HDTenEightyP/NetNews huggingface dataset:
From: israel@unixg.ubc.ca (Robert B. Israel)
Subject: Re: Logarithms - how did they start/
Organization: University of British Columbia, Vancouver, B.C., Canada
Date: Thu, 1 Oct 1992 16:48:31 GMT

In <BvG62w.IJs@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:

>In article <1992Oct1.011035.59324@cc.usu.edu> sl870@uranus.ee.usu.edu (Ullas Gargi) writes:
>>A sitter from a netter:
>>
>>How did the first guys (Napier et al) calculate the logs of numbers?
>>If I wanted to sit down and manually calculate the log of a non-trivial no.
>>(like 10), how would I do it? A Taylor expansion formula?

>There is nothing obscure about this.  Napier originally used a base
>1 + 10^(-7) and used multiplication to get the antilogarithms.  Later,
>when common logarithms came into use, one used square roots to get 
>logarithms, and then interpolation.  AC (after Calculus), Taylor 
>expansions were used to get natural logarithms, and common logarithms
>were obtained by division.

>There are many ways to compute logarithms, and to compute the natural
>logarithm of 10, one way to do this would be as 3*ln(2) + ln(1.25).
>Another way would be to use (10*ln(2) - ln(1.024))/3.

Of course, you still need ln(2).  You could get this by the Taylor
series for -ln(1/2).  One that converges much faster:

ln(2) = 3 ln(1.08) - 9 ln(0.96) + 4 ln(1.024)

from which we get

ln(10) = 10 ln(1.08) - 30 ln(0.96) + 13 ln(1.024)

-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4
Newsgroups: sci.math
From: cannon@scs.unt.edu (Dustin Cannon (Lab 3460))
Subject: HELP! I got a contradiction!
Organization: School of Community Service, UNT
Date: Sat, 26 Sep 1992 16:23:18 GMT


First off, bear with me.  I'm only a senior math major.
OK, heres the deal.

	e^ix  = cos(x) + isin(x)
	e^-ix = cos(x) - isin(x)

	Let x = 

	e^i  = -1
	e^-i = -1

	e^i = e^-i
	i = -i
	i = -i
	ii = -ii
	-1 = 1

	I had one of my profs tell me the problem is in taking the log of
e^i, but I didn't quite follow.  Can anyone give me a better explanation.
Thanks in advance.
_________________________________________________________________
Dustin W. Cannon      "Just because some of us can read and write
University of          and do a little math, that doesn't mean we
North Texas            deserve to conquer the universe."
                                                -Kurt Vonnegut
Newsgroups: sci.math,rec.puzzles,comp.theory
From: torkel@sics.se (Torkel Franzen)
Subject: Re: more on .999... = 10.0
Organization: Swedish Institute of Computer Science, Kista
	<1a9qi2INN898@uwm.edu>
	<1992Sep30.172329.19466@CSD-NewsHost.Stanford.EDU>
Distribution: inet
Date: Wed, 30 Sep 1992 19:14:31 GMT

In article <1992Sep30.172329.19466@CSD-NewsHost.Stanford.EDU> casley@Tarski.
Stanford.EDU (Ross Casley) writes:

   >>and EQUIVALENT to the real number system...
   >No, it isn't.  The real number axioms (I'm using the ones that define a 
   >complete ordered field) imply that the integers are not bounded above.
   >If they were then they would have a least upper bound x.  But then it is 
   >easy to show that x-1 is also an upper bound on the integers, but less
   >than x.  Contradiction.

  Just to forestall confusion: let's keep straight what theories we
are talking about. "The real number system" is not a theory. On the
one hand, the second order axioms describing a complete ordered field
are categorical, i.e. any model is isomorphic to the real field. Or in
other words, there is only one complete ordered field (up to
isomorphism). On the other hand, whatever first order axioms we put
down, if the real field is a model of those axioms, then they admit
nonstandard models, i.e. models containing infinitesimals (or equivalently,
containing infinite numbers).

  Nonstandard analysis is based on the existence of nonstandard models
of a certain uncountable first-order theory T that is defined in terms
of the real field.
From: maciejw@math.arizona.edu (M P Wojtkowski)
Newsgroups: sci.math
Subject: Re: Poncelet's Theorem, Elliptic Curves & Dynamical Systems
Date: 26 Sep 92 23:18:21 GMT
Organization: Mathematics Department, University of Arizona
From: casley@Tarski.Stanford.EDU (Ross Casley)
Subject: Re: more on .999... = 10.0
Organization: /usr/lib/news/organization
Distribution: inet
Date: Wed, 30 Sep 1992 17:23:29 GMT

In article <1a9qi2INN898@uwm.edu> markh@csd4.csd.uwm.edu (Mark) writes:
>...
>Add a constant object (omega) to the formal system, and add an infinite set
>of axioms (omega > 0, omega > 1, omega > 2, ...).  This system is logically
>consistent 

OK.

>and EQUIVALENT to the real number system...

No, it isn't.  The real number axioms (I'm using the ones that define a 
complete ordered field) imply that the integers are not bounded above.
If they were then they would have a least upper bound x.  But then it is 
easy to show that x-1 is also an upper bound on the integers, but less
than x.  Contradiction.
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: consequences of the Axiom of Choice
Date: 6 Oct 92 14:21:24 GMT
Article-I.D.: netnews.91826
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: holmes@opal.idbsu.edu (Randall Holmes)

In article <1992Oct5.192243.17318@guinness.idbsu.edu>, holmes@opal (Randall Holmes) writes:
>Solovay showed that there is a model of ZF + DC (dependent choices) in
>which all sets of reals are Lebesgue measurable.  This requires
>existence of an inaccessible (and I believe that this consistency
>strength is unavoidable).  DC is enough for classical measure theory.

I agree with this, although one should be careful about what "classical
measure theory" covers.  In the sense used here, it obviously does not
include pathological sets.  General L^p space theory sometimes invokes
the Hahn-Banach theorem.  And general infinite products certainly need
the Prime Ideal theorem, as does compactness of the infinite [0,1] cube.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
Newsgroups: sci.math
From: minch@lotka.stanford.edu (Eric Minch)
Subject: Re: References Wanted: Emergent Behavior/"Swarming Theory"
Organization: Stanford University, California, USA
Date: Thu, 1 Oct 1992 23:54:21 GMT

In article <1992Oct1.145831.22862@news.cs.indiana.edu>  
mchui@spinner.cs.indiana.edu (Michael Chui) writes:
> 
> 	I'm looking for references to theories/models of emergent  
behaviors
> of large numbers of organisms, e.g. the phenomenon of an entire  
school
> of fish, or flock of birds, appearing to turn all at once, etc.

UCLA maintains an archive of artificial life materials in which some  
examples of this may be found. You can anonymous ftp to cognet.ucla.edu  
and look in the directories /pub/alife/papers, /pub/alife/software, and  
/pub/alife/archive.

The comp.theory.cell-automata newsgroup also touches on this territory  
occasionally.

Disclaimer: Use of the term "emergence" appears to have a polarizing  
effect. At one pole are those who believe any unpredicted result  
qualifies as emergence ("Gee, *I* didn't know the 145,932nd digit of pi  
was ...."), at the other pole are those who deny that emergence has any  
meaning in formal systems or even in computer programs. You may want to  
keep your asbestos suit handy.
--
 
Eric Minch             Any resemblance to the opinions of persons
Epistemic Artisan      or organizations other than myself--living,
Stanford Genetics      dead, or imaginary--is purely fortuitous.
Newsgroups: sci.math,rec.puzzles
From: ferenbaugh-charles@CS.YALE.EDU (Charles Ferenbaugh)
Subject: Re: Conway's Recursive Sequence
Organization: Yale University Math Dept.
Date: Fri, 2 Oct 1992 19:00:54 GMT


In article <1992Sep30.220851.15985@husc3.harvard.edu> kubo@zariski.harvard.edu (Tal Kubo) writes:

>I am looking for information about a recursive sequence
>defined by John Conway in 1988:

>   a(n)= a(a(n-1)) + a(n-a(n-1))        for n>2
   
>   a(1) = 1, a(2)=1

>         .... Pointers to relevant literature, or
>any information whatsoever, would be most welcome.


I spent several hours playing with this sequence yesterday, and came up
with the following pretty way of presenting it:

Let b(n) = a(n) - a(n-1).  Then b(n) = 0 or 1 always (easy result).
Write + for 1.  Then b(n) obeys the following pattern:

  n=3,4                         +    0

  n=5..8                     +    +0    0

  n=9..16               +    ++0      +00    0

  n=17..32          +   +++0    ++0+00    +000    0

                     4     3  2   2   2   2  3     4
  n=33..64      +   + 0   + 0+ 0+0   + 0+0 +0    +0    0

                   5                                 5
  n=65..128  +    + 0          ...etc...           +0      0

where each string is the concatenation of the two above it,
a la Pascal's triangle.

I checked with Conway by email, and he suspects that this is
equivalent to one of the combinatorial presentations already known.
But at any rate, it's prettier than the presentation in the Mallows
article.

- Charles Ferenbaugh
  crf@math.yale.edu

Newsgroups: sci.math
From: hiraga@Csli.Stanford.EDU (Yuzuru Hiraga)
Subject: Z <-> Q formula
Organization: Stanford University CSLI
Date: Sun, 27 Sep 1992 08:59:52 GMT


# This may be a FAQ, but...

The set of intergers Z and the set of rationals Q have the same cardinality,
i.e. there exist a bijection between them.  So we all know that.
But then, is there a "simple" formula that expresses any such mapping?
You may use any device you please, but note that the subjective little
"simple" is the criterion.

-Yuzuru Hiraga
From: nakada@ntttsd.ntt.jp (Hiroshi Nakada)
Newsgroups: sci.math
Subject: Re: a graph problem
Date: 7 Oct 92 01:11:58 GMT
Organization: NTT Transmission Systems Laboratories, Yokosuka, JAPAN

In article <1992Oct6.153924.23274@penny.cs.fredonia.edu> kwong@mary.cs.fredonia.edu (Harris Kwong) writes:
 >nakada@ntttsd.ntt.jp (Hiroshi Nakada) writes:
 >: Let n(d,D) be a number of nodes of a graph with a given maximum degree
 >: d and diameter D (D is the maximum distance between any two nodes).
 >: A theoretical upper bound on n(d,D) is given by Moore (see [1]) as
 >: 
 >:                  n(d,D) <= (d(d-1)^D -2)/(d-2)
 >: 
 >: for d>2. This bound is called the Moore bound.  The graphs 
 >: satisfying the equality are called the Moore graphs.
 >: Unfortunately, the Moore graphs can exist only if D=2 and 
                                    ~~~~~~~~~~~~~~~~~
 >: d=3, 7 or 57.  

 >I believe the case of D=2, d=57 is still open.

Yes. Bollobas said that if there exists a Moore graph of d>2 and
D=2 then d=3, 7 or 57. Moore graphs of D=2 and d=3 and 7 do exist
(D=2, d=3 and n=10 is known as Petersen graph) but it is not known
whether there is a Moore graph of degree 57 and diameter 2 (see [1]).

  HIROSHI NAKADA                    nakada@ntttsd.NTT.JP
  NTT Transmission Systems Labs.    Phone +81-468-59-8537
Newsgroups: sci.math,sci.physics
From: bhv@areaplg2.corp.mot.com (Bronis Vidugiris)
Subject: Re: Another Cute Math Joke  (explanation of)
Organization: Motorola, CCR&D, CORP, Schaumburg, IL
Date: Fri, 2 Oct 1992 17:27:23 GMT

In article <1992Oct2.123755.22685@ulrik.uio.no> jarleb@athena.uio.no (Jarle Brinchmann) writes:
)|>
)|>Hmm.  Well, the reason it's funny is that the joke implies you are
)|>taking a CROSS product of an elephant and a grape, but the math shows
)|>a DOT product of an elephant and a grape!
)|>
)|>(if A and B are vectors, |A| * |B| * sin(theta) is one expression for the
)|>dot product || -> absolute value, theta is the angle between the vectors).
)|>
)|>That's pretty funny :-) :-) :-)
)|>
)|>
)
)Yes, espescially since you're wrong :-)

Ooops!

Got some E-mail on this too.  Oh well. :-( :-( :-(

Newsgroups: sci.math
From: wjh@cbnewsl.cb.att.com (william.j.hery)
Subject: 1, 2, 4, 8, 13...where have I seen it before?
Organization: AT&T Bell Laboratories
Distribution: na
Date: Wed, 30 Sep 1992 03:20:15 GMT

In computing something in combinatorics, a sequence that started
with 1, 2, 4, 8, 13 (not the expected 16) came up, and I'm sure I've
seen it before, possibly in some kind of trick problem that is set
up to make you think the next term is 16. Does anyone recognize something
simple that starts like this? BTW, my sequence continues with 21, 31, 45(?)

Another (simpler) source might give some insight into the problem I'm looking
at.

Bill Hery
hery@att.COM
Newsgroups: sci.math
From: tbeke@phoenix.Princeton.EDU (Tibor Beke)
Subject: looking for an article
Originator: news@nimaster
Organization: Princeton University
Date: Tue, 6 Oct 1992 13:58:54 GMT


There exists a proof of the Jordan curve theorem using non-standard
analysis (replace the curve by a polygon of infinitesimal sides, do
the easy proof of the polygonal case and show that two standard points
of the plane are separated by the curve iff they are separated by the
non-standard polygon). Can anyone tell me where this appeared? I was
only told "very long time ago (between 20 and 30 years?) in the American
Mathematical Monthly, but it might be another journal" :-/.
Please respond directly to tbeke@phoenix.princeton.edu.
Thanks, Tibor
Newsgroups: sci.math
From: dak@messua.informatik.rwth-aachen.de (David Kastrup)
Subject: Re: Closed form solution for Sum[ 1/x , {x,1,n} ] ?
Organization: Rechnerbetrieb Informatik  /  RWTH Aachen
Date:  7 Oct 92 01:21:54 GMT

sklar@picasso.ocis.temple.edu (Dave Sklar) writes:


>	This is kind of cheesy, but Sum(1/x,{x,1,n}) does equal ln n -
>Euler's Gamma Constant, which is approx. 0.577+

It's not only slightly cheesy, but also slightly wrong. When you write
ln n PLUS gamma, you get an APPROXIMATION with an error O(1/n). See my
previous post for a much better approximation including the formula
giving the form of the factors of n^{-k}.
Newsgroups: sci.math
From: myrberger@elin.e.kth.se (Johan Myrberger)
Subject: mechanical puzzle
Reply-To: myrberger@elin.e.kth.se (Johan Myrberger)
Organization: School of EE, Royal Institute of Technolgy
Date: Mon, 28 Sep 1992 20:50:03 GMT

Have anybody seen this before?
 
           *********
            *   *   *
            *   *   *
            *   *   *
            *********
            *  /*\  *
            * /Y*Y\ *
            */YY*YY\*
            *********
            *\YY*YY/*
            * \Y*Y/ *
            *  \*/  *
*********************************
*   *  /*\  *  /*\  *  /*\  *   *
*   * /R*R\ * /B*B\ * /G*G\ *   *
*   */RR*RR\*/BB*BB\*/GG*GG\*   *
*********************************
*   *\RR*RR/*\BB*BB/*\GG*GG/*   *
*   * \R*R/ * \B*B/ * \G*G/ *   *
*   *  \*/  *  \*/  *  \*/  *   *
*********************************
            *  /*\  *
            * /P*P\ *
            */PP*PP\*
            *********
            *\PP*PP/*
            * \P*P/ *
            *  \*/  *
            *********
            *   *   *
            *   *   *
            *   *   *
            *********

This is a mechanical puzzle I once bought in Germany without any instructions.

Above is a picture of it as I think it'll look solved, but as I bought it in a
scrambled position it's only a guess (but it seems reasonable to me).

It consists of 20 squares, each of them half coloured:

e.g. *****    Y=yellow                *****
     *\  *    R=red                   *   *
     *B\ *    B=blue                  *   *   = empty place
     *BB\*    G=green                 *   *
     *****    P=pink                  *****

They are placed in a cross which allow you to move either all the squares in
the left-right arm simultaneously left or right, or all the squares in the 
up-down arm simultaneously up or down.

If anyone could give me a hint on how to solve it or any references I would
appreciate it.

Thanks

Johan Myrberger

E-mail: myrberger@elin.e.kth.se
From: chinnici@ghost.dsi.unimi.it (roberto riccard chinnici)
Newsgroups: sci.math
Subject: A question in Recursion Theory
Summary: Sections of recursive functions do always exist?
Keywords: recursion
Date: 29 Sep 92 16:56:54 GMT
Organization: Computer Science Dep. - Milan University

A question for Recursion Theorists.

If you take a partial recursive function phi and two sets I and J such that
 dom(phi) contains I and range(phi) is included in J, you can define an
 obvious set-theoretic function f:I->J.

The question is: does EVERY surjective f admit a section, i.e. a g:J->I
 obtained in the same way from a partial recursive function, say psi, such
 that
        for each j in J    f(g(j))=j       ?

People-who-know will see I'm actually defining a category and asking "does
 every good (i.e surjective) epi split?"

I suspect this to be false, but I can't manage to find a counter-example; the
 difficulties arise from the clumsiness of restricting arbitrarily the domain
 of a partial recursive function.

Thanks in advance, and feel free to e-mail me!!

----------------------------------------------------------------------
	Roberto R. Chinnici	<chinnici@hp2.sm.dsi.unimi.it>

	Dipartimento di Scienze dell'Informazione
	Universita' degli studi di Milano

	"but now
	 she stands upon southampton dock
	 with her handkerchief
	 and her summer frock clings
	 to her wet body in the rain
	 in quiet desperation knuckles
	 white upon the slippery reins
	 she bravely waves the boys goodbye again"

				southampton dock, Pink Floyd
----------------------------------------------------------------------
Newsgroups: sci.math
From: pmontgom@math.orst.edu (Peter Montgomery)
Subject: Re: Z <-> Q formula
Organization: Oregon State University Math Department
Date: Sun, 27 Sep 1992 18:33:09 GMT

In article <1992Sep27.085952.7534@Csli.Stanford.EDU>
hiraga@Csli.Stanford.EDU (Yuzuru Hiraga) writes:

>The set of integers Z and the set of rationals Q have the same cardinality,
>i.e. there exist a bijection between them.  So we all know that.
>But then, is there a "simple" formula that expresses any such mapping?

       I'll assume you already have a "simple" bijection g from
the nonnegative integers to the integers, with g(0) = 0.  Perhaps

               1 - (2n+1)(-1)^n
        g(n) = ----------------,
                      4

which takes successive values 0, 1, -1, 2, -2, ... for n = 0, 1, 2, ...

        To define f : Z -> Q, we first define f(n) for n > 0.
If the prime factorization of n is

              e1   e2   e3        ek
        n = p1   p2   p3    ... pk  ,

where the pi are distinct primes, then define

                 g(e1)   g(e2)        g(ek)
        f(n) = p1      p2      ...  pk     .

Also define f(0) = 0 and f(-n) = -f(n) for n > 0.
I'll let you judge whether this is "simple".
-- 
        Peter L. Montgomery              Internet: pmontgom@math.orst.edu
        Dept. of Mathematics, Oregon State Univ, Corvallis, OR 97331-4605 USA
``Who do you trust?''  Those whom I trust know English spelling and grammar.
Our economy needs more business, not more Bushiness.
Newsgroups: sci.math
From: leyfre@McRCIM.McGill.EDU (Frederic Leymarie)
Subject: Re: Comprehensive reference book suggestion/solicitation
Organization: McGill Research Centre for Intelligent Machines
Date: Sun, 4 Oct 92 16:23:31 GMT


In article <1992Oct2.180458.9439@nntpd.lkg.dec.com>, edp@math.zk3.dec.com
 (Eric Postpischil) writes:
|> Is there a reference book that contains the bulk of the results in most
|> branches of mathematics?  I am thinking of something that is basically
|> an extract from the textbooks in each field -- it would give the
|> definitions (e.g., a group is an ordered pair consisting of a set and a
|> closed binary operation such that . . .) and the theorems, without proof
|> or much explanation.  Thus, a person who had completed graduate work in
|> mathematics would have a place to go to refresh their memory of the
|> formal definition of the real numbers, the fundamental theorem of
|> algebra, et cetera.

Try:

@Book{Bronshtein85Handbook,
  author =	{I. N. Bronshtein and K. A. Semendyayev},
  title =	{Handbook of Mathematics},
  editor = 	{K. A. Hirsch},
  year =	1985,
  edition = 	{$3^{rd}$},
  publisher =	{Van Nostrand Reinhold},
  address =	{New York, NY, U.S.A.},
  note =	{English version.}}

-- 
Frederic Leymarie -- leyfre@mcrcim.mcgill.edu
McGill University, Electrical Eng. Dept., McRCIM,    |	Tel.: (514) 398-8236
3480 University St., Montreal, QC, CANADA, H3A 2A7.  |	FAX:  (514) 398-7348
From: clong@remus.rutgers.edu (Chris Long)
Newsgroups: sci.math,rec.puzzles
Subject: Re: The Last Word on Brown Eyes, Blue Eyes
Date: 27 Sep 92 13:28:49 GMT
Followup-To: sci.math
Organization: Rutgers Univ., New Brunswick, N.J.

In article <1992Sep27.015944.932@Csli.Stanford.EDU>, Yuzuru Hiraga writes:

>   A2. The comment is irrelevant, and the "immediate-suicides" should have/
>       could have taken place much earlier.
>      This comes from people quite familiar with the problem (or its
>      variants), including Chris Long.

Upon further reflection, I realized to my horror that the statement
of the problem, although different from Smullyan's own, also yields
the desired result.  If I had actually bothered to write out the
trivial induction argument I wouldn't have stuck my foot in my mouth.

> # Note: crosspost to "comp.theory" has been deleted on
> # request, thus overriding Chris' Followup-To line (I wonder why he 
> # did this).

I didn't do this, the posting software did.
-- 
Chris Long, 265 Old York Rd., Bridgewater, NJ  08807-2618