HDTenEightyP/NetNews huggingface dataset:
From: bts@duke.cs.duke.edu (Bruce T. Smith)
Newsgroups: sci.math,sci.logic
Subject: Solving linear Diophantine equations?
Keywords: integers equations AC-unification why-not?
Date: 16 Apr 92 12:26:45 GMT
Followup-To: poster
Organization: Duke University Computer Science Dept.; Durham, N.C.

What's the current state of the art in solving linear
Diophantine equations? (Integer coefficients and only
integer solutions are interesting, basically.) I know
some of the math involved, but what about algorithms
or even programs -- so long as in something civilized
like Pascal or Scheme?

I have not looked at anything newer then Huet's 1978
paper in Information Processing Letters.

This's posted to sci.logic because one application is
in computing AC (associative-commutative) unifiers.
______________
Bruce T. Smith
From: sichase@csa1.lbl.gov (SCOTT I CHASE)
Newsgroups: sci.math
Subject: Re: Math songs?
Date: 17 Apr 92 20:25:15 GMT
Organization: Lawrence Berkeley Laboratory - Berkeley, CA, USA
Distribution: na
Reply-To: sichase@csa1.lbl.gov
NNTP-Posting-Host: 128.3.254.196
News-Software: VAX/VMS VNEWS 1.3-4   

In article <1992Apr17.142606.920@drycas.club.cc.cmu.edu>, fenthe@drycas.club.cc.cmu.edu writes...
>Does anyone have any mathematical songs (mnemonic ideally, but

My highschool "mathletes" team used to have a cheer that started with something
like:

Tangent! Secant! Cosine! Sine!
Three point one four one five nine!
Go Team!

I can't remember the rest, and I'm not sure that I should bother.

-Scott

--------------------
Scott I. Chase
SICHASE@CSA2.LBL.GOV


Newsgroups: sci.math
From: gerry@macadam.mpce.mq.edu.au (Gerry Myerson)
Subject: Re: Large primes; largest pi(n) known.
Organization: ceNTRe for Number Theory Research
Date: Tue, 7 Apr 92 23:42:36 GMT

In article <1992Apr7.103427.4558@csc.canterbury.ac.nz>, wft@math.canterbury.ac.nz (Bill Taylor) writes:
> 
> It doesn't seem to be on the FAQ, so can someone tell us...
> 
> what is the largest known prime for which all smaller primes are known;
> 
> i.e.
> 
> how far does the list of known primes extend, without gaps;
> 
> i.e. (I presume equivalently)
> 
> what is the largest n for which pi(n) is known ?
> 
> 
> Bill Taylor             wft@math.canterbury.ac.nz 

Sorry, I don't know the answers, but I do know that the third problem is NOT 
equivalent to the other two. There are ways of computing the number of primes 
up to n without computing the primes themselves. See, for example, the 
discussion of Meissel's formula in many intro number theory or analytic 
number theory books. 

Gerry Myerson

Newsgroups: sci.math
From: allenk@harry.ugcs.caltech.edu (Allen Knutson)
Subject: FOLLOWUP to Why is Trace so easy?
Organization: California Institute of Technology, Pasadena
Date: Fri, 10 Apr 1992 17:19:08 GMT

I asked (a week back or so) what the real reason is that Trace should be so
easy to calculate (since determinant isn't), and more concretely, a formula 
for the other symmetric polynomials of the eigenvalues (equivalently, the
other coefficients in the characteristic polynomial) that is manifestly
easier for Trace.

Many people pointed out what should have been obvious: while Tr M gives the
sum of the eigenvalues, Tr M^n gives the sum of their nth powers, and using
some formulae due to Newton one can go back and forth between the power sums
and the symmetric polynomials. (Which I'd even used a few years ago, to show
that if you already know the character table of a group, knowing the power 
maps of the conjugacy classes is equivalent to knowing the eigenvalues, rather
than merely their traces.)

Many people pointed out that Trace is the only one that's linear in the 
entries. But a much more complete answer was that it comes down to the fact
that we tend to write linear transformations using their matrix coefficients,
rather than (say) as the product of elementary matrices (those differing from 
the identity in only one entry). Were this latter the standard representation 
of a matrix, I'd be asking why determinant was so easy, while trace was so 
hard.

Thanks very much to everyone who responded!			Allen K.

Newsgroups: sci.math
From: bdm@cs.anu.edu.au (Brendan McKay)
Subject: Erdos, the pronunciation
Reply-To: bdm@cs.anu.edu.au
Organization: Australian National University
Date: Sat, 11 Apr 92 07:29:01 GMT


In article 15616@wheaton.wheaton.edu, poynor@wheaton.edu (George V. Poynor) writes:
>	If an enquirer hasn't heard of Erdos, it might also be
>        that (s)he doesn't know how to pronounce it: Ur'dish.

I am afraid that you are quite wrong.  Here is the correct pronunciation
as taught me by Erdos himself.  After writing it down, I verified it
with another Hungarian mathematician who has been a close friend of
Paul's for more decades than she would admit.

  Er  : similar to the English word "air" but the "r" is rolled briefly.
        It is hard for English speakers like me to roll the "r" without
        creating a pseudo-syllable between this one and the next;
        however this is essential.  

  dos : (there is a long Hungarian umlaut on the "o", in TeX: \H)
        the sound is similar to the English "ersh" though not exactly.
        The long accent makes the "er" fairly long. 

  The accent is on the first syllable.

His first name is actually Pal (accent on the "a") pronounced "Pahl",
but practically everyone calls him "Paul" with the English pronunciation.

Brendan.
From: ender2@husc8.harvard.edu (Matt Ender)
Newsgroups: sci.math
Subject: Re: Erdos problems
Date: 11 Apr 92 23:30:46 GMT
Article-I.D.: husc3.1992Apr11.193047.10973
Distribution: world
Organization: Harvard University Science Center

In article <1992Apr10.144022.10926@husc3.harvard.edu> fry@zariski.harvard.edu (David Fry) writes:
>
>There's a famous Erdos problem involving a combinatoric
>problem P for a given integer n.  Unfortunately I can't
>remember what the problem P was, other than it was deceptively
>simple. I think P(1) and P(2) is pretty easy and P(3) has been
>solved, although with effort.
>
>The story is that aliens from another planet come to Earth and
>say they will destroy it in one year unless we can prove our
>worth by solving P(n).  Erdos says that if the aliens set n=4,
>we should stop all computers on the planet from their normal
>work, and set them to solving P(4), and possibly we'll get it
>done in one year.  But if the aliens set n >= 5, we should
>spend the time figuring out how to destroy the aliens.
>
>So does anyone know what P(n) is?
>
>David Fry                               fry@math.harvard.EDU
>Department of Mathematics               fry@huma1.bitnet
>Harvard University                      ...!harvard!huma1!fry
>Cambridge, MA  02138            

The problem is the Ramsey problem, often stated as a 'party and
handshakes' problem.  Q(k,n) = "K people go to a party.  Over the
course of the evening some of them shake hands with others.  Is it
necessarily true that (there are n people each of which has shaken hands
with all other n-1 in that group) or (there are n people each of which
has not shaken hands with all n-1 in that group)."  P(n) = least k
such that Q(k,n) is true.

Graph theoretic:  Let (G,E) represent graphs with undirected edges.  What is
the smallest number such that, for any graph G of that size, either (G,E)
has a k-clique or (G,E') where (i,j) in E' iff (i,j) not in E has
a k-clique?

Personally, I thought the bound was one higher (i.e. n => 6 would be
breakpoint), but I'm not sure on that.

-- Matt
Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: Series involving primes
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 9 Apr 1992 14:31:13 GMT

In article <1992Apr7.162849.2852@ifi.uio.no> einarr@ifi.uio.no (Einar Andreas R|dland) writes:
:
:In article <1992Mar31.200531.9836@linus.mitre.org>, bs@gauss.mitre.org (Robert D. Silverman) writes:
:> In article <1992Mar30.173346.1602@ifi.uio.no> einarr@ifi.uio.no (Einar Andreas R|dland) writes:
:> :However, I'd like to know if
:> :
:> :    oo
:> :   ---      1 
:> :   >    ---------
:> :   ---  p(i) d(i)
:> :   i=1
:> :
:> :converges.
:> 
:> Yes it does. We have the above sum equal to the following Stieltjes
:> integral:
:> 
:> integral from 2 to infty of  f(x) d[pi(x)]  where f(x) is:
				       ^^^^^

This should have been floor[x]. Typo. But the integral still converges.

:> 
:> 	1
:>  -------------------------
:> x log^2 x (1 + O(1/log^k x))
:>  
:> f(x) is obtained by using the PNT. p(i) ~ i log i   and
:>  d(i) ~ log i (1+O(1/log^k i))  for some constant k.
:>  
:> and pi(x) is the prime counting function.
:> 
:> The above integral is finite, so the sum is finite.
:> 
:Well, if d(i) ~ log i (1+O(1/log^k i)) there would, for instance, have to
:be a finite number of prime couples (which, of course, there might ... I
:don't know...). The difference d(i)=p(i+1)-p(i) does not increase regularly;
 
I left out one important and critical word in my post. The NORMAL order
of d(i) is log i (1 + O(...)). The exception set to this is sufficiently
thin as to not affect my argument. This is shown by Brun's theorem.
[sum of reciprocals of primes differing by a constant d converges].

Break the sum 1/d(i)p(i) into 2 parts. The first is where d(i) < sqrt(log i)
and the second where d(i) > sqrt(log i). The second part clearly converges.
To show that the first part converges [i.e. the set of prime gaps that are
small] one follows Brun's proof. You start by summing by parts then showing
by a sieve argument that the main term has sufficiently few terms that it
converges.

This is more or less equivalent to representing the NORMAL order [in the
sense of Hardy & Wright] of d(i) as log i (1 + O(error)). However, breaking
the sum and following Brun's argument IS more rigorous.
--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
Newsgroups: sci.math
From: bart@bluemoon.rn.com (Bart Jaszcz)
Subject: looking for a trig identitiy not easily solvable...
Organization: Blue Moon BBS ((614) 868-998[024])
Date: Fri, 17 Apr 92 00:53:17 EDT

if you have a trig identity that you could not solve or it took a lot 
of thinking to solve, please send it to bart@bluemoon.rn.com
 
by a trig identity, i mean ... 
 
(simple example)  cos^2 @ + sin^2 @ = tan@ cos@ csc@
 
thanx..

 This is from
     bart@bluemoon.rn.com
who doesn't have their own obnoxious signature yet
From: bs@gauss.mitre.org (Robert D. Silverman)
Newsgroups: sci.math
Subject: Re: Fermat's Last Theorem
Date: 31 Mar 92 17:15:29 GMT
Article-I.D.: linus.1992Mar31.171529.4532
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA

In article <1992Mar31.074735.28595@augean.eleceng.adelaide.edu.AU> dabbott@augean.eleceng.adelaide.edu.AU (Derek Abbott) writes:
>>No, Erdos only offers money for his own problems.
>
>Pardon my ignorance. Who is this Erdos dude?
 
You're joking right???  April 1 is tomorrow, not today, right???

This must be a joke. How can anyone who reads or posts to this newsgroup
not know who Paul Erdos is???

I can understand Joe Blow off the street not knowing, but how can anyone
with an interest in math [evidenced by the fact that he reads this group]
not know??



--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: Fermat's Last Theorem
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 9 Apr 1992 14:10:40 GMT

In article <1992Apr7.051902.4889@rata.vuw.ac.nz> harper@kauri.vuw.ac.nz (John Harper) writes:
: >Pardon my ignorance. Who is this Erdos dude?
:Several people have been abusing one another instead of answering the
:question. Paul Erdos is a very famous Hungarian mathematician (aged around
			    ^^^^^^^^^^

This is why I expressed surprise. 

--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
Newsgroups: sci.math
From: erick@fraser.sfu.ca (Erick Bryce Wong)
Subject: Re: Prime numbers (Was Re: Lackey again)
Organization: Simon Fraser University, Burnaby, B.C., Canada
Date: Wed, 8 Apr 1992 01:18:28 GMT

bs@fatima.mitre.org (Robert D. Silverman) writes:
>gjschmid@descartes.waterloo.edu (Greg Schmidt) writes:
>
>:Actually, the _order of magnitude_ (ie exponent of ten when the number is
>:in scientific notation) is approximately four times that of the old record
>:(somewhere near 10^227 or 10^272, I'm not sure which).

The old record had 58 digits?  Gee, good thing we have this new prime, or we'd
never get to implement RSA public-key encryption...  :-)

>Want to try again??  The new prime is near 10^227831. You are off by a factor
>of 1000 in your EXPONENT.

To be more precise... :)...around:
1.74135906820087097325163599245903327890779363690507030974654735538382721562066
2576319147974364224616106351300713682936607281597090545867723690494911429347720
2089620405024218873003497567737597556640892789979850725619057310321637108470694
6529168988544530722380248547797941846968948877581472117196096521071301381477836
5553675674358992096753406551200742920360681239094095454312630905781679734461358
8213522035352461072027970989987749208699507241369141881560320836818547291247759
8626046460960216257228388273893989189010461721931215554593257013799532456881185
7599667820430751438563819872265046778990753886146840591627960356117462730111874
0917331778061439712232526149228237359258800387983350478603205086462606622810603
0298325285057443402167635765093406904801961800958032801840943252236944301654241
x 10^227831

Oh, and the corresponding perfect number?
1.51616570220270280278989635525124694991858659922135385078738030466756180166369
8741760279877416301155230868871045913875461925674304437972810503434985062455133
4479221500362244432814245739110634613759900194287076334767196094656346430551170
8455319548680020795670864835385109738817616848580643282118305278567712397626791
0810791376724060597119363034916279756615039945129739254215116974286415588534805
7191214741104027105797434721583861247327352780911664881001798794412847348158580
3323455442993857367813444972623970719057103348996384611054998367280999325520081
1149296191738743271972353341588927226602875426950286648209339489730437775309039
7598363641252686411687776566642824594706905432718640281181016510156395300965917
7589470433334598229347756367378987925284567302328047307614324192388323983273329
x 10^455662

What can I say?  I got bored one day and computed both of them...(Anyone want
the rest of the digits? ;)

-- 
 -- Erick (erick@sfu.ca, erickw@sfu.ca, or erick@cs.sfu.ca, take your pick)
From: rose@fsu1.cc.fsu.edu (Kermit Rose)
Newsgroups: sci.math
Subject: fractional derivatives
Date: 3 Apr 92 03:50:36 GMT
Article-I.D.: mailer.1992Apr2.225037.9980
Reply-To: rose@fsu1.cc.fsu.edu
Organization: Florida State University
News-Software: VAX/VMS VNEWS 1.3-4



Hi. I saw a refrence here to fractional derivatives.  A long time ago, I 
tried to define fractional derivatives, and did not come up with consistent 
results.  Could someone please e-mail me a short definition of fractional 
derivative?
Kermit@FSU or rose@fsu1.cc.fsu.edu.bitnet
If you want to be sure I see your response, please e-mail me.
---------------------------------------------------------------------------- 
This is the signature file of Kermit Rose.

Believing in dichotomies leads to hardening of your logic.

What determines the existence of Free Will?

Peace is the way to peace.  Hostility is the way to hostility.
From: v068pzyd@ubvmsb.cc.buffalo.edu (Vance A Arocho)
Newsgroups: sci.math
Subject: Request for info
Keywords: numerical analysis, solar system
Date: 7 Apr 92 10:11:00 GMT
Distribution: na
Organization: University at Buffalo
News-Software: VAX/VMS VNEWS 1.41

I'm taking my 2nd course in numerical analysis and I'm looking for articles
and texts for information that would help me model the solar system. Please send
email if you know of any.

Thanks in advance. 
Newsgroups: sci.math
From: mkrogers@amherst.edu
Subject: Re: fun with primes
Organization: Amherst College, Amherst, MA.
Date: Wed, 15 Apr 1992 21:02:11 GMT

In article <1992Apr6.182451.12004@linus.mitre.org>, bs@gauss.mitre.org (Robert D. Silverman) writes:
> Let me try to shed a little light on the confusion around Euclid's proof that
> there are an infinite number of primes.
> 
> Suppose there are only a finite number of primes. (2,3,5,7,11,.....N) for
> some N. Consider the number S =  2*3*5*7*11*.... *N + 1.  This is NOT N!+1,
> but only the product of the primes up to and including N.
>  
> Now S is not divisible by any of 2,3,5,7,11,.....N since it leaves a
> remainder of 1. Therefore if S is composite, then it is divisible by
> a prime not in our presumed finite set of primes. And if S is prime, then
> it is not in our list. 
> 
> We have therefore shown that if we assume there is a finite list of primes,
> there must be another prime that is not included in our list.
> 
> Contradiction.

Someone else asked whether the infinitude of primes can be proved without
reference to the order ('<') of the primes.   Robert's excellent proof
can be altered in this way:

Suppose there are finitely many primes  p1, p2, ... , pn .  Let  S  be one
plus the product of these primes:  S = 1 + p1 * ... * pn .  The rest of the
proof above yields a contradiction.

	-mike rogers
	mkr@cs.amherst.edu
From: parson_r@cubldr.colorado.edu (Robert Parson)
Newsgroups: sci.math
Subject: Re: eigenvalue PDE
Date: 15 Apr 92 20:19:41 GMT
Organization: University of Colorado, Boulder

In article <12070.29eb1a18@ohstpy.mps.ohio-state.edu>, luk@ohstpy.mps.ohio-state.edu writes:
> Does anybody have the experience solving the following partial differential
> eigenvalue equation:
> 
> L[\phi(x,y)]=E*\phi(x,y)   where E is eigenvalue and L is operator defined by 
> 
> A*(\partial_{x,x}+\partial_{y,y})+ i*B*(x*\partial_{y}-y*\partial_{x})+
> C*(x**2+y**2)=0
> 
> A,B, and C are real constant and i is imaginary sqrt(-1). 

This looks like the Schrodinger equation for a two-dimensional isotropic
harmonic oscillator perturbed by a uniform field that couples to the
vibrational angular momentum (the term involving B.)  Take a look at
Cohen-Tannoudji et al, _Quantum Mechanics_, VOl. 1, Complement D-6.
Or, use plane-polar coordinates.

Robert
Xref: sparky sci.math:6072 sci.math.symbolic:1269
From: steve@maths.warwick.ac.uk (Steve Rumsby)
Newsgroups: sci.math,sci.math.symbolic
Subject: Re: Announcement of GAP 3.1
Keywords: Computational Group Theory
Date: 15 Apr 92 10:06:25 GMT
Organization: Geometry Group, Maths Institute, Warwick University, UK.

In article <martin.703247020@bert>, martin@math.rwth-aachen.de (  Martin Schoenert) writes:
>     For users in the United Kingdom with only Janet access, neither 'ftp' nor
>     the mail server will work (please do *not*  try to  use the mail server).
>     Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
>     has kindly offered us to distribute GAP in the United Kingdom.
>
For UK people *with* Internet access, the distribution is also available
for ftp from ftp.maths.warwick.ac.uk:/pub/gap.

Steve.
-- 
UUCP:	 ...!uknet!warwick!steve	Internet: steve@maths.warwick.ac.uk
JANET:	 steve@uk.ac.warwick.maths	PHONE:	 +44 203 524657
From: afarrell@maths.tcd.ie (Andrew Farrell)
Newsgroups: sci.math
Subject: Re: On knowing who's who (was: Fermat's Last Theorem)
Date: 13 Apr 92 14:22:30 GMT
Organization: Dept. of Maths, Trinity College, Dublin, Ireland.

In <29E4E96C.13732@ics.uci.edu> jeff@madeleine.ics.uci.edu (Jeff Erickson) writes:

>a_rubin@dsg4.dse.beckman.com (Arthur Rubin) writes:

>| Erdos is probably the best known active mathematician, possibly the
>| best known living mathematician.

>Within the mathematical community, probably so.  But outside it, I
>would expect Benoit Mandelbrot, John Conway, and Roger Penrose to be
>better known than Paul Erdos, in about that order.  I hadn't even
>heard of Erdos until last year, but I would've recognized the other
>three names ten years ago (in high school).

>Then there's Doug Hofstadter and Martin Gardner.  I'm not sure where
>to put them in the familiarity standings (certainly above Erdos), or
>whether they're properly classified as active mathematicians.

If You count Gardner, I'd place him at #1. Benoit Mandlebrot's name may be
more famous, since Fractals were last years Neat Mathematical Subject (TM)
but I'd disqualify him because no-one knows his first name :-)

>| I don't think Lederman is the best known active physicist.

>I'd guess Steven Hawking, but I'm not a physicist.  [I would've said
>Feynman, but he's dead.]

Definitly Hawking. Penrose should be a candidate for this, rather than for
most famous mathematician, but his most famous book (The Emperor's New Mind)
is atrociously written, and compares really badly to Hofstadter's GEB:EGB anyway
By the way, if dead physicist are allowed, the overall Grand Champeen has to be
the spirit of Albert Einstein, still famous after all these years of deadness.


Andrew. (The current incarnation of pop culture.)
Newsgroups: sci.math
From: cyeomans@ms.uky.edu (Charles Yeomans)
Subject: Re: New Mersenne prime discovered
    <1992Apr9.132549.24451@linus.mitre.org>
Date: Thu, 9 Apr 1992 14:53:15 GMT
Organization: University Of Kentucky, Dept. of Math Sciences
X-Bytes: 

In article <1992Apr9.132549.24451@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
>In article <pete.02ff@bignode.equinox.gen.nz> pete@bignode.equinox.gen.nz (Pete Moore) writes:

>Let me quote John Selfridge, who originally put forth this 'conjecture'.
>
>It is certainly true, and equally certainly it will never be proved.
>
Of course Hilbert made a statement not unlike this about a problem
in transcendental number theory; Gelfond and Schneider then settled
the problem.

>Counter examples are beyond the range of current computers. There is no known
>method that will allow us to prove (2^p+1)/3 is prime for p much above 5000.
>[There is no equivalent to the Lucas-Lehmer test for 2^p+1/3]
>
>--
Charles Yeomans
From: jk87377@cc.tut.fi (Juhana Kouhia)
Newsgroups: sci.math
Subject: Re: the domain of a rational function
Keywords: discontinuity, L'Hospital
Date: 14 Apr 92 11:30:48 GMT
Organization: Tampere University of Technology


In article <1992Apr13.063539.13454@clinet.fi> dix@clinet.fi (Risto
Kaivola) writes:
>The following question was on this year's matriculation examination here
>in Finland:
>               ax + b
>Let f:  f(x) = ------
>               cx + d
>
>The coefficients of this function satisfy c <> 0,  d <> 0,  and  ad = bc.
>Prove that f is a constant function for all x in Dom(f).
  [..deleted..]
>The function f is defined whenever x <> -d/c.
  [..deleted..]
>So, I see no reason to say that the domain of f isn't the set of all real
>numbers.  Comments?

Hmm... I'm not professional..
Let assume you're right and the function is defined in Real.
We want calculate f(x) with x = -d/c:

           a(-d/c)+b     0
f(-d/c) = ----------- = ---
               0         0

Ok, the function is not defined at x = -d/c, so you're wrong.

I think that you miss the point that L'Hospital rule is limit
operation, which might be (I'm not sure) equal to
 \lim_{x \to -{d/c}^+} f(x) = \lim_{x \to -{d/c}^-} f(x)
(Limits from the both sides to -d/c.)

By the way, I think we can't use L'Hospital if we don't have the same
limit from the both sides; is this true and is this included in
condition that we get f() = 0/0? No math book on hand...


I got solutions (two) by checking derivative of f(x) (it's zero)
and by dividing the polynomials:

   ax + b     a     b - ad/c
  -------- = --- + ----------
   cx + d     c      cx + d

Latter term is 0, because b-ad/c=0 (with c <> 0).


Juhana Kouhia