HDTenEightyP/NetNews huggingface dataset:
From: ecaxron@ariel.lerc.nasa.gov (Ron Graham)
Subject: Question about "sparsity" of matrices
News-Software: VAX/VMS VNEWS 1.41    
Organization: NASA Lewis Research Center
Date: 24 Dec 1992 11:26 EST  

I ask your pardon if this question is too simple, or if the subject has
been discussed already.  I am posting this question to two newsgroups,
one of which I do not currently subscribe to.

Is there a measure for the "sparsity" (or maybe I should say "sparseness"?)
of a matrix?  We know that a sparse matrix has a lot of zeroes in it.  But
who decides when the matrix is sparse?  Are there degrees of sparseness?

The answer to this question will help us determine a test plan for a 
matrix inversion algorithm we are currently testing in an engineering
application.  Pointers to references welcomed, will summarize any 
conscientious responses.

RG

Who became an engineer thinking "free-body diagrams" might be interesting.
Xref: sparky sci.math:17262 rec.puzzles:8077
From: isopi@acf9.nyu.edu (Marco Isopi)
Newsgroups: sci.math,rec.puzzles
Subject: Re: Integral Puzzle (Cute, not Evil)
Date: 21 Dec 92 14:37:36 GMT
Organization: New York University

edgar@function.mps.ohio-state.edu (Gerald Edgar) writes:


>In more flowery language: given the L_infinity and L_1 norms of f, estimate
>the L_2 norm of f.

And then Riesz-Thorin theorem shold bring you home.
Or there are other and better ways?

>  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu

Marco Isopi
Newsgroups: sci.math,sci.physics
From: tim@iss.nus.sg (Tim Poston)
Subject: Re: More on Huygens' principle
Organization: Institute of Systems Science, NUS, Singapore
X-Newsreader: Tin 1.1 PL4
Date: Thu, 24 Dec 1992 07:38:52 GMT

columbus@strident.think.com (Michael Weiss) writes:

: Did the gods give us three spatial dimensions so music would sound nice?
Could some synth sound wizard out there make a recording of what
2D or 4D piano music would sound like?
(Actually, I hate piano music, mostly.
Make it a violin, drum or trombone,
or a bamboo pipe.)

Tim


_______________________________________________________________________
              My lucky number is irrational.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Newsgroups: sci.math
From: waknispv@necs.Vanderbilt.EDU (Prashant V. Waknis)
Subject: A sequence matching puzzle
Organization: School of Engineering, Vanderbilt University
Date: Mon, 28 Dec 1992 17:56:11 GMT

Consider any two pairs of numbers. (n1, n2) and (n3, n4). 
Let (n1 * n2)/ (n1 +n2) = (n3 *n4) / (n3 + n4) = C.

For example, n1 = 12, n2  = 60, n3 = 15, n4 = 30.

Now, look at the following chart.

12 | 0    12    24    36   48      60 ....
60 | 0                             60 ....
-------------------------------------------
15 | 0       15    30     45       60  ....
30 | 0             30              60  ....

Can you see how the chart is made? Basically, starting with 0, each 
the sequences are constructed.
Now try to match the numbers  (on the right of the |s) that are above 
the horizontal line to the once that are  below  the horizontal line. 
The rule is the following.
Assume that the numbers start appearing on the scene  from left to right (in
increasing order)

So, in the above example,
all zeros appear first, get matched to other zeros above/below the line.
12 comes, waits till 15 comes (for 3 time units), they match up. 
24 comes, waits till 30 comes (for 6 time units), and matches to one 
of the 30s.
Remaining 30 waits for 6 units, and matches to 36.
45 waits for 48 (3 units) and they match up.
All 60s match. 
The cycle repeats.

My hypotheis is, in this matching, the max. waiting period (in this case 6)
can't exceed C ( =  (n1 * n2)/ (n1 +n2), in this case 10).

Is this hypothesis right? Can somebody prove/disprove it?


-- Prashant
(waknispv@vuse.vanderbilt.edu)



Newsgroups: sci.math,sci.physics
From: jim@cs.UAlberta.CA (Jim Easton)
Subject: Re: Three-sided coin
Organization: University of Alberta, Edmonton, Canada
Date: Wed, 30 Dec 1992 20:15:23 GMT

samuel@front.se writes:

>In article <jim.721702281@st-michael>, jim@cs.UAlberta.CA (Jim Easton) writes:

>> Given that it is possible to bounce from one state to another it
>> is more probable to bounce out of a high energy state to a lower
>> energy state than it is the reverse.  I claim that the lowest energy
>> state is the most probable and therefore conclude that the three states
>> should be equal in energy.
>> 
>> That implies that the thickness is equal to the diameter.

>I agree with you to a certain extent. However, if the thickness is equal to the
>diameter, then wouldn't (assuming that your energy-state-theory is correct) the
>coin behave similar to a cube, where the sides of the coin correspond to two
>sides of the cube and the 'edge' of the coin corresponds to FOUR sides of the
>cube. This would imply that the probability for head id 1/6, tail 1/6 and
>'edge' 4/6 = 2/3. I'm not sure that I'm entirely correct on this one, but I'm
>almost sure that the probability for 'edge' would be significantly higher than
>the one for head or tail, given that the thickness is equal to the diameter. 

Good point - I think you might be right.  I don't have an answer,
at least not at the moment.  I should make one and try it.

	Jim
From: lotto@math.berkeley.edu (Ben Lotto)
Newsgroups: sci.math
Subject: Re: Rounding Rules
Date: 21 Dec 92 06:28:20
Organization: U.C. Berkeley Math. Department.
NNTP-Posting-Host: math.berkeley.edu
In-reply-to: shaw@toadflax.UCDavis.EDU's message of Sun, 20 Dec 1992 00:30:18 GMT

>>>>> On Sun, 20 Dec 1992 00:30:18 GMT, shaw@toadflax.UCDavis.EDU (Rob Shaw) said:

Rob> What is the rationale behind the following rounding rule?

Rob> when dealing with 5's followed by all zero's, check the
Rob> next digit to the left. If it's even, round down; odd,
Rob> round up.

I was told way back in my undergraduate days that one should generally
"round to evens."  Someone recently explained the rationale for this
as follows.  Suppose we start with 1.49 and round in the traditional
manner twice.  First we would round to 1.5 and then to 2.  On the
other hand, if we round just once we get 1.  Rounding to evens
prevents us from getting different answers.
--
-B. A. Lotto
Department of Mathematics/Lake Forest College/Lake Forest IL 60045
708-735-5159
lotto@math.berkeley.edu
From: soubyran@ERE.UMontreal.CA (Soubyran Richard)
Newsgroups: sci.math
Subject: Real to fraction...
Date: 28 Dec 92 19:38:35 GMT
Organization: Universite de Montreal

Hello Mathers!

May be this one has already been posted... but I am taking the
chance...

Can anyone give me an algorithm to find the fractional
representation of a real number?

I thank you very much in advance,

Richard Soubyran
soubyran@alize.ere.umontreal.ca
Newsgroups: sci.math
From: aet@mullian.ee.mu.OZ.AU (bert thompson)
Subject: Re: math dictionary
Organization: Computer Science, University of Melbourne, Australia
Date: Sat, 2 Jan 1993 01:49:52 GMT

glenne@csd4.csd.uwm.edu (Neuromancer ) writes:

>Hi,
> 
>I bought _The Penguin DICTIONARY OF MATHEMATICS_.  Many times
>I find the definitions either too cryptic or lacking enough
>explanation.  Does anyone know of a matematical dictionary
>for idiots that still contains useful and comprehensive
>information?

check out the Collin's  Dictionary Of Mathematics by 
Borowski and Borwein.  The Collin's one says that each of its 
definitions is aimed at the audience most likely to look 
it up: so `sum' has a simple definition, and 
`algebraic topology' has definition intended for a more
sophisticated reader.

	bert.

---------------------------------------------------------------------------
bert thompson	aet@munagin.ee.mu.oz.au.
SITEE		school of information technology and electrical engineering
		university of melbourne
---------------------------------------------------------------------------
     g e t  t h e  p o i n t 
---------------------------------------------------------------------------
From: sundar+@cs.cmu.edu (Sundar Vallinayagam)
Newsgroups: sci.math
Subject: Help! how to compute this derivative ?
Summary: differentiating  running sum frequency
Keywords: derivative w.r.t. frequency
Date: 31 Dec 92 20:02:37 GMT
Organization: School of Computer Science, Carnegie Mellon

Howdy!

I want to reach the minimum of an error surface by a gradient
descent procedure.  The error criterion is least-squared error,
i.e., of the form ||y - X*inv(X'*X)*X'*y||.

The k-th column of X would be:  cos(2*pi*k*f*n)  n=0,1,...,N-1.
That is, the frequency is k*f.  For a gradient descent procedure
I need to calculate the derivative of the pseudoinverse, which needs
the calculation of the derivative of the k-th column of X w.r.t. f.
This is simply -2*pi*k*n*sin(2*pi*k*f*n),  n=0,1,...,N-1.
No problems with this simple case.


Next consider the more general case where for the k-th column
the frequency f is not fixed but is changing for each value of n,
i.e., f now becomes f(n).  I do not have a closed form expression
for the frequency variation but have a set of N measurements.
The k-th column of X now becomes:   cos(2*pi*k*F(n))
where F(n)--the phase--is the running sum of f(n)--the frequency.
Now if I want to differentiate the k-th column w.r.t. frequency,
how is the derivative defined and what would be a numerical procedure
to implement it ?

Thanks a lot in advance.





-- 
**************************************************
ramli@orca.ele.uri.edu
**************************************************
From: samuel@front.se
Newsgroups: sci.math
Subject: Two problems
Date: 28 Dec 92 21:04:54 GMT
Organization: Samuel Gustaf Siren

Problem 1 (This is an old problem)
----------------------------------
Suppose that a big (infinite) number of married couples are invited to a party.
All the guests are sitting around a big round table. If every guest is placed
randomly at the table, what is the probability that no man will sit at the 
LEFT of wis wife ?

Problem 2
---------
Same as above, but what is the probability that no man will sit next to his
wife at all (left or right) ?


Problem 1 is fairly easy. One solution is to denote p(m,n) = 'The probability
that exactly n men out of a total of m sits left to their wives'. The solution
is the value of p(oo,0) [oo = infinity]. I won't tell the value here in case
you want solve the problem yourself. 

Problem 2, however, have I not been able to solve.
Brute force got me the following results; p(m,n) = 'The probability that 
exactly n men out of a total of m sits next to their wives' :
    p(1,0) =      0       = 0.000000
    p(2,0) =      0       = 0.000000
    p(3,0) =     1/6      = 0.166667
    p(4,0) =     2/24     = 0.083333
    p(5,0) =    13/120    = 0.108333
    p(6,0) =    80/720    = 0.111111
    p(7,0) =   579/5040   = 0.114881
    p(8,0) =  4738/40320  = 0.117510
    p(9,0) = 43387/362880 = 0.119563
Anybody out there got any ideas ?

/SS
-- 
+----------------------+--------------------------------------+
|                      |        / \                           |
| Samuel Gustaf Siren  |      __!_/_                          |
| SAMUEL@front.se      |    / \ |  / \                        |
|                      |   |/  || | \__oW)----------          |
| All opinions and     |   !! / /\ \                          |
| questions are my     |     / /  \ \                         |
| own /SS              |    //    //                          |
|                      |   /|    ((                           |
+----------------------+--------------------------------------+
From: somos@ces.cwru.edu (Michael Somos)
Newsgroups: sci.math
Subject: Re: need proof:  (1 + 1/n)^n ==> e
Date: 25 Dec 1992 21:40:44 GMT
Organization: Computer Engineering and Science, Case Western Reserve University
NNTP-Posting-Host: lychee.ces.cwru.edu
Summary: truly elementary proof given

The tone of the question suggests that a very elementary proof of
the limit (1+1/n)^n is requested.  It is no trick to come up with
fancy proofs that obscure rather than clarify.  An elementary proof
may use a few more steps than a slick fancy proof but lead to more
insight into the problem.  While the previously posted proofs all
have some merit, so far no one has mentioned the following approach.

We consider lists of real numbers like x = [.34,.51,.12].  Define the
operator P as the product of all the numbers incremented by one.  So

    P [a,b,c] = (1+a)(1+b)(1+c) , for example.

We now require all the numbers of the list to have the same sign.
The reason is the following inequality

    P [a+b] = 1+a+b < 1+a+b+ab = (1+a)(1+b) = P [a,b]  if  ab>0 .

We now define a partial ordering of lists by requiring that any
splitting of a number is considered as being greater, for example,

    [a,b+c,d+e,f] < [a,b,c,d+e,f] < [a,b,c,d,e,f] .

Note that two lists that have the same sum also have a common upper
bound since we can keep splitting numbers of both to get a list of
numbers which can be rejoined to get either of the original lists.

This definition is designed to ensure that P is monotonic.  Since
(1+a)(1-a) = 1-a^2 < 1 , we now require that all the numbers of the
list be less than one in absolute value.  This ensures the following

    P(x) P(-x) = P(-x^2) < 1  ,  P(x) = P(-x^2)/P(-x) < 1/P(-x) .

where -x is the list of all negatives of the numbers in x.  But now

    if x < y , then  -x < -y  and P(x) < P(y) < 1/P(-y) < 1/P(-x) .

All we need now is to show that an increasing sequence of lists,
where the maximum absolute value of the numbers goes to zero, will
converge to a limit when operated by P.  A simple estimate is needed.
Define the S operator as the sum of all the numbers of a list.  To
estimate how close we are approaching a limit use the simple

    [S(x)] < x , and  1 + S(x) = P [S(x)] < P(x) .

But  S(-x^2)  approaches 0 by our last hypothesis, so the  P(x)  and
the  1/P(-x)  approach each other and are equal in the limit. 

Finally, consider the sequence of lists  [1] , [1/2,1/2] , ... ,
[1/n,1/n,...,1/n] , ...  operated by  P  giving  (1+1) , (1+1/2)^2 ,
... , (1+1/n)^n , ... .  The limit exists and is defined to be  e .

Note that this approach also defines the exponential function at the
same time.  It uses nothing beyond the concept of limits and simple
algebra.  It is truly elementary although it uses more steps.  

Shalom, Michael Somos <somos@alpha.ces.cwru.edu> (My opinions alone)
From: slow@ferrari.cray.com (David Slowinski)
Subject: checking modular arithmetic
Reply-To: slow@ferrari.cray.com (David Slowinski)
Organization: Cray Research, Inc.  Eagan, MN.
Date: 31 Dec 92 13:43:29 CST

Background:
  For my Mersenne Prime search I use the Lucas-Lehmer test to test
the primality of 2^p-1:
1)    u = 4
2)    for i = 3 to p
3)      u = u^2 mod 2^p-1 - 2
4)    next i 
5)    if u == 0 then 2^p-1 is prime

  Nearly all of the computer time is spent computing u^2 in line 3.
My current program uses a Schonhage-Strassen multiply based on
a mixed-radix FFT microtasked to run in parallel on multi-cpu
shared memory machines.  Since I often make programming errors and
run on experimental hardware, I use a residue check to verify the
correctness of each 2p bit long multiply.
  Barry Fagin and Richard Crandall have developed a clever algorithm
that directly computes the 'u^2 mod 2^p-1' using FFT's on only p bits.
This algorithm is over twice as fast as the 2p bit FFT multiply that 
I use now.  I would like to use this faster algorithm in my new
mpp based program, but I am afraid to without a check on each 
result to give me confidence.
  
Question:
  Is there a way to check the correctness of the result of 
'u^2 mod 2^p-1' where p is prime in the range 500,000 - 1,000,000?
  
Thank you kindly,

David Slowinski
slow@cray.com
Cray Research Inc
Chippewa Falls, Wisconsin 
Xref: sparky sci.math:17509 sci.physics:21885
From: greg@zaphod.uchicago.edu (Greg Kuperberg)
Newsgroups: sci.math,comp.text.tex
Subject: How do I do semidirect products in LaTeX?
Date: 3 Jan 93 18:50:44 GMT
Followup-To: comp.text.tex
Organization: Dept. of Mathematics, U. of Chicago

I need a semidirect product symbol for a paper and I can't seem to
find it in the LaTeX manual.  Does the symbol exist, and if not,
is there a standard macro to generate it?

Please send a copy of your reply by mail.  Thanks.
Newsgroups: sci.math
From: news@massey.ac.nz (USENET News System)
Subject: Re: Negative Zero
Organization: Massey University
Date: Wed, 23 Dec 92 21:15:10 GMT

In article <1992Dec21.190158.4586@galois.mit.edu>, jbaez@riesz.mit.edu (John C. Baez) writes:
> 
> 
> >All this reminds me of a student I encountered once who had written a program
> >to check whether infinity existed. He was incrementing and printing out a
> >counter in a continuous loop and sat looking at the screen enthusiastically.
> >Shame.
> 
> >Colin
> 
> Seriously?  That's great.  What happened then?  I would imagine that
> after a while he'd get bored, start thinking about the problem and come
> to some conclusion.  Maybe: "Infinity MUST exist.  If it didn't this
> program would have halted by now!"
> 
I'd better explain the joke - obviously some people need it :-)
SPOILER FOLLOWS
















After a while the counter reached 32767. On the next iteration it
became -32768.


Of course he might have used LONGINT, in which case the program
would run a little longer before cycling.

To be serious for a moment :-), a computer is a finite state machine.

Terry Moore
Newsgroups: sci.math
From: ilan@leland.Stanford.EDU (ilan vardi)
Subject: Re: Mathmatics
Organization: DSG, Stanford University, CA 94305, USA
Date: Wed, 23 Dec 92 06:54:30 GMT

In article <1992Dec22.211716.7065@primerd.prime.com> jasonp@bungie.prime.com (Jason Pascucci) writes:
>Hi,
>
>Yes, this might be a strange request, however please
>indulge me.
>
>I would like to obtain the firm grasp of 'higher-math'
>without spending time in school for a Doctorate in Mathmatics. 
>

It's not a strange request at all. As a matter of fact, some guy asked
the same thing some time ago and got the response ``there is no royal
road to geometry'' or words to that effect.
Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Negative Zero
Keywords: -0
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Mon, 21 Dec 92 19:01:58 GMT


>All this reminds me of a student I encountered once who had written a program
>to check whether infinity existed. He was incrementing and printing out a
>counter in a continuous loop and sat looking at the screen enthusiastically.
>Shame.

>Colin

Seriously?  That's great.  What happened then?  I would imagine that
after a while he'd get bored, start thinking about the problem and come
to some conclusion.  Maybe: "Infinity MUST exist.  If it didn't this
program would have halted by now!"
From: an1060@anon.penet.fi
Newsgroups: sci.math
Subject: Book Wanted...
Date: 26 Dec 92 16:58:49 GMT
Organization: Anonymous contact service
X-Anonymously-To: sci.math

X-Anon-To: sci.math


 
Hello All,
 
Does anyone have a copy of:
 
Mathematics Applied to Continuum Mechanics, by Segel & Handelman.
 
For those who attended RPI, this is the book used for Foundations of
Applied Mathematics II (FOAM II).
 
I am looking for an original copy, i.e. published by Macmillan,
the book is black hardcover with green lettering (yes, quite unattractive)...
 
If you have one in very good condition, please contact me.
 
Respond to:  mantell@ams.sunysb.edu
 
Thanks...
 
Abe Mantell
 
 
-------------------------------------------------------------------------
To find out more about the anon service, send mail to help@anon.penet.fi.
Due to the double-blind system, any replies to this message will be anonymized,
and an anonymous id will be allocated automatically. You have been warned.
From: tim@maths.tcd.ie (Timothy Murphy)
Newsgroups: sci.math
Subject: Professorship in Mathematical Physics
Summary: Professor of Math Physics in Maynooth Ireland advertised
Keywords: Mathematics Physics appointment
Date: 22 Dec 92 03:06:27 GMT
Organization: Dept. of Maths, Trinity College, Dublin, Ireland.

I've been asked to post the following advertisement
by a colleague in Maynooth College
(a constituent of the National University of Ireland).

======================================================================

           St. Patrick's College, Maynooth.

  Applications are invited for the post of Professor in Mathematical
Physics.  The salary scale is  35,506 - 44543 pounds. The person 
appointed will be Head of the Mathematical Physics Department in the
College.  Further details from:
  
  The Personnel Officer, St. Patrick's College, Maynooth, Co. Kildare,
Ireland. (Confidential fax: +353-1-628-9177).

  Applicants should forward a Curriculum Vitae, together with the names,
addresses, telephone and fax numbers of three referees, so as to arrive
not later than 27th January 1993.

======================================================================

-- 
Timothy Murphy  
e-mail: tim@maths.tcd.ie
tel: +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
Xref: sparky sci.math:17504 rec.puzzles:8165
From: lordSnooty@cup.portal.com (Andrew - Palfreyman)
Newsgroups: sci.math,rec.puzzles
Subject: Optimisation Problem With Groups
Date: Wed, 30 Dec 92 06:38:04 PST
Organization: The Portal System (TM)
Distribution: world

I am interested in finding an efficient algorithm for solving
the following problem:

System
------
A bounded Euclidean 2-D surface with discrete coordinates (0<=x<=XMAX,
0<=y<=YMAX) is assigned numbers z(x,y) at each point. Nothing is known 
_a priori_ about this assignment. The range of z (0<=z<=ZMAX) is far
higher than the total number of points; 
	i.e. XMAX*YMAX / (dx*dy) << ZMAX/dz, where d(i) is the granularity.

Problem
-------
Given a fixed range of z-values RZ (RZ > dz), the problem is:
group contiguous points within the range such that, after covering
every point on the surface, the final grouping minimises the total 
number of groups.

Notes
-----
* "Contiguous" is taken to mean +/- dx or dy of the point under
  consideration, but not both (i.e. no diagonal connections).
* The minimum size of a group is one point.
* Note that RZ is a range = zmax-zmin, and we are free to choose the
  absolute value of one of {zmax, zmin} for any group.
 --------------------------------------------------------------------------
| lord snooty @the giant |  Would You Like Space Potatoes With That?       |
| poisoned electric head |              andrew_-_palfreyman@cup.portal.com |
 --------------------------------------------------------------------------
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: complemented subspaces!
Date: 30 Dec 1992 13:38:42 -0500
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu

In article <amirishs.725734227@acf9> amirishs@acf9.nyu.edu (shaahin amiri sharifi) writes:
>Is that true, every closed subspace of C[0,1], with sup-norm topology,
>has a closed complement?


Certainly not.  The only (up to isomorphism) Banach spaces where every
closed subspace has a closed complement are the Hilbert spaces.
Lindenstrauss & Tzafriri, Israel Journal 9 (1971) 263--269.

I believe the first example of a non-complemented subspace was a subspace
of C[0,1].  Banach & Mazur, Studia Math 4 (1933) 100--112.
-- 
  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)
Xref: sparky sci.math:17406 sci.logic:2479