HDTenEightyP/NetNews huggingface dataset:
From: markd@locus.com (Mark Dubinsky)
Subject: Solutions to a cubic equation
Organization: Locus Computing Corporation, Los Angeles, California
Distribution: usa
Date: Mon, 16 Nov 1992 22:15:27 GMT

OK, this bugged me back in high school, and then I was sick last week
with nothing better to do, and I tried to figure it out, and it's 
bugging me again.  What is the formula for the solutions of a cubic
equation

x3 + ax2 + bx + c = 0    ?

The quadratic formula is so trivial, but I couln't make any progress at
all with this.  Actually, I saw the formula for the solutions a long time
ago in a book, and they involved sines and cosines, which didn't make 
much sense, but I didn't pursue it further.  So, how is the solution
derived?

-Mark
Newsgroups: sci.math
From: glue@ellis.uchicago.edu (brian wesley glueck)
Subject: Re: 1+1/2+1/3+1/4+...+1/n
Keywords: partial sums of the harmonic series
Reply-To: glue@midway.uchicago.edu
Organization: University of Chicago
Date: Thu, 19 Nov 1992 16:32:23 GMT

In article <92324.132329K3032E2@ALIJKU11.BITNET> Mutter Christoph Johannes <K3032E2@ALIJKU11.BITNET> writes:
>I've a problem. I have to calculate the sum 1+1/2+1/3+1/4+...+1/n.
>The result should be 100. But my computer is far too slow, to solve this
>problem. I need the index n when the sum > 100.0
>And that exactly.
>Has anyone calculated this? (Perhaps on a CRAY|) ?

Since log(x) is the integral of the reciprocal function from 1 to x, the
partial sums of the harmonic series can be used to give upper and lower
bounds for log(x).  Draw a picture if you need to in order to see that

    1/2 + 1/3 + ... + 1/n <= log(n) <= 1 + 1/2 + 1/3 + ... + 1/(n-1)

So if log(n) is larger than 100, then the sum on the right is also larger
than 100, which suggests that you might want to look at values of n that
are close to exp(100).  Of course, exp(100) is a pretty big number (on the
order to 10^43, I think), but a little analysis helps you get an idea of
how big n should be without doing any of the summations at all, with or
without a Cray.

Brian Glueck
b-glueck@uchicago.edu
Organization: Mathematics, Carnegie Mellon, Pittsburgh, PA
Newsgroups: sci.math
Date: Thu, 19 Nov 1992 15:36:00 -0500 
From: Oswald Wyler <ow0a+@andrew.cmu.edu>
Subject: Re: 1 + 1/2 + 1/3 + ... + 1/n

Call the sum 1/n.  Then by integration
h(n) - 1 < ln n < h(n-1).
So the n for which the sum becomes 100 is between e^99 and e^100.
This makes it clear that numerical computation, with roundoff errors,
is useless.

Newsgroups: sci.math,sci.physics,sci.astro,sci.bio,sci.chem,misc.education
From: pehoushe@CS.Stanford.EDU (Dan Pehoushek)
Subject: Re: Most important event in the history of mathematics
Organization: Computer Science Department, Stanford University
Date: Sat, 21 Nov 1992 10:41:59 GMT

How about: "It hasn't happened yet!"

Dan Pehoushek

Bandwidth? What's that?
Newsgroups: sci.math,sci.physics,sci.astro,sci.bio,sci.chem,misc.education
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: What can we have for an educational system?
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 19 Nov 1992 00:47:34 GMT

In article <BxrGCs.4tp@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
:In article <1992Nov15.122659.129397@zeus.calpoly.edu> mhorning@zeus.calpoly.edu (Mark E. Horning (Captain Neutrino)) writes:
:>clin@eng.umd.edu (Charles Lin) states
:
:>>In article <ARA.92Nov11034458@camelot.ai.mit.edu>, ara@zurich.ai.mit.edu (Allan Adler) writes:
:
:
:>>>As I have pointed out on other occasions, if they would improve the
:>>>working conditions, they could have PhD's teaching in the public schools.
:>>>I don't mean improve them a little. I mean offer working conditions that
:>>>would make a PhD happy and the glut of unemployed and underemployed
:>>>PhD's will beat down the doors of the schools trying to get in.
 
Actually, at least in my experience, the conditions that would make 
most PhD's happy are essentially impossible to obtain. Nothing the
schools or school systems could do would provide what they want.

What they want of course is motivated students and students who
are *interested* in learning. This is the responsibility of parents.
Currently, cultural conditions in the U.S. are such that most people
do not place a high value on education. The result? Students who
don't give a shit.
--
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
Subject: closed forms
From: <YUKQC@CUNYVM.BITNET>
Date: Thursday, 19 Nov 1992 22:31:38 EST
Organization: City University of New York/ University Computer Center

Let S be a set and f1,...,fn be functions over S. I want to know
which branch of mathematics has dealt with questions of the following kind.

Given a bunch of recursive equations that purport to define
a function f, how to decide the existence of, or find, a closed formula
for f in terms of f1,...,fn (I mean a finite closed formula).
The recursive equations only contain f1,...,fn.

Example: Does a closed form of +, -, * exist for a factorial function?
         Does a closed form of +, -, *, /, sqrt exist for a factorial
         function over reals?

References to books are greatly appreciated.
(What's the answer to the above question anyway?)
Xref: sparky sci.math:15449 sci.physics:19530
From: baez@guitar.ucr.edu (john baez)
Newsgroups: sci.math,sci.physics
Subject: Re: SU(4), SO(6)
Date: 24 Nov 92 01:03:07 GMT
Followup-To: sci.math
Organization: University of California, Riverside

In article <COLUMBUS.92Nov23103624@strident.think.com> columbus@strident.think.com (Michael Weiss) writes:
>SU(2) is the double cover of SO(3), and thereby hangs a tale, ably told in
>many places (e.g., Kaufmann's "Knots and Physics".)
>
>SU(2) x SU(2) is the double cover of SO(4).  Am I correct in suspecting
>that this is related to the Dirac equation?

Probably, but there are plenty of things it's good for that I'm more
familiar.  For example, Schrodinger's equation for the hydrogen atom is
exactly solvable but most central potentials are not.  Why?  The 1/r potential
has a hidden so(4) symmetry - not just the obvious SO(3) symmetry.  At
the level of Lie algebras, so(4) = so(3) + so(3) by the facts you cite
above.  The "extra" so(3) symmetry of hydrogen atom Hamiltonian
is most easily seen from the presence of 3 extra conserved quantities in addition to angular momenta -- these are the components of the "Runge-Lenz vector".

So as it turns out, there is a deep relationship between the hydrogen atom
Hamiltonian and the Laplace equation on S^3.  There is a nice 1-1 correspondence
between the eigenfunctions of the two (if one counts only the bound
states of hydrogen, not the ionized states) since they are isomorphic as
representations of so(4).  

For more variations on this theme take a peek at Guillemin and Sternberg's
nice tour of mathematics, Variations on a Theme by Kepler.

>SU(4) is the double cover of SO(6).  Has nature made use of this delightful
>fact?

Probably, but I don't know where.  I hadn't remembered this one!  There are
some other nice Lie group coincidences and they all lead to beautiful
connections between seemingly disparate things.

Newsgroups: sci.math
From: nissim@mary.fordham.edu (Leonard J. Nissim)
Subject: Re: A Question About Fundamental Group
Organization: Fordham University
Date: Wed, 18 Nov 1992 13:58:00 GMT
News-Software: VAX/VMS VNEWS 1.41    
X-Posted-From: mary.fordham.edu
NNTP-Posting-Host: sol.ctr.columbia.edu

In article <amirishs.722059588@acf9>, 
amirishs@acf9.nyu.edu (shaahin amiri sharifi) writes...
>Consider a bunch of infinitely many (countable) circles with a
>point in common. The radii of these circles make a sequence like
>{1/n}. What is the fundamental group of this space? Any comments
>or referenc would be so helpful!

Sometimes known as the "Hawaiian earing space".
Its fundamental group has coutably many generators (one for each circle)
and no relations.  (I.e., it is a free non-Abelian group on countably 
many generators.)  

"Algebraic Topology: An Introduction" by Massey is a good reference for 
the definition and early theorems about the fundamental group.

This is cute:  take the cone over your original space; that is, put all 
the circles in the xy-plane tangent to the y-axis at (0,0) and connect 
each point of every circle to the point (0,0,1) in R^3.  
Now you have a space that is contractable (homotopic to a point), but 
*not* locally simply connected.  In fact, every small neighborhood of (0,0,a)
(where 0<=a<1) has a fundamental group with infinitely many generators. 

-------------------------------------------------------------------------------
Leonard J. Nissim  (nissim@mary.fordham.edu)   
Disclaimer:  "I speak only for myself."
-------------------------------------------------------------------------------
From: rgep@emu.pmms.cam.ac.uk (Richard Pinch)
Newsgroups: sci.math
Subject: Re: Help with Algebraic Number Theory
Keywords: hilbert algebraic number
Date: 23 Nov 92 12:07:15 GMT
Organization: Department of Pure Mathematics, University of Cambridge

In article <1epji1INN66u@manuel.anu.edu.au> molrchon@mehta.anu.edu.au writes:
>[...]
>The second problem I have is much more minor, but I can't for the life of me
>see how it.  Suppose J is some number field, and x \in O(J).  Then the claim
>is that:
>
>      x != 0  <==>  (\exist v, y \in O(J))[xy = (2v-1)(3v-1)]
>
>It apparently follows from the Chinese Remainder Theorem, but I can't see the
>trick.  No doubt it is embarassingly easy.
>molrchon@mehta.anu.edu.au

Write the ideal (x) as A_1 A_2 A_3 where A_1 is the product of the prime
power ideals not dividing 6, and A_2 resp A_3 is the product of the prime 
power ideals dividing 2 resp 3.  Solve 2V_3 == 1 mod P_3; 3V_2 == 1 mod P_2 
and (say) 2V_1 == 1 mod P_1.  Each is soluble since 2 is a unit mod P_3, 
3 is a unit mod P_2 and both 2 and 3 are units mod P_1.  By the CRT, find 
v == v_i mod P_i.  Then (2v_i)-1)(3v_i-1) == 0 mod P_i for i=1,2,3,and so 
(2v-1)(3v-1) == 0 mod (x). 

Richard Pinch
From: psalzman@sfsuvax1.sfsu.edu (Peter Salzman)
Newsgroups: sci.math
Subject: Re: Arch algorithm
Date: 16 Nov 92 22:45:56 GMT
Article-I.D.: csus.1992Nov16.224556.9922
Organization: California State University, Sacramento



I think I know what you're talking about.  I forget what the engineers
call it, but I think you're referring to the hypercosine function:

             x     -x 
            e  +  e
   f(x) =   -------
               2


This function describes the shape of the curve made by any flexable wire
held at 2 points.


peter
From: eppstein@wormwood.ics.uci.edu (David Eppstein)
Subject: Re: Erdos number
Newsgroups: sci.math
Reply-To: eppstein@ics.uci.edu (David Eppstein)
Organization: Univ. of Calif., Irvine, Info. & Computer Sci. Dept.
Date: 19 Nov 92 18:16:06 GMT

chisnall@cosc.canterbury.ac.nz (The Technicolour Throw-up) writes:
> Interesting to see the name of one of sci.math's frequent posters show up in
> the tables that Tompa uses.  Apparently Vaughn Pratt's Erdos number is at
> worst 4.

In fact it's at most 2.

    F.R.K. Chung, P. Erdos, R.L. Graham, S.M. Ulam, and F.F. Yao,
    Minimal decompositions of two subgraphs into pairwise isomorphic subgraphs,
    10th Southeastern Conf. Combinatorics, Graph Theory, and Computing, vol I.,
    Congressus Numerantium XXIII, Utilitas Math., Winnepeg (1979) 3-18.

    V.R. Pratt and F.F. Yao, On lower bounds for computing the ith
    largest element, 14th IEEE Symp. Found. Comp. Sci. (1973) 70-81

Vaughan?  You have any papers with E himself?
--
David Eppstein    UC Irvine, Info & Computer Science    eppstein@ics.uci.edu
From: mlevin@husc8.harvard.edu (Michael Levin)
Newsgroups: sci.math
Subject: help me take a derivative of this complex function
Date: 17 Nov 92 17:41:51 GMT
Distribution: usa
Organization: Harvard University Science Center


    I have a set of functions in complex variables that I need to take
a derivative of, for a fractals project I am going to do. I've looked
up everything in a book, except one thing. All the other functions
(comon arithmetic, powers, etc.) are "derived" like real functions,
but I can't find this one anywhere.  Here it is:
f(c) for a complex c, is equal to c', where the real part of c' is
equal to the imaginary part of c, and vice versa. In other words, f()
simply switches the real and imaginary parts of the variable it's
applied to. so, what would be f'()? If anyone has any clues, please
send email to mlevin@husc8.harvard.edu. Thanks in advance.
BTW, I am a biology, not math, major, so take it easy if this involves
very complex stuff (or if it is a stupid question...).

Mike Levin
Xref: sparky sci.math:15442 comp.edu:1939
Newsgroups: sci.math,comp.edu
Subject: Re: Minimum standards for math "competency"
From: news@massey.ac.nz (USENET News System)
Date: Mon, 23 Nov 92 23:23:00 GMT
 <1992Nov23.140653.13522@scott.skidmore.edu> <By6E16.JFx.2@cs.cmu.edu> <1992Nov23.162755.9636@galileo.cc.rochester.edu>
Organization: Massey University

In article <1992Nov23.162755.9636@galileo.cc.rochester.edu>, landsbur@troi.cc.rochester.edu (Steven E. Landsburg) writes:
> 
> In article <By6E16.JFx.2@cs.cmu.edu> jmount+@CS.CMU.EDU (John Mount) writes:
> >In article <1992Nov23.140653.13522@scott.skidmore.edu>, pvonk@scott.skidmore.edu (Pierre VonKaenel) writes:
> >|> Perhaps you haven't visited a technical university lately.  I recall
> >|> professors complaining that most of their students are oriental, and
> >|> where are the American kids?  I'm not sure about the statement above,
> >
> >I trying to imply you said the above (I even left a bit of your "I'm not
> >sure about that").  But I think this kind of racist crud is intolerable- 
> >a good number of the Asian students ARE American kids.
> > 
> I applaud the denunciation of the racist crud.  But it doesn't go far
> enough.  So what if they AREN'T American kids?
> 
I think the point intended is that American schools are not producing
enough good university students. Whether or not you agree with this,
there is nothing racist about it. The fact that there are still able foreign
students is praise to the education system in other countries and evidence
that it is possible to produce students of a sufficient standard. I have no
idea how many Asian students ARE American, but if the situation is
similar to New Zealand, you will be getting a lot from Asia. The aim
was not to complain about getting too many of these, rather, it was to
complain that the American education system is not producing enough
local students. Again this is in no way racist - it is a simple statement
of the perception of the writer. (This may be fact or fiction).

It seems to me that any mention of race seems to raise a howl from
the politically correct. Perhaps the politically correct readers can
try to read the context and not just pick out isolated sentences.
Newsgroups: sci.math
From: vhansen@ipfs.bau-verm.uni-karlsruhe.de (Wolfgang von Hansen)
Subject: Re: Square root of a number
Organization: IPF, University of Karlsruhe
Date: Wed, 18 Nov 1992 11:47:10 GMT

In article <1992Nov15.194049.1@woods.ulowell.edu> buxamusaa@woods.ulowell.edu writes:
>
>Please could someone help me with an assignment that I have to do.
>It is a very general question, though the answer could be quite lengthy.
>"Discuss methods for finding the square root of a number."
>
>What I am looking for are all possible methods/alogrithms for this.

My preferred method for such algorithms is Newtons iteration:

Problem: For a given function f find x, so that f(x) = 0.

                   f(x)
Solution: x <- x - -----
                   f'(x)

The starting value of x must be chosen near enough to the desired
result. Otherwise you might get a different result or need quite
a lot of iterations. Repeat the iteration until the accuracy e

e = |x - x   |
      i   i-1

is reached. Turning to square root we seek x with

                 2         2
x = sqrt(a) <=> x = a <=> x - a = 0.

                          2
From this we get f(x)  = x - a
                 f'(x) = 2x.

Filling this into our iteration formula we receive

          2        2   2
         x - a   2x - x + a   1 /    a\
x <- x - ----- = ---------- = - |x + -|
          2x         2x       2 \    x/

         1 /    a\
<=> x <- - |x + -|
         2 \    x/

I hope that this is helping.

- Wolfgang
Newsgroups: sci.math
From: jk87377@cc.tut.fi (Juhana Kouhia)
Subject: Re: a first year grad student freaks out.
Organization: Tampere University of Technology
Date: Wed, 18 Nov 92 19:17:25 GMT


In article <BxKu9s.BAw@news.udel.edu> cornwall@bach.udel.edu (Ray J
Cornwall) writes:
>
>For those of us who wish to do research (cause we'll get our names in
>really cool journals! :) can you give us some pointers on how to come
>up with such topics?
>I've been trying to be more aggressive with my studies in math, to
>look for such ideas, but I have a feeling I'm barking up the wrong
>tree looking in textbooks and looking for my favorite topics here.

Could it help if you just read abstracts from the journals
and then try do the same job as far as you can.
And maybe then check the article entirely and check
if you were close -- things will improve greatly if
you get small 'aha'-experiments sometimes.

If you just read the texts, then you know how it is done and you
will not bother yourself to make it differently (at least, mostly).

When I read something interesting I think could that work
in the areas I'm interested in, or such; and so, I have compined
two ideas to get third idea without pain.
Sometimes you hit the nail.

Juhana Kouhia
Newsgroups: sci.math,tum.questions
From: gwoeging@iaik.wu-wien.ac.at (Guestaccount G.Woeginger)
Subject: Re: Minimum number of lines that cover a set of points
Organization: Angewandte Informationsverabeitung und Kommunikationstechnologien, TU-Graz
Date: Wed, 18 Nov 92 10:59:45 GMT

In article <1992Nov17.163800.21271@news.lrz-muenchen.de> Matthias Sauer writes:
>
>I am looking for an algorithm for the following problem:
>
>    Given a set P of points in the plane find a minimal 
>    number of straight lines such that every p in P is
>    on at least one of these lines.
>
>I am almost sure that there is some standard algorithm for this problem,
>but I could not find any. Please send an e-mail to:
>     sauer@nws.e-technik.tu-muenchen.de
>or post an article in sci.math or tum.questions

This problem was shown to be NP-complete in the beginning of the 1980ies
by van Emde Boas.  I think his paper was called "Another NP-completeness
result" or something like this, and it is also mentioned in one of 
Johnson's NP-completeness columns.

Gerhard Woeginger
From: baez@guitar.ucr.edu (john baez)
Newsgroups: sci.math
Subject: Approximation and Estimation (HELP!!)
Date: 22 Nov 92 22:34:03 GMT
Organization: University of California, Riverside

In article <1ejbebINNber@roundup.crhc.uiuc.edu> hougen@focus.csl.uiuc.edu (Darrell Roy Hougen) writes:
>I need to estimate a 'hill' function between -pi/2 and pi/2.  It must
>be 0 at -pi/2 and pi/2 and it must be 1 at 0.  It must increase
>monotonically from 0 to 1 on [-pi/2,0] and decrease monotonically from
>1 to 0 on [0,pi/2].  It must be smooth, ie., differentiable everywhere
>in (-pi/2,pi/2), and have a derivative of 0 at 0.
>
>My first question is, does there exist a basis for functions of the
>above form?

Sure, infinitely many.  It's best to say what *kind* of functions  In
other words, smooth, continuous, or L^2 functions.  But in any of the case
I just mentioned, there is a countable basis.

>One possible basis that I've been thinking about is (cos(x))^k with k
>ranging from -inf to +inf.  All the members of this set satisfy the
>conditions given above and it appears that any weighted sum of such
>functions will also although it would be nice to prove that.

Nope, it's not true, because all the functions you listed are "even"
(satisfy f(x) = f(-x)), so you can't approximate anything but even 
functions.    Also for k negative the functions you list are unbounded,
which is technically a pain in the butt.  

The best basis to use is  sin(2kx) and cos(2kx), taken together.  
This basis is used in the Fourier transform and is well-known to
be a complete basis.  I recommend this basis because a huge amount
is known about it and there is a huge amount of software written to
deal with it.

>My second question is, if I have a potential basis, how can I prove
>that it is a complete basis?  In other words, how can I prove that any
>f of the above form can be approximated arbitrarily well by a sum of
>functions in the basis set?

There are lots of ways.  For L^2 functions 
it is sufficient for the functions to "separate
points and span an algebra" to be a basis, IF they are linearly 
independent.  I can explain these terms if it matters - they are 
properties that hold for the basis I suggested, but the "separates
points" property doesn't hold for the basis you suggested.
   
>My third question is, can the results be extended to two or three
>dimensions?  More specifically, assume that the function I am trying
>to estimate, f(x,y,z), satisfies the above conditions in all three
>dimensions.  If necessary, assume additional smoothness conditions.
>The same goes for the 1-D case above.
>
>In this case, I would like to use seperable functions as a basis.  For
>example, let g_ijk(x,y,z) = (cos(x))^i(cos(y))^j(cos(z))^k.  Is this a
>basis?  Why or why not?

If you have a basis f_i(x) of the smooth/continuous/L^2 functions
on the interal (take your pick!), then one can form a basis 
f_i(x)f_j(y)f_k(z) for the same class of functions on the cube.

Again, these are 3 theorems in analysis that I could prove for you,
but I am too lazy right now.  It takes a fair understanding of
functional analysis to follow the proofs.
From: ekg90@ecs.soton.ac.uk (E K Gottesman)
Newsgroups: sci.math
Subject: Re: Office Hours
Date: 18 Nov 92 14:18:11 GMT
Distribution: sci

How about doing something really radical such as posting your 'office
hours' notice to the undergrads ?  OK, so most of them will ignore it,
but you may be lucky and find a bright student who wants to find out
about/discuss areas of maths not directly covered in courses. I know
that most of us can't even tie our own shoelaces, but amazingly there
are a few undergrads who are actually interested in their subject and
are suffering the same sort of isolation you are (it's uncool to be
too enthusastic). I'm always hanging around after lectures, chatting
about stuff I've read with the lecturers. I even got an academic
summer job simply by asking a professor a few 'awkward questions'.

--Emma.
From: kubo@zariski.harvard.edu (Tal Kubo)
Newsgroups: sci.math
Subject: Re: Counting Families of Subsets
Keywords: Dedekind, unsolved
Date: 16 Nov 92 03:08:35 GMT
Organization: Dept. of Math, Harvard Univ.

In article <1992Nov15.152951.44802@kuhub.cc.ukans.edu>
kinnersley@kuhub.cc.ukans.edu (Bill Kinnersley) writes: 
>Take a base set S with N elements.  There are 2^N subsets of S.
>Let F be a family such that all elements of F are distinct subsets,
>and no element of F is a subset of another.
>
>How many families like this are there, for given N?

You are asking for D(n), the number of antichains of subsets of 
an N-element set. 

  1) Computing D(n) is an unsolved problem that goes back to 
     Dedekind, if not earlier.

  2) It has been proved that for large N, almost all the antichains
     have all their elements in the middle four layers (i.e. consist
     of sets of size  n/2 - 2 to  n/2+2).

There is a journal called "Order", devoted to similar problems 
involving partially ordered sets.  Check there for more recent
information.

-tk    kubo@math.harvard.edu