HDTenEightyP/NetNews huggingface dataset:
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: What is a knot?
Organization: MIT Department of Mathematics, Cambridge, LA
Date: Sat, 7 Nov 92 21:25:57 GMT

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
ambient isotopy class of the knot, i.e., where one forgets about
everything except its topology.  As you note this can be defined as
either
1) equivalence class of knots under diffeomorphisms of R^3 that are
connected to the identity in the group of diffeomorphisms (i.e.,
reflection is a no-no).
2) equivalence class of knots under the following: two knots
(embeddings) are equivalent if they can be connected by a continuous
1-parameter family of embeddings.
3) equivalence class of knot diagrams (2d pictures of knots, which I
won't define) under the Reidemeister moves and diffeomorphisms of the
plane that are connected to the identity in the group of
diffeomorphisms.

These are all equivalent.

>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.

4) equivalence classes of braids under the Markov moves
5) Hom(0,0) in the category of unframed unoriented tangles.


Actually, it turns out to be very good to work with framed oriented links.
Then we have the marvelously erudite

5') Hom(0,0) in the free tortile tensor category generated by a single
object.

This is the best definition for getting link invariants from quantum
group representations.

All the above definitions involve equivalence relations, although in
some cases this is hidden.
~
From: baez@guitar.ucr.edu (john baez)
Newsgroups: sci.math,rec.puzzles
Subject: Math puzzles - answer
Date: 5 Nov 92 20:30:37 GMT
Followup-To: sci.math
Organization: University of California, Riverside

Robert Southworth, bog@ugcs.caltech.edu, asked me to post this for him:

John Baez (jbaez@riesz.mit.edu) posted two math puzzles 
<1992Oct28.170601.19561@galois.mit.edu>, to which he later gave
partial answers. He writes:

> The second question concerned the sum from 1 to infinity of o(2^n)/2^n.
> Here o(2^n) is the number of odd digits in the decimal representation of
> 2^n.  The sum is surprisingly simple - it's 1/9.

(1) Clearly the kth digit (the multiplier of 10^k in the decimal
representation) of 2^n is odd iff 

        [2^n/10^k] is odd,

where [] denotes the greatest integer function. 

(2) Similarly, the (-n)th bit (the multiplier of 2^(-n), in the binary
representation) of 10^(-k) is odd (i.e. equal to 1) iff

        [10^(-k)/2^(-n)] = [2^n/10^k] is odd.


Let d_nk be congruent to [2^n/10^k] mod 2, d_nk = 0 or 1. Then by (1),

        o(2^n) = sum_k d_nk,

and by (2),

        10^(-k) = sum_n (d_nk * 2^(-n)).

The result follows by reversing the order of summation, as follows:

        sum_n o(2^n)/2^n = sum_n (sum_k d_nk)/2^n
                         = sum_k (sum_n d_nk/2^n)
                         = sum_k 10^(-k)
                         = 1/9.

(Note that 10^0 is not included in the final sum over k, because the
0th digit of 2^n is even for all n>=1.)

This problem is easy to generalize. The particular values 2 and 10 are
not important; their only crucial property is that 10 is divisible by 2.


    Robert Southworth      bog@ugcs.caltech.edu

Newsgroups: sci.math
From: tim@maths.tcd.ie (Timothy Murphy)
Subject: Re: group theory for HS students
Organization: Dept. of Maths, Trinity College, Dublin, Ireland.
Date: Tue, 10 Nov 1992 19:00:36 GMT

weemba@sagi.wistar.upenn.edu (Matthew P Wiener) writes:

>In article <ARA.92Nov6191321@camelot.ai.mit.edu>, ara@zurich (Allan Adler) writes:
>>I would be interested in having the names and authors and publishers
>>of books that teach abstract algebra and which say explicitly that they
>>are aimed at high school students.

>There are not too many that can even come close.  G Papy GROUPS is an
>obscure book, with lots of color to illustrate symmetry.  Presumably
>some of the Rubik's cube books may have what you want.  A motivated
>high school student should be able to follow Fraleigh's text.

>There is also, off to the side, Knuth SURREAL NUMBERS and E Landau
>FOUNDATIONS OF ANALYSIS, and Lang's two Springer-Verlag paperbacks.

>And read the introduction to Dieudonne LINEAR ALGEBRA AND GEOMETRY.

This is a really weird list of recommendations!

Group theory _is_ taught in schools in many countries,
in fact I would suspect most countries in the western world.
So there must be many school textbooks dealing with groups.

I remember the British SMP (School Maths Project)
had a very good series of school textbooks
including quite sophisticated algebra
some 20 years ago.
I don't know if that project is still going?


-- 
Timothy Murphy  
e-mail: tim@maths.tcd.ie
tel: +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
Xref: sparky sci.math:14758 alt.politics.elections:23437
From: mccolm@darwin.math.usf.edu. (Gregory McColm)
Newsgroups: sci.math,alt.politics.elections
Subject: Re: Electoral college (was Re: Bill Clinton and Complex Analysis : -))
Date: 7 Nov 92 22:28:00 GMT
Organization: Univ. of South Florida, Math Department

In article <1delmqINNngv@master.cs.rose-hulman.edu> brock@NeXTwork.Rose-Hulman.Edu (Bradley W. Brock) writes:
>In article <israel.720991427@unixg.ubc.ca> israel@unixg.ubc.ca (Robert B.  
>Israel) writes:
>> To get a plurality, rather than a majority, in the electoral college,
>> you just need about 1/9, rather than 1/6. 
>> 
>> This is not entirely a fanciful exercise. It could quite easily happen
>> that a candidate with fairly strong appeal in one half of the country and
>> little in the other would be elected despite finishing a poor third in
>> popular vote. 
>
>Which is not that far removed from how Lincoln won in 1860 with a plurality of  
>only 40%.  I might add that Rutherford B. Hayes and Benjamin Harrison both won  
>despite finishing second in the popular vote.
>


Hayes was a special case.  Tilden (Democrat) actually won a 
majority in the electoral college, but irregularities led 
to the formation of a Commission (essentially, the Supreme 
Court), to determine how the electoral votes should be 
assigned.  The Commission consistently assigned all the 
contested votes to Hayes, and a "secret" agreement was 
made between the Democratic and Republican parties to have 
Hayes's election accepted --- provided that the Reconstruction 
ended.  And you thought Iran-Contra was bad ...

-----Greg McColm
From: mckay@alcor.concordia.ca (John McKay)
Newsgroups: sci.math
Subject: Re: symbolic derivation of polynomial roots
Date: 9 Nov 92 12:10:29 GMT
Organization: Concordia University, Montreal, Quebec
Originator: mckay@alcor.concordia.ca


My understanding is that the roots of ANY polynomial in Q[x] 
can be expressed in terms of radicals. What is the trick?
You may need infinitely many of them.
From: levine@symcom.math.uiuc.edu (Lenore Levine)
Newsgroups: sci.math
Subject: Re: Bozos Synonymous
Date: 13 Nov 92 20:19:45 GMT
Organization: University of Illinois at Urbana

kosowsky@phoneme.harvard.edu (Jeffrey J. Kosowsky) writes:


>Talking about making money from "crop circles"...
>Here at Harvard, I have seen numerous posters plastered on bulletin
>boards throughout the campus advertising the following:

>				Crop Circle
>			  Lecture / Slide Show
>			 with George Wingfield
>			Director, Field Research
>		Center for Crop Circle Studies, Somerset England

>The event is *not* being held on camput but rather at a local church.
>Furthermore, they are *charging* 10 dollars a ticket!

>Who is this most Bozo of Bozos who can command 10 dollars for the
>chance to hear him drivel psycho-pseudo-science. Also what is this
>so-called "Center for Crop Circle Studies?" 
>Do they have any openings for post-docs or for junior faculty tenure
>track positions? :.)

>Jeff Kosowsky

You think that "crop circles" character is something. Wait till
you read "Sexual Personae" by Camille Paglia! Lots of complete
nonsense about how mathematics is done, and mathematicians 
think! 

I defy this lady to explain why the derivative of x^2 is 2x. Meanwhile
she's making zillions, and I'm scrambling for pennies in graduate
school.
Newsgroups: sci.math,sci.physics,sci.astro,sci.bio,sci.chem,misc.education
From: onar@hsr.no (Onar Aam)
Subject: Re: 2-dimensional grading system (was Re: What can one do for the bright?)
Organization: Rogaland University Centre
Date: Thu, 5 Nov 1992 07:03:42 GMT

>In article <1d83n8INNlh1@uniwa.uwa.edu.au> waohe@uniwa.uwa.edu.au (WA Office of Higher Education) writes:
>> The 2D-grading provides more information for EMPLOYERS etc. The idea is do
>>: integrate the education system with the real world around (at the moment they
>>: poorly coincide)
>
>
>>: Onar.
>
>>At the risk of sounding like Attilla the Hun, or one of his many clones,
>>have you ever known anyone who got through their study/ professional
>>training by working much harder than anyone else, only to find
>>that they then had to struggle for the rest of their lives, holding their
>>own in an environment in which they remained almost out of their 
>>depth?  In other words, they obtained their credential as a prize 
>>for working harder rather than as an index of their ability to cope.
>
>>Maybe there's a case for making it achievement/labour ?
>
>Even worse, you get people who attained a position by dint of extremely
>hard work, and do not have the abilities needed?  Do you want your car
>worked on by someone who has become a mechanic by working very hard, but
>does not understand the problems of the automobile?  Hard work is not 
>always desirable; a little use of electrochemical computers (brains)
>can often do much more than hours of work.
>
>We have had postings of people being sent from medical specialist to 
>medical specialist, even 30 or more, when someone with a little brainpower
>diagnosed the problem fairly quickly.  This ability to think rather than
>to react, and to put apparently unrelated things together, is what any
>educational system should emphasize, with those who can only grind things
>out being recognized as second-rate.  Or if they are the norm, the ones
>who can think should be considered supernormal, and treated correspondingly.
>-- 
>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)


What the hell are you guys talking about?(!) This is EXACTLY why I proposed the
2D grading system in the first place. With today's grading system you only get
one grade which includes both effort and grasp. With this system the employer
wouldn't know whether his employee has brains or not. With a 2D system, however,
the employer gets that vital information. 


Onar.

Newsgroups: sci.math
From: mokie@cco.caltech.edu (Michael L. Brundage)
Newsgroups: sci.math
Subject: Re: group theory for HS students
Date: 10 Nov 1992 01:45:24 GMT
Organization: California Institute of Technology, Pasadena
Distribution: sci
NNTP-Posting-Host: punisher.caltech.edu

sibley@math.psu.edu (David Sibley) writes:

>>If math courses had the training of mathematicians as their only
>>purpose, then the ordering would be more like what you suggest.  On
>>the other hand, there is an urgent need to provide, say, physics
>>students, with calculus at an even faster rate than is usually done.

>When I was a grad student at Caltech many years ago, there was a rumor
>that the math department had received a request along these lines from
>the physics department.  It seems they were reorganizing freshman
>physics and could do it the way they really wanted to, providing only
>that the calculus courses be reorganized a bit as well.  They wanted
>freshman calculus taught so that Green's theorem in the plane would be
>covered by the end of the second week.

Yes, unfortunately the Caltech physics department runs about a year ahead
of the math department (generally).  They want all sophomores to know Fourier
analysis by first term, but of course this isn't covered in Ma2a.  Then of
course, there are the higher level chemistry courses which assume you know group
theory, and the many classes that assume you know probability and statistics.
Ugh.  There's a committee this year to (hopefully) suggest some core curriculum
changes, but we'll see.  It'll be interesting to see how (or if) they modify
the math program.


>David Sibley
>sibley@math.psu.edu

-- 
mokie@cco.caltech.edu        |  "Let not your heart be troubled,
        or                   |    believe in God, believe also in Me."
michaelb@iago.caltech.edu    |       John 14:1 (NASVB)
Newsgroups: sci.math
From: cath@dyrham.inmos.co.uk (Catherine Barnaby)
Subject: Re: Bozos Synonymous
Organization: INMOS architecture group
Date: Fri, 13 Nov 92 12:14:25 GMT

There was a program on british television about crop circles
recently. I'll try and summarise...

Crop circles are not new, they have been noted way back, but
are only "trendy" recently. They are also a lot more common
than they used to be. There is a physicist who has done a lot
of work on them, and believes them to be due to the effects of
wind. There are lots of people who wander round with dousing 
rods twitching, and others who feel the energies and colours
(not sure of the collective here). There are also a 
number of people who go round faking them, and farmers who
make quite a bit of money charging #1 a person to go into 
their field if it has one. 

Crop circles are somewhat different to fairy rings: evidently
there are specific ways that the crop is laid down, with
the "swirls" and pretty patterns thus made being very important.
Forging a crop circle was described as being not-as-easy as one
might think. 

The physicist came over as believing that the straightforward
circle, and the circle-with-a-number-of-smaller-circles-round-it
were the "natural" phenomena, with the wierder patterns being 
fakes. However, all of the groups (physicist, dowsers, colour-
spotters...) diagnosed a circle as "real" when it had been 
specially faked the night before by a group of faking experts.

Corn circles are big business!

catherine
Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: How to respond to enthusiasm [Was Re: Help X^2 == Y mod N]
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 5 Nov 1992 13:11:25 GMT

In article <1da1ggINNk95@gap.caltech.edu> mokie@cco.caltech.edu (Michael L. Brundage) writes:
:bs@fatima.mitre.org (Robert D. Silverman) writes:
:
:>In article <1992Nov3.042721.2035@vela.acs.oakland.edu> phkahler@vela.acs.oakland.edu (KAHLER PAUL H.) writes:
:>:I would like to know if any one knows of a (good) method of finding X in
:>:the following congruence:
:>:
:>:X^2 == Y mod N        given Y and N
:>:
:>:I know how to determine if X exists and how to find it if N is prime.
   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:>:However, my N is not prime.
:>:
:>:My math prof told me there is a way but he didn't know it off the top of his
:>:head. I checked the library and can't find it. I'm not convinced that anyone
:>:has a way, or if it can be done in general. (oh boy! an unsolved problem?!)
:> 
:>Its trivial.
:
:This is being picky, but when someone says "Oh boy!" about a problem, and
:someone else replies with "It's trivial"  it (at first, at least)
:either (a) sounds egotistical or (b) squelches the enthusiasm of the poser.
:Of course, you probably just meant "It's not a hard problem, see, here's the
:way to do it," but it sounds much less kind than that on a first (and
:second, for that matter) reading.
 
As you may recall, the original poser said that he *already knew* how
to solve square roots modulo a prime. The best algorithm(s)
for doing this are variations of a random method due to Shanks, that 
searches for a quadratic non-residue.  Direct methods are known when the prime
is 3 mod 4, or 5 mod 8, but in other cases there does not seem to be a
fast *deterministic* method.

For someone who knows this much, factoring the modulus and using the
Chinese remainder theorem should indeed be trivial, because understanding
the first problem itself isn't trivial.
 
Of course, maybe the original poser really did NOT know how to find 
square roots modulo a prime and was doing it by some stupid method, like
direct search. However, when someone says that he knows how to do something,
I have no reason to doubt him/her.

In any event, I tend to use the word 'trivial' to refer to something
that is elementary. In this case, understanding the method requires
nothing more than high school algebra.

Implying that the answer to the original question is NON-trivial would
be insulting to the reader, rather than the other way around.

--
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: rgep@emu.pmms.cam.ac.uk (Richard Pinch)
Newsgroups: sci.math
Subject: Re: definition of topological space
Keywords: Topology; Open sets; Continuity
Date: 5 Nov 92 09:44:04 GMT
Organization: Department of Pure Mathematics, University of Cambridge

In article <1992Nov5.033835.5180@leland.Stanford.EDU> 
ledwards@leland.Stanford.EDU (Laurence James Edwards) writes:
>The definition of a topological space is:
>
>    :a set with a collection of subsets satisfying the conditions that 
>    both the empty set and the set itself belong to the collection, the 
>    union of any number of the subsets is also an element of the collection, 
>    and the intersection of a finite number of the subsets is an element 
>    of the collection
>
>What is the purpose of this definition? To the naive reader (such as myself)
>it would seem that just about any set along with one of its subsets and the
>empty set would be a topological space, e.g. it would semm to me that:
>
>{1,2,3} {} {1}
>
>is a topological space. What am I missing here? In one math dictionary
>it is stated that this definition allows one to establish the notion of
>continuity as it applies to functions between topological spaces ...
>I don't see how. Can anyone clue me in?
>
Well, here goes.  The sets in the family are called "open" and you should
think of them as being rather like the open intervals in the real line.
A subset of R is open iff it is the union of a collection of open intervals
(a,b); equivalently if it contains an open interval round any of its points.

The epsilon-delta definition of continuity says that f is continuous iff
for all x, for all e > 0, there exists d > 0 such that 
     |x-x'| < e => |f(x) - f(x')| < d
i.e.
for all x, for all e > 0, there exists d > 0 such that
     f*(f(x)-d, f(x)+d) contains (x-e,x+e)
where f*(Y) is the set of x such that f(c) is in Y.
i.e.
Y open => f*(Y) open

 and this last is the topological definition of continuity.

Incidentally, {1,2,3} {} {1} is a perfectly good family of open sets
for a topology on {1,2,3}: but it has nothing to do with epsilons and
deltas.

Richard Pinch
Xref: sparky sci.math:14634 misc.education:4208
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Newsgroups: sci.math,misc.education
Subject: Re: Is Math Hard?
Date: 9 Nov 92 13:17:08 GMT
Organization: Purdue University Statistics Department

In article <9NOV199206213246@comet.nscl.msu.edu> burtt@comet.nscl.msu.edu (BRIAN BURTT) writes:
>In article <1dku2bINNdar@darksttar.UCSC.EDU>, ask@ucscb.UCSC.EDU (Andrew Stanford Klingler) writes...
>>  P.S.  Burtt does seem perilously close to whining here.  Actually, we
>>   all do, on this and related threads.  It was my impression that this
>>   newsgroup represented a slice of the mathematical "elite".  It would
>>   be very surprising if, given normal childhoods, most of us were *not*
>>   bored and, in retrospect, "underutilized".  Sure, it's a problem -
>>   just not a very shocking one.
>
>	If I seemed to be whining, I apologize profusely.  I consider 
>myself very lucky because I survived the system, with my goals and dreams
>intact.  I've had many friend, however, who were not as lucky.  For them,
>let me make the point, students should be able to realize their full
>potential with the full assistance and support of the educational system,
>rather than having to fight it tooth-and-nail all the way.  The word
>"graduate" should sound a lot less like "battle-worn veteran."
>--Brian Burtt
>


-- 
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)
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math
Subject: Re: What is a knot?
Date: 10 Nov 92 21:08:38 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <COLUMBUS.92Nov9100248@strident.think.com> columbus@strident.think.com (Michael Weiss) writes:
>In article <1992Nov7.212557.24399@galois.mit.edu> jbaez@riesz.mit.edu (John
>C. Baez) writes: 
>
>   Actually, it turns out to be very good to work with framed oriented links.
>   Then we have the marvelously erudite
>
>   5') Hom(0,0) in the free tortile tensor category generated by a single
>   object.
>
>Now that's more like it.  (How come they didn't tell me about this at
>Community Boating when I learned to tie a bowline?)

This reminds me. A while back, I saw a television show where a guy called
'Mr. Knot' demonstated the 'flying bowline'. He tossed a short length of
rope in the air and caught it at both ends. Sure enough, it had tied itself
into a bowline. It was not clear that this wasn't a camera trick, but it
looked like an incredibly smooth camera trick.

So can this be done?

If so/if not, then what are the 'knots' which can be tied in this way?

An interesting aspect of this question is that this is one problem about
knots which knot theory looks powerless to solve. 

Later,
Andrew Mullhaupt
Newsgroups: sci.math,misc.education