HDTenEightyP/NetNews huggingface dataset:
From: edgar@math.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: Non-Standard Analysis and philosophy
Date: 21 Jan 1993 16:04:47 -0500
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu

In article <1993Jan21.180359.21766@ulrik.uio.no> solan@smaug.uio.no (Svein Olav G. Nyberg) writes:
>
>Under which philosophies of mathematics is non-standard
>analysis possible, and to what extent?

Here is an answer to a guess at what is meant by the question:

Robinson's nonstandard analysis is "equivalent" to the Boolean algebra
prime ideal theorem.  It is a consequence of the axiom of choice, but
does not imply the axiom of choice.  I found this in an old paper
of W. A. J. Luxemburg.



-- 
  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: cet1@cus.cam.ac.uk (C.E. Thompson)
Subject: Re: Distribution of primes mod 4
Organization: U of Cambridge, England
Date: Mon, 25 Jan 1993 15:47:09 GMT

In article <1993Jan24.123639.19744@husc3.harvard.edu>, elkies@ramanujan.harvard.edu
(Noam Elkies) writes:

[Many thanks to Noam for sorting out the confusion rampant in previous postings]

|>                                                Now in the Prime
|> Number Theorem, when we pass from the sum of \Lambda(m) to the
|> count of primes <x the first thing we do is ignore the contribution
|> of all prime powers higher than the first, which contribute
|> "negligibly" to the sum.  However in our case this "negligible"
|> contribution is asymptotically sqrt(x) [note that chi(p^2) is
|> always 1 for p>2!].  Thus we find that pi(x;1,4)-pi(x;3,4)
|> is approximated by sqrt(x)/log(x) times an almost periodic
|> function of log(x) whose mean is -1!

Maybe it is worth pointing out that the situation isn't all that different
from PNT (estimating pi(x)=pi(x;0,1)). The function which is "centred" on 
li(x) is not pi(x), but \sum_{n<=x}{\Lambda_1(n)} = pi(x) + (1/2)pi(x^{1/2}}
+ (1/3)pi{x^{1/3)} + ... Again the oscillations can become larger than any
constant multiple of sqrt(x)/log(x), so that pi(x) > li(x) sometimes 
(although not for any explicitly computed x, as yet). One expects (given
RH + suitable independence constraints on the zeroes of the zeta function)
that the set of x such that pi(x) > li(x) has a non-zero (rather small)
limiting *logarithmic* density, but no limiting natural density.

Chris Thompson
JANET:    cet1@uk.ac.cam.phx
Internet: cet1@phx.cam.ac.uk
From: mapaj@csv.warwick.ac.uk (Mr A Scott)
Newsgroups: sci.math
Subject: f(f(x)) = exp(x)
Date: 21 Jan 1993 19:00:00 -0000
Organization: Maths Institute, Warwick University, UK.
Distribution: world
NNTP-Posting-Host: edam.csv.warwick.ac.uk
Keywords: f(f(x)) = exp(x)


	Is anything known about the existence of solutions f ( f:reals ---> reals ) of
the equation;    f ( f ( x ) )  =  exp ( x )  for all real x   ?  Examples of such f or 
proofs of nonexistence of f would be appreciated.  This problem is just for "fun"!  
I've tried induction on x, but that doesn't work.  :-)  :-)  :-)  :-)
Tony Scott   email:  as@maths.warwick.ac.uk
-- 
Tony Scott   email: as@maths.warwick.ac.uk
Newsgroups: sci.math,misc.education
From: jweiss@ra.cgd.ucar.edu (Jeffrey Weiss)
Subject: Re: Why algebra and what is important?
Organization: Climate and Global Dynamics Division/NCAR, Boulder, CO
Date: Sat, 23 Jan 1993 20:48:16 GMT

In article <00727673896@elgamy.jpunix.com> elg@elgamy.jpunix.com (Eric Lee Green) writes:

>BTW: Herman notes in another message that he discounts the "basics". Well
>-- unfortunately, all mathematics taught at higher grade levels assumes
>that the kid can do basic arithmetic without the aid of a calculator. As
>much as I wish otherwise, basic arithmetic is thus what I have to teach.
>I play with math with the kid who can play math well, but otherwise,
>well... how can a kid pass the TAAS (required to go to the next grade
>level) if he can't add, subtract, multiply, or divide? Answer: He can't.

For me the important distinction is how one teaches the basics.  I
have taught math in every elementary grade, partly through a wonderful
program called Project SEED.  Their technique is to do advanced
conceptual mathematics.  But, as Eric points out, this does require
that the kids do basic arithmetic.  I see no contradiction in this,
since it motivates the students to do, and thus learn, basic
arithmetic.  For example, one curriculum used with 4-6th graders dealt
with a new (for them) binary operation, exponentiation.  This of
course requries that they multiply, and the kids were very eager to
multiply long strings of number to find, say, 4^10, since it had a
context where they cared about the answer.

I like an analogy with carpentry.  Our current math teaching is like
giving someone a pile of nails and wood, making them drive nails for a
few years until they learn to do it perfectly, and only then letting
them do something interesting, i.e. build something.  How boring!  I
think its much better to let them build something fun (and kids do
find conceptual math fun, if taught right) even if its a little
rickety because they are not great at driving nails.  In the process
they not only learn the basics, but learn why they should care about
the basics.
--
Jeff
jweiss@ncar.ucar.edu
From: mnewman@comlab.ox.ac.uk (Matthew Newman)
Newsgroups: sci.math
Subject: Square root alg.
Date: 22 Jan 93 16:05:39 GMT
Organization: Oxford University Computing Laboratory, UK
Originator: mnewman@client41.comlab

Someone asked for a square root algorithm.

To work out what the algorithm is observe
that if N = a1 + 100a2 (+ 10000a3 + ...)
and Sqrt(N) = b1 + 10b2 (+ 100 b3 + ...)
then N = (b1 + 10b2)(b1 + 10b2) = b1^2 + 100b2^2 + 20b1b2

So, suppose you want to calculate the square root of N, we keep
track of three numbers, Res (the result), Rem (a remainder type
variable) and Add (something else!).

From the decimal point, pair off the digits of N (both left and
right e.g. N = 127.345 -> 01 27 . 34 50)

Initially Res = 0
          Rem = Left most pair from N
          Add = 0

Recursively choose x largest such that (Add + x)x <= Rem

Next, let Res' = Res*10 + x
          Rem' = 100*(Rem-(Add + x)x) + next pair of digits from N
          Add' = 10*(Add + 2x)

and repeat 'till done (i.e. until Rem=0 or Res is long enough). You have
to keep track of where the decimal point comes into Res, but it's fairly
obvious (!).

An example might help !

  N = 4096  = 40 96
  Res = 0
  Rem = 40
  Add = 0
 So the first x is 6 (since 6*(0+6)=36, and 7*(0+7)=49)
  Res' = 6
  Rem' = 400 + 96 = 496
  Add = 120
 The next x is thus 4, since 4*(120+4)=496
  Res'' = 64
  Rem'' = 0
  Add = 256
and we're done

  N = 2 = 02 . 00 00 00 00 ...
  Res = 0
  Rem = 2
  Add = 1
 so x is 1
  Res' = 1
  Rem' = 100
  Add' = 20
 so x is 4
  Res'' = 1.4 (note that the decimal point comes in here ...)
  Rem'' = 400
  Add'' = 280
 so x is 1
  Res''' = 1.41
  Rem''' = 11900
  Add''' = 2820
 so x is 4
  Res'''' = 1.414
  Rem'''' = 60400
  Add'''' = 28280
 and so on for ever ...
Newsgroups: sci.math
From: pamaga@biobase.aau.dk (Paulo Magalhaes)
Subject: Best fit to a curve
Organization: The Danish BioBase
Date: Thu, 21 Jan 1993 15:31:08 GMT

Hi,

The subjet line says it all... Just like one can use linear regression
to find a formula to represent a line based on a given set of points, I
need to do the same thing, only in my case it's a curve. I'll explain
better. Given a set of points, I can plot them on good old graph paper
and obtain a very simple curve; this can then be used to calculate, say,
an unknown Y for a given X. The problem is, I can't get a machine to do
this!!! Is there a formula (as in LR) which I can introduce in a simple
program, so that I can start saving on graph paper ;-)

Better still, is there a program which already does this? I mean, I
supply a set of (x,y), the program works out the formula for the best
fit curve, and then I can ask "C'mon slave, gimme the value of 'y' when
'x'=..." :-)

Please email your replies, as I don't usually read this newsgroup.

Thanks for the attention

Regards

Paulo
pamaga@biobase.aau.dk

Newsgroups: sci.math
From: ilan@leland.Stanford.EDU (ilan vardi)
Subject: Re: Compositum of quadratic fields
Organization: DSG, Stanford University, CA 94305, USA
Date: Fri, 22 Jan 93 09:10:11 GMT

In article <1993Jan18.044419.26095@ariel.ec.usf.edu> eclark@gauss.math.usf.edu. (Edwin Clark) writes:
>In article <1j4mh2INN2t3@uwm.edu> litow@csd4.csd.uwm.edu (Bruce E Litow) writes:
>>If p_1,...,p_n are distinct primes, what is a good upper bound on
>>the degree of Q(sqrt(p_1),...,sqrt(p_n))? Clearly one can get a
>>bound via cyclotomic extensions but that seems horrendous, or is
>>it best possible?
>>
>The exact degree is 2^n. See the Monthly article by RL Roth
>"On extensions of Q by square roots", vol 78 (4) 1971
>pp 392-393.
>

It was studied by Swinnerton-Dyer, who showed that the minimal polynomial
is simply 

  product (x (+ or -) sqrt(2) (+ or -) sqrt(3) ... (+ or -) sqrt(p_r))

where the product is over all 2^r choices of sign. Here is a Mathematica
implementation

SwinnertonDyerP[n_, x_]:= 
Fold[Expand[(#1 /. x->x+#2) (#1 /. x->x-#2)]&, x, Sqrt[Prime[Range[n]]]]
Newsgroups: sci.math
From: ilan@leland.Stanford.EDU (ilan vardi)
Subject: Re: Non-number theoretcial proofs in number theory
Organization: DSG, Stanford University, CA 94305, USA
Date: Thu, 21 Jan 93 13:04:59 GMT

In article <C15H03.A3q@cs.bham.ac.uk> ard@cs.bham.ac.uk (Antoni Diller) writes:
>It's years since I've done any proper maths, but what I'm after are
>examples of proofs of statements about the natural numbers or integers
>that use ideas and methods external to number theory, like for example
>results from the theory of complex numbers.  Ideally I'd like an example

If p=3 (mod 4) is prime, then 

   1*(1/p) + 2*(2/p) + 3*(3/p) + ... + (p-1)*((p-1)/p) < 0

where (a/p) = 1 if there is an x such that x^2 = a (mod p), and -1
                if no such x exists.
Organization: Penn State University
Date: Fri, 22 Jan 1993 11:48:46 EST
From: <DCC117@psuvm.psu.edu>
Newsgroups: sci.math
Subject: Combinatorial Problems

   Has anybody worked with, or know the solution to, the
following problem ?

   A 6-by-6 chessboard is perfectly covered with 18 dominoes.
 Prove that it is possible to cut it either horizontally or
 vertically into two non-empty pieces without cutting through
 a domino, that is, prove there must be a fault-line.
From: fc03@ns1.cc.lehigh.edu (Frederick W. Chapman)
Newsgroups: sci.math
Subject: Re: why is pi irrational
Date: 21 Jan 93 17:58:40 GMT
Organization: Lehigh University

In article <pete.03vb@bignode.equinox.gen.nz>,
pete@bignode.equinox.gen.nz (Pete Moore) writes:
>Frederick W. Chapman (fc03@ns1.cc.lehigh.edu) wrote:
>
>>Actually, the Biblical passage merely describes a circular shaped
>>pool, saying that it was 30 cubits around and 10 cubits across (if
>>memory serves me).  This is a far cry from saying that "pi = 3", for
>>the following reasons.  [...]
>
>The relevant passage is 1 Kings 7 v 23, if anyone cares.
>
>The `King James' Bible says:
>
>"...ten cubits from one brim to the other: it was round all about...and a
>line of thirty cubits did compass it round about."
>
>and the `Revised Standard Version' varies only in saying "from brim to
>brim" and "a line of thirty cubits measured its circumference."
>
>I don't think there is any doubt that it is describing a circle.

Yes, it describes a man-made circular object, but not an ideal,
mathematically perfect circle.  A man-made circular object only resembles a
perfect circle to within some tolerance, which is unspecified in this case.
My point is that without knowledge of the tolerance to which the object was
constructed to resemble the ideal, and without knowledge of the precision
and claimed accuracy of the measurements, we have no grounds for complaint.

All measurements are approximate and all man-made objects are imperfect.
So the Bible contains what I presume to be a perfect report of the
approximate measurements of an imperfect object.  No contradiction there!
If I write a report accurately describing someone else's sub-standard lab
work and/or sub-standard craftsmanship, that does not make *me* responsible
for the errors; if I accurately report what someone else did, I have done
my job.

Quite frankly, I think it is silly to expect any piece of literature to
follow modern-day standards of laboratory practice when dealing with
quantitative information.  The measurements were included in the Biblical
passage TO HELP THE READER IMAGINE THE SIZE OF THE POOL; both diameter and
circumference were included AS AN AID TO CONVEYING THIS MENTAL PICTURE TO
THE READER, not as part of a science experiment to empirically determine an
approximation of pi.  Let's not confuse literature with experimental
science!


>When it says the Earth is flat, however, it is inaccurate, rather than
>imprecise.

Where does the Bible say that?  (Response via e-mail would be more
appropriate.)


Fred Chapman
sci.math.religion.regular
:-)>>>   <--- long beard, symbolizing patriarchy

-- 

o ------------------------------------------------------------------------- o
|  Frederick W. Chapman, User Services, Computing Center, Lehigh University |
|    Campus Phone:  8-3218     Preferred E-mail Address:  fc03@Lehigh.Edu   | 
o ------------------------------------------------------------------------- o
|             Ecstasy is transitory, but a theorem is forever!              |
o ------------------------------------------------------------------------- o
Newsgroups: sci.math.research
From: cnews@umr.edu (UMR Usenet News Post)
Subject: square roots of par. der.
Originator: dan@symcom.math.uiuc.edu
Reply-To: raza@ee.umr.edu (Md Ishfaqur Raza)
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Missouri - Rolla
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Thu, 21 Jan 1993 21:37:00 GMT

Hi ( this is my first posting to this group) :


I am working with the second order wave equation -....

  ( d =  say is the partial derivative  )

(   2     2     2    2   2  )      
( d   + d   + d  - co *d    )W(x,y,z,t) =  0
( dx    dy    dz       dt   )


My question is could I take the square root of the summation of the squares of
two or more partial derivatives? 

It is true that I cannot work with the square root on the derivatives, but what
I need that for is to factorize the above equation and later using the Taylors 
series unleash the squre root.

Md Ishfaqur Raza


Newsgroups: sci.math.research
From: stevef@juliet.ll.mit.edu ( Steve Finch )
Subject: A problem of Erdos
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: M.I.T. Lincoln Lab - Group 43
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 25 Jan 1993 22:00:58 GMT


What is the status of the following problem due to Erdos?  Thank you.
Please respond via e-mail.


Denote by f(n) the largest integer so that, for every sequence of real
numbers

		a(1), a(2), ... , a(n)

with a(k) nonzero for all k, one can always select f(n) of them

		a(j(1)), a(j(2)), ... , a(j(f(n)))

with the property that

	a(j(p)) + a(j(q)) != a(j(r))   for all  1 <= p < q < r <= f(n).

PROVE THAT:  f(n) = [ (n + 2)/2 ]


NOTE:  != means "not equal to" and [x] means "greatest integer <= x".


REFERENCE:

P. Erdos, "Extremal Problems in Number Theory", Proc. Symposia in Pure Math.,
Theory of Numbers, vol. 8, AMS, pp. 181 - 189.



 

Newsgroups: sci.math.research
From: HSNEV@ROMEO.CALTECH.EDU
Subject: conjecture about graphs
Mime-Version: 1.0
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Originator: dan@symcom.math.uiuc.edu
Content-Transfer-Encoding: 7BIT
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Organization: University of Illinois at Urbana
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Date: Thu, 28 Jan 1993 01:20:35 GMT

Def. Let G be a graph and let v be in V(G). Let S_v = bigsum d(u,v)
for all u not= v ( u in V(G)). G is called self-median if 
S_v = S_u for all u and v in V(G).
Def. A graph G is called self-centered if every vertex is in the center.
(The eccenticity of a vertex v is max d(u,v) for all u in V(G). 
v is in the center if v has minimum eccentricity. It is well-know that 
a tree has at most two vertices in its center.)

CONJECTURE: Every self-median graph is self-centered.
Note that every vertex transitive graph is self-median. There are 
examples of self-centered graphs that are not self-median, so the 
above inclusion would be strict. There is no known characterization of 
self-median graphs. There are non-regular self-medain graphs - 
see the recent book by Buckley and Harary "Distances in Graphs" for 
more information.
H.S. Snevily

Newsgroups: sci.math.research
From: guyard@dollar.unice.fr (Frederic Guyard)
Subject: Generalisation for Puiseux theorem ?
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Nice Sophia-Antipolis
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 25 Jan 1993 13:14:58 GMT


Hi,

Is there some generalisation for Puiseux Theorem ?

With Puiseux theorem, you know than you can determine branches of solution  
for P(x,y)=0 i.e. you can find parametrisation y(x) with P(x,y(x))=0.

Is there some generalisation if there is 3 variables : P(x,y,z)=0. For an  
example, can i find a parametrisation : P(x,y,z(x,y)). 

In the same way, is there a generalisation for Newton Polyhedron method  
with 3 variables ?

Thank .. 

Newsgroups: sci.math.research
From: eliezer@physics.llnl.gov (David A Eliezer)
Subject: kernel of the inverse wavelet transform
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Illinois at Urbana
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Fri, 22 Jan 1993 02:02:02 GMT



Does anyone know a characterization of the kernel of the
inverse wavelet transform?  Of particular interest is the
continuous wavelet transform, or, even better, the
semi-continuous one, in which the scale parameter is 
left continuous, but the translation parameter may be
discrete.  However, any information at all would be helpful,
even the fully discrete case.  
I am referring in particular to the affine 
group in 1 dimension, but two dimensions would also be 
interesting.  Any references to literature would be helpful,
otherwise this could be a long detour on my program of 
research.  Thanks for any help,

				Dave Eliezer
				eliezer@physics.llnl.gov
From: edgar@FUNCTION.MPS.OHIO-STATE.EDU (Gerald Edgar)
Newsgroups: sci.math.research
Subject: Re: Set-free measure theory
Date: 26 Jan 93 21:11:06 GMT
Article-I.D.: function.1k49daINNi1n
Organization: The Ohio State University, Dept. of Math.
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu

In article <C1GuEp.M99@sugar.neosoft.com> claird@NeoSoft.com (Cameron Laird) writes:
>I propose this definition:  a "set-free measure" is
>a triple <X, m, T> with
>1.  X a lattice (boolean algebra),
>2.  m a function defined on the members of X, and
>3.  T an equivalence relation on X, for which
>4.  m(x + y) = m(x) + m(y) - m(x.y) and
>5.  xTy implies m(x) = m(y).
>

A place to start your reading is Chapter X of Birkhoff's _Lattice Theory_
(3rd edition).  You have described a lattice with valuation (if you
omit 3 and 5).

-- 
  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.research
From: pjp9q@honi2.acc.Virginia.EDU (Paul Joseph Perrone)
Subject: Number Theory
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Virginia
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 25 Jan 1993 19:28:44 GMT

I am looking for references or direction in solving the following
problem:

Given a set of prime numbers, whose largest element is less than
1 million. Generate a set of these primes such that NO sum or
difference between any two of these primes is equal to any other
sum or difference between any other combination of these primes.

Is there such a set?

I'd appreciate any comments regarding this.

Email: pjp9q@virginia.edu

Thank you very much.

Paul Perrone

Newsgroups: sci.math.research
From: pjp9q@honi2.acc.Virginia.EDU (Paul Joseph Perrone)
Subject: Re: Number Theory
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Virginia
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 26 Jan 1993 02:20:40 GMT

This is posted in correction of my first posting (#616) regarding
prime number theory. I apologize for my mistatement of the problem,
but here is the correct statement of the problem:

What is the LARGEST set of prime numbers with the following criteria:
      1) No Prime in this set can be greater than 1 million;
      2) No sum nor difference between any two primes in this set
         can equal any other sum or difference between any other
         combinations of primes in this set?

And how can I generate these numbers?

I would appreciate any references or direction that might be given to
me toward this problem. Thanx again to any who might reply to this
posting.

Sincerely,

Paul Perrone

pjp9q@virginia.edu



Newsgroups: sci.math.research
From: kgrant@alfred.ccs.carleton.ca (Ken Grant)
Subject: Group Theory Question
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Carleton University
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Thu, 28 Jan 1993 04:18:25 GMT

Can anybody tell me what is known about the function l(S_n)=maximum
length of a chain of subgroups of the symmetric group S_n.
I don't even see how to show that l(S_n)-n --> infinity.

---
Ken Grant
Department of Mathematics
University of Ottawa
From: tokero@bode.eng.ohio-state.edu (Onur Toker)
Newsgroups: sci.math.research
Subject: A question about Nevallina-Pick theory
Date: 27 Jan 93 01:43:13 GMT
Article-I.D.: ee.1993Jan27.014313.19660
Organization: The Ohio State University Dept of Electrical Engineering
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu


	Let m(z) be an inner function defined on the unit disc D, and
let z_1,...,z_n be finitely many points on D. It is known that there
exists a rational inner function m_f(z) which interpolates m(z) at
z_1,..,z_n (This is a classical result in Nevallina-Pick interpolation
theory).
	My question is about the parametrization of all such m_f's? 
Does anybody know anything about 

	(1) parametrization of all such m_f's? 

	(2) given N, is there an m_f of order N?

Remarks: The parametrization of all analytic functions with norm <= 1
and satisfying the above interpolation conditions is known.
Furthermore if we know the definition of the function m(z) and the
numerical values of points z_1,...,z_n then question (2) can be
answered by reducing it to a linear algebra problem and checking
whether  a system of equations has a solution or not. But I don't know
whether (2) holds for all sufficently large N values (or for N > 2*n).

	O.Toker