HDTenEightyP/NetNews huggingface dataset:
Newsgroups: sci.math
Subject: Re: Bill Clinton and Complex Analysis : -)
From: gadberry@neurocog.lrdc.pitt.edu (Mike Gadberry)
Date: 5 Nov 92 05:31:57 GMT
 <1992Nov3.135411.1621@magnus.acs.ohio-state.edu> <9067@blue.cis.pitt.edu.UUCP> <1992Nov4.211359.14309@massey.ac.nz>

In article <1992Nov4.211359.14309@massey.ac.nz>, news@massey.ac.nz (USENET News System) writes:
|> In article <9067@blue.cis.pitt.edu.UUCP>, gadberry@neurocog.lrdc.pitt.edu (Mike Gadberry) writes:
|> > 
|> > 
|> > 	There is anecdotal evidence that Abraham Lincoln was profoundly influenced
|> > by the beauty of Euclid's Elements, and the existence of the electoral college
|> > indicates that some of the framers of the American Way (notably Alexander 
|> > Hamilton) had a sophisticated understanding of the mathematical subtlties of 
|> > voting systems far ahead of their times.  (Disenfranchisement and fair voting
|> > models have only been systematically studied using mathematical tools quite 
|> > recently.)
|> 
|> According to Martin Gardner's mathematical games column,
|> Scientific American, October 1974, the subtlties were realised in 1785
|> by the Marquis de Condorcet.
|> It could be that a lot more work has been done on this in recent years.
	Thanks for pointing me to Condorcet.  I'll look him up.  Vive la 
France!  (I'll also dig up the SciAm article.  See my reply to Igor Rivin
for more info about where I think this thread on this heavily trafficked 
newsgroup could go.)
	-Mike gadberry@neurocog.lrdc.pitt.edu
	 Neural Processes in Cognition- University of Pittsburgh
Newsgroups: sci.math
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Subject: Re: Teaching Calculus
Organization: Purdue University Statistics Department
Date: Mon, 9 Nov 1992 16:28:44 GMT

In article <1992Nov9.005052.3384@news.cs.indiana.edu> johnl@spinner.cs.indiana.edu (John Lacey) writes:
>Benjamin.J.Tilly@dartmouth.edu (Benjamin J. Tilly) writes:

>>I know that many people feel that using limits to teach calculus
>>confuses the students and it might be better to have an alternate
>>approach. How about the following one which uses continuous functions
>>to replace limits?

>I, for one, feel that it is not limits but how they are taught that is
>confusing.  I have yet to see a teacher that doesn't totally confuse
>people about when you can substitute the limit in, and when you cannot.

When teaching anything in mathematics, it is essential that precision be
used.  Much can be done without using limits; one can essentially set
up differential algebras of functions.  With this, it is possible, for
example, to show that, using ' for derivative, that 

	sin'(x) = sin'(0)*cos(x),   cos'(x) = -sin'(0)*sin(x).

I agree that it is how limits are taught, with excessive formalism, such
as using the explicit epsilon-delta definition rather than the equivalent
interval definition, which makes it harder.  I have seen a definition of
tangent to a curve which does not explicitly use limits, although it is
formally equivalent.  But the usual incorrect intuitive "definition" is
not the way to do it.

>But, to side-step the issue entirely, I agree with Tom Apostol's view
>that pedagogically, historically, and perhaps even mathematically,
>integration more natually precedes differentiation.

What does integration even have to do with differentiation?  The 
"Fundamental Theorem of Calculus" states that two totally different
concepts, that of integral and that of antiderivative, give the
same answers.  Integration with respect to measures other than
length goes back 5000 years, and the notion of discrete integral
is easy high school material.  Then it is only necessary to pass
to the limit to get almost all of the rest.
-- 
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)
Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Trivial! (was: Re: Help X^2 == Y mod N)
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Thu, 5 Nov 92 20:40:18 GMT

In article <1992Nov5.061236.3595@nuscc.nus.sg> tim@iss.nus.sg (tim poston) writes:

>No mathematics is trivial;
>the trivium was grammar and rhetoric and I forget what;
>roughly, the mediaeval equivalent of a Bachelor's.
>
>Mathematics (along with harmony and two others I forget)
>was part of the quadrivium;
>equivalent of that peculiar US invention,
>the PhD.

Yes, if I remember correctly the trivium was logic, rhetoric and
grammar, while the quadrivium was arithmetic, geometry, harmony and
women's studies.  (Guess which one doesn't fit - and should be
astronomy.)

So - logic is trivial!  Mathematics is quadrivial!
From: neil@dehn.mth.pdx.edu (John Neil)
Newsgroups: sci.math
Subject: Re: What is a knot?
Date: 10 Nov 92 20:02:40 GMT
Article-I.D.: dehn.neil.721425760

jbaez@riesz.mit.edu (John C. Baez) writes:

>In article <COLUMBUS.92Nov6105242@strident.think.com>
>columbus@strident.think.com (Michael Weiss) writes:

>>How would one define a (tame) knot, intrinsically?  Definitions I am
>>familiar with either involve modding out by ambient isotopy (in fact
>there
>>are subtle points here, I believe-- perhaps someone more knowledgeable
>>would like to post), or by Reidemeister moves.

>Well, a (tame) knot is just a circle embedded in R^3, or, if you prefer
>something less intuitive, S^3.  But you seem to be speaking of the

[stuff deleted...]

Actually, if you are talking about a tame knot, you must be MUCH more
specific than that.  All knots are embeddings of S^1 in S^3.  The tame
variety are those embeddings which have a FINITE simplicial structure.
A wild knot, while still an embedding of S^1 in S^3 will not have a FINITE
simplicial structure.  Frequently, we construct the notion of tame knots
via PL (piece-wise linear) constructs (polygons in S^3).  That makes everything
nice and linear and allows one to construct ambient isotopy in a well-defined
combinatorial fashion.  In a 3-manifold, the notion of diffeomorphism is
much more complicated than it needs to be since in dimension 3, to be
diffeomorphic is to be homeomorphic.

>>Is the field just too young to have a suitably slick and (on first
>>encounter) unintuitive definition?

>If you want some definitions that are less intuitive, you could try the
>following.  These are actually definitions of isotopy classes of *links*
>- it soon becomes clear that there's no point in studying knots without
>studying links too.

[more stuff deleted...]


Again, this can (for tame knots) all be described in a nice, orderly,
combinatorial fashion.  While all the algebra and topology fits in well with
everything that is defined in this manner, the field can be studied from a
purely geometric, combinatorial fashion without any of these "exotic" (I use
the term to denote the change in categories) definitions.

As an example, in Dale Rolfsen's book "Knots and Links", you will find no
less than 10 different definitions of linking number.  These run the gambit
from algebraic to differential to geometric perspectives.  They all mean
exactly the same thing.  You can choose, then, as your starting point any
of the methods of describing these spaces and show equivalence at each step
to any of the other approaches.

--John Neil



--
John Neil, Graduate Teaching Assistant           e-mail:  neil@math.mth.pdx.edu
Mathematics Department                         NeXTMail:  neil@dehn.mth.pdx.edu
Portland State University
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Newsgroups: sci.math
From: mkrogers@unix.amherst.edu (MICHAEL K ROGERS)
Subject: Re: Problem in Category Theory (elementary)
Organization: Amherst College
X-Newsreader: TIN [version 1.1 PL6]
Date: Sun, 8 Nov 1992 07:02:40 GMT

robert j kolker (rjk@world.std.com) wrote:
: Find a category with an arrow (morph) that is epic and monic but not an
: isomorphism.

What about  objects = { A, B } and morphisms are the identity
morphisms on  A  and  B  and a morphism  A -> B  (with the
obvious law of composition)?
From: mccolm@darwin.math.usf.edu. (Gregory McColm)
Newsgroups: sci.math
Subject: Re: A question on binary relations
Date: 5 Nov 92 21:07:28 GMT
Organization: Univ. of South Florida, Math Department

In article <1992Oct31.181604.18857@mp.cs.niu.edu> rickert@mp.cs.niu.edu (Neil Rickert) writes:
>In article <rkaivola.720548141@mits> rkaivola@mits.mdata.fi (Risto Kaivola) writes:
>>  On brushing up my knowledge of discrete mathematics, I came across the
>>following question in a book on the subject.  The words inside brackets
>>are my comments.
>>
>>7. Consider the equivalence relation ~ defined on Z [the set of integers]
>>   by m ~ n if m is even and  n=m+1, or m is odd and m=n+1.  Describe the
>>   equivalence classes.  Is the natural addition well defined [natural
>>   addition having been defined earlier on]?
>>
>>I can't read this question in a way that would make the relation ~
>>reflexive.  Can you?
>>In my opinion, it is not an equivalence relation at all.
>
>Sure, its an equivalence relation.
>

It is niether reflexive (it is NOT true that 1 ~ 1) 
nor transitive (even though 1 ~ 2 and 2 ~ 3, it is 
NOT true that 1 ~ 3).  It is symmetric, and in fact 
defines the infinite graph

   ...  --(-1)--(0)--(1)--  ...  .

-----Greg McColm
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math
Subject: Re: 4x4 matrix
Keywords: matrix detrminant invers
Date: 10 Nov 92 03:30:15 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <2054@tfd.tfd.com> bret@tfd.COM (Bret Weinraub) writes:
>Help!

	[e-mail unavailable]

>
>I have the following problem.  Any input on how to solve it is greatly
>    job:  Invert the following matrix
>
>                                     3 4 1 8
>                                     1 1 2 0
>                            [A] =    2 7 3 7
>                                     1 0 6 5

>        formula 1) first find the determinant of A (det A)
>                2) to get the inverse, the formula is (1/det A) * [A]

Let's not and say we did. For many reasons, this approach to inversion
is better left alone. There are many simple and effective ways of computing
the inverse, and if you do lots of this kind of stuff you want to study
linear algebra and then perhaps some numerical analysis. To get a quick answer
to this question, you might use some machine to calculate it, or better yet,
you can calculate it as follows. Write down your matrix next to a 4x4 identity
matrix:


	3 4 1 8  |  1 0 0 0
	1 1 2 0  |  0 1 0 0
	2 7 3 7  |  0 0 1 0
	1 0 6 5  |  0 0 0 1


Now the object is to apply row operations to the two matrices so that the left
side becomes the identity - the right hand side will be the desired inverse.
By row operations, I mean you take a row in the matrix and subtract some
constant multiple of it from another row, possibly itself. Notice that this
includes just multiplying any row by some constant. You do the same operation
on both sides at once. You decide what operation to do by trying to get the
zeros and ones in the right places on the left side. You can make this into an
algorithm by first multiplying a row by the reciprocal of it's element in the
same column of the right hand side and then successively 'eliminating' the
nonzero elements in each row of that column. You do this once for each row
and you should be done. For more on this, look up "Gauss-Jordan Elimination"
(and you can find it in that ubiquitous tome _Numerical Recipes_. Note that
Gauss-Jordan is not the best way to invert a matrix, nor is inversion the
best way to solve linear equations, and in the final analysis _Numerical
Recipes_ is not the ultimate authority on the subject. Your milage may vary.

Later,
Andrew Mulhaupt

From: ric@hpspdla.spd.HP.COM (Ric Peregrino)
Date: Mon, 9 Nov 1992 18:45:35 GMT
Subject: Re: Simple Diff Eq
Organization: HP Stanford Park - Palo Alto, CA
Newsgroups: sci.math


Thanks to all that responded (a lot of you). Boy do I feel silly.
I looked at a series solution and of course I could not recognize
the closed form, which so many of you provided.

Thanks again, Ric P.
Newsgroups: sci.math,sci.math.symbolic,sci.math.num-analysis
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Subject: Re: Where to find PD symbolic math packages (FQA)
Organization: U of Cambridge, England
Date: Fri, 6 Nov 1992 17:59:58 GMT

In article <1ddsahINNfq9@sol.deakin.OZ.AU> huang@deakin.OZ.AU (Weiguang Huang) writes:

>	I suggest you consider SymbMath.
>	It has three versions: Shareware, Student, and Advanced. The 
>shareware version is available from SIMTEL20 archives on anonymous FTP sites 
>(wsmr-simtel20.army.mil as /calculator/sm211a.zip, rana.cc.deakin.oz.au as 
>/huang/sm211a.zip, etc) or by e-mail from listserv@ndsuvm1.bitnet. 

I find it interesting that this claims to be Re: something, but there have
been no posts with that subject field for the last three weeks, and I'm sure
the last one I saw was also posted by Mr Huang.

If you're going to post gratuitous advertisements you should do so honestly
rather than pretending your advertisement is a response to someone else's
post.

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
Newsgroups: sci.math
From: gsmith@urania.uucp (Gene W. Smith)
Subject: Re: Univariate polynomial equations and the FAQ
Organization: IWR, University of Heidelberg, Germany
Date: Tue, 10 Nov 92 10:09:03 GMT

In article <1dks02INNo3b@mozz.unh.edu> dvf@kepler.unh.edu (David V Feldman) writes:

>>>Fix an integer m.  Let K be the extension of Q obtained by adjoining
>>>all roots of all polynomials of the form
>>>     n     m    
>>>	  x + a x + ... a
>>>		   m         0

>While it is true that I did not ask the question that I intended to,
>I don't see any reason that  K = Q-bar.  Remember that  m  is fixed.

At this point I don't see the reason anything is anything.  I can't read
the polynomial equation above, and suggest you write exponents in ascii
form (x^n, etc.) so that we don't have to guess about what you've written.

Are you saying, fix some m, and consider all polynomials of the 
form

x^n + a_m x^m + a_(m-1) x^(m-1) + ... + a_0, a_i in Q?  

If so, then for m > 1 and n > 4, we have nonsolvable extensions of Q.

If I am reading what you wrote below correctly, you are interested in
extensions which have Tschernhausen transformations into a form with
a_i beyond a certain point 0.  We can see that over Q there is an
immediate problem getting rid of the a_2 term.  If the extension in
question is totally real, the sums of the squares of the roots is
positive.  If we have eliminated the trace term already, we see
immediately that a_2 must be positive.

Some questions like this can be approached via classical invariant
theory; it also makes a difference whether we can transform over Q or
over a larger field.  I still can't figure out exactly what it is you
are asking, though.

-- 
     Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University 
               gsmith@kalliope.iwr.uni-heidelberg.de
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: Relatives with Crackpot Proofs
Date: 13 Nov 1992 15:50:47 -0500
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu


Underwood Dudley, MATHEMATICAL CRANKS.  Mathematical Association of
America, 1992, 300pp., paperbound.  ISBN 0-88385-507-0.
List $25.00 [less for members, of course: catalog number: CRANKS].
-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
Newsgroups: sci.math
From: jones@ils.nwu.edu
Subject: New Ph.D. Program
Organization: institute for the learning sciences
Date: Mon, 9 Nov 1992 20:21:51 GMT

T h e   L  E  A  R  N  I  N  G    S  C  I  E  N  C  E  S    P h. D.   P r o g r
a m


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today's society is a complex  challenge for schools and corporations. To meet
this challenge, Northwestern University has established a unique Ph.D. program
in the Learning Sciences. Its purpose is to advance the research and
development of innovative educational structures and technologies.

Research in the Learning Sciences is aimed at creating environments that offer
all students a rich variety of opportunities for learning to think effectively,
both within and outside of formal learning settings. Graduate student and
faculty research efforts are supported by a multimedia, networked computing 
infrastructure and an unusual commitment to developing technologies that
function effectively within their intended settings. 

The Learning Sciences is a synthesis of computational and social science
disciplines including artificial intelligence, cognitive and developmental
psychology, educational research, linguistics, computer science, anthropology
and fields that develop strategies for teaching complex subject matter. 

Through research, field work and apprenticeships, the Ph.D. program will
prepare its graduates to assume leadership roles in advancing the scientific
understanding and systematic practices of learning and teaching.

Students are eligible for competitively-awarded multi-year funding. To qualify
for financial aid, candidates should apply by January 15, 1993. Applications
received after that date may be  considered. Individuals with relevant teaching
experience or a strong academic  background in a field related to the Learning
Sciences are encouraged to apply. For  application materials or for further
information, contact:

Professor Roy Pea
Ph.D. Program Chair 
The Learning Sciences 
Northwestern University
1890 Maple Avenue
Evanston, IL  60201
phone: 708-491-8152
fax:    708-491-5258
email: jones@ils.nwu.edu
  
Newsgroups: sci.math
From: uds@trace.eng.wayne.edu (Seetamraju Udaybhaskar)
Subject: Re: Functions for n-valued logic
Keywords: n-valued logic
Reply-To: uds@trace.eng.wayne.edu (Seetamraju Udaybhaskar)
Organization: Wayne State University, Detroit
Distribution:  world
Date: Thu, 12 Nov 1992 21:38:14 GMT

In article <1992Nov12.161637.20049@ghost.dsi.unimi.it> lombardo@ghost.dsi.unimi.it (franco lombardo) writes:
>
>i have a function f:{0,1/2,1}^3--->{0,1/2,1}^3 completely described
>by a table and i  want to express it using the operators of
>a three-valued Lukasiewicz logic: and, or, not and
>Lukasiewicz implication.

The function maps its domain onto itself...

Draw up a lattice, with 3^3 points.  Draw the arrows, and check if
every point can be represented as variations of the supremum
and infremum operators...  The variations will be clearly based on
and/OR/NOT operators...

This is a generalized idea for any sort of table U have...

(For lattices, look up partially ordered set theory (PO-set theory))...
Xref: sparky sci.math:14926 misc.education:4289
From: bdsm@poincare.WPI.EDU (Billy Don McConnell)
Newsgroups: sci.math,misc.education
Subject: Re: Is Math Hard?
Date: 13 Nov 1992 17:32:07 GMT
Organization: Worcester Polytechnic Institute
Distribution: world
NNTP-Posting-Host: poincare.wpi.edu
Originator: bdsm@poincare.WPI.EDU


In article <1992Nov11.213414.174@hubcap.clemson.edu>, mjs@hubcap.clemson.edu 
(M. J. Saltzman) writes:
|> In article <1992Nov10.083705.2187@news.Hawaii.Edu> lady@uhunix.uhcc.Hawaii.Edu 
|> (Lee Lady) writes:
|> >
|> >I think that card games are also very useful in teaching young children
|> >mathematics.  Some of them, such as blackjack, require a certain amount
|> >of arithmetic and a lot, like canasta, involve keeping score.  But I
|> >think that almost any card game does some good by focusing a child's
|> >attention on numbers as something worth paying attention to. 
|> 
|> I have a card/numbers game called Krypto(tm), which is played as follows:
|> There are 52 cards, numbered 1, 2, ..., 10 (3 each), 11, 12, ..., 17 (2 each),
|> and 18, 19, ..., 25 (1 each).  You deal five cards in your hand (playing
|> solitaire) and one card face up.  The object of the game is to combine the
|> five cards in your hand, using the four arithmetic operations (+, -, *, /), 
|> to form an expression with value equal to the face value of the sixth card.  
|> (All intermediate results must be integral).  You can play against an egg 
|> timer or against other players in various ways.
|> 
|> If that doesn't help your kid to learn arithmetic skills, I don't know
|> what will!
|> 
|> I don't know if they still exist, but the company was Products of the
|> Behavioral Sciences, PO Box 1176, Palo Alto CA 94302.

A related game is known as '24'.  There's a deck of cards, but each one is really
its own puzzle.  Each card has four numbers on it, like  8, 8, 4, 2, and the
object is to use  +,-,*,/  to arrive at the result 24. Some of the number
combinations are easy (6,4,1,1) and some are really hard (can't think of any).
But they work:  where I worked, students who would normally rush out of class
to go smoke cigarettes when the bell rang stuck around 'to do just one more' or
'to figure THAT one out...it's driving me nuts!' One nice thing is that the goal
is always the same...make 24...and we see that there are LOTS of ways to do just
that; 24 isn't _just_ 2 times 12, or 3 times 8, or 4 times 6.  Moreover, many of
the number combinations allow for different solutions...and a few students
would gloat (justifiably) if they found a more obscure solution, or at least a 
solution that the teacher didn't get. 

I don't know where '24' comes from...another teacher had brought the cards to
schools, and I'm not there anymore.  But if anyone out there has info on where to
order a set of these cards, please it my way.

Thanks,

Don Mc Connell
Newsgroups: sci.math
From: djoyce@black.clarku.edu (Dave Joyce)
Subject: Re: Well-known people who majored in math (other than mathematicians)
Organization: Clark University (Worcester, MA)
Date:  9 Nov 92 14:21:30 GMT

In <1992Oct28.183222.9321@ariel.ec.usf.edu> mccolm@darwin.math.usf.edu. (Gregory McColm) writes:

>In article <1992Oct20.090735.11632@infodev.cam.ac.uk> rgep@emu.pmms.cam.ac.uk (Richard Pinch) writes:
>>In article <81909@ut-emx.uucp> tim@ccwf.cc.utexas.edu writes:
>>>
>>>Hmm, Mr. Khayyam died in 1123 AD, several years before the first university
>>>was founded at Oxford.  I wonder where one could major in math back in those
>>>times, particularly in Persia.
>>>
>>Oh dear.  The University of Oxford was not even the first university
>>in Europe, let alone in the world.  The Islamic countries had a much
>>higher standard of learning than Europe in the 12th century: they had
>>[...]
>
>
>I was under the impression that the oldest university-like 
>institution was Plato's Academy, which was similar to that, 
>er, cult that Pythagoras had set up in South Italy.  The 
>oldest university---according to some undergrad texts---was 
>the Museum at Alexandria.
>
>But were there universities in Medieval Islam?  I thought 
>that it was like pre-Academy Greece, with various "enlightened 
>monarchs" setting up large courts inhabited by painters, 
>writers, astrologers, theologians, and the like.  Certainly, 
>that was the environment that Khayyam lived in----a world 
>closer politically to Mozart's than to Newton's.
>
>-----Greg McColm

Yes, the great universities at the time included those at Baghdad, the Bait
al-hakma (House of Wisdom), Cairo, and Cordova, and there were others.

For a nice intro to the subject read J.L. Berggren's _Episodes in the
Mathematics of Medieval Islam_, Springer-Verlag, 1986.

-- 
David E. Joyce				Dept. Math. & Comp. Sci.
Internet:  djoyce@black.clarku.edu	Clark University
BITnet:    djoyce@clarku		Worcester, MA 01610-1477