HDTenEightyP/NetNews huggingface dataset:
From: rwc124@rsphy7.anu.edu.au (Roderick Vance)
Subject: Matrix Logarithms
Reply-To: rwc124@rsphy7.anu.edu.au (Roderick Vance)
Organization: Optical Sciences Centre, Australian National University
Date: Wed, 26 Aug 92 05:05:44 GMT

Let U be a unitary, symmetric matrix; U is expressible as

U = exp(i H), where H is real and symmetric.

I should like to know whether there is a simple method of finding
H given U, aside from diagonalising U. My aim is to find out, as simply as
i can, whether there are noughts in certain given positions of the H-matrix.

My present method is to diagonalise U = P L P^-1, where L is diagonal, and
then i get H = -i P log(L) P^-1. This method is forbiddingly unwieldy for
matrices larger than 4x4. I have thought of writing U as a power series in
H (the power series is finite, owing to the Cayley-Hamilton Theorem) but
cannot seem to make further headway from there.

A related problem is the following:

Let U(z) = exp(i H z) where H is real and symmetric and KNOWN and z is real. 
Let x0 be a constant complex element vector. Then consider the set:

T = {a U(z) x0: a is complex, z is real}

This time, GIVEN H, is there a simple test to find out whether another
given vector
xf belongs to T. 

Note: It is dead simple to show that T is NOT a linear space, so the
problem isn't
as easy as finding a basis spanning T.

Anyone interested in this problem: please post answers on the net or
write to me
directly, as you will.

Thanks in anticipation of your answers.


Roderick Vance
Optical Sciences Centre
Australian National University
From: mueller@schaefer.math.wisc.edu (Carl Douglas Mueller)
Newsgroups: sci.math
Subject: Re: God's Existence
Date: 2 Sep 92 21:10:07 GMT
Reply-To: mueller@schaefer.UUCP (Carl Douglas Mueller)
Organization: Univ. of Wisconsin Dept. of Mathematics

In article <1992Sep2.235433.89970@vaxc.cc.monash.edu.au> kevin@vaxc.cc.monash.edu.au writes:
>I can't seriously believe that someone can assert God's existence by
>claiming it is an axiom - written in the bible.

>At first, when I read this in a posting I thought it was just
>a joke. But it keeps on popping up? Am I missing the humor? 

I too thought it was a joke ... it's absurd to expect a mathematical proof
of God's existence (or non-existence for that matter).  My reply was to 
post that the only proof I knew of depended on the axiom of choice.  This
was supposed to be a joking reply (and I thought it was reasonably funny).
Unfortunately, there seem to have been actual historical attempts to use 
mathematics to prove the existence of God, and (also unfortunately) at
least some of these seem to have made actual reference to the axiom of
choice.  Thus my joke was lost.  Ah, well.
(Unless all these replies have been jokes -- in which case, it
has clearly gone on long enough.)

>Regards,
>Kevin Davey
>
           Carl Mueller (mueller@math.wisc.edu)
Newsgroups: sci.math
From: amc1@cec2.wustl.edu (Adam M. Costello)
Subject: Re: Godel and the US constitution
Keywords: Constitution states
Organization: Washington University, St. Louis MO
Date: Wed, 2 Sep 1992 23:21:18 GMT

In article <92Sep2.183019edt.207@neuron.ai.toronto.edu> radford@cs.toronto.edu (Radford Neal) writes:
>
>Ahhh! But in Godel's time, there were also _territories_. Surely Alaska
>was plenty big enough for hundreds of states?
>
>For that matter, do states have to occupy any land? 

I seem to remember something about a territory needing 50,000 inhabitants
(or was that 50,000 adults, or adult males, I can't recall) to even apply
for statehood.

AMC
Newsgroups: sci.math
From: radcliff@csd4.csd.uwm.edu (David G Radcliffe)
Subject: Re: A Math Induction Problem
Organization: UW-Milwaukee Math Department
Date: Mon, 31 Aug 1992 15:20:32 GMT

bubai@matt.ksu.ksu.edu (P.Chatterjee) writes:
: I have solved/near-solved a math induction problem. Was wondering if somebody could point out whether it's flawed or not.
: 
: I am trying to prove that n^3 + 5n is divisible by 6.

n^3 + 5n = 6 [ C(n+1,3) + n ].  

---
David Radcliffe                                
radcliff@csd4.csd.uwm.edu            
Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: Polygon point enclosure
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 27 Aug 1992 18:00:24 GMT

In article <thompson.714934114@daphne.socsci.umn.edu> thompson@atlas.socsci.umn.edu writes:
:gjm11@cus.cam.ac.uk (G.J. McCaughan) writes:
:
:>In article <7122@bigbird.hri.com.hri.com>, mps@sparc68.hri.com (Mark Stockley) writes:
:
:>> I'm looking for a method to determine if a point is enclosed
:>> by a polygon. Any help would be greatly appreciated.
:
:>Well, here's one approach. Imagine a ray (=half-line) coming off from the point
:>in some direction. Count how many of the sides of the polygon it goes through.
:>(If it goes *along* any of those sides you're in trouble and should try another
:>direction.) If it's an odd number the point is inside the polygon; if it's an
:>even number the point is outside the polygon.
:
:This algorithm is in _Algorithms_ by Robert Sedgewick, Addison-Wesley
 
I have a question.

I will assume that the polygon is given by a set of points.

Counting how many sides it intersects seems to require computing the
equation for the line containing each side and determining whether
the ray intersects the line between the two points. This requires
computing O(N) equations and intersections.

Why is this any faster than simply representing the interior of
the polygon as a set of linear inequalities and simply checking
whether the desired point satisfies each inequality?? This is also O(N).
This is essentially equivalent to whether a solution is feasible
in linear programming.
--
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: zukicn@wl.aecl.ca
Subject: COMPUTER MODELLER Position - R&D
Organization: AECL RESEARCH
Date: Thu, 27 Aug 1992 15:16:06 GMT

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Thank you,

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        An advanced degree in Mechanical, Chemical or Nuclear
        Engineering, Applied Mathematics, or Physics is preferred,
        although a bachelors degree with high academic standing and
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        A strong background in heat transfer and scientific
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                 Pinawa, Manitoba
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From: lordSnooty@cup.portal.com (Andrew - Palfreyman)
Newsgroups: sci.math
Subject: Re: Proof of God's Existence
Date: Tue,  1 Sep 92 04:11:07 PDT
Organization: The Portal System (TM)
  <1992Sep1.090011.1@tnclus.tele.nokia.fi>

P.Chatterjee writes:
> I was told by somebody that there is a mathematical proof of God's existence;
> was wondering if somebody could shed some light on the same.

Hannu:
: I think that God's existence is an axiom. He says [...]

Clearly, my friend. Now answer the question, if you would be so bold.
The alternatives are rude, or even worse, I would hazard.
 --------------------------------------------------------------------------
| lord snooty @the giant | Would You Like Fries With That?                 |
| poisoned electric head |              andrew_-_palfreyman@cup.portal.com |
 --------------------------------------------------------------------------
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math
Subject: Re: coskeleton
Date: 3 Sep 92 04:24:29 GMT
Distribution: sci
Organization: M.I.T. Artificial Intelligence Lab.

SOrry about the typo. The first question should read: "How do I construct
the k-th coskeleton of X?".

Allan Adler
ara@altdorf.ai.mit.edu
From: mad19@seq1.keele.ac.uk (Andy Daley)
Newsgroups: sci.math
Subject: Re: Proof of God's Existence
Date: 2 Sep 92 19:34:22 GMT
Organization: University of Keele, England

In article <17ui6kINNsft@matt.ksu.ksu.edu>, bubai@matt.ksu.ksu.edu (P.Chatterjee) writes:
> I was told by somebody that there is a mathematical proof of God's existence; was wondering if somebody could shed some light on the same.
> 
> Thanks. Looking forward to the response.

Don:t know about a mathematical proof, but for the best proof just read
the Gospels, found in the Bible.

Jesus is Alive! And he has the power to save people from death into
eternal life!!
From: loeb@greco-prog.fr (Daniel LOEB)
Newsgroups: sci.math
Subject: Summer Problems
Date: 2 Sep 92 10:47:41 GMT
Organization: GRECO Programmation du CNRS - Bordeaux,France

I think the first version of this posting got lost, so I am trying
again.

Last July, I ran a summer math camp here in Bordeaux. Some of my
students came up with some pretty interesting problems. Here is a
sample of them along with the results that they got.

(DMITRI) - A door has a very strange lock. It contains n holes
arranged in a regular polygon. There are buttons at the bottom of the
holes. When all buttons are up or all down, the door opens. Buttons
can only be examined or changed by sticking your arms in the holes.
When your arms are removed from the holes, the lock spins very quickly
and then stops at an unknown point. How many hands h(n) must you have
in order to be SURE to open the doors after a finite number of
manipulations. 
Partial results: if p is prime, then h(p)=p-1. Otherwise, h(4)=2 and
h(1)=0.
Is there a general rule? Generalizations?

(VINCENT) - 100 points are arranged in a 1m square. In the worst case,
how long is the shortest path connecting them all.
Partial result: Between 11 and 21 meters.
Can an exact result be found?

(STEPHANE) - 2 congruent rectangles intersect in exactly 8 points, how
small can their interection be?
Partial result: Over half the area of the rectangle.
Can you find a tighter bound, or a series of examples approaching 1/2
the area?

And here is an open problem from me (DANNY): 
A voting scheme for a set of n voters is a collection of winning
coallitions such that all supersets of a winning coallition is still a
winning coallition, and given any set of voters either it or its
compliment is a winning coallition.
The number of voting schemes is given by 
c(0)=0, c(1)=1, c(2)=2, c(3)=4=2^2, c(4)=12=2^3*3, c(5)=81=3^4,
c(6)=2646=2*3^2*7^2, c(7)=1422564=2^2*3*11*13*829.
Is there an easy way to calculate these numbers?

Send me your problems, comments, or answers and I'll reply and/or
forward them to my students.

Thanks.

  
 


 
  
-- 

Yours, Daniel Loeb				loeb@geocub.greco-prog.fr
HOME 	150, cours Victor-Hugo; Appt D45; 33000 Bordeaux France
WORK	LABRI; Universite de Bordeaux I; 33405 Talence Cedex France
Newsgroups: sci.math
From: asimov@wk223.nas.nasa.gov (Daniel A. Asimov)
Subject: Re: Godel and the US constitution
Organization: NAS, NASA Ames Research Center, Moffett Field, CA
Date: Tue, 1 Sep 92 17:58:44 GMT

In article <TORKEL.92Sep1191401@bast.sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <a_rubin.715364793@dn66> a_rubin@dsg4.dse.beckman.com 
>(Arthur Rubin) writes:
>
>   >This reminds me of a story I heard at a memorial seminar on Godel, on which
>   >I would like confirmation.
>
>
>
>   The story is told as follows by Hao Wang, in Reflections on Kurt Godel.
>Wang refers to H.Zemanek and E.Kohler.
>
	[...]
>
>          The judge began, 'You have German citizenship up to now'. G
>        interrupted him, 'Excuse me sir, Austrian.' 'Anyhow, the wicked
>        dictator! but fortunately that is not possible in America.' 'On
>        the contrary,' G interjected, 'I know how that can happen.' All
>        three joined forces to restgrain G so as to turn to the routine
>        examination.
>
--T. Franzen
-------------------------------------------------------------------------

Out of curiosity, does anyone know how Godel envisioned a
consitutional dictatorship possibly arising in the U.S. ???


--Dan Asimov
asimov@nas.nasa.gov

Mail Stop T045-1
NASA Ames Research Center
Moffett Field, CA 94035-1000
(415) 604-4799
From: cet1@cus.cam.ac.uk (C.E. Thompson)
Newsgroups: sci.math
Subject: Re: Why don't we split the group?
Date: 4 Sep 92 13:52:55 GMT
Organization: U of Cambridge, England

In article <4066@svin09.info.win.tue.nl>, pwk@wsinfo01.win.tue.nl 
(Pieter Kuppens) writes:
|> I would like to see:
|> sci.math.coding        (with error correcting/cryptology/information theory)
|> sci.math.group         (group theory)
|> sci.math.misc          (whatever...)
|> sci.math.puzzles       (well, puzzles appear sometimes)
|> and many more.

You have forgotten

sci.math.existence-of-god
sci.math.marriage
sci.math.us-constitution

which recent contributions to sci.math seem to indicate is what the punters
really want.

Chris Thompson
JANET:    cet1@uk.ac.cam.phx
Internet: cet1@phx.cam.ac.uk
Newsgroups: sci.math
From: israel@unixg.ubc.ca (Robert B. Israel)
Subject: Re: x 2^x = 2^n ?
Organization: University of British Columbia, Vancouver, B.C., Canada
Date: Fri, 28 Aug 1992 07:37:44 GMT

In <israel.714951978@unixg.ubc.ca> israel@unixg.ubc.ca (Robert B. Israel) writes:

>In <17gnuqINNr7k@darkstar.UCSC.EDU> soeren@cse.ucsc.edu (Soren Soe) writes:

>>Can anyone tell me what `x' is in this equation:

>>	x 2^x = 2^n

>>Is there a general solution, or do I have to approximate `x' to something like
>>x = n - lg n + a_little_more ?
>>In fact I can do with an approximation to x, iff 
>>		floor(x) <= app(x) <= x

>>I need the "floor" of `x' to solve the following summation, where k 
>>would be equal to floor(x):

>>	i=n-1
>>	SUM   min(2^i, 2^2^(n-i))
>>	i=0

>>	i=k	     i=n-1
>>    =   SUM  2^i  +  SUM   2^2^(n-i)
>>	i=0	     i=k+1

>>--soeren

>>Soren Soe,
>>soeren@ce.ucsc.edu

>I assume n is a positive integer.  For n = 1, x = 1, so suppose n >= 2.

>Write n = 2^m k where m is a positive integer and 1 <= k < 2, i.e. 
>m = floor(lg n).

>Let x1 = n - m - 1.  Then x1 2^x1 < n 2^x1 = k 2^(n-1) < 2^n.  So x1 < x.

>Let x2 = n - m + 1.  Then x2 2^x2 = 2^(n+1) k x2/n >= 2^(n+1) (1 - (m - 1)/n)
>                                 >= 2^(n+1) (1 - (m-1) 2^(-m)) > 2^n

>since  m - 1 < 2^(m-1) for all m.  So x > x2.  Thus floor(x) is either
>x3 = n - m or x1 = n - m - 1.

>Now x3 2^x3 = (n-m) 2^(n-m) = (n-m)k/n 2^n.  To have floor(x) = x3 we need
>n <= (n-m) k = (n-m) n 2^(-m), i.e. n >= 2^m + m.  Thus we conclude that

Oops! Reverse these inequalities: we need n <= 2^m + m.

>floor(x) = n - m - 1 if 2^m <= n < 2^m + m for some m
>floor(x) = n - m     if 2^m + m <= n < 2^(m+1).

Should be:
floor(x) = n - m if 2^m <= n <= 2^m + m
floor(x) = n - m - 1 if 2^m + m < n < 2^(m+1).

>  
>-- 
>Robert Israel                            israel@math.ubc.ca
>Department of Mathematics             or israel@unixg.ubc.ca
>University of British Columbia
>Vancouver, BC, Canada V6T 1Y4
-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4
Xref: sparky sci.math:10983 sci.psychology:3637
From: wsadjw@rw7.urc.tue.nl (Jan Willem Nienhuys)
Newsgroups: sci.math,sci.psychology
Subject: God's and Godel's marrying ages.
Date: 4 Sep 92 11:41:44 GMT
Reply-To: wsadjw@urc.tue.nl
Followup-To: sci.psychology
Organization: Eindhoven University of Technology, The Netherlands

Please, can we have this extraneous material out of sci.math?
I mean, the inexhaustible God-stuff, Godel and the U.S. Constitution
and the endless marrying ages.  
Please mind the follow-up line.

JWN
Newsgroups: sci.math
From: bhonsle@bhonsle.iss.nus.sg (Shailendra K Bhonsle)
Subject: Re: u(v^n)w prime puzzle - HINTS
Organization: Institute of Systems Science, NUS, Singapore
Date: Mon, 24 Aug 1992 02:44:08 GMT

In article <1992Aug20.161823.20996@wri.com>, roach@bikini.wri.com (Kelly Roach) writes:
|> In article <1992Aug20.043232.5087@nuscc.nus.sg> bhonsle@bhonsle.iss.nus.sg  
|> (Shailendra K Bhonsle) writes:
|> > |>      (2) Let a=|v|=length of v, b=|w|=length of w.
|> > ...
|> > As I mentioned in my last article let us take prime p="uw".
|> > ...
|> > Now we show that p cannot divide 10^a -1 to complete the proof.
|> >      case 2: b <= a
|> >              SIMPLE, Can you do it ?
|> > ...
|> 
|> 
|>      Consider:
|> 
|>      u="1",v="36",w="1"
|>      11, 1361, 136361, 13636361, 1363636361, 136363636361
|> 
|> Here, a=2, p=11, 10^a-1=99, and p | 10^a-1.  So, yes, it
|> is possible for p to divide 10^a-1.  Some more work on this
|> proof needs to be done before it is complete I think.
|> 
|> 				Kelly
|> 


I see the mistake. The key is to choose n correctly. 
Let p^i exactly divides  10^a - 1   {where i = 1, 2 ,3 ...}
then choose n= (p^(i-1))(p-1)

else if p does not divide 10^a-1 then choose n=p-1


This will correctly show the required result. 


Shailendra
--
From: bardo@ief-paris-sud.fr (Bardo Muller)
Newsgroups: sci.math
Subject: Re: Directed Graph Display Software - LOOKING?
Keywords: graph theory
Date: 31 Aug 92 09:19:58 GMT
Reply-To: bardo@ief-paris-sud.fr
Organization: Universite de Paris


>I want the nodes and directed links of a given directed
>graph to be displayed in a pleasing way - where clusters
>of tightly connected nodes are automatically arranged on
>the diagram in a local region.
>
>I am sure there are programs out there which display 
>directed graphs nicely, but where?
>
>I need to get out a paper on the evolution of random
>directed graphs, my graphs are evolving, but currently
>I transcribe the directed matrix to xfig and muck around
>with it four a couple of hours.  I could write a program
>- but I want to shake the bushes first.



Try archie on LEDA-<version>.tar.Z

( A Library of Efficient Data Types and Algorithms, including 
display of all sort of graphs )


SITE:  ftp.cs.uni-sb.de (134.96.252.31)
For a faster ftp link, email to

                             Stefan N"aher
                   Max-Planck-Institut f"ur Informatik
                  Im Stadtwald, 6600 Saarbr"ucken, FRG     
                          stefan@mpi-sb.mpg.de
  
>>>>>>>>>>>> .de doesn't mean Denmark... <<<<<<<<<<<<<<<<

Good luck !

Bardo 

___________________________________________________________________________
| Bardo MULLER                          | phone  : [33] [1] 69 41 78 50    |
| Institut d'Electronique Fondamentale  | fax    : [33] [1] 60 19 25 93    |
| Bat. 220 Universite Paris Sud         | e-mail : bardo@ief-paris-sud.fr  |
| 91405 ORSAY CEDEX FRANCE              |             THE NET WORKS !      |
|_______________________________________|__________________________________|
From: mccolm@darwin.math.usf.edu. (Gregory McColm)
Newsgroups: sci.math
Subject: Re: Beloved Books + Request
Date: 24 Aug 92 15:31:59 GMT
Organization: Univ. of South Florida, Math Department


I am posting this because my computer insists that 
wilson@ctron.com is a figment of my imagination.

Off the top of my head....
Some set theory books:

Set theory nowadays is divided into 3 major fields:

1.  Combinatorial set theory:  use of transfinite structures 
    to do transfinite arithmetic.  This is what is meant by 
    "classical set theory".  This is (distantly) related to 
    the combinatorics of finite sets, which is part of 
    (finite) combinatorics.

2.  Forcing:  use of boolean models to demonstrate the 
    independence of given axioms from axiomatic systems.
    (This is where AC was proven independent of ZF, and 
    CH from ZFC, and GCH from ZFC + CH, etc.)

3.  Descriptive set theory:  logical expressibility of 
    topologically definable sets (open, closed, unions 
    of closed, etc, borel, etc).  This are is involved 
    (entangled) with analysis;  since its roots go back 
    to Eudoxus, it really is the oldest branch.

Some surveys:

Drake's Set Theory
  A good book, out of date and out of print;  tilted towards 
  combinatorics

Halmos Naive Set Theory
  Some love it, some hate it.  No ZFC stuff.  I found it 
  incomprehensible

Jech's little book
  A paperback (lecture notes?) of about 150 pages.  Very 
  accessible.

Jech's big book
  An encyclopedia almost, full of EVERYTHING, including typos

Kunen's Set Theory and Independence Proofs
  A short survey tilted towards forcing



Other sources

Cohen's Set Theory and the Continuum Hypothesis
  The grand original on forcing:  opaque and not recommended
  (Shoenfeld wrote a decent paper on forcing--I forget where 
  it is.  Burgess's article in the Handbook of Math Logic is 
  error-riddled.  The accounts in Jech's books are okay;  Kunen
  is too fond of complex notation)

Erdos, Hajnal, Mate, Rado
  The grand encyclopedia of combinatorial set theory.  Prepare 
  to get hopelessly lost

Kunen's article on Combinatorial Set Theory
  In the Handbook of Math Logic (ed by J Barwise), a good 
  introduction

Kuratowski's Set theory & topology
  mostly topology and naive set theory

Moschovakis' huge book on Descriptive Set Theory
  A very good book & reference

Neil Williams' little book on Combinatorial Set Theory
  A VERY good little book;  the imperialists at Elsevier have 
  let it go out of print


===================================================================
                          |
Greg McColm               |    I see it, but I don't believe it.
Dept of Mathematics       |
Univ of S Florida         |        -----Georg Cantor
mccolm@math.usf.edu       |
Newsgroups: sci.math
From: gerry@macadam.mpce.mq.edu.au (Gerry Myerson)
Subject: Re: Einstein's Erdos number
Organization: CeNTRe for Number Theory Research
Date: Wed, 2 Sep 92 05:33:32 GMT

In article <Btwt5H.A6n@watserv1.uwaterloo.ca>, shallit@graceland.uwaterloo.ca (Jeffrey Shallit) writes:
[...]
> In the preface to "A Tribute to Paul Erdos", Cambridge
> University Press, 1990, it is stated that Einstein's
> Erdos number is 2.  That means there exists a person
> who has written a paper with Einstein, and a paper with
> Erdos.  Anybody know who it is?

Just a guess, but Ernst Straus was an assistant to Einstein in the late 40s, 
and surely a co-author of Erdos'. 

> Jeff Shallit
> University of Waterloo

Gerry Myerson
Newsgroups: sci.math