HDTenEightyP/NetNews huggingface dataset:
From: platt@watson.ibm.com (Daniel E. Platt)
Subject: Re: tensors
Date: Thu, 20 Aug 1992 20:10:22 GMT
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: IBM T.J. Watson Research Center

In article <1992Aug20.190041.6215@pellns.alleg.edu>, frisinv@alleg.edu writes:
|>   I was reading a book on general relativity and the author began talking  
|> about tensor analysis, something I'd never heard of. I get the feeling  
|> tensors are some sort of number, but I'm not sure. Could anyone enlighten  
|> me as to what they are and how to use them?
|> ----
|> Vince

A Tensor is a list (array) of numbers indexed by some number of indices.
The elements of the array are called components.  A Tensor of rank 1 is
called a vector... just like from vector analysis.  A rank 2 tensor is
very much like a matrix.  

Tensors are defined by their transformation properties.  Ie, going
from rectangular coordinates to other rectangular coordinates or
from rectangular to spherical, etc, involves transformations.  
Tensor notation is a way of talking about components of vectors
that does not depend on the details of the coordinate systems.  For
example, covariant and contravariant derivatives is a way of
taking into account the effects of taking the derivatives of the
basis vectors; in spherical coordinates, the derivatives of the
basis vectors contribute to the differentiation (note the expressions
for acceleration in spherical coordinates in classical mechanics).
Tensor notation would express the coeficients that arise in a very
general way, that tracks the role those coeficients play rather
than the incidental form that arrises from the coordinate system or
the details of spacial curvature.  Lastly, since the notation does not
focus on the details of coordinate system, it makes a nice language
to try to pull out invariant structures and measures, such as curvature,
which is what General Relativity focused on.


Dan
Newsgroups: sci.math
From: chepan@gsusgi1.gsu.edu (Parthasarathy Nambi)
Subject: Re: mathematician's marrying ages.
Organization: Georgia State University
Date: Thu, 13 Aug 1992 14:21:19 GMT

hubert@spica.ucsb.edu (Chang;Hung-Hsien;H.;;;COMS;Graduate;930630;;;8-402525-19906-7;500) writes:

>My friend (yu@math.harvard.edu) keep advocating that marrying
>early is good for mathematician.  He gave several examples
>( most of them are number theorist, according to his statemnet)
>Examples he gave are Delinge, Gauss, Milnor, Serre, Zariski.
>He told me that he got these information from bibliography
>(maybe from introduction to authors...etc)


	Ramanujan married early (22) and died young. If I remember
	correctly his wife is still alive.

	What does marrying early and being a good mathematician
	has anything to do with each other? I think this is nonsense.

	This reminds me of a similar story that was published recently
	in the Chemical & Eng. News. Apparently there was story running
	around that chemists have a disproportionate number of baby
	girls than boys - based on a few well known US chemists.
	When the matter was looked carefully with a large number of chemists
	the situation changed and it was found there was no valid
	basis to the observation at all!!

	Later,

	P.Nambi
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math
Subject: Re: Math for Physics References
Date: 16 Aug 92 22:51:08 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <4540011@hpcc01.corp.hp.com> flower@hpcc01.corp.hp.com (Graham Flower) writes:
>     in question. So I guess what Im looking for is a book on mathematical
>     physics or two that will give me the framework to answer most of these 
>
>       Morse and Feshbach          

This is a bit dated, and mostly a reference. It has all the orthogonal
coordinate systems in which the Laplacian is separable, however, and not too
many other books have this.

>       Courant and Hilbert           

Actually, Hilbert had nothing to do with volume II. In fact, volume I was
really mostly written by Courant. volume II was written mostly by guys at
(what later became) the Courant Institute, such as Garabedian, Lax, etc.
There is a point of view (a little quaint in it's approach to functional
analysis) which these books are great on.

>       whittaker and watson          

Peter Lax once said of this book that there were two kinds of mathematicians
those who loved it and those who hated it. He hated it. He said this to
me while I was carrying a copy. (?) It has some good stuff, especially if
you like to read the original Hardy and Littlewood stuff in number theory,
but it is clearly dated. So is Watson's definitive treatise on the Bessel
functions. To some extent the Bateman project eclipses these, but I think
that's a bit old, too.

I would _urge_ you to consider the book _Mathematical Methods of Classical
and Quantum Physics_ by Byron and Fuller. It's based on T. D. Lee's course
notes, and it is a very nice complement to Courant and Hilbert.

Now special functions are nice and all, but modern physics needs a lot of
other stuff too, especially Calculus on manifolds and stochastic processes.
I'm not aware of a book that 'does it all', but Reed and Simon's four volumes
are as close as I know of. If you added to this Abraham and Marsden's
_Foundations of Mechanics_, you'd have a good Math shelf for physics. But
you'd have a bunch of homework, too.

A final thought - physicists have often 'absorbed' their mathematics without
the use of books explicitly labelled as mathematics. If you wanted to go
this route, you'll find some useful stuff on special functions in Jackson's
_Classical Electrodynamics_. You can learn a lot of applied math from the
classical fluid dynamics texts (Batchelor's Introduction, Lighthill's _Waves
in Fluids_ and Chorin and Marsden's book.) You can learn a good deal of
Calculus on manifolds from Misner Thorne and Wheeler's _Gravitation_.

Later,
Andrew Mullhaupt
From: dmagagno@mcs.drexel.edu (David Magagnosc)
Newsgroups: sci.math
Subject: A combinatorial question
Date: 14 Aug 92 14:27:36 GMT
Organization: Drexel University, Dept. of Math. and Comp. Sci.

For fixed n, let  f  and  g  be a pair of functions from
{1,2,...,n} to itself (not necessarily bijections).  Define
a distinguished pair to be a pair (i,j) for which f(i)=j and g(j)=i.

Question:  for how many pairs of functions (from the n^(2n) such pairs)
are there exactly k distinguished pairs?

I'd be curious about closed expressions, generating functions,
asymptotics, ..., references, anything.

D. Magagnosc
-- 
496620796F752063616E207265616420746869732C20796F752063616E206265636F6D65206120
636F6D70757465722070726F6772616D6D657220616E6420676574206120676F6F64206A6F622E
Newsgroups: sci.math
From: roach@bikini.wri.com (Kelly Roach)
Subject: u(v^n)w prime puzzle - HINTS
Organization: Wolfram Research, Inc.
Date: Wed, 19 Aug 1992 19:41:49 GMT




      Prove or disprove:  There are three non-empty
      strings of digits u,v,w such that all the
      numbers in
           L = {u(v^n)w | n is a natural number}
             = {uw, uvw, uvvw, uvvvw, uvvvvw, ...}
      are prime numbers.


			HINT TIME!!!

     This is the puzzle I posted Monday.  Today I'll
give out some hints.  On Friday, I'll explain the
solution if I can't get anyone to solve the problem
before then.

     HINTS:
      
     (1) Fermat's Little Theorem says
     
     c^(p-1) = 1 (mod p)

if p is prime and (c,p)=1.
     (2) Let a=|v|=length of v, b=|w|=length of w.
Then

     u(v^n)w = u*10^(a*n+b) + v*Sum[10^(a*i+b),{i,0,n-1}] + w

Consider the two pieces

     u*10^(a*n+b) + w
     
     v*Sum[10^(a*i+b),{i,0,n-1}]

separately.
     (3) If prime number p does not divide 10, then
     
     10^(a*i+b) (mod p)

is a periodic function of i.
     (4) If uw is not a prime, there is not much to prove.
Might as well assume uw is a prime number.



Newsgroups: sci.math
From: andyc@eliot.uucp (Andy Collins)
Subject: Continued Fractions
Keywords: Help, Continued Fractions
Organization: Systems Center, Inc., Reston, VA, USA
Distribution: na
Date: Thu, 20 Aug 1992 16:56:23 GMT

I am looking for information on continued fractions. I would
hope that there are books on the subject out there, but have
not had the time to do a full-out search. I would appreciate
any information.

Thanks

andy

PS (not a trademark)
Where are continued fractions used?
Who has done the most work on these things?

If you can could you post replies, my mail will be straigtened out
by next week.



Newsgroups: sci.math
From: erick@fraser.sfu.ca (Erick Bryce Wong)
Subject: Re: u(v^n)w prime puzzle
Organization: Simon Fraser University, Burnaby, B.C., Canada
Date: Wed, 19 Aug 1992 21:02:43 GMT

roach@bikini.wri.com (Kelly Roach) writes:
>rmurphy@aludra.usc.edu (Bob Murphy) writes:
>> If there were such values of u, v, and w, then one could construct
>> arbitrarily large prime numbers.  Since I have often heard
>> reference to "the largest known prime number", I would guess that
>> there are no such values.  Now if only I had a proof.
>
>     Not quite.  The largest known prime number is 2^756839-1.
>But emphasize the word "known" here.  In 1985, 2^216091-1 was
>the largest known prime number.  It takes a few years to
>break each new "largest known prime number" record.

[...]

>     P = 2*3*5*7*11*13+1 = 30031 = 59 * 509
>
>gets you two new prime numbers 59 and 509.  The process
>can be repeated ad infinitum.

The key difference here is that factoring P takes a *LONG* time, whereas if
we had u, v and w, we could trivially construct a prime greater than any given
number.  But I don't think this proves that they don't exist (though I'm pretty
sure they don't) because if they did exist, it doesn't necessarily mean we can
actually *find* them in any reasonable amount of time.  :-)

-- 
 -- Erick, the perfect square  :-)
Newsgroups: sci.math
From: wjcastre@magnus.acs.ohio-state.edu (W.Jose Castrellon G.)
Subject: Big primes puzzles
Keywords: Ackermann function/Puzzle/Primes
Organization: The Ohio State University,Math.Dept.(studnt)
Date: Thu, 13 Aug 1992 05:00:21 GMT

First I'd like to thank the people that emailed me quite elegant solutions
to the problem I posted last week. I was surprised to learn how many eminent
mathematicias are silent netters. Thank you again.

Now a puzzle that has to do with the Ackermann function and big primes:
The Ackermann function is a computable map that grows so fast, that for each
primitive recursive function it eventually majorizes it; it came out as a solu-
ion to a problem of Hilbert, who asked whether all intuitively computable
functions could be defined by a primitive recursive scheme.

Fast growing functions were the essence of the first arithmetic statement (not
directly related to logic) shown to be independent of Peano Arithmetic (Paris &
Harrington, and yes!, naturally...Solovay has theorems in that area too).
The following variant of Ackermann's original function is usually given
that name:

=========================================================================
A(0,m) = m+1
A(n+1,0) = A(n,1)
A(n+1,m+1) = A(n, A(n+1,m))
=========================================================================

I learned of the puzzle from S.Camargo who was an undergrad T.A. at Univ. de
Los Andes, and was taking a course in Recursion Theory. He posted a sign
offering a 5.0 (A+) to any of his precalculus students that would:

=================================================
1. Find a closed form for A(c,n), c in {1,2,3,4}
2. Decide whether A(4,4) is a prime or not.
=================================================

Of course, everybody knew the grade offer was a joke, but the question sparked
interest among the sci-eng students, and fairly quickly a cosci major came up
with the solution; he soon switched majors.
Newsgroups: sci.math
From: steve@hubcap.clemson.edu ("Steve" Stevenson)
Subject: Did Gauss ever say ...
Organization: Clemson University
Date: Mon, 17 Aug 1992 17:28:58 GMT

Bartlett's Familar Quotations give Gauss credit for the statement

"Mathematics is the queen of the sciences."

I heard anecdotally that he also said "And number theory is the queen
of mathematics." 

Can someone give me a references to the statement, please?

thanks.

steve


-- 
===============================================================================
Steve (really "D. E.") Stevenson           steve@hubcap.clemson.edu
Department of Computer Science,            (803)656-5880.mabell
Clemson University, Clemson, SC 29634-1906
Newsgroups: sci.math
From: uds@parvathi.cs.wayne.edu (Seetamraju)
Subject: Fourier series programs ....?
Organization: Wayne State University, Detroit, MI
Date: Mon, 17 Aug 1992 22:17:07 GMT


(This notice is for the confusion-master s.
 I am posting a request for Fourier SERIES, not Transforms).

Does anyone have (Free -- no cost-- obligations ok) programs
in C, pascal, fortran, matlab, (I do not have access to mathematica
, but I shall happily accept such help),   OR 
OR     cc4(now called XPLORE),....

that, take an arbitrary polynomial (in vector form or symbolic)
and spits out the fourier series coefficients upto some nth term
(uptill 5th or 6th is sufficient for now).  The coeffs may be for
the complex-exp form of fourier series or the cosine-sine form.
I can manage. with both.

(NOTE: THE POLYNOMIAL IS INTERVAL DE-LIMITED. I.E.,
THE PERIODIC FUNCTIONS I AM DEALING WITH ARE PERIODIC
WITH PERIOD = 2*PI(FOR SIMPLICITY) AND WITHIN EACH
PERIOD, THE FUNCTIONIS A POLYNOMIAL).

I wrote one in CC4, transfered it to matlab.  Both are not working
correctly.  I am unable to debug them.  So pl. help me.



Seetamraju Udaya Bhaskar Sarma
(email : seetam @ ece7 . eng . wayne . edu)


P.S. My by-hand-calculations verify that my programs
	have errors.  The moment I look at the program
	I fall asleep.

P.S.2 (No, its not IBM PS/2) If you write one now,
	put it on sci.math for others benefit too.  thanx.
From: mapsj@gdr.bath.ac.uk (Simon Juden)
Newsgroups: sci.math
Subject: Re: Is Card(R)=Card(R^2)?
Date: 13 Aug 92 15:44:22 GMT
Organization: School of Mathematics, University of Bath, UK

In article <1992Aug13.131518.2982@gdr.bath.ac.uk> mapsj@gdr.bath.ac.uk (Simon Juden) writes:
>In article <1992Aug13.011522.11161@informix.com> proberts@informix.com (Paul Roberts) writes:
>>In article <1992Aug13.000928.12631@unidus.rz.uni-duesseldorf.de> heisen@convex.rz.uni-duesseldorf.de (Henner Eisen) writes:
>>>In article <1992Aug12.102140.5231@nntp.hut.fi> samu@lammio.hut.fi writes:
>>>>I have been wondering... It seems first that there are more pairs of
>>>>real numbers than real numbers, but is it really so? And if not, I would
>>>>like to see a bijective mapping f:R->R^2.     ...
>>>
>>>Such mappings exist! To construct a mapping f(x,y) = z  choose a
>>>unique decimal representation of x and y and then merge the digits.
>>>
>>>i.E.:
>>>x =  1 0 0 2. 7 1 8 2 8 2 . . . . . .
>>>y =   3 0 3 .1 4 1 5 9 3 . . . . . . 
>>> 
>>>z =  1300032.174118529832............
>>>
>>>This will be bijective for non-negative real numbers. In order to obtain a
>>>bijective mapping R <-> R^2 this has to be combined with a bijection
>>>which maps real numbers to non-negative real numbers (right now, I
>>>don't remeber by heart how this is done, but it is possible).
>>
>>I believe that there is even an everywhere-continuous 
>>bijective mapping from the unit line to the unit square.
>
>
>Indeed so - an elementary version of Peano's construction of such may be found
>on page 283 of the American Mathematical Monthly, April 1983.
>
>Simon

Indeed not - I misread you. Peano's map is a cts surjection but of course as
has been pointed out an elementary argument yields that there can be no
bijection. Apologies for the inaccuracy, due to my reading too fast!

Simon
Newsgroups: sci.math
Subject: Re: POLLY NOMIAL, Humor
From: chisnall@cosc.canterbury.ac.nz (The Technicolour Throw-up)
Date: 23 Aug 92 21:34:50 +1200
Distribution: world
Organization: Computer Science,University of Canterbury,New Zealand

From article <1992Aug10.035041.16514@odin.corp.sgi.com>, by shreiner@arioch.csd.sgi.com (Dave Shreiner):
> 	"Exchlf!" he swore, using the vilest oath he knew.  His patience was

I've never understood this part of the story.  What exactly is the
significance of "exchlf"?  I've always assumed that its an acronym or
mnemonic from some branch of maths.  Does anybody know?

--
Just my two rubber ningis worth.
Name: Michael Chisnall  (chisnall@cosc.canterbury.ac.nz)
I'm not a .signature virus and nor do I play one on tv.
From: bhonsle@bhonsle.iss.nus.sg (Shailendra K Bhonsle)
Newsgroups: sci.math
Subject: Can you do it
Keywords: number theory
Date: 21 Aug 92 02:00:53 GMT
Followup-To: Can you do it?
Organization: Institute of Systems Science, NUS, Singapore



>>In an article in sci.math, you said:

>>:Prove that for primes p & q, q<p
>>:
>>:if 2^(p-1) == 1 (mod q^2)
>>:then either 2^(q-1) == 1 (mod q^2) or q | p-1  { q divides (p-1}
>>:
>>:I have some sort of solution but I am not very sure.
>>::Shailendra
>>:
>>:(bhonsle @ iss.nus.sg)

>>Bob Silverman (bs@gauss.mitre.org) replied:

>>> We have that 2 has order p-1 in the group of units of Z/q^2Z.
>>> Hence, p-1 | phi(q^2) by Lagrange's theorem.

>That's not correct.  We only know that  2^(p-1)  is 1, not that  p-1  is
>the minimal exponent such that  2^e  is 1.  E.g. p=13 and q=3 satisfy the
>hypothesis, but not Bob's conclusion.

>Here's a correct proof:  Let s be the order of 2 mod q; i.e. s is the
>smallest positive integer such that  2^s == 1 (mod q).  Note that
>s | p-1  and  s | q-1,  since both  2^(p-1)  and  2^(q-1)  are congruent
>to 1 mod q.
>
>Let  2^s = 1 + qk.
>
>If k is divisible by q, then  2^s == 1 (mod q^2).  Since  s | q-1,  it
>follows that  2^(q-1) == 1 (mod q^2).
>
>If k is not divisible by q, let  p-1 = sr.  Then  1 == 2^(p-1) =
>(2^s)^r = (1+qk)^r == 1 + qkr (mod q^2).  So  q^2 | qkr.  Since k is
>not divisible by q, it follows that  q | r,  so  q | p-1.
>
>(Note that the assumption that p is prime is not needed.)
>
>Dean Hickerson
>drhickerson@ucdavis.edu


Absolutely correct, (I think). I had exactly the same proof.

It is very interesting to note that p may not be prime.

Thanks a lot.

Shailendra

N.B.: I will post this article on the net.



--
From: mccolm@darwin.math.usf.edu. (Gregory McColm)
Newsgroups: sci.math
Subject: Re: You know, the integers (was: Re: Stupid question about FLT)
Date: 11 Aug 92 19:01:19 GMT
Article-I.D.: ariel.1992Aug11.190119.23518
Organization: Univ. of South Florida, Math Department

In article <1992Aug4.174928.1071@sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <1992Jul29.173257.5041@ariel.ec.usf.edu> mccolm@darwin.math.usf.
>edu. (Gregory McColm) writes:
>
>   >slightly less plausible than the claim that the 
>   >integers (whatever they are) (if they are) form a 
>   >model of PA.
>
>  A model of what? Any uncertainty you may feel concerning what "the integers"
>are carries over in a routine fashion into an uncertainty concerning what
>"PA" is.


There are two problems here.  First, do we really believe 
that the integers (model <N,+,.,0>) `exists' in some 
sense?  Realists say YES, materialists say NO.  
Nowadays, it is presumably clear what we mean by 
the term "integer", but whether this notion represents 
a real object, or some commonly accepted fiction, (or 
both:  a commonly accepted fiction might "become real" 
in some sense by virtue of popularity).

From there, we get more problems.  Suppose that you 
believe that the model of the integers exist.  Is PA 
true on this model?  Remember that this model would be 
an infinite object, which undermines any claim that 
PA is "obviously" true on it.

On the other hand, suppose that this model does not 
exist.  Is the assumption "the model exists and 
satisfies PA" going to lead to contradictions?  Is 
the answer to this obvious?

I believe that the model `exists' and satisfies PA;  
the point of my note was that a Cartesian approach to 
mathematics (ie, use only absolutely true facts) leads 
to instant impotence:  if one is to move the world, 
one must ASSUME a place to stand.

-----Greg McColm
From: mccolm@darwin.math.usf.edu. (Gregory McColm)
Newsgroups: sci.math
Subject: Re: Bell's _Men of Mathematics_ (was: Great and Inspirational Math books?)
Date: 13 Aug 92 19:38:48 GMT
Organization: Univ. of South Florida, Math Department

In article <1992Aug6.220654.28014@cl.cam.ac.uk> cet1@cl.cam.ac.uk (C.E. Thompson) writes:
>In article <1992Aug6.095915.1@lure.latrobe.edu.au>,
>matap@lure.latrobe.edu.au (Andrej Panjkov, La Trobe Maths) writes:
>|> In article <1992Aug4.194324.27258@news2.cis.umn.edu>,
>|> thomasc@ricci.geom.umn.edu (Thomas Colthurst) writes:
>|> > 
>|> > _Men of Mathematics_, E.T. Bell 
>|> 
>|> That book pissed me off.  I found it was written in a sneering and arrogant 
>|> tone.  I agree that the lives and achievements of the greatest 
>|> mathematicians are worth reading about, but I'd try another book.
>
>Does anyone else have this reaction? Bell has his strong opinions, and is
>(of course) very far from being PC, but I am slightly amazed at quite so
>negative a report.
>
>Chris Thompson
>

Eric Temple Bell is what you might call a member of the 
mathematical left (very pure and supports abstraction), 
while Morris Kline was what you might call a member 
of the mathematical right (very applied and opposed 
to abstraction).  Both have written interesting books 
on the history of mathematics, and these books have 
been criticised for being opinionated and error-infested.
In particular, Bell despises all politicians (he's not 
alone in this), and has a tendency to think that good 
mathematics is a mark of good character.  He also 
repeats gossip incautiously.

I don't know of a conventional, responsible, moderate 
book containing short biographies of mathematicians.  
As for histories of mathematics, Carl Boyer is a very 
good conventional, responsible, and moderate writer 
of mathematical history;  his book is quite reasonable.  
Edna Kramer's is more ahistorical.  Other nominations?

-----Greg McColm
Newsgroups: sci.math
From: jwr@lentil.princeton.edu (Jaroslaw Tomasz Wroblewski)
Subject: Re: Primes in x_{n+1} = ax_n+b (was Re: u(v^n)w prime puzzle)
Organization: Princeton University
Date: Fri, 21 Aug 1992 18:06:14 GMT

In article <1992Aug21.103132.29967@ecrc.de> jeanmarc@ecrc.de writes:
>Let a,b be integers, and (x_n) be a sequence s.t. x_{n+1} = a x_n + b
>
>My question is: does it contain infinitely many primes ?
>
>There are trivial cases, where the answer is no:
>1/ if (a x_0 + b = x_0) the sequence is constant.
>2/ if (a = -1) the sequence alternates between two values x_0 and x_1.
>3/ if ((a,b) > 1 or (x_0,b) > 1) then the sequence clearly contains only composite numbers.
>
>What about the other cases ?

For any odd b we can see easily that x_n=2^n-b satisfies the relation
x_{n+1} = a x_n + b with a=2 and is not a trivial case above.

For b=1 we get Mersene numbers. Most people will bet there are infinitely
many primes here, but I would not expect any proof coming.

However there exist odd b such that the sequence x_n=2^n-b contains finitely
many primes. That is the case because one can pick b in such a way that
For n=1(mod 2)    3|2^n-b
For n=2(mod 4)    5|2^n-b
For n=0(mod 3)    7|2^n-b
For n=4(mod 12)  13|2^n-b
(So far everything except 8(mod 12) is covered)
For n=0(mod 8)     17|2^n-b
For n=20(mod 24)  241|2^n-b

Then for every n the number 2^n-b will be divisible by at least one of the
primes 3,5,7,13,17,241.

Therefore 1 and the above determined b will give different answer to your
question. This shows that even making a good guess when such sequences
contain infinitely many primes is tricky.

Once we pick b as above I do not see a reason why other sequences
satisfying x_{n+1} = 2 x_n + b should also contain only finitely many
primes. Knowing a and b alone may not be enough to decide how many primes
does a sequence have.

--

Jarek (Jaroslaw Tomasz Wroblewski) ,   E-mail jwr@math.Princeton.EDU
Newsgroups: sci.math