Back


Newsgroup sci.math.research 6259

Directory

Subject: Re: _FIELDS -- From: steiner@bgnet.bgsu.edu (Ray Steiner)
Subject: Re: Fixed-point question -- From: asimov@phnom-penh.berkeley.edu (Daniel Asimov)
Subject: Re: HELP: e^(-x^2) -- From: wall@phys.chem.ethz.ch (Ernst U. Wallenborn)
Subject: Re: Sets with the 0-1 Intersection Property -- From: william@pacm.Princeton.EDU (William Schneeberger)
Subject: Re: One combinatorial problem -- From: aburshte@mail2.sas.upenn.edu (Alexander Burshteyn)
Subject: Re: tails of distributions which obey central limit theorem -- From: jedhudson@cix.compulink.co.uk ("John Hudson")
Subject: 3D-Filmstrip Version 6.2 is now available. -- From: palais@math.brandeis.edu (Richard S. Palais)

Articles

Subject: Re: _FIELDS
From: steiner@bgnet.bgsu.edu (Ray Steiner)
Date: Fri, 15 Nov 1996 14:14:11 +0500
In article <328C7D22.E48@univ-tln.fr>, Philippe Langevin 
 wrote:
> It is not a research question, but probably only specialists know the
> answer to my question :
> 
> Mathematicians use the word field to design
> an algebraic object like (C and R).
> 
> Accordind to Franz Lemmermeyer, lemmermf@star.CS.Uni-SB.DE, Dedekind
> introduced the word KOERPER for the algebraic object, 
> 
> translated in CORPS (in french),
>  
> REALM or FIELD (in english) because CORPSE was unacceptable, but
> gradually REALM went out fashion.
> 
> Do you know more about the story of this terminology ?
> why Dedekind chose the word KOERPER ?
> why FIELD in english ? 
> 
> -- 
>  _______
> |   |   | Universite de Toulon et du Var
> | G | E | Groupe d'Etude du Codage de Toulon
> |___|___| B.P. 132, 83957 LA GARDE CEDEX
> |   |   | TEL: 94.14.20.55 FAX: 94.14.24.79
> | C | T | E-MAIL: langevin@univ-tln.fr
> |___|___| URL: http//www.univ-tln.fr/~langevin
Actually, I  have some very old papers, where the term
used was CORPUS(pl. CORPORA). Why was it changed to field?
Ray Steiner
Return to Top
Subject: Re: Fixed-point question
From: asimov@phnom-penh.berkeley.edu (Daniel Asimov)
Date: 15 Nov 1996 22:57:34 GMT
Expires: 
References: 
Sender: 
Followup-To: 
Distribution: 
Organization: U.C. Berkeley Math. Department.
Keywords: 
Cc: 
In article ,
Nick Halloway   wrote:
>
>If a topological space X is compact and contractible, must it satisfy a 
>fixed point theorem?  That is, is there a compact contractible space
>X and continuous f: X --> X which has no fixed points?
>
>Please explain your terms, I'm a beginner at topology.
>
>
Perhaps Borsuk's counterexample:  a 3D cellular space (i.e., an intersection
X of homeomorphs of the 3-disk D_1 containing D_2 containing D_3 ... D_n ...)
such that X does not have the fixed-point property (the property you refer to).
I am not sure, however, whether or not X is contractible.
The Borsuk space X is roughly as follows: consider 2 (single-nappe) cones, each
missing its vertex, lying next to each other in opposite directions (i.e.,
in "69" position), but each with its vertex-end wrapped about the other in an 
infinite  spiral which approaches the boundary of the other cone in the limit.
X is the union of these 2 vertexless cones (which intersect in a curve C that
is homeomorphic to the real numbers).  X is compact and has a continuous map 
(in fact a homeomorphism) h: X -> X without fixed point obtained by sliding X 
along itself (so that the restriction of h to C resembles a translation of the
reals).
--Dan Asimov
Return to Top
Subject: Re: HELP: e^(-x^2)
From: wall@phys.chem.ethz.ch (Ernst U. Wallenborn)
Date: 16 Nov 1996 14:28:20 GMT
In article 3mt@news.ox.ac.uk, Brian Stewart  writes:
>voloch@max.ma.utexas.edu (Felipe Voloch) wrote:
>>magix (magix@dibe.unige.it) wrote:
>>: Dear all,
>>: we're facing a problem with a simple Gaussian function.
>>: Does anybody know the closed form of:
>>: (LATEK) \frac{d^m}{dx^m}(e^{-x^2})
>>: (Visual)
>>: .             2
>>: .        m  -x
>>: .       d  e
>>: .      --------
>>: .           m
>>: .        d x
>>
>>: for any m in N?
[snip]
>>Faa di Bruno's formula strikes again!
>>
>>This is the second posting on the same topic belonging to
>>Calculus (!) that skips past the moderators.
[snip]
>I must confess that I tried to use Faa di Bruno and couldn't find any 
>reasonable way to simplify the coefficients. But if I'm missing something 
>then please enlighten me. 
Look under Hermite Polynoms.
You'll find something like
$H_n(x) = (-1)^n \exp\{\frac{x^2}{2}\}\frac{d^n}{dx^n}\exp\{-\frac{x^2}{2}\}$
where the H are the Hermite polynomials given as
H_0 = 1
H_1 = x
H_{n+1} = x H_n -H_{n-1}
simple rearrangement gives what the original poster wants.
---
-ernst wallenborn.
- Du haelst die Fernbedienung verkehrtherum, Dad!
 - Nicht, wenn es einen Gott gibt, Dumpfbacke...
Return to Top
Subject: Re: Sets with the 0-1 Intersection Property
From: william@pacm.Princeton.EDU (William Schneeberger)
Date: 16 Nov 1996 16:17:41 GMT
In article <56h3l6$hfm@senator-bedfellow.MIT.EDU> lones@lones.mit.edu (Lones A Smith) writes:
>Let S(x,c) be a closed Borel subset of [0,1], for any x in [0,1] & real c. 
>
>Suppose S has the "0-1 intersection property": For any c1 and c2, and for 
>all x1 <> x2, S(x1,c1) and S(x2,c2) have either zero or one point in common.
>
>CLAIM: {x in [0,1]|union of S(x,c) over all real c has measure >0} is countable
It would appear that something is missing.
If S(x,c)={c}\intersect [0,1] , it is closed, Borel, a subset of
[0,1], and has the 0-1 intersection property.
But for any x, the union of S(x,c) over all c is [0,1], which has
measure 1.
-- 
Will Schneeberger
william@math.Princeton.EDU
http://www.math.princeton.edu/~william
Return to Top
Subject: Re: One combinatorial problem
From: aburshte@mail2.sas.upenn.edu (Alexander Burshteyn)
Date: 16 Nov 1996 17:39:37 GMT
In article ,
Gareth McCaughan   wrote:
>Alexander Burshteyn wrote:
>> I wonder if anyone can offer any solution (or even a suggestion) about the 
>> following problem which came up in the course of my research:
>> 
>> For every positive integer $n>1$ and every permutation $\tau \in S(n)$ 
>> (i.e. of {1,...,n}, the following inequality holds:
>> 
>> $
>> \sum_{j=1}^{n} {
>> 	\sum_{k=1}^{n} {
>> 		\binomial{j+k-2,j-1} \times \binomial{2n-j-k,n-j} \times 
>> 		\binomial{ \tau (j) + \tau (k) - 2, \tau (j) - 1} \times
>> 		\binomial{ 2n - \tau (j) - \tau (k), n - \tau (j)}
>> 		}
>> 	} 
>> > \binomial{2n-1,n} ^ 2
>> $
>> 
>> Even a proof or a pointer for the case $\tau = id(n)$ would be great.
>Actually, it would be more than great: it would be enough, by
>Chebyshev's[1] inequality (if (a[i]) are increasing then 
>sum of a[i].b[tau(i)] is maximal when tau is chosen so that
>(b[i]) are increasing too).
Please correct me if I misunderstood you, but I'm not looking the maximum of 
these sums over all tau (which clearly happens at tau=id, by 
Cauchy-Schwartz inequality), but rather want to prove that this result 
holds even for their minimum.
>I don't have a proof, though.
There is a rather roundabout proof of this inequality with > replaced by 
greater-or-equal sign. But it's exactly the strict inequality that I need.
>[1] If that happens not to be your preferred spelling, too bad.
My preferred spelling is the original one in Russian, since that's my native 
language. So there. As far English spellings go, I prefer the same one 
you do.
>-- 
>Gareth McCaughan       Dept. of Pure Mathematics & Mathematical Statistics,
>gjm11@dpmms.cam.ac.uk  Cambridge University, England.
Alex Burstein
alexb@math.upenn.edu
aburshte@sas.upenn.edu
-- 
AB
******************
145 = 1! + 4! + 5!
Return to Top
Subject: Re: tails of distributions which obey central limit theorem
From: jedhudson@cix.compulink.co.uk ("John Hudson")
Date: Sat, 16 Nov 1996 18:10:33 GMT
>Vince Darley wrote:
>> In my recent research, I'm trying to estimate the shape of 
distribution 
> of
> the average of n i.i.d U(0,1) random variables. 
>snip
>ut I want to look at the 
> 
> lim_{n -> infinity} [2^n * Prob(average < x)]
> 
I am not the world's best statistician but I think the Chernof bound
might be applicable for this problem though the bounds may not be very
good.
If Yn = sum X(i),   i=1,...,n
then
p[Yn > n.h'(s0) ] < exp(-n[s0.h'(s0)-h(s0)] )
where  h(s) = ln[(M(s)] = ln[E{exp(s.X)}] is the log moment generating 
function.  (there is probably a similar formula with Yn 
Return to Top
Subject: 3D-Filmstrip Version 6.2 is now available.
From: palais@math.brandeis.edu (Richard S. Palais)
Date: Sat, 16 Nov 1996 18:06:45 -0500
                                          November 16, 1996
  Version 6.2 of 3D-Filmstrip has just been released for general 
distribution. This is the first update since June,1996. 
  Below is the first part of the ReadMe file for 3D-Filmstrip.
=======
A.)   WHAT IS IT?
  3D-Filmstrip is a tool that aids in the visualization of
mathematical objects and processes. It runs under version 7 or
later of the MacOS and requires either a 680x0 cpu with an FPU
or else a PowerPC cpu (where it will run native).
  3D-Filmstrip has algorithms for displaying mathematical objects from 
many different "categories" (plane and space curves, surfaces, conformal
maps, polyhedra, ODE, waves) and for displaying various mathematical
processes associated with these categories. In addition, each category 
has a "Gallery" of many pre-programmed objects from the category, and
also a way for the user to enter User Defined obects of the category. The
Gallery items are selected from a menu, while the user defined objects
are entered by editing algebraic expressions that define an object of the
category in a dialog.
   Filmstrip was designed and programmed (in Object Pascal) by: 
       Richard S. Palais
       Department of Mathematics
       Brandeis University
       Waltham, MA 02254
       palais@math.brandeis.edu
       Home Page: http://rsp.math.brandeis.edu/pub
  The program is copyrighted, but there is a free license
to use it for non-commercial purposes in education and research.
B.) WWW and FTP Availability.
  3D-Filmstrip has a Home Page on the Web, at the URL:
   http://rsp.math.brandeis.edu/3D-Filmstrip_html/3D-FilmstripHomePage.html 
and the latest released version is available by anonymous ftp at the URL:
 ftp://rsp.math.brandeis.edu/pub
(which has a link on the 3D-Filmstrip Home page).
  The ftp distribution is in Binhex format and includes the complete 
documentation in TeX and html formats. It also includes many "settings" 
files that illustrate objects and animations of special interest.
  There is a "fat" compilation that will run both on 680x0 based Macs with
a floating point unit (fpu)  and on PowerMacs, as well as slimmer "FPU" and 
"PPC" compilations. 
(Version 6.2 does not have the SANE compilation of earlier versions, 
and therefor cannot run on non-PowerPC Macintoshes without floating 
point hardware.)
   There are currently four mirror ftp sites for the distribution of 
3D-Filmstrip, in Europe, Australia, Taiwan, and Hong Kong. 
The URL's and site administrators are:
ftp://hensel.mathp6.jussieu.fr/dist/3D-Filmstrip/  
   Dominique Bernardi --- bernardi@mathp6.jussieu.fr
ftp://maths.adelaide.edu.au/pure/mmurray/3DFilmstrip
   Michael Murray ---  mmurray@spam.maths.adelaide.edu.au
ftp://math.ntu.edu.tw/pub/mac/mirror.rsp  
   Ai Nung Wang --- wang@math.ntu.edu.tw
ftp://ftp.math.cuhk.edu.hk/pub/mac/ 
   Law Wai Kuen (Keith) --- keith@math.cuhk.edu.hk
------------
C) Documentation.
 The documentation for 3D-Filmstrip consists of:
    1) This file
    2) 3D-Filmstrip_Doc.tex  (Complete documentation in TeX format)
    3) Hypertext documentation in html format, consisting of:
        a) 3D-FilmstripHomePage.html
        b) 3D-Filmstrip_Doc.html (a Table of Contents page)
        c) IndexPage.html (an Index Page)
        d) Twenty-five html pages of actual documentation
        e) Nine .GIF files.
  All of the above documentation is included with the standard ftp 
distribution and in addition there is a link to an online version of 
the html documentation starting from the 3D-Filmstrip Home Page.
  Most of the information in the above documentation can also be accessed 
as Help panels while 3D-Filmstrip is running. These can be selected from 
the Balloon Help menu.
  ====================================================================
   The remainder of this file is a version history, describing what 
is new in the current version and what was new in earlier versions.
                       Version History
                       ---------------
What's new in Version 6.2 of 3D-Filmstrip.      November 15, 1996  
  With this version of 3D-Filmstrip, the user interface has gotten a
major facelift.  I would like to thank Mike Epstein and Xah Lee for
pushing me to make these changes, and working very hard with me in
the design and testing of the many new features.
   There are also important mathematical additions to the program---but 
most were made by my collaborators, Christopher Anand and Angel Montesinos.
Christopher considerably beefed up the Anand-Ward soliton segment of the 
program, and added User Defined... solitons as well as documentation.
I am very pleased with this example of one extreme in the spectrum of 
applications I had hoped that 3D-Filmstrip would be useful for, namely 
as a tool in advanced mathematical research. Angel extended his marvelous 
algorithm for implicitly defined plane curves to also work for implicit 
space curves, and I needed to make only minor adaptions for his code to 
work within 3D-Filmstrip. I added routines for displaying geometric objects
in the 3-Sphere. They are simply stereographically projected, and then
treated like any other object in R^3. In particular you will now find
the Hopf fibration in the Space Curve menu, and the Clifford torus in
the Surface menu.
  Here is a list of changes to the user interface:
1)  There is a new application icon (Mike Epstein helped with the design).
     You may have to remake your desktop file to see it. (Have a look with 
     your stereo glasses on).
2)   Settings files and Grand Tour files now have their own distinctive icons,
     and double-clicking will open them (after launching 3D-Filmstrip if it 
     isn't already running).
3)   Command-O = Open Settings File
    Command-S = Save Settings 
    Command-W closes Help Window and dialog windows
    Command-U opens User Defined... dialogs
4)   Cancel buttons have been added to dialogs.
5)   Surface rendering now defaults to color.
6)   Many changes to the View menu, to make it more intuitive for new users.
7)   You can now omit the * for multiplication when writing expressions in
     User Defined... dialogs (but remember to put in spaces to avoid ambiguity).
8)   You can now also use "split definitions" in expressions, giving them
different
     values on different parts of the domains of their variables. The syntax is:
     case  :  ;   :  ;  . . . end 
     For example:    
     case  x>0: ln(x);  x<=0: 1 end   
     (The final ; is optional.) 
9)   You can now signal a fast abort either with Command-period as before,
     or by pressing the Escape key.
10)  The "Virtual Sphere" button that used to inhabit the lower left 
     hand corner of the screen is gone.
    That is because the program is now always in Virtual Sphere mode.
    For example, after drawing a Surface you can immediately click on 
    it and start rotating it. The rotation happens in wireframe by default, 
    but if you put down Caps-lock it happens in patch mode (at reduced
    resolution). Moreover if, while you are rotating, you hold down the 
    Control key, the cursor changes to four arrows and you can now translate
    the object in a plane parallel to the screen. If instead you hold down 
    Shift, then the cursor changes to an up-down arrow and you can "zoom" the
    object toward or away from you. In essence, you can move along an
arbitrary  
    path in the Euclidean group, and have the object "follow along". 
    The same thing works in the Plane curve and Conformal map categories,
    except the default movement is translation instead of rotation. Morever,
    in these two categories, if you hold down Command and then drag out a
    rectangle in the usual Mac way, then when you release the mouse (with
    Command still down) your selection rectangle will zoom to the entire 
    window.
11) There used to be a special version of Virtual Sphere mode in which 
    an object spins about an axis that you could change interactively by 
    rotating it with the mouse. The old way of entering that mode was by
    putting Caps-lock down before clicking on the Virtual Sphere button,
    and you stayed in that mode until the Caps-lock key was released.
    This has now been renamed to Spin mode. You enter it by selecting
    Spin at the bottom of the Animate menu, and exit it by fast abort
    (i.e., either Command-period or Escape).  For surfaces, the spin will 
    be in wireframe if Caps-lock is up and in Patch mode if it is down.
12) If you are connected to the Internet and have the System Extension 
    Internet Config installed in your Extensions Folder, then you will 
    have two extra items at the bottom of your Balloon Help menu when 
    3D-Filmstrip is active; namely "Download Latest Version" and 
    "3D-Filmstrip Home Page". Of course, for these to work, you must have
    available an ftp program (such as Fetch or Anarchie) and a Web Browser 
    (such as Netscape Navigator) and you must have chosen one of each as
    your preference using the Helper Dialog of the Internet Config program.
-- 
Richard S. Palais
Dept. of Mathematics, Brandeis Univ.
Waltham, MA 02254
palais@math.brandeis.edu
http://rsp.math.brandeis.edu
Return to Top

Downloaded by WWW Programs
Byron Palmer