HDTenEightyP/NetNews huggingface dataset:
From: hanche@ams.sunysb.edu (Harald Hanche-Olsen)
Subject: Re: Smooth manifolds and function extensions
Organization: University at Stony Brook, NY
Date: Thu, 28 Jan 1993 02:45:19 GMT

>>>>> In article <1993Jan27.022648.23237@galois.mit.edu>,
>>>>> jbaez@riesz.mit.edu (John C. Baez) writes:

John> In article yeomans@austin.onu.edu (Charles Yeomans) writes:
>In article <2828@eagle.ukc.ac.uk>, mrw@ukc.ac.uk (M.R.Watkins) writes:
>> 
>> Given a manifold M with a smooth (C-infinity) structure, it is possible to
>> define E(M), the linear space of all smooth functions on M.  Now is it
>> possible to reconstruct the smooth structure (that is the atlas on M) from
>> the knowledge of E(M) alone?
[...]
>But for a manifold with a C-infinity structure, what you ask might be
>possible. However, I think you need more structure on X, say that of a ring or
>algebra.

For the sake of saving network bandwidth, I am not going to quote John
further.  Interested readers should look up the referenced article...

Assuming that you work with real functions, the natural ordering on
E(M) is easily recovered from the algebraic structure: The positive
elements are simply the squares.  Then the bounded functions in E(M)
are characterized by -Me<=f<=Me for some constant M, where e is the
unit of E(M) [the constant function e(x)=1].  Now define the norm of f
to be the smallest such M, and you have a normed space.  The
completion of this is B(M), the space of bounded continuous functions
on M.  However the spectrum of B(M) [the space of maximal ideals, or
equivalently, the space of scalar homomorphisms] is the Stone-Cech
(sp?)  compactification of M, a rather large and strange beast (when M
is not compact, that is).  It is not completely clear to me how to
separate the wheat (the points in M) from the chaff (the rest of the
spectrum).

If M is compact that is of course no problem, and we have now
reconstructed M as a topological space.  Since the vector fields are
the derivations of E(M) it should be possible to work out the
differentiable structure as well.

But for noncompact M, consider this: The space of functions in E(M)
with compact support is an ideal in E(M), thus is contained in a
maximal ideal I, but there is no point in M on which all members of I
vanish.  Gee, I won't even swear that the field E(M)/I has to be just
the reals, but it would be nice wouldn't it.  This little problem
might well stop you cold.  Incidentally, this is also why I am not
convinced that John's idea for recovering the conjugation in case the
starting point was the complex functions will work (but then, I am not
convinced of the opposite either (I am just hard to convince)).

- Harald
From: gore@pepper.rutgers.edu (Bittu)
Newsgroups: sci.math
Subject: Combinatorial? problem
Keywords: binomial coefficients
Date: 25 Jan 93 18:57:37 GMT
Organization: Recreation Center, Busch Campus


A few weeks ago I posted the following problem:

             (2m)! (2n)!
Prove that  ------------- is an integer where m,n are integers >=0
             m! n! (m+n)! 

I wanted a combinatorial proof. I still haven't found one (in fact I
believe this is *open*) but I found a very simple proof by induction.

Let F(m,n) denote the above quantity. It is clear that F(m,0) is an
integer for all m >= 0. Now we only need to observe that

F(m,n) = 4F(m,n-1) - F(m+1,n-1) to complete the proof.

--Bittu
From: eliezer@physics.llnl.gov (David A Eliezer)
Newsgroups: sci.math
Subject: kernel of inverse wavelet transform
Keywords: wavelets
Date: 22 Jan 93 02:55:06 GMT
Reply-To: eliezer@physics.llnl.gov (David A Eliezer)
Organization: Lawrence Livermore National Laboratory

Does anyone know anything about the kernel of the Inverse Wavelet
Transform?  Discrete or Continuous (or best of all, semi-continuous --
translation parameter discrete and dilation parameter continuous), 
any literature reference would be appreciated.  Even a large
subspace of the kernel would do me a power of good.  thanks

				Dave Eliezer
				eliezer@physics.llnl.gov



Newsgroups: sci.math
From: an1060@anon.penet.fi
Subject: Lotteries???
Organization: Anonymous contact service
X-Anonymously-To: sci.math
Date: Thu, 28 Jan 1993 18:12:45 GMT

X-Anon-To: sci.math

Hello,

I am curious to know some "strategies" for increasing one's expected
winnings in a state lottery.

Just for converstion sake let me explain the New York State Lotto 54.
One chooses any 6 numbers from 1 through 54 (inclusive).  The State
randomly selects there 6 (plus a supplemental number, but we need not
discuss this) by placing ping pong balls, numbered 1 - 54, in a hopper,
then the balls are "spit" out one at a time...etc...  The prizes are 
determined by how many numbers a player matches.  All 6 win 1'st, any 5
of 6 win 2nd, 4 of 6 for 3rd, and 3 of 6 plus the supplemental win 4th.

Anyhow, basic probability theory suggest to me that any one of the
C(54,6), i.e. 25827165, different combinations is equally likely to
win 1st prize.  However, I have read/heard of people who have some
kind of system that yields a greater pay-off than just random selection.
In fact, I recall reading about a math professor, from Florida I believe,
who as a demonstration to his class won many times (perhaps not the 1st 
prize, but 2nd or 3rd repeatedly)!!!  Does anyone know about this??
Who is this person??  What is his strategy??   Or any strategy, supported
by probability/statistics theory, that yields better than average payoffs??

I would like to present such a mathematical treatment to my class.
Thus far, I only discuss the combinatorics of the system, and keep
telling my classes that all selections are equally likely...but feel
I am not telling them the whole story!!

Many thanks in advance!!!

Abe Mantell

mantell@ams.sunysb.edu


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Newsgroups: sci.math
From: guckes@math.fu-berlin.de (Sven Guckes)
Subject: Re: Travelling Salesman Problem
Originator: guckes@medusa
Organization: Free University of Berlin, Germany
Date: Mon, 25 Jan 1993 08:40:50 GMT

nharvey@probitas.cs.utas.edu.au (Nicole Harvey) writes:

>Is there an algorithm used in maths to solve this problem?

Yes:  Calculate all possibilities - then choose the best result.

-- 
Sven@answers.answers.answers | You asked for ANY solution, right ? ;-)
Newsgroups: sci.math
Subject: Query about Egyptian fractions.
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: Sun, 24 Jan 1993 03:38:21 GMT
Organization: Department of Mathematics, University of Canterbury

This matter was raised a month or so ago, but I don't recall seeing a reply.
Apologies if I missed one.

There are some conjectures about Egyptian fractions, by (I think) Erdos, or
perhaps Seirpinski.  They relate to whether there is a global upper bound on
the number of reciprocals of positive integers required to sum to an
arbitrary positive rational.

Can anyone please give 

(i)  the author(s) and statement(s) of the conjecture(s).
(ii) their current status.

Thanks,   Bill Taylor.    wft@math.canterbury.ac.nz
Newsgroups: sci.math
From: victor@watson.ibm.com (Victor Miller)
Subject: Re: A strange recurrence and RESULT ABOUT PI
Date: Tue, 26 Jan 1993 19:26:15 GMT
Reply-To: victor@watson.ibm.com
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: IBM, T.J. Watson Research Center

>>>>> On 26 Jan 93 00:12:56 GMT, plouffe@greco-prog.fr (Simon Plouffe [melancon]) said:


Simon>    Let a(1)=1/2 and iterate the following recurrence :

Simon>                                              a(n)
Simon>                              a(n + 1) = 2 ---------
Simon>                                                   2
Simon>                                           1 - a(n)
Simon> This gives the numbers

Simon>                                       336  354144
Simon>                      1/2, 4/3, -24/7, ---, ------, ....
Simon>                                       527  164833

Simon> Now define this sign function to be,

Simon> sign(x) = 0 if x <= 0
Simon>           1 if x >  0

Simon>    If you take the sum 

Simon>                              infinity
Simon>                               -----
Simon>                                \      sign(a(i))
Simon>                        v =      )     ----------
Simon>                                /           i
Simon>                               -----       2
Simon>                               i = 1

Simon> This real number appears to be

Simon>                                   arctan(1/2)
Simon>                         v =   1 - -----------
Simon>                                        Pi


Simon> The questions are :

Simon> 0)  Is this known ?
Simon> 1)  Can we explain this fact ?

This isn't that hard to explain: Suppose \theta is real, then tan
\theta is >=0 if and only if {2 \theta/\pi} is >=0 and < 1/2, where
{x} denotes the fractional part of x.  This condition occurs exactly
when the expansion of the fractional part of 2\theta/\pi in binary
starts .0....  Thus, you see, that sign(a(i)) is 1-d(i+1) where d(i)
is 0 or 1 and 2 \theta/\pi = \sum_i d(i) 2^{-i} (assuming that a(i) is
not zero). Here \theta = arctan(1/2).
is irrational (this makes matters slightly easier).
Simon> 2)  Are the distribution of binary digits of v random ?
By Weyl's criteria, the binary digits of should be random (uniform).

Simon> 3)  Can we find another formula that will give the decimal digits of 1/Pi ?
Simon> 4)  Can formula be used to calculate v efficiently ?
Simon> Note , for an angle A we have : 


Simon>                                  tan(A)
Simon>                             2 ----------- = tan(2 A).
Simon>                                         2
Simon>                               1 - tan(A)



Simon> Any comment, references are welcome.


Simon>   Simon Plouffe.

Simon> e-mail: plouffe@lacim.uqam.ca
Simon> presently in Bordeaux at plouffe@geocub.greco-prog.fr

Simon> Ps : the real number was found experimentally.
--
			Victor S. Miller
			Vnet and Bitnet:  VICTOR at WATSON
			Internet: victor@watson.ibm.com
			IBM, TJ Watson Research Center
From: mccolm@darwin.math.usf.edu. (Gregory McColm)
Newsgroups: sci.math
Subject: Re: Let's add some sci.math.* groups
Date: 19 Jan 93 19:29:36 GMT
Organization: Univ. of South Florida, Math Department

In article 190124U53644@uicvm.uic.edu, <U53644@uicvm.uic.edu> ( Joseph Dunphy)
writes:
>Simple enough...........
>
>        Continue maving sci.math for whatever postings don't fit in one of the
>math subgroups (I'd suggest
> ...

In view of *some* of the recent posts on success in mathematics, the 
irrationality of pi, jobs in mathematics, and especially the complete 
uselessness of algebra, I suggest that we also create a newsgroup 
sci.math.flame.

-----Greg McColm
--
SHAMELESS AD:  Math PhDs in Analysis, Discrete Math./Th. of Comp., Num. Anal., 
Prob., are invited to apply to the Math. Dept. of Univ. of S. Florida for 
research/teaching positions that may be available: contact K. L. Pothoven at 
mathdept@math.usf.edu. Deadline 2/1/93. USF is a politically correct employer.
From: palais@binah.cc.brandeis.edu
Newsgroups: sci.math
Subject: Re: Frobenius Thm. on real dision algebras
Date: 27 Jan 93 14:16:48 GMT
Reply-To: palais@binah.cc.brandeis.edu
Organization: Brandeis University

>  As far as I know there is a theorem due to Frobenius, which
>states that the only associative divison algebras over real numbers
>are complex numbers and quaternions (if you drop the requirement
>of associativity octonions are also added to this list.)
>
> Here are my questions...
>1- Where can I find an explicit proof of this and related theorems?
>(In Porteus' (Topological Geometry) , there is some info about the subject
>but if there other sources, I'd like to know them.)

If you'll settle for the associative case, there is a short and elementary 
proof in:

  "The classification of real division algebras", Amer. Math. Monthly  v.75
  (1966) pp 366--368
(by one R. Palais).
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Newsgroups: sci.math
Subject: Re: f(f(x)) = exp(x)
Keywords: f(f(x)) = exp(x)
Date: 22 Jan 93 19:50:44 GMT
Organization: Beckman Instruments, Inc.

In <1jmrrgINN566@edam.csv.warwick.ac.uk> mapaj@csv.warwick.ac.uk (Mr A Scott) writes:


>	Is anything known about the existence of solutions f ( f:reals ---> reals ) of
>the equation;    f ( f ( x ) )  =  exp ( x )  for all real x   ?  Examples of such f or 
>proofs of nonexistence of f would be appreciated.  This problem is just for "fun"!  
>I've tried induction on x, but that doesn't work.  :-)  :-)  :-)  :-)

I posted the same question last year (attributing it to the late Richard
Feynmann).  Concensus seemed to be that it is clear there are C^{\infinity}
solutions, but that a real-analytic solution does exist.  If noone else
gives detailed information, I can send you want I've retained.
--
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.
My interaction with our news system is unstable; please mail anything important.
Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: Distribution of primes mod 4
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 21 Jan 1993 14:18:00 GMT

In article <ARA.93Jan21081239@camelot.ai.mit.edu> ara@zurich.ai.mit.edu (Allan Adler) writes:
>
>How would one formulate and prove analogous theorems for primes in
>arithmetic progressions other than 4n+/-1, 6n+/-1? One could take
 
stuff deleted....

>In view of all this potential generality, is 4n+/-1, 6n+/-1 really
>the state of the art?
 
Knapowski and Turan have settled special cases, but the question
of sign changes for pi(x,a1,b) - pi(x,a2,b)   a1 != a2 mod b 
for general a1,a2, and b, is still wide open.

N.B. pi(x,a,b) == #{n <= x; n is prime and congruent to a mod b}
 
It can be shown that pi(x,1,4) - pi(x,3,4) 

(a) changes sign infinitely often
(b) in an appropriate sense is positive as often as it is negative.

The way to interpret (b) is as follows:

let u(x) = #{n <= x;  pi(n,1,4) < pi(n,3,4)}

Then one would expect that u(x) = x/2 for almost all x. That is to say,
for large x, about 1/2 the integers less than x have pi(n,1,4) < pi(n,3,4)
and for about 1/2 the integers the inequality is reversed. This can be
made more precise;

u(x) = x/2 + O(x^{1-epsilon})  for any fixed epsilon.
--
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"
Xref: sparky sci.math:18778 sci.physics:23503 sci.bio:4985
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math,sci.physics,sci.bio
Subject: Re: Ian Stewart's "Mathematical Recreations" column
Date: 26 Jan 93 15:07:41 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Followup-To: sci.bio
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: mccolm@darwin.math.usf.edu. (Gregory McColm)

Note followups are redirected to sci.bio.

In article <1993Jan21.232017.2305@ariel.ec.usf.edu>, mccolm@darwin (Gregory McColm) writes:
>Not any more.  In fact, not for a while---otherwise, St. George's
>planet would be orbiting the sun.  Both astronomy and biology have
>committees of experts who approve the names.  This becomes apparent
>when some eccentric (like the late Dr. Leakey) gets in a fight with
>them.  But we needn't bow to them: when the committees decided that
>the boa constrictor should be called "constrictor constrictor", Will
>Cuppy, a reviewer for The New Yorker, decided that it should not,
>saying "two can play that game."

Just to clarify, it isn't a case of some nomenclature committee deciding
it's time to rename _Boa constrictor_ to something less euphonius, just
a realization that the agreed upon rules give some other name priority.
The committees are empowered to make exceptions, which they did with BC.
Why they haven't done this with Brontosaurus or Eohippus, I haven't the
foggiest.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
From: victor@watson.ibm.com (Victor Miller)
Subject: Re: Ancient Number Systems (Was: why is pi irrational)
Date: Fri, 22 Jan 1993 15:18:26 GMT
Reply-To: victor@watson.ibm.com
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: IBM, T.J. Watson Research Center

>>>>> On 22 Jan 93 00:28:27 GMT, fc03@ns1.cc.lehigh.edu (Frederick W. Chapman) said:

Frederick> In article <VICTOR.93Jan21101856@terse.watson.ibm.com>,

...

Frederick> This is very impressive!  *proudly bestows the sci.math Biblical
Frederick> Scholarship Award upon Victor Miller*  :-)

Thank you (blush)!

Frederick> I've heard that letters of the Hebrew alphabet all have
Frederick> numerical equivalents, which were used in ancient
Frederick> scriptures as the basis of a number system.  I would be
Frederick> interested in knowing more about this, since it must surely
Frederick> qualify as a very early example of a number system.
Frederick> Doesn't some ambiguity occasionally arise as to whether a
Frederick> given string of characters should be interpreted as a word
Frederick> or as a numerical quantity?

I'm not sure how old it is, but it doesn't stretch the imagination to
believe it's quite old.  Basically, the numerical equivalents come
from looking at the 22 letters of the Hebrew Alphabet in order: the
first 9 get assigned 1 through 9, the next 9, get assigned 10 through
90 (by ten), and the last 4 get assigned 100,200,300,400.  Religious
scholars revel in the ambiguity.  The numerical equivalents are called
Gematria and are part of the "Hidden knowledge".  You'll find many
examples of adding up the values of the letter in one name (or phrase)
and finding that they add up to the same as another name or phrase,
and "thus" showing equivalence.

Frederick> Didn't classical Greek also assign numerical values to the
Frederick> letters of the alphabet and employ a similar number system?
Frederick> (For example, there is a New Testament passage that speaks
Frederick> of "the *number* of his name" in reference to the
Frederick> anti-Christ.)

Frederick> Thanks,

Frederick> Fred Chapman
Frederick> Student of Ancient Number Systems in Religious Writings
Frederick> :-)>>>
Frederick> -- 

Frederick> o ------------------------------------------------------------------------- o
Frederick> |  Frederick W. Chapman, User Services, Computing Center, Lehigh University |
Frederick> |    Campus Phone:  8-3218     Preferred E-mail Address:  fc03@Lehigh.Edu   | 
Frederick> o ------------------------------------------------------------------------- o
Frederick> |             Ecstasy is transitory, but a theorem is forever!              |
Frederick> o ------------------------------------------------------------------------- o
--
			Victor S. Miller
			Vnet and Bitnet:  VICTOR at WATSON
			Internet: victor@watson.ibm.com
			IBM, TJ Watson Research Center
From: plouffe@greco-prog.fr (Simon Plouffe [melancon])
Newsgroups: sci.math
Subject: A strange recurrence and RESULT ABOUT PI
Date: 26 Jan 93 00:12:56 GMT
Organization: GRECO Programmation du CNRS - Bordeaux,France


   Let a(1)=1/2 and iterate the following recurrence :

                                             a(n)
                             a(n + 1) = 2 ---------
                                                  2
                                          1 - a(n)
This gives the numbers

                                      336  354144
                     1/2, 4/3, -24/7, ---, ------, ....
                                      527  164833

Now define this sign function to be,

sign(x) = 0 if x <= 0
          1 if x >  0

   If you take the sum 

                             infinity
                              -----
                               \      sign(a(i))
                       v =      )     ----------
                               /           i
                              -----       2
                              i = 1

This real number appears to be

                                  arctan(1/2)
                        v =   1 - -----------
                                       Pi

The questions are :

0)  Is this known ?
1)  Can we explain this fact ?
2)  Are the distribution of binary digits of v random ?
3)  Can we find another formula that will give the decimal digits of 1/Pi ?
4)  Can formula be used to calculate v efficiently ?
Note , for an angle A we have : 


                                 tan(A)
                            2 ----------- = tan(2 A).
                                        2
                              1 - tan(A)



Any comment, references are welcome.


  Simon Plouffe.

e-mail: plouffe@lacim.uqam.ca
presently in Bordeaux at plouffe@geocub.greco-prog.fr

Ps : the real number was found experimentally.
Newsgroups: sci.math,sci.logic,comp.text.tex
From: knappen@obelix.kph.uni-mainz.de (Joerg Knappen)
Subject: Re: inverted iota
Organization: Institute for Nuclear Physics, Uni Mainz, FRG
Date: Sun, 24 Jan 1993 01:11:54 GMT

For the oldstyle integral sign, you can use the euler font euex** from the ams-LaTeX distribution. In the
ams-LaTeX distribution (usually under a directory like amsfonts) are also additional symbol font, which
include more symbols.

Yours, J%org Knappen.
From: jeffv@physics.ubc.ca (Jeff Vavasour)
Newsgroups: sci.math
Subject: Series expansion of a functional
Date: 22 Jan 1993 22:46:48 GMT
Organization: The University of British Columbia
Distribution: world
NNTP-Posting-Host: physics.ubc.ca
Keywords: Help?

I am interested in doing a perturbation theory-type analysis of a functional.
It is a scalar functional F of five periodic fields.  I wish to expand
the functional about the fields which minimise F, subject to some constraints.

What I am looking for is a Taylor-like expansion of F in terms of the 
functional derivatives of the fields.  What would such a series look like?
I have a qualitative idea, but I'd like something a little more rigorous.
Is there a good reference on this?

Please reply by e-mail if you can help me.  Thanks.

- Jeff