Newsgroup sci.math 156577

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Jacob or Stewart Martin wrote:
> 
> I was given a problem at my Cambridge interview recently involving the
> function x^x. I was wondering whether this function is defined for x=0.
> It's just that 0^0 seems a bit unusual and although I reasoned that it must
> be 1 I'm still not sure whether this is correct.
> 
> Any help welcome.
to understand this indeterminante form we must understand what the
terminology is.  when something is put to a power, that means it is
multiplied by itself that many times.  when it is to the first power,
that means it is left alone.  when it is to the zero power, when it is
to fractional powers, that means what number multiplied by itself this
many times makes this other number.  ex: 
4^(1/2) is a number that solves the equation x^2 = 4.  
when something is to the zero power, that just means that it is divided
by itself once.  so, 0^0 = 0/0 which is an indeterminante form.
electronic monk

Michael The Roach Janszen wrote:
> 
> Figure out the following steps...
> 
> -20 = -20    (obviously)
> 
> 16 - 36 = 25 - 45  (just the same)
> 
> 16 - 36 + 81/4 = 25 - 45 + 81/4   (just added 81/4)
> 
> (4 - 9/2)^2 = (5 - 9/2)^2  (using the binominal rules)
> 
> 4 - 9/2 = 5 - 9/2          (square root)
> 
> 4 = 5     (euh.....)
this is one of those cases when it is important to use the plus or minus
square root.  
(4 - 9/2)^2 = (5 - 9/2)^2
(4 - 9/2) = -(5 - 9/2)    (took the negative square root of one and the
positive of the other.)
4 - 9/2 = -5 + 9/2        add the five, add the 9/2
9 = 9
this also works with:
-(4 - 9/2) = (5 - 9/2)
-4 + 9/2 = 5 - 9/2
9 = 9
i do think it is weird that one must take the positive square root of
one, and the negative of the other, but it works.
> Michael "The Roach" Janszen
> 
> It might look as if I don't take life seriously. But I do.
> I just don't see why life should take me seriously...
> 
> Spammer trap - when replying by e-mail, drop the last letter
> of the address...
electronic monk

Hello
Please excuse me first fo my english.
I can understand this language quite well, but speak rather bad.
So, I have a big problem !!
If you know the response, please write me.
If you have a segment AB, that you know, could you tell me what are all the
point P, so that AP-BP=k, where k is a constant that you know?
For exemple, I have a segment AB that measures 5 centimeters.
Were are the points P so that PA=PB+3 centimeters?
It's very important for me.
Thank you !

In article <59u8vk$e6f@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
>   So tell me David, is ....0002 the one and only one adic (take any
> adic) which solves this encoding----   (2+2)^1/2 = (2X2)^1/2 = 2  ???
Is there a p-adic analogous to the Real 2.00...  ?
Every p-adic represented as ...002, say the 3-adic or the 5-adic or the
19-adic are all different. But is there a special and unique p-adic
number which is analogous to the Real 2.00... and which satisfies  
    (2+2)^1/2 = (2X2)^1/2 = 2
And ,     ((N+N)^1/N) = ((NxN)^1/N) = N
   reduces to     (N+N) = (NxN) = N^N = M, provided if proper p-adic
definition of exponential and logarithmic
Anyone know that geometrical picture results when one takes only
p-adics, no n-adics, just p-adics and keeps the digits fixed and then
makes a geometrical explanation of these numbers.
  For example take the p-adic number of ....0002 in 3-adics, then
5-adics, then 7-adics ad infinitum. Keeping the _2_ fixed and varying
the adic. What is the geometrical result? 

In article <59u8vk$e6f@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
> >> (2+2)^1/2 = (2X2)^1/2 = 2
> 
> I know that, but tell me is ...0002 the only adic with that encoding.
> The encoding
> ((N+N)^1/N) = ((NxN)^1/N) = N  is ...00002 the only adic with that
> encoding?
> 
> Back in 1993, I _learned_ that ...00002 was not the only adic of the
> encoding
> 
>    k N = N^k as evinced here:
> 
> >>FLT; however, you may be interested to know that other solutions 
> >>are possible if you allow those left-infinite decimal strings that 
> >>we discussed earlier. When k=4, there is a unique nonzero solution 
> >>to N+N+N+N = N*N*N*N = M. Here is the answer, worked out to 60 
> >>
> >>  N = . . .8217568575974462578891103859665245689398767183
> >>            82655349981184
> >>  M = . . .2870274303897850315564415438660982757595068735
> >>            30621399924736
    ((N+N)^1/N) = ((NxN)^1/N) = N
of course that has a unique solution in Reals/Complex of 2.000...
    (2+2)^1/2 = (2X2)^1/2 = 2
And ,     ((N+N)^1/N) = ((NxN)^1/N) = N
   reduces to     (N+N) = (NxN) = N^N = M
  Is there a p-adic, forget all composite adics.
which satisfies (N+N) = (NxN) = N^N = M and where there is not a unique
solution?  If the answer is that there exists no unique p-adic then the
Riemann Hypothesis is False.
  The proof of Riemann Hypothesis as a true theorem depends on 2.00...
being the unique solution to (N+N) = (NxN) = N^N = M. If there are no
p-adic unique solution means that RH was false all along.
  The Euler formula is a multiplication and use of prime integers. IN
the P-adics there are an infinitude of primes , and for 2-adics it is
2, for 3-adics it is 3 and 5-adics it is 5 and so on ad infinitum.
  I posed this question to David Madore before start of the holidays,
and I pose it again. Can you adequately define exponential and
logarithm in p-adics?
  What solutions exist for (N+N) = (NxN) = N^N = M in p-adics?

sorry, the curve of the problem is actually:
      2
 -2(x)
e
I think that only the x is squared so it is not 4x^2 but it is -2x^2

In article <32CA9D09.4667@efgh.net>, Anonymous   wrote:
>The limit of 1/x as x --> 0 is infinity.
>
>
>-X
If x is real and approaches zero through positive numbers,
then the limit is +infinity; however, if it approaches
zero through negative numbers, then the limit is -infinity.
Herb
-- 
 Herbert I Brown  hibrown@math.albany.edu  (518) 442-4640
 Math Dept,  The Univ at Albany,  Albany, NY 12222  
----------------------------------------------------------

tleko@aol.com wrote:
>If there is an interest in the results where MATLAB sofrware is not
>available
>please request a fax- transmission.
I tell you what, I need the first five hundred thousand digits of pi.
Why don't you email them to me at  feedback@www.whitehouse.gov  ?
Simon

In article <5ael96$9v1@mtinsc01-mgt.ops.worldnet.att.net>, Richard Miao
 wrote:
:sorry, the curve of the problem is actually:
:
:
:      2
: -2(x)
:e
:
:I think that only the x is squared so it is not 4x^2 but it is -2x^2
OK, then this one has a solution.
Note that exp(-2x^2) is decreasing on [0,\infty) and increasing on
(-\infty,0], and that it is an even function.  Thus when one draws the
inscribed rectangle between the x-axis and the curve it really does meet
the curve at points of the form (-x,0) and (x,0), and its area is therefore
2 x exp(-2x^2).  So this is
your area function,
   a(x) = 2x exp(-2x^2).
This is a function which goes to zero as x goes to infinity, and since it's
nonnegative on [0,\infty), it must take a maximum on [0,\infty).  (This is
a standard principle, but you should think it through.)  a has only one
critical point (compute it!) in [0,\infty), which must therefore be the
maximum point.
You also wanted the average value?  This is the limit of 1/(2L) \int_{-L}^L
a(x) dx.  This has to be zero; any function which tends to zero as x tends
to infinity must have an average value of zero.
Now:  was this a homework problem, or not?  I doubt it; not on the first of
January...
--Ron Bruck
-- 
--Now 100% ISDN from this address

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.
Let us define a p-adic, p is prime and there are an infinitude of
primes. These primes come from the Real+i+j system in 1.00..., 2.00...,
3.00..., 4.00..., ad infinitum of 2.0..., 3.0...,5.00... ad infinitum.
  P-adic is a Series defined as such (where the radix point and the
finite portion is finite since p-adic is prime)
......... (a_2)p^2 + (a_1)p^1  + (a_0)p^0 +  (a_-1)p^-1 + ... +
(a_-r)p^-r  
where a_i element {0,1,..,p-1}
For example ....231.4 in 5-adics is 
  ....... 2x5^2 + 3x5^1 + 1x5^0 + 4x5^-1
 YOU CAN REPRESENT EVERY P-ADIC AS A SERIES
..........+ 5^3 + 5^2 + 5 + 1 
Peano Successor Axiom is a Series of adding 1 endlessly
Peano Successor Axiom
....... + 1 + 1 + 1 + 1 + 1 + 1
Both the definition of a p-adic and the Successor Axiom are identical  

Antoine Mathys (mmathys@bluewin.ch) wrote:
> If you have a segment AB, that you know, could you tell me what are all the
> point P, so that AP-BP=k, where k is a constant that you know?
The form an hyperbola with foci at A and B..
Miguel A. Lerma

Paul Nahay (pnahay@sprynet.com) wrote:
> The closed sheaf integrates some compact semigroups.  One Hausdorff equatio=
> n injects all open monoids.  Only one Gorenstein ring projects most Notheri=
> an categories.  It can be proven that a Abelian field commutes several Arti=
> nian polynomials, and sheafifies specific Tauberian subminors.  Thus, we ca=
> n see that a non-linear module blows up few linear matricies.  This leads u=
> s to conclude that the projective valuation differentiates no injective she=
> afs.  A minimal ideal multiplies many Cohen-Macaulay equations.  Neverthele=
> ss, one prime group adds almost all maximal rings, or subtracts some non-tr=
> ansitive fields.  =
[...]
I cannot find even one meaningful sentece in the above article. 
It looks like a sequence of automatically generated random sentences. 
What is its purpose?
Miguel A. Lerma

Bob Silverman (numtheor@tiac.net) wrote:
> guest2@thphys.irb.hr (General Guest User) wrote:
> >does anybody know why mathematicians always use 
> >m for the slope of a straight line, for example
> >y=mx + b
> >is the usual slope-intercept form of a straight line.
> >Thanks 
I have heard that it comes from a French word (perhaps "montant", 
uphill?).
> Why do they use Pi for Pi? 
Pi is the initial of the greek word "perimetros" (perimeter). 
It was introduced by William Jones in 1706 in his book "Synopsis 
Palmariorum Matheseos, or A New Introduction to the Mathematics."
Its definitive use is due to Euler. 
> Why do they use e for e? 
Also due to Euler. I have not quite clear why he chose this 
letter, although I heard that he was using different letters 
a, b, c,... for different constants and "e" was the one he used 
for the base of the system of natural logarithms.
> Why do they use 1 for 1?
We should ask ancient Hindus, but it makes sense to use a single 
stroke for a unit.
> Why does it matter?
Why not?
Miguel A. Lerma

Hello everybody
I’ve got a big problem, I have to do an assignment for university, it's
got to be handed in on the 6th of January, and don’t understand a thing.
The first assignment (Standard Deviation) was easy as the lecturer
really went into it but he just rushed through the chi square-test and
now I find myself not being able to do the assignment even though I sat
down with about 6 Statistics books...  I’m not lazy or anything, I did
do well in the last assignment and got 97% and I wouldn’t ask for help
if it wasn’t necessary so please don’t flame me.  So if anybody could
help me, I’d be really, really grateful.
Ulrike
************************************************************************
The assignment is as follows:
1.  What is meant by a "test of significance"?
2.  What is meant by "degrees of freedom"?
3.  What is meant by a "null hypothesis"?
4.  What is meant by "goodness of fit"?
5.  The management of a firm wants to know how their employees feel
about working conditions, particularly whether there are differences in
sentiment between various departments.  A study based on random samples
of the employees of four departments yielded the results shown in this
table:
Working      Department  Department  Department  Department  Total 
Conditions       A           B           C           D  
Very Good       65          112          85          80        342
Average         27           67          60          44        198
Poor             8           21          15          16         60
Total          100          200         160         140        600
a)  What would be the null hypothesis in this example?
b)  Using chi square-test, would you reject or accept the null
hypothesis (at 0.05 level).
6.  Assuming that the expected normal curve frequencies given below were
calculated using the mean and standard curvation of the observed
frequencies, test for goodness of fit at a level of significance of
0.05:
ObservedFrequencies     Expected NormalCurve Frequencies
       29                             25
      160                            156
      314                            312
      202                            215
       42                             40
        3                              2
-- 
Ulrike Hassold

Marnix Klooster (marnix@worldonline.nl) wrote:
: OK.  I accept that the proofs you gave are nicer than Van
: Gasteren's.  But that was merely an example I gave.  
: The questions that are still open for me are:
I try a partial answer to these questions: 
:  * How much agreement is there among mathematicians about beauty
: and elegance of theorems and proofs?  Where does this agreement
: come from?
IMHO there is some, but certainly no complete agreement. As far as proofs
are concerned,  I guess (almost) everybody would agree that the way
difficult results like the Fundamental Theorem of Algebra can be derived
in a few lines in the framework of complex function theory is very
elegant. Thus, I think, a part of the answer is: If a difficult problem
can be embedded into a general framework and proven to be just a
specialization of some more abstract theorem, it is certainly elegant.    
As far as theorems are concerned, there is certainly an agreement that
e.g. Cauchy's integral theorem is beautiful, because there is a kind of
balance between the level of abstraction and the relevance of the theorem:
On one hand, it is not very "abstract" (contrary to theorems which are of
similar importance for a certain theory, e.g. the Hahn-Banach Theorem for
Functional Analysis)  and on the other hand it is so fundamental for
function theory that almost everything else can be derived very easily
from it. 
What I tried to describe here is certainly not the only criterion for
"elegance", but maybe it is the one on which most people would agree. 
:  * Is a more elegant or beautiful theorem or proof `better' than
: a clumsy or ugly one?  Why?
In view of my interpretation of "elegance" given above, I would say an
elegant proof is better since it avoids a lot of work caused by technical
details, but focuses on the concepts which are the really important ones
for a statement to hold true. 
:  * What is the use of formula manipulation in finding and
: presenting proofs?  Does it help or does it obstruct?  Generally,
: is a formula-manipulation proof better (or more elegant) than a
: mostly-text one, or is it worse (or clumsier)?
This is IMHO the most difficult question: I usually find both pure
formula-manipulation proofs and pure text-based proofs hard to read.
At least for my taste, there apparently exists an "optimal" mixture of
formula and text. But I have no idea _why_ I find a particular style of  
presentation of a proof more elegant than others.
-- 
Ulrich Lange                       Dept. of Chemical Engineering
                                   University of Alberta
lange@gpu.srv.ualberta.ca          Edmonton, Alberta, T6G 2G6, Canada

What do you mean when you write that the exp(-2x^2) in increasing on 
-infinity? Is it not also decreasing.
Could you explain just how you got your area function or could someonelse 
explain it. I understand everything past that but I don't understand how 
you led up to it.

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Miguel Lerma wrote:
> 
> Paul Nahay (pnahay@sprynet.com) wrote:
> > The closed sheaf integrates some compact semigroups.  One Hausdorff equatio=
> > n injects all open monoids.  Only one Gorenstein ring projects most Notheri=
> > an categories.  It can be proven that a Abelian field commutes several Arti=
> > nian polynomials, and sheafifies specific Tauberian subminors.  Thus, we ca=
> > n see that a non-linear module blows up few linear matricies.  This leads u=
> > s to conclude that the projective valuation differentiates no injective she=
> > afs.  A minimal ideal multiplies many Cohen-Macaulay equations.  Neverthele=
> > ss, one prime group adds almost all maximal rings, or subtracts some non-tr=
> > ansitive fields.  =
> [...]
> 
> I cannot find even one meaningful sentece in the above article.
> It looks like a sequence of automatically generated random sentences.
> What is its purpose?
> 
> Miguel A. Lerma
A most elegant proof. Mathblab lives!! Thanks Paul.
John

"Antoine Mathys"  wrote:
>If you have a segment AB, that you know, could you tell me what are all the
>point P, so that AP-BP=k, where k is a constant that you know?
>For exemple, I have a segment AB that measures 5 centimeters.
>Were are the points P so that PA=PB+3 centimeters?
Using that example (I'll leave the more genral solution to you ...same
method):
Let AP = x, PB = y, then
x+ y = 5
x - y = 3
You now have two simultaneous equations which you can solve.  I think
that you can see only one solution there (x = 4, y = 1).
Geometrically, you have any of two points one unit from either end.

i have a friend working in this area who would be very interested to 
communicate. He has done similar work and i believe as an under grad 
he produced some work on the black scholes eqn
contact chritopher.waddel@strath.ac.uk

Bob Silverman (numtheor@tiac.net) wrote:
: 
: Why do they use Pi for Pi? Why do they use e for e? Why do they use 1 for 1?
: Why does it matter?
Don't talk!  Doesn't matter!  Who cares!  Get back to work!  Stop asking 
inconsequential questions!  
$$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666
		       hetherwi@math.wisc.edu
$$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666

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saouter@irisa.fr (Saouter Yannick) writes:
: 
: jms4@po.CWRU.Edu (James M. Sohr) writes:
: > 
: >    I remember a few years ago when I was a participant in
: > the PROMYS (PROgram in Mathematics for Young Scientists)
: > at Boston U., when a guest lecturer presented the topic of
: > Riemman zeta functions.
: >    In his lecture, he proved that the simple zeta function
: > 
: >         1     1     1     1     1
: >   z(2)=--- + --- + --- + --- + --- + ...
: >         1     4     9     16    25
: > 
: >     converges to (Pi^2)/6.  However, my memory being as it--
: > I've forgotten how the proof (it really wasn't a _proof_,
: > as much as a demonstration of why it was true).  Anyone
: > out there recall seeing it?
: 
: Here is an attached DVI file with several proofs. ...
There is a newer version of this charming paper of Robin Chapman's 
titled "Evaluating zeta(2)". The latest version has 3 more proofs
for a total of 14. It is available directly via 
ftp://euclid.exeter.ac.uk/pub/rjc/etc/zeta2.dvi.Z
or indirectly through a link in Robin's home page at
http://www.maths.ex.ac.uk/~rjc/rjc.html
which also contains other interesting papers and notes on number theory 
(e.g. his twenty pages of course "Notes on Algebraic Numbers"). I've
appended below an extract from Robin's home page.
-Bill Dubuque
ROBIN CHAPMAN'S HOME PAGE
I'm a lecturer in the [[Department of Mathematics]] at the [[University of
Exeter]]. My research student is [[Ray Miller]]. My mathematical interests
include number theory, algebra, combinatorics and problem solving. [[This]] is
what I looked like in 1992.
------------------------------------------------------------------------------
This is my [[teaching page]] where you'll find details of courses,
undergraduate projects etc.
------------------------------------------------------------------------------
I have the following manuscripts available:
    o Lecture notes:
        * {{A Guide to Arithmetic}} (dvi, 55k)
        * {{Notes on Algebraic Numbers}} (dvi, 120k)
    o Recent preprints:
        * [[Automorphism Polynomials in Cyclic Cubic Extensions]] (dvi, 31k)
        * [[Generalized Bianchi Groups]] (dvi, 37k)
        * [[Completely Normal Elements in Iterated Quadratic Extensions of
          Finite Fields]] (dvi, 34k)
        * [[Ideals in Quadratic Extensions of Imaginary Quadratic Fields of
          Class Number One]] (dvi, 11k)
        * [[Universal Codes and Unimodular Lattices (with Patrick Solé)]]
          (dvi, 29k)
        * {{Quadratic Residue Codes and Lattices}} (dvi, 87k)
    o Other stuff:
        * [[Evaluating zeta(2)]] (dvi, 40k)
          This gives (so far) fourteen proofs that the sum of the reciprocals
          of the squares of the natural numbers equals pi squared over six.
All of these are available by anonymous ftp from "euclid.exeter.ac.uk". Please
read the file "pub/rjc/00index".

electronic monk wrote:
> 
> Michael The Roach Janszen wrote:
> >
> > Figure out the following steps...
> >
> > -20 = -20    (obviously)
> >
> > 16 - 36 = 25 - 45  (just the same)
> >
> > 16 - 36 + 81/4 = 25 - 45 + 81/4   (just added 81/4)
> >
> > (4 - 9/2)^2 = (5 - 9/2)^2  (using the binominal rules)
> >
> > 4 - 9/2 = 5 - 9/2          (square root)
> >
> > 4 = 5     (euh.....)
> 
> this is one of those cases when it is important to use the plus or minus
> square root.
> 
> (4 - 9/2)^2 = (5 - 9/2)^2
> 
> (4 - 9/2) = -(5 - 9/2)    (took the negative square root of one and the
> positive of the other.)
> 
> 4 - 9/2 = -5 + 9/2        add the five, add the 9/2
> 
> 9 = 9
> 
> this also works with:
> 
> -(4 - 9/2) = (5 - 9/2)
> 
> -4 + 9/2 = 5 - 9/2
> 
> 9 = 9
> 
> i do think it is weird that one must take the positive square root of
> one, and the negative of the other, but it works.
What about this... it's really basic but it caught me out...
a = b
3a - 2a = 3b - 2b
3a - 3b = 2a - 2b
3(a-b)  = 2(a-b)
3 = 2
-- 
The Attack Dog: Red Alert
http://home.hkstar.com/~luibr
Three strategy pages.

hi math folks -
I'm working on my dissertation and have stumbled upon a too vague 
cite in a book that I'm hoping one of you might be able to clear
up for me. My work includes the numerical solution of stochastic
de's, and I'm currently working on a variance reduction technique
which is covered in _Numerical Solution of SDEs_ by Kloeden and 
Platen, 1995. In chapter 16 section 2 K&P; review a method of
variance reduction using Girsanov's theorem, which they credit to
Milstein. In the bibliography they list 4 articles from the SIAM
Theory of Prob and its Applications by Milstein, and 1 book. I've
checked the 4 articles - from my brief (and possibly faulty) skimming,
there is no mention in them of Milstein's variance reduction 
technique. Does anyone have a copy of the book by Milstein (which was
translated into english in 1995 by Kluwer Academic Press): The Numerical
Integration of Stochastic Differential Equations. Can you spare the
time to see if the topic is indeed covered in the book (as i suspect
it is, but cant seem to get my hands on a copy in less than 6 weeks)
and if so - perhaps a brief book review/technique review for how
straightforward it is to implement the method based on the book's
description of the method. 
The reason I'm asking is that, despite the overall excellence of 
Kloeden and Platen's book, this particular topic is somewhat poorly
reviewed and discussed, and this particular method seems that it
*might* not have been well understood or ever implemented by the authors.
(No slagging of the book or authors intended - its an outstanding 
book, practically my bible of late.) 
Anyway, I've spent an inordinate amount of time coding the method, 
error checking, bug checking and reviewing my own theory to algorithm
to code translation - and still I cant catch where I'm missing some
crucial step (laughably, my program perfectly *increases* the variance
in all situations, instead of decreasing it). 
Sorry to be longwinded. If you have done any work on these methods or
have a copy of the above book, I'ld greatly appreciate some numerical
sde prowess. Its hard to come by down here and I'm in that re-inventing
the wheel cycle. 
Thanks for any potential help!
-Mary Beth

Ronald Bruck wrote:
>
> [Snipped bunch of stuff.  What *were* you talking about?]
>
> Historically, the representation of real numbers as decimals, or to any
> base, is a very late addition to the question.  By the way, how DOES one
> write a real number to base pi?  Using WHAT as digits?
Maybe we use base 10 digits, like they (we) do with sexigesimal (base 60).
So (all numbers base pi):
1+1=2 still
and
1+2=3
but
2+2=10.220122...
On the other hand, Pi = 1.0, exactly.
-- 
D.
mentock@mindspring.com
http://www.mindspring.com/~mentock/index.htm

I have given a great deal of thought to the following problem but I have 
had no success:
An oil storage tank is obtained by revolving the curve y=(9/625)(x^4) from 
x=0 to x=5 about the y-axis where x and y are measured in feet. Oil 
weighting 50 pounds per cubic foot flowed into an initially empty tank at 
a constant rate of 8 cubic feet per minute. When the depth of the oil 
reached 6 feet, the flow stopped.
1.Let h be the depth, in feet, of oil in the tank. How fast was the depth 
of the oil in the tank increasing when h=4? Indicate units of measure.
2. Find to the nearest foot-pound, the amount of work required to empty 
the tank by pumping the oil back to the top of the tank.

;
       Polynomials over finite Galois field
       spread so evenly across their finite affine space
       I wish for a network of friends
       to count on
                               H.New Mexico
                               1996-03-05/06

             Does Apple (Apple) = Apple?
	Let's explore how the numbered word solution form can 
be used to explore situations usually taken for granted or 
solved mechanically. Let's think afresh.
Given:        3 Apples
         2    O   O   O    2
     Apples   O   O   O   rows
Find: Find the number of apples using: 
 ûû         1. numbered words not in solution form
          2. numbered words in solution form.
Solution: 1. 3 apples (2 apples) = 6 apples^2 = 6 apples
or        2. by the numbered word solution form:
             3 apples   2 rows
             --------           = 6 apples
               row
Comment: Solution 2 above is clearly a correct form in terms
         of dimensional analysis, but is 1 above acceptable?
	After some consideration, and using the principle that 
if a new form works without creating contradictions or 
problems of clarity, then the new form is a candidate for 
being a proper form. Therefore, on first consideration, I 
accept solution 1 above as a correct use of numbered words. 
	However, I would never solve problems this way, accept 
as follows:
Given:  distance d = [d] meters/atom in a square layer
             |  d  |
             |<--->|
    |-->  O     O     O      <---3 atoms in a row
    | d 
    |  
    |-->  O     O     O
û
Find:  The number of atoms in a meter square (m^2)
Solution: The quick solution is by squaring the inverse of d
          as follows:
          1 atoms        atoms
          -------  =  ----------
          [d] m        [dd] m^2
Comment: Note that atoms (atoms) = atoms
	It seems that squaring an item (not a unit of 
measurement) equals the item.  What do you think?
	It works for atoms and is used in chemistry to solve an
important problem of how many atoms per m^2 given d.
ûûû
-----------------------------------------------------------
Challenge: 1. Use the numbered word solution form to find  
              the atoms/m^2 from 1/d in the above problem. 
Use a method that doesn't require any squaring of atoms.
Hint: This is a 3 variable problem requiring 3 variables in 
      the numbered word fraction.  I learned this just by
      playing around with words in a fraction to aid my 
      thinking. It takes a little time and understanding of
      division of a fraction by word "row" to justify the 
      solution.
Challenge: 2. Same problem as 1 above, except that
              1/d = [1/d] bond/m
Hint: This only requires that a bond^2 = 1 atom to avoid 
      squaring the atom. 
-----------------------------------------------------------
	For the last analysis, the area of a rectangle is 
explained more clearly than usual by using the following 
numbered word approach:
Given: Rectangle: 3 meters by 2 meters
               3 meters
           _________________
          |     |     |     |
          |     |     |     |
          |_____|_____|_____| 2 meters
          |     |     |     |
          |     |     |     |
          |_____|_____|_____|
Find: Find the area using the numbered word solution form.
Solution:   3 m(m)   2 rows 
            ------          =  6 m(m)
             row
Comment: Why should 3 meter (2 meters) = 6 m(m)?
         Why should distance times distance give an area?
	The above solution form starts with the existence of 
squares within the rectangle and makes more sense to me. 
However, math books have adopted: Area = 3 m (2 m) = 6 m^2 
because it works, is concise and solves a useful problem.
	The numbered word solution form in the problem above 
allowed me to examine area of a rectangle in a new way that 
I never saw done anywhere else. This is not only true for 
the problems presented above, but for numerous other 
situations. For example, consider my analysis of the meaning
of a number in the my current post, "The Power Of Numbered 
Words Revealed".
	So far we been solving constant rate problems with 
numbered words. The axioms for such a system and their 
limitations have yet to be discussed (which I will do). I 
have never seen a axiomatic presentation anywhere on solving
constant rate problems with dimensional analysis. Have you?
	The use of numbered words to create and understand 
algebraic equations is very important in using computers to 
solve or research scientific problems. This too should be 
part of the K-12 curriculum and will be discussed in other 
posts.
	Good luck on this important undertaking to introduce 
numbered words (to solve constant rate problems) into the 
K-12 curriculum. 
-----------------------------------------------------------
    C by David Kaufman,                    Jan. 1, 1997
    Remember: Appreciate Each Moment's Opportunities To
          BE Good, Do Good, Be One, And Go Jolly.
-- 
                                             davk@netcom.com