HDTenEightyP/NetNews huggingface dataset:
From: dik@cwi.nl (Dik T. Winter)
Newsgroups: sci.math
Subject: Re: Rounding Rules
Date: 5 Jan 93 23:54:15 GMT
Organization: CWI, Amsterdam

(Thrown away all attributions to not confuse :-).)

 > >> The problem with many of the "rounding rules" is that they are biased.
 > >> In the IEEE round of rules, an attempt is made to make rounding as unbiased
 > >> as possible. Round to even is specified as the rule of choice, since it
 > >> avoids many of the problems of "round to infinity" or "round towards zero".
 > 
 > >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.
 > 
 > Well, it IS biased.  Rounding n places is biased by +1/2 10^-n.
 > 
It depends on what context you are using.  It is true when rounding real
numbers (as in mathematics).  It is not true when rounding floating point
numbers.  When considering floating point arithmetic (with finite precision)
you should not consider all infinite precision results before rounding with
equal probability.  Rather you ought to consider the range of possible
inputs and the operation involved and conclude from that point whether the
rounding is biased or not.

However, this leads to a more interesting case: is unbiased rounding
in floating point arithmetic superior to biased rounding?  The answer
is *no*, there are unbiased roundings that are worse than simply
truncating to zero.  While the mean is correct, an actual result can
be worse.  An example provides the multiplication on the Cray.  (Now
bear with me, this is from memory so some numbers may be wrong, but
the way it works is like this.)  When multiplying two numbers (with
48 bit mantissa) together, the Cray does not calculate all 48*48 bits
of intermediate results to add them together.  Rather in the 48*96
array to hold the intermediate results it skips the last 16(?)
positions and adds 9 to the 80(?)th position.  The number 9 is indeed
the mean value of the omitted bits considering all inputs equally
probable.  When all is added together and additional 3 is added to
positions 50 and 51.  Next, if the leading result bit is zero the
number is shifted one position (adjusting the exponent).  Finally
the result is truncated to 48 bits.  It is not easy to see, but the
algorithm delivers an unbiased result, but it can be wrong by more
than 1 bit in the least significant position.

What considering the IEEE standard, it is easy to see that with regards
to biasedness they could have chosen either round to even or round to
odd.  The reason round to even was chosen is that it actually diminishes
the need to round on subsequent operations.
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland
home: bovenover 215, 1025 jn  amsterdam, nederland; e-mail: dik@cwi.nl
From: mstankus@oba.ucsd.edu (Mark Stankus)
Newsgroups: sci.math
Subject: Let's add some sci.math.* groups
Date: 8 Jan 93 15:19:59 GMT
Organization: Mathematics @ UCSD

I recently read a post asking that a sci.math.graph group be started.

I think that we should start several new newsgroups so that 
one does not have to wade through many messages each day on 
topics which are not of interest to us. Here are my suggestions:

sci.math.beforecalculus  algebra, geometry,... which is learned
                         before learning calculus
sci.math.rec      recreational math
sci.math.culture  all of the Nobel prize stuff could be psoted here
                  and similar topics
sci.math.number   number theory, Pi,Fermat's Last Thm,...
sci.math.calculus calculus stuff
sci.math.algebra  junior year math major (college) algebra 
sci.math.analysis junior year math major analysis 
sci.math.discrete combinatorics,graph theory,...

Please suggest others if I missed some. Perhaps reading the 
FAQ to come up with an appropriate breakdown would be good.
(No flames please if I missed some potential group names or 
some interesting parts of math.)

It seems to me that either 
(a) the new newsgroups are not used very much (does not cost net much)
(b) the net is used the same amount for postings , but you don't
    have to even consider the ones which are irrelevant.
(c) more postings occur in the specific areas OR
(d) lots of cross posting occurs.

(d) is bad (this costs the net more money, right?). I recommend that we keep 
sci.math or start sci.math.misc
and get on people's cases if they cross post too much.

All of the computer oriented news groups are broken up into multiple groups,
this one should be also.

Mark Stankus
-- 
mstankus
mstankus@oba
From: landman@hal.COM (Howard Landman)
Newsgroups: sci.math
Subject: Re: 2 questions (GRE Subject test)
Date: 10 Jan 93 05:43:59 GMT
Organization: HaL Computer Systems, Inc.
NNTP-Posting-Host: bang.hal.com

In article <1992Dec25.040617.26522@news.columbia.edu> sla7@cunixa.cc.columbia.edu (Seth Louber Antiles) writes:
>In general, for f a polynomial, for what g will f(g(x))=f(x) for all x
>*not* imply that f is a constant? Appreciate simple examples.

Well, clearly g(x) = x places no restriction on f at all, and g(x) = k - x
for any real k only implies that f is symmetric around k/2 (and therefore of
even degree).

	Howard A. Landman
Newsgroups: sci.math
From: dbradley@symcom.math.uiuc.edu (David Bradley)
Subject: Re: Euclidean Domain
Organization: Math Dept., University of Illinois at Urbana/Champaign
Date: Tue, 12 Jan 1993 16:02:02 GMT
Keywords: Elementary Proof

In article <HAMMOND.93Jan3134111@annemarie.albany.edu>, hammond@csc.albany.edu (
>William F. Hammond) writes:
>
>Hungerford's problem #8 Chapter III section 3 (in my edition) asks to show
>a + b(1 + sqrt(19)) a,b \in Q is a principal ideal domain but not a
>Euclidean domain.  

For the record, that should read {a + b(1 + sqrt(19))/2 : a,b \in Z} or
equivalently, R := { a + b sqrt(19) : a,b \in (Z + 1/2) }.  See the
November 1988 issue of the Monthly for a short, elementary proof that 
the above ring is a PID but not an ED (not just not Euclidean in the
norm, but that no Euclidean function works).

>Any information netters know about this, I would be interested in hearing
>about.  What is the "folklore" on the subject?

I've been told that making the above an exercise, with no hints or
similar disproofs in the text was "irresponsible".   

-d
Newsgroups: sci.math
From: shivakl@ctrvx1.vanderbilt.edu (LAKSHMANAN SHIVAKUMAR)
Subject: Need help with Hopscotch Process
News-Software: VAX/VMS VNEWS 1.41    
Organization: Vanderbilt University
Date: Sun, 10 Jan 1993 02:45:00 GMT

Hi,

I am working on a project that requires a numerical solution
to a second order pde in two space dimensions and one time
dimension.  The pde involves a mixed derivative.  A friend
of mine recommended the use of Hopscotch process for solving
this pde.  As I am not a technical person in Math I am finding
it hard to follow the algorithm for this process.  I would be
extremely thankful if anyone out there could educate me on how
this algorithm works.  References to text books discussing this
method would also be very helpful.

Thanks in advance.
Regards
Shiva
From: arodgers@dcs.qmw.ac.uk (Angus H Rodgers)
Newsgroups: sci.math
Subject: Re: Closed forms for sums
Date: 9 Jan 93 16:37:32 GMT
Organization: Computer Science Dept, QMW, University of London

In <964.165.uupcb@hal9k.ann-arbor.mi.us>
james.jones@hal9k.ann-arbor.mi.us (James Jones)  writes:

>Some things I remember:

>1.      All closed forms had  n  and  (n + 1)  as a factors.

>                            2              2
>2.      All odd j > 1 had  n   and  (n + 1)   as a factors.


>3.      Add even j > 2 had interesting coefficients on the terms in
>        the last factor.
>        (at least the last factor the way Derive factored it)

> [...]  my question is, how does one go about determining a closed
> form for a sum? [...]

>Any explanation or references (explanation preferred) would be
>appreciated.

Perhaps this should be an FAQ?

I got interested in the problem when I was at school in the 60s,
and wrote it up then. Unfortunately I later destroyed my notes,
but I still kept copies of the general formulae for these
factorisations, in terms of the Bernoulli numbers; and when the
topic came up in sci.math recently, I reconstructed some of the
proofs, and additionally reassured myself that the simplicity of
the coefficients was at least partly a consequence of von Staudt's
theorem on the Bernoulli numbers (see Hardy & Wright).

I've got a LaTeX file with most of this stuff in it, but it's a
bit long to post to sci.math, and it still needs some work.

I've also got some pari-gp code which you can use to calculate
enormous coefficients (expressed as products of powers of primes)
to your heart's content.

Briefly, you need to prove, first, that the power sum, satisfying
a 1st-order difference equation, is a polynomial in n, and second,
that it is also an even or odd polynomial in n + (1/2). You can then
get a pair of recurrence relations for the coefficients in the
factorisation -- which is more neatly expressed in terms of n(n + 1)
than in terms of n itself. (Something that Derive didn't notice.)

(I can't face trying to work through the messy details of this part
of the work again; and in any case there may be some more elegant
approach than the pedestrian route which I took, using only school
algebra.)

The recurrence relations can be solved, in two ways. One way seems
useful for estimating the numerical size of the coefficients (which 
can be proved to alternate in sign, a fact which you didn't mention),
but I haven't followed this up, and it may not be easy. The other
solution is in terms of the Bernoulli numbers, and is useful for
getting at the more number-theoretic properties of the coefficients.

I'll post the document to you when I've edited it a bit. I suppose
it's not absolutely too long to post to sci.math, if there's enough
interest.
--
Gus Rodgers,  Dept. of Computer Science, | Cinema's worst romantic dialogue? --
Queen Mary & Westfield College, Mile End | "Mechagodzilla's brain is  installed
Road, London, England.   +44 71 975 5241 | in my stomach!" "I don't care if you
E-mail (JANET):   arodgers@dcs.qmw.ac.uk | are a cyborg  --  I still love you!"
Xref: sparky sci.math:17665 rec.puzzles:8206
From: rivin@SAIL.Stanford.EDU (Igor Rivin)
Subject: Re: Help find optimum display of directed graphs
Organization: Computer Science Department,  Stanford University.
Date: Thu, 7 Jan 1993 17:35:43 GMT

This is a vast area, below are a couple of recent references that may
have back pointers that you can use.

See a paper by Tamassia et al in Journal of Discrete and Computational
Geometry (one or two issues ago) on displaying planar directed graphs.

Spring-based schemes have received a lot of attention; Steve Skiena
(of SUNY Stony Brook) has a Mathematica implementation described in
his  book "Combinatorica," I believe he also distributes the code.


Newsgroups: sci.math
From: scotj@pinet.aip.org (scot_shinderman)
Subject: Points on Sphere
Organization: American Institute of Physics
Date: Mon, 11 Jan 93 18:45:45 GMT


  Given n points that exist on the unit sphere in 3-space (s2), what is
their distribution such that the sum of distances from each other is
maximized.
   | Pn | = 1.0;
   maximize   sum(j = 1,n)  sum(i = 1,n)  | Pi - Pj |

  any ideas?  If the points are considered charges, does the answer
correspond to a point of equilibrium?
- thanks!
- scot shinderman (scotj@aip.org)
Newsgroups: sci.math
From: dunne@math.okstate.edu (Ed Dunne)
Subject: Re: Fields Medal Winners
Organization: Oklahoma State University, Math Department
Date: Fri, 8 Jan 1993 17:52:32 GMT

erwin@trwacs.fp.trw.com (Harry Erwin) writes:

>At one time, I was under the impression that Bott and Artin had won the
>Fields Medal. It seems I'm wrong. Can someone confirm or deny? 

>This impression is associated with one of those stories. I understand
>Artin, Atiyah, and Bott were the first three at the math tripos one year
>at Oxford (1950?). Apparently it made quite an impression.

I do not know about the tripos, but it is true that Bott never won a
Fields Medal.  He may have been the advisor to more Fields medalists
than anyone else, however.

Edward Dunne
Math Dept
Oklahoma State University
Newsgroups: sci.math
From: eric@cadkey.cadkey.com (Eric Holtman)
Subject: Game theory / Probability Question
Organization: cadkey
Date: Tue, 12 Jan 1993 16:12:50 GMT



O.K., here's a probability/game theory question...... being mostly
ignorant of such things, I'll toss this out to the net.

There's a game with N players.  Each player antes one unit into the kitty.
Each player is randomly dealt a card from a deck with the numbers 1 to 1000
on it.  Then, each player decides whether to stay or drop out.  If you
stay and have the highest number of those who stayed, you win the kitty.
If you stay and are beaten by someone else who stayed, you match the kitty.
If you drop out, there is no penalty.  Everyone decides to stay or drop
or simultaneously, so you don;t know how many stayed until after you make
your choice.

You never put more than 20 units into the kitty, even if you're matching
a higher amount.  The kitty grows (obviously) when 3 or more players stay,
since one rakes in the winnings, and two others match (up to 20 units each).

To complicate matters (ignore this rule if you wish), if *no player* stays,
the player with the highest card matches the kitty (subject to 20 unit
limit).

My (uneducated) analysis:

The pot, when below 20 units is always offering you even money, i.e.
you risk 4 units to win 4 units.  So, how high does your number have to 
be to justify staying in?

In the cases where the pot has grown beyond 20 units, your getting
odds on, i.e. risking only 20 to perhaps win 50 units.  Does this mean the
number you draw might not need to be so high to stay?  Again, how high should
your number be to justify staying in??

Also, obviously, the number of players should effect your choice (i.e.
a number such as 700 is very good when N=2, not so great when N=100).

Why do I care?

Well, the game is actually played with a real deck of cards, but I'm just
toying with finding the best strategy for winning.  I assume I can rank the
different hands to get to 1 to 1000 (or some similar spread).  Just curious,
and all.

Any probability gurus out there have some insights??
-- 
Eric J. Holtman       |
Product Development   | "You fat, bloated, Eeediot!!!"
Cadkey, Inc.          |                      
                      |    - Ren Hoek ("that's Ho-ak, not HOKE!!")
Newsgroups: sci.math
From: yeomans@austin.onu.edu (Charles Yeomans)
Subject: Re: Frankly,my dear......was: Fermat's Last Theorem
Date: Fri, 8 Jan 1993 16:14:25 GMT
Distribution: usa
Organization: Ohio Northern University

In article <1993Jan7.021308.10566@nuscc.nus.sg>, matmcinn@nuscc.nus.sg (.) writes:
> 
> edgar@function.mps.ohio-state.edu (Gerald Edgar) writes:
> : 
> : * I thank those who were polite about it.
> : 
> : Mathematicians receive so many of these amateur attempts (with elementary
> : mistakes in them) that they tend to get impatient with them...
> : 
> I wonder if this might not be an occasion for some soul- searching on the
> part of professional mathematicians. X^n+ y^n = Z^n has no solutions....  
> frankly, who cares? It's recreational mathematics, right? The only reason

Actually, the equation x^n + y^n = z^n does have solutions, for every n.

Charles Yeomans
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math
Subject: Re: Looking for help
Date: 4 Jan 93 21:28:20 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <1993Jan4.140128.20155@choate.edu> jburne@spock.uucp (John Burnette) writes:
>"Give two numbers, x and y, whose sum is rational and whose product is 
>irrational."

>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.

It isn't entirely clear but I suppose that x is a Liouville* number, in
which case it is transcendental. In particular no equation of the form

	x * (1/9 - x) = r

holds for rational r. Note that you don't need transcendence - suppose
that irrational x and y have rational sum s and product p. Then

	x * y =  x * (s - x) = p

and x is a quadratic irrationality. So you can produce an answer to your
question by choosing x to be any nonquadratic irrationality. 

Later,
Andrew Mullhaupt

*Or a Thue-Siegel-Roth number, etc. A lot of stuff might apply here and
Mahler's book on Transcendental Numbers (Lecture Notes in Math)
Newsgroups: sci.math
From: muttiah@thistle.ecn.purdue.edu (Ranjan S Muttiah)
Subject: Re: Bayes' theorem and QM
Keywords: infinite
Organization: Purdue University Engineering Computer Network
Date: Tue, 5 Jan 1993 16:09:42 GMT

In article <5915@osc.COM> jgk@osc.COM (Joe Keane) writes:
>If you think so, please show me some pieces that you could make out of
>something real like wood, such that that they almost fit together into either
>one sphere or two.  Also please show me a computer chip that can store an
>infinite amount of information.  In fact i'll be generous, you only have to
>store an arbitrarily large amount of information.

Actually, there was an interesting snippet in the Sci. Am ('88 I believe)
about a mathematician who had molded goldbars into a sphere and had
successfully disassembled and then reassembled it into two more spheres using
very sensitive robots.  Supposedly he had automated the process and has built 
a factory somewhere deep in the South American forests.  If you have been 
observing the gold prices lately, you wouldn't be surprised!

:-).
From: hougen@focus.csl.uiuc.edu (Darrell Roy Hougen)
Newsgroups: sci.math
Subject: Re: Generalization of binomial theorem wanted
Date: 5 Jan 1993 03:08:45 GMT
Organization: Center for Reliable and High-Performance Computing, University of Illinois at Urbana-Champaign
NNTP-Posting-Host: focus.csl.uiuc.edu
Keywords: binomial

hougen@focus.csl.uiuc.edu (Darrell Roy Hougen) writes:

% In terms of m and n, how many unique terms are there in
% 	(x_1 + x_2 + ... + x_m)^n
% when it is expanded?

Thanks to everyone who reminded me about multinomial coefficients.  It
should have been obvious.  I've included one response below:

# The number of terms equals the number of (a1,...,am) where each a is a
# non-negative integer and a1+...+am = n. (Obviously.) Less obviously, this
# number equals the binomial coefficient (m+n-1 choose m-1).
# 
# Why? Well, convert (a1,...,am) to a row of symbols like this: write down
# a1 dots, then a slash, then a2 dots, then a slash, ..., then am dots. This
# produces a row of m+n-1 symbols, of which m-1 are slashes and n are dots.
# Conversely, given any such row we can get a sequence (a1,...,am) by counting
# the number of dots between each pair of slashes (and at the start, and at
# the end). So the number of (a1,...,am) equals the number of such rows of
# symbols, and this is just (m+n-1 choose m-1) because we can choose m-1
# "positions" to contain slashes and put dots in the other n.
# 
# The coefficient of x1^a1.x2^a2...xm^am, incidentally, is n!(a1!a2!...am!):
# a so-called multinomial coefficient.
# 
# -- 
# Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
# gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
 

I should have stated in the original post that I was looking for the
number of terms in the complete polynomial of degree n by which I mean
the polynomial with terms of all degrees less than or equal to n.
This can easily be obtained from the multinomial coefficients by
writing sum (i=1 to n) of (i+m-1 choose m-1).  This is equal to my
original formula (and much simpler):

% The formula I have is
%                 n 
%             sum                 sum                 1  
%             i = 1    k_1 <= k_2 <= ... <= k_i <= m

The second sum is just the multinomial coefficient.  This can be seen
by noticing that there are still m-1 "slashes" and the number of
"dots" between the (p-1)st pair of slashes can be computed by counting
the number of k_j's corresponding the pth variable.  The number of
k_j's is equal to 'i' which completes the proof.

Thanks again to all who responded.

Darrell R. Hougen
From: brock@NeXTwork.Rose-Hulman.Edu (Bradley W. Brock)
Newsgroups: sci.math
Subject: Re: High Prices of Math Books. I am pissed.
Date: 11 Jan 1993 01:30:51 GMT
Organization: Computer Science Department at Rose-Hulman
Reply-To: brock@NeXTwork.Rose-Hulman.Edu (Bradley W. Brock)
NNTP-Posting-Host: g210b-1.nextwork.rose-hulman.edu

In article <C0nI4K.KzD@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman  
Rubin) writes:
> It is not that uncommon that I would like to use as a text a book which
> is out of print.  It used to be that a copyright was for 44 years; I
> believe it is now lasts until 50 years after the death of the author.

I thought copyrights lasted for 28 years and were renewable, and that  
photocopying an out-of-print book was legal.  Can anyone verify this?

--
Bradley W. Brock, Department of Mathematics
Rose-Hulman Institute of Technology  | "Resist not evil.... Love your
brock@nextwork.rose-hulman.edu       |  enemies."--Jesus of Nazareth
Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Let's add some sci.math.* groups
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Fri, 8 Jan 93 15:52:11 GMT

In article <43180@sdcc12.ucsd.edu> mstankus@oba.ucsd.edu (Mark Stankus) writes:
>I recently read a post asking that a sci.math.graph group be started.
>
>I think that we should start several new newsgroups so that 
>one does not have to wade through many messages each day on 
>topics which are not of interest to us. Here are my suggestions:
>
>sci.math.beforecalculus  algebra, geometry,... which is learned
>                         before learning calculus
>sci.math.rec      recreational math
>sci.math.culture  all of the Nobel prize stuff could be psoted here
>                  and similar topics
>sci.math.number   number theory, Pi,Fermat's Last Thm,...
>sci.math.calculus calculus stuff
>sci.math.algebra  junior year math major (college) algebra 
>sci.math.analysis junior year math major analysis 
>sci.math.discrete combinatorics,graph theory,...

The main problem with this sort of split is that it leaves out a LOT of
interesting areas in math.  What about topology?  Differential geometry?
Algebra?  (Not baby algebra, grownup algebra.)  Etc..

As usenet continues its rapid growth people *will* have to address the
issue of expanding newsgroups.  I hope people develop more intelligent
newsreaders.  You might also check sci.physics, where we are trying to
set up sci.physics.research and are hoping it gets more traffic than
sci.math.research.