HDTenEightyP/NetNews huggingface dataset:
From: jburne@spock.uucp (John Burnette)
Subject: Looking for help
Organization: Choate Rosemary Hall
Date: Mon, 4 Jan 1993 14:01:28 GMT

I'm a high school teacher.  Several years ago I asked the following 
question on a math test:

"Give two numbers, x and y, whose sum is rational and whose product is 
irrational."

I got a few answers of the type +pi and -pi, but the one I really liked 
was from a very average student who answered:

x = .10100100010000....
y = .01011011101111....

Obviously x+y=1/9 (and, please, let's not start that thread again...)
but I've always been at a lost about x*y.

Over the years, I've come to *believe* that x*y is irrational, but I've 
been stumped by the proof.  If anyone could direct me to something which
would get me moving again on this problem it would be appreciated.

Thanks.
-- 
John Burnette           // The Opinions expressed above are shareware, they //
Choate Rosemary Hall    // are <<not>> freeware. If you like them and use   //
203-949-0914            // them please send $5 for documentation.           //

Newsgroups: sci.math
From: trahern@telperion.ssc.gov (Garry Trahern)
Subject: homotopy groups for products of spheres into unitary groups
Reply-To: trahern@fremont.ssc.gov
Organization: SSC Laboratory, Dallas, TX
Distribution: sci.math
Date: Wed, 6 Jan 1993 17:53:40 GMT


 I would like to know if there are results for the homotopy group for
the following class of mappings:

 The map of the cartesian product of a two sphere and a three sphere
(S2xS3) into the unitary groups U(N). 

 I know that the 5th homotopy group of U(N) is the set of integers, Z,
for N > 2, but I do not know how to find the results (if they exist)
for maps which are not just spheres. 

 An integral which I believe is a representative of this homotopy
group arises in a physics problem I am thinking about, and if there
are general results, I would appreciate knowing them. 

 Please send replies by email to trahern@fremont.ssc.gov (or to the net).

 thanks, 

 garry trahern
Newsgroups: sci.math
From: muelleru@Mathematik.TU-Chemnitz.DE (Ulrich Mueller)
Subject: Re: proof wanted 2
Organization: University of Technology Chemnitz, FRG
Date: Mon, 11 Jan 1993 13:59:30 GMT

In article <1993Jan8.195646.1694@cc.umontreal.ca> cazelaig@ERE.UMontreal.CA (Cazelais Gilles) writes:


>                                                       n
>Is it true that if  C  is a nonempty closed subset of R  and x is a point not
>in  C  that there exists a point c in C that is closest in C to x.
> 
>i.e. such that:   |x-c'| >= |x-c| for all c' in C.
> 
>If it is true I would appreciate if someone could give me a proof
>of the result.
> 
>                Thanks in advance.

There exists a supremum m of all the distances between x and any other 
element of C. 
For every n hence there is a point x_n from C with d(x_n,x)<1/n.
{x_n} has infinite elements and it is bordered. Therefore it has an 
"agglomeration point" p ( Haeufungspunkt in German ).
It's distance to x is m. (To show indirect).
C is closed, therefore p in C.  

Ciao, Ulrich
From: byrde@imhfhp16.epfl.ch (Olivier BYRDE)
Newsgroups: sci.math
Subject: sum(i=1 to n, i^3) = ??   URGENT !!
Date: 7 Jan 93 13:52:15 GMT
Organization: Ecole Polytechnique Federale de Lausanne


Could someone give me the general formula for this sum ?

Thanks, Olivier
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: Finding best mate
Date: 4 Jan 1993 15:47:43 -0500
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu

 (Jesse Goldman) writes:
>
>     This weekend I saw an interview on T.V. with a mathematician who
>stated that if a woman assumes she is to have 100 suitors in her life then
>her best
>chances of finding an ideal mate would be to reject the first 37 and then
>choose the next one better than the preceeding 37.
>     I am not a mathematician or physicist but I understand that this has
>something to do with e (root of natural log).  Could any of you math dudes
>or dudettes explain this to me?  Thanks in advance.
>

100/e = 37 (approx.).  This is sometimes known as the "secretary problem".
It is Example 8.16 in _Probability and Measure_ by Billingsley.  I won't
write the solution here, since it is a page of computations.  The general
topic that includes problems like this is "optimal stopping".  The classic
text on the subject is _Great Expectations_ by Chow, Robbins, & Siegmund.


-- 
  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)
From: arodgers@dcs.qmw.ac.uk (Angus H Rodgers)
Newsgroups: sci.math
Subject: Re: probability = 0
Date: 8 Jan 93 21:57:07 GMT
Organization: Computer Science Dept, QMW, University of London

In <1icir2INNn12@master.cs.rose-hulman.edu>
goddard@NeXTwork.Rose-Hulman.Edu (Bart E. Goddard) writes:

>Even though the probability of an event is zero, it can happen.
>(The probability that I would attain this exact physical stature was zero,
>and yet, here I am, living proof...) [...]

Mathematical models of people may have exact physical statures, but
people don't.

It's not that the event in question has probability 0; it just isn't
well defined.
--
Gus Rodgers,  Dept. of Computer Science, |  [Profound thought for 1993]
Queen Mary & Westfield College, Mile End |     "Philishave is about
Road, London, England.   +44 71 975 5241 |      personal choice" 
E-mail (JANET):   arodgers@dcs.qmw.ac.uk |             -- TV advert
From: mstankus@oba.ucsd.edu (Mark Stankus)
Newsgroups: sci.math
Subject: sci.math.* groups
Date: 9 Jan 93 04:48:48 GMT
Distribution: usa
Organization: Mathematics @ UCSD


In article <C0Jn9z.GqK@news.cso.uiuc.edu> chappell@symcom.math.uiuc.edu (Glenn Chappell) writes:
>[munch]
>>sci.math.algebra  junior year math major (college) algebra 
>>sci.math.analysis junior year math major analysis 
>
>Whoa...careful, there. Algebra & analysis don't end at the junior year.
>Lot's of people make a living studying these subjects. Perhaps what
>you're really thinking of is "sci.math.undergrad"?



Yes, I meant "sci.math.undergrad. I never intended to imply that algebra
or analysis is just an undergraduate activity. In fact, I am just
finishing a PhD in analysis.



>
>[munch]
>
>Well, here's a completely different idea:
>
>There is enough cross-over between the various fields of math, and (at
>this point, IMHO) light enough traffic to make splitting sci.math along
>algebra, etc. lines not very worthwhile, and perhaps even
>counter-productive.
>
>On the other hand, I have noticed that there are essentially two types
>of threads on this group: those that deal with mathematics and those
>that deal with people (e.g. the teaching of math, US vs. Japan, quirks
>of famous mathematicians, etc.) The latter is somewhat like your
>"sci.math.culture" idea, although there is probably a better name for
>it.

Sounds good to me.





>
>So, at this point it might be worthwhile to split off *one* group.
>
>				Glenn Chappell  <><


Mark Stankus







-- 
mstankus
mstankus@oba 
Newsgroups: sci.math
From: tao@mace.Princeton.EDU (Terry Tao)
Subject: Help with Tchebyshev polynomials, please... 
Originator: news@nimaster
Organization: Princeton University
Date: Mon, 4 Jan 1993 15:00:52 GMT


I've been toying with the problem of trying to maximize certain semi-norms
on the space of polynomials of degree n or less which are bounded in
modulus by 1 on the interval [-1, 1], i.e. the set of all f of degree \leq
n such that |f(x)| \leq 1 for all x in [-1, 1].  The semi-norms I have in
mind are things like the following: the L^p norm of f on the interval [1,
1], the derivative of f at a point x_0, the value of f at a point x_0
(where x_0 need not be inside [-1, 1]), various coefficients of f, and so
on.  (actually some of these are not quite seminorms, you have to take the
modulus of them first, but this is not important).  In most of these cases
I found that a maximum exists, because the space in question is convex,
closed, and "bounded" in a sense, and the extremal value of f must attain
the values -1 and 1 at least n times in the interval [-1, 1] and no more
than n+1 times.  If f attans the values -1 and 1 n+1 times, then I can show
f is essentially a Tchebyshev (sic?) polynomial, f(x) = cos(n arccos(x)),
which is nice.  But how does one characterize the polynomials f which only
attain the values -1 and 1 n times on the interval [-1, 1]? Some of these
polynomials are merely dilations of the Tchebyshev, but others are slightly
different.  I have some characterizations of them, but they involve cosines
of elliptic functions and this is not a friendly characterization to work
with.  I also have a recursion formula for the coefficients but they are
also very messy to work with.  Anyone know of a nice characterization of
these polynomials, or at least be able to give me a reference?

Email is preferred.  Thank you.

Terry
From: cadet@spade.rutgers.edu (Uniquely TiJean)
Newsgroups: sci.math
Subject: knots
Date: 7 Jan 93 05:44:03 GMT
Distribution: sci.math
Organization: Rutgers Univ., New Brunswick, N.J.

How does one distinguish knots of different types in euclidean sapce of dim
3 like R3 ?

Say you have 2 non-intersecting circles ( they're not inside each other)
and two circles, one inside the other.

What about when one has two circles, one is inside the other and loops around
it n times? 

JC
Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: proof wanted 2
Organization: Computer Science Department,  Stanford University.
Date: Sat, 9 Jan 1993 22:24:15 GMT

In article <1993Jan8.195646.1694@cc.umontreal.ca> cazelaig@ERE.UMontreal.CA (Cazelais Gilles) writes:
>
>                                                       n
>Is it true that if  C  is a nonempty closed subset of R  and x is a point not
>in  C  that there exists a point c in C that is closest in C to x.
> 
>i.e. such that:   |x-c'| >= |x-c| for all c' in C.

I've been trying to understand quantum mechanics lately, and knowing
how to prove things like the above seems to be a prerequisite, so let
me try my hand at it.

Let t be the distance of some point of C from x, and let r be the
infimum of all such distances.  If t=r we are done.  Otherwise let B_s
denote the set of points distant at most s from x, a closed ball, and
K_s its open complement.  No finite set of K_s's with s>r can cover
C&B_t, whence by compactness of C&B_t the union of all such K_s's
leaves some point of C&B_t uncovered, the desired closest point.
-- 
Vaughan Pratt				There's safety in certain numbers.
Newsgroups: sci.math
Subject: Re: Number of numbers (Was Re: PI)
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 5 Jan 93 23:33:07 GMT
 <1993Jan5.020509.18268@infodev.cam.ac.uk> <1993Jan5.192626.10661@oracle.us.oracle.com>
Organization: Beckman Instruments, Inc.

In <1993Jan5.192626.10661@oracle.us.oracle.com> mfriedma@uucp (Michael Friedman) writes:

>In article <1993Jan5.020509.18268@infodev.cam.ac.uk> gjm11@cus.cam.ac.uk (G.J. McCaughan) writes:
>>In article <1993Jan4.232723.3685@smsc.sony.com> markc@smsc.sony.com (Mark Corscadden) writes:
>>>1)  ========== primitive recursive functions
>>>    The are the functions which you can compute in a typical general-
>>>    purpose computer language, except that before you enter any loop
>>>    you must first compute an upper bound on the maximum number of
>>>    times you will be allowed to pass through the loop, and must not
>>>    exceed this pre-computed upper bound.

>>  [description of "partial recursive", "total recursive"]

>>>Suppose you define a real number to be "primitive recursive" if there is
>>>a primitive recursive function which computes the Nth decimal digit as
>>>a function of N.  Same for a "total recursive" real number.

>>>Does anyone know whether PI is primitive recursive by the definition above?
>>>If not, this fact seems to capture a bit of Samuel's "gut feeling" that
>>>his example SUM(10^-(n!)) is "much simplier to calculate" than PI, since
>>>SUM(10^-(n!)) is clearly primitive recursive.

>>Sorry; pi is primitive recursive. Take any of the squillions of formulas
>>for pi which give error terms which actually give bounds on how far wrong
>>a given approximation is. (For instance, though I wouldn't dream of
>>actually computing pi with this, pi=4(1-1/3+1/5-1/7+1/9...) where the
>>error incurred by stopping at any given point is at most equal to the
>>size of the next term.) Then compute how many terms you need to take to
>>get the first N decimal places, and what sort of precision you need to
>>use; and run the algorithm that far. No unbounded loops in sight.

>Is this really true?  After all, we know that there are arbitrarily
>long strings of consecutive 0s and 9s in the decimal expansion of PI.
>What precision to I need to compute the Nth digit of PI if that Nth
>digit is at teh beginning of such a sequence?  After all, you can
>change the value of digit N by adding (for 9s) or subtracting (for 0s)
>and arbitrarily small value if the string of identical digits is long
>enough.

Well, actually, we don't know that there are arbitrarily long strings of
consecutive 0s and 9s in the decimal expansion of Pi.
--
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.
Newsgroups: sci.math
From: balden@wimsey.bc.ca (Bruce Balden)
Subject: Re: sum of eigenvalues of a covariance matrix = trace ?
Organization: Wimsey Information Services
Date: Sat, 9 Jan 1993 23:23:23 GMT

In article <1993Jan9.194845.8567@noose.ecn.purdue.edu> kavuri@lips1.ecn.purdue.edu (Surya N Kavuri ) writes:
>  Is the sum of the eigenvalues of a covariance matrix equal to 
>   its trace ?
Properly interpreted, yes.

For any two square matrices A, B of the same size, tr(AB) = tr(BA)

Therefore tr((inv(A)*B)A) = tr(A*inv(A)*B) = tr(B).

Thus, tr(B) is invariant under similarity transformations.
The diagonalization of B contains the eigenvalues (repeated according
to multiplicity) and is similar to B. Therefore, it has the same trace.
Newsgroups: sci.math
From: cjh@tinton.ccur.com (Christopher J. Henrich)
Subject: Re: Frankly,my dear......was: Fermat's Last Theorem
Organization: Concurrent Computer Corp., Tinton Falls, NJ
Distribution: usa
Date: Fri, 8 Jan 1993 22:19:34 GMT

In article <1993Jan7.054017.25511@leland.Stanford.EDU> ilan@leland.Stanford.EDU (ilan vardi) writes:
[Why Fermat's Last theorem, the twin primes conjecture, etc. are 
considered important]
>
>In the case of 3x+1, one understands why it should hold but a proof
>would probably explain a lot about what is going on.
>
Does one understand why the 3x+1 conjecture should be true?  This
is news to me.  Can anyone explain the heuristic reasoning, or
point me to a reference?

Thanks in advance,
Chris Henrich
Newsgroups: sci.math,rec.puzzles
From: karr@cs.cornell.edu (David Karr)
Subject: Re: Marilyn Vos Savant's error?
Organization: Cornell Univ. CS Dept, Ithaca NY 14853
Distribution: na
Date: Mon, 4 Jan 1993 06:13:03 GMT

In article <1992Dec31.203934.1@stsci.edu> zellner@stsci.edu writes:
>  A.  "I have two children, and one of them is right over there."  
>[...]
>
>You look and see a boy.  What's the chances that they are both boys?
>Enumerate the cases:
>
>   Boy - Boy  [1]
>   Boy - Girl [2]
>   Girl- Boy  [3]
>   Girl- Girl [4]
>
>[...] But in case A you can only rule out the 
>fourth combination, and the probability is 1/3. 

Bad analysis.  Certainly either case 2 or case 3 might be true, but
taking into account *all* encounters in which statement A is made and
case 2 is true, half the time you will see a girl (assuming that which
child is "right over there" is not correlated with birth order).
Likewise for case 3.

That is, you will encounter case 1 twice as often as either one of
case 2 or 3, yielding P(other=boy)=1/2 overall.

I assume you're just trying to rephrase the problem so that the
encounter produces nothing other than an indicator variable for
the condition "at least one boy," but I don't think you succeeded.

>But suppose someone else looks, and says "at least one of them 
>is a head."  Then the probability that BOTH are heads is 1/3.
>If you don't believe it, try a few hundred tosses.

You'd better not try this if I'm the person who gets to look.  Or if
Monty Hall is.  I might choose to just say, "Nice day, isn't it," in
all the cases where there are *not* two heads, changing your desired
conditional probability from 1/3 to 1.  (And given Monty Hall's
response to Marilyn vos Savant, I suspect he'd do something similarly
perverse.)

-- David Karr (karr@cs.cornell.edu)



Newsgroups: sci.math
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Subject: Re: transfinite real numbers? (...probably not. :( )
Keywords: well defined?
Organization: Purdue University Statistics Department
Date: Tue, 12 Jan 1993 17:46:30 GMT

In article <1993Jan12.162012.4950@maths.tcd.ie> tim@maths.tcd.ie (Timothy Murphy) writes:
>edgar@math.ohio-state.edu (Gerald Edgar) writes:

>>*  As Cantor claimed, his transfinite numbers exist concretely. I wonder if
>>* a process similar to the construction of real number (e.g.Dedekind cut)
>>* can be extended.


>>This sounds like J. H. Conway's "surreal numbers".  His text on it is
>>called _On Numbers and Games_.  (Part Zero is on numbers.)

>I've never understood why Conway's definition of number has not caught on.
>Why isn't it taught in universities everywhere,
>instead of Dedekind sections or Cauchy sequences?
>Is it just that mathematicians are conservative?

>With Conway's definition of number,
>infinitesimals are real quantities, not limits.
>Surely this would make the teaching of calculus much easier?

The problem is that not everything can be done with it.  This differs from
non-standard analysis, in which nothing really new can be done, but the old
things can be looked at in a new light.  But in either, one still has the
problems of limits, Dedekind cuts, and Cauchy sequences.  There are different
definition of theses, and the limit of a sequence includes using the added
integers as well.
-- 
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: hibrown@sarah.albany.edu (Herb Brown)
Subject: Re: sum(i=1 to n, i^3) = ??   URGENT !!
Organization: State University of New York at Albany
Date: Thu, 7 Jan 93 16:00:28 GMT

In article <1993Jan7.145215@imhfhp16.epfl.ch> byrde@imhfhp16.epfl.ch (Olivier BYRDE) writes:
>
>Could someone give me the general formula for this sum ?
>
>Thanks, Olivier


 sum( i^3 , i = 1 .. n )  =  (n^4 + 2*n^3 + n^2)/4

Herb


-- 
Herb Brown  Math Dept  The Univ at Albany  Albany, NY 12222  (518) 442-4640
Director,                                          hibrown@math.albany.edu  
Computer Assisted Instruction in Mathematics       hibrown@csc.albany.edu
--------------------------------------------------------------------------
Newsgroups: sci.math
From: cazelaig@ERE.UMontreal.CA (Cazelais Gilles)
Subject: proof wanted
Organization: Universite de Montreal
					               n		
Date: Fri, 8 Jan 1993 19:53:23 GMT

i.e. such that:   |x-c'| >= |x-c| for all c' in C.

If it is true I would appreciate if someone could give me a proof 
of the result.

		Thanks in advance.


Newsgroups: sci.math
From: pfennige@scsun.unige.ch (PFENNIGER Daniel)
Subject: Projections of "semi-transparent" fractals
Keywords: fractal
Organization: University of Geneva, Switzerland
Date: Fri, 8 Jan 1993 14:12:01 GMT

Suppose a fractal set embedded in some metric space 
(concretely I'm thinking about a cloud in 3D space).  

First I want to determine the typical properties (say the fractal 
dimension) of a *projection* of the set (not a section).  

Second I would do the same for a semi-transparent fractal, i.e.
the density of the projection is a funtion of the previous 
projection. 

Could somebody point me towards the appropriate mathematical 
tools to calculate the above projections?

Thanks, Daniel
From: goddard@NeXTwork.Rose-Hulman.Edu (Bart E. Goddard)
Newsgroups: sci.math
Subject: spell this?
Date: 11 Jan 1993 23:29:00 GMT
Organization: Computer Science Department at Rose-Hulman
Reply-To: goddard@NeXTwork.Rose-Hulman.Edu (Bart E. Goddard)
NNTP-Posting-Host: g214-1.nextwork.rose-hulman.edu


The raging debate here is the correct spelling of (choose one and 
supply a reference):

a) nilpotentcy
b) nilpotency
c) other

As in "the degree of nilpotency of a thing."

--
Bart Goddard   
Math. Dept. Rose-Hulman Inst. of Tech.
goddard@nextwork.rose-hulman.edu
From: rvanhorn@nrtc.nrtc.northrop.com (Bob Van Horn <rvanhorn>)
Newsgroups: sci.math
Subject: Help Needed on 10 Serious Math Problems
Date: 4 Jan 93 02:41:42 GMT
Reply-To: rvanhorn@nrtc.nrtc.northrop.com (Bob Van Horn <rvanhorn>)
Distribution: usa
Organization: Northrop Research & Technology Center, Palos Verdes, CA

Not sure of circumstances but am sure that they are way over my
mead.  Anybody out there who'd consider helping, please e-mail me
at address below, or call at number shown below.

bob van horn
phone (213)600-3991 (have voice mail on that line)
e-mail: rvanhorn@nrtc.northrop.com
From: louis@aplpy.jhuapl.edu (Louis Vasquez)
Newsgroups: sci.math
Subject: probability = 0
Date: 5 Jan 1993 10:55:19 -0600
Organization: UTexas Mail-to-News Gateway
NNTP-Posting-Host: cs.utexas.edu

Hello,
	My friend and I got in a debate about the probability of 
something happening.  The event was that another friend of ours
could win the superbowl.  Not that he would play on a winning team
in the superbowl but that he would win it.
	My thinking is that anything at all is possible and there are
ways that this could happen however unlikely they are.  For example
the rules of the superbowl could change to allow a single person to 
be a team and this friend of ours would by some incredible luck, win it.
I also say that there are many many other ways that it is possible.  
Therefore I say there is a probability of greater than zero of this
happening.
	His arguments (although I must admit I probably don't fully 
understand them) are that by definition a single person cannot win
the superbowl.  And the case that I gave is a finite case in an 
infinite number of possible cases and therefore the probability is
n devided by infinity, a limit which approaches zero.  Therefore he
says there is a probability of exactly zero that this will happen.
	Anyone care to comment?
	Lou,
	louis@aplpy.jhuapl.edu
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math
Subject: Re: Rounding Rules
Date: 5 Jan 93 04:55:12 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <1993Jan4.142226.9639@iscsvax.uni.edu> kraai4712@iscsvax.uni.edu writes:
>My personal favorite rounding rule is to round anything that has {0,1,2,3,4} in
>the "rounding place" "down" and anything that has {5,6,7,8,9} "up".  So as in
>the previous example, 1.13500 would be rounded to 1.14 and 1.14500 would be
>rounded to 1.15.

>To this my teachers always gasped that this method was biased, and no amount of
>pursuasion using the above-mentioned sets could pursuade them.

Nor should it. As in all cases like this, Knuth's _Art of Computer Programming_
is definitive. 


It is somewhat more work to round to nearest even, but it is often preferred.
Given that most computers can do floating point arithmetic in one cycle, the
work of rounding is not onerous.

Later,
Andrew Mullhaupt
Newsgroups: sci.math