HDTenEightyP/NetNews huggingface dataset:
From: kfl@access.digex.com (Keith F. Lynch)
Newsgroups: sci.math
Subject: Re: Philosophy of Pi
Date: 29 Dec 1992 02:47:40 GMT
Organization: Express Access Public Access UNIX, Greenbelt, Maryland USA
NNTP-Posting-Host: access.digex.com

In article <1992Dec11.155846.10861@ee.ubc.ca> bdow@ee.ubc.ca (DOW BRUCE PAUL) writes:
> But in Electrical Engineering we see Pi popping up in *so* many places!

Only because people insist on measuring frequencies in cycles per second
rather than in radians per second.

Keith Lynch, kfl@access.digex.com
Newsgroups: sci.math
From: seeta@eng.wayne.edu (Seetamraju UdayaBhaskar Sarma)
Subject: Re: Philosophy of Pi
Reply-To: seeta@eng.wayne.edu
Organization: College of Engineering, Wayne State University, Detroit Michigan, USA
Date: Wed, 23 Dec 1992 20:28:47 GMT

-+>brumleve@iboga (Dan Brumleve) writes:
-+>>dat92oma@ludat.lth.se (Ola Martensson) writes:
-+>>
-+>>>A friend and I discussed if you can find any integer anywhere
-+>>>in the expansion of PI?
-+>>
-+>>Hmm...  I'd think that you would be able to find any integer in an irrational
-+>>number.  Not sure though.


What kind of number is : a irrational non-terminating number made up of the
following construct :-

a sequence of positive integers, each separated by a 0 (zero) ...


i.e.,   0:1.2.3.4.5.6.7.8.9.10.11.12.13.----101.102.103.---


where `:' is the decimal point  (!!!)
where `.' is the zero, while the `---' is supposed to be `...' the ellipsis...

U will clearly find every integer (+ve) in it...  But can ```it exist''' ???

Seetamraju Udaya Bhaskar Sarma
(email : seetam @ ece7 . eng . wayne . edu)
From: brock@NeXTwork.Rose-Hulman.Edu (Bradley W. Brock)
Newsgroups: sci.math
Subject: Re: Lapl. Transf. of Product of Functions
Date: 2 Jan 1993 05:42:11 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 <92366.132439CCB104@psuvm.psu.edu>  writes:
> Hello!
> Does anyone know what the Laplace transform of a product of functions is,
> please?

Because the Laplace transform is just a rotated Fourier transform the phrase
"the transform of the convolution is the product of the transforms" works both  
ways.  In particular if f=g h and F, G, and H are their Laplace transforms,  
then F(s)=1/(2Pi I)Integrate[H(r)G(s-r),{r,c-I Infinity,c+I Infinity}] in  
pseudoMathematica notation, where c is chosen so that the path of integration  
is to the right of all singularities of H.
--
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
Newsgroups: sci.math
From: umatf071@unibi.hrz.uni-bielefeld.de (sio)
Subject: the mode of an occupation problem
Date: Tue, 22 Dec 92 19:08:45 GMT
Organization: Universitaet Bielefeld
Keywords: occupation, Stirling 2nd, mode

occupation problem:
  If you distribute
    n balls into m urns (each urn is equally likely)
  the probability that exactly t urns contain at least
  one ball is:

  p(n,m,t) = (m)_t S(n,t) / m^n

  with:  (m)_t the falling factorial and
         S(n,t) the Stirling number of the 2nd kind.

  Question:
    where is the mode of p(n)?
    The parameters m and t are fixed.

  Conjecture:
  ===========
    z = log( m / (m-t) ) / log( m / (m-1) )
    The mode is ceiling(z) or floor(z).
    It is not each time the nearest integer.

Torsten Sillke
Xref: sparky sci.math:17440 sci.engr:2421
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math,sci.engr
Subject: Re: Question about "sparsity" of matrices
Date: 27 Dec 92 16:19:43 GMT
Followup-To: sci.math
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <24DEC199211261595@ariel.lerc.nasa.gov> ecaxron@ariel.lerc.nasa.gov (Ron Graham) writes:
>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?

A matrix may be usefully considered sparse if so many of it's elements
are zero that it pays to keep track of the nonzero elements only, and this
obviously depends on what kind of operations you are doing. A simple measure
closely related to this is the fraction of nonzero elements, and another
similarly notion of sparsity is for families of matrices, i.e. some people
consider a family of matrices to be sparse when the fraction of nonzero
elements tends to zero as the size of the matrix tends to infinity.

There are other important exploitable structures for matrices, which can
be observed in sparse matrices. Most important are polydiagonal and band
matrices, and then block polydiagonal matrices. Special algorithms and
data structures are used to exploit sparseness in each case.

So the bottom line is that _you_ get to decide what sparseness is by
showing how to exploit it. 

Later,
Andrew Mullhaupt
From: tim@maths.tcd.ie (Timothy Murphy)
Newsgroups: sci.math
Subject: Re: I imagine this comes up all the time...
Keywords: unique factors
Date: 3 Jan 93 01:27:14 GMT
Organization: Dept. of Maths, Trinity College, Dublin, Ireland.

ronl@tessi.com (Ron Lunde) writes:

>It seems odd that the first ones are all fairly small, but I can't
>find the 15th (at least nothing smaller than 614889782588491410):

Actually, with your definition, 2^{17} (approximately 130,000)
has 15 factors,
If M = p_1^{e_1}...p_r^{e_r}
is the factorisation of M into primes then
the number of factors of M is (e_1+1)...(e_r+1) - 2.
So if N+2 is a prime,
the only numbers with exactly N factors
are the prime powers p^{N+2}.
(In your case, N+2 = 17 is a prime.)



-- 
Timothy Murphy  
e-mail: tim@maths.tcd.ie
tel: +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
From: swillden@news.ccutah.edu (Shawn Willden)
Newsgroups: sci.math
Subject: Christmas tree visiting...
Date: 23 Dec 92 19:08:55 GMT
Organization: University of Utah Computer Center
X-Newsreader: Tin 1.1 PL3

An interesting problem...

A few weeks ago, my grandmother invented a "Christmas tree visiting
party" that she wanted to make a new family tradition.  The idea is to
pick an evening close to Christmas in which each family within the
extended family will visit the home of each other family to see their
Christmas tree and visit for a few moments.  One obvious constraint on
scheduling is that when family X shows up to the home of family Y,
family Y must be present.

The problem then, is stated:  Given N families and a graph representing
the distances (travel times) between homes, devise a schedule of
visits that minimizes travel time and ensures 
	(1) that in a single evening all families will visit all other
	families and 
	(2) each family will be home when each of the others comes to
	visit.

The schedule devised by my grandmother was decidedly suboptimal
(though I'd say she did quite well) so I (the mathematician) offered
to create a better schedule for her, thinking the problem would not be
that tough.

After several hours of (enjoyable) struggle, I'm not much closer to a
real solution, though I have devised a fairly good schedule, so I
thought I'd pass this on and let some of you play with it...

Shawn

--
Shawn Willden
swillden@icarus.weber.edu
Newsgroups: sci.math
From: ara@zurich.ai.mit.edu (Allan Adler)
Subject: Sums of reciprocals
Organization: M.I.T. Artificial Intelligence Lab.
Distribution: sci
Date: Tue, 22 Dec 1992 00:38:39 GMT


Let f(n)=1/n for every positive integer n. Let x be a positive
real number. Let g(n,x) be defined as follows:

(1) g(n,x) is a strictly increasing sequence of positive integers, possibly
     a terminating sequence.
(2) for all n, g(n,x) is the smallest integer consistent with (1)
    such that the sum of the reciprocals of g(k,x) for k=1,...,n
    does not exceed x.

I have been told that if x is rational then this leads to a way of writing
x as a finite sum of reciprocals of distinct positive integers. Let the number
of terms in this sum be N=N(x). I have been told that N is also the number of
terms in a minimal representation of x as a sum of reciprocals of positive
integers allowing repetitions.

Is this true and if so how or where is it proved?

Allan Adler
ara@altdorf.ai.mit.edu
Xref: sparky sci.math:17577 comp.theory:2778 sci.logic:2526
From: cet1@cus.cam.ac.uk (C.E. Thompson)
Newsgroups: sci.math
Subject: Re: TO ALL !!! -- End of Discussion, I hope
Date: 22 Dec 92 14:56:12 GMT
Organization: U of Cambridge, England

In article <a_rubin.724979263@dn66>, a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
writes:
|>
|> May I suggest, in this newsgroup, a graphic of an apple (not Apple (TM)), with
|> an announcement:  Happy Newton Day.
|> 
|> (Some readers may recall that Sir Issac Newton was born on December 25.)
                                     [sic]

But only in the Julian calendar. Maybe we really ought to wait until 4 January?

Chris Thompson
JANET:    cet1@uk.ac.cam.phx
Internet: cet1@phx.cam.ac.uk
Xref: sparky sci.math:17372 rec.puzzles:8111
From: mrob@world.std.com (Robert P Munafo)
Subject: Smallest integer with N distinct factors
Summary: Apparently not discussed before 1972
Keywords: unique factors number-theory primes composite
Organization: The World Public Access UNIX, Brookline, MA
Date: Mon, 4 Jan 1993 00:30:55 GMT

Ron, your sequence does not appear in Sloane's _Handbook of Integer
Sequences_, either in the form you gave it (6, 16, 12, 64, 24, 36, ...) or in
the form (4, 6, 16, 12, 64, 24, 36, ...) which includes the M for N=1. This
means that it was probably not discussed in journals or books before 1972.
 
As others have pointed out, the statement "since we can construct at least
one by multiplying the first N primes." should be replaced with "since
P^(N+1) for any prime P qualifies."
 
If you define M in a slightly different way "the smallest positive integer
with N distinct factors, including 1 but not including M" you get the
sequence (1, 2, 4, 6, 16, 12, 64,...) which also is not listed in Sloane's
book.  Other similar definitions of M produce the same sequence.
-- 
mrob's .sig #3 |     the Mandelbrot Set.             _,,___~-    '     Ra   _~
collect all 5! |                                           ~^___.-,          \,
Newsgroups: sci.math
From: ags@seaman.cc.purdue.edu (Dave Seaman)
Subject: Re: Measures, and Measurability
Organization: Purdue University
Date: Mon, 21 Dec 1992 14:28:38 GMT

In article <1992Dec21.053850.13489@news.media.mit.edu>  
minsky@media.mit.edu (Marvin Minsky) writes:
> [...]  Starts out with nice
> proof of metric density theorem, which says that if a set of an
> Euclidean space is measurable, then it contains rectangles whose
> measures are arbitrarily close to 1.

Without looking it up, I would venture to guess that it says if S is a  
measurable subset of a Euclidean space, then for each epsilon > 0 there is  
a rectangle D such that the measure of (D intersect S), divided by the  
measure of D, is greater than (1 - epsilon).

But then, I suppose a measure theorist would say that the two statements  
are equivalent, except the first one just leaves out the obvious details.

--
Dave Seaman
ags@seaman.cc.purdue.edu
Newsgroups: sci.math
From: Jim.Buddenhagen@launchpad.unc.edu (Jim Buddenhagen)
Subject: How's your geometric intuition?
Organization: University of North Carolina Extended Bulletin Board Service
Date: Thu, 31 Dec 1992 22:23:28 GMT

Start with a regular tetrahedron circumscribed by a sphere.  Subdivide
each triangle into 4 triangles by connecting edge midpoints, and project
to the sphere (radially), to obtain a new polyhedron with all vertices
on the sphere and with 16 triangular faces.

Repeat this procedure with the new polyhedron to get a third polyhedron.

True or false:  this third polyhedron is convex.

(I found the answer surprising, what about you?)
--
   The opinions expressed are not necessarily those of the University of
     North Carolina at Chapel Hill, the Campus Office for Information
        Technology, or the Experimental Bulletin Board Service.
           internet:  laUNChpad.unc.edu or 152.2.22.80
Xref: sparky sci.math:17564 sci.physics:21977
From: bhoughto@sedona.intel.com (Blair P. Houghton)
Newsgroups: sci.math,sci.physics
Subject: Re: Bayes' theorem and QM
Date: 31 Dec 1992 22:45:27 GMT
Organization: Intel Corp., Chandler, Arizona
NNTP-Posting-Host: stealth.intel.com

In article <1992Dec31.024039.2740@smsc.sony.com> markc@smsc.sony.com (Mark Corscadden) writes:
>Blair, there's an excellent discussion of this in the sci.math faq:
>17Q.- Cutting a sphere into pieces of larger volume  

Actually, not excellent enough.

What would really help me at this point is if someone
could email me a decent explanation of this (or at least
a direct pointer to a sufficient reference):

>       The esiest decomposition "paradox" was observed first by Hausdorff:
>
>     6) The unit interval can be cut up into COUNTABLY many pieces which,
>        by *translation* only, can be reassembled into the interval of
>        length 2.
>
>        This result is, nowadays, trivial, and is the standard example of a
>        non-measurable set, taught in a beginning graduate class on measure
>        theory.

				--Blair
				  "Something about needing more
				   gaps in my knowledge..."
Newsgroups: sci.math
From: sla7@cunixa.cc.columbia.edu (Seth Louber Antiles)
Subject: 2 questions (GRE Subject test)
Organization: Columbia University
Date: Fri, 25 Dec 1992 04:06:17 GMT

Saw this question in a sample GRE math subject test:

If f(1+x)=f(x) for all real x. If f is a polynomial and f(5)=11 then
what is f(15/2)? 

In general, for f a polynomial, for what g will f(g(x))=f(x) for all x
*not* imply that f is a constant? Appreciate simple examples.

What if we remove the restriction that f is a polynomial?

Another GRE question in the same sample:

"No element (other than identity) is its own inverse." What does this
tell you about a group?

Thanks.
Xref: sparky sci.math:17282 sci.physics:21554
From: bhoughto@sedona.intel.com (Blair P. Houghton)
Newsgroups: sci.math,sci.physics
Subject: Re: Bayes' theorem and QM
Date: 22 Dec 1992 00:26:57 GMT
Organization: Intel Corp., Chandler, Arizona
NNTP-Posting-Host: stealth.intel.com

In article <1992Dec18.134107.24536@oracorp.com> daryl@oracorp.com (Daryl McCullough) writes:
>The second statement is unwarranted, in my opinion. Bell's Theorem
>shows that there is no deterministic completion of QM that uses
>classical probability theory. However, Bell's Theorem does *not* say
>that there are no deterministic completions of quantum mechanics, if
>one is willing to give up classical probability theory.

How would a "non-classical probability theory" look?  What is
the limitation of "classical probability theory," and just
what is a "classical probability theory?"

				--Blair
				  "I'll have a scoop of Double
				   Fudge Underdog and a scoop
				   of Pine Nut Passion..."
Newsgroups: sci.math
From: dbradley@symcom.math.uiuc.edu (David Bradley)
Subject: Re: John Kemeny,66,Computer Pioneer and Educator is dead
Organization: Math Dept., University of Illinois at Urbana/Champaign
Date: Thu, 31 Dec 1992 00:24:15 GMT

In article <1hklnfINN1rp@usenet.INS.CWRU.Edu> bz223@cleveland.Freenet.Edu (Danny Guy Frezza) writes:
>
>As reported in the New York Times Today, John Kemeny died yesterday
>from a heart attack.  Dr. Kemeny was a Mathematican at Dartmouth College
>and later served as President of Dartmouth.  He is remembered by computer
>programmers as a co-inventor of "Basic", a computer language.
>     Full text in the NYT.

I don't get the Times.  Wasn't he one of a small group of gifted
high-school students recruited to work at Los Alamos during the War?

-D.M. Bradley
Newsgroups: sci.math,sci.philosophy.tech
From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Numbers and sets
Organization: Boise State University
Date: Mon, 28 Dec 1992 16:52:03 GMT

I'm avoiding nested quotations here.

On "purports to mean"; that was a slip of the metaphorical tongue.
Zeleny certainly did succeed in meaning what he said.

Certainly Foundation asserts that a _nonempty_ set is disjoint from
one of its elements :-(   Sorry about that.

My profession is relevant; Zeleny is claiming that the _definition_ of
a concept within the sphere of my work implies certain things.  Zeleny
undermines his own position by pointing out an alternate approach to
the notion of "set", that which regards sets as extensions of
predicates.  Obviously, I regard stratified comprehension (the
comprehension axiom of NF or NFU) as an acceptable version of the
latter approach; the mere existence of the alternate approach casts
doubt on the claim that Foundation is an analytic property of sets.

It is quite true that if one regards "set" as being defined in terms
of the iterative hierarchy, Foundation becomes analytic.  Well-founded
sets are well-founded for the same sort of reason that bachelors are
unmarried.  But Choice remains open to doubt (I don't doubt it,
myself, but I don't regard it as analytically true of sets, either).


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: sci.math,sci.physics
Subject: Re: Bayes' theorem and QM
Date: 28 Dec 92 15:52:56 GMT
Followup-To: sci.math
Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070

In article <TED.92Dec27145845@lole.nmsu.edu>, ted@nmsu.edu (Ted Dunning) writes:
>[...] 
> this is called the hidden variable interpretation of quantum
> mechanics.
> 
> it has been shown to be incompatible with experiments.  the basic
> problem is that of correlation of wave functions and whether or not
> correlated wave functions can communicate faster than the speed of
> light. 
> 
> basically, you have to either give up the hidden variables or
> causality.  (you pick which one you prefer).

This is not true. You are no doubt referring to tests of Bell inequality.
First Bell's result does not rule out deterministic causal theories. It
only rules out *local* theories. In the early 50's Bohm constructed a
non local causal model consistent with all the predictions of quantum
mechanics. 

Second the experiments that confirm the predictions of quantum mechanics
that Bell's inequality is violated are not conclusive. Aspect's experiment
was widely considered to be conclusive at the time but Franson showed
that Aspect did not prove that the correlated events he observed had a
space-like separation. See the following for more details about Aspect's
experiment:

J. D. Franson, Physical Review D, pgs. 2529-2532, Vol. 31, No. 10, May 1985.
A. Leggett, Foundations of Physics, V 17, p. 875.

Paul Budnik
From: kedlaya@husc8.harvard.edu (Kiran Kedlaya)
Newsgroups: sci.math
Subject: Re: Need: prpndcular dist from pt to line
Date: 30 Dec 92 03:48:44 GMT

If you can express the line in the form ax + by + c = 0,
then the distance from the point (p, q) to the line is just
|ap + bq + c| divided by the square root of a^2 + b^2.

More generally, in n-space the distance from the point
(z_1, ..., z_n) to the line a_1x_1 + ... + a_nx_n + c = 0
is just |a_1z_1 + ... + a_nz_n + c| divided by the square
root of a_1^2 + ... + a_n^2.

Anyone got a quick proof of this?

Kiran Kedlaya
Harvard University
"Lo \'unico peor que la mala salud es la mala fama."
     -Gabriel Garc\'ia M\'arquez, "El amor en los tiempos del c\'olera"
From: haddadi@sipi.usc.edu (Navid Haddadi)
Newsgroups: sci.math
Subject: Re: Lagrange successor rule (Law of succession)
Date: 23 Dec 1992 05:24:45 -0800
Organization: University of Southern California, Los Angeles, CA
NNTP-Posting-Host: sipi.usc.edu

In article <3207@devnull.mpd.tandem.com> garyb@anasazi.UUCP (Gary Bjerke) writes:
>
>I was thumbing through an old statistics textbook when I came across the 
>Lagrange successor rule. The example given was that of a collection of coins
>for which the probability of flipping a head is uniformly distributed over
>the set of values {1/N, 2/N, ..., N/N} for N some arbitrary integer. The rule
>states that the probability of getting a head on the (n+1)th flip given that
>the first n flips were heads, is n/(n+1).
>...
>I followed the proof, but I have absolutely no intuition for this result at
>all. I even fail to see how it applies to the rising of the sun (in what sense
>does the unconditional probability of its rising meet the uniform-distribution
>requirements?) Can somebody help me get a gut feel for what this result means?

I'm not sure what you are asking, but here is more explanation.
Let p be the probability of head when tossing a coin.
What is p?
Well, if for example p=1/2 then p=1/2 regardless of how many times
we flip a coin and how many times we observe heads or tails. But
lets say p is not deterministic. Then we can "guess" the value of
p based on our observations. In fact, we can treat p as a random variable.
If we make n observations A={hhttthhhh....} and see k heads in the
sequence, what can be said about value of p?  How about taking the
best "guess" to be E(p|A)? (where E is the expectation)
This value turns out to be (k+1)/(n+2) under appropriate assumtions
(see the proof!)
More explanation can be found in most probability books
e.g. Papoulis, "Probability, Random Variables, and Stochastic Processes,"84,65.

Navid
Newsgroups: sci.math,alt.books.technical
From: dwr2560@zeus.tamu.edu (RING, DAVID WAYNE)
Newsgroups: sci.math,sci.physics
Subject: Re: More on Huygens' principle
Date: 25 Dec 1992 22:03 CST
Organization: Texas A&M University, Academic Computing Services
Distribution: world
NNTP-Posting-Host: zeus.tamu.edu
News-Software: VAX/VMS VNEWS 1.41    

columbus@strident.think.com (Michael Weiss) writes...
>If one assumes that Huygens' principle holds in 3 spatial dimensions, one
>can deduce that it should fail in 2.  Hadamard called this argument the
>"method of descent".  Briefly: consider a pulse disturbance at t=0 along
>the entire z-axis.  It creates spreading waves that obviously will have no
>dependence on z, and hence satisfy the 2-dimensional wave equation.  Now we
>apply Huygens' principle in 3 dimensions--- that is, we add up spherical
>waves spreading from each (0,0,z) (i.e., integrate over z).  An observer
>positioned at (x,y,0) will of course "hear" the pulse at t = sqrt(x^2+y^2),
>due to the spherical wave spreading out from (0,0,0) but will also hear
>something at t = sqrt(x^2 + y^2 + z^2), thanks to the spherical wave from
>(0,0,z). QED (I was hoping there was a similarly intuitive argument for why
>Huygens' should work for 3 dimensions, but I guess not.)

I feel like something is missing. Wouldn't a similar line of reasoning
show that HP in 5 dimensions implies no HP in 3?

Dave Ring
dwr2560@zeus.tamu.edu
From: choy@skorpio.usask.ca (I am a terminator.)
Newsgroups: sci.math
Subject: Can I prove metalogical statements using logic and set theory?
Date: 24 Dec 1992 20:24:43 GMT
Organization: University of Saskatchewan, Saskatoon, Canada
Distribution: world
NNTP-Posting-Host: skorpio.usask.ca

Suppose that I represent logic statements as strings or sets. I can define
functions corresponding to logical inference from a set of logic statements
(or a set of sets of logic statements) to a set of (sets of) logic
statements. I can try to prove things about these functional representations
of logic inferences. Would this work? I'll bet I can read all about it
someplace if someone could tell me where to look.


-- 

Henry Choy
choy@cs.usask.ca

Hey kids, start jumping up and down like idiots.

Looking for LOG for a good price?

Dial 1-800-byBLAMO !!!!

"Log rolls down stairs. Rolls over in pairs. 
 Rolls over your neighbor's dog.
 It's good for a snack. It fits on your back.
 It's LOG LOG LOG!!!!"

             -- Ren * Stimpy