HDTenEightyP/NetNews huggingface dataset:
From: rmurphy@aludra.usc.edu (Bob Murphy)
Newsgroups: sci.math
Subject: Re: Pi in beatiful form --lost
Date: 16 Dec 1992 02:14:38 -0800
Organization: University of Southern California, Los Angeles, CA
NNTP-Posting-Host: aludra.usc.edu

In article <BzBt9q.2HM@unccsun.uncc.edu> sbardhan@uncc.edu writes:
>I  hope Mathematicians especially  Pi'ans  could help me remember this beautiful form of pi.
>I  can only faintly remember this beautiful form  invented by Ramanujam
>pi = (1*2)/(3*4) +(5*6)/(6*7)....
>
>All I remember is that it is very close to this representation ....it involved multiplication of two numbers
>and divsion by two very close numbers .. it was indeed  one of the fascinating forms.
>
>Thanks
>--bardhan
>
 
John Wallis came up with the following in the middle of the 17th century.

pi/2 = 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...


There are many other beautiful formulas involving pi.
Here are a few of them.

pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...

(pi - 3)/4 = 1/(2*3*4) + 1/(4*5*6) + 1/(6*7*8) + ...

4/pi = 1 +  1^2
           ------------------------------
            2 +  3^2
                -------------------------
                 2 + 5^2
                    --------------------
                      2 + 7^2
                         ---------------
                           2 + 9^2
                              ----------
                                2 + ...

and I guess I should include Euler's:

(pi^2)/6 = 1 + 1/(2^2) + 1/(3^2) + 1/(4^2) + 1/(5^2) + ...


              Bob Murphy (rmurphy@aludra.usc.edu)
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math
Subject: Re: Alleged shortage of mathematicians (was Re: M
Date: 19 Dec 92 21:30:11 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <1992Dec17.001527.18598@husc3.harvard.edu> kubo@boucher.harvard.edu (Tal Kubo) writes:
>In article <1413@kepler1.rentec.com>
>andrew@rentec.com (Andrew Mullhaupt) writes: 

>  Now, your own job might not be fraught with too many of
>these distractions, but if anything what I have described is a lot more
>representative than the rather peculiar conditions of your employment.

Actually, I could have said many of the things about my present in reference
to the previous job I had with a major Wall Street Firm. It wasn't quite
the same of course, but it was pretty enlightened. I know other people
in other places with roughly similar jobs.

>  From what I've heard the best work environments
>are not in business or academia, but in independent institutes such as IDA.

Actually, the place I'm at now was started by some people from IDA. I think
that they're probably somewhat 'purer' then us.

Later,
Andrew Mullhaupt
From: choy@skorpio.usask.ca (I am a terminator.)
Newsgroups: sci.math
Subject: Orbital motion question
Date: 13 Dec 1992 23:02:52 GMT
Organization: University of Saskatchewan, Saskatoon, Canada
Distribution: world
NNTP-Posting-Host: skorpio.usask.ca


Let the vector r be the position of a mass m orbiting in a plane around
another mass M (which doesn't move) at the origin. The mass m is
accelerated such that

   2
  d r     KMm
m --- = - ---- r
    2        2
  dt      |r|
                                           r
                                         d---
                                          |r|
What is the scalar product of r/|r| and ------ ?
                                          dt

If m moved in an elliptical path, can the scalar product be zero?

Henry Choy
choy@cs.usask.ca
From: dane@nermal.SantaRosa.Edu (Dane Jasper)
Newsgroups: sci.math
Subject: Finding primes/pi - how to on Dos or Unix box?
Summary: I would like to search for high primes - how?
Date: 19 Dec 92 05:19:49 GMT
Organization: Santa Rosa Junior College, Santa Rosa, CA

Is there a free/shareware program for either a Linux based unix box or a
Dos/Windows/Os/2 based intel system that will allow me to search for digits
of pi or e, or to search for prime numbers without interupting other work. 
The point is, I guess I like to make my computer sweat - I don't want it to
have any free time, and would like to fill it's free cycles with time
comsuming math.  

If anyone has any ideas, please feel free to mail me - that way I'm sure to
get it.

Dane Jasper
Computing Services, Santa Rosa JC
From: mccolm@darwin.math.usf.edu. (Gregory McColm)
Newsgroups: sci.math
Subject: Re: nonstandard analysis
Date: 9 Dec 92 20:07:22 GMT
Organization: Univ. of South Florida, Math Department

In article <ByvFuC.AIs@acsu.buffalo.edu> v104n55k@ubvmsd.cc.buffalo.edu (David I Schwartz) writes:
>After the past couple of posts, I must ask:
>what is nonstandard analysis? Is there a decent reference that deals with it? 

Nonstandard analysis is essentially a refurbished version of 
Liebniz's infinitesimal calculus.  Liebniz tried to make sense 
of the calculus by having dy and dx being infinitely small 
positive numbers.  This approach was abandoned after limits 
appeared, but was revived around 1950 by Abraham Robinson, who 
came up with a rigorous formulation.  Since there are no limits, 
some people thought it would be a great idea to teach freshman 
calculus this way (see M. Henle's INFINITESIMAL CALCULUS for 
an example).  Also, some applications to dynamics have been 
found.  There is an article on the stuff in Barwise's HANDBOOK 
OF MATHEMATICAL LOGIC.

-----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.
Newsgroups: sci.math
From: wvanweer@wsuaix.csc.wsu.edu (Wayne VanWeerthuizen)
Subject: Pi base e, and e base Pi.
Organization: Washington State University
Date: Tue, 15 Dec 92 04:15:07 GMT

Is there an efficient method for calculating pi base e, or e base pi 
to an arbitrary number of digits?

What are the first 40 digits of each?

-- 
 ---------------------------------------------------------------------------
        Wayne VanWeerthuizen    205 Scott Hall     Pullman, WA  99163
           wvanweer@wsuaix.csc.wsu.edu      (509) 335-1835 (voice)   
 ---------- Only YHVH is God.  Jesus is the Messiah, Son of YHVH. ----------
From: ph11aha@sbusol.rz.uni-sb.de (Edgar-Carlo Pitzen)
Newsgroups: sci.math
Subject: wavelet transformation
Date: 18 Dec 1992 16:23:27 GMT
Organization: Universitaet des Saarlandes,Rechenzentrum
NNTP-Posting-Host: sbusol.rz.uni-sb.de
Keywords: wavelet transformation

I am interested in review articles on the wavelet transformation.
Please mail references to my e-mail adress given below.

Stephan Mueller (Saarbruecken)

e-mail adress: ph11ahsm@sbusol.rz.uni-sb.de
Newsgroups: sci.math,comp.edu,misc.education
From: clin@eng.umd.edu (Charles Lin)
Subject: Re: BreakUp Freshman Calculus ?WAS Re: Student attitudes
Date: Sat, 12 Dec 92 07:35:26 GMT
Organization: College of Engineering, Maryversity von Uniland, College Park
Originator: clin@feedback.eng.umd.edu


In article <1992Dec11.233845.10013@magnus.acs.ohio-state.edu>, regeorge@magnus.acs.ohio-state.edu (Robert E George) writes:
[Brian Burtt suggest splitting of calculus courses]
>
>What about the engineering students who decide near the end of their sophomore
>year that they just can't abide any more lab work and want to switch to
>mathematics? What about the mathematics students who decide near the end of
>their sophomore year that they just can't abide any more abstract work and
>want to switch to engineering or physics? Can (a) and (c) be close enough
>to substitute for each other?
>
>One "advantage" of the current system is that it doesn't really force
>students to make a firm decision as to what they're studying until fairly
>well into their undergraduate career. As I understand, most European
>and Asian students know what they want to study when they enter university?
>Not so for Americans, Americans seem to like it this way.
>
>If "we" decide we want to force students to make the decision earlier in their
>career, I acknowledge that there are good arguments for doing so (and good
>arguments against -- mainly that the entire secondary curriculum isn't
>designed for people to decide that early). But it's necessary to at least
>think about this issue before separating up freshman calculus.
>
    I think this reorganization of calculus is meant to address the
following:

    (1) Students who take calculus are not as motivated when they
        do not see applications to their field of study

    (2) One should not only be taught calculus (i.e. intergrals,
        derivatives, diff eq'n), but more importantly, how to set
        up these equations for given problems in an area such as
        physics.

   If we choose to keep things the way they are, then one ought
to find ways of making students more motivated in (1), or not
to assume that the teaching of calculus, in general, will be able
to accomplish (2).   Therefore, those who teach courses in physics
or economics or whatever, should perhaps make it clear to their
majors what are the important aspects to remember out of those
courses.   Then, in the physics and economics courses, the
students ought to be taught how to apply calculus to their
problem solving.   By having this kind of tracking, as suggested
by Brian, fewer courses may have to be required, at the expense
of not being easily able to switch out.  If we keep the system the
same, then I think the *application* of calculus to problems needs
to be taught and not assumed taught in a calculus course

--
Charles Lin
clin@eng.umd.edu
Newsgroups: sci.math
Subject: Re: '-' operation
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 15 Dec 92 22:01:46 GMT
Organization: Beckman Instruments, Inc.

In <92350.145501B7D@psuvm.psu.edu> <B7D@psuvm.psu.edu> writes:

>Recently my daughter brought back Problem Set II of Wisconsin Mathematics
>Science and Engineering Talent Search, and Question 4 in it is as following:
>         Operation @ satisfies the conditions that
>                X @ (Y @ Z) = X @ Y + Z and X @ X = 0 for any real numbers
>          X, Y, Z. Show that @ must be subtraction.

A "one line proof":

X @ Y + Y = X @ (Y @ Y) = X @ 0 = X @ (X @ X) = X @ X + X = 0 + X = X, so

X @ Y = X - Y


Well, maybe two lines.
--
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.
Xref: sparky sci.math:17125 rec.games.abstract:641
From: nates@ll.mit.edu ( Nate Smith)
Newsgroups: sci.math,rec.games.abstract
Subject: Re: Game of pentominos
Date: 17 Dec 92 22:14:10 GMT
Organization: MIT Lincoln Laboratory

In article <BzF56E.7wE@csugrad.cs.vt.edu> dbush@csugrad.cs.vt.edu (David Bush) writes:
>nates@ll.mit.edu ( Nate Smith) writes:
>
>>a LONG time ago, i was given a game called "Pan-Kai".  this is a
>
>Oh, yeah? Very interesting. Back in the late sixties Parker Brothers came out
>with "Universe," a pentomino game for 2-4 players. The rules were similar. For
>2 players, the board is 10x10. In addition, there are four regions of
>25 cells each appended to the 4 sides of the main square. The entire
>board has 4-way rotational symmetry. For 3 players, use any two extra regions
>in addition to the 10x10 grid. 4 players use the whole board. On your move,you
>may not place a pentomino so as to "fence off" a region of < 5 cells connected
>horizontally and/or vertically. (Ofcourse only the active portion of the board

 i forgot that rule!  it was in the earlier Pan-Kai as well.

>is considered.) If you have no move, you're out of the game.
>The last player to move wins.
>
>Trivia: the game box contained publicity stills from "2001:A Space Odyssey."
>One shot shows an astronaut playing Hal a game of something that looks like
>Universe. Apparently that scene never made it to the movie. :)

yes - i remember seeing that come out.  it was an upgrade to Pan-Kai, which
was also Parker Bros., but earlier.  there was a screen image of pentominoes
in "2001", but it wasnt being played by anyone, just in the background along
with other flashing lights & panels...hard to pick up, cawz its panned past
rather quickly....

- nate
Newsgroups: sci.math
From: ilan@leland.Stanford.EDU (ilan vardi)
Subject: Re: Large numbers
Organization: DSG, Stanford University, CA 94305, USA
Date: Fri, 18 Dec 92 20:59:38 GMT

There is a possibly interesting game consisting of trying to name
larger numbers than your opponent, where the rules are that you can
operate on his number and the point is that you use few operations to
name a bigger number. For example:

A: names 5 

B: names 5 + 1

But B could have said 2*5 and done better, or else 2^5. So in general
you would get

A: names n

B: ``I successor you,'' i.e., n+1

   ``I double you'', i.e., 2*n.

   ``I exponentiate you,'' i.e., 2^n, 

   ``I tower you,'' i.e., 2^2^2...^2, n times. 

In fact he could have said: ``I Ackerman you,'' which really means 
``I diagonalize you''.

So in some sense, the game can be viewed as happening on the recursive
hierarchy with a move taking you from one limit to another.  For
example, after B says ``I diagonalize you'', A responds with ``I
diagonalize you'', which means ``I diagonalize your diagonalization''.
E.g., you diagonalize the using the Ackerman function, by
diagonalizing the sequence

B_1(n, k) = Ackerman(Ackerman...(Ackerman(n)))   (k Ackerman's)

B_2(n, k) = B_2(n, B_1(n, k-1))

i.e., the function you end up with is essentially

  C(n) = B_n(n, n) 

up to some fussing about whether n -> n-1 or not.

Such moves seem to be optimal, in some sense.

-ilan



Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Pi in beatiful form --lost
Organization: Computer Science Department,  Stanford University.
Date: Wed, 16 Dec 1992 20:17:41 GMT

In article <1gmvieINN7al@aludra.usc.edu> rmurphy@aludra.usc.edu (Bob Murphy) writes:
>There are many other beautiful formulas involving pi.
>Here are a few of them.
>
>pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...
>
>(pi - 3)/4 = 1/(2*3*4) + 1/(4*5*6) + 1/(6*7*8) + ...
>
>4/pi = 1 +  1^2
>           ------------------------------
>            2 +  3^2
>                -------------------------
>                 2 + 5^2
>                    --------------------
>                      2 + 7^2
>                         ---------------
>                           2 + 9^2
>                              ----------
>                                2 + ...
>
>and I guess I should include Euler's:
>
>(pi^2)/6 = 1 + 1/(2^2) + 1/(3^2) + 1/(4^2) + 1/(5^2) + ...

And then there is the oldest formula, Vieta's product from 1593:

2/pi = sqrt(1/2) * sqrt((1+sqrt(1/2))/2) * ...

where each succeeding factor is the square root of the arithmetic mean
of 1 and the preceding factor.

Vieta's product is just the arithmetic behind Archimedes' computation
of progressivey better lower bounds on pi, but starting from a square
instead of the hexagon Archimedes started with.  The product of the
first n factors in Vieta's expression yields square-Archimedes' n-th
lower bound as obtained by using the perimeter of the 2^{n+1}-gon to
estimate the circumference of the circumscribed circle.

Now square-Archimedes also obtains a matching series of upper bounds
on pi by using the perimeter of the 2^{n+1}-gon to estimate the
circumference of the *inscribed* circle.

## When the sqrt operation is dropped from the n-th factor of Vieta's
## product, the product of those first n factors yields Archimedes' n-th
## upper bound.

My post of Oct. 16 about old methods of computing pi included a proof
of this fact.  I have not seen this connection with Archimedes' upper
bounds pointed out elsewhere, so if anyone knows of a prior source I
would very much appreciate hearing about it.

I should also make the connection here with my post of a few days back
showing that the "average" value of pi for the square is within .1% of
that for the circle.  The square of course corresponds to stopping
Vieta's product at n=1, and gives 2 sqrt(2) and 4 as respectively lower
and upper bounds on pi.
-- 
Vaughan Pratt				There's safety in certain numbers.
Xref: sparky sci.math:16923 sci.physics:21175 sci.bio:4580 sci.chem:5408 misc.education:5268
From: det@phlan.sw.stratus.com (David Toland)
Newsgroups: sci.math,sci.physics,sci.bio,sci.chem,misc.education
Subject: Re: TIME HAS INERTIA - NewScientist Article
Date: 14 Dec 92 14:20:59 GMT
Organization: Stratus Computer, Software Engineering
NNTP-Posting-Host: phlan.sw.stratus.com


A "theory" without predictions that can be measured and that will
differentiate it from another "theory", is a conjecture.  If it is to
be taken seriously, it needs to contain no contradictions with observed
facts.  If it violates an accepted principle, such as the Laws of
Thermodynamics, there must be an explanation why it can do so with
impunity.

I'm not saying anything specific about the subject article, which I
have not read; it didn't sound terribly facinating at first, and I've
only repeated these principles in the hopes of putting some perspective
to what has been an unexpectedly (to me) long thread.

I don't want to be *too* quick to call someone a crackpot because his
or her theory is unorthodox, but on the other hand any Vegas gambler
could probably get rich by betting heavy odds against every unorthodox
theory that pops up.  Revolutionary breakthroughs in thinking *do*
occur, but they are extremely rare and are invariably accompanied
by quantitative (read mathematical) derivations.  Numbers can lie,
but numerical lies do not stand up well to scrutiny.

Enough pontification.

-- 
--------------------------------------------------------------------------
All opinions are MINE MINE MINE, and not necessarily anyone else's.
det@phlan.sw.stratus.com   |  "Laddie, you'll be needin' something to wash
                           |  that doon with."
Newsgroups: sci.math
From: markc@smsc.sony.com (Mark Corscadden)
Subject: Re: The Continuum Hypothesis:  Must it be {True or False}, or Not?
Keywords: farkle,true,false
Organization: Sony Microsystems Corp, San Jose, CA
Date: Thu, 17 Dec 92 22:09:09 GMT

In article <1992Dec14.203726.6303@news.cs.indiana.edu> "Norman Danner" <ndanner@kiwi.ucs.indiana.edu> writes:
>In article <1992Dec14.200024.6435@nas.nasa.gov> asimov@wk223.nas.nasa.gov (Daniel A. Asimov) writes:
>>	ASSUME that Fermat's Last Theorem is undecidable from the 
>>	usual axioms of arithmetic. 
>
>I'm reading this to mean FLT is unprovable from the Peano Axioms (PA).

Clearly you're reading Dan Asimov incorrectly.  Undecidable means that neither
FLT nor its negation is provable.  Even trivially false statements satisfy
your reading, since they are "unprovable", but that's not very interesting.

Dan is correct in saying that FLT being undecidable in PA would imply
that FLT is true.  To relate this back to the Continuum Hypothesis, there
you have a case where a proposition has been proven to be undecidable in
a certain formal system.  Some people want to use that fact alone to
"demonstrate" that it is not meaningful to ask whether CH is really true
or false.  But using undecidablily in a formal system alone to reach such
a conclusion is bogus, as Dan has shown, because the undecidability of FLT
in PA actually settles the question of its truth or falsehood (in favor
of truth).

Of course, just because one can give a bogus argument for believing that
it's not meaningful to ask whether CH is really true or false doesn't mean
that it necessarily *is* meaningful to ask the question.

By the way, it would be very, very nice to have a universally accepted way
to say that a specific mathematical proposition "is definitely either true
or false, whether or not we can determine its truth or falsehood".  You
could say, for example, that such a statement was "farkle".  Then you'd say
that FLT is absolutely farkle, whether or not its undecidable in PA, but
you might say that CH may or may not be farkle - which is what I'd like
to be able to say, if only there were a word like "farkle" ...

Mark Corscadden
markc@smsc.sony.com
work: (408)944-4086
From: news@massey.ac.nz (USENET News System)
Newsgroups: sci.math
Subject: Re: Philosophy of Pi
Date: 20 Dec 92 21:27:07 GMT
Organization: Massey University

In article <1992Dec18.114450.22206@lth.se>, dat92oma@ludat.lth.se (Ola Martensson) writes:
> 
> A friend and I discussed if you can find any integer anywhere
> in the expansion of PI?
> 
> 1415 , 92 etc is obvious BUT can you really find ALL integer
> even unlimited ones???

A number is said to be 'normal' in base 10 if every finite string of digits
occurs eventually and with the same limiting relative frequency as any
other sequence of the same length. E.g. 25369 should appear with frequency
--> 10^(-5) in the first n digits as n --> oo. Similarly for other bases.
A normal number is one which is normal in all bases.

Normal numbers behave like random numbers. In fact a random sequence of
digits will form a normal number. This implies that non-normal numbers
will be rare. (To be more precise, almost all numbers - in a measure
theoretic sense - are normal). Despite this, we do not know many normal
numbers. Rational numbers are obviously non-normal.

We do not know whether or not Pi is normal, but since it is not special
in the way rational numbers are, it seems likely that it is.

If Pi is normal, then the answer to your question is 'yes'. Pi would also
hold all possible knowledge in a suitable code - e.g. the works of
Shakespeare - as well as all possible mis-information (e.g. Shakepeare
with every conceivable error).
Newsgroups: sci.math
Subject: Re: Interesting Discrete Math Question
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: Wed, 16 Dec 1992 22:40:42 GMT
Organization: Department of Mathematics, University of Canterbury

[Draw integers randomly from 1 to 100. We want to get a pair differing by 3.]

Several people have already observed that 51 draws is insufficent to
guarantee a pair three apart.

e.g.   1 2 3, 7 8 9, 13 14 15, .....   91 92 93,  97 98 99  might be drawn.

It has not actually been proved that 52 draws is certain to give the desired
result.

Here is the proof. (Pigeonhole principle).
~~~~~~~~~~~~~~~~~
Partition the integers into 51 subsets:

{1,4}   {2,5}   {3,6}
{7,10}  {8,11}  {9,12}
.....
{91,94} {92,95} {93,96}
{97,100}
{98}
{99}

Now when 52 numbers are drawn, at least 2 must come from one subset, and thus
be 3 apart.

--------------------------------------------
Bill Taylor.       wft@math.canterbury.ac.nz 
Bill Trylor. que  rwft@maih.casterkury.aa.n! 
Tiel Tryloco quer rwst@maihuc sterkery.ga.n!
Thelworyd co quer rwsi@mvihus strikesy.gain!
The world conqueror sig-virus strikes again!
--------------------------------------------
Newsgroups: sci.math