HDTenEightyP/NetNews huggingface dataset:
From: ttyytt@mixcom.com (Adam Costello)
Newsgroups: sci.math
Subject: Re: Stupid question about FLT
Date: 20 Jul 92 23:40:36 GMT
Article-I.D.: mixcom.1992Jul20.234036.9686
Organization: Milwaukee Internet Xchange BBS, Milwaukee, WI U.S.A.

In article <1992Jul20.080932.24570@sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <1992Jul20.035836.4789@mixcom.com> ttyytt@mixcom.com
> (Adam Costello) writes:
>
>   >Isn't it?  I can think of a non-recursive set of axioms which would admit
>   >only models elementarily equivalent to the standard model N, namely the
>   >set Th N.
>
>  Yes. However, Th N also has nonstandard models, and I thought that this
>was what you meant by "models which aren't exactly like the natural
>numbers".  Anyway it is a good idea, I believe, to clarify this point.

Okay, I see that you could imagine "non-standard" models of Th N, but since
these models make exactly the same sentences true that N does, this brand
of non-standardness isn't very upsetting.  It's the possibility of a
non-standard model that actually disagrees with N about the truth of some
sentences that provokes discussions like the one we're in the middle of.

AMC
Newsgroups: sci.math
Subject: Geometry - Euclidean/elliptic/hyperbolic.
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: 30 Jul 92 17:25:15 +1200
Distribution: world
Organization: Department of Mathematics, University of Canterbury

I have a small list here of simple characteristic differences between the
three types of homogeneous geometry.

If anyone can add further items of interest, I would be most grateful.
---------------

      Euclidean                               Elliptic            Hyperbolic
      =========                               ========            ==========

One parallel to a line through a given pt.   No parallels.      Many parallels. 
 
Triangle angles add to 180.                  More than 180.     Less than 180. 
 
Triangles have arbitrary area.               Upper limt to area.   <-- ditto
 
Whole space has infinite area.               Finite.            Infinite. 
 
Minimum distance between parallel lines      <inapplicable>     At a particular
 achieved by every point on either line.                        point on each.

----------------
Bill Taylor.         wft@math.canterbury.ac.nz

Newsgroups: sci.math
From: ceblair@ux1.cso.uiuc.edu (Charles Blair)
Subject: The Lorentz transformation
Organization: University of Illinois at Urbana
Date: Thu, 30 Jul 1992 15:09:21 GMT

   Some recent posts have suggested Poincare had the Lorentz transform
equations.  I don't know about this, but there's a book by O'Rahilly
called Electromagnetics or Electromagnetic  Theory which refers to
special relativity as ``Einstein's use of Voigt's tranformation.''
Newsgroups: sci.math
From: abbasi@star.enet.dec.com (My name is...)
Newsgroups: sci.math
Subject: MAPLE: Limit of (1/x^n)*e^x as x->oo
Date: 29 Jul 92 15:08:51 GMT
Organization: Digital Equipment Corporation


Why do you think this MAPLE stopped giving the limit at 7? :
I dont get it?

> limit((1/x^6)*exp(x),x=infinity);
                                    infinity

> limit((1/x^7)*exp(x),x=infinity);
                                            exp(x)
                              limit         ------
                              x -> infinity    7
                                              x


/Nasser
Newsgroups: sci.math
From: friedman@ieor.berkeley.edu (Eric Friedman)
Subject:  Convex Dual Question
Reply-To: friedman@ieor.berkeley.edu (Eric Friedman)
Organization: U. C. Berkeley
Date: Tue, 21 Jul 1992 16:44:26 GMT

Given a convex set $ G \in R^n$  we can define a map
$T : G \rightarrow S^{n-1}$ the n-1-sphere by mapping the normal to the surface
of the convex set in the obvious manner.
This is mentioned in the standard books on convexity under the name
of convex duality.
 
I'd like to know if any serious study of this map has been done.
For example:
 
1) The map is typically injective but not surjective. It seems straightforward
to prove that we can construct a G' which is arbitrarily close to G (in some
natural metric) such that the map is surjective.
 
2) Consider the intersection of a hyperplane H  with G. The map of H\intersection G
 (which is a convex set) forms a subset of S^n. What are the properties of this
 subset? For example let n=3, then the intersection of a plane with
 G maps into a loop on S^2. I don't think that this loop can be too complicated,
 but i`m not sure exactly how to define this.
 
 Presumably, somewhere this has been studied. Any pointers, hints, or
 proofs would be greatly appreciated.
 
 thanks
 eric
From: elkies@ramanujan.harvard.edu (Noam Elkies)
Newsgroups: sci.math
Subject: Re: An interesting limit problem.
Date: 30 Jul 92 19:06:54 GMT
Article-I.D.: husc3.1992Jul30.150656.14324
Organization: Harvard Math Department

In article <MARTIN.92Jul29125203@lyra.cis.umassd.edu>
martin@lyra.cis.umassd.edu (Gary Martin) writes:
>And have we even had a proof that the limit is 1 that didn't use a machine?

The proof I posted only used the machine to substitute one power series
into another, which is just straightforward arithmetic --- it could have
been done easily by hand in a few minutes, but why bother?

>There was one posting that made the substitution  y=tan(sin x) (or something
>similar) and claimed that the result was the reciprocal of the original
>expression, but I don't think that claim was correct.

Me neither --- and even if it was, one would still have to show that the
limit exists and does not equal -1.  But that posting did contain the kernel
of an idea that does provide a valid proof along the lines indicated in
V.Miller's recent posting.  Basically we are comparing the commutators
[f,g] and [F,G] in the infinite-dimensional Lie group of power series
a1*x+a2*x^2+... (a1<>0)  under composition, with F,G the inverse functions
of f,g; these commutators either both reduce to the identity (the power
series x) or are both of the form x+an*x^n+... with the same n.  This
also leads to an explanation of the fact that in our case f=sin,g=tan
(with both f,g odd functions with leading coefficient 1) the difference
f(g)-g(f) [and so also F(G)-G(F)] vanishes to 7th order at the origin.

>      2        2
>   sin      sin
>   ---  x - --- x
>   cos      cos             0
>----------------------- =  --- = 0.  (Though they might leave out the
>   2   2       2   2        0         equal signs, too.  :( )
>Arc sin     Arc sin
>------- x - ------- x
>     2          2
>  cos        cos

                                    5 2
Ouch.  Do they automatically write 2 9  = 2592 too?  ;-)

--Noam D. Elkies (elkies@zariski.harvard.edu)
  Department of Mathematics, Harvard University
Newsgroups: sci.math
From: jmborwei@orion.waterloo.edu (Jonathan Borwein)
Subject: random triangles
Keywords: asymptotics, acute triangles
Organization: University of Waterloo
Date: Fri, 24 Jul 1992 04:21:02 GMT

	What is the probability that a random triangle is acute (has all angles
less than Pi/2)?  

	More precisely, if random lattice triples are picked in [0,n],[0,n],
what is the behaviour of the number of triples corresponding to acute
triangles as n->oo?


*************************
Jonathan Borwein (jmborwei@orion.uwaterloo.ca)

Newsgroups: sci.math
From: kellum@sjsumcs.sjsu.edu (Ken Kellum)
Subject: Re: Factorial program...
Organization: San Jose State University - Math/CS Dept.
Date: Wed, 22 Jul 1992 22:26:58 GMT

In article <1992Jul20.160312.10723@infonode.ingr.com> "Vilva Natarajan <vilva@madras.b23b.ingr.COM>" writes:
>Hi,
>
>	I am looking for source code that prints out all the factorial
>	combinations of a set of numbers.
>
> 
>        Eg.
> 
>        factorial combo of 1 2 3 4 would be, (there are 24 of them)
> 
>        1 2 3 4
>        1 2 4 3
>        1 3 2 4
>        1 3 4 2
>        1 4 2 3 
>        1 4 3 2
> 
>        2 1 3 4
>        2 1 4 3
>        2 3 1 4 
>        2 3 4 1
>        2 4 1 3 
>        2 4 3 1
>        
>        3 1 2 4
>        3 1 4 2
>        3 2 1 4 
>        3 2 4 1
>        3 4 1 2
>        3 4 2 1
> 
>        4 1 2 3
>        4 1 3 2
>        4 2 1 3
>        4 2 3 1
>        4 3 1 2
>        4 3 2 1
> 
>
>	Any help would be appreciated. Thanks>
>
>vn

The following (pseudo) Pascal procedure prints all the permutations
of a one dimensional array.  It can be modified to do what you want.

Procedure Permute (var A : array [1..n] of whatever)

   Procedure Inner (i : integer);
  { prints the permutations of a[i] .. a[n] }
   var j : integer
       temp : whatever;

   begin
      if i > n then
         writeln(a)
      else 
         for j := i to n do begin
            temp := a[i]; a[i] := a[j]; a[j] := temp;
            Inner (i + 1);
            temp := a[i]; a[i] := a[j]; a[j] := temp;
         end;
    end; { Inner }

begin {permute}
   Inner(1);
end;

It's an amusing exercise to construct an inductive proof that this
procedure works.

Kenneth Kellum
___________________________________________

Those who give up claims of uniqueness only become more unbearable
when they claim to be representative.
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: Choice and Measurability
Date: 22 Jul 92 13:59:48 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: carabalo@phoenix.Princeton.EDU (David G. Caraballo)

In article <1992Jul21.183209.8629@Princeton.EDU>, carabalo@phoenix (David G. Caraballo) writes:
>"Solovay in 1964 proved that the assertion 'all sets are Lebesgue
>measurable' is consistent with ZF and a restricted version of choice.
>This restricted version of choice, called the countable axiom of choice,
>asserts that every countable set of non-empty sets has a choice function."
>                         Malitz, _Introduction to Mathematical Logic_, p.49

>I had something similar in my own notes (which is why I posted something to 
>this effect -- included above).  I am not prepared to disregard my notes 
>(and now, the above reference) without seeing an actual proof that my claim 
>"M is consistent with ZF and Countable Choice" is false.  If someone has a 
>proof, I would love to see it.  Thank you.  

You and Malitz (assuming there is no redeeming context) are both inaccurate.

That's all.  Referring to less than DC, and leaving out the role of the
inaccessible, is an inaccuracy.  As you now know, Con(ZFC)=>Con(ZF+DC+M)
is false, so leaving out I is inaccurate.  Getting worked up over other
putative charges is strawman silliness.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
From: cet1@cl.cam.ac.uk (C.E. Thompson)
Newsgroups: sci.math
Subject: Conway's permutation sequence
Keywords: Conway Collatz Hailstone
Date: 24 Jul 92 00:19:07 GMT
Reply-To: cet1@cl.cam.ac.uk (C.E. Thompson)
Organization: U of Cambridge Computer Lab, UK

In article <1992Jul22.134601.6735@nmt.edu>, jefu@akbar.nmt.edu 
(Jeff Putnam) considers inverting the order of the Collatz sequence
(I would still like to know where this "Hailstone" tag comes from)
and points out that there are either 1 or 2 possible immediate
predecessors for each integer.

This reminded me of the following function defined on the integers:

   f(2n)   = 3n
   f(4n+1) = 3n+1
   f(4n-1) = 3n-1

due to J.H.Conway. Obviously, this f does have an inverse; therefore
under iteration of f every integer either lies in a cycle or in a
doubly-infinite chain. The first few lie in cycles:

   (1)
   (2, 3)
   (4, 6, 9, 7, 5)

Do the iterates of 8,

     ..., 287, 215, 161, 121, 91, 68, 102, 153, 115, 86, 129, 97, 73,
     55, 41, 31, 23, 17, 13, 10, 15, 11, 8, 12, 18, 27, 20, 30, 45, 34,
     51, 38, 57, 43, 32, 48, 72, 108, 162, 243, 182, 273, 205, 154, 231, 
     173, 130, 195, 146, 219, 164, 246, ...

form a doubly-infinite chain? Probably...

There is one more known cycle, of length 12, which I leave as an exercise
for the interested reader (its elements are less than 300). Are there any
others? M.J.T.Guy has proved that if so, they have length greater than 320.

Chris Thompson
Cambridge University Computing Service
JANET:    cet1@uk.ac.cam.phx
Internet: cet1@phx.cam.ac.uk
From: rose@fsu1.cc.fsu.edu (Kermit Rose)
Newsgroups: sci.math
Subject: Re: A puzzle!
Summary: A puzzle!!
Keywords: Math, puzzle
Date: 31 Jul 92 15:03:10 GMT
Reply-To: rose@fsu1.cc.fsu.edu
Organization: Florida State University
News-Software: VAX/VMS VNEWS 1.3-4

In article <chepan.712011867@gsusgi1.gsu.edu>, chepan@gsusgi1.gsu.edu (Parthasarathy Nambi) writes...
> 
> 
> Here is something for puzzle lovers:
> 
> Below is a sequence of numbers, can you find the next two numbers and 
> the general formula for generating numbers in this sequence:
> 
> The numbers are:
> 
> 4, 72, 21600, 190512000, 58090919040000,   etc 
> 
> 
> If you know the answer please post a response in this group.
> 
> 
> P.Nambi
> 

2^2, 2^3  3^2,  2^5  3^3  5^2,  2^7  3^5  5^3  7^2, etc


rose@fsu1.cc.fsu.edu          To be sure I see your response, use e-mail.
-----------------------------------------------------------------------
Be of good cheer, for it is much more fun than being depressed.
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Newsgroups: sci.math
Subject: Summation of "finite difference powers"
Date: 21 Jul 92 14:15:48 GMT
Organization: Purdue University Statistics Department


>Recently, there was a question about 'molecules' which really asked for
>the sum of the first N triangular numbers, where N=10^8. I misread the
>problem and posted the N_th triangular number. the correct solution is:

>	 N
>	sum k*(k+1)/2 = S_N
>	 1

			.....................

The various responders seem to be ignorant of the "standard" 
result, easily proved by induction, that

	 N
	sum k*(k+1)* ... *(k+j-1) = k*(k+1)* ... *(k+j)/(j+1)
	 1

This is the finite difference analog of int(x^j) = x^(j+1)/(j+1),
and the extensions to the gamma functions versions are also valid,
and likewise proved by induction.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!pop.stat!hrubin(UUCP)
From: ycchin@tartarus.uwa.edu.au (Chin Yih Chong)
Subject: Another puzzle
Summary: Another Puzzle
Keywords: Another Puzzle
Organization: University of Western Australia
Date: Thu, 30 Jul 1992 07:34:26 GMT

Here is another fun puzzle......

Try to crack this


				10
				11
				12	
				13
				14
				15
				16
				17
				20
				22
				24
				31
			       100
			       121
					      10000
		  1111111111111111 (16 ones)

Can you observe any relationship between each numbers?
Can you find a general formula for it?

(Hint: any numbers below 16 1's and above 10 is undefined)

Anyone who knows the answer, please send a respond to this group.

	^ ^
	O O
	 ~
	\_/

ycchin@tartarus.uwa.edu.au
From: cliff@WATSON.IBM.COM ("Cliff Pickover")
Newsgroups: sci.math
Subject: Hailstone (3n+1) numbers
Date: 24 Jul 92 17:37:44 GMT

A few of you were interested in hailstone numbers.  The following book
shows detailed graphic characterizations of this sequence.
Pickover, C. (1990) Computers, Pattern, Chaos and Beauty.  St. Martin's
Press:  NY.
Newsgroups: sci.math
Subject: Please help name this book.
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: 27 Jul 92 11:27:16 +1200
Distribution: world
Organization: Department of Mathematics, University of Canterbury

I would dearly love it if anyone can help me with this math text.

I don't know EITHER the author OR the title, (what a great start!).
Yes, I *do* remember the subject. It was the topology of manifolds, written
in a fairly non-technical style, for the general mathematical reader. The 
topic was the Thurston work on 3-dimensional manifolds, and the problem of
their classification, (which I gathered from the book was approaching 
completion). 

The book was written in a rather chatty way (which I found rather engaging),
with plenty of diagrams and pleasantly down-to-earth explanations. There
was a lot of talk about the (abstract) glueing together of spaces. There
was a lot of attention given to spaces made from regular solids, with faces
glued together after the corners had either been inflated or deflated so as
to fit them together correctly, providing the final manifold with positive
or negative curvature respectively.

I think the book was published in the mid-80's; certainly sometime in the
80's.  If anyone thinks they recognize it from my garbled description, I
would greatly appreciate it if they could let me know the details. 

Thanks,    Bill Taylor         wft@math.canterbury.ac.nz
From: rose@fsu1.cc.fsu.edu (Kermit Rose)
Newsgroups: sci.math
Subject: Re: Stupid question about FLT
Date: 22 Jul 92 22:34:03 GMT
Reply-To: rose@fsu1.cc.fsu.edu
Organization: Florida State University
News-Software: VAX/VMS VNEWS 1.3-4

In article <1992Jul17.023246.7915@galois.mit.edu>, tycchow@riesz.mit.edu (Timothy Y. Chow) writes...
>I have a really stupid question for which I will undoubtedly be flamed, but
>here goes anyway.
> 
>Assume ZFC is consistent.  If Fermat's Last Theorem is undecidable then
>it is true, right?                  
  ^^^^^^^^^^^^^^^
I do not know if this is true or not.  Suppose FLT is false, but with only 
a finite number of counter examples.  Suppose that the smallest 
counterexample is so large that it is not possible in practice to discover 
it.  Perhaps FLT is decidable in principle, but not computable within the 
lifetime of humanity.  Do we say that FLT is undecidable?

So, if we define undecidable to mean that there ARE NO COUNTEREXAMPLES, 
and THERE IS NO WAY TO PROVE the theorem, then I would agree that 
undecidable implies true.

>So if someone tomorrow produces a proof that FLT is
>undecidable, then FLT itself will be a corollary.  In other words the
>person will have proved FLT.  However, his proof will not be translatable
>into ZFC, for if it were, FLT would be decidable.  So our hypothetical
>genius will not only have proved FLT, but will also have found a method
>of proof that is beyond the capabilities of ZFC.  So my question is, is
>what I have just said correct?

I do not see any flaw your reasoning.  However, it suggests that the proof that 
FLT is undecidable is impossible.

>-- 
>Tim Chow     tycchow@math.mit.edu
>Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
>30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
>only 1 1/2 tons.                               ---Popular Mechanics, March 1949


rose@fsu1.cc.fsu.edu          To be sure I see your response, use e-mail.
-----------------------------------------------------------------------
Be of good cheer, for it is much more fun than being depressed.
From: cet1@cl.cam.ac.uk (C.E. Thompson)
Newsgroups: sci.math
Subject: Re: random triangles
Keywords: asymptotics, acute triangles
Date: 28 Jul 92 14:10:20 GMT
Reply-To: cet1@cl.cam.ac.uk (C.E. Thompson)
Organization: U of Cambridge Computer Lab, UK

In article <1992Jul28.131046.6090@csc.canterbury.ac.nz>,
wft@math.canterbury.ac.nz (Bill Taylor) writes:
|> >	What is the probability that a random triangle is acute  
|> 
|> I think this problem might have appeared here before. It is more elegant to
|> formalize the concept of a random triangle in R^2 , as that formed by three
|> lines whose angles are independemtly chosen from uniform[0,2pi].
|>
|> [proof that the answer is 1/4 in this case omitted]

More elegant, or just easier to solve?

In his original posting, jmborwei@orion.waterloo.edu (Jonathan Borwein)
wrote
>     More precisely, if random lattice triples are picked in [0,n],[0,n],
> what is the behaviour of the number of triples corresponding to acute
> triangles as n->oo?

More generally, what is the probability that the triangle formed by 
3 points selected randomly and independantly from a convex region K
of R^2 is acute? JB's formulation is the case when K is a square.
Numerical simulation suggests that the probability is 0.274... if
K is a square, 0.280... if K is a circle.

I suspect that the required reference is [1], but as I haven't looked
it up yet, I could be quite wrong.

[1] M.G.Kendall, "Exact distributions for the shape of random triangles
    in convex sets", Advances in Applied Probability 17 (1985) 308-329.

Chris Thompson
Cambridge University Computing Service
JANET:    cet1@uk.ac.cam.phx
Internet: cet1@phx.cam.ac.uk
Newsgroups: sci.math
From: tycchow@riesz.mit.edu (Timothy Y. Chow)
Subject: Associated primes
Organization: None.  This saves me from writing a disclaimer.
Date: Thu, 23 Jul 92 03:40:00 GMT

Given a polynomial ring k[x_1,...,x_n] and an ideal in the ring (specified
by a finite set of generators, how can I compute the associated prime
ideals?  Both algorithms suitable for hand calculation and pointers to
relevant software would be appreciated.
-- 
Tim Chow     tycchow@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons.                               ---Popular Mechanics, March 1949
Newsgroups: sci.math
From: palais@binah.cc.brandeis.edu
Subject: Re: Gordon & Breach suit against AMS
Reply-To: palais@binah.cc.brandeis.edu
Organization: Brandeis University
Date: Fri, 24 Jul 1992 19:49:08 GMT

>>   Gordon & Breach filed suit in Germany 
>>   (based on their law prohibiting comparative advertising)
>>   against the American Mathematical Society.  G & B had 
>>   previously filed a similar suit against the American 
>>   Physical Society and the American Institute of Physics.  

...and lost (it was thrown out of court).
From: fc03@ns1.cc.lehigh.edu (Frederick W. Chapman)
Newsgroups: sci.math
Subject: Re: You know, the integers (was: Re: Stupid question about FLT)
Date: 21 Jul 92 13:25:54 GMT
Organization: Lehigh University

In article <1992Jul20.173716.6310@galois.mit.edu>,
tycchow@riesz.mit.edu (Timothy Y. Chow) writes:

>Perhaps you might try to use the fact that in ZFC one can formulate a
>proof that any two Peano structures are isomorphic.  In that case,
>consider nonstandard models of ZFC...

If I am not mistaken, there are no *KNOWN* models for ZFC, standard or
otherwise!  The consistency of ZFC set theory is not known; if a model
for ZFC were to exist, then ZFC would be consistent.

I can't speak for anyone else, but I find the notion that the
consistency of ZFC has not yet been established to be the most
singularly disturbing mathematical news to ever reach my ears, given
that ZFC is intended to serve as a foundation for the rest of
mathematics.
-- 

o ------------------------------------------------------------------------- o
|  Frederick W. Chapman, User Services, Computing Center, Lehigh University |
|    Campus Phone:  8-3218     Preferred E-mail Address:  fc03@Lehigh.Edu   | 
|  "I do comedy and magic; what you don't find funny -- that's the magic."  |
o ------------------------------------------------------------------------- o
Newsgroups: sci.math