HDTenEightyP/NetNews huggingface dataset:
From: franjion@spot.Colorado.EDU (John Franjione)
Subject: Re: tensors: How about 3rd, 4th rank?
Organization: University of Colorado, Boulder
Date: Fri, 21 Aug 1992 23:01:04 GMT

Tom Pierce <pierct@rpi.edu> writes:

>What would you use a third or highr order tensor for?  Any PHYSICAL
>applications?

First, keep in mind that tensors are basically a notational convenince
for expressing linear relationships (e.g. a second order tensor is a
linear transformation from one first order vector space to another).

I do research in fluid mechanics of solids emmersed in very viscous
fluids.  Since the governing partial differential equations are
linear, tensors come in handy.  For example, the force on a particle
is linearly related to the velocity of the particle.  This is
conveniently expressed as:
	f = A . u
where f is the force vector, u is the velocity vector, and A is a
second order tensor.

Now say the particle is immersed in an ambient flow field, which is
characterized by a velocity gradient, grad(u).  The force on the
particle is linearly related to the velocity gradient:

	f = G . grad(u)

Since grad(u) is itself a second order tensor, G is a third order
tensor.  In this case, the "dot" product is actually a contraction
over 2 indices, rather than 1.

John Franjione
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math
Subject: Re: Knight's Tour
Date: 13 Aug 92 23:12:50 GMT
	<Bsw0tG.Lnn@watserv1.uwaterloo.ca> <1992Aug12.211739.3890@seas.smu.edu>
Organization: M.I.T. Artificial Intelligence Lab.

How many knight's tours are there, with no assumption on where they start?

Allan Adler
ara@altdorf.ai.mit.edu
From: "ian tuck" <ian.tuck@canrem.com>
Newsgroups: sci.math
Subject: please help!!!
Date: 18 Aug 92 20:33:35 EST
Reply-To: "ian tuck" <ian.tuck@canrem.com>
Distribution: sci
Organization: Canada Remote Systems

Hi all.  My first post here.  I have a problem I would like a computer
to solve, but can't seem to find a solution.  The problem is as follows
I wish to find out the most efficient solution to the following:
I have a box 20'X8'X8'.  I have other boxes of various dimensions.
If I have three types of boxes: 1) 1'x1'x1', 2) 2'x1'x1', and 3) 2'x2'x2'
how do I determine a method for filling up the large box with a number of
the smaller boxes with minimum volume left over.  There may be a restraint
(i.e. minimum 10 of box 1 type, 20 of box 3 type, any # of box 2 type).
I tried solving this problem with Excel using the solver and volumes, but
of course the volumes I calculated don't actually take into account
dimensions, (i.e the volume & the dimensions matter, so that no box sticks
out the top of the larger box).  Is there anyone out there who can give me
some method by which I can calculate this?  I understand calculus, but
couldn't come up with the minimax equations either.  Thanks in advance.

Ian
---
  DeLuxe/386 1.25 #8528  
--
Canada Remote Systems  - Toronto, Ontario/Detroit, MI
World's Largest PCBOARD System - 416-629-7000/629-7044
Newsgroups: sci.math
From: jac@cs.WPI.EDU (Jeffrey A Choate)
Subject: Mapping points in space to a plane
Originator: jac@penny.WPI.EDU
Organization: Worcester Polytechnic Institute
Distribution: usa
Date: Thu, 20 Aug 1992 16:20:04 GMT


Hello,

I am having a problem mapping a point to a plane, 
which I hope someone can help with.

The problem is as follows:

I have a point which lies on a plane with the
equation: x + y + z = 1 (point represents 
probabilities).  After I perform some non-linear 
transformation, the point no longer necessarily lies 
on the "probabilistic" plane.

I have to map this resulting point, which lies 
somewhere in 3-space back to the probabilistic 
plane.

My question is how to best map this point back to
the plane?

One suggestion is to find the orthogonal projection 
of the point on the plane.  Another suggestion is to 
use the normal to the plane (1,1,1), and starting at 
the point follow the normal vector back until it 
intersects the plane, and use the intersection point.

Are any of these ideas good?  Is there an algorithmic 
way of doing this? (I have to perform this mapping for 
every point in a very large data set) Any help would be 
greatly appreciated.

Thank you,

Jeff Choate

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~ Jeff Choate         | (508) 831-5006    | Computer Science Department     ~
~ jchoate@cs.wpi.edu  | AI Research Group | Worcester Polytechnic Institute ~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Newsgroups: sci.math
From: wsadjw@rw7.urc.tue.nl (Jan Willem Nienhuys)
Newsgroups: sci.math
Subject: Re: Is Card(R)=Card(R^2)?
Date: 13 Aug 92 13:26:23 GMT
Reply-To: wsadjw@urc.tue.nl
Organization: Eindhoven University of Technology, The Netherlands

In article <1992Aug13.011522.11161@informix.com> proberts@informix.com (Paul Roberts) writes:
>>
>>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.

Two remarks.  The first mapping was thought up by Cantor, but it is
wrong, because some numbers have two representations.
Trick: exclude representations that end in an infinite sequence of
zeros. Chop these in pieces not of a single digit, but of a sequence
of zeros (possibly of length 0) followed by a single non-zero.
Then mix these.

Second remark.  A continuous bijective map from a compact set onto
a compact set has a continuous inverse, i.e. a homeomorphism. The 
unit interval is not homeomorphic to the unit square.

Peano constructed a continuous map from the unit interval onto the
unit square (a fractal of dimension 2 actually).  But it is not injective,
it has many double points. At that time people were indeed worried that 
such a map (and then bicontinuous) might be found.  It would destroy
the idea of dimension.  But L.E.J. Brouwer proved around 1910 that
dimension is invariant under homeomorphism, so dimension is a topological
concept.

JWN
From: sundar+@cs.cmu.edu (Sundar Vallinayagam)
Newsgroups: sci.math
Subject: Re: Littlewood's Three Principles
Summary: Quote from Royden's "Real Analysis"
Keywords: Littlewood's Three Principles
Date: 12 Aug 92 20:24:40 GMT
Followup-To: sci.math
Organization: School of Computer Science, Carnegie Mellon

Gerald Edgar queries,

>Where do I find the original (published) formulation of Littlewood's
>three principles, or at least a verbatum statement of them?

The following is from Royden's "Real Analysis" (3rd edition), p. 72

6.  Littlwood's Three Principles

Speaking of the theory of functions of a real variable, J.E. Littlewood
says, ^6  "The extent of knowledge required is nothing like so great
as is sometimes supposed.  There are three principles, roughly expressible
in following terms: Every (measurable) set is nearly a finite union of
intervals; every [measurable] function is nearly continous; every convergent
sequence of [measurable] functions is nearly uniformly convergent.  Most of
the results of [the theory] are fairly intuitive applications of these ideas,
and the student armed with them should be equal to most occasions when real
variable theory is called for.  If one of the principles would be the obvious
means to settle the problem if it were `quite' true, it is natural to ask
if the `nearly' is near enough, and for a problem that is actually solvable
it generally is."

Footnote 6 gives the reference as [20], p. 26.  I think it is a typo
and the reference should actually be [21], which is: J.E. Littlewood, "Lectures
on the Theory of Functions", Oxford, 1944.

Hope this helps.

Ramli.




-- 
**************************************************
ramli@orca.ele.uri.edu
**************************************************
Xref: sparky sci.math:10509 sci.physics:13316
From: wsadjw@rw7.urc.tue.nl (Jan Willem Nienhuys)
Newsgroups: sci.math,sci.physics
Subject: Re: tensors: How about 3rd, 4th rank?
Date: 22 Aug 92 14:02:47 GMT
Reply-To: wsadjw@urc.tue.nl
Followup-To: sci.math
Organization: Eindhoven University of Technology, The Netherlands

In article <mcirvin.714433381@husc8> mcirvin@husc8.harvard.edu (Mcirvin) writes:
>pierct@rpi.edu (Tom Pierce) writes:
>
>The Riemann curvature tensor is fourth-order.  It has 4x4x4x4
>components in GR, though most of them are related by symmetries so
>it has many fewer independent components.  This describes the
>curvature of four-dimensional spacetime in general relativity.
>

The curvature tensor can be described without too much transformation
laws and indices.  
Take any two tangent vectors a,b, in a point p. They span a plane.  Take
a third tangent vector c.  Now carry this third vector around on the
manifold, but stay in the plane spanned by a and b (let's say in the 
submanifold consisting of geodesics emanating p from whose initial 
direction is a linear combination of a and b. After carrying around,
c will have changed a little Dc.  Of course Dc will be very small
if the path along which you have carried c is a very small circle or
other closed loop.  But if you carry c around a geodesic parallellogram
with tangent vectors a and b at p, then Dc is in first approximation
a multilinear function of a, b and c. Or you could say, you have a 
map c -> Dc that is a bilinear function of a and b.  This "trilinear
vector valued function" is what is usually called the curvature 
tensor.

For example on a sphere of radius R, a parallell transport of a
tangent vector around a closed loop results in a rotation through
an angle equal to the area enclosed by the loop (divided by R^2).
So in the above, Dc is always just the determinant of a and b divided
by R^2, times a c rotated through pi/2.  The larger R is, the closer
this is, the closer this is to the zero map.

If you try to compute the curvature tensor in concrete cases, then
you have to work with parametrizations of your manifold, and if you
want to compare different parametrizations, you get to deal with those
transformation laws. Just like when you do computations with vectors,
you represent them as rows or columns of numbers with respect to
a certain basis.  But for understanding what's going on, you don't
really need these gory details.

JWN
Xref: sparky sci.math:10154 sci.math.stat:1639 sci.math.num-analysis:2449 sci.chem:3000
From: unregistered@newcastle.ac.uk (Margarida Pereira-Leite)
Newsgroups: sci.math,sci.math.stat,sci.math.num-analysis,sci.chem
Subject: arma data
Summary: where to get arma data
Keywords: arma models, box and jenkins
Date: 10 Aug 92 16:06:34 GMT
Organization: Chemical & Process Engineering Dept, University of Newcastle, UK.

Hi!

I am very interested in finding real data following arma(1,1), arma(2,2),
ar(1)+noise or even ar(2)+noise models. If you know where I can get it or 
if you can send me some with the description of the process generating it
I would be very gratefull. My e-mail is

M.pereira-leite&newcastle.uk.ac

Cheers.
Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: A Non-Cantorian Set Theory question
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Thu, 13 Aug 92 17:31:37 GMT

In article <1992Aug12.113415.1648@gacvx2.gac.edu> kiran@gacvx2.gac.edu writes:
>Quite a while ago, I read Martin Gardner write in one of his _Mathematical 
>Games_ column in _Scientific American_ that on a plane, a letter like _O_ can 
>be written--allowing smaller O's to be written inside larger O's-- _c_ 
>times where _c_ is the cardinality of the continuum. On the other hand, he
>pointed out, letters like _T_ can only be written aleph-nought times.
>
>A question that has remained in my mind for a long time since is the following:
>
>Since we know that non-Cantorian set theories are possible, is there a
>one-dimensional shape which can be written some aleph times where that aleph is
>between aleph-nought and _c_? If so, what would the shape be?

Since the continuum hypothesis is undecideable in ZFC it will be
impossible to construct such a shape in ZFC.  Thus I don't think you
will ever get a very clear picture of what such a shape would look like
even if one "exists".  ("Exists" is in quotes here since it really means
"can be proved to exist in ZF together with some axioms that contradict
the continuum hypothesis".)  Nonetheless it's an interesting question.
Newsgroups: sci.math
From: wjcastre@magnus.acs.ohio-state.edu (W.Jose Castrellon G.)
Subject: Independence and rapidly growing functions
Organization: The Ohio State University,Math.Dept.(studnt)
Date: Tue, 18 Aug 1992 01:50:34 GMT


I have received a few requests for references on the connection between fast
growing functions and independence results in Peano Arithmetic, which I
mentioned in a casual comment on one of my previous postings. As it turned 
out, that comment sparked an interest in the topic, and I realized my posting
was rather incomplete, thus this brief note.

People with some background in logic could try Smorynski's _Lectures on non-
standard models of Arithmetic_ (Logic Colloquium '82 pp.1-70), after that
I am told that Ketonen-Solovay's _Rapidly growing Ramsey functions_ (Annals
of Math. 1981, pp.267-314) is the way to go (it deals with the combinatorial
issues of the Paris-Harrington incompleteness phenomena).

For further developments in this area  (as well as for the connections between
large cardinals and real numbers, which were recently mentioned in an article
by W.Boshuck @macgill)  you can look up the book _Harvey Friedman's Research
on the Foundations of Mathematics_  (Harrington et al. North Holland.1986),
which contains some of the most striking independence results,  obtained by
Prof. Harvey Friedman, here at Ohio State University. It also contains at the
end very readable accounts from Craig Smorynski, and one entirely accessible to
non-experts by Gina Kolata (NY Times Math Editor), which is reprinted from 
_Science_ magazine (1980-1983?,anyway all her articles make great reading). 

You might also try his lecture at the AMS Centennial entitled _The Incomplete-
ness Phenomena_  (in _Mathematics into the Twenty-first Century_ F.Browder ed.
AMS. 1991). By the way, IMO this volume is a *must have*: it contains all the
plenary addresses given by: Michael Aschbacher, Luis Cafarelli, Persi Diaconis,
Charles Fefferman, Michael Freedman, Harvey Friedman, Benedict Gross, Roger 
Howe, Vaughan Jones, Victor Kac, Andrew Majda, Karen Uhlenbeck and Edward
Witten... at just $99 its a bargain!.
From: bruck@mathj.usc.edu (Ronald Bruck)
Newsgroups: sci.math
Subject: Re: Did Gauss ever say ...
Date: 17 Aug 1992 15:07:03 -0700
Organization: University of Southern California, Los Angeles, CA
NNTP-Posting-Host: mathj.usc.edu

In article <1992Aug17.203632.21234@kth.se> TordM@VanD.PhySto.SE writes:
>In article <1992Aug17.172858.3795@hubcap.clemson.edu>, steve@hubcap.clemson.edu ("Steve" Stevenson) writes:
>
>>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." 
>
> not a reference, but I heard he made the statement "Arithmetics is the Queen
>of Mathematics"....
>

From Chapter 1 of "Mathematics, Queen and Servant of Science", by Eric Temple
Bell, published by the Mathematical Association of America:

	Mathematics is Queen of the Sciences and Arithmetic the Queen
	of Mathematics.  She often condescends to render service to
	astronomy and other natural sciences, but under all circumstances
	the first place is her due.

     So said the master mathematician, astronomer, and physicist C. F.
     Gauss (1777-1855).  Whether as history or prophecy, Gauss's declaration
     is far from an overstatement.

(End quote)

Bell doesn't give any references to the quote.  (Considering what people
had to say about Bell in another thread, I'll leave it at that.  Bell's
"Men of Mathematics" introduced me to mathematics as a sophomore in
high school; prior to reading that book I had no conception that
mathematics was a **research** area.  Similarly, "The New World of Math",
by George Boehm, casually mentioned in a footnote that "The schools most
highly rated by mathematicians are:  Chicago, Harvard and Princeton."  On
no more evidence than that I made up my mind, while still a sophomore,
that I would go to the University of Chicago.  Books like these can be
VERY influential on impressionable minds.)

--Ron Bruck
bruck@mtha.usc.edu preferred



Newsgroups: sci.math
From: roach@bikini.wri.com (Kelly Roach)
Subject: Re: u(v^n)w prime puzzle - HINTS
Organization: Wolfram Research, Inc.
Date: Thu, 20 Aug 1992 16:18:23 GMT

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
From: bill@waikato.ac.nz
Newsgroups: sci.math
Subject: Re: Primes
Date: 18 Aug 92 09:16:19 +1200
Organization: University of Waikato, Hamilton, New Zealand

In article <9208171021.aa01492@Paris.ics.uci.edu>, kibler@turing.ICS.UCI.EDU (Dennis Kibler) writes:
> In mathematica the number of primes less than n or equal to n
> is given by PrimePi[n].
> 
> The following took about 1 second to compute.
> 
> 
> 
>  Do[Print[i," ",PrimePi[2^i]],{i,30}]
> 1 1
> 2 2
> 3 4
> 4 6
> 5 11
> 6 18
> 7 31
> 8 54
> 9 97
> 10 172
> 11 309
> 12 564
> 13 1028
> 14 1900
> 15 3512
> 16 6542
> 17 12251
> 18 23000
> 19 43390
> 20 82025
> 21 155611
> 22 295947
> 23 564163
> 24 1077871
> 25 2063689
> 26 3957809
> 27 7603553
> 28 14630843
> 29 28192750
> 30 54400028
> 

Does anyone know how Mathematica stores its table of primes, and how efficient
it is in terms of space and access time? What's the minimum space a table
of primes can be compressed into if constant access time to find the nth prime
is required?

Bill Teahan,
Systems Programmer,
University of Waikato,
Hamilton, New Zealand.
"I have never been lost, but I will admit to being confused for several weeks."
-- Daniel Boone.
Xref: sparky sci.math:10348 sci.math.symbolic:2226 sci.math.num-analysis:2479
Newsgroups: sci.math
Subject: Re: u(v^n)w prime puzzle - HINTS
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 21 Aug 92 17:09:33 GMT

In <1992Aug20.233027.7862@wri.com> roach@bikini.wri.com (Kelly Roach) writes:

>In article <a_rubin.714331997@dn66>  writes:
>> uvw can't divide 10^a-1; it's larger!

                                                                    p
Essentially the same proof shows that if p=uvw is prime, then p | uv w.
--
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.
My interaction with our news system is unstable; if you want to be sure I see a post, mail it.
Newsgroups: sci.math
From: victor@watson.ibm.com (Victor Miller)
Subject: Re: Primes in x_{n+1} = ax_n+b (was Re: u(v^n)w prime puzzle)
Date: Fri, 21 Aug 1992 17:00:02 GMT
Reply-To: victor@watson.ibm.com
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: IBM, T.J. Watson Research Center

>>>>> On Fri, 21 Aug 1992 10:31:32 GMT, jeanmarc@ecrc.de (Jean-Marc Andreoli) said:

Jean-Marc> Let a,b be integers, and (x_n) be a sequence s.t. x_{n+1} = a x_n + b

Jean-Marc> My question is: does it contain infinitely many primes ?

Jean-Marc> There are trivial cases, where the answer is no:
Jean-Marc> 1/ if (a x_0 + b = x_0) the sequence is constant.
Jean-Marc> 2/ if (a = -1) the sequence alternates between two values x_0 and x_1.
Jean-Marc> 3/ if ((a,b) > 1 or (x_0,b) > 1) then the sequence clearly
Jean-Marc> contains only composite numbers.

Jean-Marc> What about the other cases ?

Jean-Marc> I've seen a "theorem of Dirichlet" in Hardy&Wright which
Jean-Marc> handles the case a=1. Can anyone give me a pointer to a
Jean-Marc> proof of that theorem (Hardy&Wright don't give it and the
Jean-Marc> only reference they give is in German).

You can find proofs of Dirichlet's Theorem in lots of books on Number
Theory.  If you just want to see elementary proofs of special cases,
you could, for example, look in Riesel's book "The Book of Prime
Number Records", Springer-Verlag, pp. 209ff.  There is a proof of the
whole theorem, in Apostol's book "Analytic Number Theory", in
Davenport and Montgomery's book "Multiplicative Number Theory" and
scores of others.  Riesel also recommends Hasse's Vorlesungen ueber
Zahlentheorie (now available in English Translation, Springer-Verlag).

The general question is certainly open.  For example, if x_0=1, a=2,
b=1, then x_n = 2^n - 1.  In this case the question boils down to
whether or not there are an infinite number of Mersenne Primes.  If
you take x_0=2, a=2, b=-1, you get the same question about Fermat
Primes.

Jean-Marc> ---

--
			Victor S. Miller
			Vnet and Bitnet:  VICTOR at WATSON
			Internet: victor@watson.ibm.com
			IBM, TJ Watson Research Center
Newsgroups: sci.math,sci.psychology
From: xcpslf@decsrv.NoSubdomain.NoDomain (stephen l favor)
Subject: Re: mathematician's marrying ages.
Reply-To: xcpslf@oryx.com
Organization: Oryx Energy Co., Dallas, Texas.
Date: Wed, 19 Aug 1992 21:15:42 GMT

In article <1992Aug19.185215.303@cc.tut.fi>, jk87377@cc.tut.fi (Juhana Kouhia) writes:
|> 
|> I have read from the newspapers about animals in zoos, how they
|> are lonely (alone) and actually die for their loneliness (also after
|> their partner has died).
|> I have also read about how some elder people die soon after
|> the dead of their wife/husband.
|> 
|> So, the researchs I have read about tells that
|> people married lives longer and are more healty than
|> people not married.

But do you have a comparison of death ages of people who never married
vs. married people?  How about those who have been through divorces
(maybe multiple times)?  What is their levels of social interaction?

|> (And the case is not simply as: "In almost every study I know of, this is
|> simply a correlation.  How many people do you know of that want to get
|> married to someone with a "mental" problem?" as suggested.)

All people have "mental problems" in a loose sense.  Pick up and read any
book on the subject and from time to time you'll say, "I do that," or
"Harry does that."  For the general population, these "problems" don't
hamper daily existence, but I think it's acceptable to say that they exist and
people don't want to deal with them.  How many mother-in-laws would qualify
as narsissistic (sp?)?  Therefore, I assert that one of the many factors
in choosing a mate is that he/she doesn't step on any of your "mental
problems."

|> Having many friends is not the same than having a caring living
|> partner.

Being married isn't the same as having a caring, living partner.
Newsgroups: sci.math