HDTenEightyP/NetNews huggingface dataset:
From: bs@faron.mitre.org (Robert D. Silverman)
Subject: Re: Calculating the # of primes < n
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 2 Apr 1992 11:53:08 GMT

In article <1992Apr2.014052.1514@cunixf.cc.columbia.edu> pvl2@cunixb.cc.columbia.edu (Priscilla V Loanzon) writes:
:I was trying to compute what fractions of the positive integers are
:prime. I was doing this by getting the fraction left if from the
:positive integers we remove everything divisible by the primes
:p_1,p_2,p_3,...  
:
:I get the following infinite series that gives the fraction of integers
:left: 
:1 - (1/p_1) - (1/p_2)(1-1/p_1) - (1/p_3)(1-1/p_1)(1-1/p_2) -
:(1/p_4)(1-1/p_1)(1-1/p_2)(1-1/p_3) - ...
:
:From the formula n/log n for # of primes less than n I think the
:infinite series must converge to 0 and the partial sums to 1/log n. Is
 
There is something I don't understand. You say above you are trying
to figure out what fraction of the integers are prime. Yet in the
sentence just above you say you know the Prime Number Theorem.
The two statements are incompatible. If you know the latter, you
have the former automatically, so what is it that you are REALLY
trying to do??

If you are trying to verify the PNT by sieving out primes (as your
inclusion/exclusion argument seems to show), it will not work.
You must carry the sieving up to sqrt(n) and in such a case the
accumulated errors in the sieve swamp the main term. The fundamental
lemma of the sieve prohibits one from sieving out too many primes
(i.e. not more than about log^2 n).  There is no known proof of PNT
by sieve methods.

 
The partial sums converge to exp(-gamma)/log n. This is known as
Merten's theorem. See Hardy & Wright. The proof is simple.
 
:Is there a computer program that can calculate what a series like that
:goes to, including the partial sums?
 
Why on Earth would one want to do such a thing??

Write your series as an infinite product instead.

(1-1/2)(1-1/3)(1-1/5)(1-1/7)..........

And trying to show the behavior of a slowly diverging or converging
series by computer is an exercize in futility.



--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
Newsgroups: sci.math
From: mroussel@alchemy.chem.utoronto.ca (Marc Roussel)
Subject: Re: shape functions
Organization: Department of Chemistry, University of Toronto
Date: Thu, 16 Apr 1992 00:13:13 GMT

In article <DAVIS.92Apr14195341@pacific.mps.ohio-state.edu>
davis@pacific.mps.ohio-state.edu  (John E. Davis) writes:
>   I am looking for a way of quantizing shapes.

     I believe that R.F.W. Bader has made extensive use of such methods
in his work on molecular topology.  An author search for his work
through (for instance) Chemical Abstracts should lead you to much
relevant literature.

				Marc R. Roussel
                                mroussel@alchemy.chem.utoronto.ca
Newsgroups: sci.math
From: gindrup@math.okstate.edu (Eric `'d'kidd' G..)
Subject: Re: Prime numbers (Was Re: Lackey again)
Organization: Oklahoma State University, Math Department
Date: Sat, 11 Apr 1992 17:55:57 GMT

In article <1992Apr8.011828.19169@sfu.ca> erick@fraser.sfu.ca (Erick Bryce Wong) writes:
>
  [A few things deleted...]
>
>What can I say?  I got bored one day and computed both of them...(Anyone want
>the rest of the digits? ;)
>
Well, if it wouldn't be too much trouble, *I* wouldn't mind having them...

-- Eric Gindrup ! gindrup@hardy.math.okstate.edu
From: piche@fuksi.cc.tut.fi (Robert Piche)
Newsgroups: sci.math
Subject: Re: ideas to prove f_n --> 0
Date: 15 Apr 92 10:41:27 GMT
Organization: Tampere University of Technology


Last month, zwang@pop.stat.purdue.edu (Zhongcheng Wang) wrote:

>Greetings all,     I am looking for any idea to prove that  
>f_n(x) goes to 0 as n increasing, where f_n(x) is a function
>on [0,1] and is defined by 
>
> f_n(x) = 0.5*( g(x/2)*f_(n-1)(x/2) + g(1-x/2)*f_(n-1)(1-x/2) )
>
>starting with f_0(x) being a constant on [0,1]....

Here are some thoughts I have had about this mapping.

First, there is the inequality
   ||f_n||_1 <= ||f_(n-1)||_1 * ||g||_2
using the usual L1 and L2 norms, so that convergence in the mean
is assured if the L2 norm of g is less than 1.  However
this does not help here since the L2 norm of Wang's g function
is more than 1.

If one restricts oneself to a finite-dimensional space, e.g. functions
defined by an N-term Fourier series or by N samples, then the recursion
is of the form f_n = A f_(n-1) where A is an NxN matrix.  The recursion
converges to the zero vector for all f_0 iff the spectral radius of A is 
less than 1.  Using Wang's data, I constructed an A matrix corresponding
to approximation of the original recursion by filling in missing data 
points with linear interpolation.  The spectral radius of the resulting A 
matrix was 0.5.  However, the problem remains how to show that the finite 
dimensional approximation is close enough to ensure that the original 
recursion converges too.

I have obtained some advice from Mark Krasnosel'skii of the Russian 
Academy of Science.  He tells me that the spectral radius r of the 
operator is bounded by the inequality
    r <= alpha_n := (sup { f_n(t): 0<=t<=1 } )^(1/n)
Using the data provided in Wang's posting I compute (again using linear
interpolation) the following estimates for alpha_n:
    n     alpha_n
    1    1.0000
    2    2.0000
    3    1.5572
    4    1.2191
    5    1.0282
    6    0.9134
    7    0.8384
    8    0.7861
    9    0.7476
   10    0.7182
So, the sixth iterate already gives the proof of contractiveness.
To complete the proof, it remains only to prove that the error
of these estimates is not too large; this could be done using
the usual error bounds for linear interpolation, I suppose.

According to Wang, this mapping arises in a time series analysis problem.
Interestingly, the finite-dimensional version of this same mapping is 
studied in a recent paper by Timo Eirola (timo@spazio.hut.fi) where it 
arises in multiresolution (wavelet) analysis.


----------------
Robert Piche                                
Tampere University of Technology          Internet:  piche@fuksi.cc.tut.fi
Dept of Mathematics                       FAX:       +358 31 161549
P.O. Box 692, SF-33101 Tampere, FINLAND   Phone:     +358 31 162169
From: spencer@panix.com (David Spencer)
Newsgroups: sci.math
Subject: Re: distribution of wealth
Date: 10 Apr 92 14:14:04 GMT
Organization: PANIX Public Access Unix, NYC

In article <1992Apr6.061033.11933@augean.eleceng.adelaide.edu.AU> dabbott@augean.eleceng.adelaide.edu.AU (Derek Abbott) writes:
>
>If you added up everbody's bank balances, in the world, and then
>divided by the world population, what would we all get??

on average, $0, of course (before taxes...)

What we'd all _have_ is a tougher question.
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Newsgroups: sci.math
Subject: Re: closed form solution to d.e.
Date: 9 Apr 92 19:09:28 GMT
Organization: Beckman Instruments, Inc.

In <a_rubin.702845575@dn66> a_rubin@dsg4.dse.beckman.com (Arthur Rubin) writes:

>In <1992Apr9.042657.3710@nuscc.nus.sg> fbagohkh@nuscc.nus.sg (Mark Goh (Dr)) writes:

>>Is there a closed form solution for this d.e.

>>y' + a(y)^b=1, 0<a,b<1 ?

>yes

Oh, you wanted to know what it is:

y[x] == ((1 - b) (C + a x)) ^ 1/(1 - b)
--
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.
Ich bin ein Virus.  Mach' mit und kopiere mich in Deine .signature.
Newsgroups: sci.math
Subject: Re: Paul Erdos
From: harper@kauri.vuw.ac.nz (John Harper)
Date: Mon, 13 Apr 1992 21:22:36 GMT
Organization: Victoria University of Wellington, New Zealand

In article <ARNDT.92Apr13153433@napoleon.c3consult.comm.se> arndt@c3consult.comm.se (Arndt Jonasson) writes:
>
>> >Pardon my ignorance. Who is this Erdos dude?
>>Several people have been abusing one another instead of answering the
>>question. Paul Erdos is a very famous Hungarian mathematician (aged around
>>80 I'd guess) who occasionally offers cash prizes of a few hundred dollars
>>for solutions to problems that he hasn't solved himself and that interest him.
>>The problems I have seen are HARD and are on combinatorics and number theory.
>
>Since no one else did: let me just add that the correct spelling of
>his name isn't "Erdos", but since the proper glyph doesn't have an
>ASCII representation (nor an ISO Latin-1 one), "Erdos" will have to do
>in many circumstances. The proper spelling has two "grave accents" on
>top of the 'o'. I think (but am not certain) it would be "Erd\Hos" in
>TeX. TeX's \"o (which does exist in ISO Latin-1) is a separate letter
>in Hungarian, so using that isn't correct either. Pronunciation?
>Something not very close to "Airdersh" (second syllable long).
>
>            //
>      E R D O S
>
>Does anyone have a machine-readable list of Erdos's unsolved problems
>(and solved, for that matter, if there are any)?
>--
>Arndt Jonasson				Communicator C3Consult AB
>arndt@c3consult.comm.se
From: holerbch@phoenix.Princeton.EDU (Uwe Hollerbach)
Newsgroups: sci.math
Subject: polyominoes, anyone?
Summary: how many?
Date: 10 Apr 92 17:54:34 GMT
Organization: Princeton University
Originator: news@ernie.Princeton.EDU

G'day, folks!
	Since pentominoes have come up recently in one or two threads
in this newsgroup, I thought I'd ask a question which I have been
wondering about for some time. Is there a formula for the number of
n-ominoes, for general n, other than enumerating all possible n-ominoes,
then counting how many you have? I have done just that up to 10 (not by
hand; you think I'm crazy!?); see the table below. But something more
general would be nice; that table is starting to get quite difficult
to extend...

1  2  3  4  5   6   7    8    9     10
1  1  2  5  12  35  108  369  1285  4655

Failing an exact formula, is anything known about the asymptotics?

regards, Uwe

-- 

Uwe Hollerbach
uh@acm.princeton.edu   <-----------------\
uh@alumni.caltech.edu   (forwards to) ---/
From: allenk@harry.ugcs.caltech.edu (Allen Knutson)
Subject: Why should Trace be so easy to calculate?
Organization: California Institute of Technology, Pasadena
Date: Wed, 8 Apr 1992 04:19:23 GMT

Given how difficult the product of the eigenvalues is to calculate, why should
the sum be so easy?

Please DON'T post a proof that the sum along the diagonal is unaffected by
change of basis. That's not what I'm asking. What I'm asking for is:

Is there a well-known way to calculate the other symmetric polynomials of
the eigenvalues? I'm interested in whether the other early terms are also
easy, for instance. That seems unlikely, though.

Again, please don't tell me how to compute the characteristic polynomial, as
that's much more work than the trace. More, I'd like a general formula where
one plugs in the matrix and the number k, and gets out the kth symmetric
polynomial, with lots of VISIBLE cancellation in the case of k=1.
Thanks!								Allen K.

Xref: sparky sci.math:6126 comp.theory:1102
From: eppstein@wormwood.ics.uci.edu (David Eppstein)
Newsgroups: sci.math,comp.theory
Subject: Re: factorial function (mod n?)
Date: 16 Apr 92 18:17:30 GMT
Reply-To: eppstein@ics.uci.edu (David Eppstein)
Organization: Univ. of Calif., Irvine, Info. & Computer Sci. Dept.

ld231782@LANCE.ColoState.Edu (L. Detweiler) writes:
> On a related note, what is the best algorithm for n! (mod m)?
> (Yes, if n>m this is trivial.  Also, try to avoid factoring m.)

abbasi@pollux.cs.uga.edu (Sarmad Abbasi) writes:
> A efficient deterministic algorithm would yield a efficient deterministic
> primality testing algorthm. (Wilson's Theorem)

It's worse than that.  If you had an algorithm for n! (mod m)
you could factor m by using binary search to find the largest n such
that n! = 0 (mod m).  So factorization is unavoidable.
--
David Eppstein    UC Irvine, Info & Computer Science    eppstein@ics.uci.edu
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math
Subject: Re: electronic math seminar?
Date: 8 Apr 92 18:58:47 GMT
Distribution: sci
Organization: M.I.T. Artificial Intelligence Lab.

Just in case I didn't make it clear in my posting, I am imagining that people
will use telnet from other sites, however remote, to participate in such
seminars.

There is another reason I am interested in this idea: being too poor to travel
to conferences, preferring to sleep in my own bed, and having limited 
enthusiasm for the social company of mathematicians in general, the idea
of electronic seminars makes it possible to attend conferences without
encountering these obstacles.

Allan Adler
ara@altdorf.ai.mit.edu
Newsgroups: sci.math
Subject: Re: An elementary, "naturally occurring" problem
From: harper@kauri.vuw.ac.nz (John Harper)
Date: Fri, 10 Apr 1992 05:09:52 GMT
Organization: Victoria University of Wellington, New Zealand

In article <1992Apr10.000446.24127@math.ucla.edu> oliver@oak.math.ucla.edu (Mike Oliver) writes:
>In article <1992Apr9.201215.1319@wdl.loral.com> mab@wdl39.wdl.loral.com (Mark A Biggar) writes:
>
>>Question: is it possible to partition the interval [0,1] into a finite
>>number of uncountable sets that are all dense on the interval [0,1]?
>>If so, can you give a rule to determine partition membership?
>
An easy way: For each x in [0,1], put x in A if its decimal expansion 
contains an infinite number of 7's, otherwise put x in B.  A and B are 
both uncountable and both dense in [0,1]. 

John Harper Mathematics Dept. Victoria University Wellington New Zealand
Newsgroups: sci.math
Subject: On consecutive primes: q<3p/2+1?
From: karl@crfm.gen.nz (Karl Scherer)
Date: 7 Apr 92 12:17:48 GMT
Reply-To: karl@crfm.gen.nz (Karl Scherer)
Organization: Household UNIX, Auckland, New Zealand

In 1845, Bertrand noted that consecutive primes were not
too far apart and conjectured q/p < 2 for consecutive primes.
Five years later this was proved by Chebychev.
In this century Erdos improved on this, showing that, in fact
q/p is not larger than 5/3.

My question is how good the value 5/3 is compared to the
actually calculated ratios q/p for known primes.
Maybe 5/3 is a good approximation only for primes < 10,
and 3/2 holds for all p >10, 6/5 for all p>100 and so on.
To be precise:

1. What is the maximum known value of some q/p , p > 10 ?
2. Does the maximum of q/p go to 1 for large p 
	(say 10^n < p < 10^n+1 and n -> infinity) ?
3. Is the inequality q < 3p/2 +1  false in any known case?

I would very much appreciate any answer to this...

 
-- 
Domain: karl@crfm.gen.nz  ------------------------------+-------------------+
Voice : 64-9-4834211                                    | This Space Sells! |
Fax   : 64-9-???????                                    |                   |
Snail : 2/63 Rangatira Rd, Birkdale, Auckland, New Zealand -----------------+
Newsgroups: sci.math
From: fbagohkh@nuscc.nus.sg (Mark Goh (Dr))
Subject: closed form solution to d.e.
Organization: National University of Singapore
Date: Thu, 9 Apr 1992 04:20:28 GMT


Newsgroups: sci.math
From: victor@watson.ibm.com (Victor Miller)
Subject: Re: Large primes; largest pi(n) known.
Date: Mon, 13 Apr 1992 02:12:56 GMT
Reply-To: victor@watson.ibm.com
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
	<1992Apr8.163734.12362@cl.cam.ac.uk>
Organization: IBM, T.J. Watson Research Center

>>>>> On 8 Apr 92 16:37:34 GMT, cet1@cl.cam.ac.uk (C.E. Thompson) said:

Chris> In article <1992Apr7.103427.4558@csc.canterbury.ac.nz>,
Chris>  wft@math.canterbury.ac.nz (Bill Taylor) writes:
Bill> It doesn't seem to be on the FAQ, so can someone tell us...
Bill> 
Bill> what is the largest known prime for which all smaller primes are known;
Bill> 
Bill> i.e.
Bill> 
Bill> how far does the list of known primes extend, without gaps;
Bill> 
Bill> i.e. (I presume equivalently)
Bill> 
Bill> what is the largest n for which pi(n) is known ?

Chris> A natural assumption, but false. There are methods of calculating \pi(x)
Chris> which do not involve determining all primes less than x. See

Chris>   "Computing $\pi(x)$: The Meissel-Lehmer method"
Chris>    J.C.Lagarias, V.S.Miller, & A.M.Odlyzko
Chris>    Math. Comp. 44 (1985) pp. 537-560

Chris> The largest x for which \pi(x) is presented in this paper is 4 * 10^16; 
Chris> I expect it has pushed quite a bit higher by now.

Actually, I haven't pushed it farther yet.  Maybe in another few
months (I'm the middle author, and did the actual calculations).

Chris> The method used is based on Meissel's work in the 1870s, and can 
Chris> calculate \pi(x) in time proportional to x^{2/3}, give or take a few
Chris> (log x)'s. The above paper mentions a different (unublished) method due
Chris> to Lagarias and Odlyzko which can achieve x^{3/5}. Does anyone know where
Chris> and when (and if) this was eventually published?

Lagarias and Odlyzko published "Calculating \pi(x): an Analytic
Method" in the Journal of Algorithms in 1986.  Since then, because of
a result of Schonhage and Odlyzko about calculating many values of
\zeta(x) at the same time, the time has been lowered to
x^{1/2+\epsilon} at the expense of space x^{1/4} (I believe) instead
of the space x^\epsilon that the x^{3/5} method uses.  I don't
remember off the top of my head where the Odlyzko and Schonhage paper
is published.

Chris> Chris Thompson
Chris> JANET:    cet1@uk.ac.cam.phx
Chris> Internet: cet1@phx.cam.ac.uk

--
			Victor S. Miller
			Internet: victor@watson.ibm.com
			IBM, TJ Watson Research Center
From: baumgart@masg1.epfl.ch (Marcel Baumgartner)
Newsgroups: sci.math
Subject: A simple question about R2 - Q2 (irrational points of the plane)
Date: 7 Apr 92 15:45:10 GMT
Organization: Ecole Polytechnique Federale de Lausanne

Keywords: In R2 - Q2, does there exist an oblique straight line ?

Newsgroups: sci.math
From: asimov@wk223.nas.nasa.gov (Daniel A. Asimov)
Subject: Re: nonexistance of counterexample -> existance of proof?
Organization: NAS, NASA Ames Research Center, Moffett Field, CA
Date: Thu, 9 Apr 92 23:38:11 GMT

In article <1992Apr9.211606.22882@pasteur.Berkeley.EDU> luzeaux@roma.berkeley.edu (Dominique Luzeaux) writes:
>In article <31475@sdcc12.ucsd.edu>, ec120wee@sdcc4.ucsd.edu (Anthony
>Minkoff) writes:
>|> He made the statement "If no counterexample exists, then a proof
>|> exists."  
	(lines deleted)

>Anyway, nonexistence of counterexamples does not imply existence of proof.
>
>As you mentioned it, Goedel built a formula which was true but could not
>be proved in the theory where this formula was built. Your problem points
>at the fact that truth and provability are different, because provability
>(with the logical meaning, I mean following Hilbert's scheme) implies a
>FINITE number of steps in your proof playing with axioms and rules of 
>inference and theorems which were already proved in this manner.
>
Dominique Luzeaux



A long time ago I read an excellent article, "Goedel's Proof"
(or is it Goedel's Theorem" ?) in the 4-volume set, 
"The World of Mathematics," by, I think Kasner and Newman.

In a nutshell, the idea of the proof is as follows:

------------------------------------------------------------
A proposition in arithmetic is constructed which can also be
understood to say "There is no proof of me."  (This is quite
complicated and uses a system for numbering symbols, statements,
and proofs called "Goedel numbering.")

Now suppose the proposition is false.  Then the claim
"There is no proof of me" must be false, so the proposition
must have a proof.  But then it must be true--CONTRADICTION.

Thus the proposition must be true, and so the claim that
"There is no proof of me" must be correct.  QED.
------------------------------------------------------------

Now, it is quite interesting to note that using meta-methods
outside the axiom system under discussion, the above argument
"proves" that although there is no proof of the proposition
in question, it ***is*** true.   

For example, suppose someone came up with a variant method of 
Goedel numbering for which the proposition which says "There
is no proof of me" turns out to be the Goldbach conjecture.
Then I think many people would be convinced that the Goldbach 
conjecture is true because it would have a "meta-proof."

QUESTION:	Is it known whether all examples of undecidable
propositions must have such "meta-proofs" ?   Or is this question
impossible to make rigorous?

--Dan Asimov
asimov@nas.nasa.gov
turned
out 






Newsgroups: sci.math,sci.physics,sci.comp-aided
From: semmett@gmuvax2.gmu.edu (Steve Emmett)
Subject: Complex Hermitian eigenvalues 
Organization: George Mason University, Fairfax, Va.
Date: Mon, 6 Apr 1992 16:52:55 GMT

(Seems my submission over the weekend got eaten by our system here)

I've got a problem using CH from netlib.  CH is supposed to compute the
eigenvalues and/or the eigenfunctions of a complex hermitian matrix.  I
have found that it gives "incorrect" answers when the trace of the
matrix is non zero.

Is anyone familiar with CH?  Can anyone point me to a routine (and the
FTP site) that computes the eigen values/vectors of a complex hermitian.
My preference is to understand why CH doesnot give the results I expect.

The trivial examples is as follows:  let the matrix be

            |  0   i  |
            | -i   0  |

CH correctly gives the eigenvalues as +/- 1.  Now, put 1, say, in the
trace.  You would expect the eigenvalues to be 2 and 0.  However, CH
gives you -.618 and 1.618 (the golden mean).  There is something going
on that I don't understand.  

If you have some experience with CH I'd really appreciate some help.
E-mail or newsgroup response is fine.  

Steve Emmett


Xref: sparky sci.math:6111 comp.theory:1100
From: abbasi@pollux.cs.uga.edu (Sarmad Abbasi)
Newsgroups: sci.math,comp.theory
Subject: Re: factorial function (mod n?)
Date: 16 Apr 92 06:04:32 GMT
Organization: University of Georgia, Athens

In article <1992Apr16.052404.15782@yuma.acns.colostate.edu> ld231782@LANCE.ColoState.Edu (L. Detweiler) writes:
>- - -
>
>On a related note, what is the best algorithm for n! (mod m)?
>(Yes, if n>m this is trivial.  Also, try to avoid factoring m.)
>--
>
>ld231782@longs.LANCE.ColoState.EDU


A efficient deterministic algorithm would yield a efficient deterministic
primality testing algorthm. (Wilson's Theorem)



Abbasi.
Newsgroups: sci.math