Newsgroup sci.math 151826

Directory

reaton@goodnet.com wrote:
> 
> It has been 34 years since college for me, and I have reached a time that I can
> start letting up a little in my chosen career and start looking around for some
> fun things to do.  While in college, my exposure to calculus was about 1 year, and
> although it was a tough pull, it was fascinating and enjoyable and somewhat of
> a diversion from other science courses that I took.  I have always enjoyed
> mathematics, math games and such and occasionally find myself thinking about
> my environment in some mathematical form.  I would like to go back and recapture
> calculus (& analytical geometry), just for the fun of it.  I've tried this a
> couple of times in the last few years, but because of interruptions (read other
> demands) and a lack of a good text (read difficult; does not read easily and
> not a page turner) I have not succeeded.  Would anyone that frequents this
> newsgroup have any suggestions for me concerning appropriate texts or teaching
> and learning aids.  Although it has been a long time since I have had to really
> use "higher mathematics," I had a good foundation in all mathematics in high school
> and I have surprised myself on my recall of nearly all material up to the college
> calculs level, so I'm pretty sure this is a do-able thing.
> 
> Any suggestions would be appreciated.
> 
> Ron Eaton
> 
> -----------------------------------------------------------------------> This article was posted to Usenet via the Posting Service at Deja News:
> http://www.dejanews.com/           [Search, Post, and Read Usenet News]
This is a test from sci.math

Well, this is as silly a thread as appears semi-regularly, and I can't
believe I'm actually going to hit the send button for such nonsense, but
then nobody's perfect.  The debate all stems from the usual confusion
between science and maths, as Ilias pointed out, (though not quite in so
many words, perhaps alas).  "Can possibly happen" appears to be scientific, 
(logico-observational); and "probability zero" is purely mathematical.
However.
Let me contribute some nonsense of my own.  I hereby assert that the two
phrases are IDENTICAL in meaning.  It is said that there are events with
probability zero, that "can happen".   I hereby deny it.  I claim, all events
of probability zero CAN NOT HAPPEN, by definition, if you like.
"But but", I hear you leaping to cry out, scrambling over one another in
haste to be first with the usual alleged counter-example, gibbering and
drooling on one another!  (Isn't usenet *fun*!)  So let me just choose one
of you, and have it out.
You: But what about the experiment we've just done, whereby we observed some
     continuous random variable, and got an observed answer?  Say uniform[0,1]
     for definiteness.  We just observed X1 = 0.341888721.... 
Me: (grinning diabolically)  But that's a *past* experiment, and probability
    only applies to the *future*, doesn't it?
You: (jumping up and down):  Oy, hoy!  That's just a fudge; surely you're not
     going to stand on a technicality like *that*.
Me:  Well I might; but anyway, there's no future event in your scene, is there?
You: OK OK. We just observed X1 = 0.341888721....  and are just about to observe
     an i.i.d X2.  Now I admit that that *particular* value won't occur again...
Me:  How very sweet and generous of you. 
You: ...but it *did* happen the first time (for X1), didn't it?"
Me:  No it didn't!  
You:  What!?
Me:  Or rather:- just what *is* it you are claiming happened the first time? 
You: We got  X1 = 0.341888721... ! 
Me:  But those dot-dot-dots are the root of the problem. If you just mean we 
     observed  0.3418887205 < X1 < 0.3418887215, then that event had a
     strictly positive probability of, let me see, ten to the minus, ah...
You: No no NO!  I didn't mean that.  I meant that X1 was equal to some
     actual infinite sequence of decimal numbers, beginning with 0.341 etc
Me:  OK; but WHAT infinite sequence of decimals?
You: Well, I wrote down the first several, I'm not going to bother writing down
     any more, I haven't got the time!
Me:  But I insist!  You had the time to insist it was an experiment with
     an outcome of that form, so I insist on being told the outcome!
You: Well yes, all right, all right.  But I don't actually have to *tell*
     you surely?  I mean, the outcome was there on the dial, after all.
Me:  But that crumby dial only told me that X1 was between 0.335 and 0.345,
     and that event has probability...
You: Yeah yeah!  But we just couldn't read it very exactly.  The fully 
     precise outcome was *there*.
Me:  No it wasn't.
You: Yes it *was*.  I set up the experiment, so I should know!
Me:  But all you set up was a situation where we were told that X1 was betwe-...
You: Yes but now I'm using a much more precise dial-...
Me:  But that just means X1 will be between...
You: ...that measures things to INFINITELY many decimal places.  So what do
     you say if I told you the outcome was X1 = 0.341888721 + pi/(ten-to...)
Me:  I'd say you were lying, because it would never come *exactly* to that,
     unless you're a miracle worker and you've been cheating.  Nor to any
     other definable number.
You: Definable?  Well, whatever that means is irrelevant, because it *has*
     come to *some* actual number, with infinitely many dec places.
Me:  How do you know?
You: Because I can measure it with this infinite-precision dial here!  Hah!
Me: "Hah" to you!  You know perfectly well there's no such thing.  As you
     know from Heisenberg and information theory and all that, you can't 
     possibly get infinite precision on a continuous measurement, can you?
You:  Oh well!  If you're just going to fall back on a *scientific* matter...
Me:  Well wasn't that what I said right at the start?  *You* described the
     experiment in those terms, scientific terms; not me.  You ought to know
     that all continuous math models are just continuous approximations to
     incredibly complex discrete observations we only ever actually *make*.
You: Yeah but...  Oh hell!  Forget the continuous dial reading then. 
     Make it an endless sequence of coin tosses.   Now:- 
Me:  You know perfectly well that I can say all of the above exactly the same
     in that case.  I'd still insist on knowing the exact sequence.
You: But, but... Heisenberg wouldn't apply!  There's no "h-bar" in the
     set-up this time!
Me:  No; but if you want to observe all those outcomes, there's quite a bit 
     of big-G and c, isn't there!  Didn't you see John Baez' latest post on-
You: -Well; maybe if we made the coins smaller and smaller, and tossed 
     them faster and fa...  OK OK!  No need to slobber; I take that back.
     Hmmm... maybe it *is* intimately involved with science rather than
     just math.  Still, I can't help feeling there's *really* an actual
     outcome there, that had probability 0, and actually *did* occur.
Me:  You "can't help feeling" it!?   Hah.  Well *I* can't help feeling
     you're just a religious nutter in disguise, grasping at straws in
     a desparate attempt to find any silly & meaningless "counter-example"
     to what is obviously true:- events of probability zero NEVER HAPPEN.
     Never WILL happen, and never DID happen!
You (stomping off):  Mutter... stupid argument... mumble... twerp... grumble...
================================.
Bill Taylor                     |       The unrefined and sluggish mind
Math Department                 |             Of Homo Javanensis,
University of Canterbury, NZ    |       Could only treat of things concrete
W.Taylor@math.canterbury.ac.nz  |             And present to the senses.
fax  (0064-3-) 364 2587         |
=============================================================================| 

|| >Have I implemented the correct algorithm?
|| 
|| Yes.  You may also start with u := 10, but nobody does because then
|| we couldn't compare results.  There are other, alternative starting
|| values depending on the characteristics of p.
Really?  Cool.  I have to try that out sometime...
|| There's quite a difference between a 31 bits and 61 bits.  Plus you're
|| squaring -- 62 bits and 122 bits.  Sounds like overflow.  You have
|| implemented the correct algorithm incorrectly :-)
Well, my code *should* be able to handle integers up to 24000 bits.
Whether it actually does or not is another question.  :)
Well, since the algorithm is correct, it must be a problem with my code...
Turbo Debugger here I come... 
|| A P133 should take about 37 hours to test 2^1257758-1 -- the largest
|| known Mersenne prime weighing in at about 375K digits.
|| 
|| BTW, the Lucas-Lehmer test has been ported to the PowerMac by John
|| Sweeney.  Look for the "Free Software!" link while visiting the
|| above URL.  John did an outstanding job.  Written entirely in 'C',
|| the program really screams.  Impressive.
Well, once I get this version working and "algorithmically-optimized", I'm
going to rewrite the whole thing in flat-mode assembler, optimized for the
Pentium's dual pipelines.  That should be a fun weekend...  :)
Thanks for the response,
-Mike
___________________________________________________________ 
Michael Anttila aka PsychoMan of Craw Productions 
2A Pure Math / Computer Science at U. of Waterloo, Canada
E-Mail:   manttila@undergrad.math.uwaterloo.ca
Homepage: http://www.undergrad.math.uwaterloo.ca/~manttila/
Craw Productions: craw@magi.com, http://www.magi.com/~craw/

dean@psy.uq.oz.au (Dean Povey) writes:
>
>Hmm, well there is another experiment that would allow a decision on the fate 
>of the SR v AD debate once and for all.
 Right, the obvious one: use the same magnetic spectrometer and 
 calorimeter to measure the power in a monochromatic beam from a 
 small van de Graaff and from a hot source of betas.  Getting a 
 monochromatic beam from a beta-decay source when working near 
 the endpoint is going to have huge experimental uncertainties, 
 however, due to the low flux.  
>This is referred to as the "New RaE experiment", 
 Get a clue, and write Bi-210 like everyone else. 
 Since Bi-210 has an alpha branch, you have to be very careful.  A 
 pure beta source would be preferable so you can simply avoid that 
 kind of problem.  One with a higher Q value for decay would be 
 even better.  Having sat on a PAC for three years, I can guarantee 
 that these would be the easy suggestions.  Estimates of count rate 
 and expected errors would also be expected; I did not see those 
 discussed on the cited web page. 
>I would be interested if people think this is a definitive test of both 
>theories. (And if there are any experimental physicists about who might be
>interested in performing it).
 Why don't you guys get together with Carezani and do it? 
-- 
 James A. Carr        |  "The half of knowledge is knowing
    http://www.scri.fsu.edu/~jac/       |  where to find knowledge" - Anon. 
 Supercomputer Computations Res. Inst.  |  Motto over the entrance to Dodd 
 Florida State, Tallahassee FL 32306    |  Hall, former library at FSCW. 

In article <96111018004521718@hottips.com>, ralph.huff@hottips.com (Ralph
Huff) wrote:
|| At what rate and at what upward angle must a golf ball leave the driver
|| in order to travel 200 yards before it touches the ground? How about
|| 300 yards by a profesional contestant who is strong enough?
|| Can you figure it out?
I'm no physict or engineer, but it seems that there are lot more variables
involved in the problem than just the rate (of what?) and the angle.  In
other words:  No, I can't figure out.
-- 
  ,_____.     ASARI Hirotsugu
  )     (     Graduate Student/Teaching Assistant
 /       |    Department of Mathematics
(        |    The University of Illinois at Urbana-Champaign
|      * |    1409 West Green Street
 \       |    Urbana, IL  61801
  \_     |    Voice: +1.217.333.6329  Fax: +1.217.333.9576
    )   /     http://www.math.uiuc.edu/~asari/
    \_,/      Usual disclaimers apply.

In article <5669r7$df7@news-central.tiac.net> numtheor@tiac.net
(Bob Silverman):
>"gdm"  wrote:
>>            [...] A toy-plane flies well. I want to build a plane twice
>> bigger. I thought candidly that the engine had to be 2*2*2=8 times more
>> powerful. I was wrong: the engine had to be 8*root(2) more powerful [...]
>
> Kinetic energy of the plane is  1/2 m v^2.   
> What happens next depends on how you measure 'size'.  If twice as
> big means increasing all linear dimensions by 2, then the mass of the
> plane will increase by 8 (mass is proportional to volume).
>
> Now ask yourself: If  v must remain the same,  and m increases by 8x,
> what must happen to the kinetic energy?
Ah, but why should v remain the same?
By doubling the size, weight increases by a factor of 8, but the wing
area increases only by a factor of 4, which means the wing loading
(Weight/Area) has increased by a factor of 8/4 = 2.
Without going into details, a higher wing loading means the margin
above stall speed has decreased, or even gone negative, which violates
an implied condition of the problem -- namely that the larger airplane
flies as well as the smaller one.
I suppose I should be careful here -- it's easy to make a bunch of
implied assumptions about the airplane. For example, some airplanes
are capable of hovering, in which case v does indeed stay constant,
at zero. I'd go back and look at the numbers in that case, if I wasn't
feeling so lazy at the moment...
-- Frank Manning
-- Chair, AIAA-Tucson Section

In article <5669r7$df7@news-central.tiac.net> numtheor@tiac.net
(Bob Silverman):
>"gdm"  wrote:
>>            [...] A toy-plane flies well. I want to build a plane twice
>> bigger. I thought candidly that the engine had to be 2*2*2=8 times more
>> powerful. I was wrong: the engine had to be 8*root(2) more powerful [...]
>
> Kinetic energy of the plane is  1/2 m v^2.   
> What happens next depends on how you measure 'size'.  If twice as
> big means increasing all linear dimensions by 2, then the mass of the
> plane will increase by 8 (mass is proportional to volume).
>
> Now ask yourself: If  v must remain the same,  and m increases by 8x,
> what must happen to the kinetic energy?
Ah, but why should v remain the same?
By doubling the size, weight increases by a factor of 8, but the wing
area increases only by a factor of 4, which means the wing loading
(Weight/Area) has increased by a factor of 8/4 = 2.
Without going into details, a higher wing loading means the margin
above stall speed has decreased, or even gone negative, which violates
an implied condition of the problem -- namely that the larger airplane
flies as well as the smaller one.
I suppose I should be careful here -- it's easy to make a bunch of
implied assumptions about the airplane. For example, some airplanes
are capable of hovering, in which case v does indeed stay constant,
at zero. I'd go back and look at the numbers in that case, if I wasn't
feeling so lazy at the moment...
-- Frank Manning
-- Chair, AIAA-Tucson Section

   Paul Turkstra <6pt@qlink.queensu.ca> wrote:
>Suppose that A^3(a standard square matrix) = 0.  Verify by matrix
>multiplication that:
>
>	(I-A)^-1 = I + A + A^2
>
Hi !
it doesn't even matter whether A is a square matrix or an ordinary C-number:
1+A^3=(1-A)(1+A+A^2),since A^3=0 this means 
1    =(1-A)(1+A+A^2).now multiply with (1-A)^-1 from the left.
-> (1-A)^-1 = 1+A+A^2 !
(1, in my notation, stands for the unit square matrix, of course)
greetings
godzilla

In article <55okv5$t7t@ccshst05.cs.uoguelph.ca>, devens@uoguelph.ca (David
L Evens) wrote:
   The idea of an empty set is quite 
> well defined.
Yes please! I would love to see your definition of the empty set.
Terry Moore, Statistics Department, Massey University, New Zealand.
Imagine a person with a gift of ridicule [He might say] First that a
negative quantity has no logarithm; secondly that a negative quantity has
no square root; thirdly that the first non-existent is to the second as the
circumference of a circle is to the diameter. Augustus de Morgan

dylan@wam.umd.edu (Dylan Greene) writes:
:
: Given a convex polygon with N sides of arbitrary lengths -
: does the point A, B fall inside or outside of that polygon?  
:
: Not too hard, but here's the pusher - is there a formula for
: this, or even a tight C function (better than O(N)), that can
: do this?
:
: The method I'm using right now is horizontal line scans,
: counting how many times the side of a polygon is crossed. [...]
:
There is a cheap (i.e. constant time, based on integer
computations and positive-zero-negative testing) method of
finding out if a point is to the left-of, on, or to the right-of
a given directed line segment on the 2-d plane.
If a point is inside a convex polygon (which is described as a
sequence of vertices or edges in counter-clockwise order), then
this point is always "to the left of" all of the edges of the
polygon.  There is no need to project rays and check boundary
crossings (this is needed for non-convex polygons).
If you want to know about cheaper approaches (or a proof of
whether an O(N) is a necessity) let me suggest either the various
comp.graphics FAQs or Joseph O'Rourke's excellent introduction to
computational geometry.  O'Rourke is also the one who maintains
some of these FAQs, is on the net, and will probably answer
questions (if you ask politely enough and have previously checked
the FAQs :-).
In general, if you are willing to spend O(N.lgN) time in
preprocessing (i.e. doing a "sort"), you can amortize various
kinds of tests into O(lgN) time by divide-and-conquer or binary
search approaches.  What you actually want depends on the context
in which you will be using the above test.
But I really recommend the FAQs.  You will even find C code to do
what you want.
--Krishna

In article <566d6k$ms8@cantuc.canterbury.ac.nz>,
Bill Taylor  wrote:
@Well, this is as silly a thread as appears semi-regularly, and I can't
@believe I'm actually going to hit the send button for such nonsense, but
@then nobody's perfect.  The debate all stems from the usual confusion
@between science and maths, as Ilias pointed out, (though not quite in so
@many words, perhaps alas).  "Can possibly happen" appears to be scientific, 
@(logico-observational); and "probability zero" is purely mathematical.
@
@However.
@
@Let me contribute some nonsense of my own.  I hereby assert that the two
@phrases are IDENTICAL in meaning.  It is said that there are events with
@probability zero, that "can happen".   I hereby deny it.  I claim, all events
@of probability zero CAN NOT HAPPEN, by definition, if you like.
@
@"But but", I hear you leaping to cry out, scrambling over one another in
@haste to be first with the usual alleged counter-example, gibbering and
@drooling on one another!  (Isn't usenet *fun*!)  So let me just choose one
@of you, and have it out.
@
@
@You: But what about the experiment we've just done, whereby we observed some
@     continuous random variable, and got an observed answer?  Say uniform[0,1]
@     for definiteness.  We just observed X1 = 0.341888721.... 
@
@Me: (grinning diabolically)  But that's a *past* experiment, and probability
@    only applies to the *future*, doesn't it?
@
@You: (jumping up and down):  Oy, hoy!  That's just a fudge; surely you're not
@     going to stand on a technicality like *that*.
@
@Me:  Well I might; but anyway, there's no future event in your scene, is there?
@
@You: OK OK. We just observed X1 = 0.341888721....  and are just about to observe
@     an i.i.d X2.  Now I admit that that *particular* value won't occur again...
@
@Me:  How very sweet and generous of you. 
@
@You: ...but it *did* happen the first time (for X1), didn't it?"
@
@Me:  No it didn't!  
@
@You:  What!?
@
@Me:  Or rather:- just what *is* it you are claiming happened the first time? 
@
@You: We got  X1 = 0.341888721... ! 
@
@Me:  But those dot-dot-dots are the root of the problem. If you just mean we 
@     observed  0.3418887205 < X1 < 0.3418887215, then that event had a
@     strictly positive probability of, let me see, ten to the minus, ah...
@
@You: No no NO!  I didn't mean that.  I meant that X1 was equal to some
@     actual infinite sequence of decimal numbers, beginning with 0.341 etc
@
@Me:  OK; but WHAT infinite sequence of decimals?
@
@You: Well, I wrote down the first several, I'm not going to bother writing down
@     any more, I haven't got the time!
@
@Me:  But I insist!  You had the time to insist it was an experiment with
@     an outcome of that form, so I insist on being told the outcome!
@
@You: Well yes, all right, all right.  But I don't actually have to *tell*
@     you surely?  I mean, the outcome was there on the dial, after all.
@
@Me:  But that crumby dial only told me that X1 was between 0.335 and 0.345,
@     and that event has probability...
@
@You: Yeah yeah!  But we just couldn't read it very exactly.  The fully 
@     precise outcome was *there*.
@
@Me:  No it wasn't.
@
@You: Yes it *was*.  I set up the experiment, so I should know!
@
@Me:  But all you set up was a situation where we were told that X1 was betwe-...
@
@You: Yes but now I'm using a much more precise dial-...
@
@Me:  But that just means X1 will be between...
@
@You: ...that measures things to INFINITELY many decimal places.  So what do
@     you say if I told you the outcome was X1 = 0.341888721 + pi/(ten-to...)
@
@Me:  I'd say you were lying, because it would never come *exactly* to that,
@     unless you're a miracle worker and you've been cheating.  Nor to any
@     other definable number.
@
@You: Definable?  Well, whatever that means is irrelevant, because it *has*
@     come to *some* actual number, with infinitely many dec places.
@
@Me:  How do you know?
@
@You: Because I can measure it with this infinite-precision dial here!  Hah!
@
@Me: "Hah" to you!  You know perfectly well there's no such thing.  As you
@     know from Heisenberg and information theory and all that, you can't 
@     possibly get infinite precision on a continuous measurement, can you?
@
@You:  Oh well!  If you're just going to fall back on a *scientific* matter...
@
@Me:  Well wasn't that what I said right at the start?  *You* described the
@     experiment in those terms, scientific terms; not me.  You ought to know
@     that all continuous math models are just continuous approximations to
@     incredibly complex discrete observations we only ever actually *make*.
@
@You: Yeah but...  Oh hell!  Forget the continuous dial reading then. 
@     Make it an endless sequence of coin tosses.   Now:- 
@
@Me:  You know perfectly well that I can say all of the above exactly the same
@     in that case.  I'd still insist on knowing the exact sequence.
@
@You: But, but... Heisenberg wouldn't apply!  There's no "h-bar" in the
@     set-up this time!
@
@Me:  No; but if you want to observe all those outcomes, there's quite a bit 
@     of big-G and c, isn't there!  Didn't you see John Baez' latest post on-
@
@You: -Well; maybe if we made the coins smaller and smaller, and tossed 
@     them faster and fa...  OK OK!  No need to slobber; I take that back.
@     Hmmm... maybe it *is* intimately involved with science rather than
@     just math.  Still, I can't help feeling there's *really* an actual
@     outcome there, that had probability 0, and actually *did* occur.
@
@Me:  You "can't help feeling" it!?   Hah.  Well *I* can't help feeling
@     you're just a religious nutter in disguise, grasping at straws in
@     a desparate attempt to find any silly & meaningless "counter-example"
@     to what is obviously true:- events of probability zero NEVER HAPPEN.
@     Never WILL happen, and never DID happen!
@
@You (stomping off):  Mutter... stupid argument... mumble... twerp... grumble...
	As Bill's Dialogue Concerning Two New Sciences underscores, observing
   an infinite sequence of Bernoulli trials (i.e. a real number) is an abstra-
   ction.  All physical experiments and observations yield rationals; none
   has ever produced a non-rational real.  Forget P = 0; such a thing is simply
   _impossible_... it cannot happen (!).  Never has, and never will.
	If we grant this abstraction, it is math, not physics.  The n-th trial
   takes time 1/2^n ?!  Yeah, right.  Infinitely many coin-flippers... march-
   ing off to infinity?  Maybe we keep them in our vicinity by making them
   smaller and smaller...  But even then they have a limit point!  (Turing
   used a similar argument in his analysis of computability).
	We just _assume_ B^w, for B Bernoulli with  p + q = 1.  If 0 < p < 1
   and x is in B^w, "x can not occur" is logically false.  It also has pro-
   bability 1.  So some are tempted to assert it.  After all, modifying the
   measure at one point, or countably many, does not alter any results.  Of
   course the fact that P({x}) = 0 for all x says very little about P; it
   is the values on intervals (sets of x with initial segment s) that deter-
   mine which P we have.  For B^w it comes down to p.
	The remarkable part is focusing on _one_ countable set, the definable
   (recursive) x's, and deeming it "more justified".  A little-known fact:
   coins have tiny cellular telephones and talk to each other; and they all
   have access to an oracle for zero jump as well.  They will not be caught
   doing anything recursive, or recursively enumerable...   Clever little
   devils.
							Ilias

In article ,
Bernd Petrovitsch   wrote:
>"Christopher Anderson"  writes:
>
>> While in an interview I was asked why when a value of x is placed into  x^3
>> -x it is divisible by 6. Fortunately I was not asked to show this using
>> algebra. However I was told to simplify it thus x(x-1)(x+1) and place
>> values of x in. Assuming each term is separate we got list of numbers
>[stuff deleted]
>You had the solution already here (rewritten so that it is even more clear):
>x^3-x = (x-1)x(x+1)
>i.e. you have the product of three consecutive numbers => 
>therefore at least one is dividable by 2 and at least one is dividable by 3 =>
>the product is at least dividable by 6. q.e.d.
>
[...]
 Agreed: the author of the original post (C. Anderson) did include a
proof, clarified by B. Petrovich.
 What may disturb a person indoctrinated by the vague school distinction
between "algebra" (perhaps misunderstood as thoughtless symbol
manipulation?) and unspecified "non-algebra" is that the proof by
inspection "divided the problem into cases", and that is looked down upon,
in favour of an unbroken chain of equations or so. In this respect, I can
assure the audience that "distinguishing cases" (if done correctly) in
mathematical proofs is just as legitimate as a one-shot
symbol-manipulating proof, and sometimes inevitable. 
 Having said that, I can offer two smoother proofs:
 (1) Observe that (x+1)*x*(x-1)/(3*2*1) is "(x+1) choose 3", the number 
of three-member subsets of an (x+1)-member set. Hence
     x^3 - x = ((x+1) choose 3) * 6. Fertig.
 (2) Denote a(x) = x^3-x. Then a(0)=0, a(1)=0, both divisible by 6, and 
then we check mechanically that
   a(x+2) = 2 * a(x+1) - a(x) + 6 * (x+1)
so that the result follows by mathematical induction.
Have fun, ZVK (Slavek).

In article <32857B1C.73EB@pl.jaring.my>,
Tang Chi Yan   wrote:
[...]
>Anyone there knows how to do this?
>
>  |\
>  |    x^2
>  | --------  dx
> \| x^4 + 1
[...]
Let s be a symbol for sqrt(2), then  
    x^4 + 1 = (x^2 + 1)^2 - 2 * x^2 
            = (x^2 - s * x + 1)*(x^2 + s * x + 1)
and then you apply standard partial fractions procedure (it is still a 
lot of symbol manipulation, but routine).
Good luck, ZVK (Slavek).

Let C be Cantor set.
can you find a subset D of R that is omeomorph with C, and such that
Lebesgue measure of D isn' t zero?

ksbrown@seanet.com (Kevin Brown) wrote:
>So here are the smallest null sums for 1st through 5th
>powers:
>  1+2-3 = 0
>  1+4-9+16-25-36+49 = 0
>  1+8-27+64-125-216-343+512+729-1000-1331+2744 = 0
                                           ^^^^
You mean 1728, of course.
>  1^4 + 2^4 + 3^4 + 4^4 - 5^4 - 6^4 - 7^4 + 8^4 + 9^4 + 
>             + 10^4 + 11^4 + 12^4 - 13^4 - 14^4 - 15^4 + 16^4  =  0
>  1^5 + 2^5 - 3^5 - 4^5 + 5^5 + 6^5 - 7^5 - 8^5 - 9^5
>      + 10^5 - 11^5 - 12^5 - 13^5 + 14^5 + 15^5 + 16^5 - 17^5
>             + 18^5 + 19^5 + 20^5 - 21^5 - 22^5 - 23^5 + 24^5  =  0
>Letting f(n) denote the least integer k such that at least one signed
>sum of the first k nth powers equals zero, we have f(0)=0, f(1)=3,
>f(2)=7, f(3)=12, f(4)=16, and f(5)=24.  What is f(6)?
Here are some additional results, in a more compact form:  S is
a subset of {1, 2, ... , f(n)} the nth powers of which have the
same sum as the nth powers of its complement.  [Note that my
solution for n=5 differs from yours.]
n  f(n)  S
1   3    {3}
2   7    {3, 5, 6}
3   12   {1, 2, 4, 8, 9, 12}
4   16   {5, 6, 7, 13, 14, 15}
5   24   {3, 4, 6, 8, 9, 11, 12, 16, 17, 18, 22, 24}
6   31   {1, 6, 7, 8, 11, 15, 18, 21, 24, 27, 29, 31}
7   39   {2, 4, 5, 7, 8, 11, 12, 13, 15, 22,
          23, 24, 29, 31, 32, 35, 36, 39}
8   47   {1, 2, 3, 8, 10, 13, 14, 15, 16, 17, 18, 25,
          29, 31, 32, 33, 36, 38, 39, 40, 42, 43, 47}
9   44   {3, 5, 9, 10, 14, 19, 20, 21, 25, 26,
          28, 31, 35, 36, 37, 38, 40, 41, 42}
10  60   {2, 3, 4, 11, 13, 14, 17, 24, 25, 26, 32, 35,
          38, 39, 41, 42, 43, 46, 48, 49, 58, 59, 60}
So f(n) isn't monotonic; but its growth appears to be roughly
quadratic.
--
Fred W. Helenius	

I'm taking a course in category theory, and the term 'natural
bijection' often pops up. But I can't seem to find a nice definition
of what is meant with the phrase. 
Let's say we've got two categories, C and D, with a bijection (*not* a
functor, just bijection) F between the arrows/morphisms of the two
categories. Then the idea of the 'naturalness' seems to mean that the
bijection (much like a functor) preserves the identity of the arrows
under arrow-composition. More formally, (the way I understand it, in
any way!) for
   p: X1-->Y, q: X2 --> Y arrows in C, that is we've got the diagram
	   p	            q
   X1------> X2 --------> Y
the 'naturalness' of F means that
  F( q o p ) = F( q ) o F( p )
for all arrows p and q in C. (where the notation F(p), for p an arrow
in D, refers to the arrow in D that is associated with p under the
bijection F)
But this only makes sense when there is a 'total' (don't know the
proper term bijection F associating *all* the arrows in C with arrows
in D (and of course visa vers). What if there is only a subsection
(also not a formal term) of the arrows in C that is bijected with a
subsection of the arrows in D? For example, what if we've got the
following:
  For X a specific object in category C and W a specific object in
category D, let F be a bijection mapping arrows (in C) to X to arrows
(in D) to W. 
Now what is meant with the phrase 'let F be 'natural bijection in X' ?
What bothers me is that the bijection is only defined on arrows with X
and W as codomains (in C and D respectively), so the above definition
is not really of much use.
I have an idea that the 'naturalness' of a bijection is only of
relevance when one also has a functor defined from C to D, and that it
has got something to do with the interactions between the mappings of
the bijection and the mappings of the functor.
  Thanks a lot for any tips.
  Judy Retief
  judy.retief@epiuse.co.za

Ralph Huff wrote:
> 
> At what rate and at what upward angle must a golf ball leave the driver
> in order to travel 200 yards before it touches the ground? How about
> 300 yards by a profesional contestant who is strong enough?
> Can you figure it out?
This is somewhat of an easy solution.  Disregarding you're normal
mathematical concepts, and looking at the question as being more physics
related, there are two angles that will give you the same displacement
in the x-direction.  This displacement is given by the equation:
	Dx = [(v0)^2*sin2@]/g
ThereFORE ;), substituting the given distances of both 200 and 300 yards
separately into the afore mentioned equation, and using an arbitrary
angle, of say 30 degrees (which is a good approximation for a regular
drive):
	The 200 yard drive would require having an initial velocity of 47.5732
m/s at an angle of 30 degrees to the horizontal.  The 300 yard drive
would require having an initial veloctiy of 58.2651 m/s at an angle of
30 degrees to the horizontal.  However, there are numerous solutions
that entail different angles to the horizontal which thus change the
required velocity.  Thus, it can be concluded that the required
component of velocity is dependent on the angle that the ball is driven
to the horizontal axis.  Hope this answer is sufficient!!!
		P.T

Greetings, John. 
I believe your sum of squares sequence is an integer 
at least up to denominator 19. 
Don. 
: From: mckay@cs.concordia.ca (MCKAY john) 
: Newsgroups: sci.math  
: Subject: Re: Law of small numbers  [ Sequence problem.] 
: Date: 25 Oct 1996 08:58:28 GMT  
: Organizn: Computer Science, Concordia University, Montreal, Quebec 
: Message-ID: <54pvfk$1co@newsflash.concordia.ca> 
: 
: a[0]:=1;a[1]:=1; for n>=1, a[n+1]:=sum(a[k]^2,k=0..n)/n 
: What is the smallest value of a[n] not an integer?  
 ...  
 #Don. 1, 1, 2/1, 6/2, 15/3, 40/4, 140/5, 924/6, 24640/7, ?, ? 
 --   REM  5*11*13 *128 *829*3527 = 2.67593772E11 
rem > .smallnos.block1 
    Prog "donmcd.calc.profile.smallnos.anann/n2", 8.11.96, 20:40 pm. 
    multiprecision Integer multiply/divide, 
              SEQUENCE..  (mean sum of squares..) 
              #####################   
    an*(an+n)/(n+1) = 2*(2+1)/2,  3*(3+2)/3,  5*(5+3)/4 , ... 
    678e5319  /18 =  integer   
    1e10641   /19 =  074,767,494,499, .. = integer.**   
    REM FOR count = 1 TO 16..     cont.    
                      SEQUENCE.   
                      *****   
    6e0 /2  cofactor = 003,  remainder 0   
                       **** 
    15e0 /3  cofactor = 005,  remainder 0   
                        *** 
    40e0 /4  cofactor = 010,  remainder 0   
    140e0 /5  cofactor = 028,  remainder 0   
    924e0 /6  cofactor = 154,  remainder 0   
    24e3 /7  cofactor = 003,520,  remainder 0   
**24,000+       
    12e6 /8  cofactor = 001,551,880,  remainder 0   
**12,000,000+.       
    2e12 /9  cofactor = 000,267,593,772,160,/cont. /   
      remainder 0        ********************** 
                         shown by previous email. 
    71e21 /10  cofactor = 007,160,642,690,122,/cont. /   
    633,501,504,  remainder 0   
    51e42 /11  cofactor = 004,661,345,794,146,/cont. /   
    064,133,843,098,964,919,305,264,116,096,  remainder 0   
    21e84 /12  cofactor = 001,810,678,717,716,/cont. /   
    933,442,325,741,630,275,004,084,414,865,420,898,591,223,   
    522,682,022,447,438,928,019,172,629,856,  remainder 0   
    3e168 /13  cofactor = 000,252,196,724,522,/cont. /   
    541,410,812,579,097,160,361,769,690,288,016,260,160,445,329, 
059,280,814,119,090,545,214,689,630,356,941,057,966,683,002,706, 
243,045,823,080,829,174,307,488,383,892,818,892,448,294,858,132, 
739,063,481,087,616,  remainder 0   
    63e333 /14  cofactor = 004,543,084,847,135,/cont. /   
    617,156,827,912,757,388,745,495,368,280,304,614,840,838,595, 
881,866,184,916,862,484,568,611,323,251,237,466,514,400,298,666, 
035,505,442,921,252,351,741,476,408,723,848,134,651,603,051,301, 
482,732,426,317,279,967,824,692,473,590,004,311,667,893,669,816, 
825,659,451,437,945,264,179,017,683,087,851,534,875,769,612,520, 
430,358,764,794,779,896,148,056,440,854,180,983,745,517,181,150, 
456,925,894,414,444,744,263,895,766,823,050,176,  remainder 0   
    20e666 /15  cofactor = 001,375,974,661,884,/cont. /   
    883,593,958,307,872,179,178,705,368,405,647,167,653,708,277, 
552,968,028,912,473,812,609,102,217,376,850,169,111,765,710,515, 
773,835,599,302,526,014,250,241,161,260,301,717,399,820,768,970, 
184,134,876,804,192,150,886,854,801,356,030,366,186,182,424,436, 
442,996,889,021,994,405,615,153,205,582,903,467,344,478,792,123, 
106,618,114,102,235,565,032,186,504,968,849,726,350,885,599,336, 
533,205,922,352,410,334,758,695,941,004,389,844,907,248,384,651, 
429,484,082,872,799,202,174,077,593,625,353,904,681,667,598,612, 
880,451,668,712,691,329,300,699,545,740,348,437,987,114,107,645, 
533,752,280,969,067,680,573,611,264,214,603,078,561,454,961,628, 
228,293,064,187,407,527,797,705,091,082,562,348,153,851,826,709, 
309,999,245,224,806,527,712,609,283,567,103,740,647,842,272,786, 
590,546,895,644,500,236,443,431,510,130,695,525,780,542,823,497, 
682,908,835,486,798,511,724,130,649,088,896,  remainder 0   
    1e1332 /16  cofactor = 000,118,331,641,884,/lines deleted cont. /   
    068,353,612,205,856,207,165,427,016,  remainder 0   
    14e2661 /17 ** 
    14.,,,, x 10^2661      / 17. ##  equals INTEGER. 
    cofactor = 000,823,669,263,002,/cont. lines deleted/ 
708,015,852,736,  remainder 0   
    Product is quite big.  #   
    I did not multiply any more. ### In this program.   
    Except by modulo arithmetic.  
    In program anann/n1, I multiplied as ##   
    far as denominator = 19.  ##   
    I looked to see if the next factor 18, 19, 20, ..., was  
(hopefully going to come up in time)  spare.#   
    18  cofactor = 000,045,759,403,500,/cont. lines deleted/   
    706,000,880,707,  remainder 10   
    count =17, n2 = 10*27 = 0 MOD (n1) Prospective = 18  cont.   
                    10*(10+17) 
                    *****************   
    i.e. remainder*(remainder + 17 )  MOD  (count +1) =? 0 
    =========================================================   
    Denominator 18 is going to be o.k.# 
    Verified by multiply not shown here. 
    19  cofactor = 000,043,351,013,842,/cont. lines deleted /   
    721,474,518,565,  remainder 1   
    count =18, n2 = 1*19xxxx = 0,MOD (n1) Prospective = 19  cont.    
                    ******************   
    Factor 19 is going to be o.k. to divide evenly !##? NO!  NO! 
    Should be 1*(1+17) MOD 19.. <> 0  xx 
    However Proved by multiply, not shown = 
    1.,,x 10 ^10641  /  19  = INTEGER. 
          ********************** 
don.mcdonald@welcom.gen.nz 

darla@accessone.com (Darla) writes:
> You cannot judge the importance or the value of a thing by the number and
> weight of the tools needed to produce or complete it.  It takes a lot of
> heavy equipment to haul garbage, but only a heart to fall in love.  And
> which is the greater endeavor? Which has more directly enhanced the lives
> of men and the survival of the planet?
Well, how long do you think you'll continue to love if your love
refuses to take the garbage away?
And how long do you think you are going to *live* if the garbage keeps
piling up in your rooms?  Can be pretty unhealthy.
-- 
David Kastrup                                       Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de         Fax: +49-234-709-4209
Institut fuer Neuroinformatik, Universitaetsstr. 150, 44780 Bochum, Germany

numtheor@tiac.net (Bob Silverman) writes:
> lv54308@ltec.net (Lyle VonSpreckelsen) wrote:
> 
> >Three Pipes supply an oil storage tank.  The tank can be filled by
> >pipes  A and B running for 10 hours, by pipes B and C running for 15
> >hours, or by pipes A and C running for 20 hours.  How long does it
> >take to fill the tank if all three pipes run?
> 
> I will give a hint:
> 
> Think "harmonic mean". 
I'll offer you half of that.
-- 
David Kastrup                                       Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de         Fax: +49-234-709-4209
Institut fuer Neuroinformatik, Universitaetsstr. 150, 44780 Bochum, Germany

Nick Halloway (snowe@rain.org) wrote:
: 
: This brought up the question:  are there some conditions on homology 
: groups for a space such that a fixed point theorem holds?  If a 
: space is contractible, does a fixed point theorem hold, i.e.
: for f continuous: X --> X, does X have a fixed point? 
: 
No. Take X:=]0,1[ = {x\in R | 0 X,  x |---> x^2.
d.A.

eun woo park (ewpark@undergrad.math.uwaterloo.ca) wrote:
: What is the difference between two concept other than the definition ?
: 
Closedness is a topological property, openness is a metric one.
: Can we have a set that is complete but not closed ?
A complete subset of a metric space is necessarily a
closed subset, containing as it does, its derived set.
: 
: Can we have a set that is closed but not complet ?
Yes. Take the rational numbers as your metric space
(with the usual metric) and it is closed but not complete
as a subset of itself.
: 
: Is closed set in complet set complete ?
Yes. The complete subsets of a complete metric space
are precisely the closed subsets.
: 
: Is complete set in closed set closed ?
: 
Yes
d.A.

"goldbach"  writes:
> Religion does not even require the assumption of a god. Religion is
> distinguished by the concept of faith which is the acceptance of
> a belief without reason to do so. It is the believing ' in ' something
> being true as opposed to believing ' that ' something is true. The 
> latter points to some kind of reason.
> Religion can rot your whole life if you let it get a hold on you.
Yes, it might keep you awake at night after you killed some stupid
neighbour of yours.
-- 
David Kastrup                                       Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de         Fax: +49-234-709-4209
Institut fuer Neuroinformatik, Universitaetsstr. 150, 44780 Bochum, Germany

>: First, I recently posted a disclaimer to two of my earlier postings,
>@> but now I want to disclaim the disclaimer. :-) Someone had posted
>@> saying that const qualification (in C) does not convey non-alias
>@> information, because the const qualifier guarantees only that no
>@> attempt will be made to write to the underlying addresses *through the
>@> const-qualified pointer*.
C9X, the revision of the C language that we (the C standards committee)
hope to have done by year end 1999, has added a new type qualifier: restrict.
It applies to pointers and indicates that there are no aliases to the 
pointed at object (unless the programmer gives up that property by doing
an assignment of the restricted pointer to another pointer).  In effect,
a restricted pointer acts like a pointer to malloc'ed memory.  A good place
to use restricted pointers is the parameters to a function.  The compiler
vendors who have already implemented this new feature report great performance
improvements.
Fred Tydeman          +49 (7031) 288-964  Tydeman Consulting
Meisenweg 20          tydeman@tybor.com   Programming, testing, C/C++ training
D-71032 Boeblingen                        Voting member of X3J11 (ANSI "C")
Germany               Sample FPCE tests:  ftp://ftp.netcom.com/pub/ty/tydeman

Hi,
	Does anybody know what should be the constraint to force a bivariate
implicit polynomial to be closed (bounded) ?
	(e.g. x^2+y^2-25=0 is a closed one since it represents a circle).
	Thanx...

Does 0.99... equal 1? I should think so, but can anyone confirm this?
	Dieter

jday@csihq.com (John Day) wrote:
>Can anyone recall why (in Abstract Algebra) the compositions of mappings
>of sets into their permutations are called  "symmetric" groups? Since each
>element of the group is a permutation, why not call it a permutation group?
We say a function F(x_1, ..., x_n) is symmetric if we have
f(x_1, .. , x_n)=f(x_1s, ..., x_ns) for all permutations s.
The "symmetric" (like "alternating" in the corresponding case) has become 
detached from the functions and attached to the group of all the 
transformations which preseerve this "symmetry". So it's not ulike what 
the grammarians call "transferred epithet".
But Rule 0 of mathematics is "don't think the name is the definition".  
********************************************************
* Dr W B Stewart            phone +44 1865 279628      *
* Exeter College            fax   +44 1865 279630      *
* Oxford                                               *
* OX1 3DP                                              *
* UK                        home            60629      *
********************************************************

"Jack W. Crenshaw"  wrote:
>Here's what I have so far, mathematically.  If the resistor connects two 
>_INTERNAL_ nodes (neither at power or ground), the solution resolves into 
>finding the determinant of the matrix:
>
>     [ a+z   -z                 ]
>     [ -z    b+z    other stuff ] 
> P = [ ...                      ]
>     [   other stuff            ]
>     [                          ]
>
>where z = 1/R. The matrix is symmetric, with the off-diagonal elements 
>The question is, what's the determinant of the matrix?  From the presence 
>of z in the four terms shown, it would seem that we could get quadratics 
>in z.  However, each time I try it with an example, I find the 
>determinant is only linear in z, which would prove my conjecture.  
>Somehow, the z^2 terms are cancelling.
>
You can see what you are asserting is true if you perform the Laplace 
expansion of the determinant in terms of the first two rows; you know,
the sum of all the 2x2 cofactors from the top two rows times the 
corresponding (n-2)x(n-2) cofactor.
[see any ancient book on determinants, such as Aitken's Determinants and 
Matrices.
********************************************************
* Dr W B Stewart            phone +44 1865 279628      *
* Exeter College            fax   +44 1865 279630      *
* Oxford                                               *
* OX1 3DP                                              *
* UK                        home            60629      *
********************************************************

(Ilias wrote: ...)
very interesting comments, for instance about "Henkin witnesses", "Skolem 
functions" and the "Herbrand expansion", thanks. 
Meanwhile I wonder how to `format' those posts - anybody still with us?
At least I see that the algorithm I posted earlier was flawed in not 
relating the constructed names in Def( S ) to each other properly (they were 
all equivalent).
So I `just' post it again - corrected (sorry for this `dragging game' :),
but without having done a lot of reading (sorry again). 
Nevertheless, trying to address the given and `anticipated' comments:
>    By the way, this model will not necessarily be recursive... or even
> recursively enumerable.  And of course this only works if S is consistent
> to begin with!  
Where >>in the given algorithm<< is something (I suppose the maps/selections 
or instanciations) not recursively enumerable - >>except<<, of course, `you' 
having to provide `your list of acceptable names' and `criteria of 
acceptibility' to the algorithm?
> The problem is that you cannot tell
> whether  Ex Ay Ez R(x, y, z) holds or not if you only have some designa-
> tion for x; you need the entire universe U that x, y, z range over, and
> some relation R_U(.,.,.) (i.e. subset of U x U x U) to interpret the re-
> lation symbol R.  There might be many such ; there might be none.
I'm sure that I don't understand fully what this implies, however:
That's what pointed me to the `correction' - to instanciate U using all 
`names' already accepted in Def( S ) and to `construct' new ones enforcing 
consistency by check - accept or discard. Of course S and the `universe of 
theorems' U it `creates' remain the same throughout.
Algorithm (to construct Def( S ) from a universe U( S ). (Z denotes sets): 
0) Identify `initial operation axioms' - having one All-quant. or ~Ex. 
For P those are: Succ( x ), Add_0( x ), Mul_0( x ). From them collect the 
`constructive non-tautologies' in Z_C, for P this would be {Succ( x )}.
1) Select `initial number-names' Z_Ni (not necessarily distinct) and a 
bijective map to all i elements of Z_C (both the argument and the `result').
2) Form the `initially instanciated universe of S' IUi replacing the free 
variables of All-quant.s in U( S ) successively with Z_Ni-elements in all 
possible ways, producing the sequence U( S ), IUi( 1 ), ..., IUi( i ) = IUi
(please, read on :)
3) Check for `typographical inacceptabilities' in IU e.q. for an expression 
"A = B" and an expression stating that A <> B resp. ~Ex | "A( x ) = B( x )" or
any (of a rec. enum. set of) absurdities. In case try another selection (1)
resp. (5) with `more different names', otherwise Def( S ) = Z_N
4) Identify more `generating operations or constructive non-tautologies' from 
the IUk( k ) resp. IUk( k - 1). Those are expressions of the form:  
Ex O_yz( x ), or Ax ~O_yz( x ) where O_yz is an `symbolic operation' with 
argument x (or several, in case of Eu...Ex O_yz( x )) and instanciated 
variable(s) y (, z). Include new distinct ones in Z_C.
5) Select `number-names' Z_Nk (not necessarily distinct) and a bijective map 
to all k elements of Z_C (both the argument(s) and the `result', where 
applicable).
6) Form the `instanciated universe of S' IUk replacing the free 
variables of All-quant.s in U( S ) successively with Def( S ) elements in all
possible ways, producing the sequence U( S ), IUk( 1 ), ..., IUk( k ) = IUk
- and continue 3) - 6) (please, read on :) 
Comments:
Definitly, 3), 4) and 6) are `critical' wrt. the `size of the entities in
question', however they are part of an iterative algorithm which should have
no problem to start. The `technique of number-generating operations' is 
(obviously) modelled after the Peano Axioms (for which I have a reference :), 
but it seems powerfull. The `typographical' check to decide consistency also 
relies on rec. enum. criteria.
Thanks again for your continuing interest,                     Frank  W ~@) R

Posit a truly random  expansion of integers infinitely to the left and 
right of (for instance) a decimal point. Remove the decimal point.
Now posit a second such expansion also truly random.
The Polaner hypothesis states that these two infinite random expansions 
are identical! Reasoning that either must contain the other to any size 
without limit by the very definition of randomness.
Is this true? Has any one any clear pop of this?
The Polaner hypothesis states that there is only one infinite random 
expansion so expressed. That in some sense one defines the other.
I get dizzy thinking about it.  

In article <5672s1$8tj@news.ox.ac.uk> Brian Stewart  writes:
>From: Brian Stewart 
>Subject: Re: Where's the symmetry?
>Date: 11 Nov 1996 11:32:17 GMT
>jday@csihq.com (John Day) wrote:
>>Can anyone recall why (in Abstract Algebra) the compositions of mappings
>>of sets into their permutations are called  "symmetric" groups? Since each
>>element of the group is a permutation, why not call it a permutation group?
>We say a function F(x_1, ..., x_n) is symmetric if we have
>f(x_1, .. , x_n)=f(x_1s, ..., x_ns) for all permutations s.
>The "symmetric" (like "alternating" in the corresponding case) has become 
>detached from the functions and attached to the group of all the 
>transformations which preseerve this "symmetry". So it's not ulike what 
>the grammarians call "transferred epithet".
>But Rule 0 of mathematics is "don't think the name is the definition".  
>********************************************************
>* Dr W B Stewart            phone +44 1865 279628      *
>* Exeter College            fax   +44 1865 279630      *
>* Oxford                                               *
>* OX1 3DP                                              *
>* UK                        home            60629      *
>********************************************************
Probably the group of  a l l  permutations on n objects is called the 
symmetric group because a function of these objects is symmetric exactly if it 
is invariant under  a l l  permutations, that is, invariant under the 
symmetric group. For example, the determinant of n vectors is not a 
symmetric function because it changes its sign under an odd permutation. 
It is, however, invariant under the group of all even permutations, also 
known as the alternating group. Quite consistently, the determinant of n 
vectors is called an alternating function of these vectors. There are many 
permutation groups different from the symmetric groups; in fact, all subgroups 
os symmetric groups are permutation groups. So there is a definite difference 
between the two concepts. 

In article <3284C8AB.71C8@cdf.toronto.edu>, Peter Kanareitsev  writes:
> Does anyone know of a real-world situation where a tensor product of
> vector spaces occurs "naturally" (apart from quantum mechanics)? Thanks.
Any time you build a multidimensional functional space out of one
dimensional ones.  Surface fitting is a common one; many 2D spline
spaces are tensor products of 1D spline spaces, for example.
---------------------------------------------------------------------------
Tim Hollebeek         | Disclaimer :=> Everything above is a true statement,
Electron Psychologist |                for sufficiently false values of true.
Princeton University  | email: tim@wfn-shop.princeton.edu
----------------------| http://wfn-shop.princeton.edu/~tim (NEW! IMPROVED!)

In article <328778BC.6657@math.unifi.it> Biblioteca matematica  writes:
>From: Biblioteca matematica 
>Subject: measure
>Date: Mon, 11 Nov 1996 11:04:28 -0800
>Let C be Cantor set.
>can you find a subset D of R that is omeomorph with C, and such that
>Lebesgue measure of D isn' t zero?
From: flor@email.kfunigraz.ac.at (Peter Flor)
Subject: Re: measure
Date: Mon, 11 Nov 1996 09:24:00 LOCAL
Keywords: Cantor sets
Of course you can, and that�s well known. The Cantor set is constructed by 
removing "middle thirds" infinitely often, and the resulting set has measure 
zero because (1-1/3)*(1-1/3)*... = 0. All you have to change is this infinite 
product; replace it by one that converges (i.e. to some positive number).

Thank you

Kralor (ms-drake@students.uiuc.edu) wrote:
: Please try not to bruise me...I'm just a naive college student who's 
: curious.  I recently started reading about set theory and all the 
: contrivances that are used to eliminate paradoxes such as Russell's.  I 
: was just wondering if the entire situation could be resolved by an axiom 
: which doesn't allow sets to be members of themselves, or does this lead 
: to other problems?  Thanks for any help--
If you would not allow sets to be elements of sets, the power set
of a given set wouldn't be a set. Still further and because of that,
you could not form e.g. the cartesian product of two sets A and B 
(which is a subset of the power set of the power set of AUB),...
I hope this is of some help.
Franka M.Brueckler

In article <55v90f$etp@rzsun02.rrz.uni-hamburg.de>
fc3a501@AMRISC04.math.uni-hamburg.de (Hauke Reddmann) writes:
> Anyone interested in p-adics may try the American
> Mathematican Monthly (Aug-Sep 1996) for a nice
> article on the theme.
> (I would have taken this into email - if Pu's
> email account hadn't been mailbombed into oblivion)
Thanks, read that article. The preamble reads:
" You can sum some of the series some of the time 
and some of the series none of the time...
but can you sum some of the series all of the time?"
Mathematicians up to the day of this writing still think the Naturals =
Finite Integers is precise and is mathematics. That system of belief
would be analogous in physics if all physicists believed that Newtonian
Mechanics was modern day physics.
Naturals = Infinite Integers = p-adics is the truth just as modern
physics is based on Quantum Mechanics, not the older and imprecise
Newtonian Mechanics.
Trouble with mathematics is that the community is far lazier than is
the physics community. Those guys can ignore and keep doing wrong
things. Whereas in physics, experiments prompt change and prod that
community into action.
What is nice about the day when p-adics are found essential in physics
is that mathematics will then be linked to science experiments, just as
the physicists are linked to every physics experimental result.

T.Moore@massey.ac.nz (Terry Moore) writes:
> In article <55okv5$t7t@ccshst05.cs.uoguelph.ca>, devens@uoguelph.ca (David
> L Evens) wrote:
>    The idea of an empty set is quite 
> > well defined.
> 
> Yes please! I would love to see your definition of the empty set.
Doubtless.  But he only stated the *idea* of an empty set is quite
well-defined, not the empty set itself.
Now the idea of an empty set is defined as the idea of a set full of
useful things I had in mind when doing this post.
I have no idea whether other definitions might apply as well.
-- 
David Kastrup                                       Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de         Fax: +49-234-709-4209
Institut fuer Neuroinformatik, Universitaetsstr. 150, 44780 Bochum, Germany

From svk@neva.ru Sun Nov 10 16:03:45 1996
From: Sergey Khruschev 
                 A REPORT FROM GENERAL MEETING
           OF DEPARTMENT OF MATHEMATICS OF RUSSIAN
                    ACADEMY OF SCIENCES
                    (October 31, 1996)
                     SERGEI KHRUSHCHEV
The meeting has been chaired by Voevodin - a Buro member.
First Faddeev presented his account for 5 years.
When Faddeev finished I was given a word.
Below is a summary of what I said.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
                 A SUMMARY OF THE TALK
                  Sergei Khrushchev
The decisions of Buro which I complained included:
1) Approval of USTAV of UCHREZHDENIE POMI shortly after the illegal
decision on stopping my membership in Buro;
2) Publication of the announcement in "POISK" on the elections of 
Director of UCHREZHDENIE POMI with unlimitted rights on undetermined
term from ONE CANDIDATE chosen by BURO headed by Faddeev;  
3) Approval of financing unnecessary expenses of POMI from the budget 
of the Department;
4) Inactivity of Buro concerning financial violations in POMI discovered
by authorized organizations.
The main reason of my disagreement with Buro was different
understanding of the aimes of our Department.
>From my may be too patriotic point of view international scientific
programs in Russia are important for russian mathematics and should
be run effectively with economy of budget money.  
>From the point of view of academic-secretary (= L.D.Faddeev) it is better
to work abroad and budget money of Department should be used for
this very goal.
Probably because of this approach round 40 of my colleagues left POMI and
got work in different countries, beeing left in the POMI staff however.
In addition a group of round 30 people headed by academic-secretary
spends abroad approximately 70-80% of time. 
On a recent meeting in Smolnyi (City Administration in St.Petersburg)
an official representative of Elzyn in St.Petersburg (former
mathematician Sergei Cyplyaev) said to the audience of heads and
professors of St.Petersburg Universities that almost all leading
mathematicians left POMI.
Long beeing abroad led Faddeev to complete loss of the feeling
of economical reality which is so important for surviving in
modern Russia. This is also the main reason for leaving by
Faddeev the Justice Space of Russia.
I am not going to discuss illegal orders of former
Director of liquidated Euler Institute. All of them are complained
by me to the court and I believe that the justice will be finally
restored.
The subject of my present talk is rather ordinary for our time but is not
so usual for academic community, since it leaves the field of academic
activity. I will talk on juridical aspects of activity of our
academic-secretary.
It is already known that on September 19, 1996 the court abolished the 
decision of Buro to stop my membership. This not so important event
however well illustrates the juridical competence of our academic-secretary.  
Unfortunately violations of Laws may lead him to more serious consequences.
Let me recall that after taking the decision on reorganization of
St.Petersburg scientific institutions many documents were issued
by Presidium of RAS. They finally led to adoption by Department of
Mathematics and by Presidium of a "new edition" of USTAV of 
UCHREZHDENIE RAS "St.Petersburg Division of V.A.Steklov Institute"
(such a combersome name was chosen for POMI).
I present only few, the most odious, statements from this document.
1. Uchrezhdenie is created in correspondence with permission of
the Council of People Komissars on January 4, 1940.
2. Uchrezhdenie has a right for external economic activity in
correspondence with its requirements.
3. The property of Uchrezhdenie is in the disposal of its Director
who appoints a REVISION COMMITTEE and determins the way it must work.
Soon after approval of this Ustav a new Uchrezhdenie had been charged by
the Main Treasury Office of St.Petersburg on the sum equal to $3000 for
using budget money in wrong way. Notice that this charge equals to
the sum of charges of all other checked academic organizations in
St.Petersburg (round 30).
Soon after completion in September of the elections announced by Buro 
in June of Director POMI from one candidate this very Uchrezhdenie
was visited by Taxes Police of St.Petersburg. Taxes Police found
at the Payroll Office of POMI several thousands of US dollars.
Since POMI is not a currency exchange office the limit of the
cash register is set to be zero. Whatever exceedes the limit is a
subject of confiscation and of a charge. The charge for such a
violation of the currency regulations is set to be 1000% of the
sum discovered. 
Even other violations will not be found the charge amounts several
dozens of thousands of US dollars.
In addition to the illegal financial activity POMI exploits a palace
on Pesochnaya quay, were the liquidated Euler Institute previously was
located, by making use of the economical methods of Brezhev's time.
It is enough to say that expences on the complex increased 2.5 times
and now are beeing covered from the salaries of scientists of POMI and MIRAN.
However even these expenses are not enough and debts on communal payment
reached 200 million rubles (= $40 000) for the SUMMER PERIOD. I think that for
actual staff of at most 40 people these are very big expences.
Nobody will cover them on the regular basis and this results in
arbitration court consideration.
Naturally all these events couldn't not affect international programs.
To begin with from 15 days they shortened up to 3-7. Next the number
of participants decreased tremendously. If in 1992-94 the Euler Institute
with its small staff accepted 300 foreign and 400 russian mathematicians
then in 1995-96 POMI decreased this ammount approximately 10 times.
At the present time POMI runs conferences by hiring commercial
organizations and pays extra commission which only increases the
expences.
I should also mention a very important point. On the last meeting of Presidium
of RAS Ustav of Uchrezhdenie was cancelled as not corresponding to 
Ustav of RAS. However it became clear that up to the last moment
POMI was considered in USTAV of MIRAN as its part. I consulted with lawyers
and they told me that any organization which wants to get its money
from POMI by using this fact may take MIRAN to solidary responsibility.
Notice please that now we are already talking about the sum which approaches
one half of billion of rubles (=$100 000).
Academic-secretary is a serious mathematician who is an author of more
than 200 scientific papers and have many awards. Cooperation with him
was fruitful for dozens of mathematicians who became his co-authors.
But his activity in the oprganization field turned out to be ruinous 
for the Department of Mathematics.
To conclude : To-day in the time of the market economy you actually make
choice not between a candidate with heavy financial violations of the Law
and other candidates but between a financial crash of Department of
Mathematics and its relatively save existence. Please take notice of this
alternative.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
At the end I handled a copy of my talk to the Chairman (Voevodin) and
said that I can defend every word of my talk at the court if
academic-secretary would make such a complain. After that another copy
was given to Faddeev. 
After my talk first vice-president A.Gonchar took a word and confirmed
that many things I talked about are correct. He told to the meeting that
USTAV of UCHREZHDENIE POMI was discussed on the last Presidium of RAS in
the presence of Deputy of State Duma Sergei Popov who is a member of
DUMA COMMITTEE on Laws and Justice reform. Sergei Popov described to
Presidium of RAS negative outcomes of this USTAV.
Next Victor Maslov took a word and said that all these financial
and juridical violations are not important because Faddeev is an outstanding
mathematician and he obtained many important results during last years
which Maslov beeing an expert in the field can confirm as such.
He added that Faddeev is not guilty. This is a result of activity of
Faddeev's deputies and advisors. He also said that now Faddeev was getting
better in administration because he learns how to do this.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
                         AN IMPORTANT REMARK
The idea of innocent Faddeev and bad deputies is already dead.
For example, deputy of Faddeev on organization questions in Buro is
Alexei Zizchenko. He is responsible for all documents Buro prepares.
In particular he is responsible for USTAV which not only was approved
by Buro but also was approved by a special order of Presidium signed
by vice-president of RAS Andreev.
Zizchenko was deputy of the following academic-secretaries:
I.M.Vinogradov, N.N.Bogolubov, A.A.Gonchar, L.D.Faddeev. This list shows
that he is an extremly competent administrator. He will never do things
without consent of his chief. Moreover he always shows the chief
possible poor results of wrong decisions. However if a chief insists he
will do what he is told. So it was unfair from the side of Maslov to put the
responsibility from the shoulders of Faddeev to the shoulders of
Zizchenko.
On the last Buro meeting where I could participate only by
the decision of the court Faddeev blamed for this USTAV Anatolii Oskolkov
who died last fall. This claim of Faddeev is another lie. Oskolkov also never
made a step without Faddeev's consent. Moreover Oskolkov died before
this USTAV was approved by Buro and Presidium.
Next, financial violations are the results of activity in the Payroll Office
of POMI of the former software engineer who at the age of 60-ies replaced
last summer, following the directions of Faddeev, a professional bookkeeper
with higher financial education who worked in POMI for many years.
Obviously this devoted to Faddeev software engineer and unexperienced 
bookkeeper would never dare to make a step without Faddeev's consent.
I cannot also agree with Maslov that Faddeev is getting better in
administration. Just to the contrary I would say that violations
of the Laws he makes are getting only stronger.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Buro didn't want to discuss the facts presented in my talk.
Therefore Voevodin claimed that there is a very important topic to discuss.
He said that Buro stopped my membership last fall. However 
after my complain to the court this decision was cancelled.
Now the Department of Mathematics may be charged 30 million
rubles (= $6000) for moral demage made by this decision of Buro
to Khrushchev. He told that Buro member Ildar Ibragimov would explain
more details.
Ildar Ibragimov said that the lawyers of POMI told him that General Meeting
of Department of Mathematics should approve a resolution saying that membership
of Khrushchev in General meeting was illegal starting from 1992 when
the General Meeting of RAS was formed. Moreover this resolution should be
approved by Presidium. If this wouldn't be done then they couldn't complain
the decision of a local court and moreover S.Khrushchev should be paid a
moral demage of $6000.
I took a word and tried to explain that we do not have time for
discussion of this nonsense. Whatever decision would be made 
it would have no influence for the court because it couldn't correct
the situation which existed at the moment of violation of the law when I
sent my complain to the court.
Moreover I said that no harm will be made to Department of Mathematics.
I said that in case I win the complain of Faddeev's lawyers I will demand
not Department of Mathematics but academic-secretary personally should pay
a moral demage.
However Buro members and especially Ildar Ibragimov insisted on
continuation of this stupid discussion.
The discussion followed occupied round 40 minutes. Many members of the
meeting however agreed that Meeting of Department shouldn't discuss
this topic because in one hour new Buro will be elected.
Academician Sergei Nikol'skii said that first in 1992 Buro members
claimed that creation of the Euler Institute was a very important
task, next they claimed that its liquidation is very important
and now they say that we should consider this juridical point.
Why we discuss this question and do not discuss the real reasons
for liquidation of the Euler Institute?
Academician Vladimirov said that members of the meeting do not
understand such subtle juridical matter and therefore a serious
mistake can be made. So he suggested that the topic should be
dropped. 
But Voevodin insisted. Voevodin mentioned that Khrushchev
provided a help to the enemy of MIRAN (that is to Professor Larry Shepp).
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
                             REMARK
This last claim deserves  a speciale discussion at the end of this
document.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Finally Voevodin insisted on voting and voted a new illegal decision
which I already complained. This was done in spite of protests of
such mathematicians like A.Vitushkin who demonstrated a very good
juridical knowledge by saying that this decision just cannot enter
the court because the decision of the first instance was already made.
He claimed that he does not want to vote this matter.
After a small break elections of a new academic-secretary and a new Buro
started. Candidature of Faddeev was put in the list first. Next candidature
of Ershov from Novosibirsk was included too by Lavrentiev. Lavrentiev with
reference to my talk said that in contradistinction to Faddeev
Ershov beeing a very good mathematician is a very good rector of Novosibirsk
University and therefore is very experienced from the administrative point
of view. Lavrentiev was interrupted by saying that this should be said 
later and shortly after that Ershov said that he did not want ballot
in this company again.
So Faddeev remained alone. Nontheless Sergei Novikov  emotionally
advocated for Faddeev and called him a hero. Sergei Novikov works
abroad and just does not understand the difficulties existing
in Russia now.  
Moreover he talked on the "achivements" of L.D.Faddeev with the
reforms at the Euler Institute (actually its liquidation) without
any study of the topic.
He claimed that after reorganization Euler Institute became more
open. This does not corresspond to the real state of things because
in reality the number of visitors decreased more than ten times.
Moreover a fence surrounding the building is at the place and
the doors are locked while inside the Palace there are only dogs
and security.
After that it turned out that Faddeev was elected from one candidature.
The result of election of Buro members  was the following.
All members remained in Buro with exception of Sergei Novikov who
just refused to be elected.
I consider this step of Sergei Novikov as a very principal step.
>From my point this shows some progress in understanding that
in difficult economic situation in Russia one should make
a definite choice between administrative and organizational
work in Russia and scientific work abroad. 
There were 45 members of General Meeting of Department of Mathematics.
On the other hand Buro incorporates 17 members. Since academic-secretary
should get at least 23 votes to be elected one can easily see that
if Buro members support academic-secretary he needs only 6 extra votes.
In Department of Mathematics membership in Buro has been always
considered as a honorable matter. However nowadays this is in a direct
contradiction with requirements of life. Department of Mathematics
needs a working Buro. All branches of mathematics and important institutions
should be present. This can be satisfied if Buro had say 10-12 people.
Now in spite of the fact that Buro includes 17 mathematicians
there is no Jewish mathematician in Buro.
The only candidate Arnold did not get enough votes.
This strange fact is in direct contradiction with what was
said by academician Victor Maslov on the Meeting.
As it is clear from the discussion scientific achivements of
Faddeev for last years were put higher than terrible consequences
of his administrative efforts.
In contradistinction to Faddeev Arnol'd has never had such
administrative achivements as Faddeev. On the other hand I hope
many people agree that research made by Arnol'd is of great
value for mathematics. So it is not clear why Department of
Mathematics cannot live (following the claim made by Maslov) without 
Faddeev as academic-secretary
but feels save without Arnol'd as a Buro member.
It is especially interesting if such a big care (following Victor
Maslov) was paid to the level of research of Buro members.
Perhaps the explaination roots in words said by not experienced
Chairman Voevodin. During the discussion on the court matter
he said that we (Buro members) operate in a friendly atmosphier
(=we are all friends) and take decisions almost unanimously.
Therefore we never take a look of Ustav and Polozhenie because we
just do not need them.
I can only ask what is a name for a group of "friends" violating Laws
and acting as officials on the basis of friend relations?
Sergei Khrushchev
PS Faddeev arived to the meeting from abroad and departed abroad
the very last day of the meeting.
Next day after departure of Faddeev our newspapers reported that
academician Vladimir Nechaev, the creator of a nuclear center
"Chelyabinsk-70", committed suicide because he as a Director
could do nothing to provide money for salary of his scientists... 
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
               ADDENDUM ON THE ENEMY OF MIRAN
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
SUBJECT: WHAT ARE RULES AND IS THIS FAIR?
I was given to understand that I am eligible to be considered for the position
of the next Director of POMI. Now I learn that the search for the position has
been closed and that my application was denied because my application
did not satisfy a certain procedural requirement.
No attempt was made to convey this procedural requirement to me and had
it been, I would have tried to fulfill it, possibly with success. I
demand that the search be reopened and that my candidacy be considered,
or that I at least be given time to solicit the necessary support.
People in Russia who know me well are aware of my deep interest in the success
of mathematics in Russia and also in the future of democratic reform in Russian
mathematics. This very example would seem to indicate how much Russia needs
such reform.
Lawrence Shepp
ENCLOSED IS A COPY OF PROF. SHEPP's APPLICATION:
To: Department of Mathematics, POMI FAX = 7-095-938-14-66
From: Lawrence Shepp, Columbia U, Dep'ts of Statistics and Operations
Research, ATT Bell Labs, US National Academy of Science
Subject: Directorship of the new organization in StP
I learned from an announcement distributed by Sergei Khruschev that the
Department of Mathematics has a new UCHREZHDENIE RAN "Sankt-Peterburgskoe
Otdelenie Matematicheskogo Instituta im. V.A.Steklova". I understand from
the announcement that the Directorship of this Institute is vacant and
that the Director need not be Russian or a Russian citizen.
As many know, I have been a frequent visitor to MIAN and POMI over the
years and maintain an ongoing interest in the Institute and especially
in the Russian mathematical community. I would like to be considered for
the Directorship. I have recently retired from Bell Laboratories and have
been actively seeking a new position. At the present time I have an offer
from Rutgers U. and expect an offer from Columbia U., but I will give serious
consideration to an offer for the POMI position as well.
I understand from Khruschev's announcement that the new organization is
not to be an Institute of RAS but an Uchrezhdenie. Please send me the
Charter of the new organization so that I can study the rules of the election
procedure and the job description of the new Director.
Please confirm the receipt of this fax by either return fax to this number
at Columbia U. or to the fax below at Bell Laboratories (where I continue to
visit on a regular basis).
Sincerely yours,
Lawrence Shepp
Lawrence A Shepp Columbia University; Statistics and Operations Research
212-854-3653    212-854-2941 shepp@stat.columbia.edu
ATT Laboratories Room 2C-374 Murray Hill NJ 07974-0636
908-582-3585 FAX = 908-582-2379  las@research.att.com
                    EXPLAINATION
>> From: Larry Shepp
>> A word of explanation as to my sincerity in seeking the position: I deem
>> it unlikely that I would accept the terms of any offer that might be
>> tendered by POMI RAN in any scenario even remotely consistent with
>> reality. I advanced my candidacy in order to promote the principles of
>> division of power in Russian mathematics which has long suffered from
>> abuses of power concentration. I have had a long history of involvement in
>> Russian mathematics and those who know me well realize my motivations are
>> genuinely on the side of the success of Russian mathematics. I have no
>> knowledge or opinion about the correctness of one or another side in the
>> current struggles in St. Petersberg over POMI and the Euler Institute,
>> but argue only that it is desirable to keep these institutions independent.
>> These opinions were expressed more fully in a letter (joint with AM Kagan)
>> see,  Notices AMS April 6, 1996 (the text is enclosed below).
>> I never received the courtesy of a reply to any of several successfully
>> transmitted faxes announcing my candidacy, even that they were received.
Letter to the editor, Notices AMS
Information published in Science (23 June 1995, p.1695) and received
recently from friends in Russia indicates that the Euler International
Mathematical Institute (EIMI) in St. Petersburg which for many years
existed as a separate unit similar to such institutes as the Banach
Center in Warsaw, Oberwolfach in Germany, and DIMACS, IMA, and MSRI in
the US may soon be subsumed into a department of the St. Petersburg
Branch of the Steklov Mathematical Institute (POMI), Russian Academy
of Sciences.
To the best of our knowledge, the decision to fold EIMI into POMI
resulted from a conflict between the director of EIMI, Ludwig
Faddeev (who is also deputy director of the Steklov Mathematical
Institute in charge of the St. Petersburg Branch) and his deputy in
EIMI, Sergei Khrushchev. Faddeev claims that folding EIMI into POMI is
in the best interests of St. Petersburg mathematicians while Khrushchev,
who is losing his job at EIMI, says that placing it under the control
of POMI will destroy EIMI as a center of international cooperation in
mathematical research.
While the final decision on the fate of EIMI rests in the hands of our
Russian colleagues, we mathematicians in the West also have an interest
in preserving EIMI and so have the right to at least make a suggestion
and to add some thoughts to the discussion. We would like to see EIMI
continue as a separate unit, with its own budget, run by an executive
director, a capable scientist and administrator, who, however, serves
at the pleasure of a Board of Trustees, which is independent of all other
institutions, in the Western style.
A number of Western organizations (the Soros Foundation, AMS, the
European Mathematical Union, etc.) are providing, in different forms,
financial support to the Russian mathematical community. It is thus
reasonable to have, say, one trustee from the United States and one
from Europe on the Board. Besides being instrumental in fund raising
for EIMI, the Western trustees will bring to the Board needed expertise
and independence.
Abram M Kagan, U Maryland, College Park
Lawrence A Shepp, Bell Laboratories, Murray Hill

On Sat, 9 Nov 1996, Joseph Edward Nemec wrote:
> 
> And that would be the limit for an ambitious, uncultured bourgeois moron
> from Newcastle. How sad...
Good job I'm not from Newcastle then.
> 
> Translation: I am not good enough at physics to get to the top.
> 
Are you not Joe? How sad.
to be honest with you though, life at the top isn't all that great. It
just means that you are researching slightly different things to other
people.
Of course, I get to have a nice well known subject such as the Higgs, but
that's about it.
> 
> Well, aside from actually getting a Ph.D. in it...
> 
I will hand in before I head off to the city. Probably. 
> 
> Soon to realize that you were duped...
Shit, man, you're right. I ought to instead have gone to some anonymous
institution. I'd havedone much better there, that's for sure. Then,
instead of ending up a particle physicist, I could have become an expert
in queuing theory. After all, it is THE fashionable subject of the day,
isn't it?
> 
> Soon to realize your country is second rate in that field...
Oops, we were ten years ahead of the field. Never mind.
> 
> Well, except for publishing distinguished work in the field...
Been there, done that. 
> 
> Please send me a copy of that report.
Please pay me 50 pounds, and I will send you a copy. You aren't getting
one for free, that's for sure.
You wouldn't understand it anyway. Peculiarly enough, it will be pretty
technical, requiring knowledge beyond degree level of high energy physics.
> 
> We don't think you are shallow. We just know that you will not make
> several million dollars per year.
No, all the people I know in the city are obvioulsy completely
unrepresentative of what's out there. I am completely deluding myself that
I will do the same as them.
Well, at least I'm happy in my ignorance.
> You are a failure at physics.
Of course I am, of course. How foolish of me to think otherwise. 
> 
> Anthony, I would LOVE to test you on your knowledge of the stochastic
> calculus...
Now why doesn't that surprise me?
> 
> Of course not: you are the sort of idiot who rails over the internet, and
> hides behind his keyboard.
That's right Joe. My boxing matches have all been carried out over the
internet. In fact, now I think back, they weren't boxing matches at all,
they were in fact just video games.
> 
> Failure. 
> 
You oughtn't to sign yourself that way. Hell, just because you aren't
going anywhere, it doesn't mean that your parents don't love you. And not
all of us can get a place on the high energy physics courses, so don't
feel too bad about yourself.
Anthony Potts
CERN, Geneva