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Science Forum Index » Cognitive Science Forum » JSH: Mathematicians' defense
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| James Harris |
 Posted: Fri Dec 05, 2003 11:40 pm |
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Guest
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Well I contacted Lagarias again, and for those of you outside of math
circles and the area of prime numbers, he's a mathematician that you
can consider to be an expert in the area of prime numbers.
His reply makes me consider what is likely to be the defense of
mathematicians I've contacted over the years when it becomes clear
just how important to the heart of mathematics my work is.
My guess is that mathematicians wish to use journals as a defense, and
claim that it was my responsibility to get a paper in format for a
journal, and present it to journal editors until *someone* published
my results.
Their own defense then being that it's just not their fault if I
couldn't manage to write a paper that other mathematicians would
consider.
The excuse is that if you find something, no matter how dramatic it
might be, if you can't write a math paper in a standard format, then
mathematicians don't have to be expected to pay attention to your
result.
Some of you may not understand just how strict of a requirement that
is, unless you read math journals, or even maybe have read sci.math
long enough, to see the withering criticisms of Ph.D's in physics
who've tried to write math papers.
Physicists have their way of writing papers; mathematicians have their
own.
Mathematicians pride themselves on a special format, that has a
certain look, feel and read particular to, you guessed it,
mathematicians.
Basically the barrier to outsiders isn't content, but style.
I want those of you outside of the math world to hear that potential
defense now, so that you can consider it before the mathematicians can
deliver it in earnest, when I put them, and their practices, on public
trial.
James Harris
"My math discoveries, found for profit"
http://mathforprofit.blogspot.com/ |
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| Stan Gula |
| Posted: Sat Dec 06, 2003 12:10 am |
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"James Harris" <jstevh@msn.com> wrote in message
news:3c65f87.0312052040.50d01a71@posting.google.com...
<typical whine snipped>
Quote: Basically the barrier to outsiders isn't content, but style.
Actually it's both. A beautiful paper showing a trivial result will
probably not get published either.
Stop whining and learn how to write a proper exposition. It's not so hard,
James. Here's a link to get you started:
http://www.mit.edu/afs/athena.mit.edu/course/other/mathp2/www/piil.html
However, you will still have that trivial result issue to deal with... |
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| David Moran |
| Posted: Sat Dec 06, 2003 12:32 am |
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"James Harris" <jstevh@msn.com> wrote in message
news:3c65f87.0312052040.50d01a71@posting.google.com...
Quote: Well I contacted Lagarias again, and for those of you outside of math
circles and the area of prime numbers, he's a mathematician that you
can consider to be an expert in the area of prime numbers.
His reply makes me consider what is likely to be the defense of
mathematicians I've contacted over the years when it becomes clear
just how important to the heart of mathematics my work is.
My guess is that mathematicians wish to use journals as a defense, and
claim that it was my responsibility to get a paper in format for a
journal, and present it to journal editors until *someone* published
my results.
Their own defense then being that it's just not their fault if I
couldn't manage to write a paper that other mathematicians would
consider.
The excuse is that if you find something, no matter how dramatic it
might be, if you can't write a math paper in a standard format, then
mathematicians don't have to be expected to pay attention to your
result.
Some of you may not understand just how strict of a requirement that
is, unless you read math journals, or even maybe have read sci.math
long enough, to see the withering criticisms of Ph.D's in physics
who've tried to write math papers.
Physicists have their way of writing papers; mathematicians have their
own.
Mathematicians pride themselves on a special format, that has a
certain look, feel and read particular to, you guessed it,
mathematicians.
Basically the barrier to outsiders isn't content, but style.
I want those of you outside of the math world to hear that potential
defense now, so that you can consider it before the mathematicians can
deliver it in earnest, when I put them, and their practices, on public
trial.
Trial? Nobody's broken the law so you can't put anyone on trial. And if you
think you can bust them on lying, I'm afraid the mathematicians have the
upper hand on this because it is SOOOOO obvious that you don't understand
the counterarguments. Plus, I'd think if you tried to put a mathematician on
trial, that they could countersue for libel. You act like you are entitled
to bash anyone you want, but nobody can bash you.
David Moran |
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| Paul R. Mays |
| Posted: Sat Dec 06, 2003 1:06 am |
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"David Moran" <ktulwxwatcher@hotmail.com> wrote in message
news:bqrpl7$262vg0$1@ID-206850.news.uni-berlin.de...
Quote:
"James Harris" <jstevh@msn.com> wrote in message
news:3c65f87.0312052040.50d01a71@posting.google.com...
Well I contacted Lagarias again, and for those of you outside of math
SNIPPED
to hear that potential
Quote: defense now, so that you can consider it before the mathematicians can
deliver it in earnest, when I put them, and their practices, on public
trial.
Trial? Nobody's broken the law so you can't put anyone on trial. And if
you
think you can bust them on lying, I'm afraid the mathematicians have the
upper hand on this because it is SOOOOO obvious that you don't understand
the counterarguments. Plus, I'd think if you tried to put a mathematician
on
trial, that they could countersue for libel. You act like you are entitled
to bash anyone you want, but nobody can bash you.
James Harris
"My math discoveries, found for profit"
http://mathforprofit.blogspot.com/
David Moran
What the heck you talking about!!!
Everybody bash's him.... Deservedly So..... |
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| Victor Eijkhout |
| Posted: Sat Dec 06, 2003 1:14 am |
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James Harris <jstevh@msn.com> wrote:
Quote: The excuse is that if you find something, no matter how dramatic it
might be, if you can't write a math paper in a standard format, then
mathematicians don't have to be expected to pay attention to your
result.
The traditional lemma/theorem/corollary style is just a very clear way
of presenting your statement. The fact that you're unable to write your
results (whichever you'r talking about now), probably means that you
haven't proved anything.
V.
--
email: lastname at cs utk edu
homepage: cs utk edu tilde lastname |
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| David Moran |
| Posted: Sat Dec 06, 2003 1:26 am |
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"Paul R. Mays" <uce@ftc.gov> wrote in message
news:JdCdnZVTNpRw7EyiRVn-uw@giganews.com...
Quote:
"David Moran" <ktulwxwatcher@hotmail.com> wrote in message
news:bqrpl7$262vg0$1@ID-206850.news.uni-berlin.de...
"James Harris" <jstevh@msn.com> wrote in message
news:3c65f87.0312052040.50d01a71@posting.google.com...
Well I contacted Lagarias again, and for those of you outside of math
SNIPPED
to hear that potential
defense now, so that you can consider it before the mathematicians can
deliver it in earnest, when I put them, and their practices, on public
trial.
Trial? Nobody's broken the law so you can't put anyone on trial. And if
you
think you can bust them on lying, I'm afraid the mathematicians have the
upper hand on this because it is SOOOOO obvious that you don't
understand
the counterarguments. Plus, I'd think if you tried to put a
mathematician
on
trial, that they could countersue for libel. You act like you are
entitled
to bash anyone you want, but nobody can bash you.
James Harris
"My math discoveries, found for profit"
http://mathforprofit.blogspot.com/
David Moran
What the heck you talking about!!!
Everybody bash's him.... Deservedly So.....
I meant that he acts like he can bash all he wants, but no one is allowed to
bash him back.
David Moran |
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| Random Letters |
| Posted: Sat Dec 06, 2003 4:04 am |
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"James Harris" <jstevh@msn.com> wrote in message
news:3c65f87.0312052040.50d01a71@posting.google.com...
Quote: Well I contacted Lagarias again, and for those of you outside of math
circles and the area of prime numbers, he's a mathematician that you
can consider to be an expert in the area of prime numbers.
His reply...
OK James, I'll bite - what is his reply?
Why don't you post it?
Are you afraid? Did he realy reply? Is his reply real?
Does his reply consist of "text"?
Well?
I'm waiting...
Where is it? Where is Lagarias' reply? Eh? What?
Jesus Fscking Christ on a stick! |
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| Maxim Stepin |
| Posted: Sat Dec 06, 2003 4:30 am |
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"James Harris" <jstevh@msn.com> wrote in message
news:3c65f87.0312052040.50d01a71@posting.google.com...
Quote: The excuse is that if you find something, no matter how dramatic it
might be, if you can't write a math paper in a standard format, then
mathematicians don't have to be expected to pay attention to your
result.
Bingo!
Quote: Some of you may not understand just how strict of a requirement that
is, unless you read math journals
But you can't read math journals, right?
You have to be an amateur, an outsider to sell you story.
You shouldn't be afraid of losing the status of outsider - after all,
you don't have to tell anyone you read the journals, right?
Think about it. |
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| David C. Ullrich |
| Posted: Sat Dec 06, 2003 8:11 am |
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On 5 Dec 2003 20:40:06 -0800, jstevh@msn.com (James Harris) wrote:
Quote: Well I contacted Lagarias again, and for those of you outside of math
circles and the area of prime numbers, he's a mathematician that you
can consider to be an expert in the area of prime numbers.
His reply makes me consider what is likely to be the defense of
mathematicians I've contacted over the years when it becomes clear
just how important to the heart of mathematics my work is.
My guess is that mathematicians wish to use journals as a defense, and
claim that it was my responsibility to get a paper in format for a
journal, and present it to journal editors until *someone* published
my results.
This would all make a lot more sense if you told us what his
reply _was_. I'm going to _guess_ what the reply was, based
on what you say here: Something like "let me know when you've
got it written up and published"?
If so, the problem is that you still haven't learned to recognize
a polite brush-off when you get one - when someone tells
you something like that what they really mean is "huh? this
stuff makes no sense at all..." but they don't want to put
it that way lest they get sucked into the sort of thing we
see on sci.math.
Quote: Their own defense then being that it's just not their fault if I
couldn't manage to write a paper that other mathematicians would
consider.
The excuse is that if you find something, no matter how dramatic it
might be, if you can't write a math paper in a standard format, then
mathematicians don't have to be expected to pay attention to your
result.
The problem is not the _format_. The problem is that you simply
don't write _correct proofs_.
Quote: Some of you may not understand just how strict of a requirement that
is, unless you read math journals, or even maybe have read sci.math
long enough, to see the withering criticisms of Ph.D's in physics
who've tried to write math papers.
Physicists have their way of writing papers; mathematicians have their
own.
Mathematicians pride themselves on a special format, that has a
certain look, feel and read particular to, you guessed it,
mathematicians.
Basically the barrier to outsiders isn't content, but style.
I want those of you outside of the math world to hear that potential
defense now, so that you can consider it before the mathematicians can
deliver it in earnest, when I put them, and their practices, on public
trial.
James Harris
"My math discoveries, found for profit"
http://mathforprofit.blogspot.com/
************************
David C. Ullrich |
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| C. Bond |
| Posted: Sat Dec 06, 2003 10:13 am |
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James Harris wrote:
Quote: Well I contacted Lagarias again, and for those of you outside of math
circles and the area of prime numbers, he's a mathematician that you
can consider to be an expert in the area of prime numbers.
His reply makes me consider what is likely to be the defense of
mathematicians I've contacted over the years when it becomes clear
just how important to the heart of mathematics my work is.
He requires no defense. It is your job to meet his requirements if you
want his attention. The importance of your work *to you* is of no
consequence if you cannot effectively communicate that importance to
others. Yelling, crying and bleating doesn't cut it.
Quote: My guess is that mathematicians wish to use journals as a defense, and
claim that it was my responsibility to get a paper in format for a
journal, and present it to journal editors until *someone* published
my results.
Bad guess. Mathematicians require no defense. If you do not approach
experts and professionals in an manner which is acceptable to them, they
have a *right* to ignore you.
Quote: Their own defense then being that it's just not their fault if I
couldn't manage to write a paper that other mathematicians would
consider.
That isn't a defense, it's a obvious statement of fact. Common sense,
really. What's wrong with their taking that position? Everyone else has to
prepare their material properly, why not you?
Quote: The excuse is that if you find something, no matter how dramatic it
might be, if you can't write a math paper in a standard format, then
mathematicians don't have to be expected to pay attention to your
result.
There's no excuse here. It's a sensible and obvious requirement. The
burden's on you to bring the material to others in comprehensible form.
Stop complaining and do *your* job.
Quote: Some of you may not understand just how strict of a requirement that
is, unless you read math journals, or even maybe have read sci.math
long enough, to see the withering criticisms of Ph.D's in physics
who've tried to write math papers.
Tough! It's survival of the fittest out there. Publish or perish!
Quote: Physicists have their way of writing papers; mathematicians have their
own.
Every technical discipline has its own basic communication framework and
conventions. The purpose is to promote clear discussion and reduce
ambiguity. How else could it be done?
Quote: Mathematicians pride themselves on a special format, that has a
certain look, feel and read particular to, you guessed it,
mathematicians.
Pride has nothing to do with it. It is common sense that basic guidelines
would be required in presenting technical material.
Quote: Basically the barrier to outsiders isn't content, but style.
It's not a barrier, it's a fundamental prerequisite. But content *does*
matter. A well written paper with wrong results will not get published.
You need content *and* style.
Quote: I want those of you outside of the math world to hear that potential
defense now, so that you can consider it before the mathematicians can
deliver it in earnest, when I put them, and their practices, on public
trial
It is you who are on trial. Either you can survive in an evironment you
have chosen to work in, or you will fail. It is not mathematicians who are
erecting a defense, it is you who are trying to bully your way into and
environment which is clearly over your head.
James, re-read your own post. You identified no barriers, no defense, no
excuses on the part of mathematicians. You only highlighted your own
obstinate and cryptic refusal to meet the requirements that all other
individuals must meet in order to gain consideration for publication. No
one has singled you out. Clearly, *you* are your own worst enemy here. You
are simply dragging your feet instead of doing your homework and then
blaming others for your inability to get anywhere.
As has been said before, put up or SHUT UP.
--
There are two things you must never attempt to prove: the unprovable --
and the obvious.
--
Democracy: The triumph of popularity over principle.
--
http://www.crbond.com |
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| Uncle Al |
| Posted: Sat Dec 06, 2003 11:29 am |
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James Harris wrote:
Quote:
Well I contacted Lagarias again, and for those of you outside of math
circles and the area of prime numbers, he's a mathematician that you
can consider to be an expert in the area of prime numbers.
His reply makes me consider what is likely to be the defense of
mathematicians I've contacted over the years when it becomes clear
just how important to the heart of mathematics my work is.
[snip]
1) The bottom line:
Hey stooopid loud troll James Harris, put up or shut up,
http://www.rsasecurity.com/rsalabs/challenges/factoring/faq.html
http://www.rsasecurity.com/rsalabs/challenges/factoring/numbers.html
http://www.crank.net/harris.html
It's not every braying jackass that gets a whole page at crank.net
2) The aesthetics of psychosis:
Ode to James Harris
Somebody said it couldn't be done,
But James with a chuckle replied
That "maybe it couldn't," but he would be one
Who wouldn't say so till he'd tried.
So James buckled right in with the trace of a grin
On his face. If he worried he hid it.
James started to sing and he tackled the thing
And James never fucking could do it.
Somebody scoffed: "Oh, you'll never do that;
At least no one has ever done it";
But James took off his coat and he took off his hat,
And the first thing we knew he'd begun it.
With a lift of his chin and a bit of a grin,
Without any doubting or quiddit,
James started to sing and he tackled the thing
And James never fucking could do it.
There are thousands to tell James it cannot be done,
There are thousands to prophesy failure;
There are thousands to point out to James, one by one,
"How hopeless that task set before you."
But just buckle in with a bit of a grin,
James take off your coat and go to it;
Just start to sing as you tackle the thing
And James, you'll never fucking do it.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
"Quis custodiet ipsos custodes?" The Net! |
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| Brian Quincy Hutchings |
| Posted: Sat Dec 06, 2003 4:44 pm |
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when you ever figure out the difference between your "content" and
your format, then you'll have sufficient grounds
to "prove" neccesity, as well -- that's my best guess.
as for teh "secret entre to math journals,"
I'd suggest looking at *Mathematics Magazine*,
which has proofs at many levels (the "Proofs without Words" one-pagers
e.g.).
a few more steam-venting posts like taht
and I'll be free of JimiHarrisitis for ever!
jstevh@msn.com (James Harris) wrote in message news:<3c65f87.0312052040.50d01a71@posting.google.com>...
Quote: Well I contacted Lagarias again, and for those of you outside of math
Basically the barrier to outsiders isn't content, but style.
--Give the Gift of Dick Cheney -- out of office!
http://larouchepub.com/pr_lar/2003/031128_iraq_statement.html |
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| Andrzej Kolowski |
| Posted: Sat Dec 06, 2003 6:45 pm |
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jstevh@msn.com (James Harris) wrote in message news:<3c65f87.0312052040.50d01a71@posting.google.com>...
Quote: Well I contacted Lagarias again, and for those of you outside of math
circles and the area of prime numbers, he's a mathematician that you
can consider to be an expert in the area of prime numbers.
His reply makes me consider what is likely to be the defense of
mathematicians I've contacted over the years when it becomes clear
just how important to the heart of mathematics my work is.
Exactly what did he say?
Quote: My guess is that mathematicians wish to use journals as a defense, and
claim that it was my responsibility to get a paper in format for a
journal, and present it to journal editors until *someone* published
my results.
Let's be straight about this. You derived an algorithm to count
prime numbers. The main point in its favor is that it appears to
work. You and others have implemented it in various languages.
Your implementation and in fact your definition of the algorithm
is recursive. You have not given a cogent argument that it works
in any form, let alone a form which would pass muster at a math journal.
Legendre's algorithm, like yours, is based on inclusion/exclusion.
Also like yours, there is a recursive version of Legendre's algorithm.
Contrary to what you have said at great length, the recursive version
of Legendre's algorithm, like yours, is a "partial difference equation".
Thus you are not the first or only to count primes in such a manner.
Inclusion/exclusion algorithms are a natural way to count primes
without actually listing all of them. Probably many people have come
up with the same idea independently. If this were the only thing
Legendre had done, he would not be famous. Moreover, Legendre gave
a nonrecursive definition of his algorithm. In general nonrecursive
definitions are easier to work with and understand and nonrecursive
computing algorithms tend to execute more quickly. You however have
not bothered to work out a nonrecursive description or implementation
of your algorithm.
In spite of never having presented a proof that your algorithm works,
you go further. You claim that by replacing increments of "1" in the
variable "y", by "dy", you can arrive at a partial differential
equation. You imply that the solution to this equation must be
a continuous function related to the distribution of primes. You hint
that it might be related to the zeta function. This idea has some merit,
but you present not even a hint of a proof to support it.
Others here, among them David Ullrich, have shown that a partial
differential equation based on a partial difference equation in general
has solutions that are not necessarily close to the original function.
This implies that your proposed and unproven methodology will not
work in general. It may or may not work for your differential
equation, but you have given no hint of proof that it might work.
You want credit for work you have not done. You hope someone
else will take your germ of an idea and flesh it out and make it
work.
Three problems with that. One, it is not clear how you pass to the
limit (with "dy") in your partial difference equation. The "floor"
function, which is discontinuous, is part of the definition of your
partial difference equation. The fact that it is discontinuous makes
it difficult to see how it can lead to a differentiable function.
Two, it should be straightforward to replace "1" by smaller
increments (for example, .5, .1, .01, etc.) in your algorithm and study
the results for agreement with the known distribution of primes
below, say, 10^6. But you have either not done this or not reported
the results here. The agreement is of course perfect when "dy" = "1".
Presumably it becomes worse as dy --> 0. The agreement, as in Ullrich's
examples, may be *very* poor for very small dy.
Three, the partial differential equation that you have presented
is not at all like those which are known to be amenable of analytic
solutions. It is a bizarre nightmare involving quotients and square
roots. Even an experienced mathematician would have a hard time solving
it.
In summary: you have done something original in finding the
algorithm. It is not an astounding breakthrough or a great feat
unlike any that came before, as you appear to believe. You have
not presented any proof that it works. You have a very dubious idea
that your partial difference equation leads to a partial differential
equation whose solution may be related to the distribution of the
primes. You seem to think that just having this idea, with no hint
of proof that it might work, should be enough to make you famous as
a mathematician. This is a delusional view of how mathematicians
work. In general they don't get credit for speculative ideas. They
get credit for great ideas followed by rigorous proofs.
Quote: Their own defense then being that it's just not their fault if I
couldn't manage to write a paper that other mathematicians would
consider.
The excuse is that if you find something, no matter how dramatic it
might be, if you can't write a math paper in a standard format, then
mathematicians don't have to be expected to pay attention to your
result.
You think the problem here is a matter of style, and that what
you have done is being rejected because you don't know how to write
mathematics in the accepted style. That is a misconception. The
core problem here is that you have not presented any logical argument
at all. If you had, someone else could take it and write it up
as a rigorous proof that would be publishable. As in the case of
your discussions of FLT and the purported error in "core mathematics"
the problem is that you simply do not have a proof. You have
speculation. You cannot write a proof in acceptable style not because
you don't know acceptable style, but because you don't have a proof.
Quote: Some of you may not understand just how strict of a requirement that
is, unless you read math journals, or even maybe have read sci.math
long enough, to see the withering criticisms of Ph.D's in physics
who've tried to write math papers.
Absurd. Where have you heard that? John Bell was a physicist. His
math was accepted as valid by both mathematicians and physicists alike.
He even found errors in the work of the great mathematician John von Neuman.
Similarly the work of people like Richard Feynmann, Dirac, Schroedinger,
Edward Witten, and many, many other physicists is accepted and admired by
mathematicians.
Quote: Physicists have their way of writing papers; mathematicians have their
own.
Mathematicians pride themselves on a special format, that has a
certain look, feel and read particular to, you guessed it,
mathematicians.
The main requirement is that it must be logically rigorous. Plenty
of physics papers meet that requirement with no objections from anyone.
The format is nothing more than rigorous proof. That and nothing else
is what forces the "style" to look the way it does. There is no other
valid way to write mathematics.
Quote: Basically the barrier to outsiders isn't content, but style.
See above. That is what some outsiders think. Insiders know that
no amount of style can rescue trivial or incorrect or missing math.
The main thing is creative ideas and rigorous proofs. Sheer speculation
cannot be dressed up to look like mathematics because it ISN'T.
Quote: I want those of you outside of the math world to hear that potential
defense now, so that you can consider it before the mathematicians can
deliver it in earnest, when I put them, and their practices, on public
trial.
A stupid idle threat. Who are you trying to fool? You would do better
to spend your time seeing and reporting what happens with small values
of "dy". Instead you keep trying to prevail by whining and kvetching,
a total waste of time.
Andrzej
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| Christian Bau |
| Posted: Sun Dec 07, 2003 8:54 am |
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Guest
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In article <3c65f87.0312052040.50d01a71@posting.google.com>,
jstevh@msn.com (James Harris) wrote:
Quote: Well I contacted Lagarias again, and for those of you outside of math
circles and the area of prime numbers, he's a mathematician that you
can consider to be an expert in the area of prime numbers.
His reply makes me consider what is likely to be the defense of
mathematicians I've contacted over the years when it becomes clear
just how important to the heart of mathematics my work is.
My guess is that mathematicians wish to use journals as a defense, and
claim that it was my responsibility to get a paper in format for a
journal, and present it to journal editors until *someone* published
my results.
Their own defense then being that it's just not their fault if I
couldn't manage to write a paper that other mathematicians would
consider.
The excuse is that if you find something, no matter how dramatic it
might be, if you can't write a math paper in a standard format, then
mathematicians don't have to be expected to pay attention to your
result.
Some of you may not understand just how strict of a requirement that
is, unless you read math journals, or even maybe have read sci.math
long enough, to see the withering criticisms of Ph.D's in physics
who've tried to write math papers.
Physicists have their way of writing papers; mathematicians have their
own.
Mathematicians pride themselves on a special format, that has a
certain look, feel and read particular to, you guessed it,
mathematicians.
Basically the barrier to outsiders isn't content, but style.
If you had content, you would get all the help you needed with the
style. But you don't, so you won't. |
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| Arturo Magidin |
| Posted: Sun Dec 07, 2003 10:07 am |
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Guest
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In article <a1fa83d9.0312061545.1c18fe1a@posting.google.com>,
Andrzej Kolowski <akolowski@hotmail.com> wrote:
[.snip.]
Quote: You think the problem here is a matter of style, and that what
you have done is being rejected because you don't know how to write
mathematics in the accepted style. That is a misconception. The
core problem here is that you have not presented any logical argument
at all. If you had, someone else could take it and write it up
as a rigorous proof that would be publishable.
While it may be that some "could" do so, I also think it is improper
to suggest that this would or might occur. The ultimate responsibility
for writing up something is up to the proposer, not to anybody
else. Hardy did not take it upon himself to write up Ramanujan's work:
eh took it upon himself to help Ramanujan write them up himself. While
someone might try to help, they would not "write it up as a rigorous
proof" for the OP.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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