HDTenEightyP/NetNews huggingface dataset:
From: rgep@emu.pmms.cam.ac.uk (Richard Pinch)
Subject: Re: Discrete Fields
Organization: Department of Pure Mathematics, University of Cambridge
Date: Fri, 20 Nov 1992 10:12:42 GMT

In article <1992Nov19.153057.18919@news.lrz-muenchen.de> 
stpa@nws.e-technik.tu-muenchen.de  (St. Paul) writes:
>I'm looking for literature on matrix computation on 
>discrete fields  (SVD, eigenvalues, QR-decomposition etc. ..).

I suggest R. Lidl and H. Niederreiter,
Finite fields,
Addison-Wesley/Cambridge University Press


Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: 1+1/2+1/3+1/4+...+1/n
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Sat, 21 Nov 1992 12:52:18 GMT

In article <michaelg.722193567@Xenon.Stanford.EDU> michaelg@Xenon.Stanford.EDU (Michael Greenwald) writes:
 
:(d) Also, since he's looking for the particular "n" that will allow
:the harmonic series to sum to 100, it will be difficult for him to
:proceed from "the smallest to the highest" without having a guess as
:to what "n" is (although one would assume that someone who had taken
 
Since no partial sum of the harmonic series is ever an integer,
it is going to be somewhat difficult to find such an 'n'.

Hint for proof: Let H = 1 + 1/2 + .... 1/n, multiply both sides by
n! and count powers of 2 dividing both sides [or use Bertrand's hypothesis,
i.e. look at the largest prime dividing both sides]

As for the original question; one can get some very good approximation
to H = 1 + 1/2 + .... 1/n  via Euler-MacLauren summation, but the error
term isn't good enough to determine the EXACT n that lets the sum first
exceed 100. You would also need Euler's constant to at least 45 decimal
places.
--
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"
Newsgroups: sci.math
Subject: Re: exp(pi*sqrt(x))
From: harper@kauri.vuw.ac.nz (John Harper)
Date: Sun, 22 Nov 1992 23:55:15 GMT
Organization: Victoria University of Wellington

In article <1992Nov22.124131.17689@husc15.harvard.edu> blom@husc15.harvard.edu writes:
>Aitken once remarked that exp(pi*sqrt(163)) differs from an integer by
>less than 10^-12.  Why and when does exp(pi*sqrt(x)) approximate an integer?
>
See Churchhouse, RF & Muir, TE, J. Inst. Maths Applics 5, 318-328, 1969
"Continued Fractions, Algebraic Numbers and Modular Invariants" for a
summary of this problem. (Should this be in the FAQ list?)

John Harper Mathematics Dept. Victoria University Wellington New Zealand
Newsgroups: sci.math,sci.physics
From: abond@rvgs.vak12ed.edu (Andrew H. Bond)
Subject: Re: Three-sided coin (ANSWER)
Organization: Virginia's Public Education Network (Roanoke)
Date: Thu, 19 Nov 92 08:54:55 GMT

matmcinn@nuscc.nus.sg  writes:
> mueller@schaefer.math.wisc.edu (Carl Douglas Mueller) writes:
> : In the DOVER book "Fifty Challenging Problems in Probability with Solutions,"
> : the 38th problem is called "The Thick Coin."  It is:  "How thick should a coin
> : be to have a 1/3 chance of landing on edge."  
> 
> Is there any country in the world that issues such thick coins so we
> can check this out experimentally? In my travels I found that the
> thickest [relative to diameter] coin is the British pound. But that
> isn't thick enough.

If you are really interested in testing this solution, why not
just get some superglue and glue a bunch of coins together in a
stack. Superglue forms very thin, rigid layers of adhesive so I
wouldn't think that this would affect the outcome
significantly. The trick here is to find coins of the right
ratio, so that a stack of n coins will be the proper thickness.

	--AHB
-- 
Andrew H. Bond, Roanoke Valley Graduate Center
Newsgroups: sci.math.research
From: Allan Adler <ara@zurich.ai.mit.edu>
Subject: Index theory?
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: M.I.T. Artificial Intelligence Lab.
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 23 Nov 1992 16:37:33 GMT



Let k be a field and M a graded vector space over k. Asssume that each
homogeneous part of M is finite dimensional over k. If T is an operator
on the M preserving degrees, then we can define the "trace" of T on M
as a formal power series in an indeterminate u whose coefficient of degree
n is the trace of the operator T acting on the degree n part of M.

Now let f be a homogeneous polynomial of degree q+1 in r variables x_1,...,x_r,
q>2, with coefficients in k and form the Koszul complex of f (or its dual),
i.e. the complex of differential forms  with polynomial coefficients
and where the differential is wedging with the total differential of f.

For each i, let a_i be a nonzero element of k and suppose that the substitution
T: x_i |--> a_i x_i  preserves the homogeneous polynomial f. Then T acts on
the terms of the Koszul complex and I will continue to refer to these
actions by T, as well as the action on the cohomology.

It is alleged that the "trace" of T on the cohomology is
the product over i of (u^q-a_i)/(u^{-1}-a_i)  in case f is nonsingular.

Can someone supply a proof? Presumably this is done by computing the "traces"
on the Koszul complex itself. 

I would also be intersted in knowing of places in the literature where
these kinds of computations are done (including these "traces")
so I can read about it. For example, I am under the impression that
they arise in certain approaches to index theory, but I don't know where.

Ignorantly,
Allan Adler
ara@altdorf.ai.mit.edu

Newsgroups: sci.math.research
From: merlet@cygnusx1.inria.fr (Jean-Pierre Merlet)
Subject: Cayley's theorem
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: INRIA, Sophia-Antipolis (Fr)
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Thu, 19 Nov 1992 08:23:24 GMT
Keywords: Cayley


I am looking for any reference on the following Cayley's theorem:

"a line such that two points C and D on this line are constrained to lie on curves
of degree nc, nd and of circularity pc, pd, the curves not lying in parallel
planes, generates a ruled surface of degree nc(nd-pd)+nd(nc-pc)"

I am especially interested in the exception case where the curves
lie in parallel planes.
From: robby@math.orst.edu (Robby Robson)
Newsgroups: sci.math.research
Subject: West Coast Number Theory Conference
Date: 16 Nov 92 22:45:31 GMT
Article-I.D.: talon.1e98abINNcrc
Organization: Oregon State University Math Department
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu

The 1992 West Coast Number Theory conference will be held in Corvallis, Oregon.
This is an informal, self-supporting, conference welcoming all with an
interest in number theory.

The conference will begin with an evening reception on Friday, December 18,
and run through noon on Tuesday, December 22.  The scientific program will be
determined "Oberwolfach style" - all those wishing to give a talk may do so.
The schedule of talks will be determined on the first night, after we have a
complete list of speakers and titles.  Talks will be held all Saturday, for
half a day on Sunday, all Monday, and for half a day on Tuesday.  These
include problem sessions.  A current list of problems, edited by Richard Guy,
is available from us.

Although the date is rapidly approaching, it is not too late to register.  If
you need registration information, or have any questions about transporation,
housing, etc., please call the LaSell Stewart Center at 503-737-2402 or send
email to robby@math.orst.edu or flahive@math.orst.edu.  
-- 
Robby Robson 			Phone: 503-737-5171
Department of Mathematics	Fax: 503-737-0517
Oregon State University		Email: robby@math.orst.edu
Corvallis, OR  97331-4605
Newsgroups: sci.math.research
From: aaronm@cybernet.cse.fau.edu (aaron meyerowitz)
Subject: Finite axioms of choice
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Cybernet BBS, Boca Raton, Florida
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Wed, 18 Nov 1992 22:52:15 GMT

I've heard this result but can't recall where. Can anyone provide a
reference or related results for other number pairs?

Theorem: Suppose that there is a choice function for 2 element sets
, then there is a choice function for 4 element sets.

Proof: Given a set {a,b,c,d} form each of the 6 2-element subsets
and apply the choice function to it. Each of the four elements is
chosen 0,1,2 or 3 times. If any is chosen 3 times, choose it (it must
be unique). If none is chosen 3 times and one is never chosen, choose
it (IT must be unique. The remaining possibility is that two are chosen 
twice and 2 chosen once. Take the two chosen once (or twice, just decide 
which) and let the chosen element be that chosen by the function for 
that pair/////

So this result might be termed AC2 implies AC4. 
Thanks,
Aaron Meyerowitz
From: volcheck@math.ucla.edu (Emil Volcheck)
Subject: reference sought for plane birational transformation (TeX)
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: UCLA Mathematics Department
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Thu, 19 Nov 1992 01:25:51 GMT

I have been investigating a plane birational transformation specified
in a paper by Max Noether in {\it Mathematische Annalen.}  Noether claims
several properties for it which relate to resolving the singularities 
of a curve into ordinary multiple points.  It is an extension of the 
standard method of quadratic transformations described by Fulton, for instance.

Noether neither justified his claims nor gave any reference to other works
which describe this transformation.  I hope that someone might be able to 
tell me a reference that might relate to this particular transformation.  
Here's the definition in homogeneous coordinates as he gave it. The notation 
might seem a little awkward.  It transforms a curve $f(x_1: x_2: x_3)=0$, 
and the new coordinate system is in coordinates $\xi_i$.  Subscripts 
denote degree.

$$
\xi_1:\xi_2:\xi_3 = x_1 \cdot S_{s-1} : x_2 \cdot S_{s-1} : S_s,
$$
where
$$
S_{s-1}  =  x_3 U_{s-2}(x_1,x_2) + U_{s-1}(x_1,x_2), 
$$
and
$$
S_s      =  x_3 V_{s-1}(x_1,x_2) + V_s(x_1,x_2),
$$
with $U_h,$ $V_h$ homogeneous polynomials of degree $h$.

Thanks in advance!
\bye
--


Newsgroups: sci.math
From: umatf071@unibi.hrz.uni-bielefeld.de (sio)
Subject: Concrete Mathematics Ex 9.67
Date: Mon, 23 Nov 92 18:59:18 GMT
Organization: Universitaet Bielefeld
Keywords: harmonic series

A real question:
  Let Q_n be the least integer m such that H_m > n.
  Find the smallest integer n such that
  Q_n <> floor( exp(n-gamma) + 1/2 ), or prove that no such n exists.

see: Boas & Wrench,
     Partial sums of the harmonic series,
     AMM 78 (1971) 864-870

Torsten Sillke
Xref: sparky sci.math:15411 comp.edu:1932
From: landsbur@troi.cc.rochester.edu (Steven E. Landsburg)
Newsgroups: sci.math,comp.edu
Subject: Re: Minimum standards for math "competency"
Keywords: math competency education university college standards
Date: 23 Nov 92 16:27:55 GMT
Organization: University of Rochester (Rochester, NY)

In article <By6E16.JFx.2@cs.cmu.edu> jmount+@CS.CMU.EDU (John Mount) writes:
>In article <1992Nov23.140653.13522@scott.skidmore.edu>, pvonk@scott.skidmore.edu (Pierre VonKaenel) writes:
>|> Perhaps you haven't visited a technical university lately.  I recall
>|> professors complaining that most of their students are oriental, and
>|> where are the American kids?  I'm not sure about the statement above,
>
>I trying to imply you said the above (I even left a bit of your "I'm not
>sure about that").  But I think this kind of racist crud is intolerable- 
>a good number of the Asian students ARE American kids.
>


I applaud the denunciation of the racist crud.  But it doesn't go far
enough.  So what if they AREN'T American kids?

Steven E. Landsburg
landsbur@troi.cc.rochester.edu
Xref: sparky sci.math:15430 sci.physics:19509
From: israel@unixg.ubc.ca (Robert B. Israel)
Newsgroups: sci.math,sci.physics
Subject: Re: RSA Encryption
Date: 23 Nov 92 21:28:19 GMT
Organization: The University of British Columbia
NNTP-Posting-Host: unixg.ubc.ca
Keywords: cryptography, primes

In <1992Nov23.014117.21496@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU
 (Vaughan R. Pratt) writes:
> John's argument can strengthened as follows (found this even earlier,
>around 1967 when I could still remember 4th year physics).  If the
>distribution of states unfolds according to a linear operator U(t),
>then the future of the universe can be predicted by raising the matrix
>U(1) to any desired power by repeated squaring.  This enables one to
>peek n steps into the nondeterministic future with only log n
>squarings.

>My conclusion at the time was that this showed that the universe can't
>be linear, or we could manufacture time paradoxes.  Now my feelings are
>(i) U(1) is a big sucker, and (ii) should one be able to make out the
>top quark just by staring at U(1)?

"Big sucker" is putting it mildly.   You want to predict the answer you'll
get from a computer that will execute n steps and then halt?  Your Hilbert
space will have to have at least one dimension for each possible state of
the computer, in order to describe its behaviour.  If the computer is not
going to go into an infinite loop, it needs at least n different states.
So just one squaring of U(1), sparse or not, will require at least n steps.
You can't win... 
-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4
Newsgroups: sci.math,comp.theory
From: frank@cs.kuleuven.ac.be (Frank Piessens)
Subject: Question about Finite Limit Sketches and Theories
Organization: Dept. Computerwetenschappen K.U.Leuven
Date: Wed, 18 Nov 1992 11:52:11 GMT

Let S be a finite limit sketch, and let T be its theory.
On page 236 of their book "Category theory for computing science",
Barr and Wells state that it would be incorrect to assume in general
(although it is true in familiar examples) that every node of T
is a limit of a diagram in the graph of S.

Could somebody please point out an example of this? (So give an example
of a sketch, that has a theory in which some node A is not the limit
of a diagram in the graph of the sketch)

Since this is probably not of general interest, please respond by email
to frank@cs.kuleuven.ac.be

Thanks in advance,

Frank Piessens.

Newsgroups: sci.math
From: tim@maths.tcd.ie (Timothy Murphy)
Subject: Re: Gnu Multi-precision Package (GMP):
Organization: Dept. of Maths, Trinity College, Dublin, Ireland.
Date: Tue, 24 Nov 1992 03:16:24 GMT

perkins@tallis.enet.dec.com (Eric Perkins) writes:


>Has anyone used the Gnu Multi-precious Package (GMP)?
>I have been trying to use it and have had not end of trouble
>trying to get simple things to work.

>If anyone else has tried it I would be curious to know if they
>find it is working or not.

I've used it and it seemed to work ok.
Could I (we) see your program.

>Are there any other multi-precision packages out there in the public
>domain?

There are a couple of C++ implementations.
I'm just looking at one in the C++ Answer Book, by Tony Hansen;
Exercise 6.11 reads, 
"Define a class implementing arbitrary precision arithmetic."
(The code from this book is available by anon FTP,
from research.att.com [username: netlib]
directory c++/answerbook .
Nb There is a lot of it!)

C++ seems the natural language for this,
with operator overloading to allow sums
to be written as x+y, etc.


-- 
Timothy Murphy  
e-mail: tim@maths.tcd.ie
tel: +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
Newsgroups: sci.math
From: holmes@garnet.idbsu.edu (Randall Holmes)
Subject: Re: Most important event in the history of mathematics
Organization: Boise State University
Date: Thu, 19 Nov 1992 16:56:33 GMT

In article <z-y1l_j@rpi.edu> fokp@rpi.edu writes:
>How about the discovery of Chaos and Fractals?

It doesn't rate much more than a footnote so far (relative to other
things brought up in this thread).


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
From: dpick@math.uci.edu (Daniel Pick)
Subject: Undergraduate textbooks
Newsgroups: sci.math
Organization: University of California, Irvine
Date: 23 Nov 92 17:48:27 GMT

#include stddisclaimer

Has anyone else noticed that the problems that beset secondary school texts
occur in undergraduate texts as well?  For example, the text for the upper
division analysis course here at UCI is Ross' Elementary Analysis: The Theory
of Calculus.  This book contains many exercises which depend on extensive
algebraic manipulations for their solution, in other words, busywork. 
More often than not, these kinds of exercises obscure the author's points
and 'turn off' otherwise interested students.  How do we get better
texts into the classroom, or conversely, how do we discourage publishers
from printing this stuff?  I, for one, refuse to buy these kinds of texts.

--Dan 8^)
-------------------------------------------------------------------------------
Daniel Pick
University of California, Irvine
Irvine, CA 92715
dpick@math.uci.edu
Xref: sparky sci.math:15302 sci.math.stat:2395
From: hougen@focus.csl.uiuc.edu (Darrell Roy Hougen)
Newsgroups: sci.math,sci.math.stat
Subject: Approximation and Estimation (HELP!!)
Date: 20 Nov 1992 18:40:11 GMT
Organization: Center for Reliable and High-Performance Computing, University of Illinois at Urbana-Champaign
NNTP-Posting-Host: focus.csl.uiuc.edu
Summary: estimation
Keywords: approximation

I need to estimate a 'hill' function between -pi/2 and pi/2.  It must
be 0 at -pi/2 and pi/2 and it must be 1 at 0.  It must increase
monotonically from 0 to 1 on [-pi/2,0] and decrease monotonically from
1 to 0 on [0,pi/2].  It must be smooth, ie., differentiable everywhere
in (-pi/2,pi/2), and have a derivative of 0 at 0.

My first question is, does there exist a basis for functions of the
above form?  In other words, I would like a set of functions {...,
g_k, ...} such that the unknown function f can be approximated
arbitrarily well by a weighted sum of functions from the above set.
In symbols, let f(x) = sum_j a_j g_i_j(x), where i_j are members of an
index set I.

One possible basis that I've been thinking about is (cos(x))^k with k
ranging from -inf to +inf.  All the members of this set satisfy the
conditions given above and it appears that any weighted sum of such
functions will also although it would be nice to prove that.

My second question is, if I have a potential basis, how can I prove
that it is a complete basis?  In other words, how can I prove that any
f of the above form can be approximated arbitrarily well by a sum of
functions in the basis set?

My third question is, can the results be extended to two or three
dimensions?  More specifically, assume that the function I am trying
to estimate, f(x,y,z), satisfies the above conditions in all three
dimensions.  If necessary, assume additional smoothness conditions.
The same goes for the 1-D case above.

In this case, I would like to use seperable functions as a basis.  For
example, let g_ijk(x,y,z) = (cos(x))^i(cos(y))^j(cos(z))^k.  Is this a
basis?  Why or why not?

If you know anything about any part of this question, I would like to
hear from you.  I will post the responses if they are sufficiently
interesting. 

Thanks in advance.

Darrell R. Hougen

Newsgroups: sci.math
From: djoyce@black.clarku.edu (Dave Joyce)
Subject: Re: Square root of a number
Organization: Clark University (Worcester, MA)
Date: 22 Nov 92 20:51:31 GMT

In <1992Nov15.194049.1@woods.ulowell.edu> buxamusaa@woods.ulowell.edu writes:

>Please could someone help me with an assignment that I have to do.
>It is a very general question, though the answer could be quite lengthy.
>"Discuss methods for finding the square root of a number."
>
>What I am looking for are all possible methods/alogrithms for this.

One of the Old Babylonian methods was to interpolate the answer from a table
of squares.  Another was to improve your guess by better guesses:

new guess = (1/2)( old guess + (number/old guess))

Accuracy grows very fast.

-- 
David E. Joyce				Dept. Math. & Comp. Sci.
Internet:  djoyce@black.clarku.edu	Clark University
BITnet:    djoyce@clarku		Worcester, MA 01610-1477
Newsgroups: sci.math
From: hanche@ams.sunysb.edu (Harald Hanche-Olsen)
Subject: Re: 1+1/2+1/3+1/4+...+1/n
Organization: University at Stony Brook, NY
Date: Sat, 21 Nov 1992 18:24:49 GMT

Dave>      1.  Does the worm ever reach the other end of the rope?

Not in its own lifetime, no.  But its descendants will probably evolve
and develop space flight and make it there long before what the
mathematics would suggest :-)

- Harald
Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Square root
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Thu, 19 Nov 92 00:31:29 GMT

In article <n9146070.722056383@waldorf> n9146070@waldorf.cc.wwu.edu (Jacob C Kesinger) writes:
>There was a previous post about ways to find the square root of a number...
>
>Here's my contribution:
>
>Find the Taylor series about a (I used a=1, for simplicity)
>
>(x-1)^.5=1+ (x-1)/2- (x-1)^2/4 +3(x-1)^3/8 -......
>
>This is nasty, messy, ugly, and very likely wrong.  However, it IS
>a method (*grin*)

It'll work, assuming you got the right coefficients in your Taylor
series, as long as |x-1| < 1.  It's pretty good if you just want a few
digits and x is near 1.  

The best use of this formula is to impress undergraduates by working out
in your head, for example, that the square root of 1.23 is about 1.115.


From: daveb@hpgrla.gr.hp.com (Dave Boyd)
Date: Fri, 20 Nov 1992 20:52:00 GMT
Subject: Re: Re: 1+1/2+1/3+1/4+...+1/n
Organization: Hewlett-Packard, Greeley, CO
Newsgroups: sci.math

In sci.math, steve@hubcap.clemson.edu ("Steve" Stevenson) writes:

>Mutter Christoph Johannes <K3032E2@ALIJKU11.BITNET> writes:

>>I've a problem. I have to calculate the sum 1+1/2+1/3+1/4+...+1/n.
>>The result should be 100.

     The fact that I'm posting this here will be a very big hint, but
this reminds me of a problem I worked on once for fun.  It's from the
magazine The Bent, somewhere around 1984.

     Here goes:

     An inchworm (actually a centimeterworm) crawls at the constant
rate of 1 cm/sec along an infinitely stretchable rubber rope.  The rope
is initially 1 kilometer long, and the worm starts from one end at 
time 0.

     At the end of the first second, the rope instantaneously and
uniformly stretches another kilometer.  The worm continues to crawl
at 1 cm/sec.  At the end of the second second, the rope instantaneously
and uniformly stretches another kilometer.  The worm continues to crawl
at 1 cm/sec.

     This process continues, the worm patiently crawling at 1 cm/sec
and the rope stretching 1 kilometer at the end of each second.

     1.  Does the worm ever reach the other end of the rope?

     and

     2.  If he does, how long does it take him to get there?


     Have fun.

     Dave Boyd
     Hewlett-Packard, Greeley, Colorado
     Standard Disclaimers Apply


Newsgroups: sci.math