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In article <328C7D22.E48@univ-tln.fr>, Philippe LangevinReturn to Topwrote: > 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
Expires: References:Return to TopSender: 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
In article 3mt@news.ox.ac.uk, Brian StewartReturn to Topwrites: >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...
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/~williamReturn to Top
In articleReturn to Top, 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!
>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 YnReturn 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: caseReturn to Top: ; : ; . . . 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
Downloaded by WWW Programs
Byron Palmer