variables, by replacing
i -> \pi^{-1}(i), and using \pi \pi^{-1}(i) = i
the result is that you have a subscript u sitting in the middle
of lots of \pi^{-1} (other subscripts). I am talking about the
subscripts of the variables x of the functions F_{n, k} on both
sides of the equation.
2. Change \pi^{-1} -> \pi everywhere.
the result is that you have a subscript u sitting in the middle
of lots of \pi (other subscripts)
But that is not what you want, unless you define
\eps_{u} = -1, rather than \eps_{\pi(u)} = -1 in the statement
of sub-sub-lemma 1.4.1
I hope this makes sense. Once again, I really think that it is
all cosmetic, and not essential.
With best wishes,
Omar.
From amo@research.att.com Wed Mar 15 22:29:44 1995
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From: amo@research.att.com
Message-Id: <199503160329.WAA05760@euclid.math.temple.edu>
Date: Wed, 15 Mar 95 22:17 EST
To: zeilberg@euclid.math.temple.edu
Subject: ASM
Status: RO
Dear Doron,
Sorry to have kept you waiting for so long. However, this
is a bad time of year for me, and has been made worse by
some health problems in the family. Thus I have been allowing
email messages to pile up in files. It turns out to be just
about as easy to do this as with paper. Also, it says something
about where we are in the transition to electronic communication
when a hard copy message is used to attract special attention.
(Indeed, these days just about the only snail mail messages
that I receive are about editorial matters, and even there
email is taking over.)
On the subject of ASM, congratulations on the progress in
checking the proof. It is nice to see this tough nut cracked
at last, even if the proof won't be in JAMS. I also feel
that your "distributed checking" is a very worthwhile
experiment, and I was interested to see that you did get
as much cooperation as you did. (I don't think this procedure
can be used for more than the most important results that
attract wide attention, and also don't think it is proper
for the author to specify who is supposed to check which
part of the proof. However, I do think that referee reports
in one form or another should become part of an electronicly
published paper, and am eager to see what variants of the
various possible arrangements turn out to be successful.)
Thus I feel you are playing an important pioneering role.
I have actually checked the little part you have allotted
to me. However, I would prefer not to be listed in the
acknowledgements. There are two reasons for this. One
is that I feel only significant contributions should be
listed. It is clear that in my case (as well as of some
others), you have carved out a tiny piece to be checked
just to have me be listed as a checker. I do not think
this serves any legitimate scientific purpose. The other
reason is that by breaking up the job into little pieces,
you introduce new sources of uncertainty into the checking
process. The interfaces between the pieces become crucial,
and it is not certain that anyone has a good overview of
what is going on, not is there assurance that different
people might not have different interpretations of various
claims. (This problem was illustrated very vividly for me
during a recent visit to the Bell Labs facility at Indian
Hill, near Chicago. One of the chaps there, who works in
switching, told of the frustrations of working with
people in transmission. Those chaps are located in
Massachusetts, but that was not the main problem. Instead,
what was causing difficulties was that the two groups
were using conventions with meanings that were often
diametrically opposite. For example, one group would
denote a line with traffic on it by a 1, the other by
a 0. Straightening out the misunderstandings this causes
takes a lot of effort.)
Best regards,
Andrew
From brenti@math.ias.edu Wed Jan 25 12:49:25 1995
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id AA12281; Wed, 25 Jan 95 12:49:37 EST
Date: Wed, 25 Jan 95 12:49:37 EST
From: brenti@math.ias.edu (Francesco Brenti)
Message-Id: <9501251749.AA12281@sevilla.math.ias.edu>
To: zeilberg@euclid.math.temple.edu
Subject: Checker's report
Status: RO
Dear Doron,
I have now completed checking Subsubsubsublemma 1.2.1.2.4.
I have found two "mistakes" in it (namely, the equality
on l. 1, p. 37, is false, and the equality on lines 9-10
on p. 37 is also false) which, however, cancel each other,
so that the final result is still correct, and with the
same proof. I have also found two misprints. All
the corrections, together with some (personal) comments for
improving the clarity of the exposition, are
described in detail below in the enclosed "checker's report".
It is a LaTeX file.
Very best wishes,
Francesco
P.S. I like the epithet you wrote for me
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle[twoside,12pt]{article}
\setlength{\topmargin}{0.0in}
\setlength{\textheight}{22.0cm}
\setlength{\oddsidemargin}{0.5in}
\setlength{\evensidemargin}{0.2in}
\setlength{\headsep}{0.1cm}
\setlength{\textwidth}{15cm}
\setlength{\parindent}{0.6cm}
\begin{document}
\pagestyle{empty}
{\large Checker's Report on }
{\bf Subsubsubsublemma 1.2.1.2.4}
by Doron Zeilberger
\begin{itemize}
\item on p.36, l. -10 and -9, substitute ``all the sign-assignments
$\varepsilon _{i}$ equal $-1$'' with ``$\varepsilon _{i}=-1$
for $i=1, \ldots ,k$'';
\item on p.36, l. -8, erase ``(or gain)'';
\item on p.37, l. 1, replace ``$(1-x_{1}x_{k}) \Psi _{k-1}$'' with ``$(1-x_{1}x_{k}) \bar{x}_{2} \ldots
\bar{x}_{k} \Psi _{k-1}$'';
\item on p.37, l. 2, replace ``$(\bar{x}_{2} \ldots \bar{x}_{k})
(1) \ldots (1)$'' with ``$(\bar{x}_{2} \ldots \bar{x}_{k})^{2}$'';
\item on p.37, l. 6 and l. 8, substitute ``$\bar{x}_{2} \ldots
\bar{x}_{k}$'' with ``$(\bar{x}_{2} \ldots \bar{x}_{k})^{2}$'';
\item on p.37, l. 10, erase ``$. 1$'';
\item on p.37, l. 11, erase ``I claim that this equals $H_{k-1}
(k;a_{2}, \ldots , a_{k})$.'';
\item on p.37, l. 13, substitute ``$\prod _{1 \leq i < j \leq k}$''
with ``$\prod _{2 \leq ia_{2} \geq a_{3} \ldots
\geq a_{1}$'' with ``$k> a_{2} \geq a_{3} \ldots \geq a_{k}$'';
\item on p.37, l. -7, substitute ``to show that'' with ``to
show that the right side of''
\end{itemize}
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From ding@math.ias.edu Wed Jan 11 23:32:29 1995
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id AA03641; Wed, 11 Jan 95 23:32:36 EST
Date: Wed, 11 Jan 95 23:32:36 EST
From: ding@math.ias.edu (Kequan Ding)
Message-Id: <9501120432.AA03641@sevilla.math.ias.edu>
To: zeilberg@euclid.math.temple.edu
Status: RO
-------begin checker's form---------------------------
Dear Doron,
I have now completed checking (sub)^[3]lemma [1.3.1.3].
To the best of my judgment, it is correct, provided
all its offspring (sub)^[4]-lemmas are correct.
Needless to say, I don't know (and don't care!) whether
the proofs of these offspring (sub)^[4]-lemmas are actually
correct.
and
I have found the following typos or other errors:
[ in page 44, line 2 and line 3 there are three $x_1$'s.
these should be replaced by three $x_{\pi(1)}$'s ]
Best Wishes,
[Kequan Ding]
P.S. I like the epithet you wrote for me
---------end checkers form----------------
From fraenkel@wisdom.weizmann.ac.il Wed Jan 4 05:20:25 1995
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From: Fraenkel Aviezri
Received: from localhost (fraenkel@localhost) by silver.wisdom.weizmann.ac.il (8.6.5/8.6.5) id MAA10632; Wed, 4 Jan 1995 12:19:26 +0200
Date: Wed, 4 Jan 1995 12:19:26 +0200
Message-Id: <199501041019.MAA10632@silver.wisdom.weizmann.ac.il>
To: zeilberg@euclid.math.temple.edu
Status: RO
Shalom Doron, before I even begin, I have to clarify a few points.
1. P5, l 8,9: it seems to me that x_k --> x_1 is missing.
2. P5, l 10,11. I don't understand: what's PI, what's sgn(PI)?
3. P5, l -3. If I follow the definition on line -4, I get x_2^2 +
2(1-x_3) + (1-x_1^3), rather than what's written there.
4. P5, middle. It seems to me that W(B_k) is not well-defined. You
write it consists of pairs (PI, EPS)...Which pairs? Presumably you
mean ALL 2^k pairs? I can then see that it's a group.
5. P52. (To save me time.) There are a few x with hats in the formula
you want me to check. Where is the hat notation explained?
Thanks, Aviezri.
plato:/home/fac/zeilberg/doar/fraenkel>
From wenchang@mat.uniroma1.it Thu Feb 2 03:20:25 1995
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(5.67a/IDA-1.5 for ); Thu, 2 Feb 1995 08:18:41 +0100
Date: Thu, 2 Feb 1995 08:18:41 +0100
From: chu wenchang
Message-Id: <199502020718.AA14404@mercurio.mat.uniroma1.it>
Attention: The domain dm.unirm1.it is obsolete since 20 Sept 93
use in mail address the current name
mat.uniroma1.it
To: zeilberg@euclid.math.temple.edu
Subject: subsubsublemma1413 checker
Status: RO
Dear Doron, I sent you my confirmation last week. Unfortunately,
when I switch on the computer today, I find out that my message
did not come to you. Now I try it again (I could not find a gap).
From: chu wenchang
Message-Id: <199501260819.AA02184@mercurio.mat.uniroma1.it>
Attention: The domain dm.unirm1.it is obsolete since 20 Sept 93
use in mail address the current name
mat.uniroma1.it
To: zeilberg@euclid.math.temple.edu
Subject: subsubsublemma1413 checker
Dear Doron,
I have now completed checking (sub)^[1.4]lemma [1413].
To the best of my judgment, it is correct, provided
all its offspring (sub)^[1.4]-lemmas are correct.
Needless to say, I don't know (and don't care!) whether
the proofs of these offspring (sub)^[]-lemmas are actually
correct.
Best Wishes,
CHU Wenchang
euclid%
From bergeron@catalan.math.uqam.ca Sun Feb 5 12:59:27 1995
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From: bergeron@catalan.math.uqam.ca (Francois Bergeron)
Message-Id: <9502051759.AA16400@catalan.math.uqam.ca>
Subject: no subject (file transmission)
To: zeilberg@euclid.math.temple.edu
Date: Sun, 5 Feb 95 12:59:44 EST
X-Mailer: ELM [version 2.2 PL14]
Status: RO
Dear Doron,
I have finaly got down to checking my part. In itself it is
perfectly OK, but I have not read in detail the entire paper.
Hence I cannot judge what is its interaction with the rest.
Is this what you expected?
Keep me posted on the status of the problem (and your solution).
Regards, Francois.
P.s.: I am very excited about beautiful work
we are doing (with Adriano)
on themes related to diagonal harmonics and Macdonald's conjecture.
We have various new very
nice conjectures with lots of algebraic and enumerative combinatorics,
as well as beautiful results.
P.P.s.: For your Checker list, you might want to know that the descent
algebra story unfolded more or less as follows:
1) First there was Adriano and Christophe's paper on the A_n case,
2) Then Nantel and I did the B_n case,
3) Finaly, with Don Taylor and Bill Howlett, Nantel and I did
the general decomposition for all Coxeter groups in an
uniform way.
4) There was also a paper of Adriano, Christophe and myself
about nice morphisms between descent algebras, and
another one with Luc Favreau on Fourier Transforms
in the setup of descent algebras, and a paper of
Nantel on Hodge Decomposition related to the B_n case.
5) Many others are also involved, but that story would be too long.
--
euclid%
From chenyc@bepc2.ihep.ac.cn Mon Feb 6 11:05:14 1995
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+0800
Date: Tue, 07 Feb 1995 00:03:42 +0800
From: chenyc@bepc2.ihep.ac.cn
To: zeilberg@euclid.math.temple.edu
Message-Id: <0098B9A6.BBDF4120.20@bepc2.ihep.ac.cn>
Subject: RE: 1.3.1.1.1
Status: RO
Dear Doron,
1.3.1.1.1 is correct. However, I found out some typos in formulas
before 1.3.1.1.1 which I am sure have pointed out by the checker.
I am traveling in China, and forgot to bring the paper with me.
So I will send you the typos (not in 1.3.1.1.1) later.
Best regards. Congratulations again on the great breakthrough.
Bill
euclid%
From okada@math.nagoya-u.ac.jp Wed Feb 8 23:17:59 1995
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Date: Thu, 9 Feb 1995 13:16:29 +0900
From: okada@math.nagoya-u.ac.jp (Soichi Okada)
Message-Id: <199502090416.NAA08702@rabbit.math.nagoya-u.ac.jp>
To: zeilberg@euclid.math.temple.edu
Subject: subsubsublemma 1.4.1.3
Status: RO
Dear Doron,
I have now completed checking (sub)^3lemma 1.4.1.3.
To the best of my judgment, it is correct, provided
its offspring \aleph'5 is correct.
Needless to say, I don't know (and don't care!) whether
the proofs of these offspring lemmas are actually
correct.
Best Wishes,
Soichi OKADA
euclid%
From foda@maths.mu.OZ.AU Sat Feb 11 02:54:28 1995
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Message-Id: <199502110754.27783@mundoe.maths.mu.OZ.AU>
To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
In-Reply-To: Your message of Tue, 24 Jan 95 09:48:41 -0500.
<199501241448.JAA22043@euclid.math.temple.edu>
Date: Sat, 11 Feb 95 18:54:53 +1100
From: Omar Foda
Status: RO
Dear Doron,
Firstly, my apologies for this delay in answering your
e-mail. These just happen to be a rather hectic times
for me.
> I believe that to show the equivalence of 1.4 and 1.4's one
> should take the signed permutation g to be
> g=(\pi^{-1}, \eps(\pi)}
^
unballanced brackets
You clearly mean:
g=(\pi^{-1}, \eps(\pi))
> where \eps(\pi) is the sign-assignemnt that has
> \eps(\pi) (i)= \eps (\pi(i)
^ ^
something's and an unballanced
missing, I bracket!
think.
I think that you mean:
\eps(\pi)_(i)= \eps( \pi(i) )
> Then, after renaming the variables x_1 to x_{\pi(1)}
> ... x_k to x_{\pi(k)} , we should get Eq. (1.4')
> Please let me know if you agree now.
At this point, I do not agree. The above looks illegal:
you changed the definition of a signed permutation.
I think that the statement (1.4') as it stands is perf-
ectly correct. It is simply a re-labeling of the vari-
ables of (1.4).
My problem is with the notation of (1.4.1).
Why do you use \pi(u) as a subscript for the variable
whose sign-assignment will get undone?
Why not just use u?
Think of it from the point of view of the original variables:
1. Undo the relabelling of the variables, by replacing
i -> \pi^{-1}(i), and using \pi \pi^{-1}(i) = i
the result is that you have a subscript u sitting in the middle
of lots of \pi^{-1} (other subscripts). I am talking about the
subscripts of the variables x of the functions F_{n, k} on both
sides of the equation.
2. Change \pi^{-1} -> \pi everywhere.
the result is that you have a subscript u sitting in the middle
of lots of \pi (other subscripts)
But that is not what you want, unless you define
\eps_{u} = -1, rather than \eps_{\pi(u)} = -1 in the statement
of sub-sub-lemma 1.4.1
I hope this makes sense. Once again, I really think that it is
all cosmetic, and not essential.
With best wishes,
Omar.
euclid:/home/fac/zeilberg/doar/foda>
From dror@math.harvard.edu Sun Jan 15 19:20:50 1995
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From: dror@math.harvard.edu (Dror Bar-Natan)
Message-Id: <9501160020.AA05992@math.harvard.edu>
To: zeilberg@euclid.math.temple.edu
Status: RO
Dear Doron,
I have now completed checking (sub)^3 lemma 1.1.1.2. To the best of
my judgment, it is correct, provided all its offspring (sub)^4-lemmas
are correct.
I don't know whether the proofs of these offspring (sub)^4-lemmas are
actually correct.
I have found the following typos or other errors:
* In the statement of the lemma, you write n>=a_1>=a_k>=...>=a_k>=1 instead
of n>=a_1>=a_2>=...>=a_k>=1 (notice the missing a_2 in the original).
* In the proof of case (C1_i) you write (i<=i=a_k>=2". The last inequality there, a_k>=2, is misleading,
because it now sounds as if in cases (C1) and (C2) you have already
shown that ... is 0 if a_k<2. Simply drop that last >=2; it is
already included in the sentence before.
I have the following suggestion to improve the exposition:
* A general TeX comment: Writing "$Land\_Of\_Magog$" gives you incorrect
spacing between the letters. If you want to get a pretier "Land_Of_Magog"
in italics and improve your image among sophisticated TeX users, replace
$Land\_Of\_Magog$ by "{\it Land\_Of\_Magog}". This is a general remark!
Almost all names of all formulas in your paper can and should be
TeXnically improved.
* It is not necessary to define F again in sub^4lemma 1.1.1.2.2. It's
enough to recall there the definition in sub^4lemma 1.1.1.2.1.
I also have the following comments on the introduction:
* You write "A permutation \pi is better described ... matrix,...". I don't
buy the word "better" there. Why use n^2 bytes where you can use only n?
* Why does \aleph_5 appear after \aleph_2, if the proof of \aleph_2 uses
\aleph_5?
* The parenthesized word "high" in the statement of \aleph_5 is not doing
anything useful that I could see.
Best Wishes,
Dror Bar-Natan
From cfd5z@fermi.clas.virginia.edu Mon Jan 23 11:49:48 1995
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Date: Mon, 23 Jan 1995 11:50:14 -0500
From: "Charles F. Dunkl"
Message-Id: <199501231650.LAA42573@fermi.clas.Virginia.EDU>
X-Mailer: Mail User's Shell (7.2.3 5/22/91)
To: Doron Zeilberger
Status: RO
Dear Doron,
Here is my report on sub^3 lemma 1.3.1.4 case 1 (including case
1a and 1b) . I will also send it on paper to avoid electronic
misinterpretations.
The arguments are correct and believable. (I checked
important fact aleph 5 for myself, since it was crucial here).
Comments: in the material leading up to this, you defined RAT,
POLRAT, but not NUMBER , see p.45 after formula (Gil). I know
what it is. On page 41, line 3 you are using z sub R, even though
on line 1 you specifically do not define z sub R (I think you
mean Z sub R = x sub R). In case 1a (of "my" lemma) the variable
t is in the range 0 <= t <= n+k. You do not state the upper
bound, it is not really needed, but it might be helpful for the
reader to get a hold on what t means.
I looked at the exodion - my epithet looks cool (but
Cherednik might prefer no "ick").
I will await your response - tell me if you want a hard
copy of my report (with my signature).
Best wishes,
Charles
From cfd5z@fermi.clas.virginia.edu Mon Jan 23 11:51:53 1995
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Date: Mon, 23 Jan 1995 11:52:18 -0500
From: "Charles F. Dunkl"
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To: Doron Zeilberger
Status: RO
Doron,
important addendum to my message - it is subcase 1b (not
a) in which t is bounded above by n+k. Sorry for the previous
typo.
Charles
From ehrenbor@catalan.math.uqam.ca Tue Jan 24 14:54:21 1995
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Date: Tue, 24 Jan 95 14:56:41 -0500
From: ehrenbor@catalan.math.uqam.ca (Richard Ehrenborg)
Message-Id: <9501241956.AA13733@catalan.math.uqam.ca>
To: zeilberg@euclid.math.temple.edu
Subject: The ASM paper
Subsubject: Two two's missing
Reply-To: ehrenbor@lacim.uqam.ca
Status: RO
Dear Doron,
Finally after washing my hair too much I started to review your ASM
paper. By the help of two cups of hot chocolate, here is my report:
I reviewed Subsubsublemma 1.3.1.4, Case I. (This includes Subcase Ia
and Subcase Ib.)
Important: there are two 2's missing. In the first two equations in
the proof of Subsubsublemma 1.3.1.4 on page 44, the factor
$z_{i}^{n+k-i}$ appers. This factor should be $z_{i}^{n+2k-i}$.
*
*
*
(Why? Since in Equation (\tilde{A} i) on top of page 43, it appears
as $z_{i}^{n+2k-i}$ there.)
Clearly this is a missprint since in the next equation, Equation
(Doron) the factor appears with the two. Hence there is nothing to
mathematically worry about.
I have to confess that I did not check one thing in the proof.
Namely when you expand
POLRAT(x_i; x_{\pi(1)}, \ldots, \hat{x}_R, \ldots, x_{\pi(v-1)})
as a polynomial in x_i, you state in the sum that it is a polynomial
of at most degree 2k+1. This happens in both subcases. I did not check
this 2k+1. (Thus I hope it is true.) But this fact is not used in the
the proof.
Less important: I have some questions about typesetting.
(i) In the equation of Subsubsublemma 1.3.1.4, could we please
use larger square brackets ( [ and ] ). It would make
the equation easier to read.
(ii) The tilde above the A's looks so tiny. In LaTeX there is a
command \widetilde. That is, $\widetilde{A}$. The same
comment applies to hats. That is, I prefer the wide hats,
$\widehat{A}$.
This is clearly very subjective of me.
(iii) On page 48, after stating Case II of Subsubsublemma 1.3.1.4,
you use the word "below". I was told by people not use the
words "above" and "below" in a math paper, since the words
refers locally on that page.
As a last word, I say that the proof of Subsubsublemma 1.3.1.4 Case
I is correct. Congratulations to an amazing paper and an amazing result.
Yours sincerely,
Richard
From foata@math.u-strasbg.fr Sat Mar 4 09:21:13 1995
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Date: Sat, 4 Mar 1995 15:32:12 +0000
To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
Subject: Your ASM conjecture paper
Status: RO
Dear Doron,
Well, Doron, I tried to reread over and over the so-called Highest level
and could not make any further remarks. It is very frustrating. Why is it
so intricate? Perhaps, you should insist on having a fist list of Magog and
Gog trapezoids for the very small values of n and k, in particular to illustrate
the sets of inequalities between the entries in both cases. Also an example
of a trapezoid, where the reader could glance at the two different shapes
of trapezoids. In other words, anything that could attract the unwilling
reader.
The preceding exercise I had to do it for myself to have a planar
representation of those trapezoids. It is the only mathematical remark
I could make.
The other remarks are of typographical nature:
you should define:
\def\sgn{\mathop{\rm sgn}\nolimits}
\def\Res{\mathop{\rm Res}\nolimits}
to obtain those two mathematical operators in roman.
My global impression is that the paper deserves publication, but desperately
calls for a better understanding of all those countings. We miss the structure
behind it. However it is great that the conjecture is no longer a conjecture,
but a theorem, so that the right proof should follow soon. Always hard
to be the pioneer.
In any case it is a great achievement. You have mastered, first a very tricky
set of Lemmas and sublemmas, second you have shown that your superb
command of MAPLE allows to go deeply into the roots of difficult problems.
Perhaps you are already too far ahead of the other mathematicians.
Best regards,
Dominique
PS I wait for GuoNiu's remarks concerning the Graphical Major Indices
paper. But you should say to Mourad that our paper is ready any minute.
From foda@maths.mu.OZ.AU Fri Jan 20 12:26:59 1995
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To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
Subject: Re:
In-reply-to: Your message of Thu, 29 Dec 94 09:19:19 -0500.
<199412291419.JAA25072@euclid.math.temple.edu>
Date: Sat, 21 Jan 95 04:27:15 +1100
From: Omar Foda
Status: RO
Dear Doron,
> I have recently had a defection of 1.4. It would be great
> if you would be kind to volunteer to be a checker of 1.4,
Assuming 1.4.1, 1.4 is trivially true--as you know!
However, I am a bit confused by the notation:
On the 3rd and 4th lines after (1.4') on p.50,
when you say:
"..such that \epsilon_{\pi(i)} = -1."
Shouldn't that be
"..such that \epsilon_{i} = -1."
I mean, now that you write the signed permutation as
(\pi^{-1}, \epsilon}, and \epsilon) acts _after_ \pi,
\epsilon acts on objects labelled by {1, .., i, ..., k},
rather than {\pi(1), ..., \pi(i), ..., \pi(k)}. So the
variable "u" should be the smallest integer i such that
\epsilon_{i} = -1.
I noticed that you use the same notation--that I am
complaining about--in 1.4.1, and of course the whole
thing parallels what happens in act III, so I expect
that you are right.
To summarize, if I accept your notation, and 1.4.1,
then 1.4 is true.
What should I do next?
With best wishes,
Omar.
PS I wish to read the whole thing within the coming week,
and will send you a number of comments TeX matters.
From fraenkel@wisdom.weizmann.ac.il Fri Jan 6 06:13:33 1995
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From: Fraenkel Aviezri
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Date: Fri, 6 Jan 1995 13:13:03 +0200
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To: fraenkel@wisdom.weizmann.ac.il, zeilberg@euclid.math.temple.edu
Subject: Re: Understanding me easily
Status: RO
Shalom Doron, Somebody close to me prepared the following report.
With best wishes, Shabbat shalom, Aviezri.
1. The notation \hat x_i,\ldots,\hat x_R seems to mean (because of
the \ldots) that all the variables from x_i to x_R are missing.
However, the author seems to mean that only the 2 variables x_i
and x_R are missing. Is this notation explained somewhere? Is it
used consistently throughout the paper?
2.If i=1 in either 13111 or in 14111, the proof appears to be
invalid. It's nice that in 152a the author took the trouble of
specifying the range of i. It would be helpful if he would do
the same in 14111 and 13111. (Perhaps i =/= R is meant. If so,
state it.)
3. The proof hinges on Sub...lemma 152. Who checks that Sub...lemma
152 doesn't depend on Sub...lemma 14111?
4. This referee didn't check the validity of the first 2 sentences
of the proof.
5. In the penultimate line of the proof, (proved below.) should be
(proved below).
From janef@pwa.acusd.edu Thu Dec 29 22:38:43 1994
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Date: Thu, 29 Dec 1994 19:35:34 -0800 (PST)
From: Jane Friedman
Subject: Re:
To: Doron Zeilberger
In-Reply-To: <199412191354.IAA27943@euclid.math.temple.edu>
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Status: RO
Hi,
Happy new year.
I am about to go out of town for a few days and am looking at your paper
again. I am going to visit my parents and then to san francisco for the
meetings. I am really trying to do this well and quickly. But
I'm certainly not doing it quickly. And I'm sure it is ok modulo typos'
In fact, Its a great paper.
But I have some other questions. When you say on page 54 that the
POLRAT(xi;x(tilde)R) in (A(tilde)i1) etc all have degree <= 2k+1 in x_i, do
are you considering z_i and z(bar)_i to be separate variables, since if
I understand correctly z_i and z(bar)_i are really either x_i or
x(bar)_i. And if not is this 2k+1 taking into account the
(z_i-2)(1-z_i^2)(1-z_iz(bar)_i) in the numerator of (Ai1) and similar
thingss in the numerator of (Ai2),(Bi1),(Bi2)?
Also in (B(tilde)i2) Should the first POLRAT be a POL.
I'll be back on the 8th or so and I will finish this. But meanwhile it
is really a great paper.
jane
On Mon, 19 Dec 1994, Doron Zeilberger wrote:
> Dear Jane,
> Thanks for all the comments. You are perfectly right about the
> need for concrete examples. I wil do it soon in
> the upcoming revision. You are also perfectly right
> about the missing bar.
>
> I am sure that I will recieve the Talmudic paper reprints soon!
>
> Best wishes,
> and keep up the good work!
>
> Doron
>
From frank@math.ufl.edu Thu Feb 16 14:17:53 1995
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From: "Frank G. Garvan"
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Date: Thu, 16 Feb 1995 14:18:16 -0500
Message-Id: <199502161918.OAA27510@blue-tongue.math.ufl.edu>
To: zeilberg@euclid.math.temple.edu
Subject: Re: Messages from DZ
Cc: frank@math.ufl.edu
Status: RO
THUR FEB 16, 1995.
Doron:
QUESTION:
On page 55 (I am not sure if my page numbers will corespond to yours)
~
at the top in eqn (Bi2) is there a typo ?
^ ^
Should the first POLRAT(x ;x ) be a POL(x ) ?
i R R
If this is a typo then everything is ok, but otherwise
there is a problem in the proof of sub^3Lemma 1.4.1.6 (see below).
REPORT:
I checked the proof of sub^3Lemma 1.4.1.4 and everything is
fine except for some typos.
I checked that the proof of sub^3Lemma 1.4.1.6 goes thro analogously.
~
Everything is fine provided (Bi2) contains the typo mentioned above.
Everything is explained very clearly and it makes me want to read
the whole paper (when I get the time).
TYPOS:
There are some typos besides the possible one mentioned above.
Some of them are very minor.
1. Two lines before sub^3lemma 1.4.1.2 change
...........rattional function ..............that does..........
to
...........rattional functions ..............that do..........
2. There are some typos/suggestion in sub^5 lemma 1.4.1.4.1.1.
(a) In this lemma you are assuming that 1 \le i < u
so this should be included in the statement of the lemma.
(b) Throughout the proof (and in the statement) you have
POL( x_1,..)
Each of these should be
POL( x_{\pi(1)},...)
(c) Half-way through the proof in
Coeff_{x_{S}^{(\alpha \,\,\, or \,\,\, \beta )-1}} [ ]
you have
\bar x_S^{(\alpha \,\,\, or \,\,\, \beta ) }
I suggest you change this to
\bar x_S^{(\beta \,\,\, or \,\,\, \alpha ) }
CHEERS
Frank G.
From george@math.nwu.edu Mon Jan 23 01:09:26 1995
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From: george@math.nwu.edu (George Gasper)
Message-Id: <9501230609.AA23335@dehn.math.nwu.edu>
To: zeilberg@euclid.math.temple.edu
Subject: ASM
Status: RO
Dear Doron,
Here are some comments on your "Proof of the alternating sign
matrix conjecture." It is a impressive preprint, but is rather difficult
to follow in parts:
Because of your different (conflicting) definitions of A_i, it would be
less confusing to at least change the A_i operator defined on p. 4 to,
say, S_i. (S for shift!)
For clarity, on page 5 replace (231,+1-1-1) [x_1^2+ 2x_2+ x_3^3] by
([231], [+1,-1,-1])(x_1^2+ 2x_2+ x_3^3). Also, it would be better to
define
(\epsilon,\pi) f(x) = \epsilon \pi f(x)
to be consistent with the ordering in your definition of the iterated
residue.
p. 40: In (1.3'), the notation x_k^{\epsilon_k} is employed, but I
couldn't find it defined until on the top of the next page with a
"Recall that ...". It could be defined on p. 5 or in the sublemma.
Also, since (1-(1-x_i)(1-x_j)) = x_i + x_j - x_i x_j = 0 when x_i = x_j =
0 it seems to me that the left hand side of (1.3') is NOT defined when
two \epsilon's equal -1, say \epsilon_i = \epsilon_i = -1, since the
function in it is then does NOT necessarily possess a formal Laurent
expansion (by your first crucial fact on p. 6 and the definition of
F_{n,k}). This seems to affect many of the statements that follow in
the rest of the paper, Subsublemma 1.3.1. , (Noam), etc. , unless you
eliminate this possibility from occurring. Then the proof of (Noam),
etc. can be replaced by a simple direct proof consisting looking at
the factors in the denominator of A_i^\tilde --- (1-x_i), (1-x_i +x_i
x_j), etc., --- to get the representation in your first crucial fact on p.
6 (and even to get the Laurent expansion when each 0 < |x_i| < 1).
p. 55: Shouldn't the three x_1 's in lines 2 and 3 of the Proof of
(Hadas_1) be x_{\pi(1)} ?
p. 56, line 2: "are identical" should be replaced by "are of the same
form" since the POL's are different polynomials.
Other than the problem mentioned above for p. 40 the proof of
(sub)^3 lemma 1.4.1.4 seems OK if you eliminate the factors
(1-z_r z_s), etc. from possibly being to the form
(1-(1-x_i)(1-x_j)) = x_i + x_j - x_i x_j = 0 .
It's almost midnight, so I'll quit and e-mail this to you.
I hope that it is helpful,
BW, George, 1-22-95
From andrew@sophie.math.uga.edu Sat Jan 21 16:48:15 1995
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Date: Sat, 21 Jan 95 16:52:22 EST
From: Andrew Granville
Message-Id: <9501212152.AA15956@sophie.math.uga.edu>
To: zeilberg@euclid.math.temple.edu
Subject: Referee's Report for subsublemma 1.4.1.4, Subcase Ia
Cc: andrew@sophie.math.uga.edu
Status: RO
Dear Doron,
I have (finally) read over the portion of the text that you assigned
to me. I am sorry for my delay in getting this done (you don't want
to hear excuses, just the report):
Anyway, there is not too much to say. Mathematically it is fine and correct.
I would quibble with a few minor points concerning the exposition; which
I guess as `referee' I could ask you to think about before final `publication'.
I think that it would have been easier to do this job if you had more
thoroughly stated the hypotheses in the lemma rather than referred to previous
statements, and if you had explained the notation in each seperate proof.
As you didn't, I had to do a lot of hunting through the paper for definitions
which could have been supplied with each proof --- I have heard others,
of your referees, complain about the same thing. Actually I should say to
you that I have talked to about five such people, all of whom think that it
is a great idea to referee this important paper in this way, but all of
whom had more difficulty than they would have liked refereeing the portion
assigned to them, because they had to familiarize themselves with a lot more of
the paper than just their portion. I don't know how many bits you have got
back from your `referees' --- but if there are a lot to go then you might
consider this point carefully if you ask them again as they might be
embarrassed to tell you why they ain't done their part (although it would
be a lot of work to consider each `chunk' seperately and add the
appropriate definitions, etc).
Specific points in 1.4.1.4. SubCase 1a:
-- You might remind people at the top of the proof of 1.4.1.4 that
$R=\pi (u)$.
-- In the fourth line of the proof of 1.4.1.4 you say that the degree of
a certain POLRAT is $\leq 2k+2$. However you don't justify this or give
a reference to where it is proved. Moreover, it does seem to be irrelevant
to the whole proof of 1.4.1.4
-- In the fifth line of the proof of 1.4.1.4 you state `So it suffices
for us to know ...' This is misleading as the reader who is a little
confused thinks you are about to prove that. Actually what you want
to say is: `Either way, we have now obtained $A_i$ written in the
form ...'
-- In the few lines above and below (Gil') there is some inconsistency
in notation of POLRAT. Sometimes you include the x_R^ and sometimes
not. Since you are evidently doing this to help the reader understand
what is happening to x_R at each step I would suggest including it
each time.
-- Line after (Gil'''): Perhaps you should restate `crucial fact
$\aleph'_5$ ' for the convenience of the reader.
Anyway these are all pedantic little points but might help with clarity.
Good luck with your bold endeavour, and sorry for delaying your
progress. Best wishes,
Andrew
From grinberg Thu Mar 9 07:42:24 1995
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From: grinberg (Eric Grinberg)
Full-Name: Eric Grinberg
Message-Id: <199503091242.HAA17112@euclid.math.temple.edu>
To: zeilberg
Subject: The check is in the email
Status: RO
March 7, 1995
Brookline, Mass.
Dear Doron,
I have checked the proof of sublemma 1.4.1.4, subcase 1b and
it appears correct to me (bottom of page 59: `forms' should be
`form'). I hope this information will be useful to you in the
large task of verifying your ultra-manuscript. If you like, I can
supply a handwritten, xeroxed version of this notice via fax.
Let me know your local fax number in this case.
Cheers,
Eric
p.s. Thanks for lunch.
From habsiege@math.u-bordeaux.fr Mon Jan 16 04:12:38 1995
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Date: Mon, 16 Jan 95 10:15:11 +0100
From: habsiege@math.u-bordeaux.fr (Laurent Habsieger)
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To: zeilberg@euclid.math.temple.edu
Status: RO
\magnification=\magstep1
\nopagenumbers
\centerline{\bf Habsieger's report on Subsubsusbsubsubsublemma
1.2.1.2.3.1.1}
\vskip 3 cm
\hskip 8 cm Bordeaux, November 30, 1994
\vskip 1 cm
\centerline{Dear Doron,}
\vskip 2 cm
\noindent
I read carefully the three-line statement of the
Subsubsusbsubsubsublemma 1.2.1.2.3.1.1 and its three-line proof.
The proof seems perfectly right to me, under the assumption that
Subsubsublemma 1.4.1.1 is true. However I would appreciate it if
you specified the sign of $A_i$.
\vskip 1 cm
\noindent
Additional remarks:
\medskip \noindent
Page 22, line -3: you forgot a $\backslash$ before ``min''.
\medskip \noindent
Page 85, Askey's biography: you mispelt his name...
\medskip \noindent
Page 88, my biography: you are very nice to me. Thank you very much.
But I would rather like to have ``transcendence measure''
replaced by number
theory, since I worked both in diophantine approximation and in additive
number theory. By the way, my last field of investigation is
the theory of covering codes.
\end
From jhaglund@kscmail.Kennesaw.Edu Mon Jan 23 10:07:54 1995
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Date: Mon, 23 Jan 95 10:08:15 EST
From: "Jim Haglund"
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To: zeilberg@euclid.math.temple.edu
Status: RO
Doron:
I have finished checking Lemma 1.4.1. Correct me if I am wrong: when
verifying this Lemma, I can assume any sublemma is true. Given the
assumption that Lemmas 1.4.1.2 thru 1.4.1.6 are true, Lemma 1.4.1 follows
fairly easily. I did not find any errata.
p.s. I also helped Andrew Granville check 1.4.1.4a, which seems fine as
well.
Best Regards,
Jim
From foata@math.u-strasbg.fr Mon Feb 27 04:29:26 1995
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Date: Mon, 27 Feb 1995 10:29:11 +0100
Message-Id: <199502270929.KAA15716@isis.u-strasbg.fr>
To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
Subject: Corrections from GuoNiu
Status: RO
Dear Doron,
GuoNiu had an earlier version (ReReRevised:July 1994).
About lemma 1.3, he is not sure, but he thinks that
everything is right, rather he thinks that the method
is right.
Last July he had checked something ahead, after
COUNTING GOG (page 12)
top of the page (check the version):
condition (v) $1 \le d_{i,j} \le i+j-1$ instead of
$1 \le d_{i,j} \le n$
Perhaps it not important, but at ``Now let" (same page),
add :
Now let, if $n\ge k+1$
Further on the page :
By the conditions (i) - (v) instead of By the conditions (i) - (iv)
In the opinion of GuoNiu it would be better to draw a figure,
to represent some trapezoid of numbers for the
Land_Of_Gog
Doron, it is very hard.
Myself, I have found nothing so far, but will report.
Best regards,
Dominique
From jacques@lacim.uqam.ca Mon Jan 16 17:19:54 1995
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Date: Mon, 16 Jan 95 15:04:07 -0500
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To: zeilberg@euclid.math.temple.edu
From: jacques@lacim.uqam.ca (Jacques Labelle)
Subject: Re:
Status: RO
>
>
>Dear Doron
> Happy New Year!
>
>I just finished checking my (very short, no I am not complaining !!) part
[1.4.1.1.1]
It is very sound; the calculation is good and I found no "typos" or mistakes
in the proof (top of page 54) of Subsubsubsublemma 1.4.1.1.1.
I found only two minor "typos" in the last lines of the next two paragraghs:
line 9 page 54: This completes the proof of (sub)^2 lemma 1.4.1.1.
SHOULD BE: This completes the proof of (sub)^3 lemma 1.4.1.1.
line 15 page 54: cients that are rational function of
SHOULD BE: cients that are rational functions of
Best wishes,
> and a happy New half-decade, AND CONGRATULATIONS IF ...
>
Jacobi Abel known as Jacques Labelle brother of Hilbert Abel.
P.S. Some remarks about the Checkers list:
1) I'd rather be called a "Species-ist" then a "formal linguist"
2) O.K. I am a master chess player but you might like to know that Ilan Vardi
(an old friend of mine originaly from Montreal) is certainly as much of
a chess master as I am.
(we both played serious tournament chess when young)
3)Jonathan and Peter Borwein are mentionned together (I know they did a lot
of join work) but Francois and Nantel Bergeron, Gilbert and Jacques
Labelle, and Dennis and Neil White are three pairs mentionned separately.
The first two pairs being brothers pairs, and not the third (as far as I
know). I dont think this point is very important. Perhaps some
standardisation on page 1 of the 3 brother pairs (Borwein, Bergeron and
Labelle) could be done but not necessarily at the end, as you wish). I'am
not mentionning the Zeiberger set of checkers
which has a much more complex family finite structure !!!!!
#############################################################################
From kalai@cs.huji.ac.il Wed Mar 8 03:19:54 1995
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Date: Wed, 8 Mar 1995 10:25:25 +0200
From: Gil Kalai
Message-Id: <199503080825.AA22938@humus.cs.huji.ac.il>
To: zeilberg@euclid.math.temple.edu
Subject: Re: ASM Checking Update
Status: RO
Dear Doron, Isabella Sheftel, a graduate student, checked the case you assigned
to me and found it ok.
She did not check one statement there "But this is identical to
the form treated by subbcase Ia..." , (p.62 l. -1) which seems to require
reading subcase Ia as well. Is it needed? Best Gil
From kaneko@math.kyushu-u.ac.jp Fri Feb 24 22:21:03 1995
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Date: Sat, 25 Feb 1995 12:24:05 +0900
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To: zeilberg@math.temple.edu
From: kaneko@math.kyushu-u.ac.jp (Jyoichi Kaneko)
Subject: From Kaneko
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Status: RO
Dear Prof. Zeilberger, Thanks for your sending recent (p-)reprints,
especially the overwhelming ASM paper!
I have been ignorant about ASM, TSSCCP etc., but now I completely agree
with Robbins that " These conjectures are of such compelling simplicity
that it is hard to understand .....".
I have checked Subsubsublemma 1.5.2.2.
Yours sincerely,
Jyoichi Kaneko
Jyoichi Kaneko
Department of Mathematics, Kyushu University
kaneko@math.kyushu-u.ac.jp
From leroux@math.uqam.ca Tue Jan 24 15:50:10 1995
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From: leroux@math.uqam.ca (Pierre Leroux)
Message-Id: <9501242050.AA17970@caillou>
Subject: Re: ASM Checking Update
To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
Date: Tue, 24 Jan 1995 15:49:59 -0500 (EST)
In-Reply-To: <199501021716.MAA29766@euclid.math.temple.edu> from "Doron Zeilberger" at Jan 2, 95 12:16:19 pm
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Status: RO
Dear Doron,
I am happy to report that I have checked Case I of sublemma 1.4.1.5
of your ASM paper, as was assigned to me, and that it is correct, provided
that Subcases Ia and Ib (which I noticed were not assigned to me)
are also true.
I am happy with your flattering description of myself in the
Exodion. There is however a mistake in the reference to my joint
paper (with Foata) in the PAMS: it should be 87 47 83, since
it was published in 1983.
I also noticed a misprint page 7 (line -3), in the proof of
Crucial fact aleph 0 : when the polynomial Q is expanded
as C + a sum of monomials, the indexing set would be better described
by replacing the two plus symbols by comma symbols since only positive
powers of each variable occur.
Sincerely,
Pierre Leroux
From e1lewis@haverford.edu Wed Jan 25 12:16:09 1995
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Date: Wed, 25 Jan 1995 12:16:51 -0500
To: zeilberg@math.temple.edu
From: e1lewis@haverford.edu (Ethan Lewis)
Subject: ASM paper
Status: RO
Dear Doron,
How are you?
I have now completed checking (sub)^[]lemma [ 1.4.1.5], Case I, Subcase 1a.
To the best of my judgment, it is correct.
If you are interested I have a few small suggestions about the introduction.
Please let me know the schedule for the Wed. seminar, I hope to come out
a some, although my schedule will make it impossible to stay for dinner unless
I have ample notification (say a week).
Best,
Ethan
From e1lewis@haverford.edu Wed Jan 25 13:34:44 1995
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Date: Wed, 25 Jan 1995 13:35:28 -0500
To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
From: e1lewis@haverford.edu (Ethan Lewis)
Subject: Re: ASM paper
Status: RO
My thoughts were just this.
Small point 1:
On page 6 there are example showing the action of pi and epsilon on a function,
and then an example showing the action of both on a function. I think
the notation in the third case could be more consistant with the first two.
In the first two pi and then epsilon appear in square brackets and the
function in parentheses and in the third case it is the other way around.
Also in the first case commas are used two seperate the members of the
permutation while in the third case they are not.
Small point 2:
I believe that the sign of a member of the Weil group is never explicitly
introduced in the intro., but it is referred to later.
Best,
Ethan
From mguest6@math.ias.edu Thu Feb 16 14:30:41 1995
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From: mguest6@math.ias.edu (mguest6 )
Message-Id: <9502161930.AA15704@sevilla.math.ias.edu>
To: zeilberg@euclid.math.temple.edu
Status: RO
Dear Doron,
I have carefully checked the proof of sublemma 1.1,
and can testify that it is correct, provided all its subsublemmas
are.
Best wishes,
Noga Alon
Dear Andrew,
Thanks for your thoughtful and very interesting message
containing very valid comments on the checking process.
Of course I will respect your anonimity, and will list the
checker of 1212 as `Anonymous'. Among the roughly eighty
checkers only one other checker asked to be anonymous.
Hope all the health problems of your family will disapper, and that
your conscientiousness in promptly answering all mail
will not take up too much time of your research.
You are right about not chopping it into too little pieces.
I think that the optimum for the number of checkers for a paper
of length n is n^(1/2) checkers, each checking n^(1/2) of it,
and not e^n checkers checking n/e^n each.
Best wishes,
DoronFrom ppaule@risc.uni-linz.ac.at Thu Feb 23 08:08:39 1995
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Date: Thu, 23 Feb 1995 14:03:59 +0100
From: Peter Paule
Message-Id: <199502231303.AA01554@zeder>
To: zeilberg@euclid.math.temple.edu
Status: RO
-------begin checker's form---------------------------
Dear Doron,
I have now completed checking (sub)^[]lemma [152]
(i.e., Subsublemma 1.5.2).
To the best of my judgment, it is correct, provided
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!THE STATEMENT !
! "$\Delta_k(x)$ divides $\Psi_k(x)$" !
!(see the proof of Sublemma 1.5) HOLDS (i.e., !
!$\Omega_k(x)$ is indeed a POLYNOMIAL), AND !
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
all its offspring (sub)^[]-lemmas are correct.
Needless to say, I don't know (and don't care!) whether
the proofs of these offspring (sub)^[]-lemmas are actually
correct.
I have found the following typos or other errors: NONE
I have the following suggestion to improve the exposition:
At the beginning of the proof of (1.5.2) I would write
something like,
"By W(B_k) symmetry, it is enough to prove (1.5.2a) for
x_1=x_2^{-1}, which also implies (1.5.2b) as in the proof
of (1.5.1)."
ADDITIONAL COMMENT: Concerning the STATEMENT above, probably
I am just too silly to see, in a "finite" amount of time, how
it goes. There might be an easy quickie-solution to it.
If yes, please tell me.
Best Wishes,
Peter
From robbins@ccr-p.ida.org Mon Jan 16 23:13:36 1995
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From: David Robbins
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Date: Mon, 16 Jan 1995 23:13:23 -0500
Message-Id: <199501170413.XAA23813@runner.ccr-p.ida.org>
To: zeilberg@euclid.math.temple.edu
Subject: asm paper
In-reply-to: Doron Zeilberger's message of Mon, 2 Jan 1995 15:07:23 -0500 <199501022007.PAA09255@euclid.math.temple.edu>
Status: RO
Dear Doron,
Against my better judgment I have read my part and sort of checked it.
At least it was a nice short part.
It contains a few small mistakes and perhaps some stylistic infelicities.
It might be better if it read:
For each term of the polynomial \PHI_k there are at least i distinct
x's that appear as factors with multiplicity (ERROR was just to the
right) >= k-i. But the denominator of the constant termand has at
k-i+1 x's with exponent (similar ERROR) <=k-i-1. By the pigeonhole
principle, these two sets must have a non-empty intersection. Say x_i
appears in the intersection. Then, regarded as a Laurent series in
x_i, our constant termand is a positive power of x_i times a power
series in x_i. Thus its constant term with respect to x_i is zero....
Hope this helps.
David
THus the contribution of each term of PHI_k has zero constant term.
David
From ROUSSEAC@hermes.msci.memst.edu Tue Jan 24 15:13:07 1995
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Organization: Mathematical Sciences
Date: 24 Jan 95 14:13:00 CDT
Subject: ASM
Priority: normal
X-Mailer: Pegasus Mail v2.3 (R5).
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Status: RO
Doron:
I agree that subsublemma 1.5.2.1, case II is correct. The only
problem I had was that in the printed version, one factor didn't
show up because that line was too long. The factor that wasn't
printed out was (1 - x_1 z_{\pi(j)}). Of course, I just checked
the TeX file and there it was. Sorry for not getting around to
this earlier. I spent just about all the available time between
semesters trying to finish up a book.
Cecil
From sagan@math.kth.se Fri Mar 3 10:12:49 1995
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From: Bruce Eli Sagan
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Date: Fri, 3 Mar 95 16:12:31 +0100
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To: zeilberg@math.temple.edu
Subject: checking
Status: RO
Shalom Doron,
I've checked Sub^3Lemma 1.5.2.2 and it's correct. I even did
Sub^4Lemma 1.5.2.2.1 by hand to make sure everything was OK.
My only comments have to do with exposition. First of all, you've
changed typeface for the epsilons and changed notation by letting z_i be
x_i^{e_i} rather than e_i(x_i). In fact I would suggest
forgetting about the z's and just putting an epsilon acting on the
products in (Bruce). Furthermore, there's no need to write out everything
again in (Bruce'). Just say:
Every term of (Bruce) has a factor
(e_1,e_2,_e_3) [ (1 - x_{pi(a)} xbar_{pi(b)}) ...]
where {pi(a),pi(b),pi(c)} = {1,2,3}.
(Of course you should fill in the dots with the rest of the factor.)
Lehitraot, Bruce
From strehl@immd1.informatik.uni-erlangen.de Sun Feb 19 13:12:24 1995
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From: Volker Strehl
Message-Id: <199502191812.AA03294@faui10.informatik.uni-erlangen.de>
Date: Sun, 19 Feb 95 19:12:33 +0100
To: zeilberg@euclid.math.temple.edu
Subject: Re: Just checking
Cc: strehl@immd1.informatik.uni-erlangen.de
Status: RO
Dear Doron,
First let meck check what messages I received from you recently:
Mon Feb 13 21:59:55 1995 : two lines "thanks for the reminder ..."
Tue Feb 14 15:15:35 1995 : you say that you system has been sick ,,,
Wed Feb 15 15:39:38 1995 : Some Announcements (Mathematical Comrades WW)
Wed Feb 15 15:44:17 1995 : Your Catalan paper
Fri Feb 17 00:29:21 1995 : Ekhad's paper
Fri Feb 17 22:20:38 1995 : Just checking
What is missing, if my interpolation from the contents from your messages
is right, is the first announcement that you are going to submit your
ASM-paper to the Foata Festschrift. I learned it indirectly from your
message of Feb 14, and then read the public announcement one day later.
Doron, as an editor I feel very much honoured by your intention to give
your 'magnum opus' to the Foata issue, but I simultaneously feel that this
loads a lot of responsibility on the shoulders of the editors of the
Festschrift and of the Journal, given the dimensions of your article,
its complexity, and the unprecedented way of presenting it together
with "local proofchecking" etc. Frankly speaking, the perspective of becoming
responsible for the proper and fair handling of this opus intimidates
me somewhat and made me dumbfounded for a while. I will certainly need
the help of an experienced editor (such as Herb Wilf) in order to master
this unusual situation appropriately.
Coming back to the refereeing proper now: I read thoroughly 1.4.1.5
and checked it, including 1.4.1.5.1. In my opinion, this is o.k., I
only found a nonserious misprint on p. 68: in the third identity,
the one after (David), the index of x on the r.h.s. is an "R" and not
an "i". Furthermoore, the next to last sentence of the proof of 1.4.1.5
(p.69) reads somewhat incomplete. I mean the place where you write:
"... one proves that analogs ...". By the way: at this place exactly,
I must confess that I have not written out a proof of analogs of
(Noam) and (\overline{Noam}), which you claim to be "exactly the same"
as case II of (sub)^3lemma 1.4.1.4. I have looked at it and I cannot see
what should go wrong, but, as I said, I have not pendantically worked it out.
I have, however, followed your suggestion and worked out a proof of
(Michal') (...left to the reader...), which indeed goes along familiar lines,
though a tiny bit different. If I were a referee in the traditional sense,
I would be satisfied with 1.4.1.5.1 as it stands, but in this particular
situation one might as well go to the end.
That is what I could do for you at the moment, Doron. Our winter term
tends towards its end and thus time is scarce presently.
Finally, let me formally acknowledge the receipt of the Shalosh+Majewicz
paper. Thank you, and give my thanks to the authors.
Best regards,
Volker
From avanissi@math.u-strasbg.fr Mon Aug 8 05:57:05 1994
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Date: Mon, 8 Aug 1994 11:57:35 +0200
Message-Id: <199408080957.AA19879@isis.u-strasbg.fr>
To: zeilberg@euclid.math.temple.edu
From: avanissi@math.u-strasbg.fr
Subject: message from R.Supper: Sublemma 1.5.
Status: RO
Dear Prof. D.Zeilberger,
I checked the proof of the sublemma 1.5. of your paper about the
alterneting
sign matrix conjecture. I only found a little typesetting error p.41 (and
p.44):
it seems to be L_k - (-1)^k S_k^2 instead of L_k - S_k^2 ... or maybe the
(-1)^k
is included in the definition of L_k ? But it is not very important and
doesn't change anything to the recurrence.
With best regards,
sincerely yours, Raphaele Supper.
From foata@math.u-strasbg.fr Tue Feb 21 09:44:51 1995
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Date: Tue, 21 Feb 1995 15:44:32 +0100
Message-Id: <199502211444.PAA20299@isis.u-strasbg.fr>
To: zeilberg@euclid.math.temple.edu
Subject: Re: Messages from DZ
Status: RO
Dear Doron,
Here are the first corrections made by Jiang Zeng
on his program:
SubLemma 1.2
The argument is correct.
Only one correction : just after the proof of Subsublemma 1.2.1
the signature of the permutation is missing in Vandermonde's
expansion.
That's all, for the moment.
Best regards,
Dominique
From labelle.gilbert@uqam.ca Sat Jan 14 17:58:50 1995
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From: labelle.gilbert@uqam.ca (Gilbert Labelle)
Message-Id: <9501132301.AA03619@mipsmath.math.uqam.ca>
Subject: my lemma checking
To: zeilberg@euclid.math.temple.edu
Date: Fri, 13 Jan 95 18:01:18 EST
Cc: gilbert@mipsmath.math.uqam.ca (Gilbert Labelle)
X-Mailer: ELM [version 2.2 PL14]
Status: RO
Dear Doron,
I have now completed checking my (sub)^5 lemma 1.4.1.4.1.1.
Everything looks sound except for the following typo errors(?) :
1) Every occurence of x_1 in the statement and proof of the
(sub)^5 lemma should (I think) be replaced by x_pi(1). There
is also one more x_1 to be changed a few line after the proof.
2) Each of the two geometric series in the proof should (I think)
have a "barred parent" with z_r replaced by zbar_r.
3) The "/" (set-difference) in the proof should be replaced
by the more traditional "\".
Best wishes, Gilbert
P.S.1. I am not familiar with TEX, so that there may be some
TEX-typo errors in the above.
P.S.2. I have done my checking from a printed hard copy that
I got from your Nov. 7 (1994) e-mail. The hard copy has been
printed for me by a post-doctoral student (I know less printing
e-mailed TEX files than I know TEX itself ...)
P.S.3. I like the epithet you wrote for me but I would like my
e-mail address to be changed to : labelle.gilbert@uqam.ca
P.S.4. Good luck for your paper and happy new half decade
to you and your family from Helene and myself.
--
Gilbert Labelle tel : (514) 987-6168
LACIM - Dept. math. et info. fax : (514) 987-8477
Universite du Quebec a Montreal gilbert@lacim.uqam.ca
C.P. 8888, Succ. Centre-ville labelle.gilbert@uqam.ca
Montreal (Quebec)
CANADA H3C 3P8
From viggo@nada.kth.se Wed Nov 16 08:29:41 1994
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Date: Wed, 16 Nov 1994 14:23:04 +0100 (MET)
From: Viggo Kann
Subject: Re: Invitation to be co-checkers of my ASM paper
To: zeilberg@euclid.math.temple.edu
In-Reply-To: <9411071553.AA10336@ralbag.math.temple.edu>
Message-Id:
Status: RO
Dear Doron,
I have now completed checking subsublemma 1.1.2.
To the best of my judgment, it is correct, provided
that subsublemma 1.1.1 is correct.
Needless to say, I don't know (and don't care!) whether
the proof of subsublemma 1.1.1 is actually correct.
Best Wishes,
Viggo Kann
P.S. I like the epithet you wrote for me
From sparnes@cecm.sfu.ca Sat Nov 12 22:01:41 1994
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From: sparnes@cecm.sfu.ca (Sheldon Parnes)
Message-Id: <9411122201.AA02523@cecm.sfu.ca>
Subject: My section
To: zeilberg@euclid.math.temple.edu
Date: Sat, 12 Nov 1994 14:01:49 -0800 (PST)
X-Mailer: ELM [version 2.4 PL22]
Content-Type: text
Content-Length: 683
Status: RO
I can't seem to find the submission form but I have read over
Sublemma 1.3.
Everthing looks fine except you seem to have changed notation from
the first chapter.
I.e., $\epsilon f(x) = f(\epsilon_1(x_1),\ldots,\epsilon_n(x_n))$
has now become
$\epsilon f(x) = f(x_1^\epsilon_1,\ldots,x_n^\epsilon_n).$
You should probably change the introduction to reflect this notation
or just stick to the original notation.
You may want to add a comment on why $f_{n,k}(x_1,\ldots,x_k)$
has a Laurent series.
Sheldon
I will try to look at Jonathon's and Peter's sections.
I seem to remember Jonathon's section had an incorrect
index for the product but I didn't print out his chapter.
From barcelo@nelligan.la.asu.edu Sun Dec 11 22:59:11 1994
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Date: Sun, 11 Dec 1994 21:02:17 -0700
From: Helene Barcelo
Message-Id: <199412120402.VAA10811@nelligan.la.asu.edu>
To: zeilberg@euclid.math.temple.edu
Subject: paer
Status: RO
Dear Doron,
I recieved a while ago your paper
with my portion to referee. I did not have the time to get to it yet.
I am know in Monrteal for 2 weeks.
I wanted to do my "share" (=lemma 2.1.1) but will certainly
not be able to get to it before midjanuary.
Is this too late for you?
Hope evrything is fine with you,
Best
Helene
From gaurav@math.ohio-state.edu Thu Dec 8 12:05:18 1994
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From: Gaurav Bhatnagar
Message-Id: <199412081705.MAA24195@math.mps.ohio-state.edu>
Subject: Checking ASM
To: zeilberg@euclid.math.temple.edu (Professor Doron Zeilberger)
Date: Thu, 8 Dec 1994 12:05:23 -0500 (EST)
X-Mailer: ELM [version 2.4 PL23]
MIME-Version: 1.0
Content-Type: text/plain; charset=US-ASCII
Content-Transfer-Encoding: 7bit
Content-Length: 1694
Status: RO
Dear Professor Zeilberger,
As you suggested when we met at the Richmond meeting in November,
I took the task of checking
(Sub)^4lemma 14151 which you had orignally assigned to my advisor
(Prof. Steve Milne).
As you know, graduate students rarely lose an opportunity to thrust
their efforts on unsuspecting gurus. I am sending a paper, joint with
my advisor, concerning Inverse Relations. We have done some examples
in the paper.
I had read your tutorial on shift operators, and that led to my using them
to manipulate recurrences. I hope you like the use we made of them.
(See the section on Krattenthaler's inversion). The TeX file is in the
next(or previous) message. That is all I have done so far, besides
the stuff we presented in the November meeting.
Please put me on your list for announcements of papers/
programs/jokes/puns/etc
Here's your Checker's form:
-----------
(Fill in the blanks [], and delete the superfluous parts)
Dear Doron,
I have now completed checking (sub)^[4]lemma [1.4.1.5.1].
To the best of my judgment, it is correct, provided
all its offspring (sub)^[]-lemmas are correct.
Needless to say, I don't know (and don't care!) whether
the proofs of these offspring (sub)^[]-lemmas are actually
correct.
and (optional)
I have found the following typos or other errors:
I found the following typo on Page 68, on the fifth line from the top:
It should be a power series in x_R, so Replace x_i^t in the sum
with x_R^t.
Best Wishes,
Gaurav Bhatnagar
The Ohio State University
gaurav@math.ohio-state.edu
P.S. I'm sure I will like the epithet you will write for me.
---------end checkers form----------------
From bjorner@math.kth.se Thu Jan 5 12:39:41 1995
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From: Anders Bjorner
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Date: Thu, 5 Jan 95 18:40:00 +0100
Message-Id: <9501051740.AA00272@virus.math.kth.se>
To: zeilberg@euclid.math.temple.edu
Subject: Re: ASM Checking Update
Status: RO
Dear Doron,
I have now done the checking you requested of sublemma 1.1.1. I am
happy to report that it is correct (to the extent I can be considered
to have a grasp of mathematical correctness).
Two minor editorial comments:
* in the 4th row of the statement of sub^2lemma 1.1.1 I would add
"$ n \ge k \ge 1 $ for completeness"
* in the last row of the statement of sub^3lemma 1.1.1.1 you should
say something about a_1, as you do in condition (C4) a little later,
e.g. "a_1=0 or 1"
A much more minor comment:
* the permutation 123546 should be applied to the spelling of my
first name in your paper
I hope all checker's reports are favorable.
Happy New Year!
Anders
From bjorner@math.kth.se Thu Jan 5 12:39:41 1995
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From: Anders Bjorner
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Date: Thu, 5 Jan 95 18:40:00 +0100
Message-Id: <9501051740.AA00272@virus.math.kth.se>
To: zeilberg@euclid.math.temple.edu
Subject: Re: ASM Checking Update
Status: RO
Dear Doron,
I have now done the checking you requested of sublemma 1.1.1. I am
happy to report that it is correct (to the extent I can be considered
to have a grasp of mathematical correctness).
Two minor editorial comments:
* in the 4th row of the statement of sub^2lemma 1.1.1 I would add
"$ n \ge k \ge 1 $ for completeness"
* in the last row of the statement of sub^3lemma 1.1.1.1 you should
say something about a_1, as you do in condition (C4) a little later,
e.g. "a_1=0 or 1"
A much more minor comment:
* the permutation 123546 should be applied to the spelling of my
first name in your paper
I hope all checker's reports are favorable.
Happy New Year!
Anders
From erc@pollux.cs.uga.edu Mon Jan 2 15:30:33 1995
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Date: Mon, 2 Jan 95 15:23:03 EST
From: erc@pollux.cs.uga.edu (E Rodney Canfield)
Message-Id: <9501022023.AA11639@ajax>
To: zeilberg@euclid.math.temple.edu
Subject: my tardy report
Status: RO
Dear Doron,
I have now completed checking (sub)^[3]lemma [1.3.1.1].
Needless to say, I don't know (and don't care!) whether
the proof of the offspring (sub)^[4]-lemma is correct.
I have found the following typos:
In (PP1), the rightmost term, the numerator, the exponent
on z_i should be {k-2}, instead of {k-1}.
Consequently, in (Ai), the term immediately after the = sign,
the denominator, the exponent on z_i should be {k+1}, instead
of k.
Best Wishes,
Rod Canfield
P.S. I like the epithet you wrote for me
AND
I would like for you to change "Canfiled" to "Canfield", line 9,
page 84, in the EXODIAN.
From scooper@math.wisc.edu Tue Nov 29 15:08:48 1994
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From: Shaun Cooper
Message-Id: <9411292003.AA26402@schaefer.math.wisc.edu>
Subject: checker's report
To: zeilberg@euclid.math.temple.edu
Date: Tue, 29 Nov 1994 14:06:15 -0600 (CST)
In-Reply-To: <9411071546.AA10292@ralbag.math.temple.edu> from "zeilberg@euclid.math.temple.edu" at Nov 7, 94 10:46:21 am
X-Mailer: ELM [version 2.4 PL21]
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Status: RO
Dear Doron,
I have now completed checking subsubsublemma 1.1.1.3.
To the best of my judgment, it is correct. The sequence
X_k ( n; a_1, ..., a_k ) is well defined.
There are no offspring sub^4 lemmas on which subsubsublemma 1.1.1.3
depends.
Best Wishes,
Shaun Cooper
p.s.
I would like to modify my "biography" to read as follows:
(Please note the typo in the spelling of my name in the earlier version.)
Shaun Cooper, Wisconsin, scooper@math.wisc.edu, is currently
completing his thesis under Dick Askey, and has proved a conjecture
of P. Forrester.
From fraenkel@wisdom.weizmann.ac.il Wed Jan 4 05:20:25 1995
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From: Fraenkel Aviezri
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Date: Wed, 4 Jan 1995 12:19:26 +0200
Message-Id: <199501041019.MAA10632@silver.wisdom.weizmann.ac.il>
To: zeilberg@euclid.math.temple.edu
Status: RO
Shalom Doron, before I even begin, I have to clarify a few points.
1. P5, l 8,9: it seems to me that x_k --> x_1 is missing.
2. P5, l 10,11. I don't understand: what's PI, what's sgn(PI)?
3. P5, l -3. If I follow the definition on line -4, I get x_2^2 +
2(1-x_3) + (1-x_1^3), rather than what's written there.
4. P5, middle. It seems to me that W(B_k) is not well-defined. You
write it consists of pairs (PI, EPS)...Which pairs? Presumably you
mean ALL 2^k pairs? I can then see that it's a group.
5. P52. (To save me time.) There are a few x with hats in the formula
you want me to check. Where is the hat notation explained?
Thanks, Aviezri.
From sparnes@rawhide.cecm.sfu.ca Wed Nov 23 16:25:24 1994
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From: sparnes@rawhide.cecm.sfu.ca (Sheldon Parnes)
Message-Id: <9411232125.AA25483@rawhide.cecm.sfu.ca>
Subject: Jonathan's report
To: zeilberg@euclid.math.temple.edu
Date: Wed, 23 Nov 1994 13:25:35 -0800 (PST)
X-Mailer: ELM [version 2.4 PL22]
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Status: RO
Jonathan looked at his section and pronounced it fit.
Sheldon
From johnson@math.psu.edu Tue Dec 13 18:43:06 1994
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From: Warren P Johnson
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Date: Tue, 13 Dec 94 18:43:20 EST
Message-Id: <9412132343.AA17233@bernoulli.math.psu.edu>
To: zeilberg@euclid.math.temple.edu
Subject: Andrews' Checking
Status: RO
Dear Doron,
George Andrews and I finally got together today, and he agrees that Lemma
1 is correct if all sublemmas, in particular 1.1''', 1.2''' and 1.5, are
correct. For my part, I will certify that sublemmas 1.2.1.2.1.1.1 and
1.2.1.2.1.1.2 are correct.
I have another list of typos also. There is a consistent problem with
sentences that end with a parenthetical phrase--the ")" should precede
the ".". Only when an entire sentence is contained in parentheses
should you have ".)". See p. 34, lines 3 and -2, for examples where
you have it right. Similarly when a parenthetical phrase is followed by
a comma, it should look like ---), rather than ---,). For typos of this
sort I'll use the abbreviations ")." and ")," respectively.
). isn't really that serious an error, but ), looks bad.
12, line -4: ).
14, line -10: ).
15, line 6: ),
16, line 7: should have (1\le i
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From: Warren P Johnson
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Date: Thu, 1 Dec 94 21:50:16 EST
Message-Id: <9412020250.AA11236@bernoulli.math.psu.edu>
To: zeilberg@euclid.math.temple.edu
Subject: Misprints
Status: RO
Dear Doron,
I just printed off your ASM paper (having waited for a slack period in
the computer lab) and looked through it. The list of checkers was great
fun to read. If all these people are really reading their epithets then
I guess they can tell you these things, but anyway:
Several words are singular in Noga Alon's bio that should be plural, e.g.
combinatorialist.
In Askey's bio, the Askey in Askey-Gasper is misspelled.
Bressoud presumably has an e-mail address at Macalester now, and the
correct title of one of his books is "A Radical Approach to Real Analysis."
Shaun Cooper's first name is misspelled both here and on the title page.
Dieudonne is misspelled in Ehrenpreis' bio, and Macdonald in Gustafson's.
In Hanlon's bio, "cohomology" is misspelled.
In McGuinness' bio, Swinnerton-Dyer is misspelled, and two occurrences of
"xxxx" need to be filled in.
In Jamie Simpson's bio, first sentence, "to" should surely be something
more like "of". I also note idiosyncratic uses of "to" in the epithets
of Leroux, Robbins, Rota and Stanley, and of "in" in Maley's.
Dennis Stanton's affiliation is listed as Wisconsin, when it should be
Minnesota (unless something's happened that nobody's told me about).
In the proof skeleton, 1.3.1.4, P. Zimmerman occurs with one "n", but
it occurs with 2 "n"'s on the title page and in his epithet. I don't
know which is right myself, but you probably do. In Subcase Ib, K. Eriksson
is missing an "s". In 1.4.1.4.1, Krattenthaler has an extra "e". Also on
p. 83, last line before the names begin, "solely" is misspelled.
Otherwise, I've looked through the introduction. On page 4, line -5,
in the definition of an n-Gog triangle, I don't doubt that you really
mean what you say, that the bottom row should be 1,2,...,n. I found
it easier to think of this as the top row, so that a_{ij} has the same
meaning it would have in a matrix.
On page 7, line 4, in the definition of W(B_k)-antisymmetric, is there
an overbar only on x_1, or on the other x_i's also?
There are, in my opinion, a few misplaced commas. On page 6, line 13,
I think it should read "...to our proof)," rather than "...to our proof,)".
If the parenthetical phrase were removed, you would want a comma after the
preceding word W(B_k). On page 10, line 10 I think it should read "People
who, like myself (and John Riordan), are horrified..." or possibly "People,
like myself and John Riordan, who are horrified..." or something like that.
In its present form, in effect, you have two consecutive commas, as they are
separated only by a parenthetical phrase.
I expect I'll have a big batch of homework to grade in the next several
days, but after that I should be able to do a little checking of the
actual mathematics. Should I infer from your telling me a few days ago
that Andrews and Askey refuse to do their part that I should check what
you have them down for? Or should I look more at Acts III and IV?
Warren
From kimmo@nada.kth.se Thu Dec 8 07:59:32 1994
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Date: Thu, 8 Dec 1994 13:59:45 +0100 (MET)
From: Kimmo Eriksson
To: zeilberg@euclid.math.temple.edu
Subject: Re: Invitation to be checkers of the ASM paper
In-Reply-To: <9411071546.AA10292@ralbag.math.temple.edu>
Message-Id:
Mime-Version: 1.0
Content-Type: TEXT/PLAIN; charset=US-ASCII
Status: RO
Dear Doron,
I have now completed checking (sub)^3-lemma 1.3.1.4, subcase 1b.
To the best of my judgment, it is correct.
I have found the following typos or other errors:
$z_i^{n+2k-i}$ in the denominator should, I think, be $z_i^{n+k-i}$
both on the first and second line of page 45.
Best Wishes,
Kimmo Eriksson
P.S. I like the epithet you wrote for me,
AND I like this whole idea of parallel-processing checking! Good
work, Doron!
From ckratten@euclid.ucsd.edu Fri Dec 9 18:29:37 1994
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Date: Fri, 9 Dec 94 15:29:51 PST
From: ckratten@euclid.ucsd.edu (Christian Krattenthaler)
Message-Id: <9412092329.AA00534@euclid>
To: zeilberg@euclid.math.temple.edu
Subject: ASM
Status: RO
Dear Doron,
I went through my part (1.4.1.4.1) of your ASM proof. It appears to me
that you have a misprint there: On pp.61/62 POL(x_1... should read
POL(x_{\pi(1)},... Am I right or did I overlook something?
All the best
Christian
From KRATT@Pap.UniVie.AC.AT Tue Dec 13 08:45:10 1994
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Date: Tue, 13 Dec 1994 14:44 GMT+0100
Subject: ASM
To: zeilberg@euclid.math.temple.edu
Message-id: <01HKLK4ZOB409FNS1M@Pap.UniVie.AC.AT>
X-VMS-To: IN%ZEILBERGER
X-VMS-Cc: KRATT
Status: RO
Dear Doron,
I have now completed checking (sub)^4-lemma 1.4.1.4.1.
To the best of my judgment, it is correct, provided
its offspring (sub)^5-lemma 1.4.1.4.1.1 is correct.
Needless to say, I don't know (and don't care!) whether
the proof of this offspring (sub)^5-lemma is actually
correct.
I have found the following typos or other errors: Throughout pages
61/62 it should read POL(x_{\pi(1),... instead of POL(x_1,...
(also once in connection with POL_t).
Best wishes,
Christian Krattenthaler
P.S. I like the epithet you wrote for me.
From KRATT@Pap.UniVie.AC.AT Tue Dec 13 08:45:10 1994
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<01HKLK4ZOB409FNS1M@Pap.UniVie.AC.AT>; Tue, 13 Dec 1994 14:44 GMT+0100
Date: Tue, 13 Dec 1994 14:44 GMT+0100
Subject: ASM
To: zeilberg@euclid.math.temple.edu
Message-id: <01HKLK4ZOB409FNS1M@Pap.UniVie.AC.AT>
X-VMS-To: IN%ZEILBERGER
X-VMS-Cc: KRATT
Status: RO
Dear Doron,
I have now completed checking (sub)^4-lemma 1.4.1.4.1.
To the best of my judgment, it is correct, provided
its offspring (sub)^5-lemma 1.4.1.4.1.1 is correct.
Needless to say, I don't know (and don't care!) whether
the proof of this offspring (sub)^5-lemma is actually
correct.
I have found the following typos or other errors: Throughout pages
61/62 it should read POL(x_{\pi(1),... instead of POL(x_1,...
(also once in connection with POL_t).
Best wishes,
Christian Krattenthaler
P.S. I like the epithet you wrote for me.
From loeb@organon Thu Dec 22 04:19:18 1994
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Date: Thu, 22 Dec 94 10:18:00 +0100
From: loeb@organon (Daniel Elliott Loeb)
Message-Id: <9412220918.AA09819@organon.labri>
To: zeilberg@euclid.math.temple.edu
In-Reply-To: zeilberg@euclid.math.temple.edu's message of Mon, 7 Nov 94 10:53:50 EST <9411071553.AA10336@ralbag.math.temple.edu>
Subject: Invitation to be co-checkers of my ASM paper
Reply-To: loeb@labri.u-bordeaux.fr
Status: RO
Dear Doron,
I have partially completed checking (sub)^[]lemma [ 1.4.1.5(1b)].
I have found the following typos or other errors:[ ]
I have the following suggestion to improve the exposition:[ ]
* In the introduction to the case, give z_i = ... instead of (or in
addition to) \overline{z_i} = ...
* Use an \mbox for Coeff... Otherwise, you get "C oe f f".
* Emphasize that RAT_t is not the RAT_t of the preceeding page,
actually it is \sum_{i+j=n-t} RAT_i NUMBER_j.
I have found the following typos or other errors:[ ]
* In the displayed equation following (Gil), I don't see why the upper
limit of summation in 2k+1. On the other hand, you don't seem to use
this later.
* In the last displayed equation of p. 65 (and the first of p. 66), I
think \overline{x_R}^{t+1} should be \overline{x_R}^{t-1}.
Following this through you get R+2t-1 \leq 2(n+R+t) after (Gil''')
which is still true, so you can still apply \aleph'_5 and complete
your proof.
Please tell me why I am wrong (or send me a nevised 1.4.1.5) and I'll
reread it.
P.S. I like the epithet you wrote for me
You might want to mention my "negative thinking" or my "new era in
umbral calculus".
I'll be with my inlaws this next week. If you still would like to
upgrade me, I would then need a version with equation numbers after
that.
Yours, Daniel Loeb, loeb@labri.u-bordeaux.fr
WWW http://www.labri.u-bordeaux.fr/~loeb/
HOME 150, cours Victor-Hugo; Appt A6; 33000 Bordeaux France
WORK LABRI; Universite de Bordeaux I; 33405 Talence Cedex France
PHONE(H)+(33) 56 31 48 26, (W)+(33) 56 84 69 05, (FAX) +(33) 56 84 66 69
Cubum autem in duos cubos, aut quadratoquadratum in duos
quadratoquadratos, et generaliter nullam in infinitum ultra quadratum
potestatem in duos ejusdem nominis fas est dividere: cujus rei
demonstrationem mirabiliem sane detexi. Hanc siginis exiguitas non
caperet.
From loeb@organon Mon Dec 5 08:05:47 1994
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Date: Mon, 5 Dec 94 14:04:59 +0100
From: loeb@organon (Daniel Elliott Loeb)
Message-Id: <9412051304.AA08836@organon.labri>
To: zeilberg@euclid.math.temple.edu
In-Reply-To: Doron Zeilberger's message of Thu, 1 Dec 1994 08:56:25 -0500 <199412011356.IAA09959@euclid.math.temple.edu>
Subject: Bernstein pols.
Reply-To: loeb@labri.u-bordeaux.fr
Status: RO
Dear Doron,
Here are some initial comments to ASM.
I haven't finished but if you respond to these I'll be able to finish
quicker. Since I'm currently finishing several papers simultaneously!
p.1: Is an equivalence relation that identifies a great many things
called "weak"... I'd say "strong". Better to use "coarser" since
everyone would agree on that.
p.3: ... by GETting ***the*** file ...
Give some examples of the objects being bijected to make sure the
reader has understood the definitions. Perhaps give the bijection for
n < 4....
"... bottom row... "
Until this point you gave no indication of what order the rows are to
be written, and in my examples (see above) I had adopted the opposite
convention. Thus, at this point it took me an extra moment to
understand what you were saying.
p.4: comma after "more generally"
p.5: why do you say "k-continuous space" as opposed to "discrete
k-space" to the same page....
add the word "fact" after "this".... (Generally, "this" is vague when
standing alone.)
Move comma out of parenthesis ")," instead of ",)"
p.6: No matter how many times you explain yourself using \overline{t}
for 1-t is bound to confuse someone. (I was.) Why not adopt \hat{t} or
some other symbol with less prejudices.
p.7: Finish your proof with a box.
Write out i.e. as that is
Follow i.e. with a comma
p.8: "High degree" has not been defined yet.
p.9: Functions like "sgn" and "Res" should be set in Roman face (like
sin log and exp in LaTeX).
Put boxes after results to simple to be proven to indicate the lack of
a proof.
p.49: Is this the part I'm supposed to check. You didn't send me the
Exodion so I don't know what I'm supposed to do.
"the residuand of (GogTotal').... "
"the polynomial defined in (Gog_1) ..."
The only way I can check such a claim is to look through every page of
your document (many pages of which I have not printed) and look for an
equation marked GogTotal'.... If you used equation NUMBERS, or gave a
page number along with the equation NAME, then I could easily verify
such a reference. Unless you do this, I'm afraid the refereeing will
take much longer than I anticipated.
"Let ... , then ..."
is ungrammatical. Replace with "If ..., then ..." or "Let ... . Then
..."
Replace "Right above" with "equation (...)" or "above".
In the displayed eqn of sslemma 1.4.1 you use \overline{x}_{\pi(u)}
for 1/x_{\pi(u)}... Doesn't this conflict with your notiation.
Equation (Ai1) should probably be (A_i 1)
p.53: Insert (...) into (...) has undesired connotations when the equations
are named after people!!!!
p.57: "right of (..)" should read "right side of equation (..)"
p.61: The above remark about Res and sgn goes doubly for "Coeff" since
LaTeX tries to separate the f in a funny way (to avoid ligature) as if
the expression was a product C x o x e x f x f. Use \mbox{Coeff}
p.70: "We have to introduce" should be "We introduce" since there is
no way to tell whether the rational function is "needed". (Perhaps
there is a proof that does not make use of this rational function.)
Yours, Daniel Loeb, loeb@labri.u-bordeaux.fr
WWW http://www.labri.u-bordeaux.fr/~loeb/
HOME 150, cours Victor-Hugo; Appt A6; 33000 Bordeaux France
WORK LABRI; Universite de Bordeaux I; 33405 Talence Cedex France
PHONE(H)+(33) 56 31 48 26, (W)+(33) 56 84 69 05, (FAX) +(33) 56 84 66 69
Cubum autem in duos cubos, aut quadratoquadratum in duos
quadratoquadratos, et generaliter nullam in infinitum ultra quadratum
potestatem in duos ejusdem nominis fas est dividere: cujus rei
demonstrationem mirabiliem sane detexi. Hanc siginis exiguitas non
caperet.
From bousquet@labri.u-bordeaux.fr Mon Nov 28 09:47:54 1994
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Date: Mon, 28 Nov 94 15:49:08 +0100
From: bousquet@labri.u-bordeaux.fr (Mireille BOUSQUET-MELOU)
Message-Id: <9411281449.AA01868@nastasia>
To: zeilberg@euclid.math.temple.edu
Cc: zeilberg@euclid.math.temple.edu
In-Reply-To: <9411071553.AA10336@ralbag.math.temple.edu> (zeilberg@euclid.math.temple.edu)
Subject: Re: Invitation to be co-checkers of my ASM paper
Status: RO
Dear Doron,
I have now completed checking (sub)^[5]lemma [ 1.2.1.2.3.1 ].
To the best of my judgment, it is correct, provided
all its offspring (sub)^[6]-lemmas are correct.
Needless to say, I don't know (and don't care!) whether
the proofs of these offspring (sub)^[6]-lemmas are actually
correct.
I have found the following typos or other errors:
the +/- sign in the statement of sub^6 lemma 1.2.1.2.3.1.1 is a bit
strange. Does it mean that you do not know (or do not care) what the
sign of A_i exactly is ? Maybe you could say a few words about that.
I have also noticed that this sign does not appear when the expression
of A_i is used is the proog of sub^5 lemma 1.2.1.2.3.1. But whatever
the sign is, the prook is OK...
Amicalement,
Mireille
From bergeron@mathstat.yorku.ca Wed Jan 4 11:59:03 1995
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Date: Wed, 4 Jan 1995 11:59:10 -0500
From: "Nantel Bergeron"
Message-Id: <199501041659.LAA21594@bayes.math.yorku.ca>
To: bergeron@bayes.math.yorku.ca, zeilberg@euclid.math.temple.edu
Status: RO
Dear Doron,
I have now completed checking (sub)^[4]lemma [1.2.1.2.1].
To the best of my judgment, it is correct, provided
all its offspring (sub)^[5]-lemmas are correct.
Needless to say, I don't know whether
the proofs of these offspring (sub)^[5]-lemmas are actually
correct.
I did not found any typos in my portion.
Best Wishes,
Nantel Bergeron
P.S. I like the epithet you wrote for me
Wg...WhREPORTSDUKHOTFrom craig@paris-gw.cs.miami.edu Wed Nov 9 15:23:51 1994
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From: Craig Orr
Message-Id: <9411092018.AA01029@paris-gw.cs.miami.edu>
Subject: ASM paper
To: zeilberg@euclid.math.temple.edu
Date: Wed, 9 Nov 94 15:18:55 EST
In-Reply-To: <9411071553.AA10336@ralbag.math.temple.edu>; from "zeilberg@euclid.math.temple.edu" at Nov 7, 94 10:53:50 am
X-Mailer: ELM [version 2.4dev PL19]
Status: RO
Dr. Z':
I have read through the proof of sub^(3) lemma 1.1.1.1.
I plan to read through it again this weekend, but it
seems pretty straightforward (once I understood all the
definitions and nomencluture). The proof makes sense and
I can find no flaws in it.
There are, however a couple of typos that I found:
(1) In the second line of the statement of the lemma,
the last sequence of inequalities has a typo. The
second a value should be a_{2} not a_{k}.
(2) In the 12th line of the proof, I believe the last
equation should be B_{1}(1;1)=1 rather than
B_{1}(1)=1.
(3) In the fourth line from the bottom of the proof, the
last expression should be (P \Delta E(B_{k})) rather
than (P \Delta E)(B_{k}).
As I said before, I will go over the proof again this
weekend.
By the way, the bio in the exodion is fine (although
"brilliant" is a stronger word than I would use), but
you did not list me (or Sheldon) in the title page.
CRAIG
From ppaule@risc.uni-linz.ac.at Wed Dec 14 14:01:27 1994
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Date: Wed, 14 Dec 1994 20:01:25 +0100
From: Peter Paule
Message-Id: <199412141901.AA02244@zeder>
To: zeilberg@euclid.math.temple.edu
Status: RO
-------begin checker's form---------------------------
Dear Doron,
I have now completed checking (sub)^[]lemma [151]
(i.e., Subsublemma 1.5.1).
To the best of my judgment, it is correct, provided
all its offspring (sub)^[]-lemmas are correct.
Needless to say, I don't know (and don't care!) whether
the proofs of these offspring (sub)^[]-lemmas are actually
correct.
I have found the following typos or other errors:
The proof needs the following
CORRECTION:
THROUGHOUT THE PROOF the denominator expression $2 x_i - 1$
(resp. $2 x_j - 1$) in the representations of $L_k(x)$ has to
be replaced by $1 - 2 x_i$ (resp. $1-2 x_j$).
TYPO: In the specification of the summation range "$\epsilon_1=1,
\epsilon_2=2$" has to be replaced by "$\epsilon_1=+1, \epsilon_2=+1"
(this occurrs twice).
I have the following suggestion to improve the exposition:
(A) It's not really an improvement, but an additional
remark to the necessary CORRECTION spelled out above. Namely,
one could keep the denominator expression $2 x_i - 1$
(resp. $2 x_j - 1$) in the representations of $L_k(x)$ by
a sign-modification. This, for instance, would correspond to dividing
the left hand side of "(Gog-Magog)" by "$(-1)^k \Delta_k(x)^2$",
instead of "$\Delta_k(x)^2$".
(B) It seems to me that for a signed-permutation
$g=(\pi,\epsilon)$ the sign $sgn(g) (=sgn(\pi) sgn(\epsilon))$
nowhere in the paper is defined explicitly.
Best Wishes,
Peter
P.S. I like the epithet you wrote for me, I only want to report
two typos arising there: replace "sorter" by "shorter", and
"Ramnujan" by "Ramanujan".
P.P.S.: These must be extremly exciting days for you. I strongly
hope that the checkers feedback is as positive as you expected,
and, of course, resulting in a carefully checked MAGNUM OPUS of
yours.
AGAIN, THANKS A LOT FOR INCLUDING ME IN YOUR HISTORICAL PARTY!
---------end checkers form----------------
From ram@math.wisc.edu Fri Dec 16 10:41:20 1994
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Date: Fri, 16 Dec 94 09:34:54 CST
From: Arun Ram
Message-Id: <9412161534.AA22388@schaefer.math.wisc.edu>
To: zeilberg@euclid.math.temple.edu
Status: R
To: zeilberg@euclid.math.temple.edu
-------begin checker's form---------------------------
(Fill in the blanks [], and delete the superfluous parts)
Dear Doron,
I have now completed checking (sub)^2lemma 1.5.1.
To the best of my judgment, it is correct.
Needless to say, I checked (by hand) the two
utterly trivial (sub)^3lemmas 1.5.1.1 and 1.5.1.2.
even though they are being checked by other checkers.
and (optional)
I have found the following typos or other errors:
1. In equation (1.5.1e) there is an "\epsilon_2=2" that should
be "\epsilon_2=1".
2. Similarly, in equation (1.5.1f), there is an "\epsilon_2=2"
that should be $\epsilon_2=1$.
3. It seems that in the first expression for $L_k$ in the beginning
of the proof of 1.5.1 the denominator contains $\prod_i (2x_i-1)$
whereas in the definition of $\Delta_k$ at the beginning of 1.5
one has $\prod_i (1-2x_i)$. It seems to me that this introduces
a factor of $(-1)^k$, . . . which affects the
definition of $L_k$.
If you use the definition of $L_k$ in the statement of 1.5.1
(rather than that used in the proof)
it seems to me that (1.5.1a) and (1.5.1b) remain the same
you get a factor of (-1) in (1.5.1c).
Outside of these minor things, I think that the argument is convincing.
If the computer verifies these identities for $k=3$ and $k=4$ then
I think that you can be sure that there are no errors.
Best Wishes,
Arun Ram
P.S. I like the epithet you wrote for me
---------end checkers form----------------
From readdy@catalan.math.uqam.ca Sat Dec 17 14:51:24 1994
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Date: Sat, 17 Dec 94 14:53:31 -0500
From: readdy@catalan.math.uqam.ca (Margaret Readdy)
Message-Id: <9412171953.AA15302@catalan.math.uqam.ca>
To: zeilberg@euclid.math.temple.edu
Subject: referee report
Reply-To: readdy@lacim.uqam.ca
Status: RO
-------begin checker's form---------------------------
Dear Doron,
I have now completed checking (sub)^3lemma 1.4.1.1.
To the best of my judgment, it is correct, provided
all its offspring (sub)^4-lemmas are correct
AND that equation (Bi2) is corrected (see below).
Needless to say, I don't know (but do hope!) that
the proofs of these offspring (sub)^4-lemmas are actually
correct.
and
I have found the following typos or other errors. Please
note that some of these corrections are actually suggestions
to make the reading a little smoother. I leave it up to you
whether you wish to use them or not.
INTRODUCTION
------------
page 5, l 4 the three dots after x_{\pi(1)} are squished next to
the comma
page 5, l 13 our proof),
page 6, l 14 delete comma after f(x_1, \ldots, x_k)
page 6, l -4 add comma after Q(0,0,\ldots, 0)
page 7, l 6 add comma after f(x)
page 7, l 8 you never defined what CT_x(fx) is, and more
generally, CT_{x_k}f(x_1, \ldots, x_k)
page 7, l 16 Let f be any rational function. Then
page 7, l -7 a polynomial in (the single variable) x
page 7, l -6 (i.e.,
page 7, l -6 = -P(x)). Then
page 7, l -1 change , to .
page 8, l 1 Then
page 8, l 3 if P is a monomial. Since a
page 8, l -6 (highest)
page 9, l -1 (highest)
ACT II
------
page 22, (Gog_1) The two products with indexes i = 1 to k and
1 \leq i < j \leq k are independent, i.e.,
the value of i in the second product
does not depend on the the value of i in the first.
Please add parentheses to eliminate this confusion.
ACT IV
------
page 49, l 4 change exponent n + 1 + i to n + i + 1
page 49, l -11 delete , after the smallest integer i
page 49, l -8 delete , after (1.4')
page 49, l -2 i \neq R).
page 50, l 1 change exponent n + 1 + j to n + j + 1
page 51, l 9 delete i.e.
page 51, l 12 in x_R, is
page 51, equation (Bi2)
-----------------------
I think there is a small mistake in the equation
for B_i, when R < i \leq k.
I've gone through the calculation by hand, and
there seems to be a missing multiplicative factor of
1
-------------
___ 3k-3 .
z_i
Also, rather than having a multiplicative factor of
(1 - z_i^2) in the numerator of eqn (Bi2), I calculated
a (1 + z_i).
In all, I am 99.99% sure that something is missing
from eqn (Bi2).
and (optional)
I have the following suggestion to improve the exposition:
Overall, the exposition of what I read is fine. My only concern is
that I am having some difficulty (when away from my computer and the
emacs search mode) finding the equations since they have names, rather
than numbers.
Best Wishes,
Margie Readdy
P.S. My original intention was to read through all of Act IV
since this is one of the sections which you expressed intense need to be
checked. However, I have been a little pressed for time, since I am
applying for jobs now. I would be happy to spend more time checking Act IV
over the holiday break, if you wish.
---------end checkers form----------------
From christo@mipsmath.math.uqam.ca Mon Nov 14 17:15:20 1994
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From: christo@mipsmath.math.uqam.ca (Christophe Reutenauer)
Message-Id: <9411141717.AA20548@mipsmath.math.uqam.ca>
Subject: check
To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
Date: Mon, 14 Nov 94 12:17:08 EST
X-Mailer: ELM [version 2.2 PL14]
Status: RO
Dear Doron,
I have checked your subsubsublemma 1.5.2.1 and think that it is correct.
Let me make some comments.
1.The fact that Omega_k is a Laurent polynomial depends on the fact that it is (before substitution) a
polynomial: this latter fact is asserted at the beginning of the proof of sublemma 1.5,
and I have admitted it.
2.You say that one may suppose that alpha < beta : this is the main difficulty in my understanding
of your proof. The symmetry is not clear, because of substitution (cf 3.). Perhaps you can say that
Phi_k in (Gog1) is symmetric in x1,...,x2, except for the sign (?); I have some problem because of 3.
I had not the courage to write down the whole argument in the case alpha > beta. Sorry.
3.I thought first (also yesterday at the airport) that the x and x bar are independant variables.
But this is not so, as indicated p.76 l.2 (which is very important for the degree calculation).
This means that the definition (Gog1) uses an abuse of language: first the variables are independant,
then apply the signed permutations, then put x bar = 1-x.
4.A small mistake, which does not affect the proof: in Case 1 and 2, the high degree of the product
of the four products times (1-x1 x2bar)(1- x1bar x2bar) is 2(k-2)+1 and not 2(k-2)+a,2 because
the factor 1-x1 x2 bar is equal to 1- x2^(-1) (1-x2) = 2 - x2^(-1) which is of high degree 0.
5.I did not see any mistake in the technical part (case 1 and 2).
Hope this helps.
Sincerely yours, Christophe
(Christophe Reutenauer, UQAM).
From reznick@symcom.math.uiuc.edu Tue Dec 13 15:46:45 1994
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Date: Tue, 13 Dec 1994 14:45:56 -0600
From: Bruce Reznick
Message-Id: <199412132045.AA03124@mira.math.uiuc.edu>
To: zeilberg@euclid.math.temple.edu
Subject: ASM ?'s
Status: RO
Dear Doron,
So I figured out the printer over the weekend. (I had to convert
the Tex-ed documents to a ps file, piece of cake in retrospect.) I looked
for my slice of your opus: 1.5.2.1 (and by the way, isn't the first "1"
superfluous because it's on everything, or is there a "2" waiting in the
wings?) One *dis*-advantage of checking by committee for you is that, with
my anonymity unmasked, I am free to bother you with queries! In particular,
and not having read the full paper, I have two questions regarding the first
sentence of the proof of (sub)^3 lemma 1.5.2.1:
(a) How do you know that \Delta_k restricted to x_1 = x_2^{-1} (which
I assume means x_1x_2 = 1) is a Laurent polynomial in x_2 of |degrees| \le
2k - 1?
(b) How do you know that the quotient is a Laurent polynomialin x_2 and
not just a Laurent series?
I am freely admitting my ignorance here, so please just give chapter and
verse! Another question reflects on your singular naming of the equations.
Is there a good mnemonic for them? Failing that, an index might be helpful.
Regarding your brief biography. My mother taught me to accept com-
pliments gracefully and without quiblle. However immodesty forces me to
suggest that the *post*-basis theorem Hilbert would have enjoyed seeing one
of his deepest results (The solution of Waring's Problem) applied to one
of his other deepest results (His work on the 17th Problem), as indicated
in the ms I sent you after you composed your encomium. Maybe not. Perhaps
when you are finished contacting J. J. Sylvester in the spirit world (as
you evidently have done in your magnificent paper), you can ask Hilbert
directly.
Yours two days before the deadline,
Bruc
From reznick@symcom.math.uiuc.edu Tue Dec 13 15:46:45 1994
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Date: Tue, 13 Dec 1994 14:45:56 -0600
From: Bruce Reznick
Message-Id: <199412132045.AA03124@mira.math.uiuc.edu>
To: zeilberg@euclid.math.temple.edu
Subject: ASM ?'s
Status: RO
Dear Doron,
So I figured out the printer over the weekend. (I had to convert
the Tex-ed documents to a ps file, piece of cake in retrospect.) I looked
for my slice of your opus: 1.5.2.1 (and by the way, isn't the first "1"
superfluous because it's on everything, or is there a "2" waiting in the
wings?) One *dis*-advantage of checking by committee for you is that, with
my anonymity unmasked, I am free to bother you with queries! In particular,
and not having read the full paper, I have two questions regarding the first
sentence of the proof of (sub)^3 lemma 1.5.2.1:
(a) How do you know that \Delta_k restricted to x_1 = x_2^{-1} (which
I assume means x_1x_2 = 1) is a Laurent polynomial in x_2 of |degrees| \le
2k - 1?
(b) How do you know that the quotient is a Laurent polynomialin x_2 and
not just a Laurent series?
I am freely admitting my ignorance here, so please just give chapter and
verse! Another question reflects on your singular naming of the equations.
Is there a good mnemonic for them? Failing that, an index might be helpful.
Regarding your brief biography. My mother taught me to accept com-
pliments gracefully and without quiblle. However immodesty forces me to
suggest that the *post*-basis theorem Hilbert would have enjoyed seeing one
of his deepest results (The solution of Waring's Problem) applied to one
of his other deepest results (His work on the 17th Problem), as indicated
in the ms I sent you after you composed your encomium. Maybe not. Perhaps
when you are finished contacting J. J. Sylvester in the spirit world (as
you evidently have done in your magnificent paper), you can ask Hilbert
directly.
Yours two days before the deadline,
Bruc
From reznick@symcom.math.uiuc.edu Thu Dec 15 12:45:36 1994
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From: Bruce Reznick
Message-Id: <199412151745.AA14458@spica.math.uiuc.edu>
To: zeilberg@euclid.math.temple.edu
Subject: And I thought April 15 was a tough deadline!
Status: RO
Dear Doron,
I have now completed checking (sub)^3 lemma 1.5.2.1.
To the best of my judgment, it is correct, provided
all its offspring (sub)^4-lemmas are correct and provided
all material up to that point is correct. One minor point.
If I understand your notation correctly (and I better!) the
expression (1- x_1\bar x_2)(1 - \bar x_1\bar x_2) appears
in both Case I and Case II. Upon setting x_1 = x_2^{-1},
this product is a Laurent polynomial in x_2 of nominal
high degree 2 and nominal low degree 2. In fact, we obtain
(1 - (1/x_2)(1-x_2))(1 - (1 - 1/x_2)(1 - x_2)) =
(2 - 1/x_2)(x_2 - 1 + 1/x_2),
which has high degree 1, not 2. This should necessitate a
small change in the writing so that the high degree of the
four products times the above is *bounded* by 2(k-2) + 2.
Of course this does not affect the truth of the proposition!
Best Wishes,
Bruce Reznick
P.S. I like the epithet you wrote for me.
P.P.S. Have you ever seen the 50's sci-fi movie "Gog"? If I
remember right, there was a battle between two computers --
Gog and (you guessed it) Magog.
From B.Salvy@newton.cam.ac.uk Wed Jan 4 07:00:16 1995
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Date: Wed, 4 Jan 95 11:59 GMT
From: "B. Salvy"
To: zeilberg@euclid.math.temple.edu
In-Reply-To: <199501021923.OAA06834@euclid.math.temple.edu> (zeilberg@euclid.math.temple.edu)
Subject: ASM
Status: RO
Dear Doron,
Thanks for the kind and subtle reminder. I had not really
understood that I had been assigned a specific section. I decided to
have a look at it yesterday night to decide whether I could do it
before Jan. 25. It turned out to be easier than what I thought. I am
able to claim:
"I AM NOW CONVINCED THAT IF SUBSUBLEMMA 1.2.1 IS WRONG, IT CAN ONLY
BE BECAUSE ONE OF ITS SUBSUBSUBLEMMAS DOES NOT HOLD."
Thus one more node of your tree is confirmed to be solid. Here are
various comments on what I've read:
On p.5 of the introduction, the example (231, +1 -1 -1) is not
consistent with the definition of how a signed permutation acts on a
function. It seems to imply that the definition is
\[(\pi,\varepsilon)f(x)=\pi\varepsilon f(x).\]
On p.22 (definition of an nxk-Gog), condition (v) does not appear on
p.3 of the introduction when the definition is first given.
On p.22 Eq. (Gog_1), sgn(g), the sign of a signed permutation has not
been defined previously.
On p.24, "Note that ($\tilde{M}_k$) still holds", I would rather say
"extends to", since $\tilde{M}_k$ is a definition.
A general comment is that a few pictures would help the understanding,
and another comment is that it is a pity it has to be so complicated.
Happy new year,
Bruno.
From @gwuvm.gwu.edu:SIMION@GWUVM.BITNET Wed Nov 9 11:52:52 1994
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Date: Wed, 09 Nov 94 11:46:50 EDT
From: rodica simion
Subject: HURRAY!
To: doron zeilberger
Status: RO
Nov. 9, 94
Dear Doron,
thank you for the enormous privilege and thrill of having your ASM paper!
it is absolutely awsome! as you might guess, i have not quite
had the time to read it all, but i did do my ... assignment!
1.2.1.2.1.1 is all great! everything checks and works out like a charm!
(including the two (sub)^6 lemmas which it uses)
i did my assignment with so much excitement that i may have overdone it.
sorry if this happened... (i am afraid i may have played editor too much;
a tiny typo in the statement of 1.2.1.2.1.1.2 may be the only change
to consider; please ignore all my comments if you wish)
below is the portion
starting at (sub)^5 lemma 1.2.1.2.1.1
ending at the end of the proof of this (sub)^5 lemma.
my comments below are a matter of a typo and a couple
of minor suggestions.
the comments are marked by %%%%%%%% and are labeled A through F.
comment D-1 is the typo.
thank you again and it is truly moving to see the beautiful
architecture of the entire paper!
rodica
------------------------------------------------------------------------
{\bf Subsubsubsubsublemma 1.2.1.2.1.1:} Let $Jamie(x_1 ,\dots , x_k)$
be the polynomial in $(x_1 , \dots ,x_k)$ inside the braces of
$(Dave)$, i.e. the polynomial:
$$
Jamie(x_1 , \dots , x_k):=
\prod_{j=1}^l \bar x_{r_j} -
\left \{ \prod_{i=1}^{k} \bar x_i \right \}
\left \{ \prod_{j=1}^{l} \prod_{i=r_{j-1}+2}^{r_{j}} x_i \right \}
\quad.
$$
%%%%%%%%%%%%%%%%%%%%% A
%%%%%%%%% possible insert here
Recall that $r_l = k$ and we put $r_0 = 0$.
%%%%%%%%%%%%%%%%%%%%%%% END A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$Jamie(x_1 , \dots , x_k)$ can be written as follows:
$$
Jamie(x_1 , \dots , x_k)=
\sum_{
{{p=2} \atop { p \not \in \{ r_1 +1 , r_2+ 1 , \dots , r_{l-1}+1 \} } }
}^{k}
POL( \hat x_{p-1} , \hat x_p ) \, \cdot \, \bar x_p \, (1- \bar x_{p-1} x_p ) ,
\eqno(Herb)
$$
%%%%%%%%%%%%%%%% original B
where $POL( \hat x_{p-1} , \hat x_p )$ is shorthand for:
``some polynomial of $(x_1 , \dots , x_k)$ that does {\it not} depend
on the variables $x_{p-1}$ and $x_p$''. (In other words,
$POL( \hat x_{p-1} , \hat x_p )$ is some polynomial in the $k-2$ variables
$(x_1 , \dots , x_{p-2} , x_{p+1} , \dots , x_k)$.)
%%%%%%%%%%%%%%%% possible replacement B
where $POL( \hat x_{p-1} , \hat x_p )$ is shorthand for:
``some polynomial of $(x_1 , \dots , x_k)$ that does {\it not} depend
on the variables $x_{p-1}$ and $x_p$, barred or not''. (In other words,
$POL( \hat x_{p-1} , \hat x_p )$ is some polynomial in the $k-2$ variables
$(x_1 , \dots , x_{p-2} , x_{p+1} , \dots , x_k)$ with or without bars.)
%%%%%%%%%%%%%%%%%%%%%%% END B %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Proof:} Since (let $\bar x_0 :=1$)
%%%%%%%%%%%%%%%%%%%% original C
$$
\prod_{i=1}^{k} \bar x_i =
\left \{ \prod_{j=1}^{l} \prod_{i=r_{j-1}+1}^{r_{j}} \bar x_{i-1} \right \}
\cdot \bar x_k
=
\left \{ \prod_{j=1}^{l} \prod_{i=r_{j-1}+2}^{r_{j}} \bar x_{i-1} \right \}
\cdot
\left \{ \prod_{j=1}^{l} \prod_{i=r_{j-1}+1}^{r_{j-1}+1} \bar x_{i-1} \right \}
\cdot \bar x_k
$$
$$
=
\left \{ \prod_{j=1}^{l} \prod_{i=r_{j-1}+2}^{r_{j}} \bar x_{i-1} \right \}
\cdot
\left \{ \prod_{j=1}^{l} \bar x_{r_j} \right \} \quad ,
$$
%%%%%%%%%%%%%%%%%%%%% possible replacement C
$$
\prod_{i=1}^{k} \bar x_i =
\prod_{j=1}^{l}
\left \{ \bar x_{r_{j-1}} \prod_{i=r_{j-1}+1}^{r_{j}-1} \bar x_{i} \right \}
\cdot \bar x_k
=
\left \{ \prod_{j=1}^{l} \prod_{i=r_{j-1}+2}^{r_{j}} \bar x_{i-1} \right \}
\cdot
\left \{ \prod_{j=1}^{l} \bar x_{r_j} \right \} \quad ,
$$
%%%%%%%%%%%%%%%%%%%%%%%%%% END C %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
we have that
$$
Jamie(x_1 , \dots , x_k)=
\prod_{j=1}^l \bar x_{r_j} -
\left \{ \prod_{i=1}^{k} \bar x_i \right \}
\left \{ \prod_{j=1}^{l} \prod_{i=r_{j-1}+2}^{r_{j}} x_i \right \}
$$
$$
=
\prod_{j=1}^l \bar x_{r_j} -
\left \{ \prod_{j=1}^{l} \prod_{i=r_{j-1}+2}^{r_{j}} ( \bar x_{i-1} x_i )
\right \}
\cdot
\prod_{j=1}^{l} \bar x_{r_j} =
\left \{ \prod_{j=1}^{l} \bar x_{r_j} \right \} \cdot
\left [ 1- \prod_{j=1}^{l} \,\,\prod_{i=r_{j-1}+2}^{r_{j}} ( \bar x_{i-1} x_i )
\right ] \quad .
\eqno(Marvin)
$$
We now need the following $(sub)^6$ lemma:
{\bf Subsubsubsubsubsublemma 1.2.1.2.1.1.1:} Let $U_i$, $i=1, \dots , l$, be
quantities in an associative algebra, then:
$$
1- \prod_{j=1}^{l} U_j =
\sum_{j=1}^l \left \{ \prod_{h=1}^{j-1} U_h \right \} (1- U_j) \quad .
$$
{\bf Proof:} The series on the right telescopes to the expression
on the left. Alternatively, use increasing induction on $l$, starting
with the tautologous ground case $l=1$. \halmos
Using $(sub)^6$lemma $1.2.1.2.1.1.1$ with
$$
U_j = \prod_{i=r_{j-1}+2}^{r_j} ( \bar x_{i-1} x_i ) \quad ,
$$
we get that $(Marvin)$ implies:
$$
Jamie(x_1 , \dots , x_k)=
\left \{ \prod_{j=1}^{l} \bar x_{r_j} \right \} \cdot
\sum_{j=1}^l \left \{ \prod_{h=1}^{j-1}
\prod_{i=r_{h-1}+2}^{r_h} ( \bar x_{i-1} x_i )
\right \} \cdot
\left ( \,\,
1- \prod_{i=r_{j-1}+2}^{r_j} ( \bar x_{i-1} x_i ) \,\, \right ) \quad.
\eqno(Marvin')
$$
We can split $(Marvin')$ yet further apart, with the aid of the
following $(sub)^6$lemma:
%%%%%%%%%%% original D
{\bf Subsubsubsubsubsublemma 1.2.1.2.1.1.2:} Let $U_j$, ($j=K, \dots , L$), be
quantities in an associative algebra, then:
$$
1- \prod_{i=K}^{L} U_i =
\sum_{p=K}^L (1- U_p) \left \{ \prod_{i=p+1}^L U_h \right \} \quad .
$$ %
%
%%%%%%%%% TYPO ? D-1 %
% U_h in the product above should be U_i? %%%%%%%%
%%%%%%%%% END D-1 %%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% possible replacement D
{\bf Subsubsubsubsubsublemma 1.2.1.2.1.1.2:} Let $U_i$, ($i=K, \dots , L$), be
quantities in an associative algebra, then:
$$
1- \prod_{i=K}^{L} U_i =
\sum_{p=K}^L (1- U_p) \left \{ \prod_{i=p+1}^L U_i \right \} \quad .
$$
%%%%%%%%%%%%%%%%%%%%%%%%%% END D %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Proof:} The sum on the right telescopes to the expression on
the left. (Note that it is in the opposite direction to the way
in which it happened in $1.2.1.2.1.1.1$.) Alternatively,
the identity is tautologous when $K=L$, and follows by decreasing
induction on $K$. This completes the proof of $(sub)^6$ lemma
$1.2.1.2.1.1.2$. \halmos .
Going back to $(Marvin')$, we use the last
$(sub)^6$lemma ($1.2.1.2.1.1.2$), with $K=r_{j-1}+2$, $L=r_j$,
and $U_i:= ( \bar x_{i-1} x_i )$, to rewrite:
$$
Jamie(x_1 , \dots , x_k)=
\left \{ \prod_{j=1}^{l} \bar x_{r_j} \right \} \cdot
\sum_{j=1}^l
\,\, \left \{ \prod_{h=1}^{j-1}
\prod_{i=r_{h-1}+2}^{r_h} ( \bar x_{i-1} x_i ) \right \} \cdot
\sum_{p=r_{j-1}+2}^{r_j} (1- \bar x_{p-1} x_p )
\prod_{i=p+1}^{r_j} ( \bar x_{i-1} x_i )
$$
$$
=
\sum_{j=1}^l \sum_{p=r_{j-1}+2}^{r_j} \,\,
\left \{ \prod_{j=1}^{l} \bar x_{r_j} \right \}
\left \{ \prod_{h=1}^{j-1}
\prod_{i=r_{h-1}+2}^{r_h} ( \bar x_{i-1} x_i )
\right \} \cdot
(1- \bar x_{p-1} x_p )
\left \{ \prod_{i=p+1}^{r_j} ( \bar x_{i-1} x_i ) \right \}
\quad .
\eqno(Marvin'')
$$
%%%%%%%% E %%%%% minor remark about the display line above: %%%%%%%%%%%
% when the product of the barred x_{r_j}'s goes inside the sum,
% the j plays double duty (it is also the summation index for the outer sum)
% It would take a bit of patience to make a change consistently,
% and you may not deem it that important. In fact i would say
% that nobody would really get confused too much by this.
%%%%%%%%%%%%%%%%%%%% END E %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%% original F
The right side of $(Marvin'')$ is a sum over all $p$ in the
range $1 \leq p \leq k$
except that $p=r_j+1$, for $j=0, \dots , l-1$, are omitted (where
$r_0:=0$.) For each such participating $p$, the summand is
$(1- \bar x_{p-1} x_p)$ times a product of $x_i$'s and $\bar x_i$'s,
neither of which is $x_p, \bar x_p , x_{p-1}, \bar x_{p-1}$ {\it except}
for a single $\bar x_p$, which comes out of the first term of
the product $ \prod_{i=p+1}^{r_j} ( \bar x_{i-1} x_i )$ in
$(Marvin'')$, when $p< r_j$ for some $j$, or from the $j$'th term
of the product $\prod_{j=1}^{l} \bar x_{r_j}$ in case $p=r_j$ for
one of the $j$'s. Thus the polynomial $Jamie(x_1 , \dots , x_k)$
can indeed be expressed as claimed in $(Herb)$. This completes the
proof of $(sub)^5$ lemma $1.2.1.2.1.1$. \halmos
%%%%%%%%%%%%%%%% possible replacement F
Switching the order of summation, the right side of $(Marvin'')$ is a sum
over all $p$ in the range $1 \leq p \leq k$,
$p \neq r_j+1$, for $j=0, \dots , l-1$ (where $r_0:=0$).
For each such participating $p$, the sum over $j$ has only one term,
namely the term corresponding to the unique $j$ such that
$r_{j-1} + 2 \leq p \leq r_j$. This term is
$(1- \bar x_{p-1} x_p)$ times a product of $x_i$'s and $\bar x_i$'s,
none of which is $x_p, x_{p-1}$, or $\bar x_{p-1}$.
A {\it single} factor equal to $\bar x_p$ occurs in this term:
it arises from the product $ \prod_{i=p+1}^{r_j} ( \bar x_{i-1} x_i )$ in
$(Marvin'')$ when $p< r_j$, or from the product $\prod_{j=1}^{l} \bar x_{r_j}$
if $p=r_j$. Thus the polynomial $Jamie(x_1 , \dots , x_k)$
can indeed be expressed as claimed in $(Herb)$. This completes the
proof of $(sub)^5$ lemma $1.2.1.2.1.1$. \halmos
%%%%%%%%%%%%%%%%%%%%%% END F %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From rstan@math.mit.edu Tue Dec 27 11:26:58 1994
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From: Richard Stanley
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Date: Tue, 27 Dec 94 11:27:11 EST
Message-Id: <9412271627.AA05359@parseval.mit.edu>
To: zeilberg@euclid.math.temple.edu
Subject: checker's report
Status: RO
Dear Doron,
Here is my Checker's Report for the portion of your ASM proof assigned
to me. I hope the rest of the proof is just as accurate.
Best regards,
Richard
___________________________________________________________________________
% LaTeX file
\documentstyle[12pt]{article}
\textwidth6.0truein
\textheight8.5truein
\topmargin-.5truein
\oddsidemargin+0truein
\evensidemargin+0truein
\baselineskip0.20truein
\pagestyle{empty}
\parskip0.20truein
\begin{document}
\begin{center}
CHECKER'S REPORT\\[.1in] of Subsublemma 1.1.3\\[.1in]
of Proof of the ASM Conjecture by D. Zeilberger\\[.2in]
\end{center}
\hspace{.2in} The subsublemma and its proof are correct.
The identity is stated almost exactly in the form given by Subsublemma
1.1.3 by I. Schur, {\em Arch.\ Math.\ Phys.\ Series 3} {\bf 27}
(1918), 163. (Presumably this is the paper referred to by Macdonald on
p. 54 of his book when he cites {\em Ges.\ Abhandlungen}, vol.\ 3,
p. 456, but I have not checked this.) It is reproduced as Problem
VII.47 in G. P\'olya and G. Sz\"ego, {\em Problems and Theorems in
Analysis II}, Springer-Verlag, 1976. These might be more appropriate
references.
A nitpicking matter of typography: Any name of more than one letter of
a function, such as det, sgn, conv, cos, etc., should be typeset in
roman type. The rationale is that this distinguishes the name of the
function from a product of its individual letters. This is why in
\LaTeX\ (I'm not sure about plain \TeX) there exist the commands
$\backslash$sin, $\backslash$exp, $\backslash$det, $\backslash$max,
$\backslash$lim, etc. Also $\backslash$dots was used three times in
Subsublemma 1.1.3 when $\backslash$cdots should have been. (Once it
looks like $\backslash$cdot $\backslash$dots $\backslash$cdot was
used, which looks even worse.)
\end{document}
From stanton@math.umn.edu Thu Dec 22 15:41:05 1994
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To: zeilberg@euclid.math.temple.edu
Subject: my report
Status: RO
Doron:
on 1.2.1.1
typo: p. 27 middle of page, Land_Of_Gog is missing sub k
First I should say that this recurrence and the boundary conditions confused me
in an early version of the paper. The extended definitions were not given,
and where the recurrence held (or not) was not explained before,
so your rewriting has helped.
Another potential confusion is in the definition of Land_Of_Gog_k.
This set depends on n, and you have to apply it to different n's in the proof.
On page 27 "apply (Bill) with n replaced by n-1",
you need to say that (Bill) holds with n replaced by n-1 if n-1>k and
(n-1;a_1,...,a_k) is in Land_Of_Gog_k,
for this we need assumed hypothesis a_1k,
for which (n-1;a_1,...,a_k) does not lie in Land_Of_Gog_k, namely
the cases n-1=k or a_1=n. These are done in the next 2 paragraphs.
Dennis
From strehl@immd1.informatik.uni-erlangen.de Sun Dec 11 04:11:52 1994
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From: Volker Strehl
Message-Id: <199412110912.AA01586@faui10.informatik.uni-erlangen.de>
Date: Sun, 11 Dec 94 10:12:02 +0100
To: zeilberg@euclid.math.temple.edu
Subject: Re: Invitation to be co-checkers of my ASM paper
Cc: strehl@immd1.informatik.uni-erlangen.de
Status: RO
Dear Doron,
Taking part in the proof-checking-enterprise that you
initiated really causes a particular sentiment: it is
if I were participating in a mathematico-sociological
field experiment (with yet uncertain outcome). When I reread
this morning the invitation to be a co-checker you sent
me about a month ago, and when I came to the part where
you speak about the 'historical' aspect, a very famous
phrase (in German, at least) by our great poet J.W. v. Goethe
(Faust! Do you happen to know this drama ?) came to my mind:
shortly after the French revolution Goethe was watching
the first big battle between the French revolutionary troops,
not yet well organized and equipped, but full of enthusiasm,
and the well organized, very well equipped troops of the
"Ancien regime", heavily supported by all these (German,
Austrian, etc) kings, dukes, noblemen who were frightened
by the French revolution (for good reasons). Now, the battle,
known as the "Kanonade von Valmy" (this happened in Lotharingian
countries!), had no clear winner - which was a sensation, because
everybody expected a clear victory of the "old professionals"
which would make an end to the revolution, or at least would
help to prevent revolutionary ideas and actions spreading out
too rapidly. What Goethe said and reported was: "From here and
from today on a new epoch of mankind emerges -- and we can say
that we have witnessed it".
Enough of dubious historical parallels. What is important
for me and counts on short range is that a can do you a favour.
And that's why I like it to do. So below you find my report
on the portion that I had to check. Actually, I read more
than the tiny piece, but I am still very far from what Dave Bressoud
seems to have done. If I had more time, I would certainly delve
deeper into the matter, because in my opinion this proof -- once
all is checked -- is monstrous, and since the underlying combinatorial
assertion is so strikingly simple to state, the need for a shorter,
more elegant proof remains as a provocation.
Best wishes,
Volker
-------begin checker's form---------------------------
(Fill in the blanks [], and delete the superfluous parts)
Dear Doron,
I have now completed checking (sub)^[3]lemma [1.4.1.4-case(I)].
To the best of my judgment, it is correct, provided
all its offspring (sub)^[]-lemmas are correct.
Needless to say, I don't know (and don't care!) whether
the proofs of these offspring (sub)^[]-lemmas are actually
correct.
and (optional)
I have found the following typos or other errors:[none]
and (optional)
I have the following suggestion to improve the exposition:[none]
Best Wishes,
[Volker Strehl ]
P.S. I like the epithet you wrote for me
yes, I really like it, but please correct the spelling
of the one german word that appears in it: it is
"Habilitationsschrift"
(capital-H in the beginning, no double-f at the end)
---------end checkers form----------------
From bsulanke@claven.idbsu.edu Thu Dec 15 15:56:09 1994
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To: zeilberg@math.temple.edu
From: bsulanke@claven.idbsu.edu (Robert Sulanke)
Subject: checked
Cc: sulanke@math.idbsu.edu
Date: Thu, 15 Dec 1994 13:56:44 +0000
Message-ID: <1424622692-548894@Claven.idbsu.edu>
Status: RO
-------begin checker's form---------------------------
Dear Doron,
I have now completed checking subsubsublemma 1.3.1.4 , Subcase Ib. To
the best of my judgment, it is correct, with the exception of two typos
which when corrected do not change the intended proof.
I have found the following typos: (I mentioned these typos to you in a
previous e-mail message.)
In the line ``(Gil)'', page 46, the rightmost `k' should be `2k'.
Three lines below (Gil), page 46, the `k' should be `2k'.
Best Wishes,
Bob Sulanke
P.S. I like the epithet you wrote for me.
---------end checkers form----------------
From taka@math.s.kobe-u.ac.jp Wed Jan 4 22:58:40 1995
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Date: Thu, 5 Jan 95 12:58:16 JST
From: Nobuki Takayama
Message-Id: <9501050358.AA07789@math.s.kobe-u.ac.jp>
To: zeilberg@euclid.math.temple.edu
Subject: ASM
Status: RO
Dear Doron,
I have now completed checking (sub)^[]lemma [1.4.1.1].
To the best of my judgment, it is correct, provided
all its offspring (sub)^[]-lemmas are correct and
<< the symbol $\pm$ means that
you do not care about the sign of the formulas. >>
Best Wishes, Nobuki
*"...*+
asm_intro.tex111.tex*%1112.tex*& 11121.tex*' 11122.tex*(1113.tex*)112.tex**113.tex
DXACTIFrom werman@cs.huji.ac.il Fri Dec 2 04:04:34 1994
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To: zeilberg@euclid.math.temple.edu
Subject: Re: Invitation to be co-checkers of my ASM paper
Date: Fri, 02 Dec 1994 11:08:02 +0200
From: Michael Werman
Status: RO
Dear Doron,
I have now completed checking (sub)^[]lemma [ 1.3.1.4 part II ].
To the best of my judgment, it is correct, provided
all its offspring (sub)^[]-lemmas are correct.
and the other references* in its proof are also correct.
Needless to say, I don't know (and don't care!) whether
the proofs of these offspring (sub)^[]-lemmas are actually
correct.
Best Wishes,
[ Michael Werman]
From wilf@central.cis.upenn.edu Thu Dec 22 15:14:39 1994
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From: wilf@central.cis.upenn.edu (Herbert S. Wilf)
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Reply-To: wilf@central.cis.upenn.edu (Herbert S. Wilf)
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To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
Subject: Re: OK, I see now.
Status: RO
I understand the answer to my previous question. It is Vandermonde's
expansion, at the top of page 3. So disregard the question.
************************************************************
I, Herbert Wilf, have checked the proof of Sublemma 1.1, in accordance
with generally accepted mathematical principles. I find that Sublemma
1.1 is true if Subsublemmas 1.1.j are true for each j=1,2,3.
Date: December 22, 1994
************************************************************
For the exposition, the extension of the domain at the bottom of page 1
seems unnecessary to me. For suppose you had defined it by means of the
sentence that follows the word "Proof:" in the middle of p.1., and that
that definition was for all integer values of the a_i's.
then that would do it automatically. For suppose you have a set of a's
that do not match the descending condition. Then the # of M. trapezoids
is zero from their definition. Similarly if you have a set of a_i's
that have a_i>n-i+1 for some i, then your definition of the B's again
would give 0.
So I suggest
(A) Omit all of the verbiage from "By the def of magog trapezoids, the
natural ..." all the way down to "... break the rules, is 0." and
(B) Add a couple of words to the def of B_k, as follows:
"Proof: Let B_k(n,a_1,...,a_k) be the number of nxk ....
c_{i,n-i+1}=a_i. In other words, the rightmost border is ... a_i.
Notice that B_k is supported on (i.e. vanishes outside of) the domain
Land_ofmagog_k:={....}."
And stop there, and go immediately to "The following subsubleamma gives
..."
Subsubleamma 1.1.3 is very pretty. The left side has the factor in the
denominator of (1-x_1...x_k), for EVERY permutation. But it cancels out
in the sum. Pretty.
Herb
From wilf@central.cis.upenn.edu Wed Sep 8 13:37:23 1993
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Posted-Date: Wed, 08 Sep 1993 13:37:21 EDT
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To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
Subject: Re: Hi
In-Reply-To: Your message of "Sat, 04 Sep 1993 20:16:48 EDT."
<9309050016.AA21234@euclid.math.temple.edu>
Date: Wed, 08 Sep 1993 13:37:21 EDT
From: Herbert Wilf
Status: RO
I kept my hands off of Walter Shur's problem as long as possible. Finally I
resolved to look at the generatingfunctionological approach to the question.
The results are very nice. In the next message is a one-page TeX document
that says it all. Now Shur was able to extract a formula, in the unordered
case, that has only a single summation in it. At this moment I don;t see
how to do that, even in the ordered case, which is easier. But no doubt
if I hack away at it for a bit, that will emerge.
From wilf@central.cis.upenn.edu Wed Sep 8 13:38:22 1993
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Posted-Date: Wed, 08 Sep 1993 13:38:20 EDT
Message-Id: <9309081738.AA06953@central.cis.upenn.edu>
To: zeilberg@euclid.math.temple.edu (Doron Zeilberger)
Subject: Re: TeX source file (1 page)
In-Reply-To: Your message of "Sat, 04 Sep 1993 20:16:48 EDT."
<9309050016.AA21234@euclid.math.temple.edu>
Date: Wed, 08 Sep 1993 13:38:20 EDT
From: Herbert Wilf
Status: RO
\nopagenumbers
\centerline{\bf Counting pairs of lattice paths by intersections}
\centerline{Herbert S. Wilf}
\centerline{University of Pennsylvania}
\centerline{September 8, 1993}
On an $r\times (n-r)$ lattice rectangle, we consider walks that begin
at the SW corner, proceed with unit steps in either of the directions
E or N, terminating at the NE corner of the rectangle. For fixed $k\ge 0$, we ask for $N_k^{n,r}$, the number of pairs of these walks that
intersect in exactly $k$ points. By {\it pairs} we will mean either
ordered pairs or unordered pairs, and will treat both cases. The
number of points in the intersection of two such walks is defined to
mean the cardinality of the intersection of the two sets of vertices,
excluding the initial and terminal vertices.
This problem has been treated by Walter Shur (p.c.), who has obtained the explicit formula
$$N_k^{n,r}={{(k+1)}\over r}\sum_j(-1)^j{{{k\choose j}{{k-j}\choose j}{{n-2-j}\choose {r-1}}{{n-1-j}\choose {r-1-j}}}\over {{{n-2-j}\choose j}}}\eqno(1)$$
in the case of {\it unordered} pairs.
We give here a generating function approach to the problem. Indeed since clearly
$$N_k^{n,r}=\sum_{q,m}N_{k-1}^{m,q}N_0^{n-m,k-q},\eqno(2)$$
we introduce the generating function $u_k(x,y)=\sum_{n,r}N_k^{n,r}x^ny^r$, and then (2) says simply that $u_k=u_{k-1}u_0$, for $k\ge 1$. Thus $u_k(x,y)=u_0(x,y)^{k+1}$.
Now $u_0$ is well known, but it's more interesting to find it by observing that the coefficient of $x^ny^r$ in $\sum_ku_k(x,y)$ is the total number of pairs of walks, since every pair has {\it some} number of interesections.
If we are considering {\it ordered} pairs of walks, then the number of pairs of walks is ${n\choose r}^2$, so in this case,
$$\eqalign{\sum_ku_k(x,y)&=\sum_{k\ge 0}u_0(x,y)^k={1\over {1-u_0(x,y)}}\cr
&=\sum_{n,r}{n\choose r}^2x^ny^r\cr
&=\sum_{n\ge 0}x^n(y-1)^nP_n({{y+1}\over {y-1}})\cr
&=(1-2x(y+1)+x^2(y-1)^2)^{-{1\over 2}},\cr}$$
where the $P_n$'s are the Legendre polynomials. Thus
$$u_0(x,y)=1-\sqrt{1-2x(y+1)+x^2(y-1)^2}.$$
It follows that the number of ordered pairs of walks that have exactly $k$ intersections is the coefficient of $x^ny^r$ in
$$u_k(x,y)=\bigl(1-\sqrt{1-2x(y+1)+x^2(y-1)^2}\bigr)^{k+1}.$$
If we count unordered pairs instead, then the total number of pairs of walks is ${n\choose r}(1+{n\choose r})/2$, and the same kind of computation shows that the number is the coefficient of $x^ny^r$ in
$$\tilde{u}_k(x,y)=\left(1-{2\over {{1\over {\sqrt{1-2x(y+1)+x^2(y-1)^2}}}+{1\over {1-x-xy}}}}\right)^{k+1}.$$
\bye
From wilf@central.cis.upenn.edu Thu Sep 8 11:46:03 1994
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To: zeilberg@euclid.math.temple.edu
Subject:
Date: Thu, 08 Sep 1994 11:46:05 EDT
From: Herbert Wilf
Status: RO
Doron: Your system usually does not supply a complete return address
string, so if I simply "reply" to your message, this is what happens.
It doesn't really bother me, but I thought you'd like to know because
some messages might never get to you because of this. - Herb
*****************************
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The original message was received at Thu, 8 Sep 1994 11:43:12 -0400
from wilf@CENTRAL.CIS.UPENN.EDU [158.130.12.2]
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To: zeilberg@euclid (Doron Zeilberger)
In-reply-to: Your message of "Thu, 08 Sep 1994 11:21:21 EDT."
<9409081521.AA10424@euclid>
Date: Thu, 08 Sep 1994 11:43:11 EDT
From: Herbert Wilf
A beautiful idea to honor a beautiful person. Please go ahead, and ask
Strehl to keep me informed from time to time about how things are
progressing. I really like the idea.
Shana tovah!
------- End of Forwarded Message
From zimmerma@uni-paderborn.de Thu Dec 15 03:24:37 1994
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Date: Thu, 15 Dec 1994 09:24:18 +0100
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From: zimmerma@uni-paderborn.de (Paul Zimmermann)
To: zeilberg@euclid.math.temple.edu
Subject: report for sub^3lemma 1.3.1.4
Status: RO
Dear Doron,
I have now completed checking (sub)^3lemma 1.3.1.4.
To the best of my judgment, it is correct, provided
all its offspring (sub)^4-lemmas are correct.
Using the ROBBINS package, I have called S1314(k,n)
for 1<=k<=n<=3 and (1<=k<=3 for n=4) and I found no error.
Needless to say, I don't know (and don't care!) whether
the proofs of these offspring (sub)^4-lemmas are actually
correct.
I have found the following typos or other errors:
In the Exodion (p. 83) : Zimmerman -> Zimmermann (two ``n'')
I have the following suggestion to improve the exposition:
In the statement of Lemma 1.3.1.4, you should remember what
is \pi, that \bar{x_R} = 1-x_R, R=\pi(u), and z_i=x_i if
\epsilon_i=1, and 1-x_i otherwise.
Furthermore, according to the two cases in the proof, it seems
that i in in {\pi(1),...,\pi(u-1),\pi(u+1),...,\pi(k)}.
You should add this in the statement of Lemma 1.3.1.4 too.
I have two more remarks:
(1) the proof of case II refers to the proof of Case II of 1.4.1.4,
which is below. There is here a risk of a loop in the proof graph.
Who will check this ?
(2) you say for Case II ``the form of \tilde(A_i} is slightly more
complicated there''. Could you explain why ? Of course it is
not my responsability to check Case II, but the referees that
have to check it have a difficult job, since they have first to
create the proof.
P.S. I like the epithet you wrote for me
From zimmerma@uni-paderborn.de Wed Jan 4 06:01:46 1995
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Date: Wed, 4 Jan 1995 12:01:43 +0100
Message-Id: <199501041101.AA05964@courant.uni-paderborn.de>
From: zimmerma@uni-paderborn.de (Paul Zimmermann)
To: zeilberg@euclid.math.temple.edu
In-Reply-To: <199501021716.MAA29766@euclid.math.temple.edu> (zeilberg@euclid.math.temple.edu)
Subject: Re: ASM Checking Update
Status: RO
Dear Doron,
Using ROBBINS, I have now checked Lemma 1.3.1.4 for every 1 <= k <= n <= 4
[this took some time, especially S1314(4,4)].
Best wishes for 1995.
Paul
From zimmerma@uni-paderborn.de Tue Dec 6 03:15:15 1994
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Date: Tue, 6 Dec 1994 09:14:15 +0100
Message-Id: <199412060814.AA19006@courant.uni-paderborn.de>
From: zimmerma@uni-paderborn.de (Paul Zimmermann)
To: zeilberg@euclid.math.temple.edu
In-Reply-To: <9411091930.AA11250@ralbag.math.temple.edu> (zeilberg@euclid.math.temple.edu)
Subject: Re: referee
Status: RO
Dear Doron,
I have a few questions about the ROBBINS package:
- when I try S1314(1,0), I got
The left side of the identity of subsubsublemmas 1.3.1.3
and 1.3.1.4 combined, with the markers ma[i] is
0
while its right side of is
2 ma[2]
Of course Lemma 1 states that 1 <= k <= n, so perhaps you should
check these conditions in S1314.
By the way, the help of S1314 is inconsistent with the definition:
> op(S1314);
proc(n,k) S1313S1314(n,k) end
> ezra(S1314);
S1314(k,n) verifies subsubsublemmas 1.3.1.4 for k
and n, by calling in S1313S1314 that does both
In which order should the parameters be given ?
It would be nice also if ROBBINS contained a procedure that produces all
alternating matrices of dimension n. It would help the referees to have
an idea of what you proved.
Best wishes for your paper.
Paul