HDTenEightyP/NetNews huggingface dataset:
From: sundar+@cs.cmu.edu (Sundar Vallinayagam)
Newsgroups: sci.math,sci.math.num-analysis
Subject: Help! How to compute this derivative (REPOST)
Summary: differentiating  running sum frequency
Keywords: derivative w.r.t. frequency
Date: 4 Jan 93 19:20:56 GMT
Article-I.D.: cs.C0CGEz.1Cr.1
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.
(Actually, 2*pi*k*F(n) is the phase and 2*pi*k*f(n) is 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: liuli@unixg.ubc.ca (Li Liu)
Newsgroups: sci.math
Subject: Re: proof wanted 2
Date: 10 Jan 1993 09:46:01 GMT
Organization: University of British Columbia, Vancouver, B.C., Canada
NNTP-Posting-Host: unixg.ubc.ca

The problem is about the existence of a nearest point in a closed set C
in R^n to a point x outside C. 

In a previous message, the author did a proof using a sequence of points
in C whose distance to x is (weakly) decreasing. This sequence trivially
exists, since if a point S_i is not the closest to x in C, then you can
find a point S_{i+1} that is closer. 

But in order for a limit point of this sequence (or more precisely, a
subsequence of this sequence) to exist, you need a compact set.

So take a closed ball centered at x and with radius the distance between x 
and the first point in the sequence. All other points in the sequence
are guaranteed to be in this ball. The intersection of this ball and
C is a compact set. This is because it is closed and totally bounded.
So now you have a sequence in a compact set. This sequence must have
a convergent subsequence. The limit of the subsequence will be the 
closest point to x. 

This will complete the proof.



Newsgroups: sci.math
From: tao@fine.princeton.edu (Terry Tao)
Subject: Re: proof wanted 2
Originator: news@nimaster
Organization: Princeton University
Date: Mon, 11 Jan 1993 21:00:12 GMT

>>In article <ARA.93Jan10172314@camelot.ai.mit.edu> ara@zurich.ai.mit.edu (Allan Adler) writes:
>>>
>>>
>>>True or false: A metric space (X,d) is locally compact if and only if
>>>for every point p of X and every closed subset Y of X, there is a
>>>point q of Y such that d(p,q) = inf {d(p,r) | r in Y}.
>>>
>>

False.  A friend suggested the following counter example:

Let X be the space of infinitely many copies of {0} union {1/n: n = 1, 2,
...} with all the 0's glued together.  the metric is the natural one
between two points on the same copy, or through 0 on different copies.  The
space is not locally compact at 0, yet every closed set and every point has
a point of closest approach.  Basically the reason for this is that the
only problematic point is 0, and the fact that Y is closed then guarantees
that 0 is in Y.

Terry
 

Newsgroups: sci.math
From: hanche@ams.sunysb.edu (Harald Hanche-Olsen)
Subject: Re: Value of the US Dollar
Organization: University at Stony Brook, NY
Date: Wed, 13 Jan 1993 00:09:51 GMT

>>>>> In article <31561@nntp_server.ems.cdc.com>, mstemper@ems.cdc.com (Michael Stemper) writes:

Michael> I give up. How do YOU compute the value of the US dollar? :->

Michael> It really depends on what metric you use. The metric that I use is
Michael> actually a variable metric. It's the famous Beers function:
Michael>   "How many beers can I get for it?"

[ snip, snip ]

Michael> And, after 01:00, the value drops to zero, because the bars are closed.

OK.  I'll send you a stamped and addressed envelope.  The next night,
after 01:00, you put your worthless $$ in the envelope, seal it and
drop in the nearest mailbox.  It's worthless, so your loss is zero,
right?  I'll spend it on beer when I get it.  Cheers!  :-)

- Harald
Newsgroups: sci.math
From: k2cheung@napier.uwaterloo.ca (William Cheung)
Subject: about transfinite real numbers...
Keywords: filling up things...well defined?
Organization: University of Waterloo
Date: Tue, 12 Jan 1993 23:00:33 GMT

  hi, maybe i didn't explain things explicit enough. i was wondering the
possibility of adding transfinite 'real' numbers between two different
transfinite 'whole' numbers.
  please pardon my ignorance if that's not a well defined concept.
william

Newsgroups: sci.math
From: neuhaus@vier.informatik.uni-kl.de (Stephan Neuhaus (HiWi Mattern))
Subject: Serial correlation of uniformly distributed integers
Summary: What is the expected serial correlation of uniformly distributed integers?
Keywords: statistics,serial correlation
Organization: University of Kaiserslautern, Germany
Date: Tue, 12 Jan 1993 16:39:31 GMT

Hello.

I am interested in the expected serial correlation coefficient of a
large number of uniformly distributed integers.  In particular, let
N = 2n be a large number, let U_i = Y_{2i} and V_i = Y_{2i+1} for
uniformly distributed random integers 0 <= Y_i <= M (the precise
value of M should be immaterial, let's say it is on the order of
10^{10}).  Compute the serial correlation coefficient C as follows:

             n \sum (U_i V_j) - (\sum U_i)(\sum V_j)
C = ---------------------------------------------------------------
       -----------------------------------------------------------
      /           2              2              2               2
     /(n \sum (U_i ) - (\sum U_i) ) (n \sum (V_j )  - (\sum V_j) )
   -v


In _The Art of Computer Programming_, Vol. II, Chapter 3, Knuth writes
that even with truly random numbers, the correlation coefficient could
not expected to be exactly zero (this I understand).  He then states
that C should lie between \mu_n - 2\sigma_n and \mu_n + 2\sigma_n 95
percent of the time, where

                                             ----------
            1                      1        / n (n - 3)
\mu_n = - -----, and  \sigma_n = -----     / ----------
          n - 1                  n - 1  - v    n + 1

He then goes on: "[These equations] are only conjectured at this time"
for uniformly distributed random variables.

My question is thus:  Have these values been proved by now, if so, are
there references to the literature, and if not, were these values
subsequently altered?

Please reply by email, since I don't regularly read this newsgroup.
If you reply, please send mail to neuhaus@informatik.uni-kl.de, not
neuhaus@vier.informatik.uni-kl.de.  This is a bug in our news
software, and I'm not sure when we'll get it fixed.  Thank you.

Thanks in advance.

-- 
Stephan <neuhaus@informatik.uni-kl.de>
sig closed for inventory.  Please leave your pickaxe outside.
PGP 2.1 public key available on request.  Note the expiration date.
Newsgroups: sci.math,sci.math.stat
From: xm1@cec1.wustl.edu (Xingdong Ma)
Subject: How to Add White Gaussian Noise to Variables?
Organization: Washington University, St. Louis MO
Date: Tue, 5 Jan 1993 19:55:05 GMT

Hi fellows!
Could anybody tell me how to add white gaussian noises to the process
measurements?
Any response is greatly appreciated!
Lots of thanks in advance!

My address: ma@wuche2.wustl.edu
Newsgroups: sci.math,sci.math.symbolic
From: dzk@cs.brown.edu (Danny Keren)
Subject: Real Algebraic Geometry
Organization: Brown University Department of Computer Science
Date: Tue, 5 Jan 1993 15:59:16 GMT

Hi everybody,

I am looking for references (books, journals) on real algebraic
geometry. In particular, I am interested in the question of
determining the number of components a real algebraic variety
(e.g. the zero set of a polynomial with real coefficients) in
2 and 3 variables has. My interest stems from the following 
problem: when fitting implicit algebraic curves and surfaces to 
scattered data, often the resulting zero set has more than one
component, which makes the fit look awkward. I am looking for a 
way to control the number of components while doing such fittings.

Thanks,

-Danny Keren.
From: hbtnn@usno06.nor.chevron.com (Brian Tanner)
Newsgroups: sci.math
Subject: Re: Help me deal w/ infinity
Date: 12 Jan 93 14:25:17 GMT
Organization: Chevron

In article <1993Jan11.233051.26558@news.media.mit.edu>, minsky@media.mit.edu (Marvin Minsky) writes:
|> In article <11664@sun13.scri.fsu.edu> jac@ds8.scri.fsu.edu (Jim Carr) writes:
|> >In article <1992Dec23.030309.125@front.se> samuel@front.se writes:
|> >>
|> >>If this argument doesn't help, why don't ask your friends exactly how much 
|> >>they think that 0.999999... differs from 1.
|> >
|> >Good suggestion.  I think this will help these poor souls examine the 
|> >unexamined assumption that are leading them to have trouble with the 
|> >proof, which is a very real difficulty with the fact that multiplying 
|> >.999.... by 10 does *not* change the number of 9s after the decimal 
|> >point!  I would love to overhear their conversation: 
|> >
|> >The difference is 1 times 10 to the something power, lets see... hmm, 
|> >it would seem that whatever power of 10 I choose, there is a larger 
|> >power I could use.... how many 9s are there again? 
|> 
|> Simpler answer: the difference is precisely 0.000000...
|> 
|> Hmm.  Or mabe it's 0.00000... or 0.0000000...
|> 
|> Not so simple after all.
|> 
"The square root of 3 is 2 for values of 3 which are very close to 4."
(source forgotten)
Newsgroups: sci.math,sci.philosophy.tech
From: hrubin@mentor.cc.purdue.edu (Herman Rubin)
Subject: Re: Numbers and sets
Organization: Purdue University Statistics Department
Date: Sun, 10 Jan 1993 16:38:56 GMT

In article <1ioee8INN9sc@bang.hal.COM> landman@hal.COM (Howard Landman) writes:
>>>>In article <1992Dec19.140927.18700@husc3.harvard.edu>,
>>>>zeleny@husc10.harvard.edu (Michael Zeleny) writes:

>>MZ:
>>>>>The Axioms of Foundation and Choice are analytically
>>>>>true of sets;

>At the risk of appearing to agree with Bill Taylor ...

>How can anyone say this when it's been known for a couple of decades that
>the Axiom of Choice is independent of the other axioms of set theory?
>That is, you can assume it's true and get a consistent theory, but you
>can also assume it's false and get a consistent theory.

It has been known for much longer that the Axiom of Foundation is likewise
independent.  
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@snap.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
From: brock@NeXTwork.Rose-Hulman.Edu (Bradley W. Brock)
Newsgroups: sci.math
Subject: Re: Real Algebraic Geometry
Date: 10 Jan 1993 03:41:40 GMT
Organization: Computer Science Department at Rose-Hulman
Reply-To: brock@NeXTwork.Rose-Hulman.Edu (Bradley W. Brock)
NNTP-Posting-Host: g210b-1.nextwork.rose-hulman.edu

In article <1993Jan5.155916.2076@cs.brown.edu> dzk@cs.brown.edu (Danny Keren)  
writes:
> I am interested in the question of
> determining the number of components a real algebraic variety
> (e.g. the zero set of a polynomial with real coefficients) in
> 2 and 3 variables has.

"Harnack proved that a smooth real algebraic curve of degree d can have a  
maximum of (d-1)(d-2)+1 components and that for every degree this maximum is  
realized."-Barry Mazur, Arithmetic on curves, Bull. A.M.S. 14, 216.  See  
A'Campo, Lecture Notes in Math. vol. 770, Springer-Verlag, on Hilbert's 16th  
problem. 
--
Bradley W. Brock, Department of Mathematics
Rose-Hulman Institute of Technology  | "Resist not evil.... Love your
brock@nextwork.rose-hulman.edu       |  enemies."--Jesus of Nazareth
Xref: sparky sci.math:17824 comp.theory:2826
From: sitarama@cps.msu.edu (Kocherlakota Sitarama)
Newsgroups: sci.math,comp.theory
Subject: Szemeredi Regularity Lemma
Summary: Lemma
Keywords: Szemeredi Regularity Lemma
Date: 8 Jan 93 08:19:44 GMT
Organization: Dept. of Computer Science, Michigan State University
NNTP-Posting-Host: indian.cps.msu.edu
Originator: sitarama@indian.cps.msu.edu


If you know of any references on the following lemma please mail me
I will summarize the mails.

"Szemeredi Regularity Lemma"

Thanks!

Sitarama
Newsgroups: sci.math,sci.logic
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Frankly,my dear......was: Fermat's Last Theorem
Organization: MIT Department of Mathematics, Cambridge, MA
Distribution: usa
Date: Mon, 11 Jan 93 18:03:10 GMT

In article <1ilihlINN6ke@shelley.u.washington.edu> petry@zermelo.math.washington.edu (David Petry) writes:
Ilan Vardi writes:
>While it is true that no one can explain why Fermat's Last theorem has to 
>be true, there are very convincing heuristics arguments that it really 
>ought to be true.  Briefly, for a given large exponent p, the set of
>numbers which are perfect p'th powers is a very sparse set of numbers,
>and the probability that some number from such a set is the sum of two
>others from the set is very small.
>
>> This is a natural phenomenon that should be explained. 
>
>Actually, it's questionable whether it should be called a phenomenon
>at all.  Usually we think of a "phenomenon" as something that occurs
>which is improbable (don't we?).  Fermat's Last theorem is far from 
>being improbable.

Of course the notion of "probability" here is even more murky than
usual.  One is approximating a deterministic system by a stochastic one
and hoping that one is not neglecting any important patterns.  

But I tend to agree with Petry.  My friend Bruce Smith has proposed that
there should be large classes of conjectures which are "probably true"
by simple heuristic arguments, but are not provable.  I keep hoping that
some good logician will construct a natural example of the following.
A sequence P_n of number-theoretic statements such that 1) all the P_n
are "true with probability 1" by some heuristic argument, 2) one can
prove (using your favorite axioms) that all but finitely many P_n *are*
true, but 3) any consistent r.e. extension of Peano arithmetic can only
prove finitely many P_n are true.  In other words, the P_n are almost
certainly true, but "only for statistical reasons"; there is no good
reason for any one to be true other than that it'd be an amazing
coincidence if it were false.  Feel free to modifsy the problem a bit,
but the key word is NATURAL; I don't want example that are cooked up by
Goedelian encoding-type tricks, I want examples that sound a lot like
the weak Goldbach's conjecture [all sufficiently large even numbers
are the sum of two primes].  Note that Fermat's last theorem is not
true with probability 1, but only with probability equal to some number
very close to 1, while the weak Goldbach conjecture is true with
probability 1 (according to a naive heuristic argument).  

Bruce Smith has also suggested that the 4-color theorem only admits long
proofs, because it is not true for any "good reason," but it admits many
equally good long proofs, because the only way to see it is to rule out
a large finite set (a "spanning" set) of things that might go wrong.


Newsgroups: sci.math
From: ara@zurich.ai.mit.edu (Allan Adler)
Subject: Re: proof wanted 2
Organization: M.I.T. Artificial Intelligence Lab.
	<1993Jan11.210012.18587@Princeton.EDU>
	<1993Jan11.222338.17682@sol.ctr.columbia.edu>
Date: Tue, 12 Jan 1993 04:11:58 GMT

In article <1993Jan11.222338.17682@sol.ctr.columbia.edu> dy@shire.math.columbia.edu (Deane Yang) writes:

   The point here is that, except for 0, X is a discrete and therefore 
   locally compact space. The conjecture needs a further assumption,
   something that makes the space everywhere nondiscrete.

   If I'm not mistaken, a reasonable assumption is that X is a length space.
   This means that the distance between two points in X is equal to the infimum
   of the lengths of continuous curves joining the two points. In particular,
   this means that given two points that are a finite distance apart,
   then you can always move one point a little bit and make the distance a little
   smaller.


What about the plane with the French Railroad metric? The origin is
Paris and every ray from the origin is a railroad line. The distance
between two points is the minimal distance you have to travel to
get from one to the other by train. In other words, if two
points x,y are collinear with 0 then the distance from x to y is the usual
distance in the plane. Otherwise, the distance is the sum of their
usual distances to the origin. The set of all points of the form
(1,y) with y>0 is closed in this metric but has no closest point
to the origin.

Anyway, my question is answered in the negative. So what IS the necessary
and sufficient condition?

Allan Adler
ara@altdorf.ai.mit.edu
Newsgroups: sci.math
From: ara@zurich.ai.mit.edu (Allan Adler)
Subject: Re: High Prices of Math Books. I am pissed.
Organization: M.I.T. Artificial Intelligence Lab.
	<1idj1gINNhah@roundup.crhc.uiuc.edu> <C0GEBz.98t@mentor.cc.purdue.edu>
Date: Thu, 7 Jan 1993 06:21:28 GMT

In article <C0GEBz.98t@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:

   The amount of effort required to produce a fairly good book when one
   does not already exist for a one-semester to one-year course is two
   or three person-years, and preferably involves more than one person
   writing it.  An active faculty member does not have the time to do
   very much writing.

Why is this so? I heard a rumor that Serge Lang can knock off a book in
a weekend.

Allan Adler
ara@altdorf.ai.mit.edu
Newsgroups: sci.math
From: mlc@iberia.cca.cr.rockwell.com ("Mike Cook, x2792, 124-211")
Subject: RE: Ramsey Theory Games
Date: Tue, 12 Jan 1993 12:18:36 CST
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          12:18:36 CST
X-To: SCI.MATH.USENET
X-Cc: mlc@iberia.cca.cr.rockwell.com

> In-reply-to: mjd@saul.cis.upenn.edu's message of 11 Nov 92 23:55:26 GMT
> 
> I don't know why it never occurred to me before, but you could take
> nearly any result from Ramsey Theory and make a game out of it. 
> 
> For example:
> 
> 	Ramsey's Theorem: For any positive integers C and S, there
> 	exists a positive integer N such that if the edges of the
> 	complete graph on N vertices are partitioned into C sets, then
> 	there is at least one complete graph on S vertices whose edges
> 	are all in one set.
> 
> When C=2 and S=6, we have N=3.  (Well-known proof left as exercise for
> the interested reader.)  So when (C) two players take turns coloring the
> edges of the complete graph on (S) 6 vertices (that is, putting the
> edges into their own sets), at least one player will have a complete
> graph with (N) 3 vertices---a triangle.  If whoever gets a triangle
> first loses, there can be no draws.  This game is `sim'.

See:
"A Strategy for the Ramsey Game of 'Tritip'",
Michael L. Cook and Dr. Leslie Shader,
Proceedings of the Tenth Southeastern Conference on
Combinatorics, Graph Theory, and Computing,
April 1979, pp 315-324.

The object of this 2-player game is to color edges of a Ramsey
graph, trying to prevent the other player from coloring edges
that would form the following graph (other orientations allowed,
of course).

                X---------------X
                 \             /
                  \           /
                   \         /
                    \       /
                     \     /
                      \   /
                       \ /
                        X
                        |
                        |
                        |
                        |
                        |
                        X

See also some other papers by Shader where graph theory plays a part.

--

Michael Cook
Internet: MLC@IBERIA.CCA.CR.ROCKWELL.COM
These are not the opinions of my employer.

RE: net.wisdom - "It's not what we don't know that hurts,
                  it's what we know that ain't so." -- Will Rogers
Newsgroups: sci.math,alt.books.technical
From: bathurst@phoenix.Princeton.EDU (Bruce Bathurst)
Subject: Re: High Prices of Math Books. I am pissed.
Originator: news@nimaster
Organization: Princeton University
Date: Tue, 5 Jan 1993 20:37:17 GMT

I graduated from Princeton years ago, Gentlemen.  My observations on
how the trebling of technical book prices has affected education are
not complaints.  The poverty of the scholar is historical.  (It
pre-dates modern political hate-labels.)

The defending of American book prices as reasonable because they are
expensive to produce misses the point that students can't buy them.
Remarks that the rest of of world has it worse, so clam up, are not
profitable remarks.

Students need to own a library, not use a public one.  Let me remind
my colleagues that at the schools where I've taught, libraries are
nothing but desks for those fleeing amplified music, and every chair
is taken.  Even if they changed their function and purchased hundreds
of copies of the same texts, they would not be viable substitutes for
a personal library.

I've clarified my point, you can return to your insightful discussions
now.

Bruce (Peripatetic Instructor)
-- 
Department of Geological and Geophysical Sciences
Princeton University, Princeton, NJ 08544
bathurst@phoenix.princeton.edu bathurst@pucc.bitnet !princeton!phoenix!bathurst
Newsgroups: sci.math
From: graham@cs.sfu.ca (Graham Finlayson)
Subject: Addition of Vector Spaces
Keywords: Vector Spaces
Organization: CSS, Simon Fraser University, Burnaby, B.C., Canada
Date: Tue, 5 Jan 1993 22:25:45 GMT

I was wondering if anyone has come across the following problem:

Given 2 mXn matrices A and B, where m>n. The columns of A and B span
n-dimensional subspaces of m-space. What are the properties of the
subspaces spanned by linear combinations of A and B.
 
For example, given a 3rd mXn matrix C how can you check if there exists
a linear combination of A and B such that the space defined by (a*A+b*B)
intersects  the space spanned by C in 
at least 1 dimension?

Any pointers you can give would be appreciated.

Please email responses to 

	grahan@cs.sfu.ca

Graham

(If there is any interest I will post a summary)
From: neilfp@astro.ocis.temple.edu (Neil F. Palumbo)
Newsgroups: sci.math
Subject: HELP Quadratic Eqns
Date: 9 Jan 93 17:25:20 GMT
Organization: Temple University
X-Newsreader: TIN [version 1.1 PL8]

I have a set of equations of the form:

			s'*Ai*s + Bi*s + ci = 0, i=1,2,...,100

The unknown is the vector s (the same unknown vector in all 100 eqns).
The Ai's are square (100x100) matrices, the Bi's are (100x1) vectors,
and the ci's are constants. s is of course a (100x1) vector. Can anyone
suggest an approach to simultaneously solving this set of equations? Please
e-mail me with your responses. Thanks in advance...

				Neil
Newsgroups: sci.math