Newsgroup sci.math 157166

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>Because the 168-wheel is a 100% guarentee.  If you absolutely,
>positively want to ensure at least a 3-win, the best way to do it is
>to use this wheel.
Are we under the assumption that the six winning numbers come from a smaller 
set? IOW, you're not talking about wheeling every number down to 168 tickets, 
right?
The statement you made above might be interpreted as stated, which being, "If 
you buy 168 tickets using this wheel, you are guaranteed a 3-win," when it 
probably should be, "If you buy 168 tickets using this wheel, AND x-amount OF 
NUMBERS APPEAR FROM THE WHEELED GROUP, you are guaranteed a 3-win."
Right?
Later!
ck

In article <5aeect$hij@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
> How many people can see that the Successor Axiom of the Peano Axiom
> System
> 
>   is the same identical Series 
> 
> as the Series of the definition of what a P-adic, (an Infinite Integer)
> is.
From           jpb@iris8.msi.com (Jan Bielawski)
Organization   Molecular Simulations Inc.
Date           Wed, 8 Jan 1997 01:38:44 GMT
Newsgroups     sci.math,sci.physics,sci.logic,alt.ascii-art
Message-ID     
References     1 2 3
In article <5aedhv$po1@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
< In article 
< jpb@iris8.msi.com (Jan Bielawski) writes:
< 
< > In article <141421356237310@einsteinium.universe> Carl Friedrich
Socrates Einsteinium  writes:
< > < In article <32BEC8B2.794BDF32@clipper.ens.fr>
< > < David A. Madore  writes:
< > < 
< > < > example, is one allowed to ask the question of whether Finite
Integers
< > < > satisfy Fermat's Last Theorem?
< > < 
< > < This is as ridiculous as asking how many angels can stand on
< > < a pin head, or how many letters will fit in a cubic centimeter.
< > 
< > Why is it ridiculous?  Tell us, does the number ...00001 exist?
< > What about ...0002 ?  If you can conceive such numbers (I have no
< > doubt that you can) then there CAN'T be anything "ridiculous" about
< > asking whether such numbers satisfy  a^n + b^n = c^n .
< 
<   Say Jan, I used to live in Coronado and could use some (free)
< pictures for my website autobio, of that Naval station situated
across
< from Hotel Del, a little ways down south of Del where I went to SWOS
< school. 
Is this the one on Silver Strand?
< I would love a picture of the waters from that Natl Monunment
< there? Cabrillo?  Also a picture of the Naval Base where all the
ships
< are docked.  If possible, send to Archimedes Plutonium, c/o Dartmouth
< College, Hanover NH 03755 and it will get to me.
Any P.O. Box or something?
< 
<   Good to see that you Jan are not one tracked, like a RR but instead
< can lift yourself up from off the track and go another direction, as
< per, Democritus Uranium and cubic cm.
Any reason why you didn't answer my question?
-- 
Jan Bielawski
You have asked the same question over for over 2 years now, I so no
reason to abruptly change that.
A. Plutonium, HB6165 , Dartmouth College, Hanover NH, 03755. If it were
a choice, my first choice is a picture of Cabrillo for that was my
favorite spot in all of San Diego. Please, a picture you made so that I
can post it to my web site without needing copyright permission. Yes it
was the Silver Strand but that picture is not as important as the
Cabrillo and if a hardship, please forget it. It is the Cabrillo which
is dearest to me, the only calm and relaxing place that I could find
while in San Diego and I want to compare it with the picture I have of
Acadia Natl. Park in Maine. Personally I liked Cabrillo better. That
would make a nice transcontinental bicycle trek someday, from Acadia to
Cabrillo or vice versa. Would a foucalt pendulum at Acadia and Cabrillo
swing the same?

In article <32d40dbe.0@news.cranfield.ac.uk>,
Simon Read   wrote:
>ags@seaman.cc.purdue.edu (Dave Seaman) wrote:
>>Although languages like Fortran and C do give 2^(1/2) = 1 (and so does
>>BASIC, if I remember correctly),
>
>I doubt it. Since you're not using exact FORTRAN, i.e. you're
>using (^) instead of (**), I assume you're not mentioning other
>details like (1./2.) is different from (1/2) because the former
>is floating-point whereas the latter is integer. If you do not
>mention this exact detail, you can't make blanket statements like
>"2^(1/2)=1" .
I tried to make it clear that I was talking about the semantics of the
expression, not the syntax.  I did write (expt 2 1/2) for the Lisp
version, but I didn't spell out the exact syntax in every case.
Nevertheless, Fortran evaluates 2**(1/2) as 1, and C evaluates
pow(2,1/2) as 1.  Try it and see.
>
>Every BASIC I have ever used has given 2^(1/2) as the square root of
>two to several decimal digits of accuracy. "Several" is a bit vague
>because some BASICs let you put a "#" sign to signify extra precision
>and not all BASICs store their default floating-point numbers to the
>same precision.
I will freely admit that I am not expert on BASIC, and the rules for
most implementations of that language are apparently not as I first
stated.
>C  I can't comment on.
>
>"languages like" FORTRAN and C ...  that includes Pascal.
>
>The original Pascal didn't even have a power statement/operator/function,
>so you had to do 2^(1/2) by taking logarithms or writing your own
>subroutine/function. This was clearly floating-point.
Leaving aside the issue of the exponent operator, I have mentioned
elsewhere that 1/2 has a different meaning in Pascal than in Fortran or
C.  
>It sounds like languages like FORTRAN and C do indeed give floating-point
>answers.
Then you obviously haven't tried it.
Fortran Expression	Value		C Expression		Value
__________________	_____		____________		_____
2**(1/2)		1		pow(2,1/2)		1.0
2.0**(1/2)		1.0		pow(2.0,1/2)		1.0
2**(1.0/2.0)		1.414214	pow(2,1.0/2.0)		1.414214
2.0**(1.0/2.0)		1.414214	pow(2.0,1.0/2.0) 	1.414214
This behavior can be predicted by anyone who understands what the
language standards say about the evaluation of mixed-mode expressions.
Note that the first of the four cases shown here is the only one I was
discussing previously, since there were no decimal points in the
expression 2^(1/2) that began this discussion.  Fortran gives integer
1, but C gives 1.0 of type double.
The second case is more interesting.  Many people assume that if at
least one operand is a floating-point number, then the entire
expression must be evaluated in floating-point mode.  Obviously, it
isn't so.  The parenthesized expression (1/2) is evaluated first, and
it evaluates to integer zero because both the operands are integers
(that's the Fortran/C rule, and it would also be the Pascal rule if it
weren't for the presence of "/", which always means floating-point
division in Pascal, regardless of the types of the operands).  Finally,
the mixed-mode expression 2.0**0 or pow(2.0,0) is evaluated, which
gives a value of 1.0.
-- 
Dave Seaman			dseaman@purdue.edu
      ++++ stop the execution of Mumia Abu-Jamal ++++
    ++++ if you agree copy these lines to your sig ++++
++++ see http://www.xs4all.nl/~tank/spg-l/sigaction.htm ++++
-- 
Dave Seaman			dseaman@purdue.edu
      ++++ stop the execution of Mumia Abu-Jamal ++++
    ++++ if you agree copy these lines to your sig ++++
++++ see http://www.xs4all.nl/~tank/spg-l/sigaction.htm ++++

In article <32D0BFB8.441F2774@maths.ex.ac.uk>,
Robin Chapman   wrote:
>
>I presume that A * C denotes what I would write as Tor_1(A, C). If
>so this statement is true for all rings R (not necessarily commutative),
>and all right modules A and B, and all left modules C. The functor
>A |--> A * C is additive, and all additive functors preserve finite
>direct products.
Theorem 8.10 of Rotman's "An introduction to homological algebra"
(ISBN 0-12-599250-5)
states that Tor_n preserves (even infinite) direct sum for all n.
Greetings,
Ørjan (preparing for his exam in the subject next Monday.)

Joseph H Allen wrote:
> 
> Here are some paradoxes:
> 
> -- black ravens
[snip]
> 
> Now the negation of "all ravens are black" is "all non-black things aren't
> ravens".  The two statements are logically equivalent.  Thus all of the
> non-black things you find which aren't ravens (your red coat, the white
> ceiling, etc.) also support your generalization that "all ravens are black".
> 
> Now this sounds silly, but it is actually logically correct.  If you lived
It's sure does.
Hmm.. 
Don't meddle with the quality but with the quantifier:
the negation of "all ravens are black" is "not all ravens are black",
namely I've found at least one that is NOT black (maybe it's brown).
:Luben:
--
Only wimps do the general case. True teachers tackle examples.
              (Parlett, see Math. Intelligencer, vol. 14, No. 1, p. 35)
Those who can - do, those who can't - teach.
              (as quoted by a very good calculus prof of mine)

Monte J. Zerger (zergerm@rmii.com) wrote:
: With all this focus on the upcoming millennium, I am curious what we
: will call the year 10,000.  Names for periods of 10, 100 and 1,000 years
: correspond to the prefexes deci, centi, and milli.  The next commonly
: used prefix in this sequence is micro, but this would correspond to a
: period of one million years.  What about 10,000 years?  100,000 years?
:
  The usage of prefixes differs for measurements: centi- means 1/100,  
milli- means 1/1,000 and micro- means 1/1,000,000 (submultiples).
  However, for periods of years, as you say, there are decades, centuries
and millennia.  There is no prefix I've heard of for 10^4 or 10^(-4), so
one is probably free to make them up.  I've seen "eon" used for 10^9
years.  For 10,000 years you might try "dekamillenium".  I'm not sure
whether "era" has been given a meaning for a fixed number of years in
addition to its usual meaning, but it might be a candidate for one or
another of the periods in question.  There's an period of several thousand
years called a "Julian cycle" -- it's in (I think) the calendar FAQ, which
might not be too bad a place to look...

In article ,
Michael A. Stueben  wrote:
>I was trying to list the different kinds of proof for my H.S.
>precalculus students. So I gave direct, indirect, math
>induction and proof by contraposition. Fine. But later I
>thought what about this: proof by example (offer am
>illustration of a situation or give directions for a
>construction)? Or proof by verification
>(substitute and complete a calculation). Then proof by cases
>could be a proof made up of a mixture of all types of proof.
>Well this is getting interesting. I classified the first four
>as the MAIN methods and the rest as MINOR methods. Are there
>more methods? Is all of this just really hair splitting and
>pedantry? Maybe, but its kinda fun.
>              +----------------------------------------------------------+\
>              | --From Michael Stueben: high school math/C.S. teacher    ||
>              |   collector of mathematical humor and education theories ||
>              |   E-mail address: mstueben@pen.k12.va.us                 ||
>              +----------------------------------------------------------||
>              \----------------------------------------------------------\|
Instead of this, how about teaching them the complete set of rules
of proof, and how to use them?  It is quite common that a proof will
use many of them. 
But proof by induction is only a method of proof in the integers, or
if one uses transfinite induction, for ordinals.  It is not a general
type of proof.  
The entire set of rules for formal proofs form a page or two.  Instead
of beating around the bush, just show them a good description of them.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (317)494-6054   FAX: (317)494-0558

Joseph H Allen wrote:
> 
> Here are some paradoxes:
> 
[snip]
> -- Zeno's paradoxes
> 
> Zeno's paradox of Achilles seems resolved.  For Achillies to cross some
> distance he has to first cross 1/2 then 1/4 then 1/8 and so on: an infinite
> number of steps.  You've had calculus in college and say that this is no
> problem at all: the infinite series 1/2 + 1/4 + 1/8... converges to a
> constant.  There are however, still a few problems.  The God Hermes joins in
Namely, converges to 1.  
1 + 1/2 + 1/2^2 + ... + 1/2^n = 1/(1 - 1/2) = 2 as n -> +oo
2 - 1 = 1.
> the race, only he goes twice the speed as Achilles but pauses after each
> step to allow Achilles to catch up.  Calculus will show that Hermes will
> arrive at the same time as Achilles, but he had to actually perform an
> infinite number of actions (starts and stops) to do it.
Achilles also has to "perform" infinte number of "moves".
> A similar demonstration of this problem is as follows:  Suppose you have a
> light with a single button: when you press the button, the light goes on and
> when you press it again, the light goes off.  Suppose you press the button
> after each fraction which Achilles crosses: at the 1/2 mark, at the 3/4 mark
> and so on.  Will the light be on or off at the end of the race?
As your mother told you: "Joseph, if you keep switching the light on and
off, you'll burn the light bulb..." - same applies here :-)
By the time Acchilles reaches his destination, the the light would've
been turned on and off infinite number of times and would be burned, so
it's OFF. :-) (hahhhahaahahaha)
:Luben:
--
Only wimps do the general case. True teachers tackle examples.
              (Parlett, see Math. Intelligencer, vol. 14, No. 1, p. 35)
Those who can - do, those who can't - teach.
              (as quoted by a very good calculus prof of mine)

In article <330c9216.56834577@netnews.worldnet.att.net>, mdmgyn@worldnet.att.net (Tom Robertson) writes:
|> > The laws of mathematics are eternal and changeless.  Like the laws of
|> > morality and physics, they can be discovered, but never invented.
It depends on what you mean by "the laws of mathematics".  Mathematics has
axioms and theorems.  Given a system of axioms, certain theorems follow.  
A fact such as "statement A is a theorem of system S" is eternal and changeless.
What can change is whether anybody is particularly interested in system S.
This is to some extent more a question of sociology than anything else.
The prevalent post-Godel viewpoint is that there is no single "correct"
system.  New axioms are, in fact, invented from time to time, and, if they
lead to interesting new mathematics and don't contradict too much old
mathematics, may eventually become generally accepted.  2+2=4 will never
change, but I wouldn't venture to guess about whether people will accept
the Continuum Hypothesis (or even consider it a relevant question) a few centuries from now.
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia            fax 822-6074
Vancouver, BC, Canada V6T 1Y4

Tom Robertson (mdmgyn@worldnet.att.net) wrote:
: I recently made the following statement, taking its truth for granted:
: > The laws of mathematics are eternal and changeless.  Like the laws of
: > morality and physics, they can be discovered, but never invented.
:
  Not invented?  What about axioms and definitions?  In what sense do you
claim the "laws of mathematics" `exist', outside of the realm of thought?
  And in what sense did these "laws", which are not claimed to govern
anything in the physical world, "exist" before they were invented?
  To take the example "2 + 2 = 4".  This contains abstract numbers (e.g.
2 and 4, as opposed to 2 or 4 of a particular thing); a binary operation,
"plus"; and the concept of equality.  All invented.  Did any of these
things "exist", say, even 20,000 years ago?  Or 20,000,000 years ago?

In article <5b098u$mjn@kodak.rdcs.Kodak.COM>,
Matthew Daly  wrote:
>It's a little wierd that the digits are loosely based on each other, like
>the symbol for 3 being three of the symbols for 1, but that not being the
>case everywhere, but that's par for the course for numbering systems.
Indeed it turns out that the shapes of the familiar digits 1,2,3
evolved from symbols not much different from I,II,III !
--Noam D. Elkies (elkies@math.harvard.edu)
  Dept. of Mathematics, Harvard University

In article <32D3ED92.31A6@cdf.toronto.edu>,
Luben Tuikov   wrote:
>Joseph H Allen wrote:
>> Here are some paradoxes:
>> -- black ravens
>[snip]
>> Now the negation of "all ravens are black" is "all non-black things aren't
>> ravens".  The two statements are logically equivalent.  Thus all of the
>> non-black things you find which aren't ravens (your red coat, the white
>> ceiling, etc.) also support your generalization that "all ravens are black".
>> Now this sounds silly, but it is actually logically correct.  If you lived
>It's sure does.
>Hmm.. 
>Don't meddle with the quality but with the quantifier:
>the negation of "all ravens are black" is "not all ravens are black",
>namely I've found at least one that is NOT black (maybe it's brown).
It was not stated that this was a negation, but an equivalence.
This is a standard one in the literature, but with crows instead of
ravens.  It is true that finding a white shoe, which is the canonical
object in the literature, does add to confirming the staement.
It is only when looked upon from a probability statement that the
apparent paradox resolves.  There are few ravens, and so looking at
one adds a modicum of information.  There are overwhelmingly many
non-black objects, so looking at one of them adds very little 
information.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (317)494-6054   FAX: (317)494-0558

newarktm@ix.netcom.com(O. ROYCE) wrote:
>CALCULATION OF pi 
>Diameter 1
>r = radius of circle
>sqrt.(r-sqrt.((r^2-(.5ans.)^2 =.195090322..     (x 16 = 3.12144515....)
There's not enough information here to understand what you're saying.
>Each side of an equilateral triangle inside a circle, diameter 1, will 
>easure ..8860254038. (half of sqrt.3).
>
>Using the above formula:
>
> sqrt.(r-sqrt.((r^2-(.5ans.)^2   = .5      x  6 = 3.00000
>   - ditto -                     = .258819 x 12 = 3.105828.. 
It would make sense if you:
(1) included enough brackets
(2) said what "ans" was, and why you are multiplying by 2, 4, 8, etc.
    or by 6, 12, etc.

Miguel Lerma wrote:
> I can see that you are posting from UK. I have spent three years
> in the USA (I am from Spain) and never imagined before the level
> of religious fanatism I would find here.
Your impression is undoubtedly biased by the part of our civilization you find yourself in. Our 
communities vary widely from those with a single community religous perspective, not always 
conservative Christian at all, to more mixed groups. Many work quietly to their own ends but quite a 
few push the limits seeking to force everyone else to accept their whisdom. In part this is a result 
of the great success that the athiests have had in terrorizing most school districts, controling 
public funding, property, etc. I am not at all sure how healthy this overall society is, but I think 
the cacophany (sp?) of public discourse is actually a good thing. I think that our press makes a big 
deal about the nut cases, the more bizar the better, so you really don't get the overall picture 
without a lot more careful study than I can give to it. While I can't speak to any direct personal 
experience with Europe, my impression is that they don't have any corner on the market for good 
sense, religous or otherwise. Remember that the US is a really big country with a lot of very 
different local communities. Ask anyone from the west coast what they think of Texas, the mid west, 
or (shudder!!) those XXX NewYorkers. Religion is not the only thing we have strong opinions on.

From: bm373592@muenchen.org (Uenal Mutlu)
>PROBABILITIES AND WHEELS, COVERING DESIGNS, LOTTO etc.
>
>Now, I made the following IMHO interesting observation: we've seen (cf.
>table below) that the probability for 6/49 type game using only 1 ticket 
>for AT LEAST 3 matching numbers is 1.86375% (= 1 in 53.6551)   and
>for EXACT    3 matching numbers is 1.76504% (= 1 in 56.6559)
Right.
>Ie. either playing the _same_ 1 ticket in 54 drawings, or equally simply 
>playing 54 _randomly_ selected _different_ tickets in 1 drawing, should 
>give AT LEAST once 3 or more correct numbers in both cases.
Not quite.  54 is the average number of tickets you need to buy in order
to expect 1 winning ticket.  Buying 54 tickets does not guarantee a win.
Actually, since there are 260,624 ways for a ticket to win, then there is
(1 - 260,624 / 13,983,816)^54 = 0.362068 probability of losing 54
consecutive drawings if you buy 1 ticket per draw.  And even after 168
draws, there still is 0.042398 probability of having lost all draws.
>Now, the connection to the wheels and covering designs: There exists 
>a wheel which assures _always_ at least once 3 correct numbers; it is 
>built up of 168 single tickets (IMHO it's the shortest known today
>for 6/49 which always guarantees a win (>= 3)).
True.
>So, the interessting question is: why should one ever play the 
>168 ticket wheel and not simply 54 randomly choosen different single
>tickets? IMHO mathematically spoken both cases should offer nearly 
>the same assurance and probability for hitting once or more at least 
>3 correct numbers. (this maybe not 100% correct, but you can imagine 
>what I mean).
But they do offer the exact same assurance!  See below.
>A further question arises: why does such a wheel have so many tickets,
>whereas the probability calculations show us that on average only 
>54 are needed for AT LEAST 3. (Ok, I also would accept 100 tickets 
>or so, but why even more than 3 times 54 ?!)
What you are forgetting Uenal is that the 168 ticket wheel guarantees
ONE OR MORE winning tickets.  If it always guaranteed EXACTLY ONE
ticket, then only 1 ticket in 168 would be a winning ticket, right?
So that would be worse than average!  And would prove wheeling is a
bad thing to do.
But, since 1 ticket in every 53.65514 is a winner, then 3.131107
tickets in every 168 should also be winners.  This is the theoretical
value obtained by simple ratio.  To prove it to yourself, all you
have to do is take the 168 ticket wheel and count how many tickets
win for each of the 13,983,816 possible combinations.  Add up all
those numbers and divide by 13,983,816 to get the average.  I'm
predicting that it will be 3.131107.
If you do come up with the same number, then it implies that wheeling
does not change, in any way, the average number of winning tickets.
And why would wheeling have done so, any way!?

John Edser wrote [in part]:
> It appears to me that you cannot have less than nothing (nought)
> but you can have less than zero. 
> Do mathematicians discriminate between these two concepts?
> [J.E. - ] strictly a non mathematician
Since I am myself `strictly a non-mathematician' allow me 
just the comment that in `real life' there is a very large number of 
>>different "Zeros"<<:
"Zero - the integer" 
(satisfying "X + Zero = X" and/or "X * Zero = Zero" with all integers "X")
"Zero - the real number"
(satisfying "x + Zero = x" and/or "x * Zero = Zero" with all real numbers "x")
"Zero - the n-by-n matrix"
(satisfying "M_nn + Zero = M_nn" and/or "M_nn * Zero = Zero" with all ...)
"Zero - the real function on the interval [0 ... 1]"
(satisfying "f + Zero = f" and/or "f * Zero = Zero" with all ...)
"Zero - the set" (in fact: "Zero - the empty set, or `naught'") 
(satisfying "S + Zero = S" and/or "X * Zero = Zero" with all sets S)
...
However, I think that mathematicians tend not to discriminate between
those concepts (if they can avoid it) since those "Zeros" are formally all 
the same - the difference is at most in how to understand the `algebraic 
symbols "+" and "*"'.
(And neither would I :).
Regards,                                                 Frank  W ~@) R

ags@seaman.cc.purdue.edu (Dave Seaman) wrote:
>Since
>(1/2) is in parentheses, and since 1 and 2 are integers, the value of
>(1/2) is 0 in Fortran, BASIC, and C.
No, no, no. Every BASIC I have ever used has evaluated (1/2) as a
floating-point quantity. One basic I know can be forced to do it
as an integer operation by writing  1 DIV 2  

In article <32D3EA4D.6DE4@mindspring.com>,
Richard Mentock   wrote:
>Herman Rubin wrote:
>> The oldest known system which uses the same symbols for multiples
>> of different powers of the base is the Babylonian base 60.  Some
>> have conjectured that the Hindus got the idea of using it for base
>> 10 from this.
>I understand that the Babylonian "digits" were base 10.  In other 
>words, instead of 60 symbols, they used 10, and their eleventh base
>60 digit was "11".  Can you verify?
What appeared in a position in a Babylonian number was a symbol for
an integral multiple of 10 with 0 usually not present, and one for an
integer from 1 to 9.  A zero integer was sometimes explicitly written,
and of course not putting it in could, and dkd, cause confusion.
So it could be looked upon as an alternation of base 10 and base 6,
using different symbols for the two bases.  The multiplication table
would have to be size 105 instead of size 45 as ours is.
>And if so, wouldn't base 10 then be as old as base 60 (if only 
>for powers 0 and 1)?
This seems to be the case.  
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (317)494-6054   FAX: (317)494-0558

Hello friends.
Do you know any facts concerning extension
of a real-valued function f: Cl(Q) -> R,
defined on closed subsets of a compact set Q,
to a Borel measure? To a regular Borel measure?
(No additional requirements are imposed on Q.)
If you prefer to deal with open sets, consider
the "dual" question about a function f: Op(Q) -> R
defined on open subsets of Q.
What if f is defined not on all closed/open sets
but on some of them? For instance, on regular ones?
What are the simplest (easily verifiable) known properties of f
that guarantee its extendibility to a (regular) Borel measure?
Could you provide me with a reference?
--
Alexander E. Gutman
Novosibirsk, Russia
root@gutman.nsu.ru

Hello, friends.
Take a look at the following statement:
  For every infinite compact space Q
  and every infinite-dimensional Banach space X,
  there exists a function from Q into X', the dual of X,
  that is weakly* continuous but not norm-continuous.
Is it true?
By the way, the latter statement is equivalent to the following:
  For every infinite compact space Q
  and every infinite-dimensional Banach space X,
  there exists a bounded linear operator from X into C(Q)
  that is not compact.
I would be happy to get an answer just for the case Q=N*,
where N* is a one-point compactification of N.
In this case, the statement under consideration
is equivalent to the following:
  For every infinite-dimensional Banach space X,
  there exists a sequence in X' that is weakly* convergent
  but not norm-convergent.
Is it true?
Where can I find a proof or a reference?
P.S. This is a copy of the original posting requested
     by some Usenet users. Sorry if you didn't need one.
--
Alexander E. Gutman
Novosibirsk, Russia
root@gutman.nsu.ru

jac@ibms46.scri.fsu.edu (Jim Carr) wrote:
>davk@netcom.com (David Kaufman) writes:
>>
>>The square root of 2 can be written in Basic computer language
>>as follows: 2^.5 or 2^(1/2) or SQR(2).
>                     =======
>
> If this is valid Basic, no wonder it is commonly said that those 
> who learn Basic first are often crippled for life as programmers. 
> That expression is equal to 1 in other high-level languages. 
rubbish nonsense rubbish nonsense drivel nonsense garbage
spew bilge tosh tripe rhubarb moonshine nonsense hogwash
2^(1/2) is the square root of two unless you deliberately do
something strange, like using integer variables. There are
probably lots of compiler options to chane the rules about this,
but that is highly system-specific and not relevant here.
There are some finer points of FORTRAN where you can ask for integers,
or possibly get things evaluated as integers
_by not using any decimal points_
but basically, 2^(1/2) is the square root of two. There is the minor
point that the power symbol is ** instead of ^ in some languages,
but that hardly cripples for life someone's programming abilities.
If you're in a supercomputer computations research institute as
it says in your header you ought to know this already.
No way does BASIC cause people to be crippled for life as programmers.
BASIC bears a strong resemblance to FORTRAN and Pascal, as a matter
of fact, and could easily pass for a simplified version of either. 
It was invented as a teaching tool. Since it was invented to
teach people programming, it was designed with general principles and
other languages in mind.

On Tue, 07 Jan 1997 14:17:14 +0100, Ulrich Frenkel
 wrote:
>Hello!
>
>Can anybody give me a link to an algorithm transforming a set of linear
>inequalities to a boundary represenation of the corresponding
>convex polygon (2D) and polyhedron (3d)?
Well, I haven't read the literature, but this is an interesting
problem, so I'll speak from the top of my head:
A straitforward approach would be to start with a rectangle (2D) or
rectangular prism (3D) that encloses all intersections of 2 lines (2D)
or 3 planes (3D) in the problem.  Then you recursively clip this shape
against each boundary (there are lots of polygon/polyhedron clipping
algorithms).
When your done, any part of the original polygon that remains can be
extrapolated to infinity.
Finding the initial polygon is sticky because the trivial algorithm to
find all the intersections is O(N^2) (2D) or O(N^3) (3D). There are
lots of better algorithms to do this, however.  If you wan't to avoid
this problem altogether (and the final extrapolation), then you can
pick a representation for your points that allows them to be
infinitely far away from the origin and just start with a polygon or
polyhedron that covers the space.
For instance you could add a boolean to your coordinates so that:
(FALSE,x,y) means the point at (x,y), while
(TRUE,x,y) means the point at the limit as h approaches 0 of
(x/h,y/h), i.e., treat (x,y) as a vector and consider the point to be
infinitely far away in that direction.
Your initial rectangle would then be described by the points
(TRUE,-1,-1), (TRUE,-1,1), (TRUE,1,1), (TRUE,1,-1).
You would use the equivalent rectangular prism in 3D, or you could use
triangles (2D) or tetrahedrons (3D), which work just as well and have
fewer points.  It's very easy to clip lines described in terms of
these points against linear inequalities, so most of the common
clipping algorithms should still work.

Matt Timmermans               | Phone:  +1 613 596-2233
Microstar Software Ltd.       | Fax:    +1 613 596-5934
3775 Richmond Road            | E-mail: mtimmerm@microstar.com
Nepean Ontario CANADA K2H 5B7 | Web: http://www.microstar.com

Michael L. Siemon (mls@panix.com) wrote:
: In article , grove@arnold.math.ku.dk
: (Henrik Christian Grove) wrote:
: 
: +Will there ever be a year which can be split in 10 different primes?
: 
: Yes, assuming you are just asking the arithmetic question, an infinite
: number of such, the first being at roughly 6.5 billion CE. But it is
: a pretty good bet that                     ??????????????
Apparently you have a different understanding of "splitting into primes"
than the poster who started this thread. In the sense of the original
poster, the question of Henrik is IMHO trivial: The year 113731, for
example, can even be split into 12 different primes:
3   7   11   13   31   37   73   113   137   373   1373   113731
In the same way, it is very easy to increase the number of primes just by
adding a few digits to known small "prime rich" years, such as the year 
1373 mentioned by the original poster. 
-- 
Ulrich Lange                       Dept. of Chemical Engineering
                                   University of Alberta
lange@gpu.srv.ualberta.ca          Edmonton, Alberta, T6G 2G6, Canada

> : --
> : Jacob Martin
> : jacobmartin@geocities.com or try jake@scientist.com
> : http://www.jmartin.home.ml.org
> 
> Are you sure this URL is correct? I get a 'file not found' error.
The server seems to be a bit fussy right now. Not my job - don't blame
me! If you really want a crack at the challenge problems then email me
at the address below and I'll send 'em to you. Same with anyone else who
wants a go. But the rules list is on the web site, so I guess you'll
have to read them there! (Its all straightforward - no group efforts,
proofs must be comprehensible, all theories used must be named (or
general case given), etc.)
	Jake
-- 
Jacob Martin
jacobmartin@geocities.com or try jake@scientist.com
http://jmartin.home.ml.org

Hi All!  How can I solve the following equation efficiently and
accurately, either analytically or numerically?
sqrt[a + b cos(x - c)] - sqrt[d + e cos(x - f)] = g
where a,b,c,d,e,f, and g are known constants, and the unknown
x is in [0,2Pi).  Note that the right-hand-side as a function of
x is periodic.  There are usually two solutions to the equation.
If you know any references that address similar problems, please
let me know.
Thanks!
Xiangdong Liu

Christopher R Volpe wrote:
> 
> Jim Carr wrote:
> >
> > ags@seaman.cc.purdue.edu (Dave Seaman) writes:
> > >
> > >You mean, in SOME high-level languages.  In common lisp, for example,
> > >2^(1/2) would be written as (expt 2 1/2),
> >
> >  And it works in APL as I recall .... but neither it nor common lisp is
> >  a particularly common language.  ;-)
> >
> > >Although languages like Fortran and C do give 2^(1/2) = 1 (and so does
> 
> In C, the expression "2^(1/2)" yields the value "2". The reason why is
> left as an exercise for the reader.
> 
Don't be cute -- the operator "^" has nothing to do with exponentiation
in C.  If you want the square root of 2 in C, you have to say "sqrt(2)"
or "pow(2, 0.5)" or something else.  "^" is a bitwise operator, as I
recall.
The original contention that 2^(1/2) = 1 shouldn't be obfuscated here. 
Although BASIC is a flawed language for high-level programming and
generally code written in BASIC cannot be modified or understood by
anyone but the programmer, the issue here is really the definition of
operators.  In most high-level languages, the division operator is
understood to be a floating-point operator.  BASIC is rather unique in
terms of defining the default precision of all arithmetic operators as
integer.
David Schaafsma, PhD
U.S. Naval Research Laboratory

In article <32D29551.428B32ED@alcyone.com>, Erik Max Francis  writes:
> electronic monk wrote:
> 
>> i do believe that -oo and +oo are the same thing.
> 
> Mathematically, they are not; there are even separate definitions for the
> two.  A limit of +oo means that the function increases without bound, and
> -oo means that it decreases without bound.  They both mean that there is no
> finite limit, but similarly they are not identical.
It all depends on the set of definitions that you use.
1. -oo and +oo can be identical.
2. -oo and +oo can be different.
3. -oo and +oo can be non-existent.
Try projecting the real line onto a circle and see 1.
Try projecting the real line onto a closed interval and see 2.
Don't bother projecting the real line and see 3.
Is it too much to ask that people state their definitions before making
unsupported claims about 1/0, limits and infinities?
	John Briggs			vaxs09@alpha.vitro.com

Hi there,
I'm stuck with the following problem.
I have a positionner (that's used to move valves) which law of
displacement obays to:
let s = speed
    x = displacement
ds/dx = constant say A for 0 < x < x1
Boundary conditions
s and x = 0 for t = 0
s = smax for x = x1
 S|       ___________ s max
  |      /
  |     /
  |    /
  |   /
  |  /    A = smax / x1
  | /         
  |/
  +------+----------------
  0      x1               x
I try to find both s and x versus time t.
General solution
----------------
I tried to introduce dt in the equation:
(ds/dt)*(dt/dx) = A or (ds/dt)*1/s = A
A general solution is s(t) = C1 * [exp(A*t) +C2]
Particular solution
-------------------
The trick is to define the constants C1 and C2.
I can write that s(t=0) = 0 = C1 * [exp(A*0) +C2] = C1 * [1+C2]
As C1 is expected to be related to smax (thus C1 not null), C2 = -1 and 
s(t) = C1* [exp(A*t) - 1]
Now what about C1? I know that s = smax for x = x1. If I find the time T
for which x = x1, I can put T in s(t) and try to solve s(T) = smax so I
find C1. Unfortunately, I don't find the solution.
Indeed, x(t) = integral (s(t) * dt)) = C1 * [(1/A) *exp(A*t) - t]
between t = 0 to t
For t = T I have x(T) = x1 = C1 * [(1/A) *(exp(A*T) - 1) - T] = C1 *
f(T)
There is no analytic way to extract T (and a solution exits only for 0 <
T < 1/A) but I try to replace C1 with the expression above in the speed
equation, which is evaluated for t = T and for which I know s(T) = smax
s(T) = x1/f(T) * [exp(A*T) -1] = smax
The problem is that I find T = 0 which is obviouly wrong. So, where did
I fail?
A possible root of confusion is that I have 2 constants C1 and C2 and I
have at least 3 boundary conditions:
2 for t = 0 where both x and s = 0
1 for s = smax for x = x1
Can you help me?
Thank you
Frederic Deboeck
brfgd@wcsmail.com

Prove that
 b    a 
a  + b  >1 with any a,b>0
-------------------==== Posted via Deja News ====-----------------------
      http://www.dejanews.com/     Search, Read, Post to Usenet

In article <5aul32$ha1@gap.cco.caltech.edu>,
lbliao  wrote:
>I have studied complex analysis several times. I can do contour integrals.
>I have not seen the proof of the Picard's great theorem. I am interested
>in seeing that, as well as understanding what analyticity really is? You
>
>Any and all insights on analyticity are welcome.
>lbliao
Abian answers:
Dear Mr. Ibliao
    Analyticity means extreme smoothness, locally almost linear, almost
constant, etc.,, etc.  
   Now, RESTRICT YOURSELF to functions FROM REALS TO REALS.  Parabolas
are not smooth enough  and thus higher degree polynomials a fortiori 
are not smooth enough  STRAIGHT LINES are Smooth enough, although
CONSTANT functions (i.e.,horizontal Lines are the best representative of
analyticity).  But (for the sake of Picard's Great Theorem) let us
compromise  and  
         RESTRICT  OURSELVES TO FUNCTIONS FROM REALS TO REALS   
                             AND
        Let  linear functions, i.e.,  whose graphs are straight lines  on
    the whole line or on intervals (including a "point" as an interval)
         be our ANALYTIC  FUNCTIONS
   Then you will feel Analyticity concept and feel it intuitively
   subconsciously, elementally and carnally by your entire existence.
eg:  If  a linear function is defined on the entire  x-axis (i.e.
     whose graph is plotted from -oo to +oo) and if it is bounded
     then it has to be a horizontal line (i.e., a constant function)
     THIS IS THE ANALOGOUS OF THE LIOUVILLE'S THEOREM
eg:  On a bounded interval the maximum of the absolute value of
     a line  occurs at the boundary point.  And  if a line on
     an interval has no zero then the minimum of its absolute value
     also occurs at a boundary point  (Maximum modulus principle)
There are more analogies.  
   NOW, how to feel intuitively PICARD'S GREAT THEOREM
    The only linear function which could possibly have a SINGULARITY
    is a line  defined  say at  one point  "a"  and whose graph is a vertical
    line:
                       .           .
                       .           .
                       |           |
                 y-axis|           |
 e,g                   |           |
                       |           |
                       |           |
                       |           |
                       |           |
               --------|-----------a--------------   x -axis--
                       |           |\
                       |           | \
                       .           .  \essential singularity
                       .           .
  Our Analytic function is defined on  {a} its graph is a vertical
  line.  At  " a" it has AN ESSENTIAL SINGULARITY at  and at "a" it
  TAKES EVERY  VALUE !!!   (This is as close as I can convey to you the
  feeling of analyticity, essential singularity  and Picard's Great Theorem)   
  All classical proof of the Picard's great Theorem still kill the
  spirit .
  Please  post (my e-mail does't work) your reaction and comments on the
above  soul-searching confession - Thanks.
  Alexander Abian
-- 
--------------------------------------------------------------------------
   ABIAN MASS-TIME EQUIVALENCE FORMULA  m = Mo(1-exp(T/(kT-Mo))) Abian units.
       ALTER EARTH'S ORBIT AND TILT - STOP GLOBAL DISASTERS  AND EPIDEMICS
       ALTER THE SOLAR SYSTEM.  REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT  
                     TO CREATE A BORN AGAIN EARTH (1990)

In article <32d40ff5.0@news.cranfield.ac.uk>,
Simon Read   wrote:
>No, no, no. Every BASIC I have ever used has evaluated (1/2) as a
>floating-point quantity. One basic I know can be forced to do it
>as an integer operation by writing  1 DIV 2  
I'll bet you haven't used Integer BASIC on an Apple ][, have you?
[Yes, my statement about BASIC was incorrect in general.  In fact, I
would be embarrassed to admit it if I knew much about BASIC.]
-- 
Dave Seaman			dseaman@purdue.edu
      ++++ stop the execution of Mumia Abu-Jamal ++++
    ++++ if you agree copy these lines to your sig ++++
++++ see http://www.xs4all.nl/~tank/spg-l/sigaction.htm ++++

Will there ever be a year which can be split in 10 different primes?
-- 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  "Og jeg troede UENDELIG var et stort tal!"
         - sagt efter en matematikforelæsning om uendelige kardinaltal.
===========================================================================
Henrik Christian Grove
grove@math.ku.dk
http://www.diku.dk/students/grove
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Norbert Kolvenbach wrote:
 > If infinity is a number, please define the Operations (+,-,*),
 > define the inferse of infinity, concerning multiplication.
infinity's inverse is zero because lim 1/x as x-->oo = 0
 > 
 > NoKo
 > "Careful with this VAX, Eugene!"
electronic monk

On Wed, 08 Jan 1997 10:56:08 -0700, Karl Schultz  wrote:
>Lotto simulation programs should be easy to write.
>The key argument opposing that approach is how
>closely your RNG matches the lotto "ball machine"
>or whatever.
?? Come on, this must be the other way around. 
Forget the ball machine. Proving the statement with mass runs
of simulations and seeds using RNG should be more than sufficient. 
>Although you can get an answer via simulation, I
>still think that there would be a statistcal
>method that would give good results.
We got a result thru a formula and need to prove it. Best/easiest 
seems IMHO simulation, but I'm not an expert in statistics.
>Uenal Mutlu wrote:
>> It would be useful if we had a simulation software which for example
>> looks something like the following:
>> 
>> LOTSIM v k b nruns fFixedTickets fFixedDraw ...
Mine will hopefully be ready this weekend. 

Hum, it seems that a few of you out there wish to discuss Roman 
numeration - via its language *and probably abacus* based Roman numerals. 
So far very good explanations of the proto-positional base 10 system has 
been provided by other USENET posters in terms of modern defintions.
My discussion of Roman numeration - not Roman numerals - is provided
by ancient unit fraction documents such as the Florentine Papyrus.
Roman fractions followed Greek base 10 unit fraction practices and improved
nothing (as my very brief analysis suggests). If anyone would like to
see it, drop me a note.
That is, need I discuss again Greek and Egyptian unit fractions for numbers 
between zero (0) and one (1) as 100% base 10? Clearly Roman numeration 
was 100% base 10 via its Roman numerals even though the language aspect 
of its written 'numeral's appears awkward to modern eyes that follow the 
positional base 10 decimal system popularized by Stevins in 1585 AD (that 
finally added zero as a positional number).
Milo Gardner
On Tue, 7 Jan 1997, Richard Mentock wrote:
> Milo Gardner wrote:
> > 
> > Roman numerals are base 10, stated in terms of the register of an
> > abacus (for example).
> > 
> > Multi-based? No base at all? Wow, what confusion.
> > 
> > I hope this note clarifies something to somebody.
> 
> Well, usually "base n" implies positional notation, which Roman
> Numerals clearly aren't.  X *does* equal ten, but V equals five,
> so why wouldn't you claim they were base 5?
> 
as others have stated powers of base 5 are not included in Roman numeral,
only powers of a base 10 system.
> -- 
> D.
> 
> mentock@mindspring.com
> http://www.mindspring.com/~mentock/index.htm
> 
> 

Darrell Ryan wrote:
 > 
 > electronic monk wrote:
 > >
 > >
 > > like i said before, we can think of zero having levels in the same
way
 > > that we can think of infinity as having levels.  if we think of
zero
 > > only as a limit, then it will make sense.  x^2 > x for all x>0, 
 > 
 > Huh?  Since when is (1/2)^2 > 1/2 ?????
whoops, haha, meant to say 2x instead of x^2.
 > >so, lim 1/(x^2) as x-->0+  >  lim 1/x as x-->0+.
 > 
 > Not so.  lim x-->0+ [1/(x^2)] = +infinity (the limit does not exist.
 > *Even if* lim x-->0- = +infinity, which it does)
 > 
 > and lim x-->0+ (1/x) = +infinity (the limit does not exist)
 > 
 > Remember, when we write +infinity or -infinity as an answer to a
limit
 > problem, we are really saying, "The limit does not exist."  The
infinity
 > part tells us the reason *why* the limit does not exist.
no, a limit does not exist when the right hand limit differs from the
left hand one.  oo is a perfectly good answer for a limit.  lim 1/(x^2)
as x-->0 IS oo, because lim 1/(x^2) as x-->0+ = lim 1/(x^2) as x-->0-. 
we can say that the number _approaches_ infinity.  we never actualy
refer to it _as_ infinity.  but again, i mean to say 2x instead of
x^2...sorry.
 > 
 > Of course, none of this really has anything to do with why 1/0 is
 > undefined.  What number can you multiply by 0 and get an answer of 1?
 > Infinity?  Nada.  Infinity is not even a number.  'Nuff said.
yer right though, oo is not a number.  it is a concept and can only be
refered to as a limit.
 > 
 > ____________________________________________________________
 > Darrell Ryan
 >   e-mail             dryan@edge.net
 >   personal website   http://edge.edge.net/~dryan
 >   company website    http://www.edge.net/stmc
electronic monk

> Roman numerals are *highly* positional.  IX is 9 and XI is 11.
> Position matters a lot.
Yes, position does matter a lot, and Roman Numerals are "positional" in
that sense. However, I think that mathematically positional means that
the rightmost digit position (in a whole number) means 1 x how much it
is , the next means your base x how much it is, the next is (base)^2 x
how much it is, etc.  
for example, the number
xyz in base n means: (x * n^2) + (y * n) + (z * 1)
In this sense, roman numerals are not positional, or really even
"close."
Brian Christiansen (just feeling a bit feisty tonight)

ear Members of SCI-MATH Newsgroup:
The Dean of the Graduate Division of the University of California,
Berkeley,
is currently undertaking a research project studying the career outcomes
of
Ph.D.'s in selected majors approximately ten years after they have
graduated. 61 U.S. institutions and nearly 6000 Ph.D.'s are involved in
this
study. The results of this study will hopefully assist in improving the
effectiveness of doctoral programs in training our future scholars and
scientists.
Since we are using a mail out questionnaire, we have been trying very
hard
to locate the most current mailing addresses for these people. We have
been
successful in locating approximately 85% of our survey population
through
alumni associations and other sources. That still leaves a siginificant
number of persons for whom we have no mailing address.
Therefore, in a last ditch effort, we are utilizing internet
resources such as web pages of academic institutions and field specific
mailing discussion lists (such as yours) in the hope that somebody will
know the whereabouts
of some of our lost Ph.D.'s. We are requesting that if anybody
recognizes a
name in the following list of math ph.d.'s, that they contact us to
provide leads on where
that Ph.D. might be found. Any leads will be greatly appreciated. 
We also request that you forward this email to any persons who might
know anything about the whereabouts of these people. Thank you very
much.
We apologize for any inconvenience this message may cause you.
Thank you very much.
Sincerely,
Graduate Division
University of California, Berkeley
"Ph.D.'s Ten Years Later" Study
424 Sproul Hall #5900
Berkeley, CA 94720
tel: (510) 643-2791
fax: (510) 642-6366
email: phd10yr@uclink.berkeley.edu
Mathematics PhD's: [Institution/field/last name/first name/middle (if
available)/year of graduation with PhD]
SUNY-Buffalo/Mathematics/Abdullah/Salem/Ali/83      
Michigan State University/Mathematics/Attele/Kapila/Rohan/83      
Yale University/Mathematics/Bhate/Hemant//83      
UC Berkeley/Mathematics/Bolfarine/Heleno//83      
University of Chicago/Mathematics/Brandt/Jorgen//83      
University of Minnesota/Mathematics/Brierley/Stephen/David/84      
Princeton University/Mathematics/Cabrera/Javier/Fernand/83      
Columbia University/Mathematics/Chang/Fu//83      
SUNY-Buffalo/Mathematics/Chen/Chang-Shan//84      
UC Los Angeles/Mathematics/Chou/Jine-Phone//84      
SUNY-Buffalo/Mathematics/El-Henawy/Ibrahim/M/83      
Stanford University/Mathematics/Fairley/David//83      
University of Utah/Mathematics/Faltenbacher/Wolfgang//84      
UC Berkeley/Mathematics/Folledo/Manuel//83      
Northwestern University/Mathematics/Frank/George/Nelson/85      
Tulane University/Mathematics/Franzen/Berthold/Werne/83      
University of Iowa/Mathematics/Gallegos-Jarpa/Griceld//83      
University of Minnesota/Mathematics/George/Adel/Aziz/83      
University of Kentucky/Mathematics/Greenwell/Catherine/El/82      
UC Los Angeles/Mathematics/Hsueh/Yuang-Cheh//83      
University of Florida/Mathematics/Ireson/Michael/John/83      
MIT/Mathematics/Kim/Dong/Yoon/85      
University of Michigan/Mathematics/Kim/Hyuk//83      
University of Maryland/Mathematics/Kramer/David/Philip/83      
University of Pittsburgh/Mathematics/Krishna/Kottekai//83      
University of Florida/Mathematics/Kurihara/Eiji//84      
MIT/Mathematics/Lasaga/Fernando/Rene/84      
University of Minnesota/Mathematics/Lau/Chi-Ping//84      
Colorado State University/Mathematics/Leiva/Ricardo/Anibal/83      
Temple University/Mathematics/Levitan/Mark/E/85      
University of Utah/Mathematics/Luminet/Denis/Laurent/83      
University of Washington/Mathematics/Mansfield/Edward/Josep/84      
Michigan State University/Mathematics/Merkle/Milan/J/84      
Temple University/Mathematics/Moore/Annette/Louise/84      
University of New Mexico/Mathematics/Pichardo-Maya/Agustin//85      
University of Minnesota/Mathematics/Ricou/Manuel/Oliveira/84      
UC Berkeley/Mathematics/Sevilla/Agustin/Ramos/83      
University of Utah/Mathematics/Sindler/Frantisek//83      
University of Washington/Mathematics/Stedman/Edward//83      
University of Florida/Mathematics/Toledo-Manzur/Juan/Ant/82      
University of Minnesota/Mathematics/Waksman/Peter//83      
SUNY-Binghamton/Mathematics/Wender/Abraham//85      
University of Arizona/Mathematics/Xaba/Busa/Abraham/84

In article <5aeect$hij@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
>   Would Colin Douthwaite or someone please draw a ascii picture of the
> professor moving a bar magnet through loop and then loop through bar
> magnet. Sincere thanks
Subject:      [18] Anniversary
From:         cfbd@southern.co.nz (Colin Douthwaite)
Date:         1996/09/17
Message-Id:   <51mfb9$9un@orm.southern.co.nz>
Organization: Southern InterNet Services
Newsgroups:   alt.ascii-Art
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~
                                _
                               '  )        /)
                                -(   __  _(/
                             (__ )  / (_(_(_
              _____
             (, /  |            ,
               /---|  __  __      _ _   _  __  _   _   __
            ) /    |_ / (_/ (__(_ (/___(/_/ (_/_)_(_(_/ (_  (_/_
           (_/                                             .-/
                                      /)                  (_/
                               ___   //
                              (_)   /(_
                                   /)
                                  (/
               /)                        ,   ,
          _   //  _/_   __   _   _   _           __   _   __ _/_
         (_(_(/_  (__       (_(_/_)_(___(_ _(_       (_(_/ (_(__
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~  ( 18 ) 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
                  The Last Skit of NP2  ( ILLUSTRATED )
                        Text by Ludwig Plutonium
                  Pictures from just about everywhere.
                                                             30/6/94
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~
> Show a long sine wave curve.
     .-.     .-.     .-.     .-.     .-.     .-.     .-.      
  --/---\---/---\---/---\---/---\---/---\---/---\---/---\---/-
         `-'     `-'     `-'     `-'     `-'     `-'     `-'  
> Show a man dressed up in a tuxedo inspecting this sine wave curve
> for fluctuations.
                                    ../ ~`~`~'~~'~ ``;; -
                                  '/~////~|\\\\\\~\\\\\~\\\~
                                ~////~//||||\\\\~\\\\\\~\\\\\~
                               ~//~//||||~||||\\\\\\~\\\\\~\\\~
                              ~///|||~||||~||||\\~\\\\\~\\\\\\\\~.
                             ~/<<<<<||<<<<<<\\\\<<<<<"|""\\\~\\\\\~
                            ;\\\\\\\\\~~\\\\\~\\\\\~/""" ""|\\\\""|\
                             |||/\\\\|\\\\~\\\\\\|/        "||"""||\~
                             ||/                           |"""""""" ~
                             |||                               """"""|
                            ==, ,,,,,,,      ,,,,,,,,,,   ,====""|"""~
                              \----------===-------------/   "|""""||~
                              |\  \\    /  \   \\       /    """''|""~
                              | \      //    \         /     ""/~ ||/
                              |  ~~~~~ /       ~~~~~~~       "/P  '~
                              \       |      ,                   |
                               \       \ ~~\_/                  "
                                \                   /       ,.""
                                 \  \............../       ,.
                                  \    \_______/         ,.
                                   \                  ,.'
                                    \              ,.'
                                     \          ,'~
                                   /  ~~~~~~~~~         \
                         _  _  _ /    ___    ____         \ _  _  _
                     /               !   \_/     !                  \
                  /                  !___/~\_____!                    \
    _  _  _  _  _  _  _  _  _  _  _  _  _  _  _
   \               _  _  _  _  _  _  _  _  _  _  \  _  _  _  _  _
    \            /                                \             /|
     \         /  .-.     .-.     .-.     .-.      \          /  |
      \      / --/---\---/---\---/---\---/---\---/- \       /    /
       \   /          `-'     `-'     `-'     `-'    \    /    /
        \/                                            \ /    /
         |  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -|   /
         |                                             | /
         |_  _  _  _  _  _  _  _  _  _  _  _  _  _  _  |/
> Show him pouting whenever a math person standing on the sidelines
> points out that there are no fluctuations in the sine curve.
                                    ../ ~`~`~'~~'~ ``;; -
                                  '/~////~|\\\\\\~\\\\\~\\\~
                                ~////~//||||\\\\~\\\\\\~\\\\\~
                               ~//~//||||~||||\\\\\\~\\\\\~\\\~
                              ~///|||~||||~||||\\~\\\\\~\\\\\\\\~.
                             ~/<<<<<||<<<<<<\\\\<<<<<"|""\\\~\\\\\~
                            ;\\\\\\\\\~~\\\\\~\\\\\~/""" ""|\\\\""|\
                             |||/\\\\|\\\\~\\\\\\|/        "||"""||\~
                             ||/   ---- _____ -- ___       |"""""""" ~
                             |||     ___        ____           """"""|
                            ==, ,,,,,,,      ,,,,,,,,,,   ,====""|"""~
                              \----------===-------------/   "|""""||~
                              |\  \\    /  \   \\       /    """''|""~
                              | \      //    \         /     ""/~ ||/
                              |  ~~~~~ /       ~~~~~~~       "/P  '~
                              \       |      ,                   |
                               \       \ ~~\_/                  "
                                \                            ,.""
                                 \       ______            ,.
                                  \     (_____ )         ,.
                                   \   (______)       ,.'
                                    \              ,.'
                                     \          ,'~
                                   /  ~~~~~~~~~         \
                         _  _  _ /    ___    ____         \ _  _  _
                     /               !   \_/     !                  \
                  /                  !___/~\_____!                    \
    _  _  _  _  _  _  _  _  _  _  _  _  _  _  _
   \               _  _  _  _  _  _  _  _  _  _  \  _  _  _  _  _
    \            /                                \             /|
     \         /  .-.     .-.     .-.     .-.      \          /  |
      \      / --/---\---/---\---/---\---/---\---/- \       /    /
       \   /          `-'     `-'     `-'     `-'    \    /    /
        \/                                            \ /    /
         |  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -|   /
         |                                             | /   -cfbd-
         |_  _  _  _  _  _  _  _  _  _  _  _  _  _  _  |/
                  //////
                  //////
                 (//////)
... ..  ..  ... . ////// . ... .())((() .. . . . .... . . . ..   . . .
             _____|____|_____  ((()(()))
            /                \ (((((((()
           |      -----       |()(()(())_
           |      MATH        |(((()()() |
           \        MAN       '\()))))(  |
            \     -----         \(((((  /
             \              .    \___. (
              \            / \    |:| ) \
              |            | |`------'   |
              \____________/  \_________/
                                               -mh-
........................................................................
> MWBR stands for the cosmic microwave background radiation. The
> cosmic MWBR is relentlessly uniform and it is QUANTUM BLACKBODY
> radiation. It is cavity radiation. So then the Atom Totality has
> no problems with explaining it. Inside an atom totality, MWBR is
> uniform and since it is an atom then it comes out measured as
> Quantum Blackbody. This also explains Olber's Paradox because an
> electron cavity is black, from blackbody.
> But to the present day science community who believe a different
> theory, a different paradigm, they believe in the Big Bang, then
> in order to continue with that fakery, the MWBR must be seen to
> fluctuate.  Since it is uniform  and the Big Bang theory is out
> the window, con artists in the physics community are set-up to
> find fluctuations by hook or crook.
> Show Smoot trying to measure the parameters of virus using a meter
> stick.
                        __
                        ||
                        ||
            ////////    ||
            |:---[.]    ||
            |(     _J   ||                                   _     _
            | ^ ( _|    ||           ,,,,,,,,,,,,,,,,,,,,,,,  \   /
           / \_____)    ||         / (  (  (  (  (  (  (  (  \( = =)
          /  _____   \  ||        <  (  (  (  (  (  (  (  (  / ( ^ )
         |  /     \   | ||         \ (__(__(__(__(__(__(__(__)   ~
         |  |     |   | ||           ^  ^  ^  ^  ^  ^  ^  ^  ^
         \  /\   /\__ | ||                     -cfbd-
         |  |        \/---
         \   \            )
          |   >_____/_____)
          \__________/  ||
          /         \   ||
          |          |  ~~
          \          \\
           \          |\
            \         | \
             \         | |
              |        |  |
              |       |   |
             |        /   |
            |________/____|
            (_________)____)       -mh-
     ==============================
> Show him finding fluctuations in his measurement.
                               //////\\
                              /        \
                             _|  _   _ |_
                            |.|-(.)-(.)+.|
                             \|    J   |/
                              \   ---  /
--------------- UUUU --------- \      / --------- UUUU
-------------------
|                               "####"                                 
 |
|                                                                      
 |
|                                                                      
 |
|                     _     _               _     _                    
 |
|    ,,,,,,,,,,,,,,,,  \   /                 \   /             -, ,-   
 |
|   /  (  (  (  (  (  \( u u)     ,,,,,,,,,, ( @ @)     ,,,,,,,(. .)   
 |
|   \__(__(__(__(__(__) ( ^ )    (__(__(__(_) ( ^ )    (__(__(__(^)    
 |
|      ^  ^  ^  ^  ^  ^   ~       ^ ^  ^  ^ ^   ~       ^ ^  ^  ^ ^    
 |
|  ____________________________________________________________________
 |
|
[_________|______|______|______|______|______|______|______|______|__]
|
|          20            40            60            80           100  
 |
                                                          -cfbd-
> Show a true scientist yell from the sidelines
> "Hey, Smoot or smut, don't you think that you have reached the
> limit of precision of your measuring devices?  That in 20 or 50
> years from now when they have even more precise measuring devices
> that MWBR will be seen uniform way below what your own
> measurements have recorded?"
    //\\\||///\\              ,------------------------------------,
   //|        |\\            (  Hey, Smoot or smut, don't you think )
  |[ | 
 | ]|          (   that you have reached the limit of   )
  ||||   .\.  ||||     o O (   precision of your measuring devices?   )
  |||<    _   >|||   o     (     That in 20 or 50 years from now      )
      \__(_)_/  o  o       (    when they have even more precise      )
   _____!___!_____         (   measuring devices that MWBR will be    )
  |  |  \   /  |  |         (    seen uniform way below what your    )
  |  | _ \./   |  |          (    own measurements have recorded?   )
  |  |[_] |    |  |           '------------------------------------'
  |  |    o    |  |
  |  |    |    |  |
  |  |    o    |  |
  |__|    |    |__|
   WW|    o    |WW
     |    |    |
     |    o    |
     |____|____|
      |   |   |
      |   |   |
      |   |   |   -cfbd-
      |___|___|
       \_/ \_/
> Show Smoot undaunted with his quest to impose his illogic unto the
> science community. Show Smoot looking at his measurement
> instruments bewildered by what was said to him.
                               .?.?.?.?
                              / ^^^^^^ \
                             _| -   -  |_
                            |.| (0^0)  |.|
                             \|   J    |/
                              \  (O)   /
--------------- UUUU --------- \      / --------- UUUU
-------------------
|                               "####"                                 
 |
|                                                                      
 |
|                                                                      
 |
|                __________/~|                  __                     
 |
|               / __     __  |                  ||                     
 |
|               |/  \___/  \_|                 ====                    
 |
|                    | |                       |  |__                  
 |
|                    | |                       |  |-.\                 
 |
|                    | |                       |__|  \\                
 |
|                    | |                        ||   ||                
 |
|                    | |                      ======__|                
 |
|                    | |                     ________||__              
 |
|                    |_|                    /____________\             
 |
|  ____________________________________________________________________
 |
|
[_________|______|______|______|______|______|______|______|______|__]
|
|          20            40            60            80           100  
 |
                                                          -cfbd-
> Show Smoot take his big bang blanket to bed with him just as Linus 
> in the Peanuts comic strip.  
 __                                            __
 ||____________________________________________||
 ||  _______________________________________   ||
 || {                                        } ||
 || {               //////\\                 } ||
 || {              /        \                } ||
 || {             _|  _   _ |_               } ||
 || {            |.|-(.)-(.)+.|              } ||
 || {             \|    J   |/               } ||
 || {              \   ---  /                } ||
 || --- UUUU ------ \      / ----- UUUU -------
 ||{                 "####"                     )
 ||{                                             \
 || .
 ||  .                                              \
 ||   .
 ||\   .                       .  '                     \
 ~~ \   .                       * >
     \   .                 ,__./                            \
      \   .               /    \
       \   .             | BOMB |                              \
        \   .             \____/
         \   .                                                    \
          \   .
           \   .                                                     \
            \   .
             \   .                                                     
\
              \  (                                                     
 )
               \
(_______________________________________________________)
                \ |                                                    
 |
\|______________________________________________________|
                    || |                                           || |
                    || |                        -cfbd-             || |
                     `~~                                            `~~
> Show Smoot with his thumb in his mouth.
Er....I'm afraid a picture of _that_ defeats me  :-)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~

In Message-ID: <5apma6$1110@pulp.ucs.ualberta.ca> Ulrich Lange            
          Dept. of Chemical Engineering
                                   University of Alberta
lange@gpu.srv.ualberta.ca          Edmonton, Alberta, T6G 2G6, Canada
writes:
>Luke Powell (lukep@montana.com) wrote:
>: Was just having some difficulty with a certain math problem, I've been
>: pounding by brain on it for about a day. Would somebody give me a hand,
>: thanks.
>: 
>: (1+sin x)/cos x + cos x / (1+sin x) = 2 sec x
>Hint:
>Multiply the equation by (1+ sin x)*cos x to get rid of the terms in
>denominator. All you will need to prove the resulting equation is the
fact
>that (cos x)^2 + (sin x)^2 = 1.
Luke, and anybody else for that matter, if you were able to solve that
problem, then try this one: What is (are) the solution(s) to (1+sin x)/cos
x  +  (cos x)/(1+sin x) =2csc x ?
Koi 
e-mail: KMorris ww@aol.com
Fiffff-ty millll-yon! Come on everybody, out to the cemetary!
Get the extra ball! Thuhhhhh Mamushhhka!
Let's go THING!... Well played THING, you're realy on the ball!