HDTenEightyP/NetNews huggingface dataset:
From: ramsay@unixg.ubc.ca (Keith Ramsay)
Newsgroups: sci.math,sci.physics
Subject: Re: tensors: How about 3rd, 4th rank?
Date: 21 Aug 92 23:40:25 GMT
Organization: University of British Columbia, Vancouver, B.C., Canada

In article <1992Aug21.191559.20170@news2.cis.umn.edu> 
benzvi@leibniz.geom.umn.edu (benzvi) writes:
>... given two vector spaces over a common scalar field,
>V and W, V(tensor)W is defined as the set of elements
>v(tensor)w (v in V, w in W). 

Not exactly. V tensor W is *spanned* by those elements.

For example, if V and W are both 3-dimensional vector spaces, with
basis {i,j,k}, then V tensor W is a 9=3*3 dimensional space with basis
{i tensor i, i tensor j,...,k tensor k}. The elements of the form v
tensor w are just the elements of the form (ai+bj+ck) tensor
(di+ej+fk)=ad(i tensor i)+ae(i tensor j)+...+cf(k tensor k). This (6
dimensional surface) misses such things as (i tensor i)+(j tensor j).

I recommend that one think of tensors in terms of the multi-linearity
properties being utilized, rather than thinking exclusively in terms
of coordinates and how the components transform as the coordinates are
changed.

An example of a natural higher order tensor is the (Riemann) curvature
tensor. Curvature is manifested by noticing that vectors which are
"parallel-transported" around closed loops don't come back the same.
(Try this on a globe. Take a pencil, and put it on the equator
pointing east. Move it eastward around the equator 90 degrees. Now
move it up to the north pole, without changing the orientation. Now
slide it similarly along a meridian back to the original place. It is
no longer pointing east. This is because the globe is curved.)

There is a corresponding curvature tensor. To specify a (tiny) closed
(parallelogram-like) loop takes two vectors, and the initial vector to
be transported around it is another one. The "result" is a vector
indicating by how much the vector has been perturbed by going around
the loop. One can show that the perturbation is linear in all three
"input" vectors (if we take the correct limit of "tinyness" of our
closed loop). Hence we have a tensor of type (3,1) (or is it (1,3)?).

The fact that it is linear in each of the components is the key thing
as far as its being a tensor. I hope this helps to dispel some of the
mystery. Now go read some books. :-)

Keith Ramsay         The measure of our intellectual capacity is the
ramsay@unixg.ubc.ca  capacity to feel less and less satisfied with our
                     answers to better and better problems.-C.W.Churchman
From: frisinv@alleg.edu (Vincent Frisina)
Newsgroups: sci.math
Subject: Power series solution to ordinary differential equation
Date: 16 Aug 92 23:10:44 GMT
Article-I.D.: pellns.1992Aug16.231044.4401
Organization: Allegheny College

To anyone who has had differential equations:
   I tried this once before but no one responded. I'm trying to learn  
differential equations. The book I'm using has a great discussion of  
everythinmg but power series solutions. My problem isn't with the series  
themselves (I do remember calc two) but with the method. Could someone  
please send me a summery of how and when to use this method of solution.  
My address is frisinv@alleg.edu
-----
Vince
From: tarokh@server.uwindsor.ca (TAROKH VAHID             )
Newsgroups: sci.math
Subject: Injective modules of K[x_1, ....., x_n]
Date: 21 Aug 1992 16:21:09 GMT

Dear Readers,

Could anybody please tell me if all the injective modules of the ring
K[x_1,.....x_n] (where K is a field) are classified or not. I know that
all the projective modules of this ring are free. Is there an injective
analogue to this result?


Thank You,

V. Tarokh
Xref: sparky sci.math:10473 sci.physics:13206
From: wsadjw@rw7.urc.tue.nl (Jan Willem Nienhuys)
Newsgroups: sci.math,sci.physics
Subject: Re: tensors
Date: 21 Aug 92 06:55:31 GMT
Reply-To: wsadjw@urc.tue.nl
Followup-To: sci.math
Organization: Eindhoven University of Technology, The Netherlands

In article <1992Aug20.201022.33682@watson.ibm.com> platt@watson.ibm.com (Daniel E. Platt) writes:
>
>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

I don't think that's a good explanation. 
You talk about mathematical objects through their representations,
which necessitates that you have to have all kinds of rules to identify
things that are actually the same, but have different representations.

An easy example of a tensor is: a multilinear map from one vector space
to another.
In some cases you must use the dual of the vector space.
For example, if x stands for a vector from a vector space X, and
f for an element of the dual of Y, then a linear map A from X to Y
can be thought of as given by the multilinear map (x,f) -> f(Ax).
Choosing proper bases for X and Y and the dual of Y gives you the
matrix for A. (Different bases give you different matrices, that's 
why there are transformatioin rules).

In case of differential geometry, if you have two tangent vectors a and b
at a point, and you carry a third vector c around the a,b-parallellogram,
the result is something that depends linearly on a, b and c (it's a
bit more complicated, but I don't have time now).  So expressing how
parallell transport works (and hence the local curvature) naturally needs
a tensor.

The word tensor originally meant something that relates forces to 
deformations in continuous materials, I believe.

JWN

Newsgroups: sci.math
From: dld@cs.monash.edu.au (David L Dowe)
Subject: int(x*log(x)*((1-x)^n), x=0..1)
Summary: int(x*log(x)*((1-x)^n), x=0..1),  positive integer n
Keywords: Hairy integral, Beta function
Organization: Computer Science, Monash University, Australia
Date: Thu, 20 Aug 1992 07:39:51 GMT

Let  I(n) = int(x*log(x)*((1-x)^n), x=0..1);
to use Maple's notation.        n is a positive integer.     log is ln  .

I seek as simple an expression as possible or, that failing, as rapidly
converging an expression as possible for I(n).

Before you respond (to me rather than the newsgroup, please), let me tell
you what I do know:
By parts,  int((x^n)*log(x), x=0..1) = -1/((n+1)^2)          (1)

Expanding (1-x)^n binomially and using (1)  enables us to get a
fairly simple expression for I(n).   But, I would like (if possible) a simpler
or more rapidly converging expression.

o   I also know that, re-writing  I(n)  as  int((1-x)*log(1-x)*(x^n)),x=0..1);
  Taylor expanding  log(x)  =  Sum(i=1,infinity) -(x^i)/i     gives
  I(n)  =  int(-(x^(n+1)) + Sum(i=2,infinity) (x^(n+i))/((i-1)*i) , x=0..1)  ;
and so     I(n)  =  -1/(n+2)  +  Sum(i=2,infinity) 1/((i-1)*i*(n+i+1)) .
  This also converges more slowly than I would like it to.

o   I also "know" that I(n) is rational.   I(0) = -1/4,  I(1) = -5/36,
  I(2) = -13/144,  I(3) = -77/1200,  I(4) = -29/600,  I(5) = -223/5880,
  I(6) = -481/15680,   and Maple Vn4.3 gave me a mammoth expression for
  I(50) in 15 seconds.
    When I gave Maple 4.3 the integral I(n)  (n unspecified), it merely
 returned some seconds later with the same initial expression I had given it.

o   I also "know" that  n*(n-1)*I(n) + log(n + 1/2)   converges to
  (I forget whether it's + or - )  +/- 0.42... ,
  the 0.42... possibly (a guess) being something to do with Euler's constant.

   I would appreciate (a reference to)
  a proof that I(n) is rational,
  a proof of the convergence result, and the value of "0.42..." ,   and
  (most importantly)  as simple an expression as possible for I(n) .
__
   Thank you.         Yours faithfully,  David Dowe.
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Littlewood's Three Principles
Date: 12 Aug 1992 14:01:37 -0400
Organization: The Ohio State University, Dept. of Math.
Distribution: world
NNTP-Posting-Host: function.mps.ohio-state.edu


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


-----
Explanation.  Littlewood claims that most of the theory of a function
of a real variable can be reduced to three heuristic principles:
  A [measurable] set is practically a finite union of open intervals.
  A [measurable] function is practically continouous.
  A convergent sequence of [measurable] functions is practically
     uniformly convergent.


-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
Newsgroups: sci.math
From: achim@unibi.uni-bielefeld.de (Achim Flammenkamp)
Subject: f: Q -> Q ,  f analytic
Date: Fri, 14 Aug 92 14:22:52 GMT
Organization: Universitaet Bielefeld

A analytic number theoretic question (or ?):

Given a function with domain and target C excepted a finite number of points.
It maps Q into Q (Q denotes the set of rational numbers) and
it is meromorph on C (this implies uniquely extendable from Q to C).

Must this function be representable as the quotient of two polynomies with
integer coefficients or are there other functions left (present a counter-
example, please) ?

achim
Newsgroups: sci.math,news.answers
From: boshuck@triples.math.mcgill.ca (William Boshuck)
Subject: Re: Expansion of set theory
Organization: Dept Of Mathematics and Statistics
Date: Thu, 13 Aug 92 17:44:08 GMT

In article <1992Aug12.182912.4659@sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <1992Aug12.164408.10547@thunder.mcrcim.mcgill.edu> boshuck@triples.
>math.mcgill.ca (William Boshuck) writes:
>
>   >An intersting facet of all of this is that large cardinal axioms
>   >actually have a bearing on relatively small things, like Borel
>   >sets of real numbers, and sets of natural numbers in the definable
>   >hierarchy.
>
>  And, it should be added, on the natural numbers themselves (in the form
>of theorems concerning the unsolvability of Diophantine equations).

    I think that this is included in my remark about sets in the 
    definable hierarchy. That is, the solvability or not of 
    Diophantine equations is a question in what might be called
    the definability theory of Sigma^0_1 sets. On the other
    hand, I don't quite know what theorems you may be talking 
    about. A big theorem in this area, that there is no uniform 
    algorithm to decide whether a given Diophantine equation has
    solutions (due to Matiesevitz (spelling?) I think) doesn't 
    use any aspects of the theory of large cardinals (an easy
    presentation of this theorem can be found in Bell and 
    Machover's book on mathematical logic).

    Maybe you want to contribute a more complete post on this
    matter. One of the things I am looking forward to learning
    is this sort of interaction between the unumaginably large
    and facts about the natural numbers. I would appreciate 
    hearing whatever you may have to say on the matter.

    WHB


Newsgroups: sci.math
From: bx304@cleveland.Freenet.Edu (Jeff Epler)
Subject: Re: u(v^n)w prime puzzle
Reply-To: bx304@cleveland.Freenet.Edu (Jeff Epler)
Organization: Case Western Reserve University, Cleveland, OH (USA)
Date: Tue, 18 Aug 92 03:06:46 GMT


In a previous article, roach@bikini.wri.com (Kelly Roach) says:

>     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.

u, v="0". w in {"2","3","5","7","11",...}

Or does "0" for V count as empty?
>
>I know the solution to this puzzle.  I think the
>solution can be understood fairly easily by anyone that
>has had a first course in number theory.  See if you
>can discover it.

I expect that this was not what I was supposed to say... 
(I anxiously await the *real* answer...)
-- 
|Jeff Epler               Additions Welcome           c(-8 ;-) >{8-) |
| :) (=( =-] (-= Celebrating the variety of faces =-> :^) {-= |-) (: |
|                                                   Lincoln, Nebraska|
Newsgroups: sci.math
From: jeanmarc@ecrc.de (Jean-Marc Andreoli)
Subject: Re: int(x*log(x)*((1-x)^n), x=0..1)
Reply-To: jeanmarc@ecrc.de
Organization: European Computer industry Research Centre GmbH.
Date: Fri, 21 Aug 1992 10:09:18 GMT

In article 714332240@dn66, a_rubin@dsg4.dse.beckman.com (Arthur Rubin) writes:
>In <dld.714296391@bruce.cs.monash.edu.au> dld@cs.monash.edu.au (David L Dowe) writes:
>
>>Let  I(n) = int(x*log(x)*((1-x)^n), x=0..1);
>>to use Maple's notation.        n is a positive integer.     log is ln  .
>
>>I seek as simple an expression as possible or, that failing, as rapidly
>>converging an expression as possible for I(n).
>
>Futher simplification shows:
>
>           n^2+n-1       (1 + 1/2 + ... + 1/n)
>I(n) = --------------- - ---------------------,
>       (n+1)^2 (n+2)^2    (n+1) (n+2)
>
>
>which can readily be proved by induction, noting the the integral for
>
>(n+1) I(n) - (n+3) I(n+1) can be written in closed form.
>
>--
>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.

Further simplification shows

          1/2 + ..... + 1/(n+2)
I(n) = - -----------------------
              (n+1)(n+2)


---
From: ahaley@eoe.co.uk (Andrew Haley)
Newsgroups: sci.math
Subject: Re: What is the EATCS?
Date: 18 Aug 92 15:00:40 GMT
Organization: EO Europe Limited, Cambridge, UK
X-Newsreader: Tin 1.1 PL3

wilson@web.ctron.com (David Wilson) writes:
: 
:     Here I have a paper in the EATCS and I still do not know what it
:     stands for.
: 
: 

European Association for Theoretical Computer Science.

Cheers, Andrew.
Newsgroups: sci.math
From: martin@lyra.cis.umassd.edu (Gary Martin)
Subject: Re: Collineations
Organization: University of Massachusetts Dartmouth
Date: Fri, 14 Aug 1992 12:52:16 GMT

In article <ARA.92Aug13180715@camelot.ai.mit.edu> ara@zurich.ai.mit.edu (Allan Adler) writes:

   In article <1992Aug12.194204.24356@uwm.edu> radcliff@csd4.csd.uwm.edu (David G Radcliffe) writes:

      Suppose f is a one-to-one function from the plane to itself
      which maps lines into lines, and suppose the image of f is 
      not contained in a line.  Must f be affine?
[Lucid proof that I wish I had written omitted]

   If one relaces R by another field, the proof shows that f is the composition
   of an affine transformation with an automorphism of the field applied to all
   the coordinates. This works in n dimensional space for all n>1.

And to tie this to another thread, isn't this known as the Fundamental
Theorem of Affine Geometry?  (And the corresponding theorem for automorphisms
of projective geometries known as the F.T.o.P.G.?)


--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
Martin@cis.umassd.edu
Newsgroups: sci.math
From: mgh3@po.cwru.edu (mike hurley)
Subject: Re: Still another problem.
Followup-To: sci.math
Organization: case western reserve u
Date: Wed, 12 Aug 92 15:10:12 GMT

In article <1992Aug11.170858.275@csc.canterbury.ac.nz>,
wft@math.canterbury.ac.nz (Bill Taylor) wrote:
> 
> Prove that
> 
>                n     n     n    n
>      1  /     2     3     4    5        \  
>      - |  1 + -- +  --  + -- + -- +....  | 
>      e  \     1!    2!    3!   4!       /   
> 
>   is an integer for all positive integer n.
> 
> ---------------------------------------------------------------------
>       Bill Taylor              wft@math.canterbury.ac.nz 
> ---------------------------------------------------------------------
>               Stamp out silly signatures!
> ---------------------------------------------------------------------


Let K(n) denote the indicated quantity;
(i.e., K(n) = \sum_{j=0}^{infty} (j+1)^{n}/j!/e )
Then 
K(n+1) = \sum_0^{n+1} B(n+1,m)*K(m-1)
where B(i,j) is the binomial coefficient i over j
and K(-1) is defined to be 1. Since K(0)=1 it follows
that K(n) is integral for every n>=0.

The proof goes by expanding (j+1)^n using the binomial
theorem, interchanging the order of summation and 
re-indexing.

mike hurley   mgh3@po.cwru.edu
dept of mathematics
case western reserve univ. 44106-7058
Newsgroups: sci.math,sci.physics
Newsgroups: sci.math



     I've been methodically working my way through Schiff's Quantum Mechanics
     text for some time now. I find it reasonable but slow, and I find his
     problem collection OK but somewhat in need of supplement. However, one
     thing I find consistently is that I have forgotten most of what I knew
     about special functions and Schiff doesnt really deal with these except
     to mention them and provide some intermediate steps in derivations. while
     I find I can provide these steps, eventually, in doing so I burn alot of
     time, and usually have questions I cannt answer about the special function
     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 
     questions and more that are upcoming. There are some obvious choices I'd
     appreciate any comments you can give me regarding the use of these books
     or point me to others. 


       Text                            Comment


       Morse and Feshbach            looks comprehensive, dont know how readable
				     and user friendly it is. Pretty advanced.

       Courant and Hilbert           appears to emphasize the mathematical 
				     side like existence. no exercizes and old
				     (1925). However both authors esp hilbert
				     are extremely reputable mathematicians.

       Arfken                        looks fairly good, does it go far enough?

       whittaker and watson          shows up in the footnotes of alot of grad
				     qm books. Schiff's being one of them. 
				     old, but appears to have good exercizes.
				     leans to old style analysis.
     
       Weinberger's PDE book         my primary current reference. leans to math                                     is terse but clear. not alot of detail.

       Bukov                         ? looks like primarily an undergrad book

       Boas                          ? 


       All comments and suggestions are welcome. Are there other texts I should
       know about? How are Morse and Feshbach and or Courant and Hilbert to 
       work with? My present inclination is to buy Arfken and one of the top 
       two classics. 

      
       Thanks,

       Graham Flower

Graham Flower ms 90-TT               |  Better to have convictions and act on 
Hewlett-Packard  350 W Trimble Rd    |  them, even if they are wrong, than to  
Microwave Semiconductor Division     |  waffle in indecision.
San Jose, California, 95131          |
 graham_flower@sj.hp.com             |

Newsgroups: sci.math
From: torkel@sics.se (Torkel Franzen)
Subject: Re: Expansion of set theory
Organization: Swedish Institute of Computer Science, Kista
	<1992Aug12.164408.10547@thunder.mcrcim.mcgill.edu>
Date: Wed, 12 Aug 1992 18:29:12 GMT

In article <1992Aug12.164408.10547@thunder.mcrcim.mcgill.edu> boshuck@triples.
math.mcgill.ca (William Boshuck) writes:

   >An intersting facet of all of this is that large cardinal axioms
   >actually have a bearing on relatively small things, like Borel
   >sets of real numbers, and sets of natural numbers in the definable
   >hierarchy.

  And, it should be added, on the natural numbers themselves (in the form
of theorems concerning the unsolvability of Diophantine equations).
From: matmcinn@nuscc.nus.sg (Mcinnes B T (Dr))
Newsgroups: sci.math
Subject: Re: anbody interested in Fermat's Last Theorem?
Date: 20 Aug 92 04:06:57 GMT
Organization: National University of Singapore

No.
Newsgroups: sci.math
From: roach@bikini.wri.com (Kelly Roach)
Subject: Re: Primes in x_{n+1} = ax_n+b (was Re: u(v^n)w prime puzzle)
Organization: Wolfram Research, Inc.
Date: Fri, 21 Aug 1992 18:45:07 GMT

In article <1992Aug21.103132.29967@ecrc.de> jeanmarc@ecrc.de (Jean-Marc  
Andreoli) 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 ?



     A different kind of trivial example:

     x_0 = 4
     x_{n+1} = 5*x_n + 1

All the x_n are divisible by 2 or 3.  Use the Chinese
Remainder Theorem to get other examples like it.  Another
one:

     x_0 = 868
     x_{n+1} = 13*x_n + 1

All the x_n are divisible by 4, 5, 7, or 17.  Neither
of these two examples fit the conditions (1), (2), (3)
above.

				Kelly
From: 8321207d@levels.unisa.edu.au
Newsgroups: sci.math
Subject: Matching algorithm
Date: 20 Aug 92 07:25:15 GMT
Organization: University of South Australia


Hello folks, this is a first for me.

Can anyone help with the following (sorry for being verbose).

Suppose you have a number of choirs with different numbers of people in
each. The voice ranges of the choir members are known. It is possible
that the ranges of some of the people overlap; for example, person A may
sing in the range 2000-2500 Hz while person B sings in the range 2300-
2650 Hz. Furthermore, some of the people in a particular choir may have 
other links between them, for instance person C could sing one octave
(whatever an octave is) higher than person A.

Suppose now that one of the choirs performed and this performance was
recorded. If the recording of the performance is analyzed then it is
possible to isolated various bands within which singing took place.

The problem is to identify which choir was singing. However, care must
be exercised during this process because when a particular voice band
gets associated with a paerson in a choir, then there are consequences
due to possible links which may exist between the people. Hence associating
a voice band of 2350-2450 Hz with person A may result in the singing
range for person D being restricted to a new range if person D sings
at say 1.5 times that of person A. This may mean that no voice band can
subsequently be matched with person D based on the initial match of
person A.

A scoring function for the quality of the match exist and the objective
is to maximise the score. It has been suggested to me that this falls
in the category of being a matching problem as found in Graph Theory.
A check through available literature had matching problems which did
not make use of additional constraining information.

My query is whether a matching algorithm which makes use of side
constraints exists and if so what are some pointers to it?
Also could this be solved (or a valiant attempt made at doing so)
by some other scheme aside from exhaustively considering every possible
pairing between person and voice band?

Any information gratefully accepted.
Contact via email or post

Cheers
Paul
From: niranjan@eng.cam.ac.uk (Mahesan Niranjan)
Newsgroups: sci.math
Subject: HELP det(Toeplitz)
Date: 18 Aug 92 13:10:47 GMT
Distribution: sci.math
Organization: Cambridge University Engineering Department, UK

Is there a recursive (by dimension) algorithm for computing determinant of
a Toeplitz matrix?

So, if I had matrices...

r0  r0 r1   r0 r1 r2   r0 r1 r2 r3  etc
    r1 r0   r1 r0 r1   r1 r0 r1 r2
	    r2 r1 r0   r2 r1 r0 r1
		       r3 r2 r1 r0

and I want the determinants of all of them, can I do this in a smart way?
(When I go from the 3x3 to the 4x4, I am adding only one new value, r3).

thanx
niranjan@eng.cam.ac.uk
Newsgroups: sci.math