HDTenEightyP/NetNews huggingface dataset:
From: srini@milton.u.washington.edu (Srini Tridandapani)
Newsgroups: sci.math
Subject: question on "spacings"
Date: 11 Oct 92 20:43:18 GMT
Article-I.D.: u.1992Oct11.204318.22917
Organization: University of Washington, Seattle


Hi:

I have a question relating to the distribution of the
maximum spacing (or difference) between successive
order statistics obtained by dropping (n-1) points
uniformly on a straight line.

I found the distribution in D.A. Darling's paper:
"On a class of problems related to the random division
of an interval," Annals of Math. Stat. vol 24 (1953),
pp. 239-253.  According to Darling the cdf

Pr{ V < v} = Sigma[Binomial[n+1,j]*(-1)^j*(1-v*j)^n,{0,j,Floor[1/v]}]

(the rhs is in Mathematica notation) is a result
going back to Whitworth (Choice and Chance, Cambridge Univ.
Press, 1897).

My question is:
Are there are any published bounds to this distribution?

Any pointers would be much appreciated.  Please email
any response that you may have to the question to:
      srini@u.washington.edu

Thanks,
Srini.


_____________________________________________________________________
Srini Tridandapani           /  email: srini@u.washington.edu
Dept. of Elec. Engr. (FT-10) /         srini@shasta.ee.washingon.edu
Univ. of Washington          /  phone: (206) 543-5918
Seattle, WA 98195            /
Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Catalan NUmbers
Organization: Computer Science Department,  Stanford University.
Date: Thu, 8 Oct 1992 02:46:54 GMT

In article <BvrvJJ.HqG@cs.psu.edu> plu@math.psu.edu (Todd Andrew Simpson) writes:
>This is described in Knuth's book, The Art of Computer Programming,
>Volume I (which is where I found it).  Another approach might be to
>associate each of a set of C_n objects (there are lots of sets
>enumerated by the C_n) with paths from (0,0) to (n,n) in such a way
>that there are n+1 paths for each of the objects.  I haven't seen such
>a proof, though it'd be interesting since one would not need to know
>_a priori_ that C(2n,n)/(n+1) was equal to C(2n,n)-C(2n,n-1).

Well, I don't know whether C(2n,n)/(n+1) = (2n)!/(n!(n+1)!) =
C(2n+1,n)/(2n+1) is any simpler than the algebra you wanted to avoid,
but let's pretend it is.  Now consider the C(2n+1,n) paths from (0,0)
to (n+1,n), aka that many strings of n+1 X's and n Y's.  Now all 2n+1
rotations (cyclic shifts) of any such string must yield 2n+1 distinct
strings, otherwise one of the 2n intermediate points along the
corresponding path from (0,0) to (n+1,n) would have to lie on the
diagonal (the straight line from (0,0) to (n+1,n)), impossible since
only the endpoints of this diagonal lie on lattice points.  It can be
seen that exactly one of the 2n+1 paths corresponding to these
rotations does not contain any point lying strictly above the
diagonal.  The string corresponding to that path must start with an X;
delete that X to yield a balanced string (every prefix has at least as
many X's as Y's).  Conversely, prefixing a balanced string with an X
yields such a path.

(Heard this one from Knuth around 1971.  I don't see it in Vol. I and
I can't remember where it's written down.)

-- 
=================================================== In short, I am floundering
Vaughan Pratt   pratt@cs.Stanford.EDU  415-494-2545 around and don't take what
=================================================== I say seriously.
Newsgroups: sci.math
From: epstein@rene.ma.utexas.edu (Paul Epstein)
Subject: Re: Another GRE question for you folks
Organization: University of Texas at Austin Mathematics
Date: Tue, 13 Oct 1992 19:26:17 GMT

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


>Given the discussion this question has generated, I would suggest to
>the original poster that he write to ETS and complain that the
>question is poorly worded.  GRE questions should not designed to be
>tricky, ie., to depend upon knowing that the positive root was
>intended.  I certainly don't remember having it impressed upon me that
>the surd always implies the positive square root.  In fact, I remember
>equations being solved by taking *the* square root of both sides.
>There were invariably a positive solution and a negative solution.

>Darrell



Why is it wrong to ask a question which depends upon the reader
understanding common mathematical notation?
From: kwhyte@dent.uchicago.edu (Kevin Whyte)
Newsgroups: sci.math
Subject: Spectum of a Lie group ...
Date: 7 Oct 92 20:20:37 GMT
Organization: Dept. of Mathematics, U. of Chicago



  Let G be a compact semi-simple Lie group.  The killing
form gives a canonical Riemannian metric, and hence a 
laplacian acting on smooth functions.  What are the eigen-values
of this laplacian?

  L (the laplacian) is a G invariant elliptic operator, on, say
L2(G).  Hence the eigenspaces  are finite dimensional and G invariant.
Peter-Weyl then tells us that each one of these is just some sum
of irreduciple reps. of G (each of which occurs as many times as
its weight).  Thus, given an irreducible rep. of G, it occurs in at
most n such eigenspaces (if G is an n dimensional rep.), so it
should have n positive real numbers associated to it. What are they?

-Kevin
kwhyte@math.uchicago.edu
Newsgroups: sci.math
From: jhgreen@cs.sandia.gov (Jethro H. Greene)
Subject: Neat geometry problem (*NOT* Homework)
Followup-To: jhgreen@cs.sania.gov
Originator: jhgreen@cs.sandia.gov
Organization: Sandia National Laboratories, Albuquerque, NM
Date: Sat, 17 Oct 92 02:15:40 GMT


I found this problem as #1 on page 1 in a Russian geometry book.
(Ironically, it only had answers for for what it considered 
to be the "hard" problems.)

Given a cube with sides of length x, a regular tetrahedron is 
placed inside so that two of its vertices are on the cube's
space diagonal and the other two are on one of the cube's faces.
Find the volume of the tetrahedron with in terms of x.

jhgreen@cs.sandia.gov (Student)
Xref: sparky sci.math:13275 sci.math.stat:2142
From: roussea@server.uwindsor.ca (ROUSSEAU PAUL            )
Newsgroups: sci.math,sci.math.stat
Subject: Help! Median Statistical Median Problem
Date: 16 Oct 1992 00:10:44 GMT
Followup-To: sci.math
Distribution: na
Organization: University of Windsor, Ontario, Canada

Could anyone tell me how to solve this problem.  It's from a
course on social statistics and I'm not a math whiz so
easy on the tech talk.

The Problem:

Find the median for the data that follows:

Age			Number of			Number of
			Males				Females

8-9			     1			         0

10-11			      5			         0

12-13			      42		         16

14-15			      68		         77

16-17			      64		         74

18-19			       72	                 70

20-21	       		       29		          32

22-23			        13		          28

24-25			        13			  12

26-27			        2	         	   3

28-29			        0		           5

30-31			        2			   2	 		       
Please send help via e-mail


Total			   311				   319	
						  		
-- 
Paul Rousseau			roussea@uwindsor.ca
(519) 735-7959			"It's too bad ignorance is not painful"
						-anonymous

Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: Does there exist a large list of primes ftpable someplace?
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Distribution: usa
Date: Fri, 9 Oct 1992 19:31:32 GMT

In article <1b4ktbINNhp5@darkstar.UCSC.EDU> dove@cats.ucsc.edu (Ray Rischpater) writes:
:
:I'm looking for a large ascii list of prime numbers, starting preferably at
:2 and going on up.
 
Why do you need such a thing? It is generally faster to generate them 
on the fly, via a sieve, than it is to read them in, formatted off a disk!

It is debatable whether a binary file could be read faster than it takes
to generate them, but ASCII? No way!
 
You also do not say how large "large" is.

--
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: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: consequences of the Axiom of Choice
Date: 9 Oct 92 15:26:06 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: holmes@opal.idbsu.edu (Randall Holmes)

In article <1992Oct8.183047.5338@guinness.idbsu.edu>, holmes@opal (Randall Holmes) writes:
>The "alas" was somewhat ironic in intent.  The mere existence of
>Solovay's model shows that _all_ constructive work can be assumed to
>keep one in the realm of the measurable, does it not?

That is true.  But assuming such is not very constructive, now is it?
A set being measurable is more than just a state of mind--it's a specific
assertion that, for example, there are inner and outer witnesses to the
set's measure that are arbitrarily close to each other.  If you're going
to assume *that*, you as might as well stick to AC in the first place.

>						        I don't think
>that applications of the prime ideal theorem are really needed for
>classical measure theory!

That's precisely the point I had brought up when I mentioned PIT.  Do
you or don't you count the compactness of the infinite unit cube and what
it means for probability theory as part of "classical measure theory"?

>			    The interesting thing is that Solovay's
>result requires consistency strength more than that of ZFC; the
>assumption that all sets of reals are Lebesgue measurable enables one
>to intepret ZFC + "there is an inaccessible"; ZF + "all sets are
>measurable" is stronger than ZFC!

All the more reason for constructivists to be unhappy.  This certainly
rules out those constructivists who are so because of contradictophobia!
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
From: hdev@dutiag.tudelft.nl (Hans de Vreught)
Newsgroups: sci.math
Subject: Re: Looking for fast methods of computing PI
Date: 15 Oct 92 00:05:27 GMT

fokp@cary112.its.rpi.edu (Pingtao (Ben) Fok) writes:

>I don't know what method you are talking about but I have been
>thinking this problem for a long time and I found a way(the hard way)
>to find PI.  I use the idea of polygon in a circle.  I first use 
>a square to start with.  Then I turn the square into a octagon and find
>the length of the side in terms of square root of 2. And then I 
>double the number of side again and again and again.....
>It works fine but the algebra is heavy.  I figure square root of 2
>is easier to find than PI.  A friend of mine used taylor's series
>to find Pi up to 5000 digit.

The method with polygons dates back to the Greeks. Taylor series (Machin's
formula) works fine with this small number of digits. But the method used for
the record is based on the AGM (arithmetic and geometric mean). The method of
Brent and Salamin is well known (which is one such a method):
a_{n+1}=(a_n+b_n)/2         (arithmetic mean)
b_{n+1}=sqrt(a_n*b_n)       (geometric mean)
c_{n+1}=(a_n-b_n)/2
          2(a_{n+1})^2
pi_n=----------------------
     1-sum(2^m*c_m^2,m=0,n)
Roughly pi_{n+1} has twice as many correct digits as pi_n. Look in the faq
list for the references.
--
Hans de Vreught                          | John von Neumann:
hdev@dutiaa.tudelft.nl                   | Young man, in mathematics
Delft University of Technology (TWI-ThI) | you don't understand things,
The Netherlands                          | you just get used to them.
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: Algebra problem
Date: 12 Oct 1992 12:59:25 -0400
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu

In article <2AD99AA9.24889@news.service.uci.edu> dpick@math.uci.edu (Daniel Pick) writes:
>Let p be a prime number and let G be a finite group without elements of order
>p.  Show that if a^p = b^p for a,b elements of G, then a = b.
>

It looks like a homework problem to me...  I hope you do not get an answer
from the net and then claim it as your own work.  Getting an answer from
the net may be okay (that is to be determined by the instructor of the
course), but then you should include that source information.
-- 
  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: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: What a weird group!!!
Date: 15 Oct 1992 08:15:00 -0400
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu


>[the Bohr compactification of the real numbers]

>But what does this group *look like*??  That's what I'd like to know!!

The construction uses the Axiom of Choice, so no one knows...

For a (perhaps) simpler warm-up, try the Bohr compactification of the
integers.

Or, before you tackle visualizing that, how about visualizing the
Stone-Cech compactification of the integers?
-- 
  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: mccolm@darwin.math.usf.edu. (Gregory McColm)
Newsgroups: sci.math
Subject: Re: consequences of the Axiom of Choice
Keywords: Axiom of choice; Cartesian product
Date: 9 Oct 92 19:20:54 GMT
Organization: Univ. of South Florida, Math Department

In article <1af9g6INN6d2@function.mps.ohio-state.edu> edgar@function.mps.ohio-state.edu (Gerald Edgar) writes:
>In article <1992Oct1.152704.16387@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>>Bertrand Russell used to explain choice this way.  Imagine you are
>>blessed with infinitely many pairs of shoes and socks.  You wish to
>>choose one of each pair.  With the shoes it is easy:  you could take
>>all the left ones, or all the right ones.  But can you choose one sock
>>from each pair?  To claim that you can is to assert the axiom of
>>choice.
>
>So, let's try to find a good mathematical example to illustrate Russel's
>explanation.  Explicitly find a countably infinite set S of (unordered) pairs,
>for which a choice function is not obvious.  [More technical details.
>The set S should be explicitly and uniquely specified in the language of ZF.
>It should be provable in ZF that it is a countable set of pairs.]
>
>An UNCOUNTABLE set of pairs with no obvious choice function is
>easy: take the set of ALL pairs of sets of real numbers.
>-- 


Won't work:  given any pair {r,s}, choose the smaller one.  
In fact, no set with a constructible linear order will work, 
and an infinite set with no constructible linear ordering 
will be a strange beast indeed.

-----Greg McColm
From: burt@aupair.cs.athabascau.ca (Burt Voorhees)
Newsgroups: sci.math
Subject: Re: Zeno
Date: 10 Oct 92 18:20:27 GMT


  Actually, Zeno had 40 paradoxes, eight of which survive.  (In, are
still known...)  These divide into four paradoxes of motion and four
paradoxes of extension.  The paradoxes of motion look at the problems
of dividing time, those of extension at the problems of dividing space.
bv
From: mapsj@gdr.bath.ac.uk (Simon Juden)
Newsgroups: sci.math
Subject: Re: consequences of the Axiom of Choice
Keywords: Axiom of choice; Cartesian product
Date: 14 Oct 92 21:44:46 GMT
Organization: School of Mathematics, University of Bath, UK

In the referenced article, mccolm@darwin.math.usf.edu. (Gregory McColm) writes:
>In article <1af9g6INN6d2@function.mps.ohio-state.edu> edgar@function.mps.ohio-state.edu (Gerald Edgar) writes:
>>In article <1992Oct1.152704.16387@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>>>Bertrand Russell used to explain choice this way.  Imagine you are
>>>blessed with infinitely many pairs of shoes and socks.  You wish to
>>>choose one of each pair.  With the shoes it is easy:  you could take
>>>all the left ones, or all the right ones.  But can you choose one sock
>>>from each pair?  To claim that you can is to assert the axiom of
>>>choice.
>>
>>So, let's try to find a good mathematical example to illustrate Russel's
>>explanation.  Explicitly find a countably infinite set S of (unordered) pairs,
>>for which a choice function is not obvious.  [More technical details.
>>The set S should be explicitly and uniquely specified in the language of ZF.
>>It should be provable in ZF that it is a countable set of pairs.]
>>
>>An UNCOUNTABLE set of pairs with no obvious choice function is
>>easy: take the set of ALL pairs of sets of real numbers.
>>-- 
>
>
>Won't work:  given any pair {r,s}, choose the smaller one.  
>In fact, no set with a constructible linear order will work, 
>and an infinite set with no constructible linear ordering 
>will be a strange beast indeed.
>
>-----Greg McColm
....not true: what if r=s?
Simon
From: darknite@camelot.bradley.edu (John S. Novak III)
Newsgroups: sci.math
Subject: Re: Electronic math notation
Date: 14 Oct 92 22:58:56 GMT
Article-I.D.: camelot.darknite.719103536
Organization: Bradley University

Andrew Mullhaupt writes:

>>If you use TeX notation, probably most people on the net will understand

>Some of us will object strenuously to any increase in this lamentable idea.

Ieck, you got that right.
It'd be simpler to bloody write it out by hand and use a
full-page scanner to put it in some reasonable format (like, say
.gif) and uuencode it.

TeX is irritating in the extreme.

--
"Look at Every Lit-tle Symbol."
          -Shrijay Dalal, 11-03-91
John S. Novak, III		darknite@camelot.bradley.edu
Newsgroups: sci.math
From: jwales@silver.ucs.indiana.edu (jimmy donal wales)
Subject: Philosophical Foundations of Probability?
Organization: Indiana University
Date: Tue, 13 Oct 1992 22:12:05 GMT

I'm looking for pointers to books or review articles on modern
views of the philosophical foundations of probability theory.

--Jimbo

Xref: sparky sci.math:13162 sci.math.num-analysis:3004 sci.math.stat:2129 sci.math.symbolic:2709
Organization: Middle East Technical University - Computer Center
Date: Wednesday, 14 Oct 1992 10:11:22 TUR
From: <C44590@TRMETU.BITNET>
Newsgroups: sci.math,sci.math.num-analysis,sci.math.stat,sci.math.symbolic
Subject: thanks.

i would like to thank all the people that helped me in finding the e-mail

address of R.Fateman.
Newsgroups: sci.math
From: cory@riemann.math.mun.ca (Cory C. Pye)
Subject: Re: Question on numerical integration
Organization: Chem Dept., Memorial University of Newfoundland
Date: Thu, 8 Oct 1992 13:13:46 GMT

In article <1992Oct7.113401.1322@rz.uni-karlsruhe.de> vhansen@ipfs.bau-verm.uni-karlsruhe.de (Wolfgang von Hansen) writes:
>How can one calculate things like
>
> a                oo
>Int f(x) dx  or  Int f(x) dx
>-oo               a
>
>numerically. I know how to do such things with finite limits,
>but I've got no idea how to deal with infinite limits.
>
>Thanks in advance, Wolfgang

 Try re-expressing the integrand using a substitution which transforms
the infinite limit to a finite number and leaves the finite number
finite. This will obviously depend on the integrand.

                                       - Cory Pye
From: mld900@anusf.anu.edu.au (Murray Dow)
Newsgroups: sci.math
Subject: Area of five sided polygon needed
Date: 14 Oct 1992 23:55:30 GMT
Organization: ANU Supercomputer Facility
Distribution: world
Reply-To: mld900@anusf.anu.edu.au
NNTP-Posting-Host: 150.203.5.3


In article <1992Oct13.040617.16321@husc3.harvard.edu>, kubo@birkhoff.harvard.edu (Tal Kubo) writes:
|> In article <1992Oct12.200125.826@altair.selu.edu> fcs$1224@altair.selu.edu writes:
|> >Given a five sided polygon, is there a formula, in terms
|> >of its sides, for the area?
|> 
Isn't there a simple formula based on the trapezoidal rule?
If the co-ordinates of the poly are (Xi,Yi) i=1,n and add
X(n+1)=X1, Y(n+1)=Y1
then
Area=0.5*sum(Yi+Y(i+1))*(X(i+1)-X(i))
--
Murray Dow                         GPO Box 4 Canberra ACT 2601 Australia
Supercomputer Facility             Phone: +61 6 2495028
Australian National University     Fax:   +61 6 2473425
mld900@anusf.anu.edu.au
From: mccolm@darwin.math.usf.edu. (Gregory McColm)
Newsgroups: sci.math
Subject: Re: Hawking's portrayal of Newton and Einstien
Date: 12 Oct 92 17:21:25 GMT
Organization: Univ. of South Florida, Math Department

In article <2AD069BB.11232@news.service.uci.edu> dpick@math.uci.edu (Daniel Pick) writes:
>In article <1992Oct4.184909.43577@kuhub.cc.ukans.edu> dchatterjee@kuhub.cc.ukans.edu writes:
>>
>> Hi !
>>
>> I was reading Stephen W.Hawking's " A Brief History of Time ". Hawking
>> comments that Isaac Newton was an unpleasant character and had always
>> (or most of the time) engaged in personal quarrels or brawls with his
>> fellow scientists. This seems very amusing, because, why people of the
>> stature of Newton be so mundane, while on a comparative scale, Einstien
>> was more pleasant and very unassuming in appearance and of humble habits ?
>>
>> Seems to me that Hawking could be biased ! 
>>
>> Comments ? 
>>
>> - DC
>
>No, Hawking is not biased, but it is not true that Newton often engaged in
>personal quarrels or brawls.  On the contrary, several historians document that
>Newton actually had a intense distaste for controversy, and often avoided
>publishing his results to avoid it.  This disgust lies at the heart of the
>priority dispute between Leibniz and Newton over the invention of calculus.  
>It was only when Hooke persuaded Newton to publish his results on gravitation
>that the world knew that Newton had discovered fluxions.
>
>For further information, see Journey through Genius by William Dunham, Men
>of Mathematics by E.T. Bell, and
>The World of Mathematics, Vol. 1, edited by Martin Gardner (I think).
>


As I recall, Newton preferred to handle controversies by 
remote control.  The special commission investigating 
whether Liebniz stole the Calculus was controlled by 
Newton so that he could win priority without having to 
appear to fight for it.  There is a story that when 
Liebniz died, Newton bragged that he had destroyed him.
Newton's later conduct as, well, special prosecutor in 
search of counterfeiters (well, what DID you think he 
did at the mint?) was energetic and a bit inhuman.  One 
Freudian explained it all as an attempt to impress his 
long-dead stepfather.  Could be.  But he was not as nice 
a fellow as E T Bell makes out.

-----Greg McColm
Newsgroups: sci.math
From: rickert@mp.cs.niu.edu (Neil Rickert)
Subject: Re: 'Perfect' numbers
Organization: Northern Illinois University
Date: Mon, 12 Oct 1992 05:26:59 GMT

In article <Bvzsy7.B9E@ux1.cso.uiuc.edu> simms@ux1.cso.uiuc.edu (dan) writes:
>guckes@math.fu-berlin.de (Sven Guckes) writes:
>>dagjo@ifi.uio.no (Dag Espolin Johnson) writes:
>>>Does someone have a list of the first numbers (as many as possible) that are
>>>'perfect'?

>>"It's in the FAQ!" - Damn, it isn't. (I just checked in news.answers.)

>                                                         does anyone
>know an easier way of telling if a number is not perfect than just
>plowing along and finding the sum of its divisors?

It probably should be in the FAQ.

Even perfect numbers are exactly those numbers of the form

          n-1  n
	 2   (2  - 1)

                       n
where n is such that  2  - 1  is prime (called a Mersenne prime).  There
are no known odd perfect numbers.