HDTenEightyP/NetNews huggingface dataset:
From: sgccifh@citec.oz.au (Ian Henderson)
Subject: turning points of polynomials - help wanted
Organization: CITEC
Date: Tue, 14 Apr 1992 14:21:05 GMT

I am working on a geographic survey project at the moment. We are
at the stage of a working prototype but speed is our big problem. The 486
handles all the maths and graphics fine except for solving a 2nd order
polynomial to determin the turning points withing a given range.

The first problem is to determine how may turning points there are. Next
we need to solve these points to get their co-ordinates. I did a subject
on this at university but that was a long time ago. I think we used
the "Regular falser" technique but this is just a guess. 

If anyone has an algorithm to do this could you share it with me. 
Alternatively if anyone could reccomend a good text book that would
help. We are currently doing a binary chop method to determine the turning
points but there must be a faster way

Thanks in advance.    Ian h.
From: gd1@cunixa.cc.columbia.edu (Gautam Dalmia)
Newsgroups: sci.math
Subject: Re: Fermat's Last Theorem
Date: 9 Apr 92 08:19:44 GMT
Reply-To: gd1@cunixa.cc.columbia.edu (Gautam Dalmia)
Organization: Columbia University

bs@gauss.mitre.org (Robert D. Silverman) says:
>This must be a joke. How can anyone who reads or posts to this newsgroup
>not know who Paul Erdos is???
[... deleted...]

wsadjw@rw7.urc.tue.nl (Jan Willem Nienhuys) supports Silverman, saying:
:The respondee, Bob Silverman is known for the somewhat caustic way
:of expressing himself (with a dash of jusitified superciliousness),
"*Justified* superciliousness"..... my foot! The net is not an
appropriate place to hammer into people the fact that one has achieved
the position of an expert in something. If one is really that
justifiably hot and supercilious then they should go be supercilious
with the other experts. 
Listen, Nienhuys, you moron who understands caustic superciliousness,  
you may find this line unjustified right now..... but call me in 10
years and you will applaud and agree that this line was a dash of
justified ... far more justified that Silverman's.

After quoting Nienhuys' above message Silverman adds:
>I am just surprised that someone with a presumed interest in math would
>not know one of the most outstanding and prolific mathematicians of
>the 20th century. If I were Erdos, I'd be insulted.
This is *ridiculous*! I don't think Erdos would even be slightly
concerned on finding out that there is a reader who has not heard his
name, let alone feel *insulted*. 

In article <STROMME.92Apr6130830@jeeves.math.utah.edu>
stromme@math.utah.edu (Stein Stromme) writes: 
]Come on, Bob, as a reader of this group you must have noticed that its
]contributors range from the professional to the not-so-professional.
]Why not just answer the question? After all, the poster says "Pardon
]my ignorance", and then he does not deserve to be run over like that.
Yes, it is truly stupid to expect everyone reading sci.math to have
heard of Erdos. Why, walk into a class of undergraduate math majors
and ask who has heard of Erdos.... only about half (or less) will
have... try it! ...... And then blast those who haven't.

For now, flame this:

===========================================================================
In high school I read number theory and realized the the beauty of
mathematics: the  simple unanswered questions and the elusive, but
simple, answers. Then, one day, I went to a library to look at
journals/ big books and alas ... mathematicians had invented abstruse,
ugly methods to attack these beautiful questions. At first I was
saddened but then I looked up, and I laughed and said "God, what fools
these mathematicians be".... and God and I have been laughing ever since!
From: heine@barsoom.nhh.no (Heine Rasmussen)
Newsgroups: sci.math
Subject: Re: Metric on groupings?
Date: 12 Apr 92 13:04:05 GMT
Organization: Norwegian School of Economics

In <92103.003114SXW103@psuvm.psu.edu> SXW103@psuvm.psu.edu (S. Wukovitz) writes:

>Any help would be appreciated:

>A friend of mine asked the following question:

>Suppose you decide to divide n objects into discrete sets in various
>ways.  Is there some metric on the set of all possible groupings?
>i.e. if you had

>     grouping 1:  ACD E BF
>     grouping 2:  E ADC BF

>the distance between them would be 0, but

>     grouping 1:  ABCD E F      and
>     grouping 2:  ABC D EF

>would be nonzero, and the metric this distance would otherwise
>behave like a metric is supposed to.

>PLEASE DO NOT SEND MAIL TO ME!!  If you need to mail my friend,
>he is SML108@PSUVM (bitnet) or SML108@psuvm.psu.edu.  Please
>try to post any thoughts you might have...

>Thank-you!

>-Stephanie

There are several ways to do this.  Here is one example:

Let G be a partition of a finite set S, and let x and y be members of
S.  Define the function F(G,x,y) to be 1 if x and y is in the same group
under the partition G, and 0 otherwise.  The function

	D(G,H) = \sum     \sum    | F(G,x,y) - F(H,x,y) |
                x \in S  y \in S

behaves like a metric.

-heine
--
Heine Rasmussen  ---  <sam_hr@debet.nhh.no> or <heine@sterling.nhh.no>
___________________________________________________________________________
Center for Research in Economics and Business Administration
The Norwegian School of Economics, N-5035 Bergen, NORWAY
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Newsgroups: sci.math
Subject: Re: fun with primes
Keywords: prime numbers
Date: 6 Apr 92 19:07:16 GMT
 <1992Apr2.205716.2004@informix.com> <1992Apr4.014908.15094@math.okstate.edu> <1992Apr6.085801.16293@uniwa.uwa.oz.au>
Organization: Beckman Instruments, Inc.

In <1992Apr6.085801.16293@uniwa.uwa.oz.au> hartley@maths.uwa.oz.au (Mike Hartley) writes:

>gindrup@math.okstate.edu (Eric `'d'kidd' G..) writes:

>>As the earlier proofs have shown,
>>(1)  p is assumed to be the largest prime,

>But what if there IS no largest prime, because we've just thrown away
>the relation '<' ? Can we still do the proof? NB you aren't allowed to
>use p! anymore, because p! is the product of all "positive" integers
>"less than" or equal to p. Are there an infinite number of primes
>in the ring Z[i] = {a+bi:a,b are integers} ?

The best one I can remember is: in any Noetherian integral domain D which
is not a field, there are infinitely many primes.

Proof sketch:
(1)  Any Noetherian integral domain has the property that any non-zero
element x can

element x can be written as u p  ... p , where u is a unit and p  are
                               1      n                         i

prime.  (Use the If x is not prime or a unit, write x = y z; continue this

process; if it continues infinitely often, we have a sequence {x }, with 
                                                                i        

x     strictly dividing x , a contradiction of the Noetherian property.
 n+1                     n

(2)  If D is not a field, there is a non-unit; hence, by (1) there is a
prime.

(3)  Suppose there are only finitely many primes; let x be the product of
all of them plus 1.  If x is 0, then each p is a unit, a contradiction.  If
p divides x, then p divides 1, and p is a unit, a contradiction.  Hence, in
any factorization of x (guaranteed by (1)), no prime can occur, again, a
contradiction.  Done.

As for a "simple" example of a domain with no primes, let R be the power
series ring over the rationals allowing non-negative rational exponents and
rational spacing.

R is clearly an integral domain with no primes, but is not a field.
--
Arthur L. Rubin: a_rubin@dsg4.dse.beckman.com (work) Beckman Instruments/Brea
216-5888@mcimail.com 70707.453@compuserve.com arthur@pnet01.cts.com (personal)
My opinions are my own, and do not represent those of my employer.
Ich bin ein Virus.  Mach' mit und kopiere mich in Deine .signature.
Newsgroups: sci.math,sci.math.symbolic
From: Victoria Bush <vbush@copper.ucs.indiana.edu>
Subject: Dynamic programming packages?
Keywords: maple, mathematica, dynamic programming, optimization
Organization: Office of Statistical and Mathematical Computer Applications
Date: Mon, 6 Apr 1992 12:50:39 -0500


I have had a barrage of requests for programs which will do dynamic 
programming (usually time variant). Most of the people would be interested 
in packages for Maple or Mathematica, but I would also like to include 
any and all programs which will handle this. 

Post if you like; otherwise send replies to vbush@copper.ucs.indiana.edu.
If enough interest exists, I'll summarize to the net. 

-Vic
-- 
Victoria Bush, a lowly graduate student| For by your words you will be   
Indiana University                     | acquitted, and by your words you
vbush@iubacs (Bitnet)                  | will be condemned.
vbush@copper.ucs.indiana.edu (Internet)|      -Matthew 12:37


-- 
Victoria Bush, a lowly graduate student| For by your words you will be   
Indiana University                     | acquitted, and by your words you
vbush@iubacs (Bitnet)                  | will be condemned.
vbush@copper.ucs.indiana.edu (Internet)|      -Matthew 12:37
Newsgroups: sci.math
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Subject: Re: factorial puzzle
Organization: The Ohio State University, Dept. of Math.
Date: Fri, 10 Apr 1992 12:42:08 GMT

In article <4782@umriscc.isc.umr.edu> ecarsten@next1.cs.umr.edu (Edward Carstens) writes:
>
>How many zeros are at the end of 1000! ?

A common homework problem...


--
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
Xref: sparky sci.math:5974 comp.theory:1084
From: lnd@csa.bu.edu (Levin)
Newsgroups: sci.math,comp.theory
Subject: Re: Motwani et al work on proof errors
Date: 12 Apr 92 15:33:55 GMT
Article-I.D.: bu.83819
Reply-To: Lnd@cs.bu.edu
Followup-To: sci.math


 In article <1992Apr7.205150.11061@sophia.smith.edu> orourke@milkyway.smith.edu
 (Joseph O'Rourke) writes:
>Does anyone know where the work of Rajeev Motwani et al on errors in
>proofs, and approximations to intractable problems, is appearing?
>STOC?  The work is discussed in the April 7th NYTimes (p.C1).

        The result as stated in NYT appeared in STOC-1991, pp.~21-31: Checking
computations in polylogarithmic time (L.Babai, L.Fortnow, L.Levin, M.Szegedy).
  The application to approximation appeared in FOCS-1991, Approximating Clique
is almost NP-complete (U.Feige, S.Goldwasser L.Lovasz, S.Safra, M.Szegedy).
   Motwani, Arora and others made an elegant improvement (decreasing the amount
of randomness to log from polylog and using only CONSTANT (!) number of bits of
the proof rather than polylog. It will not appear until FOCS in the Fall
(though preprints may be available).
From: bumby@math.rutgers.edu (Richard Bumby)
Newsgroups: sci.math
Subject: New York Number Theory Seminar
Date: 13 Apr 92 22:15:56 GMT
Distribution: ny,nj
Organization: Rutgers Univ., New Brunswick, N.J.

An update on previous posting (one new speaker):

                        NEW YORK NUMBER THEORY SEMINAR

                        City University Graduate Center
                              33 West 42nd Street
                           New York, New York 10031

                                 Spring, 1992

                      Thursdays at 3:00 p.m. in Room 732



       March 5:	Don Coppersmith, IBM
       		"Modifications to the number field sieve"

       March 12:	Phillip A. Griffiths, Institute for Advanced Study
       		"Characteristic classes of differential equations"

       March 19:	Olivier Ramare, Univ. of Bordeaux and Rutgers Univ.
       		"Short effective intervals containing primes"

       March 26:	Doron Zeilberger, Temple University
       		"Will a computer ever be able to find Ramanujan's
       		 (1/99)^4 formula for 1/pi?"

       April 2:	Andrew Odlyzko, AT&T Bell Laboratories
       		"Counting points on high dimensional spheres
       		 and lattice patterns"

       April 9:	Prasad Tetali, DIMACS, Rutgers University
       		"Probabilistic constructions of asymptotic bases"

       April 16: Peter Sarnak, Princeton University
       		Title to be announced

       April 23:	Spring vacation

       April 30: Clara Wajngurt, Queensborough College (CUNY)
                 "Rational solutions of Diophantine equations and
                  their relationship to complex multiplication"


       May 7:		To be announced
-- 
R. T. Bumby **  Rutgers Math ||   Amer. Math. Monthly Problems Editor
bumby@math.rutgers.edu       || P.O. Box 10971 New Brunswick, NJ08906-0971
Telephone: [USA] 908 932 0277 (full-time message line) FAX 908-932-5530
From: ender2@husc10.harvard.edu (Matt Ender)
Newsgroups: sci.math
Subject: Re: A Proof about Groups
Date: 13 Apr 92 04:20:38 GMT
Article-I.D.: husc3.1992Apr13.002038.11029
Organization: Harvard University Science Center

In article <X3F6iB1w164w@netlink.cts.com> teacher@netlink.cts.com (Daniel Pick) writes:
>In Joseph Landin's book, 'An Introduction to Algebraic Structures,'
>the author makes the following definitions:
>A group G which contains only a finite number of elements is a 
>finite group.
>An element x of a group G has finite order if and only if
>there is a positive integer n such that x^n = e, where e
>is the neutral element.
>Then he asks the reader to prove that if G is a finite group,
>then every element in G has finite order.
>This proof stumps me.  If you induct on the number of elements in
>the group, how do you show that each successive element has finite
>order?

Well, actually the proof can be done directly.

Let G be a finite group, let x be an element of that group.

Consider x, x^2, x^3, ..., x^(|G|+1).  As there are more elements here
than in the group, at least two must be the same:  without loss of
generality, let these be x^i, and x^j, where j > i.  Then, as
multiplication on the right by x is 1-1, we may cancel x^i from each
side, leaving e = x^0 = x^(j-i), and j-i is a positive integer such
that x^(j-i) = e.

Q.

-- Matt
From: razer@oolong.hacks.arizona.edu (THe Artful Dodger)
Newsgroups: sci.math
Subject: Edwin Beckenbach
Keywords: Edwin Beckenbach, looking for
Date: 10 Apr 92 22:52:10 GMT
Reply-To: razer@oolong.hacks.arizona.edu
Distribution: world,local
Organization: Razer IntressenGemeinschaft
News-Software: VAX/VMS VNEWS 1.4-a4


	Please pardon the intrusion.
	I'm looking for titles of books, and ISBN nubers thereof,
written or cowriten by Edwin Beckenbach.
	He was my great uncle, and a professor of math at UCLA.
These books represent an important piece of family history, and 
your help will be greatly apreciated.  Please E-mail me, I don't 
normally read this group.  Thank you.

The Artful Dodger P65331727@CCIT.Arizona.Edu|Razer@Hacks.Arizona.EDU
"Can you imagine us years from today, sharing a park bench quietly?
  How terribly strange to be seventy!"
		-Paul Simon & Art Garfunkle "Old Friends" _Bookends_
Newsgroups: sci.math
From: spencer@panix.com (David Spencer)
Subject: Re: FERMAT'S LAST THEOREM
Date: Thu, 16 Apr 1992 14:08:27 GMT
Organization: PANIX Public Access Unix, NYC

In <1992Apr15.141050.4417@sbcs.sunysb.edu> rscott@csws11.ic.sunysb.edu (Robert Scott) writes:

>RECENTLY, HOWEVER, I CAME ACROSS A CASE (NOT IN THE FIELD OF NUMBER THEORY)
>WHERE FERMAT WAS RIGHT, AND THE PREVAILING WISDOM OF 350 SUCCEEDING YEARS'S
>WORTH OF CLEVER MATHEMATICIANS (AND PHYSICISTS) WAS WRONG.

I'll bite. What's the result that F had right and everybody else has
had wrong?

spence    spencer@panix.com

ps.  As a lawyer with a math degree, I have a warm spot in my heart
for Fermat.

pps. solid caps are hard to read. Or do you only have a teletype?

Newsgroups: sci.math
From: gindrup@math.okstate.edu (Eric `'d'kidd' G..)
Subject: Re: factoid about new prime
Organization: Oklahoma State University, Math Department
Date: Tue, 14 Apr 1992 15:52:35 GMT

In article <1992Apr9.020906.20650@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
>In article <SMITH.92Apr7102215@gramian.harvard.edu> smith@gramian.harvard.edu (Steven Smith) writes:
>:
>:                756839
>:               2       - 1                                          162781
>:               -----------  ~=  2.33397019895780128135136676956 x 10
>:                216091
>:               2       - 1
>:
>And even your arithmetic is wrong. The above ratio is around 2^540748
>
>I'd like to know where 162781 came from.
>--
>Bob Silverman

Since 2^3 ~= 10, 2^540748 ~= 10^180246 (or REALLY a bit less, since 8 is a bit
less than 10.)
Howabout 2^10 ~= 10^3.  Then, 2^540748 ~= 10^162224.4 which is closer and
with the previous approximation bounds the base 10 exponent since 1024 > 1000.

-Eric Gindrup ! gindrup@hardy.math.okstate.edu
Newsgroups: sci.math
Subject: "Half or plus".
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: 10 Apr 92 18:07:44 +1200
Distribution: world
Organization: Department of Mathematics, University of Canterbury

This one has bothered me for a long time.


"HALF-OR-PLUS" is a simple discrete-time stochastic process. 
 ~~~~~~~~~~~~
Start with value 1; and at each time step, either halve or add one to the current
value, with probability 1/2 each, independently of what's happened before.


Formally....    Y   = 1
                 0

                        /   Y  + 1  .....   if  X = 0
                       /     n                   n
              Y     = <
               (n+1)   \    Y  / 2  .....   if  X = 1
                        \    n                   n

where { X  } is a Bernoulli sequence with parameter 1/2 .
         n
-----

This is a Markov process, and, as always, one is interested in the long-term
behaviour of it.  

Intuitively, as the value is bounded below by zero, but downward moves are 
more "damaging" than upward ones (at least to large values), one would expect
that it would "settle down" to a reasonable statistcal behaviour. That is,
one might hope that the probability distribution of Y(n) would tend to
some fixed distribution on R+ , seemingly with most of its weight near the
left end.

Indeed, if the transition rule were more diffuse, that's what we would get,
fairly straightforwardly.  But the trouble here is, the state space for this
process, i.e. the set of values any of the Y(n) might take, is a discrete
set, though an unusual one. Specifically, the state space is the set of
positive dyadic fractions
                           S = { m/2^n : for integer m>0, n>=0 } .

So the probability distribution for each Y(n) is discrete, supported by 
a finite subset of S .

And it seems to be exceedingly difficult to calculate what the limiting
(continuous) probability distribution is. It's not even 100% clear that 
there WILL be a limiting distribution, though I feel sure there must be !

It seems reasonably easy to calculate (numerically), coarse approximations to 
the limiting distribution; i.e. the limiting discrete distribution on the set 
of disjoint intervals of unit width; or of half-unit width; and so on. 
But it's frustrating trying to extend the results to a continuous distribution.


Anyone who can provide help on this problem will earn my gratitude.

---
Bill Taylor.                  wft@math.canterbury.ac.nz
From: luzeaux@liszt.berkeley.edu (Dominique Luzeaux)
Newsgroups: sci.math
Subject: Re: Looking for a formula
Date: 15 Apr 92 21:49:44 GMT
Article-I.D.: pasteur.1992Apr15.214944.27291
Reply-To: luzeaux@liszt.berkeley.edu (Dominique Luzeaux)
Organization: University of California

In article <1992Apr15.143234.16947@predator.vpharm.com>, dap@vpharm.com
(David A. Pearlman) writes:
|> Assume you have rotation at a fixed radius R of a point about an axis
|> whose position is fixed in space. For a given position {x,y,z} of the
rotating
|> point, I am trying to determine the cartesian
|> derivatives of that point with respect to rotation of the axis, i.e.
|> d{x,y,z}/d(phi), where "phi" is the rotation angle for the axis.
|> 
|> Does anyone happen to know the answer?

Write your formula with spherical coordinates, you can compute the derivatives
with respect to rho, phi and psi.

	Dominique Luzeaux
Newsgroups: sci.math
From: roche@isl.Stanford.EDU (james r. roche)
Subject: Re: limit of exp(exp(Psi(Psi(x))))/x
Keywords: limit
Organization: Stanford University
Date: Fri, 17 Apr 92 21:06:28 GMT

In article <1992Apr14.141625.23740@neptune.inf.ethz.ch> gruntz@inf.ethz.ch (Dominik Gruntz) writes:
>Can some one give me a tip on how to compute the limit of
>
>               exp(exp(Psi(Psi(x))))/x
>
>as x goes to infinity? (Psi is the first polylogarithm function)
>
>I know, that the limit is finite, but I have not idea on how to
>get that limit. Of course, I have a presumption what the limit
>is, but how would YOU solve that problem?
>
>Thanks in advance,
>- Dominik
>-- 
>/*            Dominik Gruntz    Institut fuer Wissenschaftliches Rechnen  *
> *                              Department fuer Informatik                *
> *        gruntz@inf.ethz.ch    ETH-Zentrum                               *
> *      Tel: +41 1 254 74 77    CH-8092 Zuerich, Switzerland              */

If we're thinking of the same function Psi(x), then 

  exp(exp(Psi(Psi(x))))/x = exp(-1/2) + O(1/ln(x)),

 where the "O" term refers to a quantity whose magnitude
 is less than C/ln(x) for all
sufficiently large x, with C some constant.

     The Psi function that I'm using is defined as the logarithmic derivative
of the Gamma function.  According to _Higher Transcendental Functions, Vol. 1_,
by Erdelyi et al., we have

     Psi(x) = ln(x) - Sum{1/(n+x) - ln(1 + 1/(n+x))},  where the sum is

          from  n=0  to  n = infinity.

     It follows readily that

          Psi(x) = ln(x) - 1/(2ln(x)) + O(1/(x^2))

                 = ln(x) {1 - 1/(2x ln(x)) + O(1/((x^2)ln(x)))}.

Then
     ln(Psi(x)) = ln(ln(x)) + ln[1 - 1/(2x ln(x)) + O(1/((x^2)ln(x)))]

                = ln(ln(x)) - 1/(2x ln(x)) + O(1/(2x ln(x))^2) 

                                             + O(1/((x^2)ln(x)))

                = ln(ln(x)) - 1/(2x ln(x)) + O(1/((x^2)ln(x))).

Then
     Psi(Psi(x)) = ln(Psi(x)) - 1/(2 Psi(x)) + O(1/(Psi(x))^2).

Substituting from above for ln(Psi(x), we get

     Psi(Psi(x)) = [ln(ln(x)) - 1/(2x ln(x)) + O(1/((x^2)ln(x)))]

                              - 1/(2 Psi(x)) + O(1/(Psi(x))^2)

                 = ln(ln(x)) - 1/(2x ln(x)) + O(1/((x^2)ln(x)))

                    - 1/{2[ln(x)] [1 + O(1/(x ln(x)))]} + O(1/(ln(x))^2)

                 = ln(ln(x)) - 1/(2 ln(x)) + O(1/(ln(x))^2).


Then
      exp(Psi(Psi(x)) = [ln(x)] [exp(-1/(2 ln(x)))] [exp(O(1/(ln(x))^2))] 

                   = [ln(x)] [1 - 1/(2 ln(x)) + O(1/(ln(x)^2))] * (continued)

                           * [1 + O(1/((ln(x))^2))] 
 
                   = ln(x) - 1/2 + O(1/ln(x)).

Then
      exp(exp(Psi(Psi(x)))) = exp(ln(x))*exp(-1/2)*exp(O(1/ln(x)))

                   = x*exp(-1/2)* [1 + O(1/ln(x))].

Finally, we obtain

   exp(exp(Psi(Psi(x))))/x = exp(-1/2) + O(1/ln(x)),

                                                      QED.

-------------------------------------------------------------

Jim Roche
roche@isl.stanford.edu
roche@research.att.com
Newsgroups: sci.math
From: jchen@tomahawk.sct.gu.edu.au (Jinghong CHEN)
Subject: Re: ARE equations
Organization: Griffith University, Brisbane, Australia
Date: Sat, 11 Apr 92 21:05:01 GMT

In article <1992Apr11.111900.17423@sabrina.dei.unipd.it> giulio@alessia.dei.unipd.it (Giulio Rigo 345123/IL) writes:
>I'm looking for something about the theory of the convergence of the Algebric
>Riccati Equation (ARE) with matrices. May anyone tell me where to find 
>somethin interesting? Thanx in advance.
>
>                                             Giulio Rigo
>
>                                     giulio@alessia.dei.unipd.it 
>                            

You may try the "IEEE Trans. on Automatic Control" because ARE was a very
popular topic in that periodical in 70's anyway.

J.CHEN (jchen@tomahawk.sct.gu.edu.au)
Newsgroups: sci.math,sci.crypt