HDTenEightyP/NetNews huggingface dataset:
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Subject: The harm of special cases   Was: Re: Undivided distaste for algebra, more fuel for the fire...
Organization: Purdue University Statistics Department
Distribution: usa
Date: Thu, 21 Jan 1993 14:16:07 GMT

In article <20JAN199311462556@mary.fordham.edu> nissim@mary.fordham.edu (Leonard J. Nissim) writes:
>In article <1993Jan19.214027.21851@schaefer.math.wisc.edu>, 
>mueller@schaefer.math.wisc.edu (Carl Douglas Mueller) writes...

>>>2:  My opinion:  (you may wish to strech your legs here and skip this part)
>>>    Our schools
>>>    teach FOIL (first, outer, inner, last) instead of the distributive
>>>    principle for doing binomial expansion.  

>>    I learned FOIL from my students the first time I was a TA.  Nearly all of
>>the students I've taught since then have told me that this is how they had 
>>learned to multiply binomials.  I think that this is a rather TERRIBLE way
>>to teach this!  It hides what is going on and doesn't allow for the
>>(rather easy -- if you learn it right) generalization to multiplication of
>>more general polynomials.
 
>I agree.  I also learned FOIL for the first time at age 37 when teaching 
>Precalculus here.  A student complained that FOIL was easier than the 
>distributive law; when I asked what FOIL was, he was shocked that I had
>never heard of it.  He was unimpressed by my statement that FOIL was a 
>special case of a more general law, and that it would be better to under-
>stand the general law rather than to memorize the special case.  

This is what happens when tricks for special cases are taught instead of
giving an understanding of far more general cases.  Learning the FOIL
trick above even obscures the understanding in this special case. 

But that is far too often done.  The general notions of measure and 
integral, NOT the precise limitations and theory, can be rigorously
taught at the level of high school algebra, and in fact the oldest use 
 know of measure must go way back, and of integration about 5000 years. 
But if a student has learned that antiderivative is integration,
it is harder for that student to learn the concept.

The concept should also be taught in such a way that it is easy to 
generalize to path integrals, product integrals, integrals where both
the generalized signed measure and the function are in arbitrary linear
spaces, etc.  This is relatively easy if the concept, and not the
calculation, is kept in the forefront.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@snap.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: pyramid volume
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 21 Jan 1993 18:53:57 GMT

In article <1993Jan21.173423.11339@sophia.smith.edu> orourke@sophia.smith.edu (Joseph O'Rourke) writes:
:In article <1993Jan21.140402.25519@mr.med.ge.com> carl@crazyman.med.ge.com (Carl Crawford) writes:
:>
:>how do show that the volume of a pyramid is 
:>
:>	1/3 * area of base * altitude 
:>	
:>without using calculus?
 
A hint: Use a little vector analysis.  Consider 3 non-colinear
vectors A,B,C  in R^3. Then consider A * (B x C).
--
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"
From: jxh@math.ksu.edu (James C. Hu)
Newsgroups: sci.math
Subject: Re: Non-number theoretcial proofs in number theory
Date: 21 Jan 1993 04:29:36 -0600
Organization: Dept. of Mathematics, Kansas State University
NNTP-Posting-Host: hilbert.math.ksu.edu

ard@cs.bham.ac.uk (Antoni Diller) writes:

>It's years since I've done any proper maths, but what I'm after are
>examples of proofs of statements about the natural numbers or integers
>that use ideas and methods external to number theory, like for example
>results from the theory of complex numbers.  Ideally I'd like an example
>which is such that no proof only involving natural numbers or integers
>is known.  A reference to a good textbook would be acceptable.
>Please reply by email and I'll post any good examples.

How about the proof to:
	If p a positive prime and p = 1 (mod 4), then p is the sum of
	two square integers.

I believe it's proof uses Gaussian Integers, but I seem to recall an
alternate proof without them too ... (using Minkowski's?  I can't
remember ...).

But you would find many results applicable if the tools of the proof
are limited to the natural numbers or integers alone (for example, the
prime number theorem: the number of primes not exceding x is
asymptotic to x/ln(x)).

-- 
James C. Hu (jxh@math.ksu.edu), 1804 Denholm Dr., Manhattan, KS 66502
I speak for me, the whole me, and nothing but for me.  So help me me.
Xref: sparky sci.math:18856 sci.physics:23677
From: pcl@ox.ac.uk (Paul C Leyland)
Newsgroups: sci.math,sci.physics
Subject: Re: Ian Stewart's "Mathematical Recreations" column
Date: 28 Jan 93 10:25:21 GMT
	<1993Jan16.035852.25619@leland.Stanford.EDU>
	<1993Jan16.194757.5541@netcom.com>
	<1993Jan21.232017.2305@ariel.ec.usf.edu>
Organization: Oxford University Computing Services, 13 Banbury Rd Oxford OX2
	6NN
In-reply-to: mccolm@darwin.math.usf.edu.'s message of 21 Jan 93 23:20:17 GMT

In article <1993Jan21.232017.2305@ariel.ec.usf.edu> mccolm@darwin.math.usf.edu. (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 

...

Funny, I hadn't heard that good old King George had been canonized.

(When Herschel discovered what is now known as Uranus, he called it
Georgium Sidus, after the reigning monarch.)

Paul
--
Paul Leyland <pcl@oxford.ac.uk>          | Hanging on in quiet desperation is
Oxford University Computing Service      |     the English way.
13 Banbury Road, Oxford, OX2 6NN, UK     | The time is come, the song is over.
Tel: +44-865-273200  Fax: +44-865-273275 | Thought I'd something more to say.
Finger pcl@black.ox.ac.uk for PGP key    |
From: ekg90@ecs.soton.ac.uk (E K Gottesman)
Newsgroups: sci.math
Subject: Re: Advice needed
Date: 28 Jan 93 12:03:08 GMT

In <1CE00001.obtkt6@tbomb.ice.com> time@ice.com (Tim Endres) writes:
>Can someone give me there views on why I should get a BA in math as
>opposed to engineering? I have had math thru diff equ. and didn't
>really enjoy it until calc3 and linear. What else does a mAth major
>get that an engineer would not?
>Thanks for the info
>Karen 
>karen@ice.com
>tim endres - time@ice.com  (UUCP: heifetz!tbomb!time)

number theory
group theory
graph theory
statistics (LOTS of sub-topics here, but I do mostly pure stuff so I
don't know the area very well)
coding theory
operational research
game theory
topology
abstract algebra
logic
etc etc etc

That's just off the top of my head.
Maths is not just calculus.
By the way, as soon as I finish my degree (BSc Maths with Computer
Science) I start a research assistantship working with engineers and
medics. They're using number theory to model certain properties of the
human ear. Trouble is, all the maths they know is calculus (+ a little
stats). I got the job because I was NOT an engineer.

Do maths, its more fun, more interesting, more challenging and is
guarented (sorry, I can't spell) to make you socially unacceptable :-)

--Emma.
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: Frobenius Thm. on real dision algebras
Date: 27 Jan 93 20:56:26 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: hanche@ams.sunysb.edu (Harald Hanche-Olsen)

In article <HANCHE.93Jan27141930@ptolemy.ams.sunysb.edu>, hanche@ams (Harald Hanche-Olsen) writes:
>begin unashamed plug alert:

> Author:         Hanche-Olsen, Harald and Erling Stoermer.
> Title:          Jordan operator algebras
> Published:      Boston : Pitman Advanced Pub. Program, 1984.

I'll second the plug.  Damned good book, and the beginning workup
includes the gritty details usually ignored, in a readable format.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
Newsgroups: sci.math
From: bs@fatima.mitre.org (Robert D. Silverman)
Subject: Re: Number Theory
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Mon, 25 Jan 1993 23:41:45 GMT

In article <1993Jan25.193813.7909@murdoch.acc.Virginia.EDU> pjp9q@honi2.acc.Virginia.EDU (Paul Joseph Perrone) writes:
:
:I'm looking for any references or direction in solving the following
:problem:
:
:Given a set of prime numbers whose largest element is less than one
:million. Find a set of these primes such that any sum or difference
:between any two of these primes are NOT equal to any other sum or
:difference between any other combination of these primes.
:
:Is there such a set?
 
Without any further criteria your problem is trivial. Here is
such a set:

2   3      1009   21319

There is a VERY large number of such sets.
--
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"
From: rmurphy@usc.edu (Bob Murphy)
Newsgroups: sci.math
Subject: 1 is NOT prime!
Date: 27 Jan 1993 02:00:38 -0800
Organization: University of Southern California, Los Angeles, CA
Distribution: world
NNTP-Posting-Host: aludra.usc.edu

I've seen a few postings in the last couple of weeks that all
said something like "If we take 1 to be prime, then ...".

Because of this, I thought it a good idea to explain why 1 
is not considered to be prime.  It all has to do with the 
fundamental theorem of arithmetic.

--------------------------------------------------------------
   Fundamental Theorem Of Arithmetic:

   Every positive integer n > 1 can be expressed as a product
   of primes; this representation is unique, apart from the 
   order in which the factors occur.
--------------------------------------------------------------

If 1 were considered a prime then unique factorization, an
important property of positive integers, would no longer be
true.

For example, we would have 6 = 2*3
                             = 1*2*3
                             = 1*1*2*3
                             = 1*1*1*2*3
                             
                             etc.

There may be other reasons but this is the most obvious one that
I can think of.

                          Bob Murphy (rmurphy@aludra.usc.edu)
From: delliott@eng.umd.edu (David L. Elliott)
Newsgroups: sci.math
Subject: Re: Smooth manifolds and function extensions
Date: 27 Jan 1993 04:24:18 GMT
Organization: Project GLUE, University of Maryland, College Park
NNTP-Posting-Host: newra.src.umd.edu

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?
>

...
Best result I've heard of is that the ring of real-analytic (C-omega)
functions F(M) determines M up to C-omega diffeomorphism, and may be found in
S. B. Myers, "Algebras of differentiable functions," Proc. Am. Math. Soc.
Vol. 5 (1954) 917-922,
cited in T. Nagano's paper on transitive Lie algebras, J.Math.Soc.Japan
Vol. 18 (1969) 398-404.  I have not see Myers' paper, so don't know if
the result holds for C-infinity... an example where it does not would be
interesting, too!






-- 
David L. Elliott                                       delliott@src.umd.edu
Institute for Systems Research/ A.V. Williams Building
University of Maryland/ College Park, MD 20742

Xref: sparky sci.math:18814 rec.arts.books:26475 sci.philosophy.meta:3172
From: tp0x+@cs.cmu.edu (Thomas Price)
Newsgroups: sci.math,rec.arts.books,sci.philosophy.meta
Subject: Re: Hare Hare?
Date: 26 Jan 93 23:00:29 GMT
Article-I.D.: cs.C1HH8u.F0n.1
Organization: School of Computer Science, Carnegie Mellon

In article <2036@heurikon.heurikon.com> gtaylor@vme.heurikon.com (Gregory Taylor) writes:
>>system@kalki33.lakes.trenton.sc.us (Kalki Dasa) writes:
>>>
>>>Nice try, demon.
>
>We all speak from time to time of demonizing one's opponent or adversary
>in discourse. It's rare that we see it actually done in quite this
>manner. It would appear that this kind of inflammatory posting would
>seem to engage those very illusions of the self's false desiring for
>the illusory world of sensory pleasures by calling forth flamery instead
>of making thy point, Krsnaphile. In my tradition we say that "a soft
>answer turneth away wrath."
>
>Gregory Taylor/Heurikon Corporation/8310 Excelsior Drive/Madison WI 52717

No no, Gregory, I believe he was accusing you of being an AI program --
a Unix daemon which flaemes Haere Krishnae posts. 


      Tom Price   |    tp0x@cs.cmu.edu   |   Free will? What free will? 
 *****************************************************************************
  plutoniumsurveillanceterroristCIAassassinationIranContrawirefraudcryptology
Newsgroups: sci.math
From: hanche@ams.sunysb.edu (Harald Hanche-Olsen)
Subject: Re: Why Algebra...Important
Organization: University at Stony Brook, NY
Date: Tue, 26 Jan 1993 00:01:24 GMT

>>>>> In article <C1FItD.AxF@mentor.cc.purdue.edu>,
>>>>> ags@seaman.cc.purdue.edu (Dave Seaman) writes:

Dave> In article <1993Jan25.164019.19356@Princeton.EDU>
Dave> tao@mace.Princeton.EDU (Terry Tao) writes:

> Claim: \exists Cup = {(x, y, z) = d((x, y), Cup_0) = r_cup, z_0 < z
> < z_0 + h_cup} \cup {(x, y, z_0): d((x, y), Cup_0) < r_cup}:d(Cup,
> me) < \epsilon_0 \and \exists Coffee = \{ f(x, y): d((x, y), Cup_0)
> < r_cup} s.t. Coffee \in \overbar {Cup} \and T_coffee > 90 F \and t
> > t_0 => Hap_me(t) > Hap_me(t_0).

Dave> I had heard that a mathematician is someone who converts coffee into  
Dave> theorems, but I had not yet seen a demonstration.

I thought the point here was to turn a theorem into coffee?

- Harald
From: ric@hpspdla.spd.HP.COM (Ric Peregrino)
Date: Thu, 21 Jan 1993 00:16:50 GMT
Subject: tensor calculus question
Organization: HP Stanford Park - Palo Alto, CA
Newsgroups: sci.math


Hello sci.math

Is it always possible to make a change of coordinates that gives a
geodisic coordinate system at any point for a general metric that can
contain an anti-symmetric part? If so, could you supply it? eg.

 i    i   i          m   m   n   n
x' = x - x + {mn,i}(x - x )(x - x ) where (0) denotes the value at pole
         (0)     (0)    (0)     (0)

is such a transformation for a symmetric metric.

Also, can anyone verify or contest the validity of a result for the
curvature tensor for a general metric below?

              k    k        k
A   - A   + (G  - G  )A  - R   A = 0
 j,np  j,pn   np   pn  j,k  jnp k

Investigating a general metric for self-learning (it's not homework),
R. Peregrino, ric@spd.hp.com
Xref: sparky sci.math:18668 sci.physics:23306
From: yee@mipg.upenn.edu (Conway Yee)
Newsgroups: sci.math,sci.physics
Subject: Monte Carlo and Neutron Transport.
Keywords: Monte Carlo, Neutron Transport, Early Behavior
Date: 22 Jan 93 18:31:57 GMT
Followup-To: sci.math,sci.physics
Organization: University of Pennsylvania

According to authors such as Chandrasekhar (1943, Rev. Mod. Phys.)
in the limit as the number of scatters approaches infinity, the probability
distribution of a random walk of a neutron approaches that of the diffusion
equation.  In practice, a number of scatters on the order of 20-25 is 
sufficient to see the limiting behavior.

I am interested in finding out about the early behavior in the Monte Carlo
simulation.  Does anybody know of a reference which can give me this in 
terms of time?  A reference which I found fits the early behavior using
the number of scatters (i.e.  the probability distribution after n scatters).
I need the probability distribution at time t where time t is small enough
that only a few scatters has taken place.

-- 
411 Blockley Hall      |                    Conway Yee, N2JWQ
418 Service Drive      | yee@ming.mipg.upenn.edu    (preferred)
Philadelphia, PA 19104 | cy5@cunixa.cc.columbia.edu (forwarded to above)
(215) 662-6780         |
Newsgroups: sci.math
From: orourke@sophia.smith.edu (Joseph O'Rourke)
Subject: Re: pyramid volume
Organization: Smith College, Northampton, MA, US
Date: Thu, 21 Jan 1993 20:51:25 GMT

In article <1993Jan21.185357.28141@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
>:In article <1993Jan21.140402.25519@mr.med.ge.com> carl@crazyman.med.ge.com (Carl Crawford) writes:
>:>
>:>how do show that the volume of a pyramid is 
>:>	1/3 * area of base * altitude 
> 
>A hint: Use a little vector analysis.  Consider 3 non-colinear
>vectors A,B,C  in R^3. Then consider A * (B x C).

	Why don't you complete the thought?   It may be that you assume
knowledge more "complicated" than the intended goal...
From: jhusdhui@math.tau.ac.il (Gurel-gurevitch Ori)
Subject: Re: Alephs again
Organization: School of Math & CS - Tel Aviv University , Tel Aviv , ISRAEL.
X-Newsreader: Tin 1.1 PL5
Date: Fri, 22 Jan 1993 09:01:15 GMT


n case you dont know 2^aleph0=aleph1 is the continum hypothesis and
of course, is independent of ZFC. There is however a sign for 2^aleph0

it is just aleph (without any index). sometimes when dealing with
models
without GCH we use beit1 to represent 2^aleph0 beit2 is 2^beit1 etc,
etc
(beit0 is aleph0).

the enumeration given by Dennis Paul Himes is good for any countable
set
wich is already well-ordered (i.e. enumerated) but in general you
cannot
choose an enumeration for a given countale set without using AC so
this
is not a proof.

Allen K. writes:
 For instance, 2^{\aleph_0} may
 be \aleph_1, \aleph_2, or anything larger, EXCEPT \aleph_\omega
 (the next cardinality after \aleph_1, \aleph_2, ...). But it can be
 \aleph_{\omega+1}, \aleph_{\omega+2}, and so on.

 this is not quite true: as far as I know 2^aleph0 (aleph) can be any
 cardinal with cofinality greater then omega. this means that aleph
 cant
 be aleph_omega but neither aleph_(omega*2) aleph_(omega*n)
 aleph_(omega^2)
 and not even aleph_(aleph_(omega)). the proof to this statment
 follows
 immediatly from konig's lemma which state that A^B wheras B>=cf(A) is
 greater then A(not greater or equal). since
	 (2^aleph0)^aleph0 = 2^(aleph0*aleph0) = 2^aleph0
	 so cf(2^aleph0)>aleph0.
				 J.
				






--

Gurel-Gurevitch Ori			jhusdhui@libra.math.tau.ac.il
From: martin@lyra.cis.umassd.edu (Gary Martin)
Newsgroups: sci.math
Subject: Re: Matrices (problem)
Date: 26 Jan 93 15:22:04 GMT
Organization: University of Massachusetts Dartmouth

In article <Jan.24.22.14.31.1993.13572@inferno.rutgers.edu> gore@inferno.rutgers.edu (Bittu) writes:

   A simple question about the rank of a special matrix (this is not a
   homework problem; I already know a solution, want something better).
   In fact, this was probably on the Math Olympiad a few years ago.

   Let A be a matrix with the following properties:

   1. A is 2n+1 x 2n+1 for a positive integer n.

   2. A has 0's on the diagonal.

   3. The remaining entries are either 1's or -1's.

   4. Every row has exactly n 1's and n -1's.

   We want to show that the rank of this matrix is 2n (over complex
   numbers, say). Clearly the rank is <= 2n (all columns sum to the zero
   vector). 

   Let I_m be the m x m identity matrix and J_m be the all 1's matrix.
   The solution I know involves showing that the matrix J_m - I_m has
   rank m if m is even and m-1 if m is odd (over GF[2]; done by showing
   that its determinant is (-1)^(m-1) . (m-1)). We are only interested
   in the case where m is odd. We now use the well-known result that if
   the rank is m-1 over GF[2], it is m-1 over rationals, reals, complexes
   etc. Note that J_{2n+1} - I_{2n+1} is the same as A over GF[2].

   I am sure there is a much easier and elegant proof of this (for the
   field of complexes). Any takers?

This is a generalization of a Putnam problem from the 70s:
Let  S  be a multiset containing an odd number of integers with the
property that no matter which element is deleted, the rest can
be partitioned into two pieces having equal sum.  Then all the elements
of  S  are equal.

I had a paper on this in the Monthly back around '86 or '87 (can't
find that issue right now).  

Forget for a moment that the elements of S are integers.  Note that
hypothesis and conclusion are invariant under translation and dilation.
Translate so one element is zero.  Note that the elements of  S (if integers)
are congruent mod 2.  Divide by 2.  Repeat.  Since the elements were
integers infinitely divisible by 2, they are 0, so the original ones were
equal.

The extension to Q is easy: just clear denominators.  R  and  C  follow
from the simple observation that if a S is a set of vectors satisfying
the hypothesis, then the projections on any 1-dim subspace do too.

If the elements of  S  come from an abelian group  G, then the problem
generalizes to G iff G has no odd torsion.

I also generalized in a different direction:  if  S  has  km+1 elements
such that whichever element is omitted the rest can be partitioned into
k classes of size m with equal sums, then the elements of  S  are equal.
My memory of the exact result is now foggy, but it was something like:
the statement is true iff the elements of S come from an abelian group
whose only torsion is p-torsion where p is a prime dividing m.  This
doesn't sound quite right, but that's the flavor.

--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
Martin@cis.umassd.edu
From: starr-daniel@yale.edu (Daniel Starr)
Newsgroups: sci.math
Subject: Re: why is pi irrational
Date: 26 Jan 1993 23:51:19 -0500
Organization: Yale University Science & Engineering UNIX(tm), New Haven, CT 06520-2158
NNTP-Posting-Host: minerva.cis.yale.edu

In article <1993Jan26.232722.9931@cheshire.oxy.edu> pursell@cheshire.oxy.edu (William Robert Pursell) writes:
>In article <fX#wH9p@quack.sac.ca.us> dfox@quack.sac.ca.us (David Fox) writes:
>>In article <7017@tuegate.tue.nl> wsadjw@urc.tue.nl writes:
>>
>>
>>The Israelites no doubt drew circles from time to time, but their holy
>>bible says pi is equal to 3.
>>
>
>Is there a reference for this?  (pi = 3 in hebrew scripture)
Well...
sort of.
There's one spot, I think in Exodus, at any rate in the Mosaic law where the
proper procedure for thus-and-so is outlined, where the Israelites are 
commanded to dig a pit 30 cubits around and 10 cubits across.  Which is,
of course, correct to one significant figure, and only mandates pi=3 if
you consider that the law ought to have read "30 and 1 and four of the 
tenth parts of one and..." in which case, as we know, the book never would
have been finished.  So for us finite minds, I think 30 cubits around
and 10 across is a reasonable approximation, especially considering how
primitive the measures of the cubit and of lengths were.  :)

But if you want something really silly, there's the bill that was introduced
(but not passed) in Indiana that would have set pi=3.1 ... documentation
in alt.folklore.urban.

of 1 and t
-- 
This is NOT a .sig virus do NOT copy me into your .sig file  This is NOT
* Daniel Starr                   *   Ta twn thewn erga                 * 
* dstarr@minerva.cis.yale.edu    *   Hoi psukhwn twn anthrwpwn logoi.  *
a .sig virus do NOT copy me into your .sig file This is NOT a .sig virus
Newsgroups: sci.math