From: "Achava Nakhash, the Loving Snake"
Subject: Conjectures still open?
Date: Tue, 24 Aug 1999 11:07:58 -0700
Newsgroups: sci.math
Keywords: existence of Hadamard matrices, projective planes, Jacobian conject.
I have been attempting to learn Projective Geometry lately. I am
finding it hard to access what I am really interested in, as writers on
this subject do not seem to be the best expositers in the known
universe. However I am having fun with the combinatorial side of things
as I mostly ignored them in my youth. Specifically I have been reading
Ryser's book on combinatorial mathematics that is one of the early Carus
monographs. It was written around 1963. He states that it is open
whether or not there exist Hadamard matrices of order 4n for every n.
He also states that the Bruck-Ryser theorem eliminating a class of
numbers from being the orders of finite projective planes and the easy
theorem that there are finite projective planes of order p^n for every
prime p and all n >= 1 are the last word on the existence question for
finite projective planes. My questions are simple.
1) Is the conjecture about Hadamard matrices still open?
2) Has there been any progress on the existence question for finite
projective planes?
Note: At least one big conjecture, the one about the
determinant/permanent (which I think was called the VanDerWaerden
conjecture, has been settled since the book appeared. Of course a
couple of other minor issues, not mentioned in the book, have been
settled as well, such as the 4-Color conjecture, the Fermat last theorem
conjecture, the Bieberbach conjecture about schlicht functions, and many
more that I don't recall, don't understand, and/or don't know about or
have forgotten.
While I am on the subject of conjectures, my friends in Algebraic
Geometry told me about some sort of conjecture involving the Jacobian
and something or other to do with polynomials in lots of variables. It
seemed a simple and natural question and astounded me that it was open.
This was during the 70's, so it has been a while. So here are some more
questions.
3) What is the Jacobian conjecture mentioned above, and is it still
open?
4) Given a specific set of trees, T_1, T_2, ..., T_n such that each
T_i has i edges, is it possible to embed them in the complete graph on
n+1 vertices in such a way that the edges do not overlap? There are
exactly enough edges for this to happen, but if true, it is kind of
amazing that you can do this no matter which trees you pick initially.
Obviously the vertices are not distinct. I asked this question a while
ago but go no meaningful feedback except for the guy that told me where
to find the person that I heard this from in 1980. But I am very shy,
so I haven't taken this route.
Please reply. Enquiring minds want to know.
Regards to all,
Achava
PS: I dug up a paper I wrote in 1981 or thereabouts and which was
rejected (sent back for extensive revision actually)because I included
way too much detail and because I inadvertantly seemed to be claiming
results as my own which were not my own. Apparently my proofs were
sufficiently different that the journal would have published them anyway
if I had included references, cleaned up my presentation (which I have
discovered is shot full of typos) and so on. I didn't consider my main
result important enough to go through the hassle of looking over what I
had done and straightening out the presentation. I am now in the midst
of doing that, The point of all this blathering is that I will first
present the main, possibly new, result, a completely insane but
interesting to me at least determinant containing powers of Fibonacci
numbers., in this very newsgroup. I simply have to find the time to
type it into a file, and since I am extremely busy at work these days,
it might be a week or two.
==============================================================================
From: gerry@mpce.mq.edu.au (Gerry Myerson)
Subject: Re: Conjectures still open?
Date: Wed, 25 Aug 1999 13:34:34 +1100
Newsgroups: sci.math
In article <37C2DF7E.D40F1F1F@hotmail.com>, "Achava Nakhash, the Loving
Snake" wrote:
I should defer to people who know much more than I do about these questions,
but I'm pretty sure I know the answers.
=> 1) Is the conjecture about Hadamard matrices still open?
Yes.
=> 2) Has there been any progress on the existence question for finite
=> projective planes?
Yes. A lot of theory and an enormous amount of computing went into proving
a few years back that there's no projective plane of order 10. I think
that summarizes the progress.
=> 3) What is the Jacobian conjecture mentioned above, and is it still
=> open?
It is still open. I couldn't give an accurate statement.
Gerry Myerson (gerry@mpce.mq.edu.au)
==============================================================================
From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: Conjectures still open?
Date: 25 Aug 1999 03:48:23 GMT
Newsgroups: sci.math
In article <37C2DF7E.D40F1F1F@hotmail.com>,
"Achava Nakhash, the Loving Snake" writes:
|2) Has there been any progress on the existence question for finite
|projective planes?
The last news I heard was the fairly well publicized computer-aided
verification that there are none of order 10 (the second smallest
positive integer, after 6, not a prime power). It wasn't just a brute
force search, but it did involve appreciable computer time.
...
|3) What is the Jacobian conjecture mentioned above, and is it still
|open?
Indeed it is, and it's expected to be pretty hard. It doesn't seem
like it should be, but people have put a lot of work into it.
Supposedly it is possible to do a lot of calculating relevant to it.
Let f:A^n->A^n be a morphism (given by n polynomials f1,...,fn in n
variables x1,...,xn). Let J=det|df_i/dx_j| be the Jacobian polynomial
of f. It's not so hard to show that if f is an isomorphism (has an
inverse morphism) then J is a nonzero constant. The Jacobian conjecture
is that if J is a nonzero constant, then f is an isomorphism.
Keith Ramsay
==============================================================================
From: James Van Buskirk
Subject: RE: initial guesses for local optimization
Date: Mon, 28 Jun 1999 22:47:27 -0600
Newsgroups: [missing]
To: "'Dave Rusin'"
Keywords: Hadamard's maximum determinant problem
----------
From: Dave Rusin
Sent: Monday, June 28, 1999 6:20 PM
To: torsop@ix.netcom.com
Subject: Re: initial guesses for local optimization
>In article <7l7cbf$17e@dfw-ixnews7.ix.netcom.com> you write:
>>One test case with lots of local minima is simply to attempt to
>>optimize the determinant of a real NXN matrix subject to the
>>constraint that all the entries are no greater than one in
>>absolute value.
>
>I believe I have seen this problem before. Is there a name I could use for
>searching Math Reviews (say) for this? I can't remember what's known here.
>
>dave
Hadamard's maximum determinant problem. Hadamard's inequality.