Newsgroup sci.math 156831

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Stephen Jesson wrote:
> 
> In message 
>         schafer@netcom.com (Lenny Schafer) writes:
> 
> > Lots easier math tricks for the great unwashed masses:
> 
> > Three guys walk into a cheap motel wanting to rent together a room for the
[blah blah blah]
> > Therefore, each guy paid $9 (x 3) = $27, plus the $2 in the clerks pocket
> > comes to $29.  Where is the missing $1?
> 
> > When you figure it out, you can pull this on others.
> 
>    I *loved* this, does anyone else have any similar problems?
> 
>    Personally, I thought the answer was: as the three men are being
> returned $5, then they would then be paying $25 between them, and 25
> divided by 3 is '8.33 recurring' - so when the clerk gave them each a
> dollar back, it was 9.33 recurring and not '9'; it's that extra '.33'
> multiplied by three that makes up the extra 1. That's right isn't it?
> I hope so, I have to admit I spent a good half an hour working that
> one out. Great conundrum!
> 
>    Stephen.
you spent a half hour on this? oh dear... maybe you should go try the
alt.music groups instead...

sfly wrote:
> 
> On Fri, 03 Jan 1997 12:26:14 -0500, Richard Mentock
>  wrote:
> 
> >sfly wrote:x^y+y^x=(1-a)^(1-b)+(1-b)^(1-a)=(1-a)/((1-a)^b)+(1-b)/((1-b)^a)
> >> and (1-a)^b<=1-ab ,(1-b)^a<=1-ab
> >This works for a=b=.5, for instance, but not for a=b=.1
> >Try again!
> su3g4196t87!!
> no,I'm no wrong!
Sorry.  Bad night.
-- 
D.
mentock@mindspring.com
http://www.mindspring.com/~mentock/index.htm

In article <5ahtbp$c3n@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
> In article <5aehs0$qgt@dartvax.dartmouth.edu>
> Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
> 
> >   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?
> 
> 
>  I believe the answer is in. The answer is that Riemann Hypothesis is
> true after all. I was informed that: Yes, 2 is still a unique solution.
>  In the n-adics the equation x+x = x*x has the obvious solutions 0 and
> 2 and also (when n is composite) some hybrids like the 10-adic
> ...3574218752, which are "partly like 0 and partly like 2". But the
> exponentiation constraint eliminates 0 and also the 0-like part of the
> hybrids, so 2 is the only solution.  For each n >= 2, the equation in
> the n-adic ring (or field) has exactly one solution.  (Which can be
> written ...0010 when n = 2, and ...0002 when n > 2).
> 
>   And I have also been told that there is a p-adic Riemann Hypothesis.
> I have never heard of this and do not know exactly what they are
> talking about. I can be sure though that what they are not talking
> about is the program that Naturals = P-adics
The above one paragraph is actually a proof of RH. It is explained in
more detail in my website, and there I also give a geometrical proof of
RH.
But the pattern is this:   If FLT were a true statement of math, then
the proof of FLT would be that if false then there exist a number N
such that N+N+N = N*N*N = M to form a building block number for a
P-triple. But of course with Naturals being the p-adics there does
exist these numbers that satisfy kN= N^k. 
  Now the pattern can be used for RH. Look at RH and ask yourself why
is 1/2 so special? And what sort of pattern is connected with the
specialness of the Real 1/2 and the Euler encoding and the Zeta
function encoding? What is the connection between 1/2 and Euler and
Zeta? And the answer is that those three are connected by the addition
of Naturals equaling the multiplication of Naturals. The linkage, or
the connection is that addition equals multiplication and is equal to
this special number of 1/2.  Here is the pattern of linkage.
         ((N+N)^1/N) = ((NxN)^1/N) = N
   reduces to     (N+N) = (NxN) = N^N = M
There, in that encoding is the Riemann Hypothesis distilled to its
primal foundation. Just as kN=N^k distills FLT to its primal
foundation.
  Sorry to say that FLT is false because in Naturals = P-adics that
primal foundation collapses.
   But look at RH. RH in primal foundation is true in both Reals+i+j
since 2 is the unique solution and also in P-adics where ....0002 is
the unique solution.
   There, Riemann Hypothesis is proved. 
  I do not know if the primal foundation above with the P-adic solution
also answers the Riemann Hypothesis for negative numbers?   Anyone
know? I am busy at the moment to give it my full attention.

In article <19970104210200.QAA28878@ladder01.news.aol.com>
tleko@aol.com writes:
> Is tan(z) an analytic function at the origin ?
> 
Since tan(z)=sin(z)/cos(z), both of which are analytic at the origin,
and cos(z) being bounded away from zero in a neighbourhood of the
origin, tan(z) is *clearly* analytic at the origin. In fact its power
series is easy to work out!
> We have for
> 
>     tan(z)=(1/i)*(e^(2*(x+iy)*i-1)/(e^(2*(x+iy)*i+1)))  ,      where
> z=x+iy.
> 
> The real part is:
> 
> R   (2*e^(-2y)*sin(2x))/(e^(-4y)+2*e^(-2y)*cos(2x)+1)
> 
> and the imaginary
> 
> I   (e^(-4y)+2*e^(-2y)*cos(2x)-1)/(e^(-4y)+2*e^(-2y)*cos(2x)+1).
> 
> Running the following MATLAB program clearly shows that tan(z) is not
> analytic at the origin.
> 
> Real part
> 
> [x,y]=meshdom(-1.2:.1:1.2,-1:.1:1);
> colormap([1 1 1]);
> R=2*exp(-2*y).*sin(2*x)./(exp(-4*y)+2*exp(-2*y).*cos(2*x)+1);
> I=-(exp(-4.y)+2*exp(-2*y).*cos(2*x)-1)./(exp(-4*y)+2*exp(-2*y).*cos(2*x)+1
> );
> A=sqrt(R.^2+I.^2).*cos(atan(I./R));
> surf(x,y,A)
> 
> Imaginary part
> 
> [x,y]=meshdom(-1.2:.1:1.2,-1:.1:1);
> colormap([1 1 1]);
> R=2*exp(-2*y).*sin(2*x)./(exp(-4*y)+2*exp(-2*y).*cos(2*x)+1);
> I=-(exp(-4.y)+2*exp(-2*y).*cos(2*x)-1)./(exp(-4*y)+2*exp(-2*y).*cos(2*x)+1
> );
> A=sqrt(R.^2+I.^2).*sin(atan(I./R));
> surf(x,y,A)
> 
> If there is an interest in the results where MATLAB software is not
> available please request a fax-transmission.
I believe that your matlab program has a mistake in it.
Ben Tilly

Ilias Kastanas (ikastan@alumnae.caltech.edu) wrote:
[...]
> >In this matter "machine" and "algorithm" mean the same thing.
> >One of the results of Godel implies that the theorems (i.e., 
> >propositions than can be proved) of Arithmetic form a set that 
> 	You probably mean "the true statements about N"; as the _theorems_
>    (of PA or some other axiomatizable T) are a prototypical recursively
>    enumerable set.   (Therefore the two sets do not coincide.)
Sorry! Of course I meant the "true statements". The theorems _are_ 
recursively enumerable.
Miguel A. Lerma

(posted & emailed)
Doug Timbie  wrote:
> 
> Hi. My name is Heather and I am a junior in high school.  I could use
> some verification of my work on 2 project problems I am doing.
> 
> Question 1:
> Graph f(x)= x, g(x)=sin x, and h(x)= x(sin x).  Compare the magnitudes
> of f(x), g(x), and h(x) when x is "close to zero."  Write a short
> paragraph explaining why the limit of h(x) is 0 when x approaches 0.
> 
>         I graphed the three funtions on my graphing calculator.  All three
> share the point (0,0), but approach it from different positions. Since
> h(x) is the product of f(x) and g(x), and both of the latter functions
> have a magnitude of 0 at 0, then does it follow that (since 0x0 = 0),
> h(0) = 0?  Does this also show that the limit of h(x) as x --> 0 is 0?
It looks to me as if you've already answered the question.  All
three functions are continuous at 0, so there isn't too much to
explain.  One difference that I notice about the functions, though, 
is that, near 0, sin(x) is very close to x.  
   lim(x -> 0)(sin(x)/x) = 1 ,
and even if x is as large as 0.1, the value of sin(x) is well within
1% of that (I got 0.0998... on my calculator).
On the other hand, h(x) is close to x^2, and for small x, this is
very flat (stays close to zero); if you look at the slopes, f(x) and
g(x) have slopes of 1 and h(x) has a slope of 0.  You weren't asked
to say anything about that, but the functions look different in that
respect.  
> Question 2:
> The number of units in inventory in a small company is
> 
> N(t) = 25(2[[((t+2)/2)]]-t), t is in the interval [0,12]
> 
> Note: [[ ]] = greatest integer function
> 
>         The real number t is the time in months.  Sketch the graph of the
> function and discuss its continuity.  How often must thsi company
> replenish its inventory?
> 
>         I thought about it and wanted to know if the inventory needs to be
> replaced when N(t)=0?  Also, do you have any tips on how to graph this
> function?
That seems like as good a number as any, since you weren't given any
other minimum (but if I were managing it, I would choose some positive
number, to avoid losing sales).  However, another way to look at that
is that you have a function that tells what the inventory *is* -- you
know when it's replenished, because it's whenever the value of the
function increases, which it does abruptly from time to time.  The
question could as easily have been "How often does this company..."
instead of "How often must...".
>From the looks of this function, the "units" in inventory are some
kind of continuous thing -- gasoline or flour, for example -- since
the function can take non-integer values.  There are ways in which
you can simplify the function, such as moving the +2 away from the t.
If your computer runs DOS or Windows, you might try writing a 
QBasic program to plot this function, or you might do that on your
graphing calculator to see how the function behaves, then see if 
you can prove that it works the way it seems to on the graph.
Once you have determined what the critical values are (such as at
the endpoints of 0 and 12, and anywhere the function is discontinuous),
try graphing a couple of points very close to those values, both above
and below if possible, to get a feeling for what's happening there.
> Thank you for your time and help.  Since I check my email more than this
> newsgroup, please email any responses to:  timbidou@injersey.com
Another newsgroup that deals with problems like this is
alt.algebra.help,
which you might want to look at.
-- 
                      -- Vincent Johns
Please feel free to quote anything I say here.

>Jeff Cronkhite wrote:
>> 
>> I have a question from back in my GR class which I have never gotten
>> around to finding an answer to, and wonder if the newsgroup could be
>> of help.
>> 
>> After deriving the Riemann curvature tensor, one typically proceeds by
>> contracting on two of the indices to arrive at the Ricci tensor, and
>> then using the Bianchi identities to define the Einstein tensor.  The
>> mathematics is straightforward enough, but I have never seen a good
>> explanation for the motivation behind this procedure.  What does
>> contracting the indices of the Riemann tensor do for us (or put
>> another way, what is the interpretation of the Ricci tensor)?
>> 
>> Any help or good references would be greatly appreciated.
>> Thanks,
>> 
>> jmc
There are really two ways to tackle this. One is the BFBI (brute force and
ignorance) method, which says that we need some combination of tensors
which are covariant, depend on the various curvatures, and are divergenceless.
This is the method that Einstein used in the first place; the result is
that G^\mu^\nu=R^\mu^\nu - R g^\mu^\nu / 2. But this is somewhat shady, as
other things could crop up, and something just seems arbitrary.
The modern approach is to ``guage the Lorentz group.'' This means that we 
take as our guiding principle that every observer _locally_ sees that the
universe is Lorentzian, and any two observers really are connected by a
Lorentz transformation; it's just that when two people have different
accelerations, the appropriate transformation varies. This ultimately means
that the parameters of the Lorentz boost to get from one observer at the
origin to another observer at a point X are a function of X. (Remember, 
acceleration = gravity, so a different gravitational field elsewhere behaves
like a different acceleration)
When you actually plough through the math, (see any fairly modern text on
advanced GR; I think Wald's book has something about this) you find that
simply postulating local Lorentz invariance is enough to get you the Einstein
equations. This is equivalent to a Lagrangian theory with 
L = (8 Pi G / c^2) R - \Lambda
	With R the scalar curvature, and \Lambda a ``cosmological constant.''
(Set to zero to get the ordinary Einstein equation) The dynamical variables
are the components of the metric; doing the variation gives the standard
Einstein equation.
	There are still some problems with this, including that the self-energy
of the gravitational field  isn't completely clear, and there is some 
nonuniqueness in the way we do some of the guaging. But that's a subject for
a whole book in itself... I'd just browse the library shelves for a bit.
Happy hunting!
Yonatan Zunger

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. 
 But the table is missing a column:
              |    1 / 2^.5   |   2^.5 / 2    |   2^{-0.5}    |
  ------------|---------------|---------------|---------------|
  concise     |     same      |     same ?    |     same      |
  ------------|---------------|---------------|---------------|
  clear       |     No ?      |     Yes       |     Yes       |
  ------------|---------------|---------------|---------------|
  efficient   |     Yes       |     No        |     Yes       |
  ------------|---------------|---------------|---------------|
 All much ado about nothing -- just some very basic arithmetic, not 
 math, if you ask me. 
-- 
 James A. Carr        |  "The half of knowledge is knowing
    http://www.scri.fsu.edu/~jac/       |  where to find knowledge" - Anon. 
 Supercomputer Computations Res. Inst.  |  Motto over the entrance to Dodd 
 Florida State, Tallahassee FL 32306    |  Hall, former library at FSCW. 

Hi All:
This issue is best seen as a historical problem. Did Egyptians, Greeks,
Romans and other cultures in the ancient world exactly convert p/q into
unit fraction tables as part of a system of numeration?
My answer is based in reading the ancient documents, the 2000 BC Moscow P.,
1800 BC Reisner P. ,1650 BC Rhind Mathematical P., 300 BC Hibeh P. and
500 AD or so Akhmim P. From what I see, and can show in fairly clear terms:
n/pq - 1/a = (na - pq)/apq                 (equation 1.0)
was the Hibeh P. n/45 table improvement of the earlier RMP 2/p rule:
2/p - 1/a = (2a -p)/ap                     (equation 2.0)
One point that I find interesting is that Ahmes used another
rule for 2/pq, usually:
2/pq = (1/q + 1/pq)2/(p + 1)                (equation 3.0)
while the author the the Hibeh P., maybe Eudoxus, used equation 1.0
for both n/pq and n/p cases by setting either p or q = 1.
Regards to all,
Milo Gardner
Sacramento
On Thu, 2 Jan 1997, Josep Lopez wrote:
> Bob Silverman wrote:
> > 
> > "LSC"  wrote:
> > 
> > >Hi, I am trying to prove that every natural number n can be expressed as
> > >the sums of several term from the Fibbonacci series u1=1, u2=2,
> > >u(n+2)=u(n+1)+u(n) (hope the spelling is correct, I forget the real
> > >spelling).
> > 
The Euclidean Algorithm also works, and produces much smaller term series
than does Fibonacci. One problem with the Euclidean Algorithm case, as
I recall was proven by Sylvester, is that it too does not create the
needed smallness in terms that the acutal historical documents use.
> > You can't prove it, because it isn't true. Not if you want the number of terms
> > to be bounded. i.e. to represent every N as the sum of at most K Fibonacci
> > numbers for some fixed K.
> 
> There is a good comment on this subject in the book "Concrete
> mathematics" by Graham-Knut & Patashnik. A book that has already been
> mentioned in this group "some" times before
> -- 
>
There have been many attempts to close the door on Egyptian fractions 
as a meaningful historical door. Hultsch in 1895 showed that the
equation one case held for the 2/p case, a point that Bruins later
independently confirmed. Yet, Neugebauer and others see3mingly refuted 
Hultsch-Bruins, forever.
I say the Egyptian fractions story should be re-opened - not to
continue talking about the Fibonacci 'red herring' but by actually
reading the historical documents.
 _______________________________________________________
> Josep M. Lopez Besora         jlopez@etse.urv.es
> Universitat Rovira i Virgili  TARRAGONA.Catalonia.Spain
> -------------------------------------------------------
> 
> 

ptwahl@aol.com (PTWahl) wrote:
>My understanding is that the hypergeometric distribution of probability is
>a discrete distribution ( defined only at isolated points ) and the other
>things are more or less smooth curves with some holes.  I'd try the
>sci.math.stat group for the code you want.
Thank you very much for responding to my post.
Let me get this straight:
Hypergeometric function  =  discreet probability distribution
Confluent Hypergeometric function  =  continuous version of the above
but with some holes in it.
Does this mean that the the second is an approximation of the first?
The newsgroup "sci.math.stats" does not appear in my newsgroup list.
Before I go to my internet provider demanding it, are you certain it
exists and the spelling is correct?
Gilles Lehoux, P.Eng. glehoux@generation.net
Mehcanical Engineer   I like kites.    Montreal, Canada
http://www.generation.net/~lehouxri/gilles

Plutonium Day = 7th Nov.
Any one know what the date was in 'Beneath the Planet of the Apes'
that they blew up the doomsday weapon? 
I know it's off thread, but hey....
  ########## Paul van den Bergen
####       # c/- Materials Eng., Monash University
#  ####    # Clayton VIC 3168 Australia
####  #    # pavan1@student.monash.edu.au
  #   ###### ph.  +613 9905 3597
  #   #      fax. +613 9905 4940
  #####      meow *cough* feathers
I feel it is my duty to warn everyone that there is an
international consortium of powerful people who get   
together and facilitate the distribution of conspiracy
theories in order to keep the minds of the masses off 
the real issues that are affecting the world (whatever
they may be...)

In a forged article <32CA9D09.4667@efgh.net> abcd@efgh.net (-X) writes:
>
>The limit of 1/x as x --> 0 is infinity.
 positive or negative? 
-- 
 James A. Carr        |  "The half of knowledge is knowing
    http://www.scri.fsu.edu/~jac/       |  where to find knowledge" - Anon. 
 Supercomputer Computations Res. Inst.  |  Motto over the entrance to Dodd 
 Florida State, Tallahassee FL 32306    |  Hall, former library at FSCW. 

Does anyone have a good example to explain why p->q (p=F and q=T)
evaluates to true?

In article , zunger@rintintin.Colorado.EDU (Yonatan Zunger) wrote:
> There are really two ways to tackle this. One is the BFBI (brute force and
> ignorance) method, which says that we need some combination of tensors
> which are covariant, depend on the various curvatures, and are divergenceless.
[...]
> When you actually plough through the math, (see any fairly modern text on
> advanced GR; I think Wald's book has something about this) you find that
> simply postulating local Lorentz invariance is enough to get you the Einstein
> equations.
I don't think this is true.  Postulating local Lorentz invariance is
certainly necessary for GR, but I can't see how it is sufficient to
derive the Einstein equation.  Wald (as well as MTW) appears to take the
BFBI method.
> L = (8 Pi G / c^2) R - \Lambda
If you can show me how this Lagrangian follows merely from local Lorentz
invariance, I will be quite intriuged.  It is certainly the _simplest_
Lagrangian with such invariance, that hardly mandates its correctness.
-- 
Nathan Urban | nurban@vt.edu | Undergrad {CS,Physics,Math} | Virginia Tech

I wrote a response to someone else's comment to "Inside" that may
interest you, maybe not.  Question from me: how did you pull up my poem
into your text? did you retype it or is there an easier way? 
Here is my response and his comment:(I'll retype it)
He wrote: The erotic tension derivable from telling your girlfriend she
gives you a woody is almost inexpressable. Almost.
I wrote:  The poem is not about her giving me a woody, but rather the
fact that she is the only one that doesn't simply give me a woody.  It's
not about sex, the poem is about love!  My penis is irrelevant for the
overall meaning of the poem.  It is the one way I can be physically
inside her, but I was writing more along the lines of being mentally
inside her.  The definition of confide: rely on, to tell something
confidentially, to tell private matters in confidence, etc.  She is
basically my support group, probably a lot like your psychiatrist( just
kidding of course).
There's more, but it pertains to something else he said(read it if you
care to). 
I think this is what is original about this poem.  The idea that I can
be inside her without having sex, but get the pleasure of an emotional
release(someone I can tell everything).  SEX, is just a way to highlight
the pleasure this confidence gives me.
P.S. I think this was a bad poem to post for my first time.  It means a
lot to me and I have reacted unfairly to it's critism.  But, I have
learned a great deal in the past week since it was posted. I still do
not think it is generic and maybe you won't after you read this.
-RBC

In article (Dans l'article) <5an7cp$kuj@geraldo.cc.utexas.edu>,
mlerma@pythagoras.ma.utexas.edu (Miguel Lerma) wrote (écrivait) :
> Jason Lee (jplee@cymbal.aix.calpoly.edu) wrote:
> 
> > That's fine, if you want to formalize it that way, but now you need to
> > define what the set of whole numbers.
> The question was about integers, not about whole numbers (which 
> I am calling "natural numbers", including zero). Anyway, you 
> can take N = finite ordinals (but, of course, now will have to 
> define "ordinal"...).
Cardinals are maybe easier to use... They are our subjective perception of
things (I mean basic counting of simple wholes).
-- 
M.TIBOUCHI
Mystical Queror of Transcendental Nums (n'other stuffs like that ;-)
>"e^(i.pi) + 1 = 0", Euler
>The shorter poem if poetry puts real together with imaginary.

In article (Dans l'article) <5abuel$nfa@seaman.cc.purdue.edu>,
ags@seaman.cc.purdue.edu (Dave Seaman) wrote (écrivait) :
> In article <32C96CFF.537B@pogonip.scs.unr.edu>,
> The Universal Heretic   wrote:
> >However, approaching along the line y=0, we have
> >f(x,0)= 0^x = 0 for all x.
> >Thus, lim (x->0) 0 = 0.
> >Therefore, the value of 0^0 is undefined generally.
> 
> No, because 0^x is not defined for x < 0.  The lim (x->0) does not
> exist, but the one-sided lim (x->0+) does exist and is equal to 0.
> 
> Otherwise, everything is correct except your conclusion.  The value of
> 0^0 is the cardinality of the collection of mappings from the empty set
> to itself, which is one.  No limits are involved.  The mere fact that a
> function happens to be discontinuous at a point does not mean the
> function is undefined there.
> 
> -- 
> 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 ++++
Excuse me sir... I am a poor little kid but I think that :
f(0,y) = y^0 = 1 for all y, reals.
lim (y->0) 1 = 1 and y^0 is defined for all reals.
Therefore 0^0 = 1... But (x->0) 0 = 0.
So f(x,y) = x^y has TWO different limits in 0+.
That's why 0^0 is undefined -- though it IS possible to considere it equal
to 0 or 1, depends on the function you're workin with (power or
exponential).
Where's my failure ?
-- 
M.TIBOUCHI
Mystical Queror of Transcendental Nums (n'other stuffs like that ;-)
>"e^(i.pi) + 1 = 0", Euler
>The shorter poem if poetry puts real together with imaginary.

In article (Dans l'article) <32CF4D41.59AE@telepath.com>, Vincent Johns
 wrote (écrivait) :
You really wanna puzzle him, don't you ? ;-)
> (posted & emailed)
> 
> Doug Timbie  wrote:
> > 
> > Hi. My name is Heather and I am a junior in high school.  I could use
> > some verification of my work on 2 project problems I am doing.
> > 
> > Question 1:
> > Graph f(x)= x, g(x)=sin x, and h(x)= x(sin x).  Compare the magnitudes
> > of f(x), g(x), and h(x) when x is "close to zero."  Write a short
> > paragraph explaining why the limit of h(x) is 0 when x approaches 0.
> > 
> >         I graphed the three funtions on my graphing calculator.  All three
> > share the point (0,0), but approach it from different positions. Since
> > h(x) is the product of f(x) and g(x), and both of the latter functions
> > have a magnitude of 0 at 0, then does it follow that (since 0x0 = 0),
> > h(0) = 0?  Does this also show that the limit of h(x) as x --> 0 is 0?
> 
> It looks to me as if you've already answered the question.  All
> three functions are continuous at 0, so there isn't too much to
> explain.  One difference that I notice about the functions, though, 
> is that, near 0, sin(x) is very close to x.  
>    lim(x -> 0)(sin(x)/x) = 1 ,
> and even if x is as large as 0.1, the value of sin(x) is well within
> 1% of that (I got 0.0998... on my calculator).
> 
> On the other hand, h(x) is close to x^2, and for small x, this is
> very flat (stays close to zero); if you look at the slopes, f(x) and
> g(x) have slopes of 1 and h(x) has a slope of 0.  You weren't asked
> to say anything about that, but the functions look different in that
> respect.  
> 
>  
> > Question 2:
> > The number of units in inventory in a small company is
> > 
> > N(t) = 25(2[[((t+2)/2)]]-t), t is in the interval [0,12]
> > 
> > Note: [[ ]] = greatest integer function
> > 
> >         The real number t is the time in months.  Sketch the graph of the
> > function and discuss its continuity.  How often must thsi company
> > replenish its inventory?
> > 
> >         I thought about it and wanted to know if the inventory needs to be
> > replaced when N(t)=0?  Also, do you have any tips on how to graph this
> > function?
> 
> That seems like as good a number as any, since you weren't given any
> other minimum (but if I were managing it, I would choose some positive
> number, to avoid losing sales).  However, another way to look at that
> is that you have a function that tells what the inventory *is* -- you
> know when it's replenished, because it's whenever the value of the
> function increases, which it does abruptly from time to time.  The
> question could as easily have been "How often does this company..."
> instead of "How often must...".
> 
> >From the looks of this function, the "units" in inventory are some
> kind of continuous thing -- gasoline or flour, for example -- since
> the function can take non-integer values.  There are ways in which
> you can simplify the function, such as moving the +2 away from the t.
> If your computer runs DOS or Windows, you might try writing a 
> QBasic program to plot this function, or you might do that on your
> graphing calculator to see how the function behaves, then see if 
> you can prove that it works the way it seems to on the graph.
>  
> Once you have determined what the critical values are (such as at
> the endpoints of 0 and 12, and anywhere the function is discontinuous),
> try graphing a couple of points very close to those values, both above
> and below if possible, to get a feeling for what's happening there.
> 
> > Thank you for your time and help.  Since I check my email more than this
> > newsgroup, please email any responses to:  timbidou@injersey.com
> 
> Another newsgroup that deals with problems like this is
> alt.algebra.help,
> which you might want to look at.
> 
> -- 
> 
>                       -- Vincent Johns
> 
> Please feel free to quote anything I say here.
-- 
M.TIBOUCHI
Mystical Queror of Transcendental Nums (n'other stuffs like that ;-)
>"e^(i.pi) + 1 = 0", Euler
>The shorter poem if poetry puts real together with imaginary.

In article (Dans l'article) <5amg4b$6ji@news.iastate.edu>,
abian@iastate.edu (Alexander Abian) wrote (écrivait) :
> [...]
Say u_n = \sum_1^n 9*10^-n
u_5 = .99999, and so on.
Say v_n = 10^-n
so u_n + v_n = 1, that's basic maths.
u_n tends to 1 when n tends to infinity.
v_n tends to 0 when n tends to infinity.

I am very happy that you posted this Santiago, 
	as it should allow for the smoke to start clearing a bit on this issue.
Santiago Arteaga  wrote in article
<5ai7mi$r8f@courant.cs.umd.edu>...
| The point is, this branching is not
| continuous but discrete; at some point, for some reason, a spike
| branches. If they branched continuously, you would get a film of ice
| covering your windshield, instead of the spiky growth.
This is correct.
| 	Well, I don't know what is it that makes one of these spikes
| to branch. I guess it is a combination of vibration/temperature/
| humidity/pressure/somethingelse. But, whatever it is, it is likely to
| happen simultaneously in the six sides of a _small_ snowflake. You
| look at your windshield and you see that those spiky crystals do not
| branch symmetrically at the scale of centimeters.
Now we are getting somewhere! 
It is very important to realize that the branching action, for all
practical purposes, can most definitely be described as occurring
"simultaneously."  This in itself is a strong indicator that the crystal
itself has changed in a symmetrical fashion, just prior to the actual
branching, in response to outside changes in its environment. 
I will say that it is the crystals "reaction" to the environment,
and NOT the environment itself that determines this.  
In other words - there are discreet changes that must be taking place, 
more or less simultaneously, in the bonding area of each growing spike.
This problem would be solved right now, if someone would just answer the
following - 
What changes in the crystal are being triggered, and how? 
This is what I want to know!!! 
There has to be at least one polymer chemist in all of Usenet who could
offer a clue -  no?
| 	Now, since the six sides in a small snowflake were
| symmetrical before they branched, the new spikes grow in symmetrical
| directions. 
This is also true.
It is also important to realize that the branching has some options as far
as, which combinations of 60 degree angles, the new branches will start
growing in.  The crystal may have many, many growing spikes that are all
capable of branching, and all of these must undergo an identical change at
very close to the same point in time, in order for the symmetry to
continue.  
| 	So there you have a mechanism which explains why small
| snowflakes are symmetric. And you are right, I don't know the fine
| details; I don't know quantum mechanics so I can't explain how
| water molecules join in a crystal, nor why/how a spike branches.
That is what I want to know - "why/how a spike branches".
I have no problem stating the following:
WE DO HAVE A MECHANISM WHICH EXPLAINS WHY SNOWFLAKES ARE SYMMETRIC!
But the mechanism itself has not been explained.
To illustrate-
Photosynthesis is the mechanism which explains how sunlight can be
converted into carbohydrates.
The "definition" of photosynthesis is a not the same as an "explanation" of
the process.
My idea of a satisfactory explanation of photosynthesis would involve
answers to the following questions:
Chloroplasts - What are they?, and how do they work?
What types of photosystems are found in plants - and how are they
different?
What is a thylakoid, and how does it function?
What are the roles of ATP and NADPH?
What about enzymes and stromas?
How does it all come together?
You get the idea - to truly "explain" something requires more than a basic
"definition" of a process.
I want an explanation of the working details - 
otherwise we are just telling one other the things we already know.
How hard can it be? - clearly the issue of snowflake symmetry is not nearly
as complex as the issue of photosynthesis.  So is it really so unreasonable
of me to look for an explanation?
I sincerely hope that this really makes it clear at last, 
what aspects of Snowflake symmetry are still begging for an explanation, 
and which ones are not.  
-- 
        Catchulater Co.
      {:o)   /|\|/|\|/\    (o:}  
    {:o)    (  @ @)    (o:}
  {:o)       (   _> )      (o:}
-------o00o---------o00o-------
*@*=*@*=*@*=*@*=*@*
 [keep seeking the Truth]
 sprout@discover-net.net
P.S. I think it might have something to do with microwaves (o;}

In a previous article, sako.6@osu.edu (Yusaku Sako) says:
>Does anyone have a good example to explain why p->q (p=F and q=T)
>evaluates to true?
>
 One way: p->q is logically equivalent to: -p V q (- = NOT, V = OR)
          and -F V T evaluates to T.
 Another: one "meaning" ascribed to -> is: if p = T, then q = T, which
          implies that p=T, q=F should cause -> to evaluate to F; other-
          wise, -> evaluates to T -- and one case of "otherwise" is p=F,
          q=T.
 Scott McC.

In a previous article, EarlGlynn@postoffice.worldnet.att.net ("Earl F. Glynn") says:
>Gary Hampson wrote:
>> 
>> In article <32C90B5C.64B1@public.ibercaja.es>, benigno
>>  writes
>> >Hi,
>> >       I need to handle Big matrix of around 800 x 800 to implement
>> >       some optimization algorithms, I would use C++ libraries if
>> >       possible to use on BorlandC++ 4.5, but if there is any shareware
>> 
>> Unless the matrix has some structure which allows many short cuts and
>> reduced storage (eg Toeplitz), then just get on with coding it. 800*800
>> is not that big (unless of course in the optimisation you need to
>> evaluate A.x many times, or you have some time critical conditions.
>> --
>> Gary Hampson
>
>If you're multiplying matrices of that size (800 x 800), you might
>want to investigate the Strassen matrix multiplication algorithm.
>
 Are you referring to the recursive algorithm that quadrisects a square
 matrix to perform simpler mat mults? If so, then make sure you request
 enough stack size.
 Scott McC.

Bob Silverman wrote:
  "adam russell"  wrote:
   Imagine two points, right next to each other.
   The second point is at the number one on the number line.
   The point just to the left of it must be 0.999...  
  There is no such thing. Suppose your two hypothesized points have values  x  and
  1.  Thus (by your argument) x is just to the left of 1. Please note that (1+x)/2
  is greater than x and less than 1.  So, no matter what you select for x,  there
  is ALWAYS at least 1 more point between it and 1.
Is that really true? It seems to me that 1 + 0.999... = 1.999... and that
(1.999...)/2 = 0.999... (just do the division). So in this case (1+x)/2 = x.
--Fred

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On Sun, 5 Jan 1997 11:59:04 -0500, "Nelson G. Rich" 
wrote:
>On Thu, 19 Dec 1996, Jan Rosenzweig wrote:
>
>> Watcharapan Suwansuntisuk wrote:
>> > 
>> > Do you know how to solve this equation?
>> > 
>> >                  X^2   =  2^X
>> > 
>> > If you know , post it or tell me( ioiwcs@nontri.ku.ac.th )
>> 
>> 
>>    How about x=2?
>> 
>> -- 
>Maple produces both x=2 and x=4, and my TI-92 produces x=4 and 
>x=-.766664695962
>===========================================================================
>Nelson G. Rich                               | E-mail: rich@naz.edu
>Department of Mathematics & Computer Science | Voice:  (716) 389-2662
>Nazareth College                             | FAX:    (716) 586-2452
>4245 East Avenue, Rochester, NY 14618-3790   | WWW:    http://www.naz.edu/
>U.S.A.                                       | "Only the educated are free."
>===========================================================================
You can solve this by x = +/-Sqrt(2^x).
When you take the + sign, there are two solutions 2 and 4.
When you take the - sign, and iterate, it will converge to
x = -0.76666469596212309311120442251031484800667534666983205846088437
69355527957248724228530292096979025... .
-Harry
--
| Harry J. Smith
| 19628 Via Monte Dr., Saratoga, CA 95070-4522, USA
| Home Phone: 408 741-0406, Work Phone: 408 235-5088 (Voice Mail)
| E-mail: HJSmith@ix.netcom.com
| Web site: http://www.netcom.com/~hjsmith
--

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
Luke S. Powell
lukep@montana.com

David Ullrich wrote:
> 
> Tarizzo Martial wrote:
> >
> > Hello,
> >
> > I am looking for informations about methods to inverse a matrix ; but only
> > the ones which are using random numbers.
> 
>         I was puzzled by this when it appeared but I figured I'd wait and
> see what people said. Since nobody's said anything so far, at the risk of
> displaying abysmal ignorance I have to ask: What makes you think there
> exist matrix-inversion algorithms that use random numbers? Offhand I can't
> see how a bunch of random numbers would help me invert a matrix.
> 
> --
> David Ullrich
Thanks, David.  I also decided to wait, assuming that someone would
point
out the fact that English is tricky and that the proposer's question
could
mean (1) the matrix was comprised of random numbers, in which case I
suspect
one could do no better than use one of the standard inversion algorithms
with no constraints on the matrix entries, or (2) the inversion
algorithm
itself used random numbers.  This latter interpretation, which I read as
a 
monte carlo/probabilistic inversion algorithm, seems to be much more 
interesting.  It's well known that some problems can be solved more
efficiently if one is willing to sacrifice certainty (prime-testing, f'r 
instance), but I must confess I haven't heard of a probabilistic matrix
inversion algorithm.  It sounds like an interesting research problem.
Any takers?
Regards,
Rick 
-----------------------------------------------------
Rick Decker                   rdecker@hamilton.edu
Department of Comp. Sci.      315-859-4785
Hamilton College
Clinton, NY  13323            =  !=  ==  (!)
-----------------------------------------------------

Michael Cohen (mcohen@cpcug.org) wrote:
: Jacob or Stewart Martin (jake@scientist.com) 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.
: 
: x^x is not defined in general for negative x.  However the limit as x goes
: to 0 from above of x^x = 1:
: 
: x^x = exp[ln(x^x)] = exp[x ln(x)] goes to exp[0] = 1 as x goes to 0 from
: above.  I have used that x ln(x) goes to 0 which is easily verified by
: L'Hospital's rule.  This does NOT show 0^0 = 1 though!
: 
: Let a > 0.  Then exp[-a/x] goes to 0 as x goes to 0 from above because the
: exponent goes to minus infinity.
: 
: But exp[-a/x]^x  = exp[(-a/x)x] = exp(-a) so 0^0 "=" exp(-a)
: for any a>0!
: 
: As Alexander Abian notes in another post,  0^0 = 1 is a useful 
: convention when working with INTEGERS.  In other situations, it can lead
: to trouble.
In the above, I have made two corrections where I used / intending ^.  The
convention 0^0=1 is indeed useful when the numbers involved are understood
to be integers (e.g. binomial expansions, counts of mappings).  Does
anyone see any advantage to 0^0=1 (as opposed to undefined) in the context
of non-negative real numbers?
-- 
Michael P. Cohen                       home phone   202-232-4651
1615 Q Street NW #T-1                  office phone 202-219-1917
Washington, DC 20009-6331              office fax   202-219-2061
mcohen@cpcug.org

I need the solution of this vector-differential equation:
(It is very urgent!!!)
  ..
  r =-c*r/(|r|^3)
r is a vector and c a constant, positive real number.
 ..
(r  is the acceleration of course)
This is the important differential equation of the twobody-problem.

In article <5an0kr$t9s@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
> But the pattern is this:   If FLT were a true statement of math, then
> the proof of FLT would be that if false then there exist a number N
> such that N+N+N = N*N*N = M to form a building block number for a
> P-triple. But of course with Naturals being the p-adics there does
> exist these numbers that satisfy kN= N^k. 
 Here I am not very clear. So let my redo that.  If the Finite Integers
were not some ficticious ghostly absurdity. If the Finite Integers
existed such as people cats and dogs and p-adics and Reals. If the
Finite Integers existed such as atoms exist, and not like Green
Martians or fire breathing dragons. But the Finite Integers do not
exist and are Green Martians and fire breathing dragons.
   Quantum Mechanics exists. Reals+i+j exists. P-adics exist. 
Newtonian Mechanics does not exist. And Naturals = Finite Integers does
not exist.
  If Finite Integers did exist, then this must be the proof of Fermat's
Last Theorem.
   For exponent 2, FLT has solutions because it has a building block to
build a smallest P-triple. That building block is N+N = N*N = M which
of course is the Finite Integer of 2 and you can see this 2 in the
smallest P-triple (3,4,5) because 2+1,  (2+2)  and the (2+2) +1 . The
FLT equation is a use of addition and multiplication and so is
N+N=N*N=M
   If the Finite Integers existed as a true mathematical entity such as
the Reals of Mathematics exists or the P-adics exist.
   Then the proof of FLT would be this:
    For exponent 3,  N+N+N=N*N*N= M,  (encoding kN=N^k), since no
Finite Integer exists to satisfy the building block, then no smallest
P-triple exists. And for exponent of 4 or larger the same follows.
Therefore FLT, QED.
   The reason that the above is not the proof of Fermat's Last Theorem,
is because FLT is a conjecture on a set of objects which do not exist.
Finite Integers is as ficticious as a conjecture on how many boilers a
fire breathing dragon possesses. As ficticious of a conjecture as
worrying about what Columbus's ships would fall off of the Flat Earth
once Columbus reached the edge.
   Ask your math professor to define "Finite". Ask him how the endless
adding of 1 in the Peano Axioms stops these numbers from going into the
Infinite Integers. The world of mathematics as of 1997 wants to ignore
all of this because it shows how awfully stupid and ignorant the math
intelligensia is. And how dictatorial that community is. Ask Andrew
Wiles to define Finite. Ask Andrew Wiles why there is no Mathematical
Induction proof of FLT when FLT asks for _All Finite Integers_ and
because it asks for All Finite Integers entails that a Mathematical
Induction proof of FLT exists.   --- My above hypothetical proof is a
Mathematical Induction proof if you, the reader,  had not noticed that.

                What Is Ethical Truth?
Introduction:
------------
	Is a holy person (who never tells a lie) lying, if they
hide a person being chased by a killer (when confronted by 
the would be murderer) say, "The person ran that way"?
	From a science or math prospective, which usually 
ignores the ethical consequences of revealing certain 
truths, the holy person told a propositional falsehood. 
	However, from an ethical prospective, the holy person 
told the ethical truth because Truth in its human dimension 
also includes not harming others. Truth creates harmony, 
peace and joy. 
Is God All Powerful?
-------------------
	One area that has caused me much mental suffering is 
the belief that I had because my spiritual teachers told me 
that God is all powerful and all loving.
	However, I still believe God is all loving, but I now 
believe God is not all powerful, yet I still believe my 
spiritual teachers told the ethical Truth as described above
in the introduction.
	How can an all powerful God allow legs to be blown off 
by land mines and all the other horrors happening right now 
as you read these lines?
	To a would be logical person, if someone has the power 
to act to correct wrongs before their eyes and doesn't, then
that person is accountable for the harm they allow to 
happen. 
	So the only conclusion about an all knowing and all 
loving God is that God can't be all powerful. Otherwise, it 
would lead a logical person to either ignore the existence 
of God or to hate God for Mankind's ongoing physical 
suffering.
How To Explain Why Holy People Say God Is All Powerful:
------------------------------------------------------
	Breaking away from accepted beliefs is not only 
difficult for true believers (as I am), but requires setting
up an alternate plausible (non-contradictory) system of 
ideas to explain, "Why Do Holy People Say God Is All 
Powerful?" 
	To help others, who might also be struggling with the 
question of how not to hate an all powerful God, I offer the
following 2 scenarios for our current condition.
	On scenario (that is used to explain out current 
situation) is that the Devil created the world and that an 
all loving God to lesson Mankind's suffering made a deal 
with the Devil to gain access to the world through God's 
Saints. 
	However, the deal with the Devil requires that all 
God's Saints claim God is all powerful and is the Creator of
all that exists. Otherwise, the Devil regains the power to 
create even greater harm to Mankind.
	The other scenario imagines that God did create the 
universe--perhaps our galaxy the Milky Way--and has a grand 
plan to stop emerging suffering in the other galaxies. 
	To stop enormous suffering elsewhere requires our 
current training in this galaxy to become fit for the tasks 
ahead to serve the universe by eventually creating peace and
joy for all sentient beings