HDTenEightyP/NetNews huggingface dataset:
From: meuwly@urz.unibas.ch
Newsgroups: sci.math
Subject: cluster-integrals
Date: 14 Apr 92 14:33:51 MET
Organization: University of Basel, Switzerland

hello there,
i'm looking for an evaluation of cluster-integrals. i need it for the
partition-function of a real gas in the mayer-cluster-expansion.

i know that there must be an explicit formula for at least the 
first 4 cluster-integrals. the first 3 ones are

 b  = 1
  1

          infinity
     2*Pi  /                    2
 b = ----  | ((exp(-V(r)/kT)-1)r  dr
  2  L^3   /
           0

                infinity
       2     1   /  3    3
 b = 2b  +  ---- | d r  d r   f  f  f
  3    2    6L^6 /    12   13  12 13 23
                 0


L is the de broglie-wavelength and f   are the [ik] clusters of the molecules
                                    ik

there must be a phd-thesis by harrison: john hopkins-university 1938 on that
subject whee the first 4 clusterintegrals are computed. who knows something
about this subject?



thanks for your help    markus
Newsgroups: sci.math,sci.math.stat
From: dow@nova2.stanford.edu (Keith Dow)
Subject: Hypershperical Coordinates
Keywords: expected value, binomial distribution
Organization: Stanford University
Date: Fri, 17 Apr 1992 03:21:01 GMT

Where can I learn about the spherical coordinate system in higher
dimension?  Please E-Mail me the references.

Also, does anyone know the answer to the following question.  In
two dimension the angular coordinate goes through a complete circle.
In three dimensions one angular coordinate goes through 2-pi, while
the other only goes through pi.  How do the angular coordinates 
behave in n-dimensions?  Do they do complete circles, i.e. 2-pi?
Do they do half circles or less?
From: tietz@eos.informatik.uni-bonn.de (Benno Tietz)
Newsgroups: sci.math
Subject: software for doing calculations in R^d
Date: 6 Apr 92 09:26:21 GMT
Distribution: sci
Organization: Institut f"ur Informatik IV, Universit"at Bonn, FRG

Hi,

I am looking for a software package for doing some calculations (convex
hull of a given set of points, volume of a polytope) in the
d-dimensional vector space (R^d).

Any help would be appreciated.

Benno
--
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                No man's life, liberty, or property are safe
                while the legislature is in session.
----------------------------------------------------------------------
    tietz@eos.informatik.uni-bonn.de    or     tietz@cs.bonn.edu

######################################################################
#         Member of the League for Programming Freedom               #
#          -- write to league@prep.ai.mit.edu                        #
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From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Newsgroups: sci.math
Subject: Re: Mersenne Composites
Date: 6 Apr 92 19:29:13 GMT
Organization: Beckman Instruments, Inc.

In <1992Apr3.185550.20987@mp.cs.niu.edu> rusin@mp.cs.niu.edu (David Rusin) writes:

>I have this memory of having looked into the multiplicative groups of fields
>once and having found a paper (probably mid 20th century) that the only
>restriction on the multiplicative group of a field is that there be at most
>n solutions to x^n=1 (obviously). Thus for example, there would be a field
>with  multiplicative group isomorphic to the additive reals. (It would
>be of characteristic 2). 
>I may be forgetting some other condition or two but the long and short of
>it is that really there will be lots of double fields.
>(It would be a lot more fun to have "amicable pairs" of fields: each isomorphic
>to the other's multiplicative group.)
>dave  rusin@math.niu.edu

References would be appreciated.

--
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.
Ich bin ein Virus.  Mach' mit und kopiere mich in Deine .signature.
Newsgroups: sci.math
Subject: Re: Best Method(complexity-wise) for Solving System over GF(2)?
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 9 Apr 92 19:02:52 GMT
Distribution: na
Organization: Beckman Instruments, Inc.

In <1992Apr9.181436.29316@uwm.edu> leranth@miller.cs.uwm.edu ( ULYSSES ) writes:



>    I am seeking information that will help me solve a problem.

>    Question:   Is there a better method [complexity-wise O[f(n)] ]

>		than  G.E. for solving a  linear system over GF(2)  ?


>		[Don't point me to a package because I am solving a huge-huge
>		 system that will be sparse; the only thing I need to 
>		 overcome is the fact that G.E. ~ O(n^3) which will take a 
>		 LONG time for  n> 10^4                                       ]

>                An extremely well-written Numerical book that contains
>		information on solving said system would be a welcome bit
>		of enlightenment.

(1)  Are you SURE that G.E. would not be fast for your system?  Sometimes
sparse matrices have sparse inverses.

(2)  I believe there is a O(n^(log 7/log 2)) method, related to the matrix
multiplication method.
--
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.
Ich bin ein Virus.  Mach' mit und kopiere mich in Deine .signature.
From: martin@lyra.cis.umassd.edu (Gary Martin)
Newsgroups: sci.math
Subject: Re: A Proof about Groups
Date: 15 Apr 92 15:36:12 GMT
Article-I.D.: lyra.MARTIN.92Apr15103612
Organization: University of Massachusetts Dartmouth

In article <neil.703279730@dehn> neil@dehn.mth.pdx.edu (John Neil) writes:

   teacher@netlink.cts.com (Daniel Pick) writes:


   >Then he asks the reader to prove that if G is a finite group,
   >then every element in G has finite order.
   >This proof stumps me.  If you induct on the number of elements in
   >the group, how do you show that each successive element has finite
   >order?

   Your mistake here is trying to prove this by induction.  Instead of that, you
   should note that the cyclic subgroup generated by each element of a group must
   be a subgroup of that group since groups are closed under multiplication.
   Thus, to have an element which was not of finite order in a finite group would
   mean that a finite group had an infinite subgroup.  OOPS.  It is usually a
   mistake to do proofs in algebra by induction.  There are some problems for
   which it is required, but most of the problems in algebra are best proved by
   contradiction or directly.

   --John Neil
This glosses over the main point - in a finite group, the cyclic subgroup
generated by  x  is the set of positive powers of  x.  This was proved in
a previous reply (pigeonhole principle + cancellation law).  (Of course
if you define the order of  x  to be the cardinality of  <x>, then it's
trivial, but that's not the definition in the book cited in the original
post.)

Just my two zuzzim. (Only kidding. :-))
--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
Martin@cis.umassd.edu
Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: factoid about new prime
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Fri, 10 Apr 1992 16:46:47 GMT

In article <michaelg.702922014@Xenon.Stanford.EDU> michaelg@Xenon.Stanford.EDU (Michael Greenwald) writes:
:Well, I was surprised that Monty Hall apparently understood why the
:solution to the Monty Hall problem was right.  Sometimes people aren't
:as dumb as they seem.  I won't be surprised again, even if the
:newscaster in question is offering to trade a live goat for the prime
:number behind curtain number 2^(2^32) + 1.
 
Uh, 1479*2^34 + 1 divides your "prime" number.



--
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: "elliott chapin" <elliott.chapin@canrem.com>
Newsgroups: sci.math
Subject: does goedel's work ap
Date: 16 Apr 92 12:33:42 EST
Reply-To: "elliott chapin" <elliott.chapin@canrem.com>
Distribution: sci
Organization: Canada Remote Systems

MA>I believe that the following problem has been shown to be undecidable in first
MA>order logic + peano axioms (which should be sufficient for number threory):

MA>Call a finite set in positive integers LARGE if the smallest member
MA>of the set is greater then the number of elements in the set.

MA>Take the set of numbers 1..N and partition them as follows: randomly
MA>select a pair out of the set and color (label) one of the pair BLUE
MA>and the other RED.  Continue selecting pairs without replacement and coloring

How is random selection defined in this context?

Elliott Chapin
---
  DeLuxe 1.12 #4315  elliott.chapin@canrem.com
--
Canada Remote Systems  - Toronto, Ontario/Detroit, MI
World's Largest PCBOARD System - 416-629-7000/629-7044
Xref: sparky sci.math:5848 sci.math.symbolic:1221
From: joe@cats.ucsc.edu (Joseph P Christy)
Newsgroups: sci.math,sci.math.symbolic
Subject: uniform distribution on 2-sphere
Date: 9 Apr 92 22:09:32 GMT
Article-I.D.: darkstar.32476
Followup-To: sci.math
Organization: University of California, Santa Cruz


 I know that this has been discussed ad nauseam before, but I didn't
 pay attention to the answer then. Would some one please Email me
 a reference, or better algorithm, or better yet Mathematica code
 fragment to compute pseudorandom vectors, approximately uniformly
 distributed on the unit two sphere? Thanks.

-- 
Joe Christy          | joe@cats.ucsc.edu
111 John Street      | Department of Mathematics
Santa Cruz, CA 95060 | University of California
(408) 425-3817       | Santa Cruz, CA 95064
** time flies like an arrow, fruit flies like a banana **
Newsgroups: sci.math
From: israel@unixg.ubc.ca (Robert B. Israel)
Subject: Re: nonexistance of counterexample -> existance of proof?
Organization: University of British Columbia, Vancouver, B.C., Canada
Date: Fri, 10 Apr 1992 08:49:47 GMT

In article <31475@sdcc12.ucsd.edu> ec120wee@sdcc4.ucsd.edu (Anthony Minkoff) writes:
>   [stuff deleted]
>Anyway, the story continues...  I then tried to present an example
>of a theorem with no proof, and no counterexamples.  First, I made
>the claim that there must be real numbers with no finite expression.
>(A finite expression does not necessarily imply a finite expansion.
>For example, sqrt(2) is a finite expression.)  I offered the
>following proof:
>
>Any finite expression can be encoded as a finite sequence of bits,
>according to the following scheme:  Begin the sequence with an ASCII
>text description of how to decode the part of the sequence following
>the description.  (Of course, the entire description may be the
>ASCII text.)
>
>Since a one-to-one correspondence can be established between finite
>sequences of bits and integers, but not between reals and integers,
>there are reals with no finite expressions.
>
>Then I proposed the theorem, "All reals *do* have some finite
>expression."  Clearly, no proof exists, since the statement is not
>true.  Then we got into an argument over whether counterexamples
>exist.  Certainly he could not give me one, and nobody ever could.
>
>Tony Minkoff

    Actually, there IS a counterexample to your "theorem".  I assume that
you have a precise definition of what it means for a "finite expression"
to "represent" a real number (otherwise the "theorem" is ambiguous).
Those "finite expressions" that represent real numbers can then be 
enumerated (e.g. in order of the positive integers that we get when 
representing ASCII strings as hexadecimal digits). 
Let r(n) be the number represented by the n'th "finite expression"
that represents a real number.  Let d(n) be the n'th decimal digit 
(after the decimal point) of r(n) (with whatever convention you want
for numbers with nonunique decimal expansion).  Let c(n) = 1 if d(n) = 2
and c(n) = 2 otherwise.  My counterexample is 0.c(1)c(2)c(3)... .

    Have I specified a unique real number?  Yes, if you have defined
"finite expression" and  "representing" (unless you are a constructivist).  My 
definition does _look_ like a finite expression representing a real number, 
but of course it can't be one according to whatever definition you used.  
It's certainly not a _computable_ expression, if your definitions are 
sufficiently powerful, because there won't be an algorithm that can decide
whether or not an expression represents a number.  
 
Robert B. Israel
Department of Mathematics
University of British Columbia
Vancouver, BC, Canada

Newsgroups: sci.math
From: kwhyte@math.uchicago.edu (Kevin Whyte)
Subject: Re: Fermat's Last Theorem
Organization: Dept. of Mathematics, Univ. of Chicago
Date: Thu, 9 Apr 1992 21:58:05 GMT

>In article <1992Apr9.021905.20917@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
>>In article <3660@tuegate.tue.nl> wsadjw@urc.tue.nl writes:
>>:In article <STROMME.92Apr6130830@jeeves.math.utah.edu> stromme@math.utah.edu (Stein Stromme) writes:
>>
>[stuff omitted]
>>
>>I am just surprised that someone with a presumed interest in math would
>>not know one of the most outstanding and prolific mathematicians of
>>the 20th century. If I were Erdos, I'd be insulted.
>>
>>Paul Erdos has been one of the movers and shakers of the 20th century.
>>The fact that he is relatively unknown is just another example of the
>>piss-poor attitude the American public has towards math and mathematicians.
>>
>>I'm sure the original poster knows who Michael Jordan is, whose only
>>claim to fame is his ability to jump high and stuff a ball through a hoop.
>>The contributions of Paul Erdos will have long lasting value to the
>>knowledge base of the human race and his work has provided stimulous for
>>many others. What long lasting value Michael Jordan's contributions have
>>would be a matter of serious debate.
>>
>>And I did answer the question in a half joking manner. The way the question
>>was posed: "who is this Erdos dude" invites such because calling someone a
>>'dude' without knowing that person is in itself somewhat flip and insulting.
>>--
>The fact that anyone knows who Erdos is is amazing.  I don't
>think that Erdos has been any more influential than, say, Richard
>Swan, but I bet that a majority of the posters to this net don't
>know who he is, either.
>
>As for appending "dude" to Erdos' name, the annoyance expressed 
>over that reminds me of how I used to get my ass beat in junior
>high school for not wishing to automatically say "sir" to 
>(male) teachers.  I am willing to make some allowance for the
>linguistic customs of other people.
>
>CHarles Yeomans
>cyeomans@ms.uky.edu
>yeomans@austin.onu.edu

   Well, a> I do know who swan is, and
         b> I doubt has been nearly as influential as
Erdos.  Erdos has personally been a great influence on 
many young mathematicians (although, FYI I am not one of
them).  This says nothing about the relative worth of 
their mathematics , but makes it hard to has much of an
interest in mathematics and not know who Erdos is.  This
is especially true since the type of  math Erdos does is
very well suited amatuer-level exposition (at least, the 
problems are).  Witness the relative # of references to
Erdos and Swan in the works of say, Martin Gardener.

The above aside ... Why are we continuing this "discussion"?

Kevin
kwhyte@math.uchicago.edu

Newsgroups: sci.math
From: burkhard@software.org (Neil Burkhard)
Subject: Code for FFT
Organization: Software Productivity Consortium, Herndon, Virginia
Distribution: na
Date: Tue, 14 Apr 1992 18:23:03 GMT

I looking for FTPable code (Pascal, C, Fortran, etc). to do 
fast Fourier transforms (either real or integer). Are there
places you can point me to to find this? Or instead, is somebody
willing to e-mail a copy? Please respond to my e-mail address
below. Thanks.

---------------------------------------------------------------
Neil Burkhard                        Internet:  burkhard@software.org
Software Productivity Consortium     Phone:     703-742-7154
2214 Rock Hill Road                  
Herndon, VA  22070

"Just when you think you're on easy street, someone rips up
 the pavement"
From: einarr@ifi.uio.no (Einar Andreas R|dland)
Newsgroups: sci.math
Subject: Re: Calculating the # of primes < n
Date: 7 Apr 92 15:49:40 GMT
Organization: Dept. of Informatics, University of Oslo, Norway
Originator: einarr@skinneggi.ifi.uio.no


In article <1992Apr2.014052.1514@cunixf.cc.columbia.edu>, pvl2@cunixb.cc.columbia.edu (Priscilla V Loanzon) writes:
> I was trying to compute what fractions of the positive integers are
> prime. I was doing this by getting the fraction left if from the
> positive integers we remove everything divisible by the primes
> p_1,p_2,p_3,...  
> 
> I get the following infinite series that gives the fraction of integers
> left: 
> 1 - (1/p_1) - (1/p_2)(1-1/p_1) - (1/p_3)(1-1/p_1)(1-1/p_2) -
> (1/p_4)(1-1/p_1)(1-1/p_2)(1-1/p_3) - ...
> 
> From the formula n/log n for # of primes less than n I think the
> infinite series must converge to 0 and the partial sums to 1/log n. Is
> this correct? 
> 
> Is there a computer program that can calculate what a series like that
> goes to, including the partial sums?

It seemes to me that this sum is equal to

      oo
    -----
     | |  ( 1 - 1/p_i ).
     i=1

Taking the logarithm, we get a sum

      oo                   oo
     ---                  ---
     >    ln (1-1/p_i) <  >    - 1/p_i  =  -oo.
     ---                  ---
     i=1                  i=1

Hence, the product converges to 0. This would apply to any sequence p_i
such that p_i=\=1, p_i -> oo.

EinAR
-- 
.__________
|  Einar Andreas R|dland (Roedland)  ---  E-mail: einarr@ifi.uio.no
|  Mathematician (student)
|  University of Oslo, Norway (Northern Europe)
From: victor@watson.ibm.com (Victor Miller)
Newsgroups: sci.math
Subject: Re: More on primes
Date: 14 Apr 92 15:31:31 GMT
Reply-To: victor@watson.ibm.com
Organization: IBM, T.J. Watson Research Center
Disclaimer: This posting represents the poster's views, not necessarily those of IBM

>>>>> On 13 Apr 92 21:07:02 GMT, pcmoan@solan.unit.no (Per Christian Moan) said:


Per> Let P3(n) be the # of primes < n of the form a^2+b^2+c^2 where
Per> a,b,c are integers.
Per> Let P2(n)              --"--                 a^2+b^2            -"-

Per> Has it been proved that :
Per>       
Per>      lim   P3(n)/P(n) ~> 3/4 and lim    P2(n)/P(n) ~> 1/2
     n->oo                       n->oo

Per> P(n) denoting the number of primes less than n.


Per> Just asking...
Per> Per Christian.

Yes, this is fairly easy (see, for example Serre's A Course in
Arithmetic). First, a  prime, p, is of the form a^2+b^2 if and only
if it is 2 or congruent to 1 mod 4.  Second, an integer, n, is of the
form a^2+b^2+c^2 if and only if it is not of the form 4^d (8e-1).
This, combined with Dirichlet's theorem on primes in progression,
shows that

lim P3(n)/P(n) = 3/4, and lim P2(n)/P(n) = 1/2.


--
			Victor S. Miller
			Internet: victor@watson.ibm.com
			IBM, TJ Watson Research Center
Newsgroups: sci.math
From: axel@vlsivie.tuwien.ac.at (Axel Jantsch)
Subject: sci.math,sci.math.research
Organization: Technical University Vienna, Austria
Distribution: sci
Date: Thu, 16 Apr 1992 13:19:00 GMT



Dear professionals,

Excuse me if it is a stupid question:

Given is a set with n elements.
What is the number of all subsets with m or less than m elements ?

Or, in other words

    m
  ----
  \
   \   ( n )
   /   ( k )   = ?
  /
  ----
  k=0


Please, send a mail, as I do not read the newsgroup regularly.

Thanks for any hint.

Axel
--
   ^  |Axel Jantsch               email: axel@vlsivie.tuwien.ac.at  
  / \ |fax: (++43 1) 569697       voice: (++43 1) 58801-8156
 /-|-\|Institut fuer Technische Informatik, Treitlstrasse 3/1822 
/  |__|Technische Universitaet Wien, A-1040 Vienna, Austria
Newsgroups: sci.math
Subject: Re: test of non-orientability of moibus strip.
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: 9 Apr 92 16:48:43 +1200
Distribution: world
Organization: Department of Mathematics, University of Canterbury

>I tried this.  I did not find it so.  Actually what I tried was drawing the 
>circles counterclockwise.  When I came back to starting point, I was still 
>drawing them counterclockwise.  

This has already been well-answered.    I will just add $0.02  .

When fiddling with maps, arcs and figures on a mobius strip or whatever; it
often helps to think of the arcs and diagrams as being IN the surface, rather
than ON it.

So, every time you draw a line or color a region on a surface, think of it
being done with ink on a highly porous surface, so the ink soaks through
instantly.  If you are physically drawing things on a paper surface, you must
draw "co-incidentally" on both sides. (It may help to use tracing paper).

---------------------------------------------------------------------
      Bill Taylor              wft@math.canterbury.ac.nz 
---------------------------------------------------------------------
  Since we got a video player,  I hardly ever read any more.
 I used to be quite a bookworm.   Now I'm more of a tapeworm.
---------------------------------------------------------------------
Newsgroups: sci.math
Subject: Re: SYMMETRIC MATRICES
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 6 Apr 92 19:01:30 GMT
 <1992Apr3.184134.26546@zaphod.mps.ohio-state.edu> <1992Apr3.223323.13038@casbah.acns.nwu.edu>
Distribution: na
Organization: Beckman Instruments, Inc.
Keywords: symmetric matrix sleepless zombie proof

In <1992Apr3.223323.13038@casbah.acns.nwu.edu> john@hopf.math.nwu.edu (John Franks) writes:

>In article <1992Apr3.184134.26546@zaphod.mps.ohio-state.edu>, edgar@function.mps.ohio-state.edu (Gerald Edgar) writes:
>|> >In article <1992Apr2.173036.5644@cs.mcgill.ca> pone@cs.mcgill.ca (Daniel (...) Shields) writes:
>|> >>
>|> >>Given nxn matrix A, A and transpose(A) are similar to each other.
>|> >>By Similar I mean that there exists a matrix P, such that 
>|> >>transpose(A) = P A P^(-1)
>|> 
>|> In article <1992Apr3.171308.28982@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes:
>|> > it's an easy consequence of the Jordan canonical form
>|> 
>|> What if you are using a field other than the complex numbers?  (Say, the
>|> rational numbers, the real numbers, or a finite field.)
>|> 

>It is false over the integers, but I don't know an example off the top
>of my head.

The integers aren't a field.
--
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.
Ich bin ein Virus.  Mach' mit und kopiere mich in Deine .signature.