HDTenEightyP/NetNews huggingface dataset:
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: ALGEBRAIC NUMBER ARITHMETIC
Keywords: algebraic numbers, arithmetic
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 30 Jul 1992 17:21:19 GMT

In article <1992Jul30.163746.21362@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
>In article <1992Jul30.151149.12688@ugle.unit.no> ap@levangerhs.no (Andrei Prasolov) writes:
>>Answering to <aet.712408532@munagin> 
>>in <1992Jul29.163830.6443@ugle.unit.no> 
>>I made a mistake. If we represent algebraic numbers
>>only by their minimal polynomials, we cannot construct
>>a correct arithmetic. Let us take the main root x of
>					^^^^
>
>There is no such animal as a 'main root'. There is only a
>root and its conjugates. If alpha is a root of a polynomial,
>then the field Q(alpha) is isomorphic to Q(alpha') where
>alpha' is any of alpha's conjugates.  The map is controlled by the
				    ^
				  insert:

				when the field is GALOIS.

>Galois group.
>
>Your assertion that one cannot construct a correct arithmetic if all
>one has is the minimal polynomial is FALSE.
 
Oops. Carelessness alert! I don't know what I was thinking about when
I typed the above. I left out several KEY words. I've added them above.
Sloppy. Clearly Q(alpha1), Q(alpha2) ... are subfields of the splitting
field and need not be isomorphic except when one of the fields contains ALL
of the roots.
 
However, when one uses a term like "main root", one must indeed be careful
in clarifying which root you mean.
 

--
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: mcfarlan@corona.math.washington.edu (Thomas J. McFarlane)
Subject: Re: Math or Maths
Organization: University of Washington, Mathematics, Seattle
Date: Thu, 23 Jul 1992 22:49:58 GMT

mckay@alcor.concordia.ca ( John Mckay ) writes:
>
> The Greek is MATHEMATIKOS - when did the abbreviations first occur?
>
My Greek Lexicon gives
    ho mathematicos     a mathematician
    ta mathematika      mathematics
It is derived from the verb manthano, which comes from the root math.

My concise Oxford dictionary lists both math and maths as colloquial
abbreviations for mathematics, but gives no date.

Tom McFarlane

mathematics.
Newsgroups: sci.math
Subject: Re: Factorial program...
From: jmatthews@desire.wright.edu
Date: 24 Jul 92 16:28:20 EST
Organization: Wright State University 

In article <1992Jul22.222658.3613@sjsumcs.sjsu.edu>, kellum@sjsumcs.sjsu.edu (Ken Kellum) writes:
> In article <1992Jul20.160312.10723@infonode.ingr.com> "Vilva Natarajan <vilva@madras.b23b.ingr.COM>" writes:
>>Hi,
>>
>>	I am looking for source code that prints out all the factorial
>>	combinations of a set of numbers.
>> [exampple deleted]

> The following (pseudo) Pascal procedure prints all the permutations
> of a one dimensional array.  It can be modified to do what you want.
> 
> Procedure Permute (var A : array [1..n] of whatever)
> 
>    Procedure Inner (i : integer);
>   { prints the permutations of a[i] .. a[n] }
>    var j : integer
>        temp : whatever;
> 
>    begin
>       if i > n then
>          writeln(a)
>       else 
>          for j := i to n do begin
>             temp := a[i]; a[i] := a[j]; a[j] := temp;
>             Inner (i + 1);
>             temp := a[i]; a[i] := a[j]; a[j] := temp;
>          end;
>     end; { Inner }
> 
> begin {permute}
>    Inner(1);
> end;

The first time I saw this algorith was in Wirth's _Algorithms_+_Data_
_Structures_=_Programs_. The question was posed as an exercise (3.2)
in a chapter on recursion. The _answer_ appeared in the following
chapter as the (otherwise meaningless) sample output of a cross-
reference generating program.

An Easter egg in a textbook:-)

o----------------------------------------------------------------------------o
| John B. Matthews, jmatthews@desire.wright.edu, disclaimer:= myViews <> WSU |
| "I'm a commensal .sig virus, indistinguishable from an ordinary organelle."|
o----------------------------------------------------------------------------o
From: jpat@hpsad.sad.hp.com (Jeff Patterson)
Date: Mon, 27 Jul 1992 17:27:48 GMT
Subject: Re: Mathematica Help
Organization: HP Signal Analysis Division - Rohnert Park, CA
Newsgroups: sci.math

In a previous post I stated that MMa did not allow direct assignment to a list.
This is wrong, assignment is allowed as long as the element for which the 
assignment is being made exists and has an immediate value. I.e.

a[[2]] = 12 results in 
Part::noval: Symbol a in part assignment does not have an immediate value.

while 
  a={1,2,3}
  a[[2]] = 12 
 
results in
  a= {1,12,3}

Thanks to Michael Somos for pointing out my error.
-Jeff
From: kubo@zariski.harvard.edu (Tal Kubo)
Newsgroups: sci.math
Subject: Re: A Divisibility Problem
Date: 28 Jul 92 19:05:48 GMT
Article-I.D.: husc3.1992Jul28.150549.14261
Organization: Dept. of Math, Harvard Univ.

In article <1992Jul28.115747.20313@news.tu-graz.ac.at>
hhassler@iaik.tu-graz.ac.at (Hannes Hassler) writes: 

>(3) A more interesting result is the following:  Given a set of 2n
>      integers, one 
>    can always choose a subset S of cardinality n and with sum
>      divisible by n.
>    The proof of this is left to the reader as an exercise.

A bit harder is the same exercise with (2n-1) integers.
[A set of (n-1) zeros and (n-1) ones shows that  (2n-2) is impossible.]
Newsgroups: sci.math,sci.chem,sci.engr.chem,sci.engr.mech,sci.physics,sci.bio,bionet.journals.contents
From: smith@gramian.harvard.edu (Steven Smith)
Newsgroups: sci.math
Subject: The Trigonometry of Small Geodesic Triangles
Date: 28 Jul 92 21:53:39 GMT
Article-I.D.: gramian.SMITH.92Jul28165339
Distribution: sci.math
Organization: Harvard Robotics Lab, Harvard University

Let M be an n-dimensional Riemannian manifold with Riemannian
structure g, c a point in M, and N_c a normal neighborhood of c which
is a normal neighborhood of each of its points.  Let p,q be points in
N_c.  Then cpq is a triangle in N_c whose angles are C, P, Q, and
whose sides are geodesics of length x = d(c,p), y = d(c,q), and z =
d(p,q).

If the triangle cpq is small, what can be said about the relationships
between its angles and sides?

There is a very complete classical treatment of this question in
Forsyth's _Intrinsic Geometry of Ideal Space_ (MacMillan, 1935).
Forsyth derives the second order approximation of Gauss's formula

                      1
     C + P + Q - PI = - x y K sin C
                      2

where K is the sectional curvature of M at c along the section
generated by the directions of the geodesics cp and cq at c.  Also,
there appears the third order (inclusive) approximation of the law of
cosines
                                   1 (             2 2   2      )
          (  2    2               )- (     1      x y sin C     )
     z  = ( x  + y  - 2 x y cos C )2 ( 1 - - K ---------------- )
          (                       )  (     6    2  2            )
                                     (         x +y - 2xy cos C )

As I said, Forsyth's treatment is (of course) very classical (!!!).  I
suppose I could translate it into modern language, but life is short.
Could anyone point me toward a modern treatment of this subject?

I know that comparison theory (a la Cheeger), Toponogov's Theorem, and
Cartan's treatment of Riemannian manifolds of nonpositive curvature
all provide some beautiful global inequalities, but I am hoping to
find pointers to sharper approximations.

Steven Smith
From: rose@fsu1.cc.fsu.edu (Kermit Rose)
Newsgroups: sci.math
Subject: Re: "proof" that sin(x) = -sin(x)
Date: 24 Jul 92 15:38:10 GMT
Reply-To: rose@fsu1.cc.fsu.edu
Organization: Florida State University
News-Software: VAX/VMS VNEWS 1.3-4

In article <1992Jul19.220735.10289@leland.Stanford.EDU>, ledwards@leland.Stanford.EDU (Laurence James Edwards) writes...
> 
>d sin(-x)             d (-x)
>--------   = cos(-x)  ------- = -cos(-x)
>   dx                   dx
> 
>and
> 
>d -cos(-x)            d (-x)
>---------- = sin(-x)  ------ = -sin(-x) = sin(x)
>   dx                   dx
> 
>so
> 
> 2
>d sin(-x)
>--------   = sin(x)                  (1)
>     2
>   dx
> 
> 
>but we also have:
> 
>d sin(-x)             d (-x)
>--------   = cos(-x)  ------ = -cos(-x) = -cos(x)
>   dx                   dx
> 
>and
> 
>d -cos(x)
>--------- = -sin(x)
>   dx
> 

^^^^^^^^^^^^^^^^^^^^^^^^^^    error is above.  d[-cos(x)]/dx = sin(x).
                                               d[cos(x)]dx   = - sin(x)

>so
> 
> 2
>d sin(-x)
>---------  = -sin(x)                   (2)
>     2
>   dx
> 
>from (1) and (2) we therefore have: sin(x) = -sin(x)!
> 
>Whats wrong with the above "proof"?
> 
>Larry Edwards
>edwards@sunrise.stanford.edu


rose@fsu1.cc.fsu.edu          To be sure I see your response, use e-mail.
-----------------------------------------------------------------------
Be of good cheer, for it is much more fun than being depressed.
From: sichase@csa2.lbl.gov (SCOTT I CHASE)
Newsgroups: sci.math
Subject: Re: An interesting limit problem.
Date: 29 Jul 92 18:03:06 GMT
Organization: Lawrence Berkeley Laboratory - Berkeley, CA, USA
Distribution: na
Reply-To: sichase@csa2.lbl.gov
NNTP-Posting-Host: 128.3.254.197
News-Software: VAX/VMS VNEWS 1.3-4   

In article <1992Jul29.000223.27339@massey.ac.nz>, news@massey.ac.nz (USENET News System) writes...
> 
>I believe that it is far more important to understand the meaning of
>the operations of arithmetic than to be able to work through the
>algorithms like trained monkeys without having a clue why they work.
>(I know some bright spark will ask how many monkeys I've trained to
>do long division - well, no actual monkeys, but a lot of great apes :-)).

This is certainly the crucial point.  As a physics instructor I am convinced
that unlike in botany, one cannot learn the concepts of math or physics without
*doing* the work.  And the only acceptable proof that a student deeply 
understands a concept is to *apply* it to solving a problem.  So I think that
the distinction you make between understanding and being able to work through
the algorithms is completely falacious.

I remember having a specific dialog once (as a physics TA) with a premed 
student who came to my office on the day they learned Gauss's law in EM.
She was accustomed to memorizing fact and considering the matter done.

ME:  Did you follow today's lecture?
HER: Sure.
ME:  You understand Gauss's Law?
HER: Sure.
ME:  OK.  Please calculate the field of an infinite uniformly charged
cylinder using Gauss's Law.
HER: But I understand Gauss's Law.
ME: OK, then just do the problem.
HER: I said I understand it!
ME: No problem.  Then just solve the problem.
HER: Hey!  I studied the chapter yesterday and understood the lecture today.
Gauss's law says that the volume integral of the enclosed charge density is 
proportional to the surface integral of the flux.
ME: That's correct.  Please (!) do the problem.
HER: What for?!  I understand Gauss's Law.
ME: You keep saying that, but you really don't. I suggest that if you expect
to pass the midterm that you make an attempt to solve the problem.
HER: You're not helping me.
ME: I can't help you until I can see what you don't understand.  I can see 
that you have memorized the textbook.  Please try to do the problem, and when
you get stuck, I'll help.
HER: I don't need help with Gauss's Law.  I understand Gauss's Law.  If you
won't help me with what I need help with, then I'll go to another TA.
ME: OK.  It's your choice.  But I promise you that until you have worked out
a dozen problems that you don't really understand Gauss's Law.
HER:  BUT I UNDERSTAND GAUSS'S LAW...

-Scott
--------------------
Scott I. Chase			"The question seems to be of such a character
SICHASE@CSA2.LBL.GOV		that if I should come to life after my death
				and some mathematician were to tell me that it
				had been definitely settled, I think I would
				immediately drop dead again."      - Vandiver
From: chavey@jalapeno.cs.wisc.edu (Darrah Chavey)
Newsgroups: sci.math
Subject: Re: Factorial program...
Date: 21 Jul 92 19:52:29 GMT
Organization: Beloit College, Beloit, Wisc.

In article <1992Jul20.160312.10723@infonode.ingr.com> "Vilva Natarajan <vilva@madras.b23b.ingr.COM>" writes:
>	I am looking for source code that prints out all the factorial
>	combinations of a set of numbers.
>

To cut down on bandwidth traffic, I mailed source code to the author
(instead of posting it).  If, for some reason, anyone else wants it, please
ask.

--Darrah Chavey
  Beloit College
From: tgw@st-andrews.ac.uk (Trevor Gordon Walker)
Newsgroups: sci.math
Subject: Re: Square roots
Date: 29 Jul 92 14:50:36 GMT
Followup-To: sci.math
Organization: St. Andrews University

In article <Bs4HD3.ID5@usenet.ucs.indiana.edu> hpassaue@silver.ucs.indiana.edu (Billy Passauer) writes:
>I ran across this algorithm for finding the square root of a number a
>long time ago.  I was wondering if someone could tell me where I could
>find more information about it.
>
>To find the INTEGER portion of a square root:
>  Take the number you want the square root of, and successively
>subtract the odd numbers (starting with 1.)  Each time the result is
>positive, add 1 to a counter.  Repeat with the new number.  When the
>subtraction yields an odd number, the counter contains the answer.
>(integer only)  For instance the square root of 24:
>
>	              Counter
>	24 - 1 = 23  :   1
>	23 - 3 = 20  :   2
>	20 - 5 = 15  :   3
>	15 - 7 =  8  :   4
>	 8 - 9 = <0 
>
>Any information about this would be greatly appreciated.
>-- 
>  _
> /_)  *  /    /    /  /     hpassaue@silver.ucs.indiana.edu
>/__) /  (__  (__  (__/      hpassaue@ucs.indiana.edu
>     Passauer    ___/

This is a consequence of the fact that the sum of the first n odd 
numbers is n^2, which is easily proved by induction.

I must admit I had never thought of using the fact in this context,
though.

		Trevor
trevor@cs.st-and.ac.uk
tgw@st-and.ac.uk
From: cjh@tinton.ccur.com (Christopher J. Henrich)
Newsgroups: sci.math
Subject: Re: Recreational Math Journals
Date: 28 Jul 92 03:24:51 GMT
Organization: Concurrent Computer Corp., Tinton Falls, NJ

In article <9207241255.AA13353@ucbvax.Berkeley.EDU> cliff@watson.ibm.com ("Cliff Pickover") writes:
>I'm interested in what mathematical journals exist that are interested
>in "light" math -- math more of a recreational and curiosity nature than
>most heavy-duty math journals will publish.
>
American Mathematical Monthly
Mathematics Magazine (an MAA publication, aimed at high schools I
think)
Mathematical Gazette (British)

L'Enseignement Mathematique (Escrivez-vous Francais?)

regards,
Chris Henrich
Newsgroups: sci.math
From: shallit@graceland.waterloo.edu (Jeffrey Shallit)
Subject: Gordon & Breach suit against AMS
Organization: University of Waterloo
Date: Fri, 24 Jul 1992 17:23:06 GMT

I've heard several times from different sources that after the AMS
published a list of journals and how much they cost, they were sued
by a publisher who tried to prevent release of this information.
Perhaps the publisher was Gordon & Breach.  

Can anyone verify this?  In particular, I would like to have a 
pointer to a newspaper or journal article that discusses the
suit.

Jeff Shallit
shallit@graceland.waterloo.edu
Newsgroups: sci.math
From: davidc@panix.com (David C. Chorlian)
Subject: Re:The Value of Science
Summary: Whittaker account of Relativity Inaccurate
Date: Sat, 25 Jul 1992 01:49:57 GMT
Organization: PANIX Public Access Unix, NYC
Keywords: Relativity, Poincare, Whittaker


	Abraham Pais, in his biography of Einstein, "Subtle is the
Lord...", has quite a different account of Poincare's contribution
to the development of the theory of Relativity, and says, 
"His [Whittaker's] treatment of special theory of relativity ..
shows how well the author's lack of physical insight matches his
ignorance of the literature." (168)
	Of Poincare's 1904 address in St. Louis, Pais gives the
following account on page 128:
	"All  phenomena seen by one observer [in relative motion to
another] are retarded relative to the other, but they all are
retarded equally (Poincare points out) and 'as demanded by the
relativity principle [the observer] cannot know whether he is at
rest or in absolute mothion.'  Poincare is getting close.  But then
he falters: 'Unfortunately this reasoning is not sufficient and
complementary hypotheses are necessary; one must assume that bodies
in motion suffer a uniform contraction in their direction of motion.'
The refference to complementary hypotheses makes clear that
relativity theory had not yet been discovered."

	According to Pais, Poincare gave a series of lectures at
Goettingen in 1909:
	"The new mechanics, Poincare said is based on three hypotheses.
The first of these is that bodies cannot attain velocities larger
than the velocity of light.  The second is that the laws of physics
shall be the same in all inertial frames.  Then Poincare introduces
a third hypothesis: ' ... a body in translational moting suffers
a deformation in the direction in which it is displaced.'  It is 
evident that as late as 1909 Poincare did not know that the
contraction of rod is is a consequence of the two Einstein
postulates.  Poincare therfore did not understand one of the most
basic traits of the special relativity.  (167-68)

	Although Poincare had the equations of the Lorentz transformation
by 1905, HE DID NOT UNDERSTAND WHAT THEY MEANT.  Thus he is rightly
not credited with the development of special relativity.

David B. Chorlian
Neurodynamics Lab   SUNY/HSCB 
Xref: sparky sci.math:9397 sci.chem:2724
From: jimbo@leland.Stanford.EDU (Godfather of Soul)
Newsgroups: sci.math,sci.chem
Subject: Fit 3-dimensional Lorentzian peaks
Date: 22 Jul 92 08:38:36 GMT
Organization: DSG, Stanford University, CA 94305, USA

I have some nuclear magnetic resonance data that I'd like to fit.  I was
wondering if anyone knew of any program that could fit a 3D Lorentzian
peak (Lorentzian shape in both dimensions).

Thanks for responding by email.

James
Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Fibre Bundle Theory
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Mon, 27 Jul 92 21:19:01 GMT

In article <s#T-+B+@engin.umich.edu> pradeep@engin.umich.edu (Pradeep Jain) writes:
>Hi,
> 
>I understand that Fibre Bundle Theory is about formulating geometric 
>problems as Fibre Bundle Theoretic problems (and then as algebraic 
>problems?). 

Well, the algebra would only come in if one combined fiber bundles with
some other techniques like algebraic topology or Lie theory, as is often done.

Can someone please give me some introductory refrences to
>Fibre Bundle Theory? I would greatly appreciate the help.

Since it seems you may be an engineer I would recommend Analysis
Manifolds and Physics by Choquet-Bruhat et al, since this gives
applications to physics, and skips the hard proofs --- which is good for
anyone wanting a first introduction to a subject, not just engineers!
(I'm in favor of knowing what something will do for you before sweating
over it too much.)

I imagine that most good books on differential geometry talk about fiber
bundles.  For example Kobayashi and Nomizu.  

Rather painful for anyone but hardcore mathematicians, but packed with
useful information, is Dale Husemoller's book Fibre Bundles.  You may
have heard of the hairy ball theorem on the impossibility of combing a
coconut in such a way that its hair lies flat everywhere without a
"part".  In other words, there is no nowhere-vanishing tangent vector
field on the 2-sphere.  A classic problem is to find how many everywhere
linearly independent vector fields on an n-sphere (for example, there is
1 on a 1-sphere, 3 on a 3-sphere, 7 on a 7-sphere, but these are the
only cases where there are n on an n-sphere).  Husemoller shows how this
problem is related to fiber bundles and how it is solved using
K-theory (a type of algebraic topology).  This particular proof is quite
sophisticated....

>Also, what 
>are the pre-requisites for gaining an understanding of Fibre Bundle
>Theory?

Fiber bundles are one of the things one studies in topology or
differential geometry; since you mention geometry you are probably more
interested in the latter approach.  In that case one needs to know
what a manifold is before messing with fiber bundles.  And you need to
be familiar with the basic examples of fiber bundles (tangent and
cotangent bundles) before the general concept seems like something
that's obviously interesting.  To really get serious about fiber bundles
(namely, to play around with connections on fiber bundles) one also
needs to learn a bit about Lie groups.  The Choquet-Bruhat and the
Kobayashi/Nomizu refs will have more than enough on these to get going.
Newsgroups: sci.math
Subject: Re: You know, the integers (was: Re: Stupid quest
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: 23 Jul 92 18:05:31 +1200
Distribution: world
Organization: Department of Mathematics, University of Canterbury

> >    >"You know, THE SETS."
|> > 
|> >   More precisely, the cumulative hierarchy of sets.
|> > Set theory is a different matter: even today we can point to Zermelo's
|> > "Ueber Grenzzahlen und Mengenbereiche" as a presentation unequaled in
|> > clarity of the cumulative hierarchy of sets as a model of ZFC.
|> 
|>   I find the cumulative hierarch of sets very *unclear*

I would partly agree - the cumulative hierarchy is beautifully clear as far
as its LENGTH goes, the backbone of ordinals. (At least provided you dont go
as far as an inaccessible, "whatever that is".)

But the cumulative hierarchy isn't quite so magnificently clear when it comes
to WIDTH; just how many sets are added in at each stage. That's the area
where a lot of fuzzy disagreement occurs between platonist set theorists.


Come to think of it; if you don't fully believe ax.choice is "really" true
for "actual" sets, then maybe the length of the hierarchy isn't all
that clear either, beyond the first nonconstructive ordinal....... 
Newsgroups: sci.math.symbolic
From: mxl5@po.CWRU.Edu (Marshall J. Leitman)
Subject: Integers
Reply-To: mxl5@po.CWRU.Edu (Marshall J. Leitman)
Organization: Case Western Reserve University, Cleveland, OH (USA)
Date: Wed, 22 Jul 92 14:24:44 GMT

> The following code accepts input, say 2[t] and gives output 2 in Version
> 1.2:
> 
> Unprotect[Integer];
> Integer/:(x_Integer)[t___]:=x;
> Protect[Integer]
> 
> In Version 2.0 it just returns 2[t]. I just can't seem to make it work.
> Something must have changed in Version 2.0

        
-- 
Marshall J. Leitman
Professor of Mathematics
mxl5@po.cwru.edu
Newsgroups: sci.math.symbolic
From: perle@nariwork15.ethz.ch (Hans-Christian Perle,ETF D111,2791,,,)
Subject: Notebook for mma on Sun
Keywords: notebook, mma
Reply-To: perle@nariwork15.ethz.ch
Organization: ETH Zurich
Date: Thu, 30 Jul 1992 12:47:30 GMT

Is there any hope for a real notebook for mathematica on sun4?
I'm missing my old Mac


Hans


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