HDTenEightyP/NetNews huggingface dataset:
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Can Anyone Solve this?????
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Sun, 24 Jan 93 23:49:37 GMT

In article <1993Jan23.210732.19327@magnus.acs.ohio-state.edu> mfreiman@magnus.acs.ohio-state.edu (Mark R Freiman) writes:
>             If W={f such that f"-2f'+f=0}
>               What is the dimension? and what is a Basis?
>I know the answer for the dimension is 2, and a basis is
>              e  and    xe
>             but how would one go about obtaining these answers?
The point is to show that any other solution to your differential
equation is a linear combination of the two you give.  There are various
ways to do this.  One way would be to prove that any solution of a
second-order differential equation like f''(x) + p(x)f'(x) + q(x)f(x) =
0, where p and q are smooth functions (for example), is determined by
f(0) and f'(0).  (A related way is to show that two solutions of this
equation, say f and g, are linearly independent if and only if the
Wronskian fg' - gf' is zero everywhere.)  Here's one way that
generalizes to differential equations of higher order.  First rewrite
the differential equation as two first-order linear differential
equations for f and f'... which we will instead call f_1 and f_2, justto
be confusing.  We get

                f_1' = f_2
                f_2' = -pf_2 - qf_1

or if we write F for the vector (f_1,f_2) and A for the (x-dependent)
matrix

                0   1
               -q   -p

we can write the two first-order differential equations as *one*
first-order vector-valued differential equation, namely just

               F' = AF.

Simpler, huh?  Now "solve" it by integrating...

F(x) = F(0) + int_0^x A(t) F(t) dt.

This "solution" is true but not really a solution, since F(t) appears in
the integral.  However, we can repeatedly substitute this equation in
itself (e.g., for the first iteration, use the equation to express F(t)
as F(0) + int_0^t  A(s) F(s) ds).  Repeating this forever, we get an
infinite sum, and I leave it to you to write this sum down and check
that it converges and actually gives the answer... that is, an
expression for F(x) in terms of A and F(0).  Thus the solution is really
determined by its initial data f(0) and f'(0).  

My argument really can be made into a 100% precise proof of the
existence and uniqueness of solutions of such a differential equation
with given initial data.  And, better yet, the technique is enormously
general and applies to give existence and uniqueness results for
nonlinear ordinary differential equations and even nonlinear partial
differential equations.  In its full-fledged glory, it's called the
method of nonlinear semigroups and is due to Irving Segal - but you are
more likely to find it in basic differential equations books under the
name of Picard's theorem on local existence and uniqueness.  In such
books, it's only done for first-order equations, but the trick I
described allows one to solve n-th order equations too.
From: allenk@ugcs.caltech.edu (Allen Knutson)
Newsgroups: sci.math
Subject: Re: Non-Standard Analysis and philosophy
Date: 22 Jan 1993 03:07:36 GMT
Organization: California Institute of Technology, Pasadena
NNTP-Posting-Host: torment.ugcs.caltech.edu

solan@smaug.uio.no (Svein Olav G. Nyberg) argues with Torkel Franzen:

>|>    >Under which philosophies of mathematics is non-standard
>|>    >analysis possible, and to what extent?
>|>   Your question makes no apparent sense. Non-standard analysis is simply a
>|> field of mathematics that exists, hence is possible.
>I am serious, Torkel. Surely finitist mathematicians will not
>allow for infinitely large integers. 

In Robert's _Nonstandard Analysis_, he makes heavy use of *illimited*,
but *finite*, integers. I forget what illimited means, but finite has
the usual meaning - an endo-injection is a surjection.

To the other people responding to this thread: don't you need some large 
cardinal axiom to prove the kosherness of nonstandard analysis?	I could
be misremembering.						Allen K.

Newsgroups: sci.math
From: victor@watson.ibm.com (Victor Miller)
Subject: Re: why is pi irrational
Date: Thu, 21 Jan 1993 15:18:56 GMT
Reply-To: victor@watson.ibm.com
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: IBM, T.J. Watson Research Center

>>>>> On 20 Jan 93 20:57:08 +1200, pete@bignode.equinox.gen.nz (Pete Moore) said:

Pete> Frederick W. Chapman (fc03@ns1.cc.lehigh.edu) wrote:

Frederick >Actually, the Biblical passage merely describes a circular shaped
Frederick> pool, saying that it was 30 cubits around and 10 cubits
Frederick> across (if memory serves me).  This is a far cry from
Frederick> saying that "pi = 3", for the following reasons. What the
Frederick> passage describes are measurements of a man-made object.
Frederick> The passage does not purport that the pool was in the shape
Frederick> of a perfect circle (more than likely, it was egg-shaped
Frederick> :-), nor does it purport that the measurements were taken
Frederick> with good scientific instruments, nor does it assure us
Frederick> that the measurements were free of systematic error, nor
Frederick> does it report the precision to which the measurements were
Frederick> supposed to be accurate.

Pete> The relevant passage is 1 Kings 7 v 23, if anyone cares.

Pete> The `King James' Bible says:

Pete> "...ten cubits from one brim to the other: it was round all
Pete> about...and a line of thirty cubits did compass it round about."

Pete> and the `Revised Standard Version' varies only in saying "from brim to
Pete> brim" and "a line of thirty cubits measured its circumference."

It always helps to go back to the original Hebrew.  In the original
Hebrew, the word for circumference is "Qav", normally spelled qof vav.
However, in this passage it is spelled qof vav heh.  The Vilna Gaon
(Rabbi Elijah of Vilna -- probably the leading Jewish religious
scholar of his day -- in the 18th century, and up to the present day)
was an avid amateur mathematician: One is forbidden to study religious
subjects while in the outhouse -- so he studied mathematics there.  He
commented that the numerical equivalent of qof vav heh is 111 (the
standard numerics of the Hebrew alphabet give qof = 100, vav = 6 and
heh = 5), of of qof vav is 106.  If one looks at 3*(111/106) it gives
pi to four decimal place accuracy!  This would certainly seem to be at
the limit of the technology of measurement of the day (i.e. King
Solomon's day).

Pete> I don't think there is any doubt that it is describing a circle.

>"pi = 3, to one significant figure" is a correct statement, of course.

Pete> An old joke about "small values of pi and large values of 3"
Pete> leaps to mind here 8-)

Frederick> So the Bible is accurate, just not very precise.  :-)

Pete> When it says the Earth is flat, however, it is inaccurate, rather than
Pete> imprecise.
--
			Victor S. Miller
			Vnet and Bitnet:  VICTOR at WATSON
			Internet: victor@watson.ibm.com
			IBM, TJ Watson Research Center
From: time@ice.com (Tim Endres)
Newsgroups: sci.math
Subject: Advice needed
Date: Fri, 22 Jan 93 19:08:13 EST
Organization: ICE Engineering, Inc.
Reply-To: time@ice.com (Tim Endres)
X-Mailer: uAccess - Macintosh Release: 1.6v2



Can someone give me there views on why I should get a BA in math as
opposed to engineering? I have had math thru diff equ. and didn't
really enjoy it until calc3 and linear. What else does a mAth major
get that an engineer would not?
Thanks for the info
Karen 
karen@ice.com


tim endres - time@ice.com  (UUCP: heifetz!tbomb!time)

USENET - a slow moving self parody... ph
From: frandag@math.ksu.edu (Francis Fung)
Newsgroups: sci.math
Subject: Re: Irrational stamp problem
Date: 21 Jan 1993 00:26:11 -0600
Organization: Dept. of Mathematics, Kansas State University
NNTP-Posting-Host: hilbert.math.ksu.edu

wft@math.canterbury.ac.nz (Bill Taylor) writes:

>In article <2338@sjfc.UUCP>, dmc@sjfc.UUCP (Cass Dan) writes:
>|> 
>|> You have an ink stamp which is so amazingly precise that, when inked
>|> and pressed down on the plane, it makes every circle of irrational
>|> radius (centered at the center of the stamp) black.
>|> 
>|> Question:  Can one use the stamp three times and make every point
>|> in the plane black?  [assume plane was white to begin with, and
>|> ignore the fact that no such stamp is physically possible]

>A very nice problem !

>Why not do it with only two stampings, where the second one is moved an
>infinitesimal distance from the first one, thus covering all the points
>left out by the first ?          JUST KIDDING !!

>Here's the real answer, that it can be done with three stampings.

(admittedly messy solutions deleted)
This was a Putnam problem, the second time I took the exam (ARGH), and I 
missed it in spite of the fact that it was in Newman's "A Problem Seminar"
which I had studied a few days prior (ARRARRAGGGHHkklghk...*)
Anyway, here is the slick solution. We exhibit three points such that
if the distance to two of them is rational, then it must be of irrational
distance to the third.
Suppse (x,y) is at rational distance from (0,0) and (1,0).
Then sqrt(x^2 + y^2)=p and sqrt((x-1)^2 + y^2)=q are both rational.
The first implies X^2 + y^2 is rational.
The second one implies that x^2 - 2x + 1 + y^2 = q^2 so that
2x-1= x^2+y^2 - q^2 is rational so that x is rational.
Therefore,
for any other point (a, 0), if sqrt((x-a)^2 + y^2) = r is rational, then
x^2- 2ax + a^2 +y^2= r^2 so that a^2 - (2x)a + (x^2 + y^2 - a^2) = 0 is
a quadratic equation that a satisfies.
Therefore, any choice of a such that this quadratic CANNOT be satisfied, 
e.g. cube root (3) , will be such that any point at rational distance
from (0,0) and (1, 0) is at irrational distance from (a,0).
Francis Fung

Newsgroups: sci.math
From: pjp9q@honi2.acc.Virginia.EDU (Paul Joseph Perrone)
Subject: Number Theory
Organization: University of Virginia
Date: Mon, 25 Jan 1993 19:38:13 GMT


I'm looking for any references or direction in solving the following
problem:

Given a set of prime numbers whose largest element is less than one
million. Find a set of these primes such that any sum or difference
between any two of these primes are NOT equal to any other sum or
difference between any other combination of these primes.

Is there such a set?

I would appreciate any responses regarding this problem.

Thanx.

Paul Perrone

please email to: pjp9q@virginia.edu
From: edgar@math.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Top math journals (was: sci.math.* groups
Date: 25 Jan 1993 08:15:07 -0500
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu

In article <C1DE7M.J3p@cmcl2.nyu.edu> hutchins@acf3.nyu.edu (Carol Hutchins) writes:
> 	Please let me know once you come to a consensus
>on which 6 are the "top math journals".
>
>			Thanks.
>			Carol Hutchins
>			Library
>			Courant Institute

A paper on this, "Pure and Applied Mathematics Journals: What They Cite
and Vice Versa", by Eugene Garfield, appeared in _Current Contents_
5(15):5-10, March 8, 1982.  He measured the "most significant" math
journals in five different ways.  Of course, the five results did not agree.
Journals ranked high in four of the five ratings were:
   Acta Mathematica-Djursholm
   American Journal of Mathematcs
   Annals of Mathematics
   Bulletin of the American Mathematical Society
   Inventiones Mathematica
   Journal of Functional Analysis
   Mathematics of Computation
   Proceedings of the London Mathematical Society

------

There is also a list of the most widely cited papers (for the period
1961 to 1980).  The winner (with 1,174 citations) was:

Cooley & Tukey, "An algorithm for the machine calculation of complex
Fourier series", Math. Comput. 19 (1965) 297--301.

-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
Newsgroups: sci.math
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Subject: Re: Alephs again
Organization: U of Cambridge, England
Date: Thu, 21 Jan 1993 05:14:06 GMT

In article <1993Jan20.155553.19920@cadkey.com>, dennis@cadkey.com (Dennis Paul Himes) writes:
> In article <1jh2epINN15k@gap.caltech.edu> allenk@ugcs.caltech.edu (Allen Knutson) writes:
> >ydavid@dxds04.cern.ch (David Yann) greets us, and asks a bunch of
> >set-theory questions. I'll state at the beginning: if one has the
> >Axiom of Choice, much cardinal arithmetic becomes trivial.
> >Without it, most things are hard (a countable union of countable
> >sets can be uncountable, for instance).
> >
>      Is this true?  What is wrong with the following enumeration?
>      Let f(i,j) be the i^th element of the j^th set, both indices starting
> at zero.  Enumerate the union as follows:
> n = 0;
> for (i0 = 0; ; ++i0) {
>     for (i1 = 0; i1 <= i0; ++i1) {
> 	   enumeration[n++] = f (i1, i0 - i1);
>     }
> }
> 	Or, for those of you who don't know C,
> f (0, 0),
> f (0, 1), f (1, 0),
> f (0, 2), f (1, 1), f (2, 0),
> f (0, 3), f (1, 2), f (2, 1), f (3, 0),
> etc.
> 	Some work would have to be done to eliminate duplicates, but I don't
> see how this depends on AC or how this would not apply to any countable
> union of countable sets.

The problem is a really annoying one. Each of the sets may be countable
(i.e. for each set there is an enumeration) without there being a *choice*
of one such enumeration for each set -- which is what you need for your
algorithm to work. This probably sounds like hopeless nit-picking, but that's
what set theory without the axiom of choice is like.

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
Xref: sparky sci.math:18606 sci.math.symbolic:3489
From: scott@fillorkill.Eng.Sun.COM (Scott Boyd)
Newsgroups: sci.math,sci.math.symbolic
Subject: Re: Which CAS have BA's in math?
Date: 21 Jan 93 20:33:23 GMT
Reply-To: bobk@decide.com
Distribution: sci
Organization: Sun Microsystems Inc., Mountain View, CA
NNTP-Posting-Host: fillorkill.eng.sun.com

[Posted for friend.]

ilan@leland.Stanford.EDU (ilan vardi) writes:

>In article <1jhjkfINN8m9@function.mps.ohio-state.edu> edgar@math.ohio-state.edu (Gerald Edgar) writes:
>>
>>I doubt there is a computer algebra system that can get a degree
>>in mathematics: they don't write proofs (yet).  However, I tried the

>They write papers though. Shalosh B. Ekhad, Doron's Zeilberger's computer,
>has written a number of papers (more than I have, I think).

Actually, one system written in Mathematica can do proofs (printed in
TeX of course).  This system created by a couple of professors from
Carnegie-Mellon can do the proof of the Wierstass continuous-nowhere
differentiable function. All of the theorems from Chapter 2 of
Ramanujan's collected works, properties of stereographic projections
and on and on. The professors are Ed Clark and Xudong Zhao and they
have an article in the current issue of the Mathematical Journal. I
recommend it highly. A BA in analysis is not as far away as some might
think.

Peace/Bob


"The range of what we think and do is limited by what we | Bob Korsan
fail to notice. And because we fail to notice that we    | Decisions,
fail to notice there is little we can do to change until |    Decisions!
we notice that failing to notice shapes our thoughts and | 2136 Lyon Ave
deeds."                  --         R.D. Laing           | Belmont, CA
INTERNET:bobk@Decide.COM    USENET:..!decwrl!decide!bobk |    94002-1639
ATT:(415) 595-1206 (Voice)///(415) 595-1206 + 4 (FAX)    |

-- 
--
Scott Boyd                               scott.boyd@eng.sun.com
SunSoft Inc.                             
Newsgroups: sci.math
From: swann@divsun.unige.ch (SWANN philip)
Subject: Reference Wanted: Vectors & Statistics
Organization: University of Geneva, Switzerland
Date: Wed, 27 Jan 1993 10:22:45 GMT

About five years ago I read a nice paper (I think in Math Intelligencer)
which introduced statistics using  geometrical vectors. I remember the
paper was by two Australians and they announced that a book was forthcoming.
Does anybody have a reference to the original paper and/or the book?

Many thanks,

Philip swann


Newsgroups: sci.math
From: gary@colossus.cgd.ucar.edu (Gary Strand)
Subject: Steffen's method
Keywords: Steffen 1-d interpolation
Organization: Climate and Global Dynamics Division/NCAR, Boulder, CO
Date: Mon, 25 Jan 1993 23:09:54 GMT


  I'm looking for advice on how to best use Steffen's method (as described in
  Astronomy and Astrophysics 239, 443-450, 1990) to interpolate some 1-d data
  that happens to contain "blocked" or "missing" data points.

  I'm not sure how to handle these flagged values, and I'd like to know how
  others who have had a similar problem managed to get around it.

  Thanks.

--
Gary Strand                      Opinions stated herein are mine alone and are
strandwg@ncar.ucar.edu            not representative of NCAR, UCAR, or the NSF
Xref: sparky sci.math:18770 misc.education:6136
From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: godel theorem and godel theorem ....(Re: Where's
Organization: ORA Corporation
Date: Tue, 26 Jan 1993 17:14:42 GMT

cole@alexia.lis.uiuc.edu (Sandra Stewart-Cole) writes:

>I think the missing links between GT and the assertitions you've seen it 
>applied to are that science and philosophy are attempts to create a fully 
>logical complete consistent system for modeling life and the universe. That is
>not really all that accurate, but I think it explains the connection for (1) 
>and (3). The fiction of truth is a bit different. There is no proveable truth 
>logically beyond ones own existence if one does not make assumptions of truth.
>For those who have sought to make only a few assumptuions and prove various 
>truths that fully represent reality consistently from them, GT undermines the 
>attempt. It shows that logical deduction is insufficient to explain anything 
>complex no matter how many starting axioms you take. For seekers of objective 
>truth backed by logic, this is a disturbing concept.

I don't have any answer for you, but I would like to point out that
there is a problem in applying logical results to informal talk about
truth. According to our informal notions, TRUTH is something out there
waiting for us to find it. After we find the TRUTH, we may try to
express it in words, or we may find that it can't be expressed in
words. In any event, we intuitively think of TRUTH as something that
exists independently of the language we use to express it. In
contrast, when a logician talks about truth, she is talking about a
property of a language (or more generally, the relationship between a
language and an intended model). There are no language-independent
truths for the logician.

Daryl McCullough
ORA Corp.
Ithaca, NY
Xref: sparky sci.math:18854 sci.math.stat:2891
From: ksanjay@cs.tamu.edu (Sanjay D Kamat)
Newsgroups: sci.math,sci.math.stat
Subject: Request for help - A Probability Problem
Date: 27 Jan 1993 19:17:37 GMT
Organization: Computer Science Department, Texas A&M University
Distribution: World
NNTP-Posting-Host: sparc29.cs.tamu.edu
Summary: Help requested on a random sums problem
Keywords: Probability, Beta distribution, Random Sums


Hi,
   I am a computer science student working towards my PhD degree. I 
   would very much appreciate your help/suggestions on the following 
   problem I recently encountered in my research.

   A random variable S is defined as the sum of n terms, each of the
   form u_i * w_i. 
   u_i and w_i are themselves random with the following properties.

   1) All u_i's are identically distributed -but not independent in the
      sense that their sum is fixed, say U. (All splittings of U
      considerted equi-probable).
   2) All w_i's are independent as well as identically distributed.
      They are also bounded between w_min and w_max.
   3) u_i's and w_i's are independent of each other.

   I can establish that 
      mean S = mean w * U
      variance S = (2/(n+1)) * U * U * variance w.

   Further, for the parameters of concern, "beta" distribution for S
   with above mean and variance seems to fit very well.

   My questions are:

   1) Is there a sound theoretical justification for "beta"
      distribution to be applicable?
      ( Also - is there some limit theorem which applies to S for large
	       n? )
   2) What is a good source for good approximations for the incomplete
      beta function?

   I would appreciate if you email your reply to me.

   Thanks  in advance,

Sanjay Kamat.
Dept of Computer Science 
Texas A&M University
(ksanjay@cs.tamu.edu)
From: mckay@alcor.concordia.ca (John McKay)
Newsgroups: sci.math
Subject: Word needed
Date: 23 Jan 93 00:23:15 GMT
Organization: Concordia University, Montreal, Quebec
Originator: mckay@alcor.concordia.ca


Is there any "correct" word for x: x**n = x ?
For n=2 we have idempotent, how about n>2 ?

Xref: sparky sci.math:18664 sci.logic:2629 comp.text.tex:15326
From: zeleny@husc10.harvard.edu (Mikhail Zeleny)
Newsgroups: sci.math,sci.logic,comp.text.tex
Subject: inverted iota
Date: 22 Jan 93 19:22:11 GMT
Organization: The Phallogocentric Cabal

Looking for tricks to typeset Russell's description symbol in AMS-LaTeX.
(Is there a general way to rotate a given character?) Also wanted: the
lattice-theoretic relation of covering (a sideways Y), and a nice XIXth
century-style integral sign.  Finally, how does TeX represent Greek
breathing marks?

cordially,
mikhail zeleny@husc.harvard.edu
"Le cul des femmes est monotone comme l'esprit des hommes."
Newsgroups: sci.math
From: ref@draco.cis.umassd.edu (R Faulkenberry)
Subject: hypocycloid
Organization: University of Massachusetts Dartmouth
Date: Thu, 28 Jan 1993 15:19:15 GMT


If a small circle rolls along the interior of a larger
circle, a chosen point on the small circle will trace a
curve called a hypocycloid.  This curve can be 
parametrized by

    x=(a-b)*cos(t)+b*cos((a-b)/b*t)
    y=(a-b)*sin(t)-b*sin((a-b)/b*t)

where a is the radius of the larger circle and b is the
radius of the smaller circle (this also assumes that the
curve is traced beginning at the point (a,0) and that the
small circle moves counterclockwise inside the large
circle).

The hypocycloid will be a closed curve if and only if the
ratio of b to a is a rational number.  In this case, the
number of times the small circle must revolve about the
center of the large circle is a function of that ratio.

What is that function?
--

--  Richard E. Faulkenberry
    Assistant Professor
    Department of Mathematics
    University of Massachusetts Dartmouth
    North Dartmouth, Massachusetts  02747
    rfaulkenberr@umassd.edu
    508-999-8438
From: corleyj@helium.gas.uug.arizona.edu (Jason D Corley )
Newsgroups: sci.math
Subject: Re: Why Algebra...Important
Date: 25 Jan 93 19:00:27 GMT
Organization: University of Arizona, Tucson

In article <1993Jan25.023340.4349@monu1.cc.monash.edu.au> khchung@monu1.cc.monash.edu.au (Mr KH Chung) writes:
>In article <7459@ucsbcsl.ucsb.edu>, unicef@epilogue.mcl.ucsb.edu (Joel Rising) writes:
>> Math is a language, but a far more powerful one.  To convince youself,  
>> write out some equation, and then try to describe all the relationships it  
>> embodies.  It's not just "x=ky" but also "x is proportional to y" or "as x  
>> goes up one, y goes up by some factor k."  Relate the variables to some  
>> real-world quantity, and the equation becomes even more rich with  
>> information.  Perhaps a whole paragraph could be gleaned from "x=ky."
>
>On the other hand, how do you order a cup of coffee using only mathematical
>notation?
>:-)

Hmmmmm....well, let C be defined as the set of all people posessing coffee...
no...wait...let c be defined as the number of coffee mugs in a...no..no...
hang on....C|{c/d^2r/dt^2} No no no...I can do this!  Really I can!

--
*******************************************************************************
"Keep an eye on the species below us...because you never know who may be next."
	----Bertha's Save The Cat Committee (Garrison Keillor)
Jason D. "corleyj@gas.uug.arizona.edu" Corley, Poet, Polymath, Scoundrel.
Newsgroups: sci.math